Sampling Rate Conversion

Collection Editor:

Denver Greene

Sampling Rate Conversion

Collection Editor:

Denver Greene

Author:

Rudolf Riedi

Online:

< http://cnx.org/content/col11529/1.2/ >

C O N N E X I O N S

Rice University, Houston, Texas

❚❤✐s s❡❧❡❝t✐♦♥ ❛♥❞ ❛rr❛♥❣❡♠❡♥t ♦❢ ❝♦♥t❡♥t ❛s ❛ ❝♦❧❧❡❝t✐♦♥ ✐s ❝♦♣②r✐❣❤t❡❞ ❜② ❉❡♥✈❡r ●r❡❡♥❡✳ ■t ✐s ❧✐❝❡♥s❡❞ ✉♥❞❡r t❤❡

❈r❡❛t✐✈❡ ❈♦♠♠♦♥s ❆ttr✐❜✉t✐♦♥ ✸✳✵ ❧✐❝❡♥s❡ ✭❤tt♣✿✴✴❝r❡❛t✐✈❡❝♦♠♠♦♥s✳♦r❣✴❧✐❝❡♥s❡s✴❜②✴✸✳✵✴✮✳

❈♦❧❧❡❝t✐♦♥ str✉❝t✉r❡ r❡✈✐s❡❞✿ ❙❡♣t❡♠❜❡r ✺✱ ✷✵✶✸

P❉❋ ❣❡♥❡r❛t❡❞✿ ❙❡♣t❡♠❜❡r ✺✱ ✷✵✶✸

❋♦r ❝♦♣②r✐❣❤t ❛♥❞ ❛ttr✐❜✉t✐♦♥ ✐♥❢♦r♠❛t✐♦♥ ❢♦r t❤❡ ♠♦❞✉❧❡s ❝♦♥t❛✐♥❡❞ ✐♥ t❤✐s ❝♦❧❧❡❝t✐♦♥✱ s❡❡ ♣✳ ✺✼✳

❚❛❜❧❡ ♦❢ ❈♦♥t❡♥ts

✶ ❋♦✉r✐❡r ❙❡r✐❡s✿ ❘❡✈✐❡✇ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶

✷ ❉✐s❝r❡t❡ ❋♦✉r✐❡r ❚r❛♥s❢♦r♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺

✸ ❋♦✉r✐❡r ■♥t❡❣r❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼

✹ ❊♥❡r❣② ❛♥❞ P♦✇❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾

✺ ❊①❛♠♣❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸

✻ ❊st✐♠❛t✐♦♥ ♦❢ ❙♣❡❝tr✉♠ ❛♥❞ P♦✇❡r ✈✐❛ ❉❋❚ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾

✼ ❙❛♠♣❧✐♥❣✿ ❘❡✈✐❡✇ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸

✽ ❉❡❝✐♠❛t✐♦♥ ❛♥❞ ❉♦✇♥s❛♠♣❧✐♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶

✾ ■♥t❡r♣♦❧❛t✐♦♥ ❛♥❞ ❯♣s❛♠♣❧✐♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺

✶✵ ❙❛♠♣❧✐♥❣ ❘❛t❡ ❈♦♥✈❡rs✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶

✶✶ ▼♦❞❡❧s ♦❢ ◆♦✐s❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸

✶✷ ❖✈❡rs❛♠♣❧✐♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼

✶✸ ◆♦✐s❡✲❙❤❛♣✐♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸

❆ttr✐❜✉t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳✺✼

✐✈

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

❈❤❛♣t❡r ✶

❋♦✉r✐❡r ❙❡r✐❡s✿ ❘❡✈✐❡✇✶

✶✳✶ ❋♦✉r✐❡r ❙❡r✐❡s✿ ❘❡✈✐❡✇

❆ ❢✉♥❝t✐♦♥ ♦r s✐❣♥❛❧ x (t) ✐s ❝❛❧❧❡❞ ♣❡r✐♦❞✐❝ ✇✐t❤ ♣❡r✐♦❞ T ✐❢ x (t + T ) = x (t)✳ ❆❧❧ ✏t②♣✐❝❛❧✑ ♣❡r✐♦❞✐❝ ❢✉♥❝t✐♦♥

x (t) ✇✐t❤ ♣❡r✐♦❞ T ❝❛♥ ❜❡ ❞❡✈❡❧♦♣❡❞ ❛s ❢♦❧❧♦✇s

∞

a0

2π

2π

x (t) =

+

ak❝♦s k

t

+ bks✐♥ k

t

✭✶✳✶✮

2

T

T

k=1

✇❤❡r❡ t❤❡ ❝♦❡✣❝✐❡♥ts ❛r❡ ❝♦♠♣✉t❡❞ ❛s ❢♦❧❧♦✇s✿

2

T /2

2π

ak =

x (t) ❝♦s k

t

dt

(k = 0, 1, 2, ...)

✭✶✳✷✮

T

−T /2

T

2

T /2

2π

bk =

x (t) s✐♥ k

t

dt

(k = 1, 2, 3, ...)

✭✶✳✸✮

T

−T /2

T

❚❤❡ ♥❛t✉r❛❧ ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ ✭✶✳✶✮ ✐s ❛s ❛ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ s✐❣♥❛❧ x (t) ✐♥t♦ ✐♥❞✐✈✐❞✉❛❧ ♦s❝✐❧❧❛t✐♦♥s ✇❤❡r❡

ak ✐♥❞✐❝❛t❡s t❤❡ ❛♠♣❧✐t✉❞❡ ♦❢ t❤❡ ❡✈❡♥ ♦s❝✐❧❧❛t✐♦♥ ❝♦s k 2π t ♦❢ ❢r❡q✉❡♥❝② k/T ✭♠❡❛♥✐♥❣ ✐ts ♣❡r✐♦❞ ✐s T /k✮✱

T

❛♥❞ bk ✐♥❞✐❝❛t❡s t❤❡ ❛♠♣❧✐t✉❞❡ ♦❢ t❤❡ ♦❞❞ ♦s❝✐❧❧❛t✐♦♥ s✐♥ k 2π t ♦❢ ❢r❡q✉❡♥❝② k/T ✳ ❋♦r ❛♥ ❛✉❞✐♦ s✐❣♥❛❧ x (t)✱

T

❢r❡q✉❡♥❝② ❝♦rr❡s♣♦♥❞s t♦ ❤♦✇ ❤✐❣❤ ❛ s♦✉♥❞ ✐s ❛♥❞ ❛♠♣❧✐t✉❞❡ t♦ ❤♦✇ ❧♦✉❞ ✐t ✐s✳ ❚❤❡ ♦s❝✐❧❧❛t✐♦♥s ❛♣♣❡❛r✐♥❣

✐♥ t❤❡ ❋♦✉r✐❡r ❞❡❝♦♠♣♦s✐t✐♦♥ ❛r❡ ♦❢t❡♥ ❛❧s♦ ❝❛❧❧❡❞ ❤❛r♠♦♥✐❝s ✭✜rst✱ s❡❝♦♥❞✱ t❤✐r❞ ❤❛r♠♦♥✐❝ ❡t❝✮✳

◆♦t❡✿ ♦♥❡ ❝❛♥ ❛❧s♦ ✐♥t❡❣r❛t❡ ♦✈❡r [0, T ] ♦r ❛♥② ♦t❤❡r ✐♥t❡r✈❛❧ ♦❢ ❧❡♥❣t❤ T ✳ ◆♦t❡ ❛❧s♦✱ t❤❛t t❤❡ ❛✈❡r❛❣❡

✈❛❧✉❡ ♦❢ x (t) ♦✈❡r ♦♥❡ ♣❡r✐♦❞ ✐s ❡q✉❛❧ t♦ a0/2✳

❈♦♠♣❧❡① r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❋♦✉r✐❡r s❡r✐❡s

❖❢t❡♥ ✐t ✐s ♠♦r❡ ♣r❛❝t✐❝❛❧ t♦ ✇♦r❦ ✇✐t❤ ❝♦♠♣❧❡① ♥✉♠❜❡rs ✐♥ t❤❡ ❛r❡❛ ♦❢ ❋♦✉r✐❡r ❛♥❛❧②s✐s✳ ❯s✐♥❣ t❤❡ ❢❛♠♦✉s

❢♦r♠✉❧❛

ejα = ❝♦s (α) + js✐♥ (α)

✭✶✳✹✮

✐t ✐s ♣♦ss✐❜❧❡ t♦ s✐♠♣❧✐❢② s❡✈❡r❛❧ ❢♦r♠✉❧❛s ❛t t❤❡ ♣r✐❝❡ ♦❢ ✇♦r❦✐♥❣ ✇✐t❤ ❝♦♠♣❧❡① ♥✉♠❜❡rs✳ ❚♦✇❛r❞s t❤✐s ❡♥❞

✇❡ ✇r✐t❡

1

1

1

1

a❝♦s (α) + bs✐♥ (α) = a

ejα + e−jα + b

ejα − e−jα =

(a − bj) ejα +

(a + bj) e−jα

✭✶✳✺✮

2

2j

2

2

✶❚❤✐s ❝♦♥t❡♥t ✐s ❛✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡ ❛t ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✹✻✽✶✼✴✶✳✺✴❃✳

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

✶

✷

❈❍❆P❚❊❘ ✶✳ ❋❖❯❘■❊❘ ❙❊❘■❊❙✿ ❘❊❱■❊❲

❋r♦♠ t❤✐s ✇❡ ♦❜s❡r✈❡ t❤❛t ✇❡ ♠❛② r❡♣❧❛❝❡ t❤❡ ❝♦s ❛♥❞ s✐♥ ❤❛r♠♦♥✐❝s ❜② ❛ ♣❛✐r ♦❢ ❡①♣♦♥❡♥t✐❛❧ ❤❛r♠♦♥✐❝s

✇✐t❤ ♦♣♣♦s✐t❡ ❢r❡q✉❡♥❝✐❡s ❛♥❞ ✇✐t❤ ❝♦♠♣❧❡① ❛♠♣❧✐t✉❞❡s ✇❤✐❝❤ ❛r❡ ❝♦♥❥✉❣❛t❡ ❝♦♠♣❧❡① t♦ ❡❛❝❤ ♦t❤❡r✳ ■♥ ❢❛❝t✱

✇❡ ❛rr✐✈❡ ❛t t❤❡ ♠♦r❡ s✐♠♣❧❡ ❝♦♠♣❧❡① ❋♦✉r✐❡r s❡r✐❡s✿

∞

1

T

x (t) =

Xkej2πkt/T

✇✐t❤ Xk =

x (t) e−j2πkt/T dt

✭✶✳✻✮

T

k=−∞

0

◆♦t❡ t❤❛t X ✐s ❝♦♠♣❧❡①✱ ❜✉t x ✐s r❡❛❧✲✈❛❧✉❡❞ ✭t❤❡ ✐♠❛❣✐♥❛r② ♣❛rts ♦❢ ❛❧❧ t❤❡ t❡r♠s ✐♥

Xkej2πkt/T ❛❞❞ ✉♣

t♦ ③❡r♦❀ ✐♥ ♦t❤❡r ✇♦r❞s✱ t❤❡② ❝❛♥❝❡❧ ❡❛❝❤ ♦t❤❡r ♦✉t✮✳ ❚❤❡ ❛❜s♦❧✉t❡ ✈❛❧✉❡ ♦❢ Xk ❣✐✈❡s t❤❡ ❛♠♣❧✐t✉❞❡ ♦❢ t❤❡

❝♦♠♣❧❡① ❤❛r♠♦♥✐❝ ✇✐t❤ ❢r❡q✉❡♥❝② k/T ✭♠❡❛♥✐♥❣ ✐ts ♣❡r✐♦❞ ✐s T/k✮❀ t❤❡ ❛r❣✉♠❡♥t ♦❢ Xk ♣r♦✈✐❞❡s t❤❡ ♣❤❛s❡

❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❝♦♠♣❧❡① ❤❛r♠♦♥✐❝s✳ ■❢ x ✐s ❡✈❡♥✱ Xk ✐s r❡❛❧ ❢♦r ❛❧❧ k ❛♥❞ ❛❧❧ ❤❛r♠♦♥✐❝s ❛r❡ ✐♥ ♣❤❛s❡✳

❚♦ ✈❡r✐❢② ✭✶✳✻✮ ♥♦t❡ t❤❛t ❜② ✭✶✳✷✮ ❛♥❞ ✭✶✳✸✮ ✇❡ ❤❛✈❡ ❢♦r ♣♦s✐t✐✈❡ k

1

T

1

Xk =

x (t) [❝♦s (2πkt/T ) − js✐♥ (2πkt/T )] dt =

(ak − bkj) .

✭✶✳✼✮

T

0

2

❋♦r ♥❡❣❛t✐✈❡

∗

k ✇❡ ♥♦t❡ t❤❛t X−k = X∗ ❜② ✭✶✳✻✮✱ ✇❤❡r❡ () ❞❡♥♦t❡s t❤❡ ❝♦♥❥✉❣❛t❡ ❝♦♠♣❧❡①✳ ❇② ✭✶✳✺✮✱ t❤❡

k

Xk ❛r❡ ❡①❛❝t❧② ❛s t❤❡② ❛r❡ s✉♣♣♦s❡❞ t♦ ❜❡✳

Pr♦♣❡rt✐❡s

• ▲✐♥❡❛r✐t②✿ ❚❤❡ ❋♦✉r✐❡r ❝♦❡✣❝✐❡♥ts ♦❢ t❤❡ s✐❣♥❛❧ z (t) = cx (t) + y (t) ❛r❡ s✐♠♣❧②

Zk = cXk + Yk

✭✶✳✽✮

• ❈❤❛♥❣❡ ♦❢ ❢r❡q✉❡♥❝②✿ ❚❤❡ s✐❣♥❛❧ z (t) = x (λt) ❤❛s t❤❡ ♣❡r✐♦❞ T /λ ❛♥❞ ❤❛s t❤❡ s❛♠❡ ❋♦✉r✐❡r ❝♦❡✣❝✐❡♥ts

❛s x (t) ✖ ❜✉t t❤❡② ❝♦rr❡s♣♦♥❞ t♦ ❞✐✛❡r❡♥t ❢r❡q✉❡♥❝✐❡s f✿

∞

∞

Zk|

= X

s✐♥❝❡

z (t) = x (λt) =

X

Z

f = k

k |f = k

k ej2πλkt/T =

k ej2πt k

T /λ

✭✶✳✾✮

T /λ

T

k=−∞

k=−∞

❚❤❡ ❡q✉❛t✐♦♥ ♦♥ t❤❡ r✐❣❤t ❛❧❧♦✇s t♦ r❡❛❞ ♦✛ t❤❡ ❋♦✉r✐❡r ❝♦❡✣❝✐❡♥ts ❛♥❞ t♦ ❡st❛❜❧✐s❤ Zk = Xk✳ ✭❋♦r ❛♥

❛❧t❡r♥❛t✐✈❡ ❝♦♠♣✉t❛t✐♦♥ s❡❡ ❜♦① ❈♦♠♠❡♥t ❇♦① ✶ ✭♣✳ ✷✮✮

❈♦♠♠❡♥t ❇♦① ✶

T /λ

T /λ

Zk

=

1

z (t) e−2πjkt/(T/λ)dt = λ

x (tλ) e−2πjktλ/T dt

T /λ

0

T

0

✭✶✳✶✵✮

T

=

1

x (s) e−2πjks/T ds = X

T

0

k

• ❙❤✐❢t✿ ❚❤❡ ❋♦✉r✐❡r ❝♦❡✣❝✐❡♥ts ♦❢ z (t) = x (t + d) ❛r❡ s✐♠♣❧②

Zk = Xkej2πkd/T

✭✶✳✶✶✮

❚❤❡ ♠♦❞✉❧❛t✐♦♥ ✐s ♠✉❝❤ ♠♦r❡ s✐♠♣❧❡ ✐♥ ❝♦♠♣❧❡① ✇r✐t✐♥❣ t❤❡♥ ✐t ✇♦✉❧❞ ❜❡ ✇✐t❤ r❡❛❧ ❝♦❡✣❝✐❡♥ts✳ ❋♦r t❤❡

s♣❡❝✐❛❧ s❤✐❢t ❜② ❤❛❧❢ ❛ ♣❡r✐♦❞✱ ✐✳❡✳✱ d = T/2 ✇❡ ❤❛✈❡ Zk = Xkejπk = (−1)kXk✳

• ❉❡r✐✈❛t✐✈❡✿ ❚❤❡ ❋♦✉r✐❡r s❡r✐❡s ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ x (t) ✇✐t❤ ❞❡✈❡❧♦♣♠❡♥t ✭✶✳✶✮ ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ s✐♠♣❧②

❜② t❛❦✐♥❣ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ✭✶✳✶✮ t❡r♠ ❜② t❡r♠✿

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

✸

∞

2πj

x✬ (t) =

Xk · k

· ej2πkt/T

✭✶✳✶✷✮

T

k=−∞

❙❤♦rt✿ ✇❤❡♥ t❛❦✐♥❣ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ s✐❣♥❛❧✱ t❤❡ ❝♦♠♣❧❡① ❋♦✉r✐❡r ❝♦❡✣❝✐❡♥ts ❣❡t ♠✉❧t✐♣❧✐❡❞ ❜② k 2πj ✳

T

❈♦♥s❡q✉❡♥t❧②✱ t❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ❞❡❝❛② s❧♦✇❡r✳

❊①❛♠♣❧❡s

• ❚❤❡ ♣✉r❡ ♦s❝✐❧❧❛t✐♦♥ ✭❝♦♥t❛✐♥✐♥❣ ♦♥❧② ♦♥❡ r❡❛❧ ❜✉t t✇♦ ❝♦♠♣❧❡① ❢r❡q✉❡♥❝✐❡s✮

j

x (t) = s✐♥ (2πt/T )

X1 = −X−1 =

✭✶✳✶✸✮

2

♦r B1 = 1/2 = −B−1✱ ♦r b1 = 1✱ ❛♥❞ ❛❧❧ ♦t❤❡r ❝♦❡✣❝✐❡♥ts ❛r❡ ③❡r♦✳ ❚❤✐s ❢♦r♠✉❧❛ ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞

✇✐t❤♦✉t ❝♦♠♣✉t✐♥❣ ✐♥t❡❣r❛❧s ❜② ♥♦t✐♥❣ t❤❛t s✐♥ (α) = ejα − e−jα / (2j) = (j/2) e−jα − ejα ❛♥❞

s❡tt✐♥❣ α = 2πt/T ✳

• ❋✉♥❝t✐♦♥s ✇❤✐❝❤ ❛r❡ t✐♠❡✲❧✐♠✐t❡❞✱ ✐✳❡✳✱ ❞❡✜♥❡❞ ♦♥ ❛ ✜♥✐t❡ ✐♥t❡r✈❛❧ ❝❛♥ ❜❡ ♣❡r✐♦❞✐❝❛❧❧② ❡①t❡♥❞❡❞✳ ❊①❛♠♣❧❡

✇✐t❤ T = 2π✿

1

❢♦r 0 < t < π

4

x (t) = { −1

❢♦r − π < t < 0

bk = 2Bk =

❢♦r ♦❞❞ k ≥ 1

✭✶✳✶✹✮

πk

c

❢♦r t = 0, π

❛♥❞ ❛❧❧ ♦t❤❡r ❝♦❡✣❝✐❡♥ts ③❡r♦✳ ◆♦t❡ t❤❛t c ✐s ❛♥② ❝♦♥st❛♥t❀ t❤❡ ✈❛❧✉❡ ♦❢ c ❞♦❡s ♥♦t ❛✛❡❝t t❤❡ ❝♦❡✣❝✐❡♥ts bk✳ ❲❡ ❤❛✈❡ ❢♦r 0 < t < π

4

1

1

1 =

s✐♥ (t) + s✐♥ (3t) + s✐♥ (5t) + ...

✭✶✳✶✺✮

π

3

5

◆♦t❡ t❤❛t ❢♦r t = 0 t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ s❡r✐❡s ♦♥ t❤❡ r✐❣❤t ✐s ✵✱ ✇❤✐❝❤ ✐s ❡q✉❛❧ t♦ x (0−) + x (0+)✱ t❤❡

♠✐❞❞❧❡ ♦❢ t❤❡ ❥✉♠♣ ♦❢ x (t) ❛t ✵✱ ♥♦ ♠❛tt❡r ✇❤❛t c ✐s✳ ❙✐♠✐❧❛r ❢♦r t = π✳

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

✹

❈❍❆P❚❊❘ ✶✳ ❋❖❯❘■❊❘ ❙❊❘■❊❙✿ ❘❊❱■❊❲

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

❈❤❛♣t❡r ✷

❉✐s❝r❡t❡ ❋♦✉r✐❡r ❚r❛♥s❢♦r♠✶

✷✳✶ ❉✐s❝r❡t❡ ❋♦✉r✐❡r ❚r❛♥s❢♦r♠

❚❤❡ ❉✐s❝r❡t❡ ❋♦✉r✐❡r ❚r❛♥s❢♦r♠✱ ❢r♦♠ ♥♦✇ ♦♥ ❉❋❚✱ ♦❢ ❛ ✜♥✐t❡ ❧❡♥❣t❤ s❡q✉❡♥❝❡ (x0, ..., xK−1) ✐s ❞❡✜♥❡❞ ❛s K−1

Θ

xk=

xne−2πjk n

K

(k = 0, ..., K − 1) ✭✷✳✶✮

n=0

❚♦ ♠♦t✐✈❛t❡ t❤✐s tr❛♥s❢♦r♠ t❤✐♥❦ ♦❢ xn ❛s ❡q✉❛❧❧② s♣❛❝❡❞ s❛♠♣❧❡s ♦❢ ❛ T ✲♣❡r✐♦❞✐❝ s✐❣♥❛❧ x (t) ♦✈❡r ❛ ♣❡r✐♦❞✱

❡✳❣✳✱ xn = x (nT/K)✳ ❚❤❡♥✱ ✉s✐♥❣ t❤❡ ❘✐❡♠❛♥♥ ❙✉♠ ❛s ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ❛♥ ✐♥t❡❣r❛❧✱ ✐✳❡✳✱

K−1

nT

T

T

f

f (t) dt

✭✷✳✷✮

K

K

n=0

0

✇❡ ✜♥❞

K−1

T

Θ

nT

K

x

k/T

k =

x

e−2πj nT

K

x (t) e−2πjtk/T dt = KXk✭✷✳✸✮

K

T

n=0

0

◆♦t❡ t❤❛t t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ✐s ❜❡tt❡r✱ t❤❡ ❧❛r❣❡r t❤❡ s❛♠♣❧❡ s✐③❡ K ✐s✳

❫

❘❡♠❛r❦ ♦♥ ✇❤② t❤❡ ❢❛❝t♦r K ✐♥ ✭✷✳✸✮✿ r❡❝❛❧❧ t❤❛t Xk ✐s ❛♥ ❛✈❡r❛❣❡ ✇❤✐❧❡ xk ✐s ❛ s✉♠✳ ❚❛❦❡ ❢♦r ✐♥st❛♥❝❡

❫

k = 0✿ X0 ✐s t❤❡ ❛✈❡r❛❣❡ ♦❢ t❤❡ s✐❣♥❛❧ ✇❤✐❧❡ x0 ✐s t❤❡ s✉♠ ♦❢ t❤❡ s❛♠♣❧❡s✳

❋r♦♠ t❤❡ ❛❜♦✈❡ ✇❡ ♠❛② ❤♦♣❡ t❤❛t ❛ ❞❡✈❡❧♦♣♠❡♥t s✐♠✐❧❛r t♦ t❤❡ ❋♦✉r✐❡r s❡r✐❡s ✭✶✳✻✮ s❤♦✉❧❞ ❛❧s♦ ❡①✐st ✐♥

t❤❡ ❞✐s❝r❡t❡ ❝❛s❡✳ ❚♦ t❤✐s ❡♥❞✱ ✇❡ ♥♦t❡ ✜rst t❤❛t t❤❡ ❉❋❚ ✐s ❛ ❧✐♥❡❛r tr❛♥s❢♦r♠ ❛♥❞ ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❜② ❛

♠❛tr✐① ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✭t❤❡ ✏❡①♣♦♥❡♥t✑ T ♠❡❛♥s tr❛♥s♣♦s❡✮✿

T

Θ

Θ

x0, ..., xK−1

= DF TK · (x0, ..., xK−1)T .

✭✷✳✹✮

❚❤❡ ♠❛tr✐① DF TK ♣♦ss❡ss❡s K ❧✐♥❡s ❛♥❞ K r♦✇s❀ t❤❡ ❡♥tr② ✐♥ ❧✐♥❡ k r♦✇ n ✐s e−2πjkn/K✳ ❆ ❢❡✇ ❡①❛♠♣❧❡s

❛r❡





1

1

1

1









1

1



1

−j

−1

j



DF T





1 =

1

DF T2 =

✭✷✳✺✮





DF T4 = 



1

−1



1

−1

1

−1 





1

j

−1

−j

✶❚❤✐s ❝♦♥t❡♥t ✐s ❛✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡ ❛t ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✹✻✽✵✹✴✶✳✷✴❃✳

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

✺

✻

❈❍❆P❚❊❘ ✷✳ ❉■❙❈❘❊❚❊ ❋❖❯❘■❊❘ ❚❘❆◆❙❋❖❘▼

√

❚❤❡ r♦✇s ❛r❡ ♦rt❤♦❣♦♥❛❧✷ t♦ ❡❛❝❤ ♦t❤❡r✳ ❆❧s♦✱ ❛❧❧ r♦✇s ❤❛✈❡ ❧❡♥❣t❤✸ K✳ ❋✐♥❛❧❧②✱ t❤❡ ♠❛tr✐❝❡s ❛r❡ s②♠♠❡tr✐❝

✭❡①❝❤❛♥❣✐♥❣ ❧✐♥❡s ❢♦r r♦✇s ❞♦❡s ♥♦t ❝❤❛♥❣❡ t❤❡ ♠❛tr✐①✮✳ ❙♦✱ t❤❡ ♠✉❧t✐♣❧②✐♥❣ ❉❋❚ ✇✐t❤ ✐ts ❝♦♥❥✉❣❛t❡ ❝♦♠♣❧❡①

♠❛tr✐①

∗

(DF TK ) ✇❡ ❣❡t K t✐♠❡s t❤❡ ✉♥✐t ♠❛tr✐① ✭❞✐❛❣♦♥❛❧ ♠❛tr✐① ✇✐t❤ ❛❧❧ ❞✐❛❣♦♥❛❧ ❡❧❡♠❡♥ts ❡q✉❛❧ t♦ K✮✳

■♥✈❡rs❡ ❉❋❚

❋r♦♠ ❛❧❧ t❤✐s ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t t❤❡ ✐♥✈❡rs❡ ♠❛tr✐① ♦❢

∗

DF TK ✐s IDF TK = (1/K) · (DF TK) ✳ ❙✐♥❝❡

∗

(e−α) = eα ✇❡ ✜♥❞

K−1

1

Θ

xn =

xk e2πjk n

K

(n = 0, ..., K − 1) ✭✷✳✼✮

K k=0

❙♣❡❝tr❛❧ ✐♥t❡r♣r❡t❛t✐♦♥✱ s②♠♠❡tr✐❡s✱ ♣❡r✐♦❞✐❝✐t②

❈♦♠❜✐♥✐♥❣ ✭✷✳✸✮ ❛♥❞ ✭✷✳✼✮ ✇❡ ♠❛② ♥♦✇ ✐♥t❡r♣r❡t Θxk ❛s t❤❡ ❝♦❡✣❝✐❡♥t ♦❢ t❤❡ ❝♦♠♣❧❡① ❤❛r♠♦♥✐❝ ✇✐t❤

❢r❡q✉❡♥❝② k/T ✐♥ ❛ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ ❞✐s❝r❡t❡ s✐❣♥❛❧ xn❀ ✐ts ❛❜s♦❧✉t❡ ✈❛❧✉❡ ♣r♦✈✐❞❡s t❤❡ ❛♠♣❧✐t✉❞❡ ♦❢ t❤❡

❤❛r♠♦♥✐❝ ❛♥❞ ✐ts ❛r❣✉♠❡♥t t❤❡ ♣❤❛s❡ ❞✐✛❡r❡♥❝❡✳

■❢ x ✐s ❡✈❡♥✱ Θxk ✐s r❡❛❧ ❢♦r ❛❧❧ k ❛♥❞ ❛❧❧ ❤❛r♠♦♥✐❝s ❛r❡ ✐♥ ♣❤❛s❡✳

❯s✐♥❣ t❤❡ ♣❡r✐♦❞✐❝✐t② ♦❢ e2πjt ✇❡ ♦❜t❛✐♥ xn = xn+K ✇❤❡♥ ❡✈❛❧✉❛t✐♥❣ ✭✷✳✼✮ ❢♦r ❛r❜✐tr❛r② n✳ ❙❤♦rt✱ ✇❡ ❝❛♥

❝♦♥s✐❞❡r xn ❛s ❡q✉❛❧❧②✲s♣❛❝❡❞ s❛♠♣❧❡s ♦❢ t❤❡ T ✲♣❡r✐♦❞✐❝ s✐❣♥❛❧ x (t) ♦✈❡r ❛♥② ✐♥t❡r✈❛❧ ♦❢ ❧❡♥❣t❤ T ✿

K/2−1

Θ

xk=

xne−2πjk n

K .✭✷✳✽✮

n=−K/2

❙✐♠✐❧❛r❧②✱ Θ

Θ

xk ✐s ♣❡r✐♦❞✐❝✿ Θ

xk= xk+K✳ ❚❤✉s✱ ✐t ♠❛❦❡s s❡♥s❡ t♦ ❡✈❛❧✉❛t❡ Θ

xk ❢♦r ❛♥② k✳ ❋♦r ✐♥st❛♥❝❡✱ ✇❡ ❝❛♥

r❡✇r✐t❡ ✭✷✳✶✮ ❛s

K/2−1

1

Θ

xn =

xk e2πjk n

K ✭✷✳✾✮

K n=−K/2

❙✐♥❝❡ Θxk ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ❢r❡q✉❡♥❝② k/T ✱ t❤❡ ♣❡r✐♦❞ K ♦❢ Θxk ❝♦rr❡s♣♦♥❞s t♦ ❛ ♣❡r✐♦❞ ♦❢ K/T ✐♥ ❛❝t✉❛❧

❢r❡q✉❡♥❝②✳ ❚❤✐s ✐s ❡①❛❝t❧② t❤❡ s❛♠♣❧✐♥❣ ❢r❡q✉❡♥❝② ✭♦r s❛♠♣❧✐♥❣ r❛t❡✮ ♦❢ t❤❡ ♦r✐❣✐♥❛❧ s✐❣♥❛❧ ✭K s❛♠♣❧❡s ♣❡r T t✐♠❡ ✉♥✐ts✮✳ ❈♦♠♣❛r❡ t♦ t❤❡ s♣❡❝tr❛❧ r❡♣❡t✐t✐♦♥s✳

❍♦✇❡✈❡r✱ t❤❡ ♣❡r✐♦❞ T ♦❢ t❤❡ ♦r✐❣✐♥❛❧ s✐❣♥❛❧ x ✐s ♥♦✇❤❡r❡ ♣r❡s❡♥t ✐♥ t❤❡ ❢♦r♠✉❧❛s ♦❢ t❤❡ ❉❋❚ ✭❝♣r❡✳ ✭✷✳✶✮

❛♥❞ ✭✷✳✼✮✮✳ ❚❤✉s✱ ✐❢ ♥♦t❤✐♥❣ ✐s ❦♥♦✇♥ ❛❜♦✉t T ✱ ✐t ✐s ❛ss✉♠❡❞ t❤❛t t❤❡ s❛♠♣❧✐♥❣ r❛t❡ ✐s ✶ ✭✶ s❛♠♣❧❡ ♣❡r t✐♠❡

✉♥✐t✮✱ ♠❡❛♥✐♥❣ t❤❛t K = T ✳

❋❋❚

❚❤❡ ❋❛st ❋♦✉r✐❡r ❚r❛♥s❢♦r♠ ✭❋❋❚✮ ✐s ❛ ❝❧❡✈❡r ❛❧❣♦r✐t❤♠ ✇❤✐❝❤ ✐♠♣❧❡♠❡♥ts t❤❡ ❉❋❚ ✐♥ ♦♥❧② Klog (K)

♦♣❡r❛t✐♦♥s✳ ◆♦t❡ t❤❛t t❤❡ ♠❛tr✐① ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✇♦✉❧❞ r❡q✉✐r❡ K2 ♦♣❡r❛t✐♦♥s✳

▼❛t❧❛❜ ✐♠♣❧❡♠❡♥ts t❤❡ ❋❋❚ ✇✐t❤ t❤❡ ❝♦♠♠❛♥❞ ❢❢t✭①✮ ✇❤❡r❡ x ✐s t❤❡ ✐♥♣✉t ✈❡❝t♦r✳ ◆♦t❡ t❤❛t ✐♥ ▼❛t❧❛❜

t❤❡ ✐♥❞✐❝❡s st❛rt ❛❧✇❛②s ✇✐t❤ ✶✦ ❚❤✐s ♠❡❛♥s t❤❛t t❤❡ ✜rst ❡♥tr② ♦❢ t❤❡ ▼❛t❧❛❜ ✈❡❝t♦r x✱ ✐✳❡✳ x (1) ✐s t❤❡

s❛♠♣❧❡ ♣♦✐♥t x0 = x (0)✳ ❙✐♠✐❧❛r✱ t❤❡ ❧❛st ❡♥tr② ♦❢ t❤❡ ▼❛t❧❛❜ ✈❡❝t♦r x ✐s✱ ✐✳❡✳ x (K) ✐s t❤❡ s❛♠♣❧❡ ♣♦✐♥t xK−1 = x ((K − 1) T /K) = x (T − T /K)✳

✷❚❤❡ s❝❛❧❛r ♣r♦❞✉❝t ❢♦r ❝♦♠♣❧❡① ✈❡❝t♦rs x = (x1, x2, ..., xK) ❛♥❞ y = (y1, y2, ..., yK) ✐s ❝♦♠♣✉t❡❞ ❛s x · y = x1y∗

1 + x2y∗

2 + ... + xK y∗ ,

✭✷✳✻✮

K

✇❤❡r❡ a∗ ✐s t❤❡ ❝♦♥❥✉❣❛t❡ ❝♦♠♣❧❡① ♦❢ a✳ ❖rt❤♦❣♦♥❛❧ ♠❡❛♥s x · y = 0✳

✸

√

q

▲❡♥❣t❤ ✐s ❝♦♠♣✉t❡❞ ❛s ||x|| = x · x = px1x∗ + x

+ ... + x

=

|x

1

2x∗

2

K x∗

K

1|2 + |x2|2 + ... + |xK |2✳

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

❈❤❛♣t❡r ✸

❋♦✉r✐❡r ■♥t❡❣r❛❧✶

✸✳✶ ❋♦✉r✐❡r ■♥t❡❣r❛❧

❈♦♥t✐♥✉♦✉s✲t✐♠❡ s✐❣♥❛❧s x (t) ✇❤✐❝❤ ❛r❡ ♥♦t ♣❡r✐♦❞✐❝ ❝❛♥ st✐❧❧ ❜❡ ✉♥❞❡rst♦♦❞ ❛s s✉♣❡r♣♦s✐t✐♦♥s ♦❢ ♣✉r❡ ♦s❝✐❧✲

❧❛t✐♦♥s ej2πft ✇❤❡r❡ ♥♦✇ ❛❧❧ ❢r❡q✉❡♥❝✐❡s ❛r❡ ♣r❡s❡♥t ✐♥ t❤❡ s✐❣♥❛❧✳ ❚❤❡ ❝♦❡✣❝✐❡♥ts X (f) ♦❢ t❤❡ ♦s❝✐❧❧❛t✐♦♥s

❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❛s ❢♦❧❧♦✇s✿

∞

X (f ) =

x (t) e−j2πftdt

❬❋♦✉r✐❡r tr❛♥s❢♦r♠❪

✭✸✳✶✮

−∞

❚❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ❛s ❛ s✉♣❡r♣♦s✐t✐♦♥ t❛❦❡s t❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✿

∞

x (t) =

X (f ) ej2πftdf

❬■♥✈❡rs❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠❪

✭✸✳✷✮

−∞

❲❡ ❝❛❧❧ X (f) t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ♦❢ x ❛♥❞ ✇r✐t❡ ❛❧s♦ F{x} (f) ✐♥st❡❛❞ ♦❢ X (f) t♦ ✐♥❞✐❝❛t❡ ❝❧❡❛r❧② ✇❤✐❝❤

s✐❣♥❛❧ ❤❛s ❜❡❡♥ tr❛♥s❢♦r♠❡❞✳ ❚❤❡ ✏❋♦✉r✐❡r s♣❡❝tr✉♠✑✱ ♦r s✐♠♣❧② t❤❡ s♣❡❝tr✉♠✱ ♦r ❛❧s♦ t❤❡ ✏♣♦✇❡r s♣❡❝tr✉♠✑

♦❢ t❤❡ s✐❣♥❛❧ ✐s t❤❡ sq✉❛r❡❞ ❛♠♣❧✐t✉❞❡ |X (f) |2✳ ❚❤✐s ✐s t❤❡ ❢✉♥❝t✐♦♥ ✉s✉❛❧❧② ♣❧♦tt❡❞✱ ✇❤✐❧❡ t❤❡ ♣❤❛s❡ ♦❢ X

✐s ♥♦t s❤♦✇♥✳ ◆❡✈❡rt❤❡❧❡ss✱ t❤❡ ♣❧♦ts ❛r❡ ✉s✉❛❧❧② ✖❛♥❞ ❡rr♦♥❡♦✉s❧②✖ ❧❛❜❡❧❡❞ ✇✐t❤ X ✐♥st❡❛❞ ♦❢ |X|2 ✭s❡❡

❋✐❣✉r❡ ✹✳✶✮✳

❆ s✐❣♥❛❧ ✐s ❝❛❧❧❡❞ ❜❛♥❞❧✐♠✐t❡❞ ✐❢ ✐ts ❋♦✉r✐❡r tr❛♥s❢♦r♠ X (f) ✐s ③❡r♦ ❢♦r ❤✐❣❤ ❢r❡q✉❡♥❝✐❡s✱ ✐✳❡✳ ❢♦r ❧❛r❣❡ |f|✳

❙✐♠✐❧❛r❧② ✇❡ s❛② t❤❛t ❛ s✐❣♥❛❧ ✐s t✐♠❡✲❧✐♠✐t❡❞ ✐❢ ✐t ✐s ③❡r♦ ❢♦r ❧❛r❣❡ t✐♠❡s✱ ✐✳❡✳✱ ❢♦r ❧❛r❣❡ |t|✳ ❇② ❍❡✐s❡♥❜❡r❣✬s

♣r✐♥❝✐♣❧❡ ❛ ❜❛♥❞❧✐♠✐t❡❞ s✐❣♥❛❧ ❝❛♥ ♥♦t ❜❡ t✐♠❡✲❧✐♠✐t❡❞✳ ❙✐♥❝❡ ❜❛♥❞❧✐♠✐t❡❞ s✐❣♥❛❧s ❛r❡ ♦❢ ❣r❡❛t ✐♠♣♦rt❛♥❝❡✱

t❤❡r❡ ✐s ❛ ♥❡❡❞ t♦ st✉❞② s✐❣♥❛❧s ✇❤✐❝❤ ❛r❡ ♥♦t t✐♠❡✲❧✐♠✐t❡❞ ❛♥❞✱ t❤✉s✱ t❤❡ ❋♦✉r✐❡r ✐♥t❡❣r❛❧✳

Pr♦♣❡rt✐❡s

• ▲✐♥❡❛r✐t②✿

F {ax + y} (f ) = aX (f ) + Y (f )

✭✸✳✸✮

• ❈♦♥✈♦❧✉t✐♦♥

F {x ∗ y} (f ) = X (f ) · Y (f )

F {x · y} (f ) = X (f ) ∗ Y (f )

✭✸✳✹✮

• ❈❤❛♥❣❡ ♦❢ t✐♠❡ s❝❛❧❡

1

t

1

f

F { x

} (f ) = X (af )

F {x (at)} (f ) =

X

✭✸✳✺✮

a

a

a

a

✶❚❤✐s ❝♦♥t❡♥t ✐s ❛✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡ ❛t ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✹✻✽✶✾✴✶✳✸✴❃✳

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

✼

✽

❈❍❆P❚❊❘ ✸✳ ❋❖❯❘■❊❘ ■◆❚❊●❘❆▲

• ❚r❛♥s❧❛t✐♦♥ ✐♥ t✐♠❡ ❛♥❞ ❢r❡q✉❡♥❝②

F {x (t − b)} (f ) = X (f ) e−j2πbf

F {x (t) ej2πbt} (f ) = X (f − b)

✭✸✳✻✮

• ❙②♠♠❡tr✐❡s ❛♥❞ ❋♦✉r✐❡r ♣❛✐rs ❚❤❡ s②♠♠❡tr② ♦❢ ✭✸✳✷✮ ❛♥❞ ✭✸✳✶✮ ❧❡❛❞s ♦♥❡ t♦ ❝♦♥s✐❞❡r x (t) ❛♥❞ X (f)

❛s ❛ ❋♦✉r✐❡r ♣❛✐r✳ ■♥❞❡❡❞✱ t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ♦❢ X ✐s ❛❧♠♦st x✿ F{X} (f) = x (−f)✳ ❈❧❡❛r❧②✱

t❤❡ s②♠♠❡tr② ✐s ♥♦t ♣❡r❢❡❝t s✐♥❝❡ X (f) ✐s ✐♥ ❣❡♥❡r❛❧ ❝♦♠♣❧❡①✱ ✇❤✐❧❡ x ✐s r❡❛❧✳ ❍♦✇❡✈❡r✿ ■❢ x (t) ✐s

s②♠♠❡tr✐❝✱ ✐✳❡✳ x (−t) = x (t) t❤❡♥ X (f) ✐s r❡❛❧✲✈❛❧✉❡❞✱ ❛♥❞ ✈✐❝❡ ✈❡rs❛✦

■♥ s✉♠♠❛r②✿ ❙②♠♠❡tr✐❝ r❡❛❧ s✐❣♥❛❧s ❤❛✈❡ s②♠♠❡tr✐❝ r❡❛❧ ❋♦✉r✐❡r tr❛♥s❢♦r♠s ❛♥❞ ✈✐❝❡ ✈❡rs❛✳ ❆s ✇❡ ✇✐❧❧ s❡❡

❜❡❧♦✇✱ t❤❡② ❛❧s♦ ♣♦ss❡ss t❤❡ s❛♠❡ ❡♥❡r❣②✳

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

❈❤❛♣t❡r ✹

❊♥❡r❣② ❛♥❞ P♦✇❡r✶

✹✳✶ ❊♥❡r❣② ❛♥❞ P♦✇❡r

❚❤❡ ❡♥❡r❣② ♦❢ ❛ ❝♦♥t✐♥✉♦✉s✲t✐♠❡ s✐❣♥❛❧ x (t) ✐s ❣✐✈❡♥ ❛s

∞

||x||2 :=

x2 (t) dt

✭✹✳✶✮

−∞

P❧❛♥❝❤❡r❡❧✬s t❤❡♦r❡♠ s❛②s ✭❢♦r ♠♦r❡ ✐♥❢♦r♠❛t✐♦♥ s❡❡ ❈♦♠♠❡♥t ❇♦① ✷ ✭♣✳ ✾✮✮✿ ■❢ t❤❡ s✐❣♥❛❧ x (t) ❤❛s ✜♥✐t❡

❡♥❡r❣② t❤❡♥ ✐ts ❋♦✉r✐❡r tr❛♥s❢♦r♠ X (f) ❤❛s t❤❡ s❛♠❡ ❡♥❡r❣②✿

∞

∞

||X (f ) ||2 =

|X|2 (f ) df =

x2 (t) dt = ||x||2

❬✜♥✐t❡ ❡♥❡r❣② ❝❛s❡❪

✭✹✳✷✮

−∞

−∞

❈♦♠♠❡♥t ❇♦① ✷ P❧❛♥❝❤❡r❡❧ t❤❡♦r❡♠ ✐s ❛ r❡s✉❧t ✐♥ ❤❛r♠♦♥✐❝ ❛♥❛❧②s✐s✱ ✜rst ♣r♦✈❡❞ ❜② ▼✐❝❤❡❧ P❧❛♥❝❤❡r❡❧✳

■♥ ✐ts s✐♠♣❧❡st ❢♦r♠ ✐t st❛t❡s t❤❛t ✐❢ ❛ ❢✉♥❝t✐♦♥ ❢ ✐s ✐♥ ❜♦t❤ L1 (IR) ❛♥❞ L2 (IR)✱ t❤❡♥ ✐ts ❋♦✉r✐❡r tr❛♥s❢♦r♠ ✐s

✐♥ L2 (IR)❀ ♠♦r❡♦✈❡r t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ♠❛♣ ✐s ✐s♦♠❡tr✐❝✳ ❚❤✐s ✐♠♣❧✐❡s t❤❛t t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ♠❛♣

r❡str✐❝t❡❞ t♦ L1 (IR) ∩ L2 (IR) ❤❛s ❛ ✉♥✐q✉❡ ❡①t❡♥s✐♦♥ t♦ ❛ ❧✐♥❡❛r ✐s♦♠❡tr✐❝ ♠❛♣ L2 (IR) → L2 (IR)✳ ❚❤✐s

✐s♦♠❡tr② ✐s ❛❝t✉❛❧❧② ❛ ✉♥✐t❛r② ♠❛♣✳

P❡r✐♦❞✐❝ s✐❣♥❛❧s ❤❛✈❡ ♦❢ ❝♦✉rs❡ ✐♥✜♥✐t❡ ❡♥❡r❣②❀ t❤❡r❡❢♦r❡✱ ♦♥❡ ✐♥tr♦❞✉❝❡s t❤❡ ♣♦✇❡r Px ♦❢ t❤❡ s✐❣♥❛❧

x (t)✱ ✇❤✐❝❤ ✐s t❤❡ ❛✈❡r❛❣❡ ❡♥❡r❣② ♦✈❡r ♦♥❡ ♣❡r✐♦❞✳ ❊♥❡r❣② ✐s ♠❡❛s✉r❡❞ ✐♥ ❏♦✉❧❡✱ ♣♦✇❡r ✐s ♠❡❛s✉r❡❞ ✐♥

❲❛tt❂❏♦✉❧❡✴❙❡❝✳

❚❤❡ ❛♥❛❧♦❣ ♦❢ P❧❛♥❝❤❡r❡❧✬s t❤❡♦r❡♠ ✐s P❛rs❡✈❛❧✬s t❤❡♦r❡♠ ✇❤✐❝❤ ❛♣♣❧✐❡s t♦ T ✲♣❡r✐♦❞✐❝ s✐❣♥❛❧s ❛♥❞ s❛②s

∞

1

T /2

P = Px =

|x (t) |2dt =

|Xk|2

❬♣❡r✐♦❞✐❝ ❝❛s❡❪

✭✹✳✸✮

T

−T /2

k=−∞

❲❡ ♠❛② ❞❡r✐✈❡ P❛rs❡✈❛❧✬s t❤❡♦r❡♠ ❛s ❢♦❧❧♦✇s✱ ✉s✐♥❣ ✭✶✳✻✮ ❛♥❞ |a|2 = a · a∗✿

T /2

T /2

∞

Px

=

1

|x (t) |2dt = 1

X

T

−T /2

T

−T /2

k=−∞

k ej2πkt/T 2dt

T /2

∞

=

1

X

X

T

−T /2

k=−∞

k ej2πkt/T

∞

n=−∞

nej2πnt/T ∗dt

T /2

∞

=

1

X

X∗e−j2πnt/T dt

✭✹✳✹✮

T

−T /2

k=−∞

k ej2πkt/T

∞

n=−∞

n

∞

∞

T /2

=

X

1

ej2π(k−n)t/T dt

k=−∞

n=−∞

k X ∗

n T

−T /2

∞

∞

=

X

=

|X

k=−∞

k X ∗

k

k=−∞

k |2

✶❚❤✐s ❝♦♥t❡♥t ✐s ❛✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡ ❛t ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✹✻✽✶✶✴✶✳✹✴❃✳

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

✾

✶✵

❈❍❆P❚❊❘ ✹✳ ❊◆❊❘●❨ ❆◆❉ P❖❲❊❘

❍❡r❡✱ ✇❡ ✉s❡❞ t❤❛t T/2 ej2π(k−n)t/T ❡q✉❛❧s T ✇❤❡♥ k = n ✭s✐♥❝❡ e0 = 1✮✱ ❜✉t ❡q✉❛❧s ✵ ✇❤❡♥ k = n s✐♥❝❡

−T /2

✭s✐♥❝❡ ejs = cos (s) + jsin (s)✱ ✇❤✐❝❤ ❛r❡ ✐♥t❡❣r❛t❡❞ ♦✈❡r s❡✈❡r❛❧ ♣❡r✐♦❞s✮✳

❆ s✐♠✐❧❛r ❝♦♠♣✉t❛t✐♦♥ ❝❛♥ ❜❡ ❝❛rr✐❡❞ ♦✉t ❢♦r P❧❛♥❝❤❡r❡❧✬s ❡q✉❛t✐♦♥✳ ❍♦✇❡✈❡r✱ s♦♠❡ ❞✐✣❝✉❧t✐❡s ❛r✐s❡ ❞✉❡

t♦ t❤❡ ✐♥t❡❣r❛❧s ♦✈❡r ✐♥✜♥✐t❡ ✐♥t❡r✈❛❧s ✭s❡❡ ❈♦♠♠❡♥t ❇♦① ✸ ✭♣✳ ✶✵✮ ❜❡❧♦✇✮✳ ❆❧s♦✱ ❛ ❥✉st✐✜❝❛t✐♦♥ ♦❢ P❧❛♥❝❤❡r❡❧

❝♦✉❧❞ ❜❡ ❣✐✈❡♥ ❜② ♣❡r❢♦r♠✐♥❣ ❛ ❧✐♠✐t ♦❢ ✐♥✜♥✐t❡ ♣❡r✐♦❞ ✐♥ P❛rs❡✈❛❧✬s ❡q✉❛t✐♦♥ ✭s❡❡ ❈♦♠♠❡♥t ❇♦① ✹ ✭♣✳ ✶✷✮

❜❡❧♦✇✮✳

❋♦r ✜♥✐t❡ ❞✐s❝r❡t❡ s✐❣♥❛❧s t❤❡ ❛♥❛❧♦❣ ✐s s✐♠♣❧② t❤❡ ❢❛❝t✱ t❤❛t ❉❋❚ ✐s ✉♥✐t❛r② ✉♣ t♦ ❛ str❡t❝❤✐♥❣ ❢❛❝t♦r✳

√

▼♦r❡ ♣r❡❝✐s❡❧②✱ t❤❡ ♠❛tr✐① DF TK ❧❡❛✈❡s ❛♥❣❧❡s ✐♥t❛❝t ❛♥❞ str❡t❝❤❡s ❧❡♥❣t❤ ❜② K✳ ■♥t✉✐t✐✈❡❧②✱ ♦♥❡ ♠❛②

t❤✐♥❦ ♦❢ t❤❡ ❉❋❚ ❛s ❛ r♦t❛t✐♦♥ ❛♥❞ ❛ str❡t❝❤✐♥❣✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t♦ ♣❡r❢♦r♠ ❛ ❉❋❚ s✐♠♣❧② ♠❡❛♥s t♦ ❝❤❛♥❣❡

√

t❤❡ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ✐♥t♦ ❛ ♥❡✇ ♦♥❡✱ ❛♥❞ t♦ ❝❤❛♥❣❡ ❧❡♥❣t❤ ♠❡❛s✉r❡♠❡♥t ❜② ❛ ❢❛❝t♦r K✳ ❚❤✉s✿

K

K

1

1

2

Θ

1

Px =

x2 =

|xk | =

P

❬❞✐s❝r❡t❡ ❝❛s❡❪✭✹✳✺✮

K

n

K2

K ❫

n=1

x

k=1

◆♦t❡ t❤❛t t❤❡ ❉❋❚ ❋♦✉r✐❡r ❝♦❡✣❝✐❡♥ts ❛r❡ ❝♦♠♣❧❡① ♥✉♠❜❡rs❀ t❤✉s✱ t❤❡ ❛❜s♦❧✉t❡ ✈❛❧✉❡ ❤❛s t♦ ❜❡ t❛❦❡♥ ✭❢♦r

❛ ❝♦♠♣❧❡① ♥✉♠❜❡r a ✇❡ ❤❛✈❡ |a|2 = a · a∗✱ ✇❤✐❝❤ ✐s ✉s✉❛❧❧② ❞✐✛❡r❡♥t ❢r♦♠ a2✖✉♥❧❡ss a ✐s ❜② ❝❤❛♥❝❡ r❡❛❧

✈❛❧✉❡❞✮✳

❈♦♠♠❡♥t ❇♦① ✸ ❆ ✏❤❛♥❞✲✇❛✈✐♥❣✑ ❛r❣✉♠❡♥t ❢♦r P❧❛♥❝❤❡r❡❧✬s t❤❡♦r❡♠ r✉♥s ❛s ❢♦❧❧♦✇s✱ ✉s✐♥❣ ✭✸✳✶✮ ❛♥❞

|a|2 = a · a∗✿

∞

∞

∞

2

||x (t) ||2

=

|x (t) |2dt =

X (f ) ej2πftdf

dt

−∞

−∞

−∞

∗

∞

∞

∞

=

X (f ) ej2πftdf

X (g) ej2πgtdg

dt

−∞

−∞

−∞

∞

∞

∞

∗

=

X (f ) ej2πftdf

X(g) e−j2πgtdg dt

−∞

−∞

−∞

✭✹✳✻✮

∞

∞

∗

∞

=

X (f ) X(g)

ej2πt(f−g)dt df dg

−∞

−∞

−∞

∞

∞

∗

=

X (f ) X(g) δ (f − g) df dg

−∞

−∞

∞

∗

∞

=

X (f ) X(f ) dt =

|X (f ) |2dt = ||X||2

−∞

−∞

❚❤❡r❡❜②✱ t❤❡ st❡♣ ∞ ej2πt(f−g)dt = δ (f − g) ✇♦✉❧❞ r❡q✉✐r❡ s♦♠❡ ♠♦r❡ ❝❛r❡✱ ❜✉t ✇❡ ❝♦♥t❡♥t ♦✉rs❡❧✈❡s ❤❡r❡

−∞

✇✐t❤ t❤✐s ✐♥t✉✐t✐✈❡ ❝♦♠♣✉t❛t✐♦♥✳

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

✶✶

✭❛✮

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

✭❜✮

✭❝✮

❋✐❣✉r❡ ✹✳✶✿ ✭❛✮ ❆♥ ✐♠♣♦rt❛♥t ❡①❛♠♣❧❡ ♦❢ ❛ ❜❛♥❞✲❧✐♠✐t❡❞ s✐❣♥❛❧✿ t❤❡ s✐♥❝✲❢✉♥❝t✐♦♥✳ ✭❜✮ ❙✐♥❝ ❛♥❞ t❤❡

✉♥✲♥♦r♠❛❧✐③❡❞ ✈❡rs✐♦♥ ✐♥ ❝♦♠♣❛r✐s♦♥✳ ✭❝✮ ❚❤❡ ♣♦✇❡r s♣❡❝tr✉♠ ♦❢ s✐♥❝ ✐s ❘❡❝t✭❢✮✳

✶✷

❈❍❆P❚❊❘ ✹✳ ❊◆❊❘●❨ ❆◆❉ P❖❲❊❘

❈♦♠♠❡♥t ❇♦① ✹ ❲✐t❤ P❛rs❡✈❛❧✬s ❡q✉❛t✐♦♥ ❡st❛❜❧✐s❤❡❞ ♦♥❡ ♠❛② ♣r♦✈✐❞❡ ❛ ❞❡r✐✈❛t✐♦♥ ♦❢ P❧❛♥❝❤❡r❡❧✬s

❢♦r♠✉❧❛ ✇❤✐❝❤ ✐s ♠♦r❡ ❝♦♥✈✐♥❝✐♥❣✱ t❤♦ ♠♦r❡ ❞✐✣❝✉❧t t♦ ♠❛❦❡ r✐❣♦r♦✉s✳ ❆ss✉♠❡ t❤❛t t❤❡ s✐❣♥❛❧ x (t) ✐s t✐♠❡✲

❧✐♠✐t❡❞✱ s❛② ❞❡✜♥❡❞ ♦♥ −

T /2

T /2 < t < T /2✳ ■ts ❋♦✉r✐❡r✲s❡r✐❡s ❝♦❡✣❝✐❡♥ts ❛r❡ Xk = 1/T

x (t) e−j2πkt/T dt✳

−T /2

❈♦♠♣❛r✐♥❣ t♦ t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ✇❡ ✜♥❞ t❤✉s T ·Xk = X (k/T )✳ ❲❡ ♠❛② ✐♥t❡r♣r❡t x ❛s ❜❡✐♥❣ ♣❡r✐♦❞✐❝❛❧❧②

❡①t❡♥❞❡❞ ❛♥❞ ✉s❡ P❛rs❡✈❛❧✬s ❡q✉❛t✐♦♥ ✇❤✐❝❤ s❛②s Px =

|Xk|2❀ t❤✐s ❛❧❧♦✇s t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♠♣✉t❛t✐♦♥

||

T /2

∞

∞

x||2

=

|x (t) |2dt = T · P

|X

1 T 2|X

−T /2

x = T

k=−∞

k |2 =

k=−∞ T

k |2

✭✹✳✼✮

∞

=

1 |X (k/T ) |2

∞ |X (t) |2dt

k=−∞ T

−∞

■♥ t❤❡ ❧❛st st❡♣ ✇❡ ✉s❡ ❛ ❘✐❡♠❛♥♥ s✉♠ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ✐♥t❡❣r❛❧✳ ■t r❡♠❛✐♥s t♦ ❡①t❡♥❞ t❤✐s t♦ s✐❣♥❛❧s

✇❤✐❝❤ ❛r❡ ♥♦t t✐♠❡✲❧✐♠✐t❡❞✳ ❚❤❡ ❛♣♣r♦①✐♠❛t✐♦♥s ♠❛❞❡ ❛r❡ ♠♦r❡ ❜❡❧✐❡✈❛❜❧❡ t❤❛♥ t❤❡ st❡♣ ✉s✐♥❣ t❤❡ ❉✐r❛❝ ❞❡❧t❛

✭s❡❡ ❈♦♠♠❡♥t ❇♦① ✸ ✭♣✳ ✶✵✮ ❛❜♦✈❡✮✳ ❆❧t❤♦✉❣❤ ♠♦r❡ ✐♥t✉✐t✐✈❡ ❛♥❞ ❜❡❧✐❡✈❛❜❧❡✱ t❤❡s❡ ❛r❣✉♠❡♥ts ❛r❡ ❤❛r❞❡r t♦ ♠❛❦❡ r✐❣♦r♦✉s t❤❛♥ ♠❛❦✐♥❣ t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❈♦♠♠❡♥t ❇♦① ✸ ✭♣✳ ✶✵✮ r✐❣♦r♦✉s ❜② ♣❛ss✐♥❣ t❤r♦✉❣❤ ❛

❝♦rr❡❝t✱ ❞✐r❡❝t ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ✐♥t❡❣r❛❧s ❛✈♦✐❞✐♥❣ t❤❡ ❉✐r❛❝ ❞❡❧t❛✳

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

❈❤❛♣t❡r ✺

❊①❛♠♣❧❡s✶

✺✳✶ ❊①❛♠♣❧❡s

• ❚❤❡ ♣✉r❡ ♦s❝✐❧❧❛t✐♦♥ ✭❝♦♥t❛✐♥✐♥❣ ♦♥❧② ♦♥❡ ❢r❡q✉❡♥❝②✮

j

sin (2πt/T )

F {sin (2πt/T )} (f ) =

(δ (−1) − δ (1))

✭✺✳✶✮

2

❚❤✐s ❢♦r♠✉❧❛ ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ✇✐t❤♦✉t ❝♦♠♣✉t✐♥❣ ✐♥t❡❣r❛❧s ❜② ♥♦t✐♥❣ t❤❛t sin (x) =

ejx − e−jx / (2j) = (j/2) e−jx − ejx ✳ ■ts ♣♦✇❡r ✐s P = 1/2 ✭s❡❡ ✭✹✳✸✮✮✳

• ❚❤❡ ♣❡r❢❡❝t ❧♦✇ ✭❢r❡q✉❡♥❝②✮ ♣❛ss ❢✉♥❝t✐♦♥✿

1

✐❢ − 1/2 < f < 1/2

s✐♥❝

sin (πt)

(t) :=

F {s✐♥❝ (t)} (f) = ❘❡❝t (f) = {

✭✺✳✷✮

πt

0

❡❧s❡✳

❛♥❞ ♠♦r❡ ❣❡♥❡r❛❧ ✭t♦ ♣❛ss ❡①❛❝t❧② t❤❡ ❢r❡q✉❡♥❝✐❡s f ∈ −fc, fc [✮

1

✐❢ − fc < f < fc

2fc · s✐♥❝ (2fct)

F {2fcs✐♥❝ (2fct)} (f) = ❘❡❝t

f

= {

✭✺✳✸✮

2fc

0

❡❧s❡✳

❇♦t❤✱ x ❛♥❞ X ❛r❡ s②♠♠❡tr✐❝ ❛♥❞ r❡❛❧✳ P❧❛♥❝❤❡r❡❧✬s ❢♦r♠✉❧❛ ❛❧❧♦✇s t♦ ❝♦♠♣✉t❡ t❤❡ ❡♥❡r❣② ♦❢ t❤❡ s✐♥❝✿

∞

1/2

||s✐♥❝ (t) ||2 =

X2 (f ) df =

1df = 1.

✭✺✳✹✮

−∞

−1/2

▼♦r❡ ❣❡♥❡r❛❧❧②✱ t❤❡ ❡♥❡r❣② ♦❢ 2fc · s✐♥❝ (2fct) ❛♠♦✉♥ts t♦ 2fc✳ ◆♦t❡ t❤❛t t❤❡ s✐♥❝ ✐s ♥♦t t✐♠❡✲❧✐♠✐t❡❞❀

✐t ❝❛♥✬t ❜❡ ❜② t❤❡ ❍❡✐s❡♥❜❡r❣ ♣r✐♥❝✐♣❧❡ s✐♥❝❡ ✐t ✐s ❜❛♥❞❧✐♠✐t❡❞✳

• ❚❤❡ ❉✐r❛❝ ❢✉♥❝t✐♦♥ δ (t) ✐s ♦❢t❡♥ s②♠❜♦❧✐❝❛❧❧② ✇r✐tt❡♥ ❛s

“∞✬✬

✐❢ t = 0

δ (t) = {

F {δ (t)} (f ) ≡ 1

✭✺✳✺✮

0

❡❧s❡✳

❈❧❡❛r❧②✱ t❤❡ ❉✐r❛❝ ❢✉♥❝t✐♦♥ ✐s ♥♦t r❡❛❧❧② ❛ ❢✉♥❝t✐♦♥✱ ❛♥❞ ✐t ❤❛s ✐♥✜♥✐t❡ ❡♥❡r❣②✳ ❍♦✇❡✈❡r✱ ♠♦st ♠❛♥✐♣✲

✉❧❛t✐♦♥s ✇♦r❦ ✜♥❡ ❛❧s♦ ❢♦r δ (t)✳ ❚❤✐s ✐s ❛❣❛✐♥ ❛♥ ✐❧❧✉str❛t✐♦♥ ♦❢ t❤❡ ❍❡✐s❡♥❜❡r❣ ♣r✐♥❝✐♣❧❡✳ ❚❤❡ ❉✐r❛❝

❢✉♥❝t✐♦♥ ✐s t❤❡ ❡①tr❡♠❡ ❝❛s❡ ✇❤✐❝❤ ✐s s❤❛r♣❧② ❧♦❝❛t❡❞ ✐♥ t✐♠❡✱ ❜✉t ❤❛s ♥♦ ❝❤❛r❛❝t❡r✐st✐❝ ❢r❡q✉❡♥❝② ✭❛❧❧

✶❚❤✐s ❝♦♥t❡♥t ✐s ❛✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡ ❛t ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✹✻✽✶✺✴✶✳✸✴❃✳

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

✶✸

✶✹

❈❍❆P❚❊❘ ✺✳ ❊❳❆▼P▲❊❙

❢r❡q✉❡♥❝✐❡s ❛r❡ ♣r❡s❡♥t ✇✐t❤ ❡q✉❛❧ str❡♥❣t❤✮✳ ❚❤❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❉✐r❛❝ ❢✉♥❝t✐♦♥ ❛r❡ ❜❡st ✉♥❞❡rst♦♦❞

✐♥ t❡r♠s ♦❢ ✐♥t❡❣r❛❧s✿

v

g (0)

✐❢ u < 0 < v

g (t) δ (t) dt = {

.

✭✺✳✻✮

u

0

❡❧s❡

❆s ❛ s♣❡❝✐❛❧ ❝❛s❡✱ t❤❡ ❝♦♥✈♦❧✉t✐♦♥ ✇✐t❤ ❛ ❢✉♥❝t✐♦♥ g ✐s ❛❣❛✐♥ g✿

∞

∞

{δ ∗ g} (a) =

δ (a − t) g (t) dt =

δ (t) g (a − t) dt = g (a)

✭✺✳✼✮

−∞

−∞

s❤♦rt✿ δ ∗ g (t) = g (t)✳ ❋♦r t❤❡ s❤✐❢t❡❞ ❉✐r❛❝ ❢✉♥❝t✐♦♥ δb (t) = δ (t − b) ✇❡ ❤❛✈❡

{

∞

∞

δb ∗ g} (a)

=

δ

δ (t − b) g (a − t) dt

=

✭✺✳✽✮

−∞

b (t) g (a − t) dt

=

−∞

∞ δ (s) g (a − (s + b)) ds = g (a − b)

−∞

s❤♦rt✱ ❝♦♥✈♦❧✉t✐♦♥ ✇✐t❤ δb ♣r♦❞✉❝❡s ❛ s❤✐❢t ❜② b✿ δb ∗ g (t) = g (t − b)✳

• ❉♦✉❜❧❡ ❡①♣♦♥❡♥t✐❛❧✿

2

x (t) = e−|t|

X (f ) =

✭✺✳✾✮

1 + 4π2f 2

◆♦t❡ t❤❛t x ❛♥❞ ✐ts ❋♦✉r✐❡r tr❛♥s❢♦r♠ X ❛r❡ r❡❛❧ ❛♥❞ s②♠♠❡tr✐❝✳ ❚❤❡ ♣♦✇❡r s♣❡❝tr✉♠ ✐s |X (f) |2 =

4/ 1 + 4π2f 2 2✳ ❙✐♥❝❡ x ✐s ♥♦t ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t ✵✱ t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ X ❞❡❝❛②s s♦♠❡✇❤❛t s❧♦✇❧②✿

❤✐❣❤ ❢r❡q✉❡♥❝✐❡s ❛r❡ q✉✐t❡ str♦♥❣ ✐♥ t❤✐s s✐❣♥❛❧ ✐♥ ♦r❞❡r t♦ ♠❛❦❡ t❤❡ s❤❛r♣ ♣❡❛❦ ❛t ✵✳ ❲✐t❤ t❤✐s ❡①❛♠♣❧❡✱

✇❡ ♠❛② ❝♦♠♣✉t❡ t❤❡ ❡♥❡r❣② ❞✐r❡❝t❧②✿

∞

2

∞

||e−|t|||2 =

e−|t|

dt = 2

e−2tdt = −e−2t|∞ = 1.

✭✺✳✶✵✮

0

−∞

0

• ❖♥❡✲s✐❞❡❞ ❊①♣♦♥❡♥t✐❛❧✿

e−t

✐❢ t > 0

1

x (t) = {

X (f ) =

✭✺✳✶✶✮

0

❡❧s❡✳

1 + 2πjf

❚❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ X ✐s ❝♦♠♣❧❡① ✇✐t❤ ♣♦✇❡r s♣❡❝tr✉♠ |X (f) |2 = 1/ 1 + 4π2f2 ✳ ❙✐♥❝❡ x ✐s ♥♦t

❡✈❡♥ ❝♦♥t✐♥✉♦✉s ❛t ✵✱ t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ X ❞❡❝❛②s ❡✈❡♥ s❧♦✇❡r t❤❛♥ ❢♦r t❤❡ ❞♦✉❜❧❡ ❡①♣♦♥❡♥t✐❛❧✿

❤✐❣❤ ❢r❡q✉❡♥❝✐❡s ❛r❡ ❡✈❡♥ str♦♥❣❡r ✐♥ t❤✐s s✐❣♥❛❧ ✐♥ ♦r❞❡r t♦ ♠❛❦❡ t❤❡ ❥✉♠♣ ❛t ✵✳ ❲✐t❤ t❤✐s ❡①❛♠♣❧❡✱

✇❡ ♠❛② ✈❡r✐❢② P❧❛♥❝❤❡r❡❧✬s t❤❡♦r❡♠✿

∞

∞

||x||2 =

e−t 2dt =

e−2tdt = −e−2t/2|∞ = 1/2.

✭✺✳✶✷✮

0

0

0

1

∞

1

1

1

||

||2 =

df =

arctan (2πt) |∞

(π/2 − (−π/2)) = 1/2.

✭✺✳✶✸✮

1 + 2πjf

−∞ =

−∞ 1 + 4π2f 2

2π

2π

• ❚❤❡ ●❛✉ss✐❛♥ ❑❡r♥❡❧ ✐s ♣r❛❝t✐❝❛❧❧② ✐♥✈❛r✐❛♥t ✉♥❞❡r t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ✭s❡❡ ❈♦♠♠❡♥t ❇♦① ✺ ✭♣✳ ✶✹✮✮

√

x (t) = e−t2/2

X (f ) =

2πe−(2πf)2/2

✭✺✳✶✹✮

❍❡r❡✱ ✐t ✐s ❡❛s② t♦ ✈❡r✐❢② ||x|| = ||X|| ✈✐❛ ❛ s✉❜st✐t✉t✐♦♥ t = 2πf✳ ❚❤❡ ❝♦♠♣✉t❛t✐♦♥ ✐s s♦♠❡✇❤❛t

√

❤❛r❞❡r ❛♥❞ ②✐❡❧❞s ||x||2 = π

1.7725✳ ❈♦♠♠❡♥t ❇♦① ✺ ❋r♦♠ Pr♦❜❛❜✐❧✐t② t❤❡♦r②✱ ✇❡ ❦♥♦✇ t❤❛t

✭s❡❡ ✏❝❤❛r❛❝t❡r✐st✐❝ ❢✉♥❝t✐♦♥ ♦❢ ❛ ●❛✉ss✐❛♥ ❞✐str✐❜✉t✐♦♥✑✮

1

eitf √

e−t2/2dt = e−f2/2

✭✺✳✶✺✮

2π

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

✶✺

√

◆♦✇ r❡♣❧❛❝❡ f ❜② 2πf ❛♥❞ ♠✉❧t✐♣❧② ✇✐t❤ 2π t♦ ✜♥❞ t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠✳ ❋♦r t❤❡ ❡♥❡r❣②✿

∞

2

√

1

∞

√

||x||2 =

e−t2/2

dt =

π √

e−t2 dt =

π

✭✺✳✶✻✮

−∞

π −∞

✇❤❡r❡ ✇❡ ✉s❡ ✐♥ t❤❡ ❧❛st st❡♣✱ t❤❛t 1

√ e−t2 ❝♦♥st✐t✉t❡s t❤❡ ♣r♦❜❛❜✐❧✐t② ❞❡♥s✐t② ♦❢ ❛ ●❛✉ss✐❛♥ ✈❛r✐❛❜❧❡

π

✇✐t❤ ✈❛r✐❛♥❝❡ 1/2✱ ❛♥❞ t❤✉s ✐♥t❡❣r❛t❡s t♦ ✶✳

• ❚❤❡ ▼❡①✐❝❛♥ ❍❛t ✭❛❧s♦ ❝❛❧❧❡❞ ❘✐❝❦❡r ❲❛✈❡❧❡t ✐♥ ●❡♦♣❤②s✐❝s✮ ✐s t❤❡ ♥❡❣❛t✐✈❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡

●❛✉ss✐❛♥✿

√

x (t) = 1 − t2 e−t2/2

X (f ) = 4π2 2πf 2e−(2πf)2/2

✭✺✳✶✼✮

❇♦t❤✱ t❤❡ ●❛✉ss✐❛♥ ❦❡r♥❡❧ ❛♥❞ t❤❡ ▼❡①✐❝❛♥ ❤❛t ❛r❡ ✈❡r② ✉s❡❢✉❧ s✐♥❝❡ t❤❡② ❛r❡ ✇❡❧❧ ❧♦❝❛t❡❞ ❜♦t❤ ✐♥

s♣❛❝❡ ❛♥❞ ✐♥ ❢r❡q✉❡♥❝② ✭s❡❡ ❋✐❣✉r❡ ✺✳✶✮✱ ♠❡❛♥✐♥❣ t❤❛t t❤❡ ♠❛✐♥ ♣♦rt✐♦♥ ♦❢ t❤❡✐r ❡♥❡r❣② st❡♠s ❢r♦♠ ❛

♥❛rr♦✇ r❛♥❣❡ ♦❢ ❧♦❝❛t✐♦♥s ❛s ✇❡❧❧ ❛s ❛ ♥❛rr♦✇ r❛♥❣❡ ♦❢ ❢r❡q✉❡♥❝✐❡s✳ ❚❤✉s✱ t❤❡ ♠❛② ❜❡ ✉s❡❞ ❛s ❧♦✇✲♣❛ss✱

r❡s♣❡❝t✐✈❡❧② ❜❛♥❞✲♣❛ss ✜❧t❡rs✳ ❚❤❡ ▼❡①✐❝❛♥ ❤❛t ✐s ❛ ✇❛✈❡❧❡t❀ ✇❛✈❡❧❡ts ❛r❡ ✉s❡❞ t♦ ❞❡t❡r♠✐♥❡ ✇❤✐❝❤

❢r❡q✉❡♥❝✐❡s ❝♦♥tr✐❜✉t❡ t❤❡ ♠❛✐♥ ♣♦rt✐♦♥ ♦❢ t❤❡ ❡♥❡r❣② ❛t ❛ s♣❡❝✐✜❝ t✐♠❡✳ ❚♦ t❤✐s ❡♥❞✱ ❛ ✇❛✈❡❧❡t ♥❡❡❞s t♦ ❜❡ ✇❡❧❧ ❧♦❝❛❧✐③❡❞ ✐♥ t✐♠❡ ❛s ✇❡❧❧ ❛s ✐♥ ❢r❡q✉❡♥❝②✳

• ❚❤❡ ❉✐r❛❝ ❈♦♠♣ ✭♣❡✐❣♥❡ ❞❡ ❉✐r❛❝✮ ♦❢ st❡♣ τ

∞

τ · ∆τ (t) := τ

δ (t − nτ )

F {τ · ∆τ } (f ) =

δ (f − k/τ )

✭✺✳✶✽✮

n

k=−∞

❚♦ ✈❡r✐❢② t❤✐s ❢♦r♠✉❧❛✱ ✇❡ ❝❤♦♦s❡ t❤❡ ✐♥t❡❣r❛t✐♦♥ ✐♥t❡r✈❛❧ [−τ/2, τ/2] ❛♥❞ ✇r✐t❡

1

τ /2

τ /2

τ /2

Xk =

τ · ∆τ (t) e−j2πkt/τ dt =

δ (t − nτ ) e−j2πkt/τ dt =

δ (t) e−j2πkt/τ dt = 1

τ

−τ /2

n

−τ /2

−τ /2

✭✺✳✶✾✮

❚❤❡ ❋♦✉r✐❡r✲r❡♣r❡s❡♥t❛t✐♦♥ ❜❡❝♦♠❡s

∞

τ · ∆τ (t) =

ej2πkt/τ

✭✺✳✷✵✮

k=−∞

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

✶✻

❈❍❆P❚❊❘ ✺✳ ❊❳❆▼P▲❊❙

❋✐❣✉r❡ ✺✳✶✿ ❚❤❡ ●❛✉ss✐❛♥ ❦❡r♥❡❧✳ ■ts ❋♦✉r✐❡r tr❛♥s❢♦r♠ ❤❛s t❤❡ ✐❞❡♥t✐❝❛❧ s❤❛♣❡✳

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

✶✼

❋✐❣✉r❡ ✺✳✷✿ ❚❤❡ ▼❡①✐❝❛♥ ❤❛t ✐s t❤❡ ♥❡❣❛t✐✈❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ●❛✉ss✐❛♥ ❦❡r♥❡❧✳

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

✶✽

❈❍❆P❚❊❘ ✺✳ ❊❳❆▼P▲❊❙

❋✐❣✉r❡ ✺✳✸✿ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ♦❢ t❤❡ ▼❡①✐❝❛♥ ❤❛t✳

❚❤✐s ❢♦r♠✉❧❛ ✇✐❧❧ ❜❡ ❝r✉❝✐❛❧ ✐♥ t❤❡ r❡❝♦♥str✉❝t✐♦♥ ❢♦r♠✉❧❛ ✐♥ t❤❡ ◆②q✉✐st✲❙❤❛♥♥♦♥ ❚❤❡♦r❡♠✳ ◆♦t❡✿

❖♥❡ s❤♦✉❧❞ ♥♦t tr② t♦ ❡✈❛❧✉❛t❡ τ · ∆τ (t)❀ ✐♥❞❡❡❞✱ ✐t ✐s ♥♦t r❡❛❧❧② ❛ ❢✉♥❝t✐♦♥✱ s✐♥❝❡ ✐t ✐s ❢♦r♠❡❞ ❜② ❉✐r❛❝

t❡r♠s✳ ❊✈❡♥ ✐ts ♣♦✇❡r ✐s ✐♥✜♥✐t❡✳

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

❈❤❛♣t❡r ✻

❊st✐♠❛t✐♦♥ ♦❢ ❙♣❡❝tr✉♠ ❛♥❞ P♦✇❡r ✈✐❛

❉❋❚✶

✻✳✶ ❊st✐♠❛t✐♦♥ ♦❢ ❙♣❡❝tr✉♠ ❛♥❞ P♦✇❡r ✈✐❛ ❉❋❚

❙♣❡❝tr✉♠ ❢♦r ✜♥✐t❡ ❡♥❡r❣② s✐❣♥❛❧s

❚❤❡ ❋♦✉r✐❡r ✐♥t❡❣r❛❧ ✐s ✐♥t❡♥❞❡❞ ❢♦r ♥♦♥✲♣❡r✐♦❞✐❝ s✐❣♥❛❧s✱ s✐♥❝❡ t❤❡ ♣❡r✐♦❞✐❝✐t② ✇✐❧❧ ❧❡❛❞ t♦ ♣r♦❜❧❡♠s ✇✐t❤

t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ✐♥t❡❣r❛❧✳ ■♥ ❢❛❝t✱ t❤❡ ❋♦✉r✐❡r ✐♥t❡❣r❛❧ ✭✸✳✶✮ ✐s ❞❡✜♥❡❞ r✐❣♦r♦✉s❧② ♦♥❧② ❢♦r s✐❣♥❛❧s

✇✐t❤ ✜♥✐t❡ ❡♥❡r❣② ✭s❡❡ ♥❡①t s❡❝t✐♦♥ ❜❡❧♦✇✮✳ ❘❡❝❛❧❧ t❤❛t ✜♥✐t❡ ❡♥❡r❣② s✐❣♥❛❧s ❝❛♥ ♥❡✈❡r ❜❡ ♣❡r✐♦❞✐❝✳ ❋♦r ❛♥

❛❧t❡r♥❛t✐✈❡ ✇❛② ♦❢ ❞❡✜♥✐♥❣ X (f) ❢♦r ♣❡r✐♦❞✐❝ s✐❣♥❛❧s s❡❡ t❤❡ ♥❡①t ❜✉❧❧❡t ❥✉st ❜❡❧♦✇✮✳

❍♦✇❡✈❡r✱ ✐♥ ♦r❞❡r t♦ ♠❛❦❡ t❤❡ ❉❋❚ ❛❧s♦ ✉s❡❢✉❧ ❢♦r ♥♦♥✲♣❡r✐♦❞✐❝ ✜♥✐t❡ ❡♥❡r❣② s✐❣♥❛❧s✱ ✇❡ ❝❛♥ ❝♦♥s✐❞❡r ❛♥②

s✐❣♥❛❧ x (t) ❛s ❜❡✐♥❣ r❡str✐❝t❡❞ t♦ t❤❡ ✐♥t❡r✈❛❧ [−L/2, L/2] ❛♥❞ t❤❡♥ ❡①t❡♥❞❡❞ ♣❡r✐♦❞✐❝❛❧❧②✳ ❉❡♥♦t❡ t❤✐s ♥❡✇

L✲♣❡r✐♦❞✐❝ s✐❣♥❛❧ ❜② ˜

x (t) ✇✐t❤ ❋♦✉r✐❡r ❝♦❡✣❝✐❡♥ts ˜

Xk✳ ❲❤❡♥ L ✐s ❧❛r❣❡✱ ❝♦♠♣❛r✐♥❣ ✭✶✳✻✮ ✇✐t❤ ✭✸✳✶✮ ❣✐✈❡s

L/2

∞

˜

1

1

1

k

Xk =

˜

x (t) e−j2πkt/Ldt

x (t) e−j2πkt/Ldt =

X

.

✭✻✳✶✮

L −L/2

L −∞

L

L

❚❤❡ ♥♦t❛t✐♦♥ ˜

Xk ✐♥st❡❛❞ ♦❢ Xk s❤♦✉❧❞ r❡♠✐♥❞ t❤❡ r❡❛❞❡r✱ t❤❛t x ✐ts❡❧❢ ✐s ♥♦t ♣❡r✐♦❞✐❝✳

■❢ ♥♦t t❤❡ ❡♥t✐r❡ s✐❣♥❛❧ x ❜✉t ♦♥❧② s♦♠❡ K s❛♠♣❧❡s xn = x (nL/K) ♦❢ t❤❡ s✐❣♥❛❧ ❛r❡ ❣✐✈❡♥✱ ✇❡ ♠❛② ✉s❡

t❤❡ ❉❋❚ ❛s ✐♥ ✭✷✳✽✮ ❛♥❞ ❝♦♠❜✐♥❡ ✐t ✇✐t❤ ✭✷✳✸✮ t♦ ❣❡t ❢♦r k = −K/2, .., K/2 − 1

✜♥✐t❡ ❡♥❡r❣②✿

k

L Θ

X

L · ˜

Xk

xk .✭✻✳✷✮

L

K

◆♦t❡ t❤❛t t❤✐s ❛♣♣r♦①✐♠❛t✐♦♥ ✐♠♣r♦✈❡s ✐♥ q✉❛❧✐t② ❛s L ❜❡❝♦♠❡s ❧❛r❣❡r❀ t❤❡ ❢❛❝t♦r L/K ✐♥❞✐❝❛t❡s t❤❛t K

✇♦✉❧❞ ❤❛✈❡ t♦ ❜❡ ♠❛❞❡ ♣r♦♣♦rt✐♦♥❛❧❧② ❧❛r❣❡r ❛t t❤❡ s❛♠❡ t✐♠❡✳ ❋♦r ❛♥ ✐❧❧✉str❛t✐♦♥ s❡❡ ❋✐❣✉r❡ ✻✳✶✱ ❋✐❣✉r❡ ✻✳✷✱

❛♥❞ ❋✐❣✉r❡ ✻✳✸✱ ❧❡❢t ❝♦❧✉♠♥✱ ✇✐t❤ L = 20✱ K = 50✳ ■♠❛❣✐♥❡ t❤❛t ✇❡ r❡❞✉❝❡ L t♦ ❤❛❧❢✱ ✐✳❡✳ t♦ L∗ = 10✱

❧❡❛❞✐♥❣ t♦ ❛ ♥❡✇ s✐❣♥❛❧ x∗✳ ❚❤✐s ✇♦✉❧❞ ♠❡❛♥ t♦ ❞✐s❝❛r❞ ❤❛❧❢ ♦❢ t❤❡ s❛♠♣❧❡s✱ t❤✉s ❧❡❛❞✐♥❣ t♦ K∗ = 25✳ ❙✐♥❝❡

Θ

t❤❡ ❞✐s❝❛r❞❡❞ s❛♠♣❧❡s ❛r❡ ♣r❛❝t✐❝❛❧❧② ③❡r♦✱ ✉s✐♥❣ ✭✷✳✽✮ s❤♦✇s t❤❛t

Θ

x∗k

x2k✳ ❚❤✐s ❛❣r❡❡s ✇✐t❤ ✭✻✳✷✮ s✐♥❝❡

L∗/K∗ = L/K ❛♥❞ X (k/L∗) = X (2k/L)✳

✶❚❤✐s ❝♦♥t❡♥t ✐s ❛✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡ ❛t ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✹✻✽✶✻✴✶✳✸✴❃✳

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

✶✾

✷✵

❈❍❆P❚❊❘ ✻✳ ❊❙❚■▼❆❚■❖◆ ❖❋ ❙P❊❈❚❘❯▼ ❆◆❉ P❖❲❊❘ ❱■❆ ❉❋❚

✭❛✮

✭❜✮

❋✐❣✉r❡ ✻✳✶✿ ❙♣❡❝tr❛❧ ❊st✐♠❛t✐♦♥ ✈✐❛ ❉❋❚✳ ❆ ✜♥✐t❡ ❡♥❡r❣② s✐❣♥❛❧ ✭●❛✉ss✐❛♥ ❦❡r♥❡❧✮✳

✭❛✮

✭❜✮

❋✐❣✉r❡ ✻✳✷✿ ❙♣❡❝tr❛❧ ❊st✐♠❛t✐♦♥ ✈✐❛ ❉❋❚✳ ❚❤❡ s❛♠♣❧❡❞ ♣❡r✐♦❞✐❝ s✐❣♥❛❧ x (t) = 1 + sin (2πt) +

sin (3 · 2πt) + sin (4 · 2πt) ✭♣❡r✐♦❞ T = 1✱ ❧❡♥❣t❤ L = 3✮✳ ❚❤❡ ❧❛r❣❡r t❤❡ s❛♠♣❧❡✲s✐③❡ K✱ t❤❡ ❜❡tt❡r

❛r❡ t❤❡ ❧♦❝❛t✐♦♥s ✭✵✱ ✶✱ ✸ ❛♥❞ ✹✮ ❛♥❞ ❛♠♣❧✐t✉❞❡s ✭✶✱ ✵✳✺✱ ✵✳✺ ❛♥❞ ✵✳✺✮ ♦❢ t❤❡ ❞❡❧t❛ ♣❡❛❦s ♦❢ X ❛♣♣r♦①✐✲

❫

♠❛t❡❞ ❜② |xk|/K✳

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

✷✶

✭❛✮

✭❜✮

❋✐❣✉r❡ ✻✳✸✿ ❙♣❡❝tr❛❧ ❊st✐♠❛t✐♦♥ ✈✐❛ ❉❋❚✳ ❚❤❡ s❛♠♣❧❡❞ ♣❡r✐♦❞✐❝ s✐❣♥❛❧ x (t) = 1 + sin (2πt) +

sin (3 · 2πt) + sin (4 · 2πt) ✭♣❡r✐♦❞ T = 1✱ ❧❡♥❣t❤ L = 3✮✳ ❚❤❡ ❧❛r❣❡r t❤❡ s❛♠♣❧❡✲s✐③❡ K✱ t❤❡ ❜❡tt❡r

❛r❡ t❤❡ ❧♦❝❛t✐♦♥s ✭✵✱ ✶✱ ✸ ❛♥❞ ✹✮ ❛♥❞ ❛♠♣❧✐t✉❞❡s ✭✶✱ ✵✳✺✱ ✵✳✺ ❛♥❞ ✵✳✺✮ ♦❢ t❤❡ ❞❡❧t❛ ♣❡❛❦s ♦❢ X ❛♣♣r♦①✐✲

❫

♠❛t❡❞ ❜② |xk|/K✳ ✭❜✮

❙♣❡❝tr✉♠ ❢♦r ♣❡r✐♦❞✐❝ s✐❣♥❛❧s

❋♦r T ✲♣❡r✐♦❞✐❝ s✐❣♥❛❧s✱ t❤❡ ❋♦✉r✐❡r ✐♥t❡❣r❛❧ ✭✸✳✶✮ ❞♦❡s ♥♦t ❝♦♥✈❡r❣❡ ✐♥ t❤❡ ✉s✉❛❧ s❡♥s❡✳ ❖♥❡ ❝❛♥ st✐❧❧ ❞❡✜♥❡

❛ ❋♦✉r✐❡r tr❛♥s❢♦r♠ X (f) ❝♦♥s✐st✐♥❣ ♦❢ ❉✐r❛❝ ❞❡❧t❛ ❢✉♥❝t✐♦♥s✿

1 Θ

X (f ) =

Xkδ (f − k/T )

xk δ (f − k/T ) .✭✻✳✸✮

K

k

k

❲✐t❤ t❤✐s s❡tt✐♥❣ ✇❡ ❤❛✈❡ x (t) = X (f) ej2πftdf ❛❧s♦ ❢♦r ♣❡r✐♦❞✐❝ s✐❣♥❛❧s✳ ◆♦t❡ t❤❛t t❤❡ ❉❡❧t❛✲❢✉♥❝t✐♦♥s ✐♥

✭✻✳✸✮ ❛r❡ ✏✐♥✜♥✐t❡❧② ❧❛r❣❡✑ ❛t f = k/T ✱ ✐♥ ❛❣r❡❡♠❡♥t ✇✐t❤ ✭✻✳✷✮ ✇❤❡r❡ L s❤♦✉❧❞ ✐❞❡❛❧❧② ❜❡ ✏✐♥✜♥✐t❡❧② ❧❛r❣❡✑✳

❍♦✇❡✈❡r✱ ✏✐♥✜♥✐t❡❧② ❧❛r❣❡✑ ✈❛❧✉❡s ❛r❡ ♥♦t ✉s❡❢✉❧ ✇❤❡♥ ♦♥❡ ✇❛♥ts t♦ r❡❛❞ ♦✛ ❛♥ ❛♠♣❧✐t✉❞❡✳ ❈♦♠♣❛r✐♥❣ ✭✻✳✸✮

❛♥❞ ✭✻✳✷✮ ✇❡ s❡❡ t❤❛t t❤❡ ❉❋❚ Θxk s❤♦✉❧❞ ❜❡ ♥♦r♠❛❧✐③❡❞ ❞✐✛❡r❡♥t❧② ❢♦r ♥♦♥✲♣❡r✐♦❞✐❝ ✜♥✐t❡ ❡♥❡r❣② s✐❣♥❛❧s t❤❛♥

❢♦r ♣❡r✐♦❞✐❝ s✐❣♥❛❧s✳ ❉✐✈✐❞✐♥❣ t❤❡ ❉❋❚ ❜② t❤❡ s❛♠♣❧✐♥❣ r❛t❡ ❢♦r ♥♦♥✲♣❡r✐♦❞✐❝ s✐❣♥❛❧s ♣r♦✈✐❞❡s ❛♥ ❡st✐♠❛t✐♦♥

♦❢ t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ✈✐❛ ✭✻✳✷✮✳ ❉✐✈✐❞✐♥❣ t❤❡ ❉❋❚ ❜② t❤❡ s❛♠♣❧❡ s✐③❡ ❢♦r ♣❡r✐♦❞✐❝ s✐❣♥❛❧s ♣r♦✈✐❞❡s ❛♥

❡st✐♠❛t✐♦♥ ♦❢ t❤❡ ♣❡❛❦ ❤❡✐❣❤ts ♦❢ t❤❡ ❉✐r❛❝ ❞❡❧t❛ ❢✉♥❝t✐♦♥s ✐♥ ✭✻✳✸✮ ✭s❡❡ ❋✐❣✉r❡ ✻✳✷ ❛♥❞ ❋✐❣✉r❡ ✻✳✸✮✳

❊♥❡r❣② ❛♥❞ P♦✇❡r ❢r♦♠ ❙❛♠♣❧✐♥❣

❋✐rst ♦❢ ❛❧❧✱ ♥♦t❡ t❤❛t t❤❡ ❡♥❡r❣② ♦❢ ❛ ♣❡r✐♦❞✐❝ s✐❣♥❛❧ ✐s ✐♥✜♥✐t❡ ❜❡❝❛✉s❡ t❤❡r❡ ❛r❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♣❡r✐♦❞s

✇✐t❤ t❤❡ s❛♠❡ ❡♥❡r❣②✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ t❤❡ ♣♦✇❡r ♦❢ ❛ ✜♥✐t❡ ❡♥❡r❣② s✐❣♥❛❧ ✐s ③❡r♦ s✐♥❝❡ ✐t ✐s t❤❡ ❛✈❡r❛❣❡

❡♥❡r❣② ♦❢ ❛♥ ✐♥✜♥✐t❡❧② ❧♦♥❣ ✐♥t❡r✈❛❧✳

❙❤♦rt✱ ♦♥❧② ♦♥❡ ♠❡❛s✉r❡ ✐s ♠❡❛♥✐♥❣❢✉❧ ❢♦r ❛ s✐❣♥❛❧✿ ❡✐t❤❡r t❤❡ ♣♦✇❡r ♦r t❤❡ ❡♥❡r❣②✱ ❞❡♣❡♥❞✐♥❣ ♦♥ ❝♦♥t❡①t✳

◆❡✈❡rt❤❡❧❡ss✱ ✇❡ ♠❛② ❝♦♠♣❛r❡ ♣♦✇❡r ❛♥❞ ❡♥❡r❣② ♦❢ t❤❡ ✈❛r✐♦✉s ✇❛②s ♦❢ r❡♣r❡s❡♥t✐♥❣ ❛ s✐❣♥❛❧✿ ❛s ❛ ✜♥✐t❡

❡♥❡r❣② s✐❣♥❛❧✱ ❛s ❛ s✐❣♥❛❧ ♦♥ ❛ ❣✐✈❡♥ ✜♥✐t❡ ✐♥t❡r✈❛❧ ✭❡①t❡♥❞✐♥❣ ✐t ♣❡r✐♦❞✐❝❛❧❧②✮ ♦r t❤r♦✉❣❤ ✐ts s❛♠♣❧❡s✳

❈♦♠♣❛r✐♥❣ t❤❡ ♣♦✇❡r ♦❢ ❛ T ✲♣❡r✐♦❞✐❝ s✐❣♥❛❧ ✇✐t❤ ✐ts s❛♠♣❧❡s ♦✈❡r ❛♥ ✐♥t❡r✈❛❧ ♦❢ ❧❡♥❣t❤ L✱ ✇❤❡r❡ L ✐s ❛♥

✐♥t❡❣❡r ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ♣❡r✐♦❞ T ✱ ✇❡ ✜♥❞ ✐♥ t❤❡ ❢r❡q✉❡♥❝② ❞♦♠❛✐♥ ✭✉s✐♥❣ ✭✷✳✸✮ ❛♥❞ ✭✹✳✺✮✮

∞

K/2

K/2

1

2

Θ

Px (❝♦♥t) =

|Xk|2

|Xk|2

|xk | = Px (s❛♠♣❧❡❞) ✭✻✳✹✮

K2

k=−∞

k=−K/2

k=−K/2

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

✷✷

❈❍❆P❚❊❘ ✻✳ ❊❙❚■▼❆❚■❖◆ ❖❋ ❙P❊❈❚❘❯▼ ❆◆❉ P❖❲❊❘ ❱■❆ ❉❋❚

❈♦♠♣✉t✐♥❣ ✐♥ t❤❡ t✐♠❡ ❞♦♠❛✐♥ ✭✉s✐♥❣ ❛ ❘✐❡♠❛♥♥ ❙✉♠✮ ✇❡ ❛rr✐✈❡ ❛t t❤❡ s❛♠❡ ❝♦♥❝❧✉s✐♦♥✿

K/2

2

K/2

1

L/2

1

L

L

1

Px (❝♦♥t) =

|x (t) |2dt

|x n

| =

|xn|2 = Px (s❛♠♣❧❡❞)

✭✻✳✺✮

L −L/2

L

K

K

K

n=−K/2

n=−K/2

❈♦♠♣❛r✐♥❣ t❤❡ ❡♥❡r❣② ❢♦r ✜♥✐t❡ ❡♥❡r❣② s✐❣♥❛❧s ❛♥❞ ❢♦r t❤❡✐r s❛♠♣❧❡s ♦♥ ❛♥ ✐♥t❡r✈❛❧ ♦❢ ❧❡♥❣t❤ L ✇❡ ✜♥❞

L/2

❊♥❡r❣② = ||x||2

|x (t) |2dt = L · Px (❝♦♥t)

L · Px (s❛♠♣❧❡❞)

✭✻✳✻✮

−L/2

✇❤❡r❡ ✇❡ ✉s❡❞ ✭✻✳✹✮ ✐♥ t❤❡ ❧❛st st❡♣✳

■♥ s✉♠♠❛r②✱ ✉s✐♥❣ s✉✣❝✐❡♥t❧② ♠❛♥② s❛♠♣❧❡s t❛❦❡♥ ♦✈❡r s✉✣❝✐❡♥t❧② ❧❛r❣❡ ✐♥t❡r✈❛❧s s❤♦✉❧❞ ❛❧❧♦✇ t♦ ❝♦♠♣✉t❡

t❤❡ ♣♦✇❡r ♦r ❡♥❡r❣② ♦❢ s✐❣♥❛❧s ✐♥ ❡✐t❤❡r ♦❢ t❤❡ t❤r❡❡ r❡♣r❡s❡♥t❛t✐♦♥s✱ t❤❡r❡❜② ❛♣♣r♦①✐♠❛t✐♥❣ ♣♦✇❡r ✇✐t❤ ❡♥❡r❣②

♣❡r t✐♠❡ ♦✈❡r ❛ ❧♦♥❣ ✐♥t❡r✈❛❧✳

❆s ❛♥ ✐♠♣♦rt❛♥t ❝♦♥❝❧✉s✐♦♥ ✇❡ ♥♦t❡ t❤❛t t❤❡ ♣♦✇❡r ♦❢ ❛ ♣❡r✐♦❞✐❝ s✐❣♥❛❧ s❤♦✉❧❞ ♥♦t ❞❡♣❡♥❞ ♦♥ t❤❡ ♥✉♠❜❡r

♦❢ s❛♠♣❧❡s t❛❦❡♥✱ ❛t ❧❡❛st ❛♣♣r♦①✐♠❛t✐✈❡❧②✱ ❛♥❞ ❛s ❧♦♥❣ ❛s s✉✣❝✐❡♥t ♠❛♥② s❛♠♣❧❡s ❛r❡ t❛❦❡♥✳ ❲❡ ✇✐❧❧ ❧❛t❡r

♠❛❦❡ ♣r❡❝✐s❡ ❤♦✇ ♠❛♥② s❛♠♣❧❡s ❛r❡ s✉✣❝✐❡♥t✳ ❙✐♠✐❧❛r ❢♦r t❤❡ ❡♥❡r❣② ♦❢ ❛ ✜♥✐t❡ ❡♥❡r❣② s✐❣♥❛❧✳

❫

❆s ❛ ✜♥❛❧ r❡♠❛r❦ ✇❡ ♥♦t❡ t❤❛t t❤❡ ♥❛♠❡ ✏♣♦✇❡r s♣❡❝tr✉♠✑ ✉s❡❞ ❢♦r |X (f) |2✱ |Xk|2 ❛♥❞ |xk|2 ✐s ❛♣♣r♦♣r✐❛t❡

s✐♥❝❡ t❤❡② ✐♥❞✐❝❛t❡ ❤♦✇ t❤❡ ♣♦✇❡r ✐s ❞✐str✐❜✉t❡❞ ♦❢ t❤❡ s♣❡❝tr✉♠ ♦❢ ❢r❡q✉❡♥❝✐❡s✱ ✐✳❡✳✱ ✇❤✐❝❤ ❢r❡q✉❡♥❝✐❡s

❝♦♥tr✐❜✉t❡ ♠✉❝❤ ❛♥❞ ✇❤✐❝❤ ❧❡ss t♦ t❤❡ ♣♦✇❡r✳

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

❈❤❛♣t❡r ✼

❙❛♠♣❧✐♥❣✿ ❘❡✈✐❡✇✶

✼✳✶ ❙❛♠♣❧✐♥❣✿ ❘❡✈✐❡✇

▲❡t ✉s ❝♦♥s✐❞❡r ❛ s✐❣♥❛❧ x (t) ✇❤✐❝❤ ✐s ✉♥✐❢♦r♠❧② s❛♠♣❧❡❞ ♠❡❛♥✐♥❣ t❤❛t t❤❡ s❛♠♣❧❡s ❛r❡ xn = x (nτ) ✭n =

... − 2, −1, 0, 1, 2, ...✮✳ ❚❤❡ s❛♠♣❧✐♥❣ ❢r❡q✉❡♥❝② fe = 1/τ ✐s t❤❡ ♥✉♠❜❡r ♦❢ s❛♠♣❧❡s ♣❡r t✐♠❡ ✉♥✐t✳

■♥ ♣r❛❝t✐❝❛❧ ❛♣♣❧✐❝❛t✐♦♥s✱ t❤❡ s✐❣♥❛❧ ✇✐❧❧ ❜❡ s❛♠♣❧❡❞ ♦♥❧② ♦✈❡r ❛♥ ✐♥t❡r✈❛❧✱ s❛② ♦❢ ❧❡♥❣t❤ L ❛♥❞ ✐♥t♦ K

s❛♠♣❧❡s✳ ❚❤❡♥ τ = L/K ❛s ❢♦r ❞✐s❝r❡t❡ s✐❣♥❛❧s✳

❙♣❡❝tr❛❧ ❝♦♣✐❡s ✭❛❧s♦ ❝❛❧❧❡❞ s♣❡❝tr❛❧ r❡♣❡t✐t✐♦♥s ♦r ✐♠❛❣❡s✮

❚❤❡ s❛♠♣❧❡❞ s✐❣♥❛❧ ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s t❤❡ ❉✐r❛❝ ❝♦♠❜ ♦❢ st❡♣ τ ♠♦❞✉❧❛t❡❞ ❜② x (t)

∞

xe (t) = x (t) · τ ∆τ (t)

=

τ x (nτ ) · δ (t − nτ )

✭✼✳✶✮

n=−∞

❚❤❡ ❢❛❝t♦r τ ✐s ❛❞❞❡❞ t♦ ♦❜t❛✐♥ ❛ ❋♦✉r✐❡r tr❛♥s❢♦r♠ t❤❛t ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ τ ❛♥❞ t♦ ♣r❡s❡r✈❡ ❛♥ ❛✈✲

❡r❛❣✐♥❣ ♣r♦♣❡rt② ✭t❤❡ ✐♥t❡❣r❛❧ ♦❢ xe ♣r♦✈✐❞❡s ❛ ❣♦♦❞ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ✐♥t❡❣r❛❧ ♦❢ x s✐♥❝❡ xe (t) dt =

∞

τ x (nτ ) ✐s t❤❡ ❘✐❡♠❛♥♥ s✉♠ ❛♣♣r♦①✐♠❛t✐♥❣ t❤❡

x (t) dt ❛s ❛ s✉♠ ♦❢ r❡❝t❛♥❣❧❡s ♦❢ ✇✐❞t❤ τ ✳✮

n=−∞

❯s✐♥❣ ✭✺✳✷✵✮ ✇❡ ♠❛② ✇r✐t❡

∞

xe (t) = x (t) · τ ∆τ (t)

=

x (t) · ej2πkt/τ

✭✼✳✷✮

k=−∞

◆♦t❡✱ t❤❛t ✭✼✳✷✮ ✐s ♥♦t ❛ ❋♦✉r✐❡r r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ xe (t) ✭t❤❡ ✏❝♦❡✣❝✐❡♥ts✑ x (t) ♦❢ t❤❡ s✐♥✉s♦✐❞s ❞❡♣❡♥❞ ♦♥ t

✐♥st❡❛❞ ♦❢ k✮✳ ❈♦♠♣✉t✐♥❣ t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ♦❢ t❤❡ s❛♠♣❧❡❞ s✐❣♥❛❧ xe (t) ♦♥❝❡ ♠♦r❡✱ ✉s✐♥❣ ✭✼✳✷✮✱ ✇❡ ✜♥❞

∞

∞

Xe (f )

=

F {x (t) ej2πkt/τ } (f ) =

X (f − k/τ )

✭✼✳✸✮

k=−∞

k=−∞

❲❡ ♦❜s❡r✈❡ t❤❛t t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠✱ ❛♥❞ ❛❧s♦ t❤❡ s♣❡❝tr✉♠ ♦❢ t❤❡ s❛♠♣❧❡❞ s✐❣♥❛❧ ✐s ♣❡r✐♦❞✐❝ ✇✐t❤ ♣❡r✐♦❞

fe = 1/τ ✱ t❤❡ s❛♠♣❧❡ r❛t❡ ✭s❡❡ ❋✐❣✉r❡ ✼✳✶ r✐❣❤t✮✳ ❍♦✇❡✈❡r✱ ✇❡ s❡❡ ♥♦✇ t❤❛t Xe ✐s ❝♦♠♣♦s❡❞ ♦❢ ❝♦♣✐❡s ♦❢ X✱

s❤✐❢t❡❞ ❜② ❛ ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ♣❡r✐♦❞✱ ✐✳❡✳✱ s❤✐❢t❡❞ ❜② k/τ = k · fe✳ ❚❤❡s❡ ❛r❡ ❝❛❧❧❡❞ s♣❡❝tr❛❧ ❝♦♣✐❡s✳

❙❛♠♣❧✐♥❣ ❛♥❞ t❤❡ ❞✐s❝r❡t❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠

❋✐♥✐t❡ ❡♥❡r❣② s✐❣♥❛❧s✿ ❯s✐♥❣ ✭✼✳✶✮ ❛❣❛✐♥ ✇❡ ✜♥❞ t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ♦❢ t❤❡ s❛♠♣❧❡❞ s✐❣♥❛❧ xe (t)

∞

∞

Xe (f )

=

τ x (nτ ) F {δ (t − nτ )} (f ) = τ

x

n=−∞

n=−∞

n · e−j2πf nτ

✭✼✳✹✮

❙❡tt✐♥❣ ♥♦✇ f = k/L ❛♥❞ r❡❝❛❧❧✐♥❣ τ = L/K ✇❡ ❣❡t ❛♣♣r♦①✐♠❛t✐✈❡❧②

✜♥✐t❡ ❡♥❡r❣②✿

K/2

Θ

Xe (k/L)

τ

x

x

n=−K/2

n · e−j2π(k/L)n(L/K) = τ

k .

✭✼✳✺✮

❚❤✐s ❛❣r❡❡s ✇✐t❤ ✭✻✳✷✮ ❢♦r ✜♥✐t❡ ❡♥❡r❣② s✐❣♥❛❧s ❛♥❞ ✐s ❛❝t✉❛❧❧② ✈❛❧✐❞ ❢♦r ❛❧❧ k✱ s✐♥❝❡ ❜♦t❤ s✐❞❡s ❛r❡ ♣❡r✐♦❞✐❝

✇✐t❤ ♣❡r✐♦❞ K✳ ❍♦✇❡✈❡r✱ ✐t ✐s ♦♥❧② ✉s❡❢✉❧ ✐❢ x ✐s ♦❢ ✜♥✐t❡ ❡♥❡r❣②✱ s✐♠✐❧❛r t♦ ✭✻✳✷✮✳

✶❚❤✐s ❝♦♥t❡♥t ✐s ❛✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡ ❛t ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✹✻✽✸✸✴✶✳✹✴❃✳

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

✷✸

✷✹

❈❍❆P❚❊❘ ✼✳ ❙❆▼P▲■◆●✿ ❘❊❱■❊❲

❋♦r ♣❡r✐♦❞✐❝ s✐❣♥❛❧s ✇❡ ❝♦✉❧❞ ✇r✐t❡ X ❛♥❞ Xe ❛s s✉♠s ♦❢ ❉✐r❛❝ ❞❡❧t❛ ❢✉♥❝t✐♦♥s ❛s ✐♥ ✭✻✳✸✮✳ ❉♦✐♥❣ s♦✱

✭✼✳✸✮ ❝♦♥✜r♠s t❤❛t ❛❧s♦ ♣❡r✐♦❞✐❝ s✐❣♥❛❧s ♣♦ss❡ss s♣❡❝tr❛❧ ❝♦♣✐❡s ✇❤❡♥ s❛♠♣❧❡❞✳

❋♦r ♣❡r✐♦❞✐❝ ❢✉♥❝t✐♦♥s✱ t❤❡ r❡❧❛t✐♦♥ ✭✼✳✺✮ s❤♦✇s t❤❛t t❤❡ t❡r♠s Θ

τ xk ✇✐❧❧ ❛tt❡♠♣t t♦ ❛♣♣r♦①✐♠❛t❡ t❤❡

❉✐r❛❝✲s❤❛♣❡ ♦❢

Θ

Xe✱ ♠❡❛♥✐♥❣ t❤❛t t❤❡ ✈❛❧✉❡s ♦❢ τ xk ✇✐❧❧ ❜❡ ❤✉❣❡ ❢♦r k/L ❝❧♦s❡ t♦ ❛ ♣❡❛❦ ❧♦❝❛t✐♦♥ ❜✉t ✈❡r②

s♠❛❧❧ ♦t❤❡r✇✐s❡ ✭s❡❡ ✭✼✳✷✮✱ ✭✼✳✸✮✱ ❛♥❞ ✭✼✳✹✮✱ ❝♦♠♣❛r❡ ❞✐s❝✉ss✐♦♥ ❛❢t❡r ✭✻✳✸✮✮✳ ■♥ ♦r❞❡r t♦ ✐❞❡♥t✐❢② t❤❡ ❛♠♣❧✐t✉❞❡

Xk ♦❢ t❤❡ ❉✐r❛❝ ❞❡❧t❛ ❢✉♥❝t✐♦♥s ❜❡tt❡r✱ ♦♥❡ ✉s❡s t❤❡ ❡❛r❧✐❡r ❛♣♣r♦①✐♠❛t✐♦♥

♣❡r✐♦❞✐❝✿

Θ

X

1

k

x

K

k .

✭✼✳✻✮

❋✐❣✉r❡ ✼✳✶✿ ❙❛♠♣❧✐♥❣ ❛t ✸✵✵ ❞♣✐ ❛♣♣❡❛rs t♦ ❜❡ s✉✣❝✐❡♥t t♦ ❦❡❡♣ t❤❡ q✉❛❧✐t② ♦❢ t❤❡ ✐♠❛❣❡✳

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

✷✺

❋✐❣✉r❡ ✼✳✷✿ ❙❛♠♣❧✐♥❣ ❛t ✺✵ ❞♣✐ ✐♥tr♦❞✉❝❡s ❛❧✐❛s✐♥❣✳

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

✷✻

❈❍❆P❚❊❘ ✼✳ ❙❆▼P▲■◆●✿ ❘❊❱■❊❲

❋✐❣✉r❡ ✼✳✸✿ ❙❛♠♣❧✐♥❣ ♦❢ ❛ ❞❡t❛✐❧ ❛t ✺✵✵❞♣✐ r❡✈❡❛❧s ❤✐❣❤✲❢r❡q✉❡♥❝② ❝♦♥t❡♥t✳ ❚❤❡s❡ ❢r❡q✉❡♥❝✐❡s ✇✐❧❧ ❧❡❛❦

❢r♦♠ t❤❡ s♣❡❝tr❛❧ ❝♦♣✐❡s ✐♥t♦ t❤❡ ❧♦✇ ❢r❡q✉❡♥❝✐❡s ♦❢ t❤❡ ♠❛✐♥ s♣❡❝tr❛❧ ❝♦♣② ✇❤❡♥ s❛♠♣❧❡❞ ❛t t♦♦ ❧♦✇ ❛

❢r❡q✉❡♥❝②✳

❆❧✐❛s✐♥❣

❆ss✉♠❡ t❤❛t t❤❡ ❝♦♥t✐♥✉♦✉s✲t✐♠❡ s✐❣♥❛❧ x (t) ✐s ❜❛♥❞❧✐♠✐t❡❞✱ ♠❡❛♥✐♥❣ t❤❡r❡ ✐s s♦♠❡ B > 0 s✉❝❤ t❤❛t X (f ) = 0 ❢♦r |f | > B✳ ■❢ t❤❡ s❛♠♣❧❡ r❛t❡ ❤❛❞ ❜❡❡♥ t♦♦ s♠❛❧❧✱ ♥❛♠❡❧②

fe < 2B

✭✼✳✼✮

t❤❡♥ t❤✐s ❝♦♣✐❡s ✇♦✉❧❞ ♦✈❡r❧❛♣ ❛♥❞ t❤❡ ♦r✐❣✐♥❛❧ s✐❣♥❛❧ ❝❛♥ ♥♦t ❜❡ r❡❝♦✈❡r❡❞✳ ❚❤❡ s❛♠♣❧❡❞ s✐❣♥❛❧ s❤♦✇s

❛rt✐❢❛❝ts ❝❛❧❧❡❞ ❛❧✐❛s✐♥❣ ✭r❡❝♦✉✈r❡♠❡♥t✮❀ t❤❡s❡ ❛rt✐❢❛❝ts ♠❛♥✐❢❡st ✐s ❡rr♦♥❡♦✉s ❧♦✇ ❢r❡q✉❡♥❝② ❝♦♥t❡♥t✳ ❙✉❝❤

❝♦♥t❡♥t s♣✐❧❧s ♦r ❧❡❛❦s ❢r♦♠ t❤❡ s♣❡❝tr❛❧ ❝♦♣✐❡s ✐♥t♦ t❤❡ ♠❛✐♥ s♣❡❝tr❛❧ ♣❡r✐♦❞ ✭s❡❡ ❋✐❣✉r❡ ✼✳✶✮✳

◆②q✉✐st ❙❤❛♥♥♦♥ s❛♠♣❧✐♥❣ t❤❡♦r❡♠✿

❆ss✉♠❡ t❤❛t t❤❡ ❝♦♥t✐♥✉♦✉s✲t✐♠❡ s✐❣♥❛❧ x (t) ✐s ❜❛♥❞❧✐♠✐t❡❞✱ ♠❡❛♥✐♥❣ t❤❡r❡ ✐s s♦♠❡ B > 0 s✉❝❤ t❤❛t X (f ) = F {x (t)} (f ) = 0 ✇❤❡♥❡✈❡r |f | > B✳

■❢ t❤✐s s✐❣♥❛❧ x ✐s s❛♠♣❧❡❞ ✉♥✐❢♦r♠❧② ❛t ❛ s✉✣❝✐❡♥t r❛t❡✱ ♥❛♠❡❧② ❛t fe > 2B✱ t❤❡♥ x (t) ❝❛♥ ❜❡ ♠❛t❤❡♠❛t✲

✐❝❛❧❧② r❡❝♦♥str✉❝t❡❞ ♣❡r❢❡❝t❧② ❢r♦♠ ♦♥❧② t❤♦s❡ ❞✐s❝r❡t❡ s❛♠♣❧❡s✳

❘❡❝♦♥str✉❝t✐♦♥ ✭■❞❡❛❧ ❧♦✇ ♣❛ss ✜❧t❡r✐♥❣✮

■❢ t❤❡ s❛♠♣❧❡ r❛t❡ ❤❛s ❜❡❡♥ ❤✐❣❤ ❡♥♦✉❣❤ t❤♦✉❣❤✱ ♥❛♠❡❧②

fe > 2B

✭✼✳✽✮

t❤❡♥✱ t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ♦❢ t❤❡ ♦r✐❣✐♥❛❧ s✐❣♥❛❧ ❝❛♥ ❜❡ r❡❝♦✈❡r❡❞ ❜② ❝✉tt✐♥❣ ♦✛ t❤❡ ♣❡r✐♦❞✐❝ ❝♦♣✐❡s✿

X (f ) = Xe (f ) · ❘❡❝t

f

✭✼✳✾✮

2fc

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

✷✼

✇❤❡r❡ fc ❤❛s t♦ ❜❡ ❝❤♦s❡♥ s✉❝❤ t❤❛t B < fc < fe − B ✭s❡❡ ❋✐❣✉r❡ ✼✳✹ r✐❣❤t✮✳ ✭❚❤❡ ✜rst ❝♦♣② t♦ ❜❡ ❝✉t ♦✛

❧✐❡s ♦✈❡r t❤❡ ✐♥t❡r✈❛❧ [fe − B, fe + B]✳✮ ❆ ♣♦ss✐❜❧❡ ❝❤♦✐❝❡ ✐s fc = fe/2✳

■♥ t❤❡ t✐♠❡ ❞♦♠❛✐♥✱ t❤✐s ❝♦rr❡s♣♦♥❞s t♦ ❝♦♥✈♦❧✈✐♥❣ t❤❡ s❛♠♣❧❡❞ s✐❣♥❛❧ ✇✐t❤ t❤❡ ✐❞❡❛❧ ❧♦✇✲♣❛ss ✜❧t❡r ✭s✐♥❝✮✳

▼♦r❡ ♣r❡❝✐s❡❧②✿ ❚r❛♥s❧❛t✐♥❣ ✭✼✳✾✮ ✐♥t♦ t❤❡ t✐♠❡ ❞♦♠❛✐♥✱ t❤❡♥ ✉s✐♥❣ t❤❡ ❧✐♥❡❛r✐t② ♦❢ t❤❡ ❝♦♥✈♦❧✉t✐♦♥ ❛♥❞ ✭✼✳✶✮✱

✇❡ ✜♥❞ t❤❡ ✐❞❡❛❧ ❧♦✇✲♣❛ss ✜❧t❡r✐♥❣ ✭r❡❝♦♥str✉❝t✐♦♥✮ ❢♦r♠✉❧❛✿

x (t)

=

xe (t) ∗ 2fcs✐♥❝ (2fct)

∞

=

τ

x (nτ ) δ (t − nτ ) ∗ 2f

✭✼✳✶✵✮

n=−∞

cs✐♥❝ (2fct)

∞

=

(τ 2fc)

x (nτ ) s✐♥❝ (2f

n=−∞

c (t − nτ ))

◆♦t❡ t❤❛t t❤❡ ♣r❡✲❢❛❝t♦r 2τfc = 2fc/fe t❛❦❡s ✐♥t♦ ❛❝❝♦✉♥t t❤❡ ♠✐s♠❛t❝❤ ❜❡t✇❡❡♥ s❛♠♣❧✐♥❣ r❛t❡ fe ❛♥❞ t❤❡

❝✉t✲♦✛ ❢r❡q✉❡♥❝② fc✳ ❲❤❡♥ ❝❤♦♦s✐♥❣ fc = fe/2✱ ✐t ❞✐s❛♣♣❡❛rs ✭❜❡❝♦♠❡s ✶✮✳

Pr❛❝t✐❝❛❧ r❡❝♦♥str✉❝t✐♦♥✿ ❘❡❝❛❧❧ t❤❛t t❤❡ s✐♥❝ ✜❧t❡r ✐s ♥♦t r❡❛❧✐③❛❜❧❡ s✐♥❝❡ ✐t r❡q✉✐r❡s ✉s✐♥❣ ❛❧❧ s❛♠♣❧❡s

♦❢ t❤❡ ♣❛st ❛♥❞ t❤❡ ❢✉t✉r❡✳ ❙✐♥❝❡ ❛ ❜❛♥❞❧✐♠✐t❡❞ s✐❣♥❛❧ ❝❛♥ ♥♦t ❜❡ t✐♠❡✲❧✐♠✐t❡❞ ❛t t❤❡ s❛♠❡ t✐♠❡ ✭❍❡✐s❡♥❜❡r❣✮✱

t❤✐s ✇♦✉❧❞ ✐♥ t❤❡♦r② ❧❡❛❞ t♦ ✐♥✜♥✐t❡❧② ♠❛♥② ♦♣❡r❛t✐♦♥s✳

■❢ fe > > 2B✱ t❤❡♥ ❛ ✜❧t❡r ❞✐✛❡r❡♥t ❢r♦♠ t❤❡ ✐❞❡❛❧ s✐♥❝✲✜❧t❡r ❝❛♥ ❜❡ ✉s❡❞✳ ■♥ ❢❛❝t✱ t❤❡ r❡❝♦♥str✉❝t✐♦♥

x (t) = xe (t) ∗ b (t) ✐s ✈❛❧✐❞ ❢♦r ❛♥② ❧♦✇✲♣❛ss ✜❧t❡r b ✭♥♦t ♦♥❧② ❢♦r b (t) = 2fcs✐♥❝ (2fct)✮ ❛s ❧♦♥❣ ❛s ✇❡ ❛ss✉r❡

t❤❛t t❤❡ s♣❡❝tr✉♠ ♦❢ t❤❡ ✜❧t❡r b ✐s ❡q✉❛❧ t♦ ✶ ❢♦r ❛❧❧ ❢r❡q✉❡♥❝✐❡s ✐♥ [−B, B] ❛♥❞ ③❡r♦ ❢♦r ❢r❡q✉❡♥❝✐❡s ❧❛r❣❡r fe − B✳ ✭◆♦t❡ t❤❛t ✇❡ ❞♦ ♥♦t s♣❡❝✐❢② t❤❡ ♣❤❛s❡✮✳ ❯s✉❛❧❧②✱ ♦♥❡ ❞❡s✐❣♥s ❛ ✜❧t❡r s✉❝❤ t❤❛t

✶✳ ❚❤❡ ✜❧t❡r✬s s♣❡❝tr✉♠ ❡q✉❛❧s ✶ ✐♥ ❛♥ ✐♥t❡r✈❛❧ [−fp, fp] ❝❛❧❧❡❞ ♣❛ss✐♥❣ ❜❛♥❞✳ ■t ♠✉st ❝♦♥t❛✐♥ t❤❡ s✐❣♥❛❧✬s

❢r❡q✉❡♥❝②✲❜❛♥❞ [−B, B]✳

✷✳ ❚❤❡ ✜❧t❡r ❛tt❡♥✉❛t❡s ❛❧❧ ❢r❡q✉❡♥❝✐❡s ❜❡②♦♥❞ fe − fp✳

✸✳ ❚❤❡ tr❛♥s✐t✐♦♥ ❜❛♥❞ ♦❢ t❤❡ ✜❧t❡r ✐s [fp, fe − fp]✳ ❚❤✐s r❛♥❣❡ ♦❢ ❢r❡q✉❡♥❝✐❡s ❝❛♥ ❜❡ ❝❤♦s❡♥ ❛s ❧❛r❣❡ ❛s

[B, fe − B] ❜② s❡tt✐♥❣ fp = B ❛♥❞ ❛s s❤♦rt ❛s ❞❡s✐r❡❞✱ ❜✉t ❛❧✇❛②s ❝♦♥t❛✐♥✐♥❣ t❤❡ ♣♦✐♥t fe/2 ❜② s❡tt✐♥❣

fp s♠❛❧❧❡r ❜✉t ❛s ❝❧♦s❡ t♦ fe/2 ❛s ❞❡s✐r❡❞✳

✭❛✮

✭❜✮

❋✐❣✉r❡ ✼✳✹✿ ✭❛✮ ❚❤❡ ❉✐r❛❝ ❈♦♠❜ ♠♦❞✉❧❛t❡❞ ❜② ❛ s✐❣♥❛❧✳ ✭❜✮ ❚❤❡ s♣❡❝tr✉♠ ♦❢ t❤❡ s❛♠♣❧❡❞ s✐❣♥❛❧ ✭♥♦t❡

t❤❡ s♣❡❝tr❛❧ ❝♦♣✐❡s✮ ❛♥❞ ❛ ♣r❛❝t✐❝❛❧ r❡❝♦♥str✉❝t✐♦♥ ✜❧t❡r✿ ■t✬s ♣❛ss✐♥❣ ❜❛♥❞ ❝♦♥t❛✐♥s t❤❡ ❜❛♥❞ [−B, B] ♦❢

t❤❡ s✐❣♥❛❧ ❛♥❞ ✐t ❛tt❡♥✉❛t❡s ❛❧❧ s♣❡❝tr❛❧ ❝♦♣✐❡s✳

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

✷✽

❈❍❆P❚❊❘ ✼✳ ❙❆▼P▲■◆●✿ ❘❊❱■❊❲

✭❛✮

✭❜✮

❋✐❣✉r❡ ✼✳✺✿ ✭❛✮ ❚❤❡ ❧♦✇✲♣❛ss ❋■❘ ✜❧t❡r ♦❢ ❧❡♥❣t❤ ✻✶ ❛♥❞ ✐❞❡❛❧ ❝✉t♦✛ ❛t fc = 1/8 = 0.125 ❛s ♣r♦❞✉❝❡❞

❜② ▼❛t❧❛❜✳ ❚❤❡ ✜❧t❡r ✈❛❧✉❡s ❛r❡ ✐♥ r❡❞✳ ❋♦r ❝♦♠♣❛r✐s♦♥ ✐♥ ❜❧✉❡ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s❛♠♣❧❡s ♦❢ t❤❡ ✐❞❡❛❧

✜❧t❡r 2fcs✐♥❝ (2fct) ❛t t = −30, .., −1, 0, 1, ..30✳ ✭❜✮ ❉❋❚ ♦❢ t❤❡ s✐❣♥❛❧s ♦♥ t❤❡ ❧❡❢t✳ ❚❤❡ ❛❝t✉❛❧ tr❛♥s✐t✐♦♥

❜❛♥❞ ♦❢ t❤❡ ✜❧t❡r ✐s [0.1, 0.15]✳ ◆♦t❡ t❤❛t t❤❡ s✐♥❝✲s❛♠♣❧❡s ♣r♦✈✐❞❡ ❛ ✜❧t❡r ✇✐t❤ s❤❛r♣❡r ❝✉t♦✛ ❜✉t s❧✐❣❤t❧②

❧♦✇❡r q✉❛❧✐t② ✐♥ t❤❡ ♣❛ss✲❜❛♥❞ s✐♥❝❡ ✐t r❡q✉✐r❡s ✐♥✜♥✐t❡❧② ♠❛♥② s❛♠♣❧❡s✳ ❚❤❡ ♣♦✇❡r ❝♦♠♣✉t❡❞ ❢r♦♠ ❡✐t❤❡r s✐❞❡ ✭r❡❝❛❧❧ ✮ ✐s Pb = 0.003✬891 = 0.237✬4/61

2fc/❧❡♥❣t❤✳

❆❧❧ t❤✐s ❛ss✉r❡s t❤❛t ✇❡ ❦❡❡♣ t❤❡ ❡♥t✐r❡ s♣❡❝tr❛❧ ✐♥❢♦r♠❛t✐♦♥ ♦❢ x (t) ❝♦♥t❛✐♥❡❞ ✐♥ [−B, B]✱ ②❡t r❡♠♦✈❡ ❛❧❧

s♣❡❝tr❛❧ ❝♦♣✐❡s ✭s❡❡ ❋✐❣✉r❡ ✼✳✹✮✳ ❙❛♠♣❧✐♥❣ ❛♥ ✐❞❡❛❧ r❡❝♦♥str✉❝t✐♦♥ ✜❧t❡r ✐s ❛ ♠❡t❤♦❞ t♦ ❛ ❞❡s✐❣♥ ❛ ❧♦✇ ♣❛ss

✜❧t❡r✳ ❍❡r❡✱ ♣♦✐♥t ✷ ✐♥ t❤❡ ❛❜♦✈❡ ❧✐st ❤❡❧♣s ❛✈♦✐❞✐♥❣ ❛❧✐❛s✐♥❣ ✐♥ s✉❝❤ ❛ ✜❧t❡r✳ ❍♦✇❡✈❡r✱ t❤❡ r❡s✉❧t✐♥❣ ✜❧t❡rs

❛r❡ ♥♦t ✐❞❡❛❧ ❛s ❋■❘ ✜❧t❡rs ♣♦ss✐❜❧❡ s✐♥❝❡ ♥♦t ❛❧❧ s❛♠♣❧❡s ❝❛♥ ❜❡ ✉s❡❞✱ r❡s✉❧t✐♥❣ ✐♥ ●✐❜❜s✲❧✐❦❡ ♣❤❡♥♦♠❡♥❛ ✭s❡❡

❋✐❣✉r❡ ✼✳✺ ♦♥ t❤❡ r✐❣❤t✮✳

▼❛t❧❛❜ ✉s❡s t❤❡ ❝♦♠♠❛♥❞ ❢✐r✶ t♦ ❝♦♠♣✉t❡ ✜♥✐t❡ ✐♠♣✉❧s❡ r❡s♣♦♥s❡ ✜❧t❡rs ✭❋■❘✮✳ ❆ ❧❛r❣❡r tr❛♥s✐t✐♦♥ ❜❛♥❞

❛❧❧♦✇s ❢♦r s❤♦rt❡r ✜❧t❡rs t♦ ❜❡ ✉s❡❞✱ t❤✉s s♣❡❡❞✐♥❣ ✉♣ ❝♦♠♣✉t❛t✐♦♥✳ ❋✐❣✉r❡ ✼✳✺ s❤♦✇s ❛ ❧❡♥❣t❤ ✻✶ ❋■❘ ✜❧t❡r

✇✐t❤ ✐❞❡❛❧ ❝✉t♦✛ ❢r❡q✉❡♥❝② ❛t fc = 1/8 = 0.125✳ ■ts tr❛♥s✐t✐♦♥ ❜❛♥❞ ✐s r♦✉❣❤❧② [0.10, 0.15]✳ ❆s ✐s ❝♦♠♠♦♥ ✐♥

❞✐❣✐t❛❧ ✜❧t❡r✐♥❣✱ t❤❡ s❛♠♣❧✐♥❣ ❢r❡q✉❡♥❝② ✐s ❛ss✉♠❡❞ t♦ ❜❡ fe = 1✳

P♦✇❡r ❛♥❞ ❊♥❡r❣② ❢r♦♠ ❙❛♠♣❧✐♥❣

❲❡ s✉♠♠❛r✐③❡ ❛ ❢❡✇ ❢❛❝ts✳

●♦♦❞ q✉❛❧✐t② ✇❤❡♥ ❛❜♦✈❡ ◆②q✉✐st ❲❡ ❛r❡ ♥♦✇ ✐♥ ❛ ♣♦s✐t✐♦♥ t♦ ♠❛❦❡ ♠♦r❡ ♣r❡❝✐s❡ ❤♦✇ ♠❛♥② s❛♠♣❧❡s

❛r❡ ❡♥♦✉❣❤ t♦ ✇❡❧❧ ❛♣♣r♦①✐♠❛t❡ t❤❡ ♣♦✇❡r ♦❢ ❛ s✐❣♥❛❧ ✈✐❛ ✭✻✳✹✮✱ ♦r ✐ts ❡♥❡r❣② ♣❡r t✐♠❡ ♦✈❡r ❛ ❧❛r❣❡ ✐♥t❡r✈❛❧

✈✐❛ ✭✻✳✻✮✱ ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ❝♦♥t❡①t✳ ❘♦✉❣❤❧②✱ t❤❡ s✐❣♥❛❧ ♠✉st ❜❡ s❛♠♣❧❡❞ ❛t ❧❡❛st ❛t ◆②q✉✐st r❛t❡ s♦ t❤❛t t❤❡ s❛♠♣❧❡s ❛♥❞ t❤❡✐r ❉❋❚ ❢❛✐t❤❢✉❧❧② r❡♣r❡s❡♥t t❤❡ s✐❣♥❛❧ ❛♥❞ ✐ts ❋♦✉r✐❡r tr❛♥s❢♦r♠✳ ❈❧❡❛r❧②✱ s❛♠♣❧✐♥❣ ❛t ❛

♠✉❝❤ ❤✐❣❤❡r r❛t❡ t❤❛♥ ◆②q✉✐st ✇✐❧❧ ✐♠♣r♦✈❡ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥✳

P♦✇❡r ✐♥❞❡♣❡♥❞❡♥t ♦❢ s❛♠♣❧✐♥❣ r❛t❡ ❲❡ ❝♦♥❝❧✉❞❡ t❤❛t ❝❤❛♥❣✐♥❣ t❤❡ s❛♠♣❧✐♥❣ r❛t❡ ✇✐❧❧ ♥♦t ❝❤❛♥❣❡ ♣♦✇❡r✱

❛s ❧♦♥❣ ❛s ✇❡ st❛② ❛❜♦✈❡ ◆②q✉✐st✱ ❛♥❞ ❛t ❧❡❛st ❛♣♣r♦①✐♠❛t❡❧②✳ ❇❡❢♦r❡ ✇❡ st✉❞② ❤♦✇ t♦ ❝❤❛♥❣❡ t❤❡ s❛♠♣❧✐♥❣

r❛t❡ ✐♥ t❤❡ ♥❡①t s❡❝t✐♦♥s✱ ✇❡ ❣✐✈❡ ❛ q✉✐❝❦ ❞❡♠♦♥str❛t✐♦♥ ♦❢ ✭✻✳✻✮ ✉s✐♥❣ ❛ s✐♠♣❧❡ ❜❛♥❞✲❧✐♠✐t❡❞ s✐❣♥❛❧✿ ❛ ✜❧t❡r✳

■❞❡❛❧ ✜❧t❡r ❛♥❞ ❋■❘ ✜❧t❡r ❚❤❡ ❡♥❡r❣② ♦❢ t❤❡ ✐❞❡❛❧ ✜❧t❡r ✇✐t❤ ❝✉t♦✛ ❢r❡q✉❡♥❝② fc ✐s ||2fcs✐♥❝ (2fct) ||2 = 2·fc

✇❤✐❝❤ ❢♦❧❧♦✇s ❡❛s✐❧② ❢r♦♠ t❤❡ ♣♦✇❡r s♣❡❝tr✉♠ ❜❡✐♥❣ ✶ ♦♥ ❛♥ ✐♥t❡r✈❛❧ ♦❢ ❧❡♥❣t❤ 2 · fc ❛♥❞ ③❡r♦ ❡❧s❡✱ ✉s✐♥❣

P❧❛♥❝❤❡r❡❧✳ ❚♦ ❝♦♠♣✉t❡ t❤❡ ♣♦✇❡r ♦❢ ❛ ❞✐❣✐t❛❧ ❧♦✇ ♣❛ss ❋■❘ ✜❧t❡rb = (b0, ..., bK−1) ♦❢ ❧❡♥❣t❤ K ✇✐t❤

Θ

Θ

❛♣♣r♦①✐♠❛t❡ ❝✉t♦✛ ❢r❡q✉❡♥❝② fc ✇❡ st✉❞② ✐ts ❉❋❚ b✿ ✇❡ ♥♦t❡ t❤❛t bk= 1 ❢♦r r♦✉❣❤❧② 2fcK ♦❢ t❤❡ ✐♥❞✐❝❡s k

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

✷✾

Θ

❛♥❞ bk= 0 ❢♦r t❤❡ ♦t❤❡r ✐♥❞✐❝❡s✳ ◆♦t❡ t❤❛t fe = 1 ❛♥❞ 0 < fc < 1/2 ❢♦r ❛ ❞✐❣✐t❛❧ ✜❧t❡r✳ ❯s✐♥❣ ✭✹✳✺✮ ✇❡ ✜♥❞

K−1

1

2

Θ

1

Pb =

|bk |

2fcK = 2fc/K = 2fc/L.✭✼✳✶✶✮

K2

K2

k=0

❚❤✐s ✜ts ♣❡r❢❡❝t❧② ✇✐t❤ ✭✻✳✻✮ s✐♥❝❡ ❢♦r ❛ ❞✐❣✐t❛❧ ✜❧t❡r fe = 1 ❛♥❞✱ t❤✉s✱ K = L✳ ❚❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❜❡❝♦♠❡s

❜❡tt❡r✱ t❤❡ s❤❛r♣❡r t❤❡ tr❛♥s✐t✐♦♥ ❜❛♥❞ ♦❢ t❤❡ ✜❧t❡r b✱ ✐✳❡✳✱ t❤❡ ❧♦♥❣❡r t❤❡ ✜❧t❡r ✐s✳

❈♦♠♣❛r❡ ❛❣❛✐♥ ✇✐t❤ ❋✐❣✉r❡ ✼✳✺ ❢♦r ❛ ❝♦♥❝r❡t❡ ✜❧t❡r ✇✐t❤ ❧❡♥❣t❤ K = 61✱ fc = 1/8 = 0.125 ❛♥❞ ♣♦✇❡r

Pb = 0.003✬891✳

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

✸✵

❈❍❆P❚❊❘ ✼✳ ❙❆▼P▲■◆●✿ ❘❊❱■❊❲

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

❈❤❛♣t❡r ✽

❉❡❝✐♠❛t✐♦♥ ❛♥❞ ❉♦✇♥s❛♠♣❧✐♥❣✶

✽✳✶ ❉❡❝✐♠❛t✐♦♥ ❛♥❞ ❉♦✇♥s❛♠♣❧✐♥❣

✽✳✶✳✶ P✉r❡ ❉♦✇♥s❛♠♣❧✐♥❣

❚❤❡ ❡❛s✐❡st ❝❛s❡ ❝♦♥s✐sts ✐♥ r❡❞✉❝✐♥❣ t❤❡ s❛♠♣❧✐♥❣ r❛t❡ ❜② s✐♠♣❧② ❞r♦♣♣✐♥❣ s❛♠♣❧❡s✳ ❚❤✐s ♣r♦❝❡❞✉r❡ ✐s ❝❛❧❧❡❞

❉♦✇♥s❛♠♣❧✐♥❣✳ ❍♦✇❡✈❡r✱ ✇❡ ♥❡❡❞ t♦ ❜❡ ❝❛r❡❢✉❧ ♦♥ t❤❡ ❡✛❡❝t ♦❢ t❤✐s ♣r♦❝❡❞✉r❡✳ ❲❡ ♥♦t❡ ✐♠♠❡❞✐❛t❡❧②✱ t❤❛t t❤❡ ♥❡✇ s❛♠♣❧✐♥❣ r❛t❡ fd = fe/M ♥❡❡❞s t♦ ❜❡ st✐❧❧ ❛❜♦✈❡ ◆②q✉✐st✱ ✐✳❡✳✱ fe > M ·2B ✐♥ ♦r❞❡r t♦ ❛✈♦✐❞ ❛❧✐❛s✐♥❣✳

• ❉♦✇♥s❛♠♣❧✐♥❣ ❆ss✉♠❡ t❤❛t fe > M · 2B✳

(y0, y1, y2, ...) →

↓ M

→ (y0, yM , y2M , ...) = (z0, z1, z2, ...)

✭✽✳✶✮

❊✛❡❝t ✐♥ t❤❡ ❢r❡q✉❡♥❝② ❞♦♠❛✐♥ ■♥ ❝♦♥t✐♥✉♦✉s t✐♠❡✱ ❞♦✇♥s❛♠♣❧✐♥❣ t❤❡ s✐❣♥❛❧ y (t) ❝♦rr❡s♣♦♥❞s t♦ ♣❛ss✐♥❣

t♦ t❤❡ s✐❣♥❛❧ z (t) = y (Mt)✳ ■♥❞❡❡❞✱ s❛♠♣❧✐♥❣ z ❛t t❤❡ s❛♠❡ ❢r❡q✉❡♥❝② ❛s y ✇❡ ♦❜t❛✐♥ t❤❡ s❛♠♣❧❡s zk = ykM

❛s ✇❡ s❤♦✉❧❞✳ ■♥ s✉♠♠❛r②✱ ✉s✐♥❣ t❤❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠

1

f

z (t) = y (M t)

Z (f ) =

Y

✭✽✳✷✮

M

M

❚❤✐s t❡❧❧s ✉s✱ t❤❛t t❤❡ s♣❡❝tr❛❧ ❝♦♣✐❡s ♦❢ Ze ✭t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ♦❢ t❤❡ s❛♠♣❧❡❞ s✐❣♥❛❧ ze (t)✮ s❤♦✉❧❞ ❤❛✈❡

t❤❡ s❛♠❡ ♦✈❡r❛❧❧ s❤❛♣❡ ❛♥❞ ❛t t❤❡ s❛♠❡ ❞✐st❛♥❝❡ fe ❛s t❤♦s❡ ♦❢ Ye✱ ❜✉t t❤❡② ❛r❡ str❡t❝❤❡❞ ✐♥ ❢r❡q✉❡♥❝② ❜② ❛

❢❛❝t♦r M ❛♥❞ sq✉❡❡③❡❞ ✐♥ ❛♠♣❧✐t✉❞❡ ❜② ❛ ❢❛❝t♦r M✳

■♥❞❡❡❞✱ ✇❡ ❛rr✐✈❡ ❛t t❤❡ s❛♠❡ ❝♦♥❝❧✉s✐♦♥ ❝♦♠♣✉t✐♥❣ ✈✐❛ t❤❡ s❛♠♣❧❡❞ s✐❣♥❛❧s✱ ✉s✐♥❣ ✭✼✳✺✮✿

1

1

f

Ze (f ) =

τ zke−j2πfkτ =

τ ykM e−j2πfkτ =

(M τ ) ykM e−j2π(f/M)k(Mτ) =

Yd

✭✽✳✸✮

M

M

M

k

k

k

❘❡❝❛❧❧ t❤❛t Yd (f)✱ t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ♦❢ y s❛♠♣❧❡❞ ❛t ❢r❡q✉❡♥❝② fd = fe/M✱ ❧♦♦❦s ❧✐❦❡ Ye ❡①❝❡♣t t❤❛t

✐ts s♣❡❝tr❛❧ ❝♦♣✐❡s ❧✐❡ ❛t ❞✐st❛♥❝❡ fd✱ t❤✉s M t✐♠❡s ❝❧♦s❡r t❤❛♥ t❤♦s❡ ♦❢ Ye✳

❋♦r t❤❡ ❞✐s❝r❡t❡ ❋♦✉r✐❡r ❚r❛♥s❢♦r♠ ✇❡ s❤♦✉❧❞ ❡①♣❡❝t t♦ s❡❡ r♦✉❣❤❧② t❤❡ s❛♠❡ ❜❡❤❛✈✐♦r✳ ❘❡❝❛❧❧✱ t❤♦✉❣❤✱

t❤❛t t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ ❝♦♥t✐♥✉♦✉s ❛♥❞ ❞✐s❝r❡t❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ✐s ♦♥❧② ❛♣♣r♦①✐♠❛t✐✈❡✳

✶❚❤✐s ❝♦♥t❡♥t ✐s ❛✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡ ❛t ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✹✻✽✵✾✴✶✳✹✴❃✳

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

✸✶

✸✷

❈❍❆P❚❊❘ ✽✳ ❉❊❈■▼❆❚■❖◆ ❆◆❉ ❉❖❲◆❙❆▼P▲■◆●

✭❛✮

✭❜✮

❋✐❣✉r❡ ✽✳✶✿ ❚❤❡ r❡s✉❧t ♦❢ ❞♦✇♥s❛♠♣❧✐♥❣ t❤❡ ✜❧t❡r ❢r♦♠ ❋✐❣✉r❡ ✼✳✺ ❜② ❛ ❢❛❝t♦r M = 2 ♦♥ t❤❡ ❧❡❢t✱ ❛♥❞ ✐ts

♣♦✇❡r s♣❡❝tr✉♠ |❉❋❚|2 ♦♥ t❤❡ r✐❣❤t ✭❜❧❛❝❦ ❛♥❞ r❡❞✮✳ ❆❣❛✐♥✱ t❤❡ ✐❞❡❛❧ s✐♥❝ ✜❧t❡r ✐♥ ❜❧✉❡ ❢♦r ❝♦♠♣❛r✐s♦♥✳

◆♦t❡ t❤❛t t❤❡ ✜❧t❡r✬s ❝✉t♦✛ ❢r❡q✉❡♥❝② ❤❛s ✐♥❝r❡❛s❡❞ ❜② M ✭str❡t❝❤✐♥❣ ♦❢ t❤❡ s♣❡❝tr✉♠ ❜② M✮ ❛♥❞ t❤❛t

✐ts |❉❋❚|2 ✈❛❧✉❡s ❤❛✈❡ ❞❡❝r❡❛s❡❞ ❜② 1/M2 = 0.25 ✭s❡❡ t❡①t✮✳ ❚❤❡ ♣♦✇❡r ❝♦♠♣✉t❡❞ ❢r♦♠ ❡✐t❤❡r s✐❞❡ ✐s

✵✳✵✵✸✽✷✾ ✭r❡❝❛❧❧ ✭✹✳✺✮✮✿ ❞♦✇♥s❛♠♣❧✐♥❣ ♦❢ ❛ s✐❣♥❛❧ ✇✐t❤ 2B < fe/M ❞♦❡s ♥♦t ❝❤❛♥❣❡ ♣♦✇❡r✳ ✭❛✮ ✭❜✮

❆❧✐❛s✐♥❣✿ ❲❡ ❣✐✈❡ t✇♦ ❡①❛♠♣❧❡s✳

❋✐rst✱ ❞♦✇♥s❛♠♣❧✐♥❣ t♦ ❛ s❛♠♣❧✐♥❣ r❛t❡ t❤❛t ✐s t♦♦ ❧♦✇ ❝❛♥ ❧❡❛❞ t♦ ❛❧✐❛s✐♥❣✳ ❉♦✇♥s❛♠♣❧✐♥❣ t❤❡ ✐♠❛❣❡ ♦❢

❋✐❣✉r❡ ✼✳✶ ❧❡❢t ❧❡❛❞s t♦ t❤❡ s❛♠❡ ❡✛❡❝t ❛s ✈✐s✐❜❧❡ ✐♥ t❤❡ s❛♠❡ ✜❣✉r❡ ❝❡♥t❡r✳

❙❡❝♦♥❞✱ ❞♦✇♥s❛♠♣❧✐♥❣ ❛ s✐♠♣❧❡ s✐❣♥❛❧ s✉❝❤ ❛s t❤❡ ✜❧t❡r b ♦❢ ❋✐❣✉r❡ ✼✳✺ ❜② M = 2 r❡s✉❧ts ✐♥ ❛ ♥❡✇ ✜❧t❡r b✬

❫

✇✐t❤ t✇✐❝❡ t❤❡ ❝✉t♦✛ ❢r❡q✉❡♥❝②✱ ✐✳❡✳ f✬ = 1/4 = 0.25✱ ❜✉t ✇✐t❤ ❛ ✈❛❧✉❡

c

b k = 1/2 = 1/M ♦✈❡r t❤❡ ♣❛ss✲❜❛♥❞✱

t❤✉s ❛ ♣♦✇❡r s♣❡❝tr❛❧ ✈❛❧✉❡ 1/4 = 1/M2 ♦✈❡r t❤❡ ♣❛ss✲❜❛♥❞ ✭s❡❡ ❋✐❣✉r❡ ✽✳✶✮✳ ◆♦ ❛❧✐❛s✐♥❣ ♦❝❝✉rs s✐♥❝❡ t❤❡

❝♦♥❞✐t✐♦♥ fe > M · 2B ✐s s❛t✐s✜❡❞✳

P♦✇❡r✿ ❲❡ ❝♦♥s✐❞❡r ✜rst t❤❡ ❝❛s❡ ♦❢ ❛ T ✲♣❡r✐♦❞✐❝ s✐❣♥❛❧ y✳ ❚❤❡♥✱ z (t) = y (Mt) ❤❛s ♣❡r✐♦❞ T/M✳

❙✉❜st✐t✉t✐♥❣ s = tM ✇✐t❤ ds = Mdt ✇❡ ❣❡t

T /M

T /M

T

P

✭✽✳✹✮

z

=

1

|z (t) |2dt = M

|y (M t) |2dt = 1

|y (s) |2ds = P

T /M

0

T

0

T

0

y

❋r♦♠ t❤❡ s✐♠♣❧❡ ♣r♦♣❡rt✐❡s ✇❡ ❦♥♦✇ Zk|

= Y

✭s❛♠❡ ❝♦♠♣❧❡① ❛♠♣❧✐t✉❞❡s✷✱ ❜✉t ❜❡❧♦♥❣✐♥❣ t♦

f =

k

k |

T /M

f = k

T

❞✐✛❡r❡♥t ❢r❡q✉❡♥❝✐❡s✮✳ ❯s✐♥❣ P❛rs❡✈❛❧ ✇❡ s❡❡ ❛❣❛✐♥ t❤❛t t❤❡ ♣♦✇❡r ❞♦❡s ♥♦t ❝❤❛♥❣❡✳ ❙❡❡ ♣r♦♣❡rt✐❡s ♦❢ ❋♦✉r✐❡r

❙❡r✐❡s✳

✷❚❤❡r❡ ✐s ♥♦ ❝♦♥tr❛❞✐❝t✐♦♥ ❜❡t✇❡❡♥

“

”

Z

f

k = Yk ❛♥❞ Z (f ) = 1 Y

✳ ❇♦t❤ ❡①♣r❡ss t❤❛t z (t) = y (Mt)✱ ❜♦t❤ ❛❧❧♦✇ t♦ ❝♦♥❝❧✉❞❡

M

M

t❤❛t ♣♦✇❡r ❞♦❡s ♥♦t ❝❤❛♥❣❡ ✉♥❞❡r ❞♦✇♥s❛♠♣❧✐♥❣✱ ❛♥❞ ❜♦t❤ ✐♠♣❧② t❤❛t t❤❡ ❉❋❚ ♦❢ z ✐s M t✐♠❡s s♠❛❧❧❡r ❛♥❞ ❝♦rr❡s♣♦♥❞s t♦

❢r❡q✉❡♥❝✐❡s ✇❤✐❝❤ ❛r❡ M t✐♠❡s ❢✉rt❤❡r ❛♣❛rt t❤❛♥ t❤♦s❡ ♦❢ y✳ ■♥❞❡❡❞✱ ✉s✐♥❣ ✭✷✳✸✮ ✇❡ ❣❡t

❫

K

1

1 ❫

z

y

k |

=

Z

KY

✭✽✳✺✮

f =

k

k =

k =

k |f= k

T /M

M

M

M

T

❛♥❞ ✉s✐♥❣ ✭✻✳✷✮ ✇✐t❤ L ❡q✉❛❧ t♦ t❤❡ ❝♦rr❡❝t ♣❡r✐♦❞ ❛♥❞ ♥✉♠❜❡r ♦❢ s❛♠♣❧❡s ✭T/M ❛♥❞ K/M ❢♦r z (t) r❡s♣❡❝t✐✈❡❧② T ❛♥❞ K ❢♦r y (t)✮ ✇❡ ❣❡t ❛❣❛✐♥

❫

K/M

„

k

«

K 1

„

1

k

«

1 K

„ k «

1 ❫

z

y

k |

=

Z

=

Y

=

Y

=

✭✽✳✻✮

f =

k

k |f= k

T /M

T /M

T /M

T M

M T /M

M T

T

M

T

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

✸✸

◆♦t❡ t❤❛t t❤❡ ❡♥❡r❣② ♦❢ ❛ ✜♥✐t❡ ❡♥❡r❣② s✐❣♥❛❧ ✇♦✉❧❞ ❝❤❛♥❣❡ ✉♥❞❡r ❞♦✇♥s❛♠♣❧✐♥❣✿ ❝♦♠♣✉t✐♥❣ ✐♥ t✐♠❡

∞

∞

∞

||z||2

=

|z (t) |2dt =

|y (M t) |2dt = 1

|y (s) |2ds = 1 ||y||2

✭✽✳✼✮

−∞

−∞

M

−∞

M

❆ s✐♠❧✐❛r ❝♦♠♣✉t❛t✐♦♥ ❝❛♥ ❜❡ ❞♦♥❡ ✐♥ ❢r❡q✉❡♥❝②✳ ❙❡❡ ❈♦♠♠❡♥t ❇♦① ✻ ✭♣✳ ✸✸✮✳

❈♦♠♠❡♥t ❇♦① ✻ ❈♦♠♣✉t✐♥❣ ✐♥ ❢r❡q✉❡♥❝② ✇✐t❤ f = Mg ❛♥❞ df = Mdg✿

∞

∞

2

∞

||Z||2

=

|Z (f ) |2df =

| 1 Y

f

| df = 1

|Y (g) |2dg = 1 ||Y ||2

✭✽✳✽✮

−∞

−∞ M

M

M

−∞

M

❋♦r ❛ ❞✐s❝r❡t❡ s✐❣♥❛❧ ✇❡ ❛r❡ ♥❛t✉r❛❧❧② ✐♥ t❤❡ ♣❡r✐♦❞✐❝ ❝❛s❡✳ ❋r♦♠ t❤❡ ❛❜♦✈❡ ✇❡ s❤♦✉❧❞ ❡①♣❡❝t t❤❛t ♣♦✇❡r st❛②s ❛♣♣r♦①✐♠❛t✐✈❡❧② t❤❡ s❛♠❡ ✉♥❞❡r ❞♦✇♥s❛♠♣❧✐♥❣ ♣r♦✈✐❞❡❞ t❤❛t fe > M · 2B✳ ❆❧s♦✱ ✇❡ ♥♦t❡❞ ❡❛r❧✐❡r t❤❛t

♣♦✇❡r s❤♦✉❧❞ ♥♦t ❞❡♣❡♥❞ ♦♥ t❤❡ s❛♠♣❧✐♥❣ r❛t❡✱ ❛s ❧♦♥❣ ❛s t❤❡ s❛♠♣❧❡s ❢❛✐t❤❢✉❧❧② r❡♣r❡s❡♥t t❤❡ s✐❣♥❛❧✱ ❛♥❞ ❛t

❧❡❛st ❛♣♣r♦①✐♠❛t✐✈❡❧②✳

❋♦r t❤❡ s✐♠♣❧❡ s✐❣♥❛❧ b✱ t❤❡ ✜❧t❡r ❢r♦♠ ❋✐❣✉r❡ ✼✳✺✱ ✇❡ ♠❛② ✈❡r✐❢② t❤✐s ❡①♣❧✐❝✐t❧②✳ ❉❡♥♦t❡ t❤❡ ❞♦✇♥s❛♠♣❧❡❞

✜❧t❡r ❜② b✬ ✭s❡❡ ❋✐❣✉r❡ ✽✳✶ ❢♦r ❛♥ ✐❧❧✉str❛t✐♦♥ ✇✐t❤ M = 2✮✳ ❙✐♥❝❡ ♥♦ ❛❧✐❛s✐♥❣ ♦❝❝✉rs ❞✉r✐♥❣ ❞♦✇♥s❛♠♣❧✐♥❣✱

Θ

t❤❡ ♣❛ss✲❜❛♥❞ ✐s ♥♦✇ M t✐♠❡s ❧♦♥❣❡r✱ ♠❡❛♥✐♥❣ t❤❛t M t✐♠❡s ♠♦r❡ ♦❢ t❤❡ b✬k ❛r❡ ❞✐✛❡r❡♥t ❢r♦♠ ③❡r♦ ✭t❤✐s Θ

♠❛❦❡s t❤❡ ♣♦✇❡r ✐♥❝r❡❛s❡ ❜② M✮✳ ❋✉rt❤❡r✱ t❤❡✐r ♣♦✇❡r s♣❡❝tr❛❧ ✈❛❧✉❡s |b✬k |2 ❛r❡ ❜② M2 s♠❛❧❧❡r ✭t❤✐s ♠❛❦❡s t❤❡ ♣♦✇❡r ❞❡❝r❡❛s❡ ❜② M2✮✳ ❋✐♥❛❧❧②✱ t❤❡ s❛♠♣❧❡ ❧❡♥❣t❤ ✐s ♥♦✇ M t✐♠❡s s❤♦rt❡r ✭t❤✐s ✐♥❝r❡❛s❡s t❤❡ ♣♦✇❡r ♠②

M ❀ r❡❝❛❧❧ ✭✹✳✺✮✮✳

❆❧❧ ✐♥ ❛❧❧✱ ♣♦✇❡r ✐s ♥♦t ❝❤❛♥❣❡❞✱ ❛t ❧❡❛st ❛♣♣r♦①✐♠❛t✐✈❡❧②✳ ❖♥❡ ✜♥❞s Pb✬ = 0.003✬829 ✇❤✐❝❤ ❤❛s t♦ ❜❡

❝♦♠♣❛r❡❞ t♦ t❤❡ ♣♦✇❡r Pb = 0.003✬891 ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ✜❧t❡r✳

❚♦ ♦❜t❛✐♥ ❛ ❧♦✇✲♣❛ss ✜❧t❡r ♦♥❡ ✇♦✉❧❞ ❤❛✈❡ t♦ ♥♦r♠❛❧✐③❡ b✬ t♦ b✬✬ = M · b✬✳

✽✳✶✳✷ ❉❡❝✐♠❛t✐♦♥

▲❡t ✉s ♥♦✇ ❞r♦♣ t❤❡ ❛ss✉♠♣t✐♦♥ fe > M · 2B✳

❚♦ r❡s❛♠♣❧❡ ❛ s✐❣♥❛❧ x (t) ❛t ❛♥ M t✐♠❡s ❧♦✇❡r r❛t❡✱ ❛ ✜rst ❛tt❡♠♣t ✇♦✉❧❞ ❜❡ t♦ ❞✐s❝❛r❞ ❛❧❧ ❜✉t ❡✈❡r② M✲t❤

s❛♠♣❧❡✿ zk = xkM = x (kMτ)✳ ❚❤✐s st❡♣ ✐s ❝❛❧❧❡❞ ❞♦✇♥s❛♠♣❧✐♥❣ ❛s ✇❡ ❤❛✈❡ s❡❡♥ ❛❜♦✈❡✳ ❍♦✇❡✈❡r✱ t♦ ❛✈♦✐❞

❛❧✐❛s✐♥❣ ❡✛❡❝ts ❝❛✉s❡❞ ❜② ❞♦✇♥s❛♠♣❧✐♥❣ ❜❡❧♦✇ ◆②q✉✐st r❛t❡ ❛ ❧♦✇✲♣❛ss ✜❧t❡r✐♥❣ ❛t ❝✉t♦✛ B∗ = fe/ (2M) ✐s r❡q✉✐r❡❞ ❜❡❢♦r❡ ❞♦✇♥s❛♠❧✐♥❣✳ ❚❤❡ ✜❧t❡r ✉s❡❞ ✐s ❝❛❧❧❡❞ ❛♥t✐✲❛❧✐❛s✐♥❣ ✜❧t❡r✳ ❚❤❡ ♥❡✇ ✉t✐❧✐③❡❞ ❜❛♥❞✇✐❞t❤ ✇✐❧❧

❜❡ ♦♥❧② B∗✳

❚❤❡ ♣r♦❝❡❞✉r❡ ♦❢ ✜rst ❛♣♣❧②✐♥❣ ❛♥ ❛♥t✐✲❛❧✐❛s✐♥❣ ✜❧t❡r ❛♥❞ t❤❡♥ ❞♦✇♥s❛♠♣❧✐♥❣ ✐s ❝❛❧❧❡❞ ❞❡❝✐♠❛t✐♦♥✳

❆❣r❡❡❛❜❧②✱ t❤❡ ✜❧t❡r✐♥❣ ❜❡❢♦r❡ ❞♦✇♥s❛♠♣❧✐♥❣ ❞❡str♦②s ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t x (t)✳ ❍♦✇❡✈❡r✱ t❤✐s ❧♦ss ♦❝❝✉rs

✐♥ ❛ ❝♦♥tr♦❧❧❛❜❧❡ ♠❛♥♥❡r✿ ■t r❡♠♦✈❡s ❤✐❣❤✲❢r❡q✉❡♥❝② ✐♥❢♦r♠❛t✐♦♥✳ ❋♦r ❛♥ ❛✉❞✐♦ s✐❣♥❛❧✱ ✇❡ ❧♦♦s❡ t❤❡ ❤✐❣❤

♣✐t❝❤ s♦✉♥❞✳ ❋♦r ❛♥ ✐♠❛❣❡ ✇❡ ❧♦♦s❡ s❤❛r♣♥❡ss ♦❢ ❡❞❣❡s✳ ❚❤✐s s❤♦✉❧❞ ❜❡ ❝♦♠♣❛r❡❞ t♦ ❛♥ ✉♥❝♦♥tr♦❧❧❡❞ ❧♦ss ♦❢

q✉❛❧✐t② ✇❤❡♥ ♥♦ ❛♥t✐✲❛❧✐❛s✐♥❣ ✜❧t❡r ✐s ❛♣♣❧✐❡❞ ✭s❡❡ ❋✐❣✉r❡ ✼✳✶✮✳

■♥ s✉♠♠❛r②✱ ❉❡❝✐♠❛t✐♦♥ ❜② M ♠❡❛♥s t♦ r❡s❛♠♣❧❡ ❛t M t✐♠❡s ❧♦✇❡r r❛t❡ fd = fe/M ❛♥❞ ❝♦♥s✐sts ♦❢

t✇♦ st❡♣s✿

✶✳ ❧♦✇✲♣❛ss ✜❧t❡r✐♥❣ t❤❡ s❛♠♣❧❡s ❛t ❝✉t♦✛ ❢r❡q✉❡♥❝② 1

2M

(x0, x1, x2, ...) →

∩ 1

→ (y

2M

0, y1, y2, ...)

✭✽✳✾✮

✷✳ ❉♦✇♥✲s❛♠♣❧✐♥❣ ✭✐♥ ❋r❡♥❝❤✿ ❞❡❝✐♠❛t✐♦♥✮

(y0, y1, y2, ...) →

↓ M

→ (y0, yM , y2M , ...) = (z0, z1, z2, ...)

✭✽✳✶✵✮

▲♦✇✲♣❛ss ✜❧t❡r✐♥❣ ✇✐❧❧ r❡❞✉❝❡ ♣♦✇❡r✱ ❛s ❤✐❣❤ ❢r❡q✉❡♥❝✐❡s ❛r❡ ❛tt❡♥✉❛t❡❞✳ ❉♦✇♥✲s❛♠♣❧✐♥❣ ✇✐❧❧ ❧❡❛✈❡ t❤❡ ♥❡✇

♣♦✇❡r r♦✉❣❤❧② t❤❡ s❛♠❡✳

❚❤❡ ♠❛t❧❛❜ ❝♦♠♠❛♥❞s ❛r❡ ❞❡❝✐♠❛t❡ ❛♥❞ ❞♦✇♥s❛♠♣❧❡✳

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

✸✹

❈❍❆P❚❊❘ ✽✳ ❉❊❈■▼❆❚■❖◆ ❆◆❉ ❉❖❲◆❙❆▼P▲■◆●

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

❈❤❛♣t❡r ✾

■♥t❡r♣♦❧❛t✐♦♥ ❛♥❞ ❯♣s❛♠♣❧✐♥❣✶

✾✳✶ ■♥t❡r♣♦❧❛t✐♦♥ ❛♥❞ ❯♣s❛♠♣❧✐♥❣ ■♥t❡r♣♦❧❛t✐♦♥

▲❡t ✉s ♥♦✇ ❧♦♦❦ ❛t ✐♥❝r❡❛s✐♥❣ t❤❡ s❛♠♣❧❡ r❛t❡✳ ❙✐♥❝❡ ✐t ✐s ❧❡ss ♦❜✈✐♦✉s ❤♦✇ t♦ ❛❝❤✐❡✈❡ t❤✐s✱ ❧❡t ✉s ✜rst ❝♦♥s✉❧t t❤❡♦r②✳

●✐✈❡♥ ❛r❡ t❤❡ s❛♠♣❧❡s xn = x (nτ) ♦❢ ❛ ❜❛♥❞✲❧✐♠✐t❡❞ s✐❣♥❛❧ x (t) t❛❦❡♥ ❛t ❢r❡q✉❡♥❝② fe = 1/τ ✐s ❛❜♦✈❡

t❤❡ ◆②q✉✐st r❛t❡ 2B✳ ❲❡ ✇❛♥t t♦ ❝♦♠♣✉t❡ t❤❡ s❛♠♣❧❡s zk ♦❢ x (t) ❛t ❛ ❤✐❣❤❡r s❛♠♣❧✐♥❣ r❛t❡ fu = N · fe✳

❚❤✐s ♠❡❛♥s t❤❛t t❤❡ ♥❡✇ s❛♠♣❧✐♥❣ st❡♣ s❤♦✉❧❞ ❜❡ 1/fu = 1/ (feN) = τ/N ♦r✱ zk = x (kτ/N)✳

❙✐♥❝❡ t❤❡ ♦r✐❣✐♥❛❧ s❛♠♣❧✐♥❣ r❛t❡ fe > 2B ✐s ❛❜♦✈❡ ◆②q✉✐st✱ ✇❡ ❝❛♥ ✐♥ t❤❡♦r② r❡❝♦♥str✉❝t t❤❡ ❡♥t✐r❡ s✐❣♥❛❧

x (t) ✉s✐♥❣ t❤❡ r❡❝♦♥str✉❝t✐♦♥ ❢♦r♠✉❧❛ ✭✼✳✶✵✮ ✉s✐♥❣ fc = fe/2✳ ❖♥❝❡ t❤❡ ❝♦♥t✐♥✉♦✉s✲t✐♠❡ ✭✜♥✐t❡ ❡♥❡r❣②✮ s✐❣♥❛❧

x (t) ✐s ♦❜t❛✐♥❡❞✱ ✇❡ ♦♥❧② ♥❡❡❞ t♦ s❛♠♣❧❡ ✐t ❛t t = kτ /N = k/fu✳ ❚❤✐s r❡❛❞s ❛s ❢♦❧❧♦✇s

(28)

∞

−nτ

∞

z

✭✾✳✶✮

k = z (kτ ) = x

k τ

=

x

=

x

N

n=−∞

ns✐♥❝

k τ

N

τ

n=−∞

ns✐♥❝

k−nN

N

❚❤✐s ❢♦r♠✉❧❛ ❛❧❧♦✇s ✐♥❞❡❡❞ t♦ ❝♦♠♣✉t❡ zk ❢r♦♠ xn✱ ❛t ❧❡❛st ✐♥ ♣r✐♥❝✐♣❧❡✳ ❍♦✇❡✈❡r✱ ❛ ❝❧♦s❡r ❧♦♦❦ ❛t t❤❡♦r② ✐s r❡q✉✐r❡❞ t♦ ✉♥❞❡rst❛♥❞ t❤❡ ❡✛❡❝t ✇❤❡♥ ✉s✐♥❣ ♦♥❧② ✜♥✐t❡ ♠❛♥② s❛♠♣❧❡s✳ ❚❤❡ r❡❝♦♥str✉❝t✐♦♥ ❢♦r♠✉❧❛ ✭✼✳✶✵✮ ✐s

❜❡st ✉♥❞❡rst♦♦❞ ✐♥ t❤❡ ❢r❡q✉❡♥❝② ❞♦♠❛✐♥✿ ✐t ❛♠♦✉♥ts t♦ r❡♠♦✈✐♥❣ t❤❡ s♣❡❝tr❛❧ ❝♦♣✐❡s ♦❢ Xe (f) ✈✐❛ ✜❧t❡r✐♥❣

✇✐t❤ ❝✉t✲♦✛ ❢r❡q✉❡♥❝② fc = fe/2✳ ❚♦ ♠❛❦❡ t❤✐s ✜❧t❡r✐♥❣ st❡♣ ✈✐s✐❜❧❡ ✇❡ ♥❡❡❞ t♦ ✇r✐t❡ ✭✾✳✶✮ ✐♥ ❢♦r♠ ♦❢ ❛

❝♦♥✈♦❧✉t✐♦♥✳ ❚♦ t❤✐s ❡♥❞✱ ✇❡ ✇r✐t❡

∞

zk =

y

= {y

} = {N · y

s✐♥❝ m }

✭✾✳✷✮

m=−∞

ms✐♥❝

k−m

N

m} ∗ {s✐♥❝

m

N

m} ∗ { 1

N

N

✇❤❡r❡ t❤❡ ♥❡✇ s❡q✉❡♥❝❡ ym ✐s ♦❜t❛✐♥❡❞ ❜② ✏✉♣s❛♠♣❧✐♥❣✑ ❛♥❞ ✐s ❣✐✈❡♥ ❛s✿

xn

✐❢ m/N = n ✐s ✐♥t❡❣❡r

ym = {

✭✾✳✸✮

0

✐❢ m/N ✐s ♥♦t ✐♥t❡❣❡r

❚❤❡ ❝♦♥✈♦❧✉t✐♦♥ ✭✾✳✷✮ ❛❧❧♦✇s ❢♦r ♠♦r❡ ❝♦♥✈❡♥✐❡♥t ❞❛t❛ ♣r♦❝❡ss✐♥❣ ✈✐❛ ❞✐❣✐t❛❧ ✜❧t❡r✐♥❣ ❛♥❞ ❢♦r ❛ s✐♠♣❧❡

s♣❡❝tr❛❧ ✐♥t❡r♣r❡t❛t✐♦♥✳

■♥t❡r♣♦❧❛t✐♦♥ ❜② N ♦r r❡s❛♠♣❧✐♥❣ ❛t N t✐♠❡s ❧❛r❣❡r r❛t❡ ❝♦♥s✐sts ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ st❡♣s✿

✶✳ ❯♣✲s❛♠♣❧✐♥❣ ✭✐♥ ❋r❡♥❝❤✿ ✏✐♥t❡r♣♦❧❛t✐♦♥✑✮





(x0, x1, x2, ...)

→

↑ N →

x

, x

, x

, x

= (y



0, 0, .., 0

1, 0, .., 0

2, 0, .., 0

3...

0, y1, y2, ...)

✭✾✳✹✮

N −1

N −1

N −1

✶❚❤✐s ❝♦♥t❡♥t ✐s ❛✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡ ❛t ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✹✻✽✷✶✴✶✳✹✴❃✳

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

✸✺

✸✻

❈❍❆P❚❊❘ ✾✳ ■◆❚❊❘P❖▲❆❚■❖◆ ❆◆❉ ❯P❙❆▼P▲■◆●

✷✳ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❜② N ❛♥❞ ▲♦✇✲♣❛ss ✜❧t❡r✐♥❣ ❛t ❝✉t♦✛ ❢r❡q✉❡♥❝② 1 ✉s✐♥❣ t❤❡ ✐❞❡❛❧ ✜❧t❡r 1 s✐♥❝ m ✿

2N

N

N

(y0, y1, y2, ...)

→

·N → ∩ 1 → {y

} = (z

2N

m} ∗ {s✐♥❝

m

N

0, z1, z2, ...)

✭✾✳✺✮

✭❛✮

✭❜✮

❋✐❣✉r❡ ✾✳✶✿ ❚❤❡ r❡s✉❧t ♦❢ ✉♣s❛♠♣❧✐♥❣ t❤❡ ✜❧t❡r ❢r♦♠ ❋✐❣✉r❡ ✼✳✺ ❜② ❛ ❢❛❝t♦r N = 3 ♦♥ t❤❡ ❧❡❢t✱ ❛♥❞

✐ts ♣♦✇❡r s♣❡❝tr✉♠ |❉❋❚|2 ♦♥ t❤❡ r✐❣❤t ✭❜❧❛❝❦ ❛♥❞ r❡❞✮✳ ❆❣❛✐♥✱ t❤❡ ✐❞❡❛❧ s✐♥❝ ✜❧t❡r ✭❞✐✈✐❞❡❞ ❜② N t♦

❛❞❥✉st ❢♦r t❤❡ ③❡r♦✲s❛♠♣❧❡s✮ ✐♥ ❜❧✉❡ ❢♦r ❝♦♠♣❛r✐s♦♥✳ ◆♦t❡ t❤❛t t❤❡ ✜❧t❡r✬s ❝✉t♦✛ ❢r❡q✉❡♥❝② ❤❛s ❞❡❝r❡❛s❡❞

❜② N ✭❝♦♥tr❛❝t✐♦♥ ♦❢ t❤❡ s♣❡❝tr✉♠ ❜② N✮ ❛♥❞ t❤❛t ✐ts |❉❋❚|2 ✈❛❧✉❡s ❤❛✈❡ ♥♦t ❝❤❛♥❣❡❞ ✭s❡❡ t❡①t✮✳ ❚❤❡

♣♦✇❡r ❝♦♠♣✉t❡❞ ❢r♦♠ ❡✐t❤❡r s✐❞❡ ✐s ✵✳✵✵✶✬✷✾✼ ❛♥❞ ❤❛s ❞❡❝r❡❛s❡❞ ❜② ❛ ❢❛❝t♦r N ❢r♦♠ t❤❡ ♦r✐❣✐♥❛❧ ♣♦✇❡r

♦❢ ✵✳✵✵✸✬✽✷✾✿ ♦♥❧② ✉♣s❛♠♣❧✐♥❣ ♦❢ ❛ s✐❣♥❛❧ ✇✐t❤ 2B < fe ❞❡❝r❡❛s❡s ♣♦✇❡r ❜② t❤❡ ✉♣s❛♠♣❧✐♥❣ ❢❛❝t♦r✳ ✭❜✮

❙♣❡❝tr❛❧ ♣✐❝t✉r❡ ♦❢ ■♥t❡r♣♦❧❛t✐♦♥ ❜② N

■♥ ❛♥❛❧♦❣② t♦ t❤❡ ❞❡❝✐♠❛t✐♦♥ ✇❡ s❡t z (t) = x (t/N)✳ ❚❤❡♥✱ t❤❡ s❛♠♣❧❡s ♦❢ z (t) t❛❦❡♥ ❛t t❤❡ s❛♠❡ r❛t❡ fe

❝♦♥st✐t✉t❡ t❤❡ s❛♠♣❧❡s ♦❢ x t❛❦❡♥ ❛t r❛t❡ fu✳ ❆♥❛❧♦❣♦✉s❧② t♦ t❤❡ ❞❡❝✐♠❛t✐♦♥ ✇❡ ✜♥❞ q✉✐❝❦❧②

z (t) = x (t/N )

Z (f ) = N X (f N )

✭✾✳✻✮

✇❤✐❝❤ ✐♥❞✐❝❛t❡s ❤♦✇ t❤❡ s♣❡❝tr✉♠ ❛t r❛t❡ fu = Nfe ✐s ♦❜t❛✐♥❡❞✿ t❤❡ s♣❡❝tr✉♠ ❛t r❛t❡ fe ❣❡ts ❝♦♥tr❛❝t❡❞

✐♥ t❤❡ ❢r❡q✉❡♥❝② ❛①✐s ❜② N ❛♥❞ ❡①♣❛♥❞❡❞ ✐♥ ❛♠♣❧✐t✉❞❡ ❜② N✳ ◆♦t❡ t❤❛t t❤❡ s♣❡❝tr❛❧ ❝♦♣✐❡s ♦❢ Ze ❛r❡ ❛t

❞✐st❛♥❝❡ fe ❥✉st ❧✐❦❡ t❤♦s❡ ♦❢ Xe✳

❲❡ ♠❛② ❜r❡❛❦ t❤❡ ♣r♦❝❡❞✉r❡ ❞♦✇♥ ✐♥t♦ t❤❡ ✐♥❞✐✈✐❞✉❛❧ st❡♣s✿

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

✸✼

✭❛✮

✭❜✮

❋✐❣✉r❡ ✾✳✷✿ ❚❤❡ r❡s✉❧t ♦❢ ✐♥t❡r♣♦❧❛t✐♥❣ ✭✉♣s❛♠♣❧✐♥❣ ❛♥❞ ✜❧t❡r✐♥❣✮ t❤❡ ✜❧t❡r ❢r♦♠ ❋✐❣✉r❡ ✼✳✺ ❜② ❛ ❢❛❝t♦r N = 3 ♦♥ t❤❡ ❧❡❢t✱ ❛♥❞ ✐ts ♣♦✇❡r s♣❡❝tr✉♠ |❉❋❚|2 ♦♥ t❤❡ r✐❣❤t ✭❜❧❛❝❦ ❛♥❞ r❡❞✮✳ ❆❣❛✐♥✱ t❤❡ ✐❞❡❛❧ s✐♥❝

✜❧t❡r ✭♥♦ ♥❡❡❞ t♦ ❞✐✈✐❞❡ ❜② N s✐♥❝❡ t❤❡ ③❡r♦✲s❛♠♣❧❡s ❛r❡ ♥♦✇ ❝♦rr❡❝t❡❞ t❤r♦✉❣❤ t❤❡ ✜❧t❡r✐♥❣ st❡♣✮ ✐♥ ❜❧✉❡

❢♦r ❝♦♠♣❛r✐s♦♥✳ ◆♦t❡ t❤❛t t❤❡ ❢✉♥❞❛♠❡♥t❛❧ ♣❡r✐♦❞ ❝♦♥t❛✐♥s ♥♦✇ ♦♥❧② ♦♥❡ s♣❡❝tr❛❧ ❝♦♣②✱ ❛s ✐t s❤♦✉❧❞✱ ❛s

❛ r❡s✉❧t ❢r♦♠ t❤❡ ✜❧t❡r✐♥❣✳ ❚❤❡ s♣❡❝tr✉♠ ✐s ❝♦♥tr❛❝t❡❞ ❜② N ✐♥ ❢r❡q✉❡♥❝② ❛♥❞ ❡①♣❛♥❞❡❞ ✐♥ ❛♠♣❧✐t✉❞❡

❜② N2✳ ❚❤❡ ♣♦✇❡r ❝♦♠♣✉t❡❞ ❢r♦♠ ❡✐t❤❡r s✐❞❡ ✐s ✵✳✵✵✸✬✽✾✶ ❛♥❞ ❛❧♠♦st ✐❞❡♥t✐❝❛❧ t♦ t❤❡ ♦r✐❣✐♥❛❧ ♣♦✇❡r ♦❢

✵✳✵✵✸✬✽✷✾✿ ✐♥t❡r♣♦❧❛t✐♦♥ ♦❢ ❛ s✐❣♥❛❧ ✇✐t❤ 2B < fe ❞♦❡s ♥♦t ❝❤❛♥❣❡ ♣♦✇❡r✳ ✭❜✮

Ye (f ) =

τ yme−j2πfmτ =

τ xne−j2πfnNτ = Xe (N f ) .

✭✾✳✼✮

m

n

✭✐✮ ❯♣s❛♠♣❧✐♥❣ ✭✐♥tr♦❞✉❝✐♥❣ t❤❡ ③❡r♦✲s❛♠♣❧❡s✮ ❧❡❛✈❡s t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠✱ ❛♥❞ t❤✉s t❤❡ s♣❡❝tr✉♠ ❛❧♠♦st

✐♥t❛❝t✱ ❧❡❛❞✐♥❣ ♦♥❧② t♦ ❛ r❡s❝❛❧✐♥❣ ♦❢ t❤❡ ❢r❡q✉❡♥❝✐❡s ✭❝♦♥tr❛❝t✐♦♥ ♦❢ X✮✿ ■♥❞❡❡❞✱ t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ Ye ♦❢

t❤❡ s❛♠♣❧❡s ym ❜❡❝♦♠❡s

❚❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s✐❣♥❛❧ y (t) ✭✇✐t❤ s❛♠♣❧❡s ym ❛t s❛♠♣❧✐♥❣ r❛t❡ fe✮ ✐s ♥♦t ♦❢ ✐♥t❡r❡st✳ ■❢ ②♦✉ ✇❛♥t t♦

❦♥♦✇ ❛❜♦✉t ✐t ❛♥② ✇❛②✱ s❡❡ ❈♦♠♠❡♥t ❇♦① ✼ ✭♣✳ ✸✼✮✳ ◆♦t❡✱ ❤♦✇❡✈❡r✱ t❤❛t t❤❡ ❢✉♥❞❛♠❡♥t❛❧ ♣❡r✐♦❞ ♦❢ Ye

❛♠♦✉♥ts t♦ fe ❛♥❞ ❝♦♥t❛✐♥s N ❝♦♣✐❡s ♦❢ X (Nf) ✭s❡❡ ❋✐❣✉r❡ ✾✳✶✮✳

❈♦♠♠❡♥t ❇♦① ✼ ❋♦r ❝❧❛r✐t②✿ t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ♦❢ y ✐s ❢♦✉♥❞ ❜② r❡♠♦✈✐♥❣ t❤❡ s♣❡❝tr❛❧ ❝♦♣✐❡s

♦❢ Ye♦✉ts✐❞❡ [−fe/2, fe/2]✳ ❚❤❡s❡ ❝♦♣✐❡s ❛r❡ ❝❛✉s❡❞ ❜② s❛♠♣❧✐♥❣✳ ❚❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ♦❢ y ❝♦♥s✐sts ♦❢

N ❝♦♥tr❛❝t❡❞ ❝♦♣✐❡s ♦❢ X ❛t ❞✐st❛♥❝❡ fe/N ♦❢ ❡❛❝❤ ♦t❤❡r ✇❤✐❝❤ ❛r❡ ❝❛✉s❡❞ ❜② ✉♣s❛♠♣❧✐♥❣✳ ❯s✐♥❣ t❤❡

r❡❝♦♥str✉❝t✐♦♥ ❢♦r♠✉❧❛ ✭✼✳✶✵✮ ✇✐t❤ t❤❡ s❛♠♣❧❡s ym✱ s❛♠♣❧❡ r❛t❡ fe ❛♥❞ ❝♦rr❡❝t ♣❛ss✲❜❛♥❞ fe/2 ②✐❡❧❞s t❤❡

s✐❣♥❛❧ y (t) =

yms✐♥❝ t−mτ ✳ ❯s✐♥❣ t❤❛t s✐♥❝ (0) = 1 ✇❤✐❧❡ s✐♥❝ (m) = 0 ❢♦r ❛❧❧ ✐♥t❡❣❡r m = 0 ✇❡ ✜♥❞

τ

q✉✐❝❦❧② t❤❛t t❤❡ s❛♠♣❧❡s ♦❢ y (t) ❛r❡ ✐♥❞❡❡❞ y (kτ) = yk✱ ✐✳❡✳✱ (..., x0, 0..0, x1, 0...0, x2, ...)✳

✭✐✐✮ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ✇✐t❤ N r❡st♦r❡s t❤❡ ❛✈❡r❛❣❡ ✈❛❧✉❡ ♦❢ t❤❡ s❛♠♣❧❡s✳ ❚❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ✐s ♥♦✇

N Xe (N f ) ✇❤✐❝❤ ❝♦♥s✐sts ♦❢ ❝♦♣✐❡s ♦❢ N X (f N )✱ ♦r Z (f )✱ ❛t ❞✐st❛♥❝❡ fe/N ✭❛s ❢♦r Ye✱ t❤❡r❡ ❛r❡ N ❝♦♣✐❡s

✐♥ ♦♥❡ ♣❡r✐♦❞✮✳

✭✐✐✐✮ ❚❤❡ ❞✐❣✐t❛❧ ❧♦✇✲♣❛ss ✜❧t❡r✐♥❣ ♦❢ {Nym} ❛t ❝✉t✲♦✛ ❢r❡q✉❡♥❝② 1/ (2N) r❡♠♦✈❡s ❛❧❧ ♦❢ t❤❡ ❝♦♣✐❡s ♦❢

N X (f N ) ❡①❝❡♣t t❤❡ ♦♥❡s ❝❡♥t❡r❡❞ ❛t ✵✱ fe✱ 2fe ❡t❝✳ ❛♥❞ ❧❡❛✈❡s ♦♥❧② ♦♥❡ ❝♦♣② ♣❡r ♣❡r✐♦❞✱ ✐♥ ♦t❤❡r ✇♦r❞s✱

♦♥❧② ❝♦♣✐❡s ❛t ❞✐st❛♥❝❡ fe✳ ❲❤❛t r❡♠❛✐♥s ✐s ❡①❛❝t❧② Ze (f) ❛s ✇❡ ❤❛✈❡ ♣♦✐♥t❡❞ ♦✉t ❡❛r❧✐❡r✳ ✭s❡❡ ❋✐❣✉r❡ ✾✳✷✮✳

P♦✇❡r ❛♥❞ ■♥t❡r♣♦❧❛t✐♦♥

❙✐♠✐❧❛r❧② ❛s ✇✐t❤ t❤❡ ❞❡❝✐♠❛t✐♦♥ t❤❡ ♣♦✇❡r ♦❢ ❛ ♣❡r✐♦❞✐❝ s✐❣♥❛❧ ❞♦❡s ♥♦t ❝❤❛♥❣❡ ✉♥❞❡r ✐♥t❡r♣♦❧❛t✐♦♥✳

■♥ ❢❛❝t✱ ✇❡ ♦♥❧② ♥❡❡❞ t♦ ✉s❡ Z (f) = NX (fN) ❛♥❞ r❡♣❧❛❝❡ 1/M ❜② N ✐♥ t❤❡ ❝♦♠♣✉t❛t✐♦♥ ❞♦♥❡ ✇✐t❤ t❤❡

❞❡❝✐♠❛t✐♦♥✳ ❲❡ ❝♦♥❝❧✉❞❡ t❤❛t t❤❡ ♣♦✇❡r ♦❢ ❞✐s❝r❡t❡ s❛♠♣❧❡s ❞♦❡s ♥♦t ❝❤❛♥❣❡ ✉♥❞❡r ✐♥t❡r♣♦❧❛t✐♦♥ ♣r♦✈✐❞❡❞

t❤❛t fe > 2B✱ ❛t ❧❡❛st ❛♣♣r♦①✐♠❛t✐✈❡❧②✳

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

✸✽

❈❍❆P❚❊❘ ✾✳ ■◆❚❊❘P❖▲❆❚■❖◆ ❆◆❉ ❯P❙❆▼P▲■◆●

❚♦ ✈❡r✐❢② t❤✐s✱ ❧❡t ✉s ♠♦✈❡ t❤r♦✉❣❤ t❤❡ ✸ st❡♣s ❛❜♦✈❡✳ ❙t❡♣ ✭✐✮✱ ✉♣s❛♠♣❧✐♥❣✱ r❡❞✉❝❡s ♣♦✇❡r ❜② ❛ ❢❛❝t♦r N

s✐♥❝❡ t❤❡ s✉♠ ♦❢ sq✉❛r❡s ♦❢ t❤❡ s❛♠♣❧❡s ✐s t❤❡ s❛♠❡ ✭t❤❡ ③❡r♦s ❛❞❞❡❞ ❞♦♥✬t ❝♦♥tr✐❜✉t❡✮✱ ❜✉t t❤❡r❡ ❛r❡ ♥♦✇ N

t✐♠❡s ♠♦r❡ s❛♠♣❧❡s✳ P♦✇❡r ✐s ❛♥ ❛✈❡r❛❣❡✳

❙t❡♣ ✭✐✐✮ ♦❜✈✐♦✉s❧② ♠✉❧t✐♣❧✐❡s ♣♦✇❡r ✇✐t❤ N2✳ ❙t❡♣ ✭✐✐✐✮✱ t❤❡ ❧♦✇✲♣❛ss ✜❧t❡r✐♥❣✱ r❡♠♦✈❡s N − 1 s♣❡❝tr❛❧

❝♦♣✐❡s ❛♥❞ ❧❡❛✈❡s ♦♥❧② ✶✱ t❤✉s ❞✐✈✐❞❡s ♣♦✇❡r ❜② N✳ ❆❧❧ st❡♣s t♦❣❡t❤❡r ❧❡❛✈❡ t❤❡ ♣♦✇❡r ❛s ✐t ✐s✳

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

✭❝✮

✸✾

■♥t❡r♣♦❧❛t✐♦♥

✭❞✮

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

✭❛✮

✭❡✮

❋✐❣✉r❡ ✾✳✸✿ ❍♦r✐③♦♥t❛❧ ❛rr♦✇s ✐♥❞✐❝❛t❡ s♣❡❝tr❛❧ ❝♦♣✐❡s✳ ❚♦ ❦❡❡♣ t❤❡ s❛♠♣❧❡ r❛t❡ ❛t t❤❡ s❛♠❡ ✈❛❧✉❡ fe

♦♥❡ ❝❤❛♥❣❡s ❢r♦♠ Xu✱ r❡s♣✳ Yd t♦ Ze ✭s❡❡ ✹t❤✮✳ ■♥t❡r♣♦❧❛t✐♦♥ ✐❧❧✉str❛t❡❞ ✐♥ t❤❡ ❙♣❡❝tr❛❧ ❉♦♠❛✐♥ ✇✐t❤

N = 2✳

❍♦r✐③♦♥t❛❧ ❛rr♦✇s ✐♥❞✐❝❛t❡ s♣❡❝tr❛❧ ❝♦♣✐❡s✳ ❚♦ ❦❡❡♣ t❤❡ s❛♠♣❧❡ r❛t❡ ❛t t❤❡ s❛♠❡ ✈❛❧✉❡ fe ♦♥❡ ❝❤❛♥❣❡s

❢r♦♠ Xu t♦ Ze ✭s❡❡ ✭❞✮✮✳ ✭❛✮ ↑ N

s✐❣♥❛❧ ✐s ❝♦rr❡❝t❧② s❛♠♣❧❡❞ ✭B/fe < 1/2✮ ✭❜✮ ∩ 1

2N

✐♥s❡rt✐♥❣ N − 1 ③❡r♦❡s ❝♦♥tr❛❝ts t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ❜② ❛ ❢❛❝t♦r N ✭❝✮ ·N

❞✐❣✐t❛❧ ❧♦✇✲♣❛ss ✜❧t❡r✐♥❣ ✇✐t❤ ❝✉t✲♦✛ ❢r❡q✉❡♥❝② ❛t 1/ (2N) ❧❡❛✈❡s ♦♥❧② ♦♥❡ ♦❢ t❤❡ ❝♦♥tr❛❝t❡❞ ❝♦♣✐❡s ♣❡r

♣❡r✐♦❞ ✶✱ ✐✳❡✳✱ ✐t ❧❡❛✈❡s Xu (Nf) ✭❞✮ ■❢ B < fe2

♠✉❧t✐♣❧②✐♥❣ ✇✐t❤ N ❧❡❛❞s t♦ NXu (Nf) = Ze (f) ✇❤❡r❡ z (t) = x (t/N)✳ ❙❛♠♣❧✐♥❣ z ❛t r❛t❡ fe ♣r♦✈✐❞❡s

t❤❡ ❞❡s✐r❡❞ s❛♠♣❧❡s ❛t r❛t❡ fu ♦❢ x✳ ✭❡✮

■❢

i

B < fe/2✱ t❤❡♥ ❛ tr❛♥s✐t✐♦♥ ❜❛♥❞ h B , 1 − B

✐s ❢❡❛s✐❜❧❡

N fe

N

N fe

✭❜✮

✭❝✮

✹✵

❈❍❆P❚❊❘ ✾✳ ■◆❚❊❘P❖▲❆❚■❖◆ ❆◆❉ ❯P❙❆▼P▲■◆●

✭❞✮

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

✭❛✮

✭❡✮

❋✐❣✉r❡ ✾✳✹✿ ❉❡❝✐♠❛t✐♦♥ ✐❧❧✉str❛t❡❞ ✐♥ t❤❡ ❙♣❡❝tr❛❧ ❉♦♠❛✐♥ ✇✐t❤ M = 2✳

❍♦r✐③♦♥t❛❧ ❛rr♦✇s ✐♥❞✐❝❛t❡ s♣❡❝tr❛❧ ❝♦♣✐❡s✳ ❚♦ ❦❡❡♣ t❤❡ s❛♠♣❧❡ r❛t❡ ❛t t❤❡ s❛♠❡ ✈❛❧✉❡ fe ♦♥❡ ❝❤❛♥❣❡s

❢r♦♠ Yd t♦ Ze ✭s❡❡ ✭❞✮✮✳

■♥t❡r♣♦❧❛t✐♦♥ ❛♥❞ ❞❡❝✐♠❛t✐♦♥ ✐❧❧✉str❛t❡❞ ✐♥ t❤❡ ❙♣❡❝tr❛❧ ❉♦♠❛✐♥ ✇✐t❤ N = 2 ❛♥❞ M = 2✳ ✭❛✮ ∩ 1

2M

1/M ♠❛r❦s t❤❡ ❝❡♥t❡r ♦❢ t❤❡ ✜rst s♣❡❝tr❛❧ ❝♦♣② ❛❢t❡r ❞❡❝✐♠❛t✐♦♥ ✭❜✮ ↓ M

❞✐❣✐t❛❧ ❧♦✇✲♣❛ss ✜❧t❡r✐♥❣ ✇✐t❤ ✭✐❞❡❛❧✮ ❝✉t✲♦✛ ❢r❡q✉❡♥❝② ❛t 1/ (2M) ❡♥s✉r❡s ♥♦ ❛❧✐❛s✐♥❣ ❛❢t❡r ❞❡❝✐♠❛t✐♦♥❀

t❤❡ ♥❡✇ ✉t✐❧✐③❡❞ ❜❛♥❞✇✐❞t❤ ✐s B∗ = fe/ (2M)❀ ❤✐❣❤ ❢r❡q✉❡♥❝② ✐♥❢♦r♠❛t✐♦♥ ♠✐❣❤t ❜❡ ❧♦st✱ r❡s✉❧t✐♥❣ ✐♥ ❛

♥❡✇ s✐❣♥❛❧ y✳ ✭❝✮ ❛❞❥✉st✐♥❣ s❛♠♣❧✐♥❣ r❛t❡

❆❢t❡r ❞❡❝✐♠❛t✐♦♥ ♦♥❧② t❤❡ s❛♠♣❧❡s ykM r❡♠❛✐♥✱ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ❛ s❛♠♣❧✐♥❣ ♦❢ y ❛t fd = fe/M✱ ✇✐t❤

❋♦✉r✐❡r tr❛♥s❢♦r♠ Yd (f)✳ ✭❞✮ ■❢ B∗ < fe ❚❤❡ s❛♠♣❧❡s z

2M

k = ykM ❝♦rr❡s♣♦♥❞ t♦ z (t) = y (M t) s❛♠♣❧❡❞ ❛t

fe✱ ❛♥❞ Ze (f ) = (1/M ) · Yd (f /M ) ✭s❡❡ t❡①t✮✳ ✭❡✮ ❈❤♦♦s✐♥❣ t❤❡ ♥❡✇ ✉t✐❧✐③❡❞ ❜❛♥❞✇✐❞t❤ t♦ ❜❡ B∗ < fe ✱

2M

t❤❡♥ ❛ tr❛♥s✐t✐♦♥ ❜❛♥❞ h

i

B∗ , 1 − B∗ ✐s ❢❡❛s✐❜❧❡

fe

M

fe

✭❜✮

❈❤❛♣t❡r ✶✵

❙❛♠♣❧✐♥❣ ❘❛t❡ ❈♦♥✈❡rs✐♦♥✶

✶✵✳✶ ❙❛♠♣❧✐♥❣ r❛t❡ ❝♦♥✈❡rs✐♦♥

■♥ ❣❡♥❡r❛❧✱ s❛♠♣❧✐♥❣ r❛t❡s ❛r❡ ❝♦♥✈❡rt❡❞ ♦♥❧② ✐♥ r❛t✐♦♥❛❧ ❢r❛❝t✐♦♥s s✉❝❤ ❛s ❢r♦♠ ✸✷ t♦ ✹✽ ❦❍③ ❜② ❛ ❢❛❝t♦r ♦❢

✸✴✷ ✇❤✐❝❤ ❛r❡ ♣❡r❢♦r♠❡❞ ❜② ❛♣♣r♦♣r✐❛t❡ s❡q✉❡♥❝❡s ♦❢ ✉♣✲ ❛♥❞ ❞♦✇♥✲s❛♠♣❧✐♥❣✳

Pr❛❝t✐❝❛❧ ❝♦♥s✐❞❡r❛t✐♦♥s✿

• ❋✐rst ✐♥t❡r♣♦❧❛t❡✱ t❤❡♥ ❞❡❝✐♠❛t❡ ✭♣r❡s❡r✈❡ s✐❣♥❛❧ q✉❛❧✐t②✮

• t❤❡ ❧♦✇✲♣❛ss ✜❧t❡rs ✉s❡❞ ❛❢t❡r ✉♣s❛♠♣❧✐♥❣ ✭✷♥❞ st❡♣ ♦❢ ✐♥t❡r♣♦❧❛t✐♦♥✮ ❛♥❞ ❜❡❢♦r❡ ❞♦✇♥s❛♠♣❧✐♥❣ ✭✶st

st❡♣ ♦❢ ❞❡❝✐♠❛t✐♦♥✮ ❝❛♥ ❜❡ ❝♦♠❜✐♥❡❞ ✐♥t♦ t❤❡ ♦♥❡ ✜❧t❡r ✇✐t❤ t❤❡ s♠❛❧❧❡r ♣❛ss✲❜❛♥❞✳

• ■♥t❡r♣♦❧❛t✐♦♥ ❛♥❞ ❉❡❝✐♠❛t✐♦♥ ❜② ❧❛r❣❡ ❢❛❝t♦rs s❤♦✉❧❞ ❜❡ ❞♦♥❡ ✐♥ st❡♣s✱ ❦❡❡♣✐♥❣ N ❛♥❞ M ❜❡❧♦✇ ♦r ❛t

✻ ❛t ♠♦st✳

• ▼❛t❧❛❜ ❝♦♠♠❛♥❞s✿ ❞❡❝✐♠❛t❡✱ ❞♦✇♥s❛♠♣❧❡✱ ✐♥t❡r♣✱ ✉♣s❛♠♣❧❡✱ r❡s❛♠♣❧❡✳ ❙❡❡ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❤❡❧♣

♠❛♥✉❛❧ ✭❤❡❧♣ ❞❡❝✐♠❛t❡ ❡t❝✮✳

• ❚❤❡ ✐❞❡❛❧ s✐♥❝ ✜❧t❡r ✐s ♥♦t r❡❛❧✐③❛❜❧❡ s✐♥❝❡ ✐t r❡q✉✐r❡s ❦♥♦✇❧❡❞❣❡ ♦❢ t❤❡ ✐♥✜♥✐t❡ ♣❛st ❛♥❞ ❢✉t✉r❡❀ ♦♥❡

✉s❡s ❋■❘ ✜❧t❡rs ✇✐t❤ ❧✐♥❡❛r ♣❤❛s❡ t♦ ❛✈♦✐❞ ❛✉❞✐❜❧❡ ❛rt✐❢❛❝ts✳

✶❚❤✐s ❝♦♥t❡♥t ✐s ❛✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡ ❛t ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✹✻✽✷✽✴✶✳✷✴❃✳

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

✹✶

✹✷

❈❍❆P❚❊❘ ✶✵✳ ❙❆▼P▲■◆● ❘❆❚❊ ❈❖◆❱❊❘❙■❖◆

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

❈❤❛♣t❡r ✶✶

▼♦❞❡❧s ♦❢ ◆♦✐s❡✶

✶✶✳✶ ▼♦❞❡❧s ♦❢ ◆♦✐s❡

❲❤✐t❡ ◆♦✐s❡ ■♥ ❛ ♥✉ts❤❡❧❧✱ ❛ s❡q✉❡♥❝❡ ( 1, 2, 3, ...) ✐s ❝❛❧❧❡❞ ✇❤✐t❡ ♥♦✐s❡ ✐❢ ✐ts ❡♥tr✐❡s k ❛r❡ ✐♥❞❡♣❡♥❞❡♥t

✐❞❡♥t✐❝❛❧❧② ❞✐str✐❜✉t❡❞ r❛♥❞♦♠ ♥✉♠❜❡rs✳

• ✏■♥❞❡♣❡♥❞❡♥❝❡✑ ♠❡❛♥s t❤❛t t❤❡r❡ ✐s ♥♦ ✐♥❢♦r♠❛t✐♦♥ ♦r r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ♠❡♠❜❡rs k❀ ✐♥ ♦t❤❡r

✇♦r❞s✱ ❦♥♦✇✐♥❣ ♦r ♦❜s❡r✈✐♥❣ t❤❡ s❡q✉❡♥❝❡ ✉♥t✐❧ t✐♠❡ t✱ ✐✳❡✳✱ ( 1, ..., t) ❞♦❡s ♥♦t ❛❧❧♦✇ t♦ ♣r❡❞✐❝t t❤❡

♥❡①t ♠❡♠❜❡r t+1 ❛♥② ❜❡tt❡r t❤❛♥ ✐❢ ♥♦t❤✐♥❣ ❤❛❞ ❜❡❡♥ ♦❜s❡r✈❡❞✳

• ✏❘❛♥❞♦♠✑ ♠❡❛♥s t❤❛t ✐♥ ♣r✐♥❝✐♣❧❡✱ ❛♥ ❡♥tr② ❝❛♥ t❛❦❡ ❛♥② ✈❛❧✉❡ ✐♥ s♦♠❡ ❣✐✈❡♥ s❡t ✇✐t❤ ❝❡rt❛✐♥ ❝❤❛♥❝❡✱

❛♥❞ t❤❛t ✐t ✐s ❛r❡ ♥♦t ✐♥ ❛❞✈❛♥❝❡ ✇❤✐❝❤ ✈❛❧✉❡ ✐t ✇✐❧❧ t❛❦❡✳

• ✏■❞❡♥t✐❝❛❧❧② ❞✐str✐❜✉t❡❞✑ ♠❡❛♥s t❤❛t ❡❛❝❤ ❡♥tr② ❤❛s t❤❡ s❛♠❡ ❝❤❛♥❝❡s t♦ ❛ss✉♠❡ ❛ ♣♦ss✐❜❧❡ ✈❛❧✉❡✳

❲❤✐t❡ ♥♦✐s❡ ✐s ❛♥ ❡①❛♠♣❧❡ ♦❢ ❛♥ st❛t✐♦♥❛r②✱ ❡r❣♦❞✐❝ s❡r✐❡s✳ ❙t❛t✐♦♥❛r✐t② ♠❡❛♥s t❤❛t t❤❡ st❛t✐st✐❝s ❞♦♥✬t

❝❤❛♥❣❡ ♦✈❡r t✐♠❡✳ ❊r❣♦❞✐❝✐t② ♠❡❛♥s t❤❛t st❛t✐st✐❝❛❧ ♠❡❛s✉r❡s s✉❝❤ ❛s ♠❡❛♥ ❛♥❞ ✈❛r✐❛♥❝❡ ❝❛♥ ❜❡ ❡st✐♠❛t❡❞

❜② ♦❜s❡r✈✐♥❣ ❡♥♦✉❣❤ s❛♠♣❧❡s ♦❢ ♦♥❡ s✐♥❣❧❡ s❡q✉❡♥❝❡ ( 1, 2, 3, ...)❀ t❤❡ ♠❡❛♥ ♦r ❡①♣❡❝t❛t✐♦♥✷ ♦❢ ❛♥ ❡♥tr②

IE [ 1] ❝❛♥ ❜❡ ❡st✐♠❛t❡❞ ❛s t❤❡ s❛♠♣❧❡ ♠❡❛♥ (1/N ) ( 1 + ... + N )✳

❆♥ ❡①❛♠♣❧❡ ♦❢ ❛ st❛t✐♦♥❛r②✱ ♥♦♥ ❡r❣♦❞✐❝ s❡r✐❡s ✐s t❤❡ ♦♥❡ ✇❤❡r❡ 1 ❡q✉❛❧s ✶ ♦r −1 ✇✐t❤ ❡q✉❛❧ ♣r♦❜❛❜✐❧✐t②✱

❛♥❞ ❛❧❧ ♦t❤❡r k ❛r❡ ❡q✉❛❧ t♦ 1✳ ❈❧❡❛r❧②✱ t❤❡ s❛♠♣❧❡ ♠❡❛♥ ♦❢ ♦♥❡ s✐♥❣❧❡ s❡q✉❡♥❝❡ ✐s t❤❡♥ ❡✐t❤❡r ✶ ♦r −1✱ ❜✉t

♥♦t ✵ ❛s ✐t s❤♦✉❧❞✳

❆♥ ❡①❛♠♣❧❡ ♦❢ ✇❤✐t❡ ♥♦✐s❡ ✐s t❤❡ ❡rr♦r k ✐♥tr♦❞✉❝❡❞ ❜② q✉❛♥t✐③❛t✐♦♥✱ ✐✳❡✳✱ k = yk − xk ✇❤❡r❡ xk ✐s ❛

✏t②♣✐❝❛❧✑ s✐❣♥❛❧ ❜❡❢♦r❡ ❛♥❞ yk t❤❡ s✐❣♥❛❧ ❛❢t❡r q✉❛♥t✐③❛t✐♦♥✳ ❈❤❡❝❦ ♦❢ ❘❛♥❞♦♠♥❡ss ❛♥❞ ■♥❞❡♣❡♥❞❡♥❝❡✿ ❛s ✇❡

❛r❡ ♦❜s❡r✈✐♥❣ t❤❡ ✜rst t s❛♠♣❧❡s ( 1, ..., t) ✇❡ ❤❛✈❡ ♥♦ ✐♥❞✐❝❛t✐♦♥ ✇❤❛ts♦❡✈❡r ♦♥ t❤❡ q✉❛♥t✐③❛t✐♦♥ ❡rr♦r ♦❢ t❤❡

t + 1st s❛♠♣❧❡ ✖ ✉♥❧❡ss t❤❡ s✐❣♥❛❧ ✐s ✈❡r② s♣❡❝✐❛❧✳ ❬❆♥ ❡①❛♠♣❧❡ ♦❢ ❛♥ ❛t②♣✐❝❛❧ s✐❣♥❛❧ ✇♦✉❧❞ ❜❡ ♦♥❡ t❤❛t ✐s

❛❧r❡❛❞② q✉❛♥t✐③❡❞✿ ❛❢t❡r ♦❜s❡r✈✐♥❣ ✺✵✵ t✐♠❡s ❛♥ ❡rr♦r ✵ ✇❡ st❛rt t♦ s✉s♣❡❝t t❤❛t t❤❡ ❢✉t✉r❡ ❡rr♦rs ✇✐❧❧ ❛❧s♦ ❜❡

✵✳❪ ❈❤❡❝❦ ♦❢ t❤❡ ❞✐str✐❜✉t✐♦♥✿ ❛ q✉❛♥t✐③❛t✐♦♥ ❞♦♥❡ ❜② r♦✉♥❞✐♥❣ t♦ t❤❡ t❤✐r❞ ❞❡❝✐♠❛❧✱ ❡✳❣✳✱ ✇✐❧❧ r❡s✉❧t ✐♥ ❡rr♦rs t❤❛t ❧✐❡ ❜❡t✇❡❡♥ [−0.00049999..., 0.0005] ✇❤❡r❡ ❛❧❧ ✈❛❧✉❡s ✐♥ t❤✐s ✐♥t❡r✈❛❧ ❛r❡ ❡q✉❛❧❧② ❧✐❦❡❧②✳ ❚❤✐s ♠❡❛♥s✱ ❡✳❣✳✱

t❤❛t k ✐s ♥❡❣❛t✐✈❡ ✇✐t❤ ❝❤❛♥❝❡ ✶✴✷ ❛♥❞ t❤❛t✱ ❡✳❣✳✱ k ✐s ✇✐t❤✐♥ [0.0002, 0.0003] ✇✐t❤ ❝❤❛♥❝❡ 1/10✳ ❙✐♥❝❡ t❤✐s

✐s t❤❡ s❛♠❡ ♦❢ ❛❧❧ ❡♥tr✐❡s k✱ t❤❡② ❛r❡ ✐❞❡♥t✐❝❛❧❧② ❞✐str✐❜✉t❡❞✳

❙♣❡❝tr❛❧ ❛♥❛❧②s✐s ♦❢ st❛t✐♦♥❛r② s✐❣♥❛❧s ❛♥❞ s❡r✐❡s

❇② t❤❡✐r ♦✇♥ ♥❛t✉r❡✱ s✐♠✐❧❛r❧② t♦ ♣❡r✐♦❞✐❝ s✐❣♥❛❧s✱ st❛t✐♦♥❛r② s✐❣♥❛❧s ❛♥❞ s❡r✐❡s ❤❛✈❡ ♥♦ ✜♥✐t❡ ❡♥❡r❣②✳ ❆s

❛ s✐♠♣❧❡ ❡①❛♠♣❧❡ ❝♦♥s✐❞❡r t❤❡ s❡q✉❡♥❝❡ k = ±1 ✇✐t❤ r❛♥❞♦♠ s✐❣♥✳ ❙✐♥❝❡ 2 = 1 ❢♦r ❛❧❧ k✱ t❤❡ ❡♥❡r❣② ♦❢ t❤❡

k

s❡q✉❡♥❝❡ ✐s ❝❧❡❛r❧② ✐♥✜♥✐t❡✳ ■♥ ♦r❞❡r t♦ ❛rr✐✈❡ ❛t ❛ ♠❡❛♥✐♥❣❢✉❧ s♣❡❝tr❛❧ ❛♥❛❧②s✐s ♦♥❡ ❞❡✜♥❡s t❤❡ ♣♦✇❡r ♦❢ ❛

✶❚❤✐s ❝♦♥t❡♥t ✐s ❛✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡ ❛t ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✹✻✽✷✷✴✶✳✹✴❃✳

✷❘❡❝❛❧❧ t❤❛t IE [X] ❞❡♥♦t❡s t❤❡ ❡①♣❡❝t❛t✐♦♥ ♦❢ t❤❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ X✳

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

✹✸

✹✹

❈❍❆P❚❊❘ ✶✶✳ ▼❖❉❊▲❙ ❖❋ ◆❖■❙❊

st❛t✐♦♥❛r② s✐❣♥❛❧ x (t) ❛s t❤❡ t✐♠❡ ❛✈❡r❛❣❡ ♦❢ t❤❡ ❡♥❡r❣②✿

1

L/2

P := lim

|x (t) |2dt

✭✶✶✳✶✮

L→∞ L

−L/2

◆♦t❡ t❤❛t ❢♦r ❛ ♣❡r✐♦❞✐❝ s✐❣♥❛❧ ❞❡✜♥✐t✐♦♥ ✭✶✶✳✶✮ ❣✐✈❡s t❤❡ s❛♠❡ ✈❛❧✉❡ ❛s ✭✹✳✸✮✳

❲❤✐❧❡ ♣❡r✐♦❞✐❝ s✐❣♥❛❧s ♣♦ss❡ss ❛ ♥❛t✉r❛❧ ❋♦✉r✐❡r ❡①♣❛♥s✐♦♥ ✐♥t♦ ❛ s❡r✐❡s✱ ✇❡ ♥❡❡❞ t♦ t❛❦❡ t✐♠❡✲❛✈❡r❛❣❡❞✱

✇✐♥❞♦✇❡❞ ❋♦✉r✐❡r tr❛♥s❢♦r♠s ❢♦r st❛t✐♦♥❛r② s✐❣♥❛❧s❀ ❛❧s♦✱ ❞✉❡ t♦ r❛♥❞♦♠♥❡ss✱ ♦♥❡ ♥❡❡❞s t♦ t❛❦❡ ❛✈❡r❛❣❡s ♦✈❡r

❞✐✛❡r❡♥t r❡❛❧✐③❛t✐♦♥s ✐♥ ♦r❞❡r t♦ ♦❜t❛✐♥ ❛ ❞❡t❡r♠✐♥✐st✐❝ ♥♦♥✲r❛♥❞♦♠ s♣❡❝tr❛❧ ❞❡s❝r✐♣t♦r✳ ▼♦st ✉s❡❢✉❧ ✐s t❤❡

♣♦✇❡r s♣❡❝tr✉♠S (f) ✇❤✐❝❤ ❝♦rr❡s♣♦♥❞s t♦ t❤❡ sq✉❛r❡ ♦❢ t❤❡ ❛❜s♦❧✉t❡ ✈❛❧✉❡ ♦❢ t❤❡ t✐♠❡✲❛✈❡r❛❣❡❞ ✇✐♥❞♦✇❡❞

❋♦✉r✐❡r tr❛♥s❢♦r♠✿

2

2

L

2L+1

1

station.

1

S (f ) := lim

IE

k e−j2πf k

=

lim

IE

k e−j2πf k

✭✶✶✳✷✮

L→∞ 2L + 1

L→∞ 2L + 1

k=−L

k=1

❙♦♠❡t✐♠❡s ✐t ✐s ✭❡rr♦♥❡♦✉s❧②✮ ✇r✐tt❡♥ ❛s S (f) = |E (f) |2✳ ❡rr♦♥❡♦✉s❧② ❜❡❝❛✉s❡ ✐t s❤♦✉❧❞ ❜❡ ❛✈❡r❛❣❡❞✿

S (f ) = IE|E (f ) |2✳ ❚❤❡ ❢❛♠♦✉s ❲✐❡♥❡r✲❑❤✐♥❝❤✐♥❡ t❤❡♦r❡♠ s❛②s✱ t❤❛t

∞

S (f ) =

r (k) e−j2πfk = F {r (k)} (f )

✭✶✶✳✸✮

−∞

✇❤❡r❡ r (k) = IE [ n · n−k] = IE [ k · 0] ✐s t❤❡ ❛✉t♦✲❝♦rr❡❧❛t✐♦♥ ♦❢ t❤❡ ♣r♦❝❡ss✳ ◆♦t❡ S (f) = S (f + 1)✳

❈♦♥♥❡❝t✐♦♥ t♦ ❉✐s❝r❡t❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ✭❉❋❚✮ ■♥ ♣r❛❝t✐❝❡✱ ❛ ✜♥✐t❡ ❧❡♥❣t❤ s✐❣♥❛❧ 0, ..., K−1 ✐s

✐♥t❡r♣r❡t❡❞ ❛s ♣❡r✐♦❞✐❝❛❧❧② r❡♣❡❛t❡❞ ✭✇✐t❤ ♣❡r✐♦❞ K✮✳ ■ts ❋♦✉r✐❡r tr❛♥s❢♦r♠ ✐s t❤❡♥ ❛ ♣❡r✐♦❞✐❝ s❡r✐❡s ❛❣❛✐♥✱

❝❛❧❧❡❞ ❉❋❚✱ ✇❤✐❝❤ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✈✐❛ ❛♥ ❛❧❣♦r✐t❤♠ ❝❛❧❧❡❞ ❋❋❚ ✐♥ ♦♥❧② Klog (K) ❝♦♠♣✉t❛t✐♦♥s ❛❝❝♦r❞✐♥❣

t♦ t❤❡ ❢♦r♠✉❧❛

K−1

❉❋❚✿

Θ

n=

k e−j2πkn/K ✭✶✶✳✹✮

k=0

❘❡❝❛❧❧ t❤❛t ▼❛t❧❛❜ st❛rts ✐♥❞✐❝❡s ❛❧✇❛②s ✇✐t❤ ✶✱ t❤✉s ✐♥t❡r♣r❡t✐♥❣ t❤❡ ✜rst ❡♥tr② ♦❢ t❤❡ ✈❡❝t♦r Θ ❛s ❝♦rr❡✲

s♣♦♥❞✐♥❣ t♦ ❢r❡q✉❡♥❝② ✵ ✭s❡❡ ♠❛t❧❛❜ ❤❡❧♣ ❢❢t✮✳

■❢ ✇❡ s❡t K = 2L + 1 ✇❡ ✜♥❞ t❤❛t Θn ❛♣♣❡❛rs ♦♥ t❤❡ r✐❣❤t s✐❞❡ ♦❢ ✭✶✶✳✷✮✳ ■❢ ✇❡ ❛ss✉♠❡ t❤❛t L ✐s ❧❛r❣❡ s♦

t❤❛t ✇❡ ❝❛♥ ♥❡❣❧❡❝t t❤❡ ❧✐♠✐t✱ ✇❡ ✜♥❞ t❤❛t S (f) ✐♥❞✐❝❛t❡s ✇❤❛t t♦ ❡①♣❡❝t ♦❢ t❤❡ sq✉❛r❡❞ ❋❋❚ ❝♦❡✣❝✐❡♥ts

✭✐✳❡✳✱ t❤❡ ❋❋❚✲♣♦✇❡r✲s♣❡❝tr✉♠✮ ♦♥ ❛✈❡r❛❣❡✿

2

Θ

n

IE n

K · S

✭✶✶✳✺✮

K

❢♦r n = 1, .., K✳ ❱✐❝❡ ✈❡rs❛✱ t❤❡ s❛♠♣❧❡s ♦❢ t❤❡ ♣♦✇❡r s♣❡❝tr✉♠ S (f) ❢♦r f = k/K ❝❛♥ ❜❡ ❡st✐♠❛t❡❞ ❢r♦♠

❛✈❡r❛❣✐♥❣ t❤❡ ❋❋❚ ❝♦❡✣❝✐❡♥ts ♦✈❡r s❡✈❡r❛❧ ♥♦✐s❡ s✐❣♥❛❧s ❡❛❝❤ ♦❢ t❤❡ ❢♦r♠ 1, ..., 2L+1✿

k

1

2

Θ

S

IE k

✭✶✶✳✻✮

K

K

❢♦r k = −L, ..., 0, ..., L ✭✐t ✐s ♠♦r❡ ♥❛t✉r❛❧ t♦ r❡♣r❡s❡♥t S ♦✈❡r ❛ s②♠♠❡tr✐❝ ✐♥t❡r✈❛❧❀ r❡❝❛❧❧ t❤❛t

Θ

Θ

Θ

Θ

−L=

L+1, ..., 0=

2L+1✮✳

❋❧❛t s♣❡❝tr✉♠ ♦❢ ③❡r♦ ♠❡❛♥ ✇❤✐t❡ ♥♦✐s❡

❚❤❡ ♣♦✇❡r s♣❡❝tr✉♠ S (f) ♦❢ ✇❤✐t❡ ♥♦✐s❡ ✇✐t❤ ③❡r♦ ♠❡❛♥ ✭IE [ek] = 0✮ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❛s ❢♦❧❧♦✇s✳ ❋✐rst

✇❡ ✜♥❞

r (0) = IE

2

= σ2

❛♥❞

r

0

k = IE [ k ] · IE [ 0] = 0

k = 0

✭✶✶✳✼✮

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

✹✺

❜② ✐♥❞❡♣❡♥❞❡♥❝❡✳ ❇② ✭✶✶✳✸✮ ✇❡ ✜♥❞

S (f ) = σ2 = P♦✇❡r

✭✶✶✳✽✮

❢♦r ❛❧❧ f✳ ◆♦t❡ t❤❛t t❤❡ ♣♦✇❡r s♣❡❝tr✉♠ ♦❢ ✇❤✐t❡ ♥♦✐s❡ ✐s ✢❛t✳ ■ts ❝♦♥st❛♥t ✈❛❧✉❡ ✐s ❡q✉❛❧ t♦ t❤❡ ✈❛r✐❛♥❝❡

♦❢ t❤❡ ③❡r♦ ♠❡❛♥ ♥♦✐s❡✳ ❚❤❡ ❝♦♥st❛♥t ✈❛❧✉❡ ♠✉st ❝❧❡❛r❧② ❜❡ ❡q✉❛❧ t♦ t❤❡ ♣♦✇❡r✳ ✭❆❧s♦✿ P = 1/K

2

n

IE

2

= σ2 ❢r♦♠ st❛t✐st✐❝s✳✮

0

❈♦♥s❡q✉❡♥t❧②✱ s✐♥❝❡ t❤❡ ♠❡❛♥ ♦❢ t❤❡ ❉❋❚✱ ✐✳❡✳

Θ

IE| n |2 ✐s ❝♦♥st❛♥t✱ t❤❡ ❛✈❡r❛❣❡ ♦❢ t❤❡ ❉❋❚ ✇❤❡♥ ❝♦♠♣✉t❡❞

♦✈❡r ♠❛♥② r❡❛❧✐③❛t✐♦♥s ♦❢ t❤❡ ♥♦✐s❡ ✐s ♥❡❛r❧② ✢❛t✸❀ ❤♦✇❡✈❡r✱ t❤❡ ❋❋❚ ♦❢ ❛♥② s✐♥❣❧❡ r❡❛❧✐③❛t✐♦♥ ✇✐❧❧ s❤♦✇ ❧❛r❣❡

♦s❝✐❧❧❛t✐♦♥s ✭s❡❡ ❋✐❣✉r❡ ✶✷✳✶ ✭◆♦✐s❡ s❛♠♣❧❡s✮ ✸r❞✮✳

◆❡✈❡rt❤❡❧❡ss✱ t❤❡ ♣♦✇❡r ❝❛♥ ❜❡ ❡st✐♠❛t❡❞ ❢r♦♠ ♦♥❡ r❡❛❧✐③❛t✐♦♥✿ ✉s✐♥❣ ✭✶✶✳✻✮✱ ✉s✐♥❣ ❡r❣♦❞✐❝✐t② t♦ ❡st✐♠❛t❡

t❤❡ ❡①♣❡❝t❡❞ ✈❛❧✉❡ IE [...] s✐♠♣❧② ❛s t❤❡ ❛✈❡r❛❣❡ ♦✈❡r ❛❧❧ s❛♠♣❧❡s✱ ❛♥❞ ✉s✐♥❣ ✭✹✳✺✮ ✇❡ ❣❡t

K−1

2

k

1

❫

1

1

❫

S

IE| k|2

·

| n| = P .

✭✶✶✳✾✮

K

K

K

K n=0

❚❤✐s ❝♦♠♣✉t❛t✐♦♥ ❛❧s♦ ❝♦♥✜r♠s t❤❛t t❤❡ s♣❡❝tr✉♠ ♦❢ ✇❤✐t❡ ♥♦✐s❡ ✐s ✢❛t ❛♥❞ t❤❛t ✐ts ❝♦♥st❛♥t ✈❛❧✉❡ ✐s ❡q✉❛❧

t♦ t❤❡ ♣♦✇❡r✳

❫

■♥ ❢❛❝t✱ ❛ ❞✐r❡❝t ❝♦♠♣✉t❛t✐♦♥ ❢r♦♠ ✭✶✶✳✹✮ s❤♦✇s t❤❛t t❤❡ ❋❋❚ ♦✉t♣✉t n ✐s ❛ ✇❤✐t❡ ♥♦✐s❡ ❛s ✇❡❧❧✱ ❤♦✇❡✈❡r

✇✐t❤ ✈❛r✐❛♥❝❡ Kσ2 ❛♥❞ s♦ ✭✶✶✳✻✮ ✭t❤❡ ✜rst ❛♣♣r♦①✐♠❛t✐♦♥ ✐♥ ✭✶✶✳✾✮✮ ❤♦❧❞s ❢♦r ✇❤✐t❡ ♥♦✐s❡ ❡①❛❝t❧②✱ ♥♦t ♦♥❧②

❛♣♣r♦①✐♠❛t✐✈❡❧②✳

■♥t✉✐t✐✈❡❧②✱ t❤❡ ❢♦r♠✉❧❛ S (f) = σ2 ♠❡❛♥s t❤❛t ❛❧❧ ❢r❡q✉❡♥❝✐❡s ❛r❡ ♣r❡s❡♥t ✐♥ ✇❤✐t❡ ♥♦✐s❡ ✇✐t❤ ❡q✉❛❧

♦✈❡r❛❧❧✲❛♠♣❧✐t✉❞❡✳ ❚❤❡ s♣❡❝tr✉♠ ❜❡✐♥❣ ✢❛t ✐s ❛ ❞✐r❡❝t ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡ ✐♥❞❡♣❡♥❞❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ♥♦✐s❡

t❡r♠s✳ ❲❡ s❤♦✉❧❞ r❡❝❛❧❧✱ t❤♦✉❣❤✱ t❤❛t str✐❝t❧② s♣❡❛❦✐♥❣ t❤❡ s♣❡❝tr✉♠ ✐s ✢❛t ♦♥❧② ❛s ❛♥ ❛✈❡r❛❣❡ ♦✈❡r ♠❛♥②

♥♦✐s❡ r❡❛❧✐③❛t✐♦♥s✳

❆♣♣❧✐❝❛t✐♦♥✿ ◗✉❛♥t✐③❛t✐♦♥ ■♥ ♠❛♥② ❛♣♣❧✐❝❛t✐♦♥✱ s✉❝❤ ❛s ✐♥❢❡r❡♥❝❡ ✐♥ ✇✐r❡❧❡ss ❝♦♠♠✉♥✐❝❛t✐♦♥ ❛♥❞ s✉❝❤

❛s ✐♥ q✉❛♥t✐③❛t✐♦♥ ✐♥ s✐❣♥❛❧ ♣r♦❝❡ss✐♥❣✱ ❛ ❝♦♠♠♦♥❧② ❛❝❝❡♣t❡❞ ♠♦❞❡❧ ♦❢ t❤❡ ❡✛❡❝t ♦❢ ❛ s♦✉r❝❡ ♦❢ ❡rr♦rs ♦♥ t❤❡

s✐❣♥❛❧ x (t) ✐s t❤❡ s♦✲❝❛❧❧❡❞ ❛❞❞✐t✐✈❡ ✇❤✐t❡ ♥♦✐s❡ ♠♦❞❡❧✿

xk → + k → yk

✭✶✶✳✶✵✮

❋♦r ❛ q✉❛♥t✐③❛t✐♦♥ ✇✐t❤ ♣r❡❝✐s✐♦♥ ∆ t❤❡ ❡rr♦r ✐s t❤❡ q✉❛♥t✐✜❝❛t✐♦♥ ❡rr♦r k = yk − xk ❛♥❞ ✐t ✐s ❞❡t❡r♠✐♥❡❞

❜② t❤❡ s✐❣♥❛❧ ✐ts❡❧❢✳ ❬❖♥❡ ❝♦✉❧❞ ❛r❣✉❡✱ t❤❛t k ✐s ♥♦t r❛♥❞♦♠ s✐♥❝❡ ✐t ✐s ❝♦♠♣❧❡t❡❧② ❞❡t❡r♠✐♥❡❞ ♦♥❝❡ xk ✐s

❦♥♦✇♥✳ ❙t✐❧❧✱ t❤❡ ♠♦❞❡❧ ✐s ✉s❡❢✉❧ s✐♥❝❡ ✇❡ ❝❛♥ ✉s✉❛❧❧② ♥♦t ♣r❡❞✐❝t k+1 ❢r♦♠ ♦❜s❡r✈✐♥❣ x1, .., xk✳❪

❋♦r q✉❛♥t✐③❛t✐♦♥ ✉s✐♥❣ r♦✉♥❞✐♥❣ ✭♠❛t❧❛❜✿ r♦✉♥❞✹✮ ✐t ❝❛♥ ❜❡ s❤♦✇♥ t❤❛t t❤❡ ♥♦✐s❡ ♣♦✇❡r ♣❡r s❛♠♣❧❡

❛♠♦✉♥ts t♦ PQuant.noise = σ2 = ∆2 ✳ ❲❤❡♥ ✉s✐♥❣ ♦♥❡ ♠♦r❡ ❜✐t ❢♦r q✉❛♥t✐③❛t✐♦♥✱ t❤❡♥ t❤❡ ❡rr♦r ∆ ✐s ❤❛❧❢ ❛s 12

❧❛r❣❡✱ t❤✉s t❤❡ ♣♦✇❡r ✹ t✐♠❡s s♠❛❧❧❡r✱ ✇❤✐❝❤ ❛♠♦✉♥ts t♦ r♦✉❣❤❧② ✲✻ ❞❇✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ♣♦✇❡r ♦❢ t❤❡

q✉❛♥t✐③❛t✐♦♥ ♥♦✐s❡ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ ✲✻❞❇ t✐♠❡s t❤❡ ♥✉♠❜❡r ♦❢ ❜✐ts ✉s❡❞✳

❯s✐♥❣ t❤❡ ❛♥❛❧♦❣ ♦❢ P❛rs❡✈❛❧✬s ❡q✉❛t✐♦♥ ❛♥❞ r❡❝❛❧❧✐♥❣ t❤❛t S (f) ✐s t❤❡ ♣♦✇❡r s♣❡❝tr✉♠ ✭❛♥❛❧♦❣ ♦❢ t❤❡

sq✉❛r❡ ♦❢ t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠✮ ✇❡ ❤❛✈❡ ✐♥❞❡❡❞✿

∆2

1

fe/2

∆2

Se (f ) =

❘❡❝t f

PQuant.noise =

Se (f ) df =

✭✶✶✳✶✶✮

12

fe

fe −f

12

e /2

❋♦r K s❛♠♣❧❡s ♦❢ ♥♦✐s❡ t❛❦❡♥ ♦✈❡r ❛ t✐♠❡ ✐♥t❡r✈❛❧ ♦❢ ❧❡♥❣t❤ K/fe ✇❡ ❤❛✈❡✱ t❤✉s✱ ❛♣♣r♦①✐♠❛t✐✈❡❧②

K−1

K−1

1

1

2

Θ

∆2

| k|2 =

| k |

PQuant.noise =

✭✶✶✳✶✷✮

K

K2

12

k=0

k=0

✸♠❛t❧❛❜ ❞❡♠♦✿ ♣❧♦t✭♠❡❛♥✭❛❜s✭✛t✭✵✳✶✯✭r❛♥❞✭✶✵✵✵✱✶✵✵✵✮✲✵✳✺✮✱❬ ❪✱✷✮✳Θ✷✮✮✮❀❛①✐s✭❬✵ ✶✵✵✵ ✵ ✵✳✵✶❪✮

✹❚❤❡ ❡rr♦r k ✐s ✐♥ t❤✐s ❝❛s❡ ✉♥✐❢♦r♠❧② ❞✐str✐❜✉t❡❞ ♦♥ t❤❡ ✐♥t❡r✈❛❧ [−∆/2, ∆/2]✳

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

✹✻

❈❍❆P❚❊❘ ✶✶✳ ▼❖❉❊▲❙ ❖❋ ◆❖■❙❊

◆♦t❡ t❤❛t t❤❡ ❋❋❚ ✐♥❝r❡❛s❡s ♣♦✇❡r ❜② K✳ ❚❤❡ r❡❧❛t✐♦♥ ✭✶✶✳✶✷✮ ❜❡❝♦♠❡s ❡①❛❝t ✇❤❡♥ t❛❦✐♥❣ ❡①♣❡❝t❡❞ ✈❛❧✉❡s IE✳ ❚❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ✐♠♣r♦✈❡s t❤❡ ❧❛r❣❡r K ✐s✱ s✐♥❝❡ t❤❡ ❧❡❢t s✐❞❡ ✐s ❛♥ ❡st✐♠❛t♦r ♦❢ t❤❡ ✈❛r✐❛♥❝❡ σ2✱ ✇❤✐❝❤

✐s ∆2/12 ❢♦r q✉❛♥t✐③❛t✐♦♥ ✇✐t❤ ♣r❡❝✐s✐♦♥ ∆✳

❆♣♣❧✐❝❛t✐♦♥✿ ■♥t❡r❢❡r❡♥❝❡ ❆ ❢✉rt❤❡r ❡①❛♠♣❧❡ ♦❢ ❛ s✐t✉❛t✐♦♥ ✇❤❡r❡ ❛♥ ❛❞❞✐t✐✈❡ ✇❤✐t❡ ♥♦✐s❡ ♣r♦✈❡s ✉s❡❢✉❧

✐s ✇✐r❡❧❡ss tr❛♥s♠✐ss✐♦♥ ♦❢ ❛ ❜✐♥❛r② s✐❣♥❛❧ ✉♥❞❡r ✐♥t❡r❢❡r❡♥❝❡✳ ❍❡r❡✱ ❜✐ts ♠❛② ✢✐♣ ❢r♦♠ ✵ t♦ ✶ ❛♥❞ ✈✐❝❡ ✈❡rs❛

s✐♥❝❡ ❞❡t❡❝t✐♦♥ ✐s ♥♦t ♣❡r❢❡❝t✳ ❚❤❡ ❡rr♦r k = yk − xk ✐s ❤❡r❡ ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ✐♥t❡r❢❡r✐♥❣ s✐❣♥❛❧✱ ❛♥❞ t❤❡

❝♦♥✜❣✉r❛t✐♦♥ ♦❢ t❤❡ ❞❡❝♦❞❡r✳ ◆♦t❡ t❤❛t t❤❡ ♣♦ss✐❜❧❡ ✈❛❧✉❡s ♦❢ ❡❛❝❤ k ✐s ❡✐t❤❡r ✵ ✭♥♦ ✢✐♣✮ ♦r ✶ ✭✢✐♣✮✳ ❲✐t❤♦✉t s♣❡❝✐✜❝ ✐♥❢♦r♠❛t✐♦♥ ♦♥ t❤❡ ✐♥t❡r❢❡r❡♥❝❡✱ t❤❡ ❝❤❛♥❝❡ ♦❢ ❛ ✢✐♣ ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ t✐♠❡ k✱ ❛♥❞ ✐♥❞❡♣❡♥❞❡♥t

♦❢ t❤❡ ♣❛st ♦❝❝✉rr❡♥❝❡ ♦❢ ✢✐♣s✳ ❚❤✉s✱ ✇❤✐t❡ ♥♦✐s❡ ✐s ❛ ✈❡r② r❡❛s♦♥❛❜❧❡ ♠♦❞❡❧ ❢♦r k✳ ❈❧❡❛r❧②✱ t❤❡ ♣r♦❜❛❜✐❧✐t②

P [ k = 0] ✇✐❧❧ ❜❡ ❝❧♦s❡ t♦ ✶ ✐❢ ♦♥❧② ❧✐tt❧❡ ✐♥❢❡r❡♥❝❡ ✐s ♣r❡s❡♥t ❛♥❞ ✇✐❧❧ ❞❡❝r❡❛s❡ t❤❡ str♦♥❣❡r t❤❡ ✐♥❢❡r❡♥❝❡✳

●❛✉ss✐❛♥ ♥♦✐s❡ ❆s ❛♥ ✐♠♣♦rt❛♥t s♣❡❝✐❛❧ ❝❛s❡ ✇❡ ♠❡♥t✐♦♥ ●❛✉ss✐❛♥ ✇❤✐t❡ ♥♦✐s❡✱ ✇❤❡r❡ t❤❡ ❝♦♠♠♦♥

❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ k ✐s ●❛✉ss✐❛♥✱ ♦r ✏♥♦r♠❛❧✑✿ ❚❤✐s ♠♦❞❡❧ ❛ss✉♠♣t✐♦♥ ✐s st❛♥❞❛r❞ ✇❤❡♥❡✈❡r ♥♦t❤✐♥❣ ✐s

❦♥♦✇♥ ♦♥ t❤❡ ❞✐str✐❜✉t✐♦♥✳ ■t ♠❛❦❡s s❡♥s❡✱ ❡✳❣✳✱ ❛s ♠♦❞❡❧ ❢♦r ❛♥ ♦✈❡r❛❧❧ ❡rr♦r ✇❤✐❝❤ ✐s ❝♦♠♣♦s❡❞ ♦❢ s❡✈❡r❛❧

s♠❛❧❧ ✉♥❦♥♦✇♥ ❡rr♦rs✳ ✭❝♦♠♣❛r❡ ❈❡♥tr❛❧ ▲✐♠✐t ❚❤❡♦r❡♠✮

❈♦❧♦r❡❞ ◆♦✐s❡ ❆ s❡q✉❡♥❝❡ (n1, n2, n3, ...) ✐s ❝❛❧❧❡❞ ❝♦❧♦r❡❞ ♥♦✐s❡ ✐❢ ✐ts t❡r♠s nk ❛r❡ r❛♥❞♦♠ ❛♥❞ ♣♦ss❡ss

❛ r❡❧❛t✐♦♥ ♦r ❞❡♣❡♥❞❡♥❝❡ ❜❡t✇❡❡♥ t❤❡♠✳ ❈♦♥s❡q✉❡♥t❧②✱ t❤❡ ♣♦✇❡r s♣❡❝tr✉♠ ♦❢ ❝♦❧♦r❡❞ ♥♦✐s❡ ✐s ♥♦t ✢❛t✱ ❜✉t

♣♦ss❡ss❡s ❝❡rt❛✐♥ ♣r❡✈❛❧❡♥t ❢r❡q✉❡♥❝✐❡s ✖ ❤❡♥❝❡ t❤❡ ♥❛♠❡ ✏❝♦❧♦r❡❞✑ ✭r❡❝❛❧❧ t❤❛t t❤❡ ❢r❡q✉❡♥❝✐❡s ♦❢ ❧✐❣❤t ✇❛✈❡s

❝♦rr❡s♣♦♥❞ t♦ ❝♦❧♦rs✮✳

❖♥❡ ♦❢ t❤❡ ♠♦st s✐♠♣❧❡ ✇❛②s t♦ ♣r♦❞✉❝❡ ❝♦❧♦r❡❞ ♥♦✐s❡ ✐s t♦ ✜❧t❡r ✇❤✐t❡ ♥♦✐s❡✳ ❋♦r ✐♥st❛♥❝❡✱ mk = k + k+1

✐s ❝♦❧♦r❡❞ s✐♥❝❡ t❤❡ ❡♥tr✐❡s mk ❛r❡ ♥♦ ❧♦♥❣❡r ✐♥❞❡♣❡♥❞❡♥t✿ m0 = 0 + 1 ❛♥❞ m1 = 1 + 2 ❝♦♥t❛✐♥ t❤❡ s❛♠❡

♥✉♠❜❡r 1 ❛s ❛♥ ❛❞❞✐t✐✈❡ t❡r♠✳ ❙✐♠✐❧❛r❧②✱ nk = k − k+1 ✐s ❝♦❧♦r❡❞✳

❆❞♦♣t✐♥❣ ❛ ❝♦♥t✐♥✉♦✉s✲t✐♠❡ ♥♦t❛t✐♦♥ ✭❢♦r ❝♦♥✈❡♥✐❡♥❝❡✮ ✇❡ ✇r✐t❡

m (t) = (t) + (t + 1)

n (t) = (t) − (t + 1)

✭✶✶✳✶✸✮

✇✐t❤ ❋♦✉r✐❡r tr❛♥s❢♦r♠

M (f )

=

E (f ) + E (f ) e−j2πf

=

E (f ) 1 + e−j2πf

✭✶✶✳✶✹✮

=

E (f ) ejπf + e−jπf e−jπf

=

E (f ) 2cos (πf ) e−jπf

❛♥❞ s✐♠✐❧❛r❧②

N (f ) = E (f ) − E (f ) e−j2πf = E (f ) 1 − e−j2πf

= E (f ) ejπf − e−jπf e−jπf =

✭✶✶✳✶✺✮

E (f ) 2jsin (πf ) e−jπf

❛♥❞ ♣♦✇❡r s♣❡❝tr✉♠✺✱ ✭✉s❡ t❤❛t 2cos2 (x) = 1 + cos (2x)✮ ❛♥❞ 2sin2 (x) = 1 − cos (2x)✮

|M (f ) |2 = |E (f ) |24cos2 (πf ) = |E (f ) |22 (1 + cos (2πf )) = 2σ2 (1 + cos (2πf )) ❘❡❝t (f)

✭✶✶✳✶✻✮

|N (f ) |2 = |E (f ) |24sin2 (πf ) = |E (f ) |22 (1 − cos (2πf )) = 2σ2 (1 − cos (2πf )) ❘❡❝t (f)

✭✶✶✳✶✼✮

✇❤❡r❡ σ2 ✐s t❤❡ t♦t❛❧ ♣♦✇❡r ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ♥♦✐s❡ k ❛♥❞ S (f) = |E (f) |2 = σ2✳ ◆♦t❡ t❤❛t ♥❡✐t❤❡r |M (f) |2

♥♦r |N (f) |2 ❛r❡ ✢❛t✳ ❱❡r✐✜❝❛t✐♦♥ ✈✐❛ ♠❛t❧❛❜ ✐s ❡❛s②✳

✺❚❤❡ s❛♠❡ ❢♦r♠✉❧❛ ❢♦r |M (f) |2 ❛♥❞ |N (f) |2 ❝♦✉❧❞ ❜❡ ♦❜t❛✐♥❡❞ ❜② ❝♦♠♣✉t✐♥❣ ✐t ❛s t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ♦❢ t❤❡ ❛✉t♦✲

❝♦rr❡❧❛t✐♦♥ ✭s❡❡ ✭✶✶✳✸✮✮❀ ✐♥❞❡❡❞✱ ❢♦r m (t)✿ r (0) = 2σ2✱ r (1) = r (−1) = σ2 ❛♥❞ r (k) = 0 ❢♦r ❛❧❧ ♦t❤❡r k❀ ❢♦r n (t) t❤❡ s❛♠❡

❡①❝❡♣t r (1) = r (−1) = −σ2✳

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

❈❤❛♣t❡r ✶✷

❖✈❡rs❛♠♣❧✐♥❣✶

✶✷✳✶ ❖✈❡rs❛♠♣❧✐♥❣

❚❤❡ ♣r✐♥❝✐♣❧❡ ♦❢ ♦✈❡rs❛♠♣❧✐♥❣ ✐s s✐♠♣❧❡✿ r✉♥ ❆❉❈ ❛♥❞ ❉❆❈ ✭❆♥❛❧♦❣↔❞✐❣✐t❛❧ ❝♦♥✈❡rt❡rs✮ ❛t ❛ s❛♠♣❧✐♥❣ r❛t❡

fe ✇❡❧❧ ❜❡②♦♥❞ ◆②q✉✐st✱ s❛② ❛t fe = βf0 ✇❤❡r❡ f0 > 2B ✐s ❛ s✉✣❝✐❡♥t s❛♠♣❧✐♥❣ r❛t❡✳

❲❡ ❝❛❧❧ β > 1 t❤❡ ♦✈❡rs❛♠♣❧✐♥❣ r❛t❡ ✭❖❙❘✮❀ ✐t ✐s ✉s✉❛❧❧② ❛♥ ✐♥t❡❣❡r✳ ❋♦r ❛✉❞✐♦ ❈❉✱ β = 4 ✭❞❛t❛ ✐s

s❛♠♣❧❡❞ ❛t fe =✶✼✻✬✶✵✵ s❛♠♣❧❡s✴s❡❝ ❜✉t st♦r❡❞ ♦♥ t❤❡ ❞✐s❝ ❛t f0 =✹✹✬✶✵✵ s❛♠♣❧❡s✴s❡❝✮✳

❚♦ ✉♥❞❡rst❛♥❞ t❤❡ ❜❡♥❡✜ts ♦❢ ♦✈❡rs❛♠♣❧✐♥❣✱ ♦♥❡ ❤❛s t♦ r❡❝❛❧❧ t❤❡ t②♣✐❝❛❧ s❝❤❡♠❛ ✭✶✷✳✶✮ ♦❢ ❞✐❣✐t❛❧ s✐❣♥❛❧

♣r♦❝❡ss✐♥❣ ✭❉❙P✮✳ ❋✐rst✱ ✐♥ t❤❡ ❆❉❈ ❛♥❞ ❉❆❈ ♣❛rts ♦✈❡rs❛♠♣❧✐♥❣ r❡❞✉❝❡s t❤❡ ❧♦ss❡s ♦❢ q✉❛❧✐t② ❞✉❡ t♦

✇♦r❦✐♥❣ t♦♦ ❝❧♦s❡ ❛t ◆②q✉✐st r❛t❡✿ s❛♠♣❧✐♥❣ ❛♥❞ r❡❝♦♥str✉❝t✐♦♥ ♦❝❝✉r ✇✐t❤ ❛ ❝✉t♦✛✲❢r❡q✉❡♥❝② fe ❢❛r ❢r♦♠

t❤❡ ◆②q✉✐st r❛t❡ 2B ❛❧❧♦✇✐♥❣ ❢♦r s✐♠♣❧❡ ✜❧t❡r ❞❡s✐❣♥ ♦r ❡✈❡♥ ♦♠✐ss✐♦♥ ♦❢ ✜❧t❡r✐♥❣✳ ❙❡❝♦♥❞✱ ❛s ❛♥ ❛❞❞✐t✐♦♥❛❧

❜❡♥❡✜t✱ ♦✈❡rs❛♠♣❧✐♥❣ r❡❞✉❝❡s ♥♦✐s❡ ❝r❡❛t❡❞ ❜② q✉❛♥t✐✜❝❛t✐♦♥✱ s❛♠♣❧✐♥❣ ♦r ♦t❤❡r s♦✉r❝❡s ♦❢ ❡rr♦rs✳

◆♦t❡✿ ❞♦✇♥✲ ❛♥❞ ✉♣s❛♠♣❧✐♥❣ ❜❡❢♦r❡ ❛♥❞ ❛❢t❡r ♣r♦❝❡ss✐♥❣ ✭❉❙P✿ ❡✳❣✳✿ ❞❡✲♥♦✐s✐♥❣✱ ❞❡t❡❝t✐♦♥✱ ❡♥❤❛♥❝❡♠❡♥t✱

st♦r❛❣❡✱ tr❛♥s♠✐ss✐♦♥✮ ❛❧❧♦✇s ❢♦r ❡✣❝✐❡♥t ❝♦♠♣✉t❛t✐♦♥ ❛♥❞✴♦r tr❛♥s♠✐ss✐♦♥ ❛t ❧♦✇ r❛t❡✳

band − limit → ADC →↓ M → DSP →↑ M → DAC → smooth

✭✶✷✳✶✮

◆♦✐s❡ r❡❞✉❝t✐♦♥ ✉♥❞❡r ♦✈❡rs❛♠♣❧✐♥❣

■♥ t❤❡ ❝♦♥t❡①t ♦❢ ♦✈❡rs❛♠♣❧✐♥❣✱ t❤❡ s✐❣♥❛❧ ❤❛s ❜❡❡♥ s❛♠♣❧❡❞ ✭❛♥❞ t❤❡r❡❜② q✉❛♥t✐③❡❞✮ ❛t fe = βf0✱ ✇✐t❤

f0 > 2B ❛♥❞ ❖❙❘ β > > 1✳ ❚❤✉s✱ ✇❡ ♠❛② ❞❡❝✐♠❛t❡ t❤❡ q✉❛♥t✐③❡❞ s✐❣♥❛❧ ❜② ❛ ❢❛❝t♦r β ❛♥❞ st✐❧❧ ❦❡❡♣

t❤❡ ❢✉❧❧ s✐❣♥❛❧ q✉❛❧✐t②✳ ❉❡❝✐♠❛t✐♦♥ ❝♦♥s✐sts ♦❢ ❞✐❣✐t❛❧ ❧♦✇✲♣❛ss ✜❧t❡r✐♥❣ ❛t ❝✉t♦✛ ❢r❡q✉❡♥❝② 1/ (2β) ✭✇❤✐❝❤

❝♦rr❡s♣♦♥❞s t♦ f0/2✮✱ ❢♦❧❧♦✇❡❞ ❜② ❞♦✇♥s❛♠♣❧✐♥❣✳

▲♦✇✲♣❛ss ✜❧t❡r✐♥❣ ❛♥❞ ❞♦✇♥s❛♠♣❧✐♥❣ ✇✐❧❧ ♥♦t ❝❤❛♥❣❡ t❤❡ s✐❣♥❛❧ s✐♥❝❡ f0 > 2B✱ ❛♥❞ t❤✉s ❞❡❝✐♠❛t✐♦♥

✇✐❧❧ ♥♦t ❝❤❛♥❣❡ t❤❡ s✐❣♥❛❧ ♣♦✇❡r ✭❝♣r❡✳ ❋✐❣✉r❡ ✶✷✳✶ ✭◆♦✐s❡ s❛♠♣❧❡s✮✱ ❋✐❣✉r❡ ✶✷✳✷ ✭◆♦✐s❡ s❛♠♣❧❡s sq✉❛r❡❞✮✱

❋✐❣✉r❡ ✶✷✳✸ ✭❊st✐♠❛t✐♦♥ ♦❢ ❙✭❢✮✮ ♠❛❣❡♥t❛✮✳ ❍♦✇❡✈❡r✱ ❧♦✇✲♣❛ss ✜❧t❡r✐♥❣ ✇✐❧❧ r❡❞✉❝❡ t❤❡ ♣♦✇❡r ♦❢ t❤❡ ♥♦✐s❡❀

t❤❡ ❢❛❝t♦r ♦❢ r❡❞✉❝t✐♦♥ ✐s β ✐❢ ❛♥ ✐❞❡❛❧ ✜❧t❡r ✐s ✉s❡❞✳ ❲❡ ♦✛❡r s❡✈❡r❛❧ ❛r❣✉♠❡♥ts ❢♦r t❤✐s ❢❛❝t✳

❙♣❡❝tr❛❧ ♣✐❝t✉r❡ ♦❢ ♥♦✐s❡ r❡❞✉❝t✐♦♥ ✭✐❞❡❛❧ ✜❧t❡r✮✳ ❉✉r✐♥❣ t❤❡ ✜rst st❡♣✱ ❧♦✇✲♣❛ss ✜❧t❡r✐♥❣✱ t❤❡ ❤✐❣❤

❢r❡q✉❡♥❝✐❡s ♦❢ t❤❡ ♣♦✇❡r s♣❡❝tr✉♠ ♦❢ t❤❡ ♥♦✐s❡ ❛r❡ ❝✉t ❛✇❛②✿

1

S (f ) = P · ❘❡❝t (f ) → ∩

ideal f ilter sinc → Sideal (f ) = P · ❘❡❝t (f β)

✭✶✷✳✷✮

2β

❆❢t❡r ❧♦✇✲♣❛ss ✜❧t❡r✐♥❣✱ t❤❡ ♥♦✐s❡ ✐s ❝♦rr❡❝t❧② ❜❛♥❞✲❧✐♠✐t❡❞ ✭❛t ❛ ❧♦ss ♦❢ ♣♦✇❡r✮❀ ❝♦♥s❡q✉❡♥t❧②✱ ❞♦✇♥s❛♠♣❧✐♥❣

✇✐❧❧ ♥♦t ❝❤❛♥❣❡ t❤❡ ♣♦✇❡r ❛♥② ❢✉rt❤❡r✳

✶❚❤✐s ❝♦♥t❡♥t ✐s ❛✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡ ❛t ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✹✻✽✸✹✴✶✳✺✴❃✳

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

✹✼

✭❝✮

✹✽

❈❍❆P❚❊❘ ✶✷✳ ❖❱❊❘❙❆▼P▲■◆●

❙♣❡❝tr❛❧ ♣✐❝t✉r❡ ♦❢ ♥♦✐s❡ r❡❞✉❝t✐♦♥ ✭❞✐❣✐t❛❧ ✜❧t❡r✮✳ ❲❤❡♥ ✉s✐♥❣ ❛ ❞✐❣✐t❛❧ ✜❧t❡r b = (b1, ..., bq) ✇✐t❤

❡♥❡r❣② Eb = b2 + ... + b2 t♦ ✜❧t❡r t❤❡ s❛♠♣❧❡s

1

q

n✱ t❤❡ s♣❡❝tr✉♠ ✇✐❧❧ ❝❤❛♥❣❡ t❤r♦✉❣❤ ✜❧t❡r✐♥❣ r♦✉❣❤❧② ❛s

1

S (f ) = P · ❘❡❝t (f ) → ∩

digital f ilter b → Sdigital (f )

P · Eb · β · ❘❡❝t (f β)

✭✶✷✳✸✮

2β

❍❡r❡✱ ✇❡ ✉s❡❞ t❤❛t t❤❡ ❉❋❚ ♦❢ ❛ ✇❡❧❧✲❞❡s✐❣♥❡❞ ❞✐❣✐t❛❧ ✜❧t❡r b ✐s r♦✉❣❤❧② ❛ r❡❝t❛♥❣❧❡ ♦❢ ✇✐❞t❤ 1/β✱ ♠❡❛♥✐♥❣

❫

t❤❛t ❛ ❢r❛❝t✐♦♥ 1/β ♦❢ t❤❡ s❛♠♣❧❡s bk ❛t t❤❡ ❝❡♥t❡r t❛❦❡ s♦♠❡ ❝♦♥st❛♥t ✈❛❧✉❡ λ✱ t❤❡ ♦t❤❡rs ❛r❡ ③❡r♦✳ ■t ✐s ❡❛s②

√

t♦ ✈❡r✐❢② t❤❛t λ = Ebβ ✐♥ ♦r❞❡r ❢♦r t❤❡ ❡♥❡r❣② ♦❢ b t♦ ❜❡ Eb ✭s❡❡ ❈♦♠♠❡♥t ❇♦① ✽ ✭♣✳ ✹✽✮✮✳ ❈♦♥s❡q✉❡♥t❧②✱

❫

t❤❡ ♣♦rt✐♦♥ 1/β ♦❢ t❤❡ K s❛♠♣❧❡s n ✇✐❧❧ ❜❡ ♠✉❧t✐♣❧✐❡❞ ❜② λ✱ t❤❡ ♦t❤❡rs ✇✐❧❧ ❜❡ s❡t t♦ ③❡r♦✳ ❯s✐♥❣ ✭✶✶✳✻✮ ✇❡

s❡❡ t❤❛t S (k/K) ✐s ♠✉❧t✐♣❧✐❡❞ ❜② λ2 = Ebβ ❢♦r −K/ (2β) ≤ k ≤ K/ (2β)✱ ❛♥❞ s❡t t♦ ③❡r♦ ♦t❤❡r✇✐s❡✳

❈♦♠♠❡♥t ❇♦① ✽ ▲❡t ✉s ❞❡♥♦t❡ ❜② q t❤❡ ♥✉♠❜❡r ♦❢ s❛♠♣❧❡s ♦❢ t❤❡ ❉❋❚ ♦❢ b✳ ❋♦r ❛ ✇❡❧❧ ❞❡s✐❣♥❡❞ ❞✐❣✐t❛❧

✜❧t❡r t❤❡r❡ ❛r❡ r♦✉❣❤❧② q/β s❛♠♣❧❡s ❡q✉❛❧ t♦ s♦♠❡ ❝♦♥st❛♥t λ✱ t❤❡ r❡♠❛✐♥✐♥❣ s❛♠♣❧❡s ❛r❡ ③❡r♦✳ ❙✐♥❝❡ t❤❡

❉❋❚ ✐♥❝r❡❛s❡s ❡♥❡r❣② ❜② ❢❛❝t♦r ❡q✉❛❧ t♦ t❤❡ s❛♠♣❧❡ s✐③❡ q✱ ✇❡ ❣❡t

Θ

Θ

qE

2

2

b =b

+ ...+ b = λ2q/β✭✶✷✳✹✮

1

q

√

λ =

Ebβ✳

❆❝❝♦r❞✐♥❣ t♦ ✭✼✳✶✶✮ ✇❡ ❤❛✈❡ Eb = qPb

2fc = 1/β ❢♦r ❛ ❞✐❣✐t❛❧ ✜❧t❡r t❤❛t ✐s ✇❡❧❧ ❞❡s✐❣♥❡❞✱ ♠❡❛♥✐♥❣ t❤❛t

c

1✳ ❚❤✉s✱ s✉❝❤ ❛ ✜❧t❡r ❛❝❤✐❡✈❡s ❛♣♣r♦①✐♠❛t✐✈❡❧② t❤❡ s❛♠❡ ♣♦✇❡r r❡❞✉❝t✐♦♥ ❛s t❤❡ ✐❞❡❛❧ ✜❧t❡r✳ ❚②♣✐❝❛❧❧②✱

❤♦✇❡✈❡r✱ Eb ✐s s♦♠❡✇❤❛t s♠❛❧❧❡r t❤❛♥ 1/β✳

◆♦✐s❡ s❛♠♣❧❡s

✭❞✮

❋✐❣✉r❡ ✶✷✳✶✿ ❇❧❛❝❦✿ ❲❤✐t❡ ◆♦✐s❡ ✉♥❞❡r ❉❡❝✐♠❛t✐♦♥✳

▼❛❣❡♥t❛✿ ❋♦r ❝♦♠♣❛r✐s♦♥✱ ❛ s✐❣♥❛❧ ✇✐t❤ ❜❛♥❞❧✐♠✐t ❇❂✵✳✵✷✵✽ ✐s s✉❜❥❡❝t❡❞ t♦ t❤❡ s❛♠❡ ♠❛♥✐♣✉❧❛t✐♦♥s ❛s

t❤❡ ◆♦✐s❡✳

❉❡t❛✐❧s✿ ∆ = 10✱ P = ∆2/12 = 8.3✱ M = 5✱ fe = 1✱ fd = 1/5 = 0.2✱ Eb = 0.16 ✭❧❡♥❣t❤ q = 20✮✳

◆♦t❡ t❤❛t t❤❡ ♥♦✐s❡ ♦✈❡r✇❤❡❧♠s t❤❡ s✐❣♥❛❧ ✐♥ r♦✇s ✶ ❛♥❞ ✹❀ ✐♥ r♦✇s ✷ ❛♥❞ ✸ s✐❣♥❛❧ ❛♥❞ ♥♦✐s❡ ❛r❡

❝♦♠♣❛r❛❜❧❡✳ ✭❚❤❡ ❛♠♣❧✐t✉❞❡ ♦❢ t❤❡ s✐❣♥❛❧ ✐s ❦❡♣t ✉♥♥❛t✉r❛❧❧② s♠❛❧❧ ❢♦r r❡❛s♦♥s ♦❢ ✈✐s✉❛❧✐③❛t✐♦♥✮✳

✭❛✮ ❲❤✐t❡ ◆♦✐s❡ k ✭❜✮ k ❛❢t❡r ❧♦✇♣❛ss ✜❧t❡r✐♥❣ ∩ 1 ✭❝✮

2M

k ✜❧t❡r❡❞ ❛♥❞ ❞♦✇♥s❛♠♣❧❡❞ ✭❞✮

k ♦♥❧②

❞♦✇♥s❛♠♣❧❡❞❀ s❛♠❡ ♣♦✇❡r ❛s ✐♥ ✶st r♦✇✱ ❙✭❢✮ ✢❛t✿ S (f) = P ✳

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

✭❛✮

✭❜✮

✭❝✮

✹✾

◆♦✐s❡ s❛♠♣❧❡s sq✉❛r❡❞

✭❞✮

❋✐❣✉r❡ ✶✷✳✷✿ ❇❧❛❝❦ ❲❤✐t❡ ◆♦✐s❡ ✉♥❞❡r ❉❡❝✐♠❛t✐♦♥✳ ✭❞❡t❛✐❧s s❡❡ ✜❣✉r❡ ✶✮

▼❛❣❡♥t❛✿ ❙✐❣♥❛❧ ❢♦r ❝♦♠♣❛r✐s♦♥✱

❇❧✉❡✿ t❤❡ ♠❡❛♥ ♦✈❡r ✶✵✵ r❡❛❧✐③❛t✐♦♥s ♦❢ t❤❡ ♥♦✐s❡✳

❘❡❞✿ t❤❡♦r❡t✐❝❛❧ ♣♦✇❡r❀ ❞❛s❤❡❞ ✇✐t❤ ❛♥ ✐❞❡❛❧ ✜❧t❡r ✭✶✷✳✷✮✱ s♦❧✐❞ ✇✐t❤ ❛ ❞✐❣✐t❛❧ ✜❧t❡r ✭✶✷✳✸✮✳

◆♦t❡✿ t❤❡ ❛✈❡r❛❣❡ ♦❢ t❤❡ sq✉❛r❡❞ ♥♦✐s❡ t❡r♠s ✭❜❧✉❡✮ ✐s r♦✉❣❤❧② ❝♦♥st❛♥t ❛t ❛ ❧❡✈❡❧ r♦✉❣❤❧② ❡q✉❛❧

t♦ ✐ts ♣♦✇❡r ✭r❡❞✮✳ ✭❛✮ ❲❤✐t❡ ◆♦✐s❡ k := 8.3 ✭❜✮ ❛❢t❡r ❧♦✇♣❛ss✿ Pideal = P/M = 1.66 ✭❝✮ ❧♦✇♣❛ss ❛♥❞

❞♦✇♥s❛♠♣❧❡❞✿ Pideal = P/M = 1.66 ✭❞✮ k ♦♥❧② ❞♦✇♥s❛♠♣❧❡❞❀ k := 8.3

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

✭❛✮

✭❜✮

✭❝✮

✺✵

❈❍❆P❚❊❘ ✶✷✳ ❖❱❊❘❙❆▼P▲■◆●

❊st✐♠❛t✐♦♥ ♦❢ ❙✭❢✮

✭❞✮

❋✐❣✉r❡ ✶✷✳✸✿ ❇❧❛❝❦✿ |❉❋❚|2/❑♦❢ ❲❤✐t❡ ◆♦✐s❡ ✉♥❞❡r ❉❡❝✐♠❛t✐♦♥ ✭❞❡t❛✐❧s s❡❡ ✜❣✉r❡ ✶✮✳

▼❛❣❡♥t❛✿ ❙✐❣♥❛❧ ❢♦r ❝♦♠♣❛r✐s♦♥✳

❇❧✉❡✿ t❤❡ ♠❡❛♥ ♦✈❡r ✶✵✵ r❡❛❧✐③❛t✐♦♥s ♦❢ t❤❡ ♥♦✐s❡✳

❘❡❞✿ t❤❡♦r❡t✐❝❛❧ ♣♦✇❡r ❛♥❞ ❙✭❢✮❀ ❞❛s❤❡❞ ✇✐t❤ ❛♥ ✐❞❡❛❧ ✜❧t❡r ✭✶✷✳✷✮✱ s♦❧✐❞ ✇✐t❤ ❛ ❞✐❣✐t❛❧ ✜❧t❡r ✭✶✷✳✸✮✳

◆♦t❡✿ t❤❡ s✐❣♥❛❧ ❛♥❞ ✐ts ♣♦✇❡r ❛r❡ ♥♦t ❝❤❛♥❣❡❞ ❜❡❝❛✉s❡ ♦❢ ❜❛♥❞✲❧✐♠✐t❛t✐♦♥✳ ❚❤❡ s✐❣♥❛❧ t♦ ♥♦✐s❡ r❛t✐♦

✐♠♣r♦✈❡s ❜② ❛ ❢❛❝t♦r ▼ ✇❤❡♥ ❧♦✇♣❛ss✲✜❧t❡r✐♥❣ ❛♥❞ ❞♦✇♥s❛♠♣❧✐♥❣ ✭s❡❡ ➞✮❀ ✐t ❞♦❡s ♥♦t ✐♠♣r♦✈❡ ✇❤❡♥

♦♥❧② ❞♦✇♥s❛♠♣❧✐♥❣ ✭s❡❡ ✭❞✮✮▼❛❣❡♥t❛✿ ❋♦r ❝♦♠♣❛r✐s♦♥✱ ❛ s✐❣♥❛❧ ✇✐t❤ ❜❛♥❞❧✐♠✐t ❇❂✵✳✵✷✵✽ ✐s s✉❜❥❡❝t❡❞

t♦ t❤❡ s❛♠❡ ♠❛♥✐♣✉❧❛t✐♦♥s✳

❇❧✉❡✿ t❤❡ ♠❡❛♥ ♦✈❡r ✶✵✵ r❡❛❧✐③❛t✐♦♥s ♦❢ t❤❡ ♥♦✐s❡✳

❘❡❞✿ t❤❡♦r❡t✐❝❛❧ ♣♦✇❡r ❛♥❞ ❙✭❢✮❀ ❞❛s❤❡❞ ✇✐t❤ ❛♥ ✐❞❡❛❧ ✜❧t❡r ✭✶✷✳✷✮✱ s♦❧✐❞ ✇✐t❤ ❛ ❞✐❣✐t❛❧ ✜❧t❡r ✭✶✷✳✸✮✳

❉❡t❛✐❧s✿ ∆ = 10✱ P = ∆2/12 = 8.3✱ M = 5✱ fe = 1✱ fd = 1/5 = 0.2✱ Eb = 0.16 ✭❧❡♥❣t❤ q = 20✮✳

◆♦t❡ t❤❛t t❤❡ ♥♦✐s❡ ♦✈❡r✇❤❡❧♠s t❤❡ s✐❣♥❛❧ ✐♥ r♦✇s ✶ ❛♥❞ ✹❀ ✐♥ r♦✇s ✷ ❛♥❞ ✸ s✐❣♥❛❧ ❛♥❞ ♥♦✐s❡ ❛r❡

❝♦♠♣❛r❛❜❧❡✳ ✭❚❤❡ ❛♠♣❧✐t✉❞❡ ♦❢ t❤❡ s✐❣♥❛❧ ✐s ❦❡♣t ✉♥♥❛t✉r❛❧❧② s♠❛❧❧ ❢♦r r❡❛s♦♥s ♦❢ ✈✐s✉❛❧✐③❛t✐♦♥✮✳ ✭❛✮

❊st✐♠❛t✐♦♥ ♦❢ S (f) ❜② sq✉❛r❡❞ ❉❋❚ ✭❜✮ ❛❢t❡r ❧♦✇♣❛ss✿ S (f) ❝✉t ♦✛ ❛t 1/2▼ = 0.1✱ P✐❞❡❛❧ = P/▼ = 1.66

✭❝✮ ❧♦✇♣❛ss ❛♥❞ ❞♦✇♥s❛♠♣❧❡❞✿ S (f) P✐❞❡❛❧ = P/▼ = 1.66 ✢❛t ✭❞✮ ♦♥❧② ❞♦✇♥s❛♠♣❧❡❞✿ S (f) P = 8.3 ✢❛t

❚❡♠♣♦r❛❧ ♣✐❝t✉r❡ ♦❢ ♥♦✐s❡ r❡❞✉❝t✐♦♥✳ ❖♥❡ ♠❛② ❛r❣✉❡ t❤❛t ♦♥❧② t❤❡ ❤✐❣❤ ❢r❡q✉❡♥❝✐❡s ♦❢ t❤❡ ♥♦✐s❡ ❛r❡

r❡♠♦✈❡❞✱ ✇❤✐❝❤ ❛r❡ ♥♦t ✉s❡❞ ❛♥②✇❛②✳ ■♥ ❛♥ ❛✉❞✐♦ s✐❣♥❛❧✱ ❢♦r ✐♥st❛♥❝❡✱ s❡tt✐♥❣ fe = β ·44kHz ❛♥❞ ❞❡❝✐♠❛t✐♥❣

❜② β r❡♠♦✈❡s ♦♥❧② t❤❡ ♥♦✐s❡ ♣♦rt✐♦♥ ✐♥ t❤❡ ❢r❡q✉❡♥❝② r❛♥❣❡ ✇❤✐❝❤ ❝❛♥ ♥♦t ❜❡ ❤❡❛r❞ ❛♥②✇❛②✳ ❙♦✱ ♥♦ ❜❡♥❡✜t s❤♦✉❧❞ r❡s✉❧t✳ ❚❤✐s ❛r❣✉♠❡♥t❛t✐♦♥ ❞♦❡s ♥♦t t❛❦❡ ✐♥t♦ ❛❝❝♦✉♥t t❤❛t t❤❡ ♣♦✇❡r ♦❢ ♥♦✐s❡ ✐s r❡❞✉❝❡❞ ✇❤✐❝❤ ♠✉st r❡s✉❧t ✐♥ s♠❛❧❧❡r ♥♦✐s❡ ✈❛❧✉❡s✳ ❈♦♠♣❛r✐♥❣ r♦✇s ✷ ❛♥❞ ✹ ✐♥ ❋✐❣✉r❡ ✶✷✳✶ ✭◆♦✐s❡ s❛♠♣❧❡s✮✱ ✇❡ ♥♦t❡ ✐♥ t❤❡ ❧❡❢t

❝♦❧✉♠♥ t❤❛t t❤❡ ♥♦✐s❡ ♠❛❣♥✐t✉❞❡ ❤❛s ✐♥❞❡❡❞ ❞❡❝r❡❛s❡❞ ❛❢t❡r ✜❧t❡r✐♥❣✱ ❜✉t ♥♦t ✉♥❞❡r ♣✉r❡ ❞♦✇♥s❛♠♣❧✐♥❣✳

❈♦♠♣❛r✐♥❣ t❤❡ s♣❡❝tr❛ S (f) ✐♥ ❋✐❣✉r❡ ✶✷✳✷ ✭◆♦✐s❡ s❛♠♣❧❡s sq✉❛r❡❞✮ ✇❡ ♥♦t❡ ✐♥ t❤❡ ❧♦✇ ❢r❡q✉❡♥❝✐❡s ❢r♦♠

✲✵✳✶ t♦ ✵✳✶ t❤❡ s❛♠❡ ❛✈❡r❛❣❡ ❧❡✈❡❧ ♦❢ t❤❡ s♣❡❝tr❛❧ s❛♠♣❧❡s ♦❢ r♦✉❣❤❧② P = ∆2/12 = 8.3❀ ❤♦✇❡✈❡r✱ ❢♦r t❤❡

❞♦✇♥s❛♠♣❧❡❞ ♥♦✐s❡ ❛❧❧ s♣❡❝tr❛❧ s❛♠♣❧❡s ❛r❡ ❛t t❤✐s ❛✈❡r❛❣❡ ❧❡✈❡❧✱ ✇❤✐❧❡ t❤❡ ✜❧t❡r❡❞ ♥♦✐s❡ ♣♦ss❡ss❡s β = M

♠♦r❡ s❛♠♣❧❡s ❛♥❞ ♦♥❧② ❛ ♣♦rt✐♦♥ 1/M ♦❢ t❤❡♠ ❛r❡ t❤❛t ❧❡✈❡❧✱ t❤❡ ♦t❤❡rs ❛r❡ ❛t ③❡r♦✳ ❚❤✉s✱ t❤❡ ♣♦✇❡r ♦❢ t❤❡

✜❧t❡r ♥♦✐s❡ ✐s M t✐♠❡s s♠❛❧❧❡r t❤❛♥ t❤❡ ♣♦✇❡r ♦❢ t❤❡ ❞♦✇♥s❛♠♣❧❡❞ ♥♦✐s❡✳ ✭❚❤✐s ❞❡s♣✐t❡ t❤❡ ❢❛❝t t❤❛t ♦♥❧②

❤✐❣❤✲❢r❡q✉❡♥❝② ♥♦✐s❡ ❤❛s ❜❡❡♥ r❡♠♦✈❡❞✳✮

❋✐♥❛❧❧②✱ ✇❡ ♠❛② ❛❧s♦ ❛r❣✉❡ ❞✐r❡❝t❧② ✐♥ t❤❡ t✐♠❡ ❞♦♠❛✐♥ ✉s✐♥❣ st❛t✐st✐❝s✳ ❆❢t❡r ✜❧t❡r✐♥❣✱ t❤❡ ♥♦✐s❡ t❡r♠s

❛r❡ nk = b1 k+1 + b2 k+2 + ...bq k+q✳ ■♥t✉✐t✐✈❡❧②✱ t❤✐s ✐s s♦♠❡ s♦rt ♦❢ ❛✈❡r❛❣❡ ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ♥♦✐s❡✱ s✐♥❝❡

b1 + .. + bq = 1✳ ❇✉t ❛♥ ❛✈❡r❛❣❡ ✐s ❛❧✇❛②s ❝❧♦s❡r t♦ t❤❡ t❤❡♦r❡t✐❝❛❧ ♠❡❛♥ t❤❛♥ t❤❡ s❛♠♣❧❡s t❤❡♠s❡❧✈❡s✱ ❛♥❞ t❤❡

♠❡❛♥ ✐s ❤❡r❡ ③❡r♦✳ ❙♦✱ ✐♥t✉✐t✐✈❡❧②✱ t❤❡ ✜❧t❡r❡❞ ♥♦✐s❡ nk t❛❦❡s s♠❛❧❧❡r ✈❛❧✉❡s✳ ❙✐♥❝❡ t❤❡ ♠❡❛♥ ♦❢ t❤❡ ✜❧t❡r❡❞

♥♦✐s❡ t❡r♠s nk ✐s ③❡r♦✱ ❛ ♠❡❛♥✐♥❣❢✉❧ ♠❡❛s✉r❡ ♦❢ t❤❡✐r s✐③❡ ✐s t❤❡✐r ♣♦✇❡r✱ ✐✳❡✳✱ t❤❡✐r ✈❛r✐❛♥❝❡ ✭❝♣r❡✳ ✭✶✶✳✶✷✮✮✳

❯s✐♥❣ s✐♠♣❧❡ r✉❧❡s ❢r♦♠ st❛t✐st✐❝s✷ ✇❡ ❣❡t IE [nk] = b2 + ... + b2 IE [ ] = E

1

q

b · P ✳ ❙❤♦rt✿ t❤❡ ♥♦✐s❡ ❧❡✈❡❧

✭✐ts ♣♦✇❡r✮ r❡❞✉❝❡s ❜② ❛ ❢❛❝t♦r Eb✱ ❥✉st ❛s ✇❡ ❤❛✈❡ s❡❡♥ ❜❡❢♦r❡✳

✷❚❤❡ ✈❛r✐❛♥❝❡ ♦❢ ❛ s✉♠ ♦❢ ✐♥❞❡♣❡♥❞❡♥t r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ✐s t❤❡ s✉♠ ♦❢ t❤❡ ✈❛r✐❛♥❝❡s❀ ✇❤❡♥ ♠✉❧t✐♣❧②✐♥❣ ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡

❜② ❛ ❝♦♥st❛♥t b t❤❡ ✈❛r✐❛♥❝❡ ✐♥ ♠✉❧t✐♣❧✐❡❞ ❜② b2✳

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

✭❛✮

✭❜✮

✺✶

■❧❧✉str❛t✐♦♥ ♦❢ ♥♦✐s❡ r❡❞✉❝t✐♦♥✳

❋♦r ❛ ✐❧❧✉str❛t✐♦♥ s❡❡ ❋✐❣✉r❡ ✶✷✳✶ ✭◆♦✐s❡ s❛♠♣❧❡s✮✱ ❋✐❣✉r❡ ✶✷✳✷ ✭◆♦✐s❡ s❛♠♣❧❡s sq✉❛r❡❞✮✱ ❋✐❣✉r❡ ✶✷✳✸

✭❊st✐♠❛t✐♦♥ ♦❢ ❙✭❢✮✮✳ ❚❤❡ ♠❛✐♥ ❝♦♠♣❛r✐s♦♥ ✐s r♦✇ ✸ ✭♦✈❡rs❛♠♣❧❡❞ ❛♥❞ ❞❡❝✐♠❛t❡❞✮ ✇✐t❤ r♦✇ ✹ ✭s❛♠♣❧❡❞ ❛t

❧♦✇❡r ❢r❡q✉❡♥❝②✮✳ ❲❤✐❧❡ t❤❡ s✐❣♥❛❧ ✐s ♣r❛❝t✐❝❛❧❧② ✐❞❡♥t✐❝❛❧✱ t❤❡ ♥♦✐s❡ ✐s r❡❞✉❝❡❞ ❞r❛♠❛t✐❝❛❧❧② ✐♥ r♦✇ ✸✳

◆♦t❡ ❛❧s♦ t❤❡ ❞❡♣❡♥❞❡♥❝❡ ✐♥ t❤❡ ♥♦✐s❡ ❛❢t❡r ❧♦✇♣❛ss ✜❧t❡r✐♥❣✿ ✐t ✐s ♣♦ss✐❜❧❡ t♦ ♣r❡❞✐❝t ❛ ❢❡✇ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣

♥♦✐s❡ t❡r♠s✳ ■♥❞❡❡❞✱ t❤❡ ♥♦✐s❡ ✐s ♥♦t ✇❤✐t❡ ❛♥❞ ✐ts s♣❡❝tr✉♠ ✐s ♥♦t ✢❛t✳ ❚❤✐s ❞❡♣❡♥❞❡♥❝❡ ❞✐s❛♣♣❡❛rs ❛❧♠♦st

❡♥t✐r❡❧② ❛❢t❡r ❞♦✇♥s❛♠♣❧✐♥❣✿ t❤❡ ♣r❡❞✐❝t✐♦♥ ❞♦❡s ♥♦t r❡❛❝❤ M s❛♠♣❧❡s ✐♥t♦ t❤❡ ❢✉t✉r❡✳

◆♦t❡ t❤❡ s❤❛♣❡ ♦❢ t❤❡ ❡st✐♠❛t❡❞ ❙✭❢✮ ✐s ♥♦t ❡①❛❝t❧② ❛ r❡❝t❛♥❣❧❡ ❜❡❝❛✉s❡ t❤❡ ✜❧t❡r ✐s ♥♦t ✐❞❡❛❧ ✭❧❡♥❣t❤ ✷✺✮✳

❆ ❧♦♥❣❡r ✜❧t❡r ✇♦✉❧❞ r❡s✉❧t ✐♥ ❜❡tt❡r s❤❛♣❡s ❜✉t ❛❧s♦ ✐♥ ❧♦♥❣❡r ❞❡❧❛②✳

❈♦♥❝❧✉s✐♦♥ ■♥ s✉♠♠❛r②✱ ❧♦✇✲♣❛ss ✜❧t❡r✐♥❣ r❡❞✉❝❡s ♣♦✇❡r ❜② ❛ ❢❛❝t♦r β✿ ❖♥❧② ❛ ♣♦rt✐♦♥ 1/β ♦❢ t❤❡

s♣❡❝tr✉♠ r❡♠❛✐♥s r♦✉❣❤❧② ✉♥❝❤❛♥❣❡❞ t❤❡ r❡st ✐s s❡t t♦ ③❡r♦✳ ❆ ❢♦r♠❛❧ ❝❛❧❝✉❧❛t✐♦♥ ❣♦❡s ❛s ❢♦❧❧♦✇s✱ s❡tt✐♥❣

fe = 1 ❛s ✉s✉❛❧ ❢♦r ❞✐❣✐t❛❧ ✜❧t❡r✐♥❣ ❛♥❞ P = ∆2/12 ❢♦r ❛ q✉❛♥t✐③❛t✐♦♥ ♥♦✐s❡ ✭✇❤② ✇❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ ♥♦✐s❡✱

❛♥❞ ♥♦t ✐♥ t❤❡ ♥♦✐s② s✐❣♥❛❧✱ s❡❡ ❈♦♠♠❡♥t ❇♦① ✾ ✭♣✳ ✺✶✮✮

1/2

1/(2β)

P

∆2

✭✶✷✳✺✮

over =

S

P df = 1 P = 1

−1/2

filtered (f ) df =

−1/(2β)

β

β 12

❈♦♠♠❡♥t ❇♦① ✾ ❲❡ ❞♦ ♥♦t ❝♦♠♣❛r❡ ♣♦✇❡r ♦❢ s✐❣♥❛❧ t♦ ♣♦✇❡r ♦❢ ♥♦✐s② s✐❣♥❛❧✱ ❜❡❝❛✉s❡ t❤❡② ❛r❡ ❛❧♠♦st

❡q✉❛❧❀ ❛❧s♦✱ ❛ t❤❡♦r② ♦❢ t❤❡ ♣♦✇❡r ♦❢ t❤❡ ♥♦✐s② s✐❣♥❛❧ ✐s ❤❛r❞ t♦ ❞❡✈❡❧♦♣ s✐♥❝❡ ♣♦✇❡r✭s✐❣♥❛❧✰♥♦✐s❡✮ ✐s ♥♦t

♣♦✇❡r✭s✐❣♥❛❧✮✰♣♦✇❡r✭♥♦✐s❡✮✱ ✉♥❧❡ss s✐❣♥❛❧ ❛♥❞ ♥♦✐s❡ ❛r❡ ✉♥❝♦rr❡❧❛t❡❞✱ ❡✳❣✳ ✐❢ t❤❡ s✐❣♥❛❧ ❝❤❛♥❣❡s s❧♦✇❧② ❛♥❞

t❤❡ ♥♦✐s❡ ❤❛s ♠❡❛♥ ✵✳ ■♥ ❝♦♥❝❧✉s✐♦♥✱ ✇❡ ❝♦♠♣❛r❡ t❤❡ ♣♦✇❡r ♦❢ t❤❡ s✐❣♥❛❧ t♦ t❤❡ ♣♦✇❡r ♦❢ t❤❡ ♥♦✐s❡✳ ❚❤❡ r❛t✐♦

✐s ❝❛❧❧❡❞ ❙◆❘ ✏❙✐❣♥❛❧ t♦ ◆♦✐s❡ ❘❛t✐♦✑✳

❙✐♥❝❡ ❞♦✇♥s❛♠♣❧✐♥❣ ♦❢ ❛ ♣r♦♣❡r❧② ❜❛♥❞✲❧✐♠✐t❡❞ s✐❣♥❛❧ ❞♦❡s ♥♦t ❝❤❛♥❣❡ ✐ts ♣♦✇❡r ✭s❡❡ ❋✐❣✉r❡ ✶✷✳✶ ✭◆♦✐s❡

s❛♠♣❧❡s✮✱ ❋✐❣✉r❡ ✶✷✳✷ ✭◆♦✐s❡ s❛♠♣❧❡s sq✉❛r❡❞✮✱ ❋✐❣✉r❡ ✶✷✳✸ ✭❊st✐♠❛t✐♦♥ ♦❢ ❙✭❢✮✮✮✱ t❤❡ ❢❛❝t♦r 1/β ✐s t❤❡

♦✈❡r❛❧❧ ❝❤❛♥❣❡ ♦❢ ♣♦✇❡r ❞✉r✐♥❣ ❞❡❝✐♠❛t✐♦♥✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❙◆❘✸ ✐♠♣r♦✈❡s ✉♥❞❡r ♦✈❡rs❛♠♣❧✐♥❣ ✇✐t❤

s✉❜s❡q✉❡♥t ❞❡❝✐♠❛t✐♦♥ ❜② β ✭✇❤✐❝❤ ✐♥❝❧✉❞❡s ❧♦✇✲♣❛ss ✜❧t❡r✐♥❣ ❛t 1/2β✦✮ ❜② t❤❡ ❢❛❝t♦r β✱ ✐✳❡✳✱ ✐t ✐♠♣r♦✈❡s ❜②

t❤❡ ♦✈❡rs❛♠♣❧✐♥❣ r❛t❡ ✭❖❙❘✮✳ ❋♦r ❛ st❛t✐st✐❝❛❧ ❛r❣✉♠❡♥t❛t✐♦♥✱ s❡❡ ❜♦① ❈♦♠♠❡♥t ❇♦① ✶✵ ✭♣✳ ✺✶✮✳

❈♦♠♠❡♥t ❇♦① ✶✵ ❙t❛t✐st✐❝❛❧ ❡①♣❧❛♥❛t✐♦♥ ♦❢ ♥♦✐s❡ r❡❞✉❝t✐♦♥✳ ❙❤♦rt✿ ✜❧t❡r✐♥❣ ✐s ❛ s♣❡❝✐❛❧ ✇❛② ♦❢ ❛✈❡r❛❣✐♥❣

t❤❡ s❛♠♣❧❡s✳ ❆✈❡r❛❣✐♥❣ r❡❞✉❝❡s ✈❛r✐❛♥❝❡✱ t❤✉s ✐t r❡❞✉❝❡s t❤❡ ♥♦✐s❡ ♣♦✇❡r✳ ■♥ ❛❞❞✐t✐♦♥✱ t❤❡ ❛✈❡r❛❣✐♥❣ ✐s ❞♦♥❡

✐♥ ❛ s♣❡❝✐❛❧ ✇❛② ❛s t♦ ❧❡❛✈❡ s♠❛❧❧ ❢r❡q✉❡♥❝✐❡s✱ ✐✳❡✳ s❧♦✇❧② ✈❛r②✐♥❣ ❝♦♠♣♦♥❡♥ts✱ ✐♥t❛❝t✳ ❚❤✉s✱ t❤❡ ❛✈❡r❛❣✐♥❣

r❡❞✉❝❡s ♦♥❧② ♥♦✐s❡✱ ✇❤✐❝❤ ✐s ❤✐❣❤ ❢r❡q✉❡♥❝②✱ ✐✳❡✳ q✉✐❝❦❧② ✈❛r②✐♥❣✱ ❛♥❞ ❧❡❛✈❡ t❤❡ s✐❣♥❛❧ ✐♥t❛❝t✳ ❙♦♠❡✇❤❛t ♠♦r❡

r✐❣♦r♦✉s❧②✿ ❋✐❧t❡r✐♥❣ ♥♦✐s❡ ♠❡❛♥s t♦ ❝♦♠♣✉t❡

M

✬ =

b

k

n k−n

✭✶✷✳✼✮

n=1

❈♦♠♣✉t✐♥❣ t❤❡ ✈❛r✐❛♥❝❡ ♦❢ t❤❡ r❡s✉❧t✐♥❣✱ ✜❧t❡r❡❞ ♥♦✐s❡✱ ✇❡ ✜♥❞✿

M

M

var

✬

=

b2 var (

b2 .

✭✶✷✳✽✮

k

n

k−n) = σ2

n

n=1

n=1

❍❡r❡✱ ✇❡ ✉s❡ t❤❛t t❤❡ ♦r✐❣✐♥❛❧ ♥♦✐s❡ ✐s ✐♥❞❡♣❡♥❞❡♥t✳ ❙❡❡ ❢♦♦t♥♦t❡ ❢♦♦t♥♦t❡ ♦♥ ♣✳ ✺✵✳ ◆♦✇✱ ❢♦r ❛ ❋■❘ ✜❧t❡r b ✐t ✐s ❡❛s② t♦ ✈❡r✐❢②✱ ❡✳❣✳ ✇✐t❤ ♠❛t❧❛❜✱ t❤❛t

M

b2

1/β✳ ❚❤✐s ❛♣♣r♦①✐♠❛t✐♦♥ ❜❡❝♦♠❡s ♠♦r❡ ❛❝❝✉r❛t❡

n=1

n

t❤❡ ❧❛r❣❡r M ✐s❀ ✐t r❡✢❡❝ts t❤❡ ❢❛❝t t❤❛t t❤❡ ♣♦✇❡r ♦❢ b ✐s ❝❧♦s❡ t♦ t❤❡ ♣♦✇❡r ♦❢ t❤❡ ✐❞❡❛❧ ❧♦✇♣❛ss ✜❧t❡r c ❛♥❞

t❤❡ ♣♦✇❡r ♦❢ c ✐s ❡①❛❝t❧② 1/β✳ ❆ s✐♠✐❧❛r ❛♣♣r♦①✐♠❛t✐♦♥ ❤♦❧❞s ❛❧s♦ ❢♦r ❛♥② ❧❡❛st sq✉❛r❡ ✜❧t❡r a✿ ❚❤✐s ✐s ❞✉❡

t♦ t❤❡ ❢❛❝t t❤❛t a ❛♣♣r♦①✐♠❛t❡s t❤❡ ✐❞❡❛❧ ❧♦✇♣❛ss ✜❧t❡r c ✐♥ t❤❡ ❧❡❛st sq✉❛r❡ s❡♥s❡✳ ▼♦r❡ ♣r❡❝✐s❡❧②✱ ❧❡t ✉s

❞❡♥♦t❡ t❤❡ ♥♦r♠ ♦❢ c✱ ✐✳❡✳ t❤❡ ✏❧❡♥❣t❤✑ ♦❢ c ❜② c ✳ ❆ ✇❡❧❧ ❦♥♦✇♥ ❢♦r♠✉❧❛ s❛②s t❤❛t c = c2 + ... + c2 ✳

1

M

✸❚❤❡ ❙✐❣♥❛❧ t♦ ◆♦✐s❡ ❘❛t✐♦ ✭❙◆❘✮ ✐s t❤❡ r❛t✐♦ ♦❢ t❤❡ ♣♦✇❡r ♦❢ s✐❣♥❛❧ ❛♥❞ ♣♦✇❡r ♦❢ ♥♦✐s❡✳ ❯s✐♥❣ P❛rs❡✈❛❧✬s ❡q✉❛t✐♦♥✱ ✐t ❝❛♥

❜❡ ❝♦♠♣✉t❡❞ ✐♥ t❤❡ t✐♠❡ ♦r ❢r❡q✉❡♥❝② ❞♦♠❛✐♥✿

R |x (t) |2dt

R |X (f ) |2df

❙◆❘ =

=

✭✶✷✳✻✮

R |n (t) |2dt

R |N (f ) |2df

✳

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

✺✷

❈❍❆P❚❊❘ ✶✷✳ ❖❱❊❘❙❆▼P▲■◆●

❚♦ ❛♣♣r♦①✐♠❛t❡ c ✐♥ t❤❡ ❧❡❛st sq✉❛r❡ s❡♥s❡ ♠❡❛♥s t❤❛t a − c ✐s ♠❛❞❡ ❛s s♠❛❧❧ ❛s ♣♦ss✐❜❧❡✳ ❇✉t ❛♥♦t❤❡r

✇❡❧❧ ❦♥♦✇♥ ❢❛❝t s❛②s t❤❛t |

a

−

c

| ≤ a − c ✳ ❚❤✐s ♠❡❛♥s✱ t❤❛t t❤❡ ♥♦r♠s ♦❢ a ❛♥❞ c✱ t❤✉s t❤❡✐r

❡♥❡r❣✐❡s✱ ❛r❡ ❛❧♠♦st ✐❞❡♥t✐❝❛❧✳ ■♥ s✉♠♠❛r②✱

var

✬

= σ2/β.

✭✶✷✳✾✮

k

■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❛✈❡r❛❣❡ ♣♦✇❡r ♣❡r s❛♠♣❧❡ ✐s r❡❞✉❝❡❞ ❜❡ ❛ ❢❛❝t♦r β✳

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

❈❤❛♣t❡r ✶✸

◆♦✐s❡✲❙❤❛♣✐♥❣✶

✶✸✳✶ ◆♦✐s❡✲s❤❛♣✐♥❣

◆♦✐s❡ s❤❛♣✐♥❣ ✐s ❛ ♣r♦❝❡❞✉r❡ ✇❤✐❝❤ ❛❧❧♦✇s t♦ ♣✉t s♦♠❡ ❝♦❧♦r ♦♥ ❛♥ ❛❞❞✐t✐✈❡ ✇❤✐t❡ ♥♦✐s❡✳ ❲❤✐❧❡ t❤✐s ❛❞❞s t♦

t❤❡ ♦✈❡r❛❧❧ ♥♦✐s❡✱ ♦♥❡ ❝❛♥ s❤❛♣❡ t❤❡ r❡s✉❧t✐♥❣ ♥♦✐s❡ s✉❝❤ t❤❛t ✐ts ♣r❡✈❛❧❡♥t ✏❝♦❧♦r✑ ❧✐❡s ✐♥ t❤❡ ❤✐❣❤ ❢r❡q✉❡♥❝✐❡s

❛♥❞ ❛t t❤❡ s❛♠❡ t✐♠❡ r❡❞✉❝❡ t❤❡ ♣r❡s❡♥❝❡ ♦❢ ♥♦✐s❡ ✐♥ t❤❡ ❧♦✇ ❢r❡q✉❡♥❝✐❡s✳

✶❚❤✐s ❝♦♥t❡♥t ✐s ❛✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡ ❛t ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✹✻✽✷✸✴✶✳✷✴❃✳

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

✺✸

✺✹

❈❍❆P❚❊❘ ✶✸✳ ◆❖■❙❊✲❙❍❆P■◆●

✭❛✮

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

✭❜✮

✭❝✮

❋✐❣✉r❡ ✶✸✳✶✿

✭❛✮ ❯s✉❛❧ q✉❛♥t✐③❛t✐♦♥ ♥♦✐s❡✳ ✭❜✮ ❲❤❡♥ ❖✈❡rs❛♠♣❧✐♥❣✱ ♦♥❧② t❤❡ ♣❛rt ✐♥ t❤❡ ❜❛♥❞

[−f0/2, f0/2] ✇✐❧❧ ❜❡ r❡❧❡✈❛♥t ❛❢t❡r ❞♦✇♥✲s❛♠♣❧✐♥❣✳ ✭❝✮ ◆♦✐s❡ ❙❤❛♣✐♥❣ ♠♦✈❡s t❤❡ ♥♦✐s❡ ❛✇❛② ❢r♦♠ t❤❡

s♠❛❧❧❡r ❢r❡q✉❡♥❝✐❡s✳

✺✺

■♥ t❤❡ ❝♦♥t❡①t ♦❢ ♦✈❡rs❛♠♣❧✐♥❣✱ ♥♦✐s❡ s❤❛♣✐♥❣ ❛❝❤✐❡✈❡s ❛♥ ✐♠♣r♦✈❡♠❡♥t ✐❢ ✐t ✐s ❞♦♥❡ s♦ t❤❛t ✐t r❡❞✉❝❡s ♥♦✐s❡

✐♥ t❤❡ ❜❛♥❞ [−f0/2, f0/2] ❛♥❞ ♣✉s❤❡s ✐t t♦ t❤❡ ✏❝♦❧♦rs✑ ✇✐t❤ ❢r❡q✉❡♥❝✐❡s ✐♥ [f0/2, fe/2] ❛♥❞ [−fe/2, −f0/2]✳

❚❤❡ ♠♦st s✐♠♣❧❡ ✈❡rs✐♦♥ ♦❢ ♥♦✐s❡ s❤❛♣✐♥❣ r❡s✉❧ts ✐♥ t✇♦ ♥♦✐s❡ t❡r♠s✱ t❤❡ ♦♥❡ ♣r♦❞✉❝❡❞ ❜② t❤❡ s②st❡♠✱

♦♥❡ ❛❞❞❡❞ ✇✐t❤ ❛ ❞❡❧❛② ♦♥ ♣✉r♣♦s❡✳ ❙❡t 0 = 0✱ t❤❡♥✿

xk

→

− k−1 → (xk − k−1) →

y

✭✶✸✳✶✮

k

♦✉t♣✉t

→ Quant. → {

k = yk − (xk −

k−1)

❢❡❡❞❜❛❝❦

❚❤✐s ❛ss✉♠❡s t❤❛t ✇❡ ❦♥♦✇ ♦r ❝❛♥ ❦♥♦✇ t❤❡ ❡rr♦r✳ ❙✉❝❤ ✐s t❤❡ ❝❛s❡ ✇✐t❤ q✉❛♥t✐③❛t✐♦♥✱ ✇❤❡r❡ ✇❡ ❝❛♥ ❝♦♠♣✉t❡

k s✐♠♣❧② ❛s t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ ♦✉t♣✉t ❛♥❞ ✐♥♣✉t ♦❢ t❤❡ q✉❛♥t✐③❛t✐♦♥ ✭s❡❡ ✭✶✸✳✶✮✮✳ ❚❤✐s ✐s ♥♦t ❢❡❛s✐❜❧❡ ✐♥

t❤❡ ❡①❛♠♣❧❡ ♦❢ ❛ ✇✐r❡❧❡ss ❝❤❛♥♥❡❧ ✇✐t❤ ✐♥❢❡r❡♥❝❡✳

❚♦ st✉❞② t❤❡ ❡✛❡❝t ♦❢ ♥♦✐s❡ s❤❛♣✐♥❣ ✇❡ ❝♦♠♣✉t❡ t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ♦❢ t❤❡ ♦✈❡r❛❧❧ ❡rr♦r nk = yk − xk✳

❲❡ ✜♥❞

nk = yk − xk = k − k−1

✭✶✸✳✷✮

♦r

n (t) = y (t) − x (t) = e (t) − e (t − τe)

✭✶✸✳✸✮

✇✐t❤ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ✭✉s❡ t❤❛t fe = 1/τe✮

N (f )

=

E (f ) − E (f ) e−j2πτef = E (f ) 1 − e−j2πf/fe

✭✶✸✳✹✮

=

E (f ) 2sin (πf /fe) e−jπf/fe

❛♥❞ ♣♦✇❡r s♣❡❝tr✉♠✱ ✭✉s❡ t❤❛t 2sin2 (x) = 1 − cos (2x)✮

2πf

|N (f ) |2 = |E (f ) |22 1 − cos

✭✶✸✳✺✮

fe

◆♦t❡ t❤❛t t❤❡ s♣❡❝tr✉♠ ✐s ♥♦ ❧♦♥❣❡r ✢❛t❀ ✐♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ♥♦✐s❡ nk ✐s ❝♦❧♦r❡❞✳ ❲❡ ♥♦t❡ t❤❛t t❤❡ ❝♦❧♦r❡❞

s♣❡❝tr✉♠ ✐s s♠❛❧❧ ❢♦r s♠❛❧❧ f✱ ❛♥❞ ✐t ✐s st✐❧❧ s♣r❡❛❞ ♦✈❡r ❛ ♣❡r✐♦❞ ♦❢ ❧❡♥❣t❤ fe✳ ❚❤✉s✱ ♥♦✐s❡ s❤❛♣✐♥❣ r❡s✉❧ts ✐♥

❛ r❡❞✉❝t✐♦♥ ♦❢ ♥♦✐s❡ ✐♥ t❤❡ s♠❛❧❧ ❢r❡q✉❡♥❝✐❡s ✭s❡❡ ❋✐❣✉r❡ ✶✸✳✶✮✳

◆♦✇ ✇❡ ❝♦♥t✐♥✉❡ ❛s ✇✐t❤ s✐♠♣❧❡ ♦✈❡rs❛♠♣❧✐♥❣✿ s✐♥❝❡ t❤❡ s✐❣♥❛❧ ❤❛s ❜❡❡♥ s❛♠♣❧❡❞ ❛t fe = βf0✱ ✇✐t❤

f0 ≥ 2B ❛♥❞ β > 1 ✇❡ ♠❛② ❧♦✇✲♣❛ss ✜❧t❡r ❛❢t❡r ♥♦✐s❡ s❤❛♣✐♥❣ ❛t ❝✉t♦✛ ❢r❡q✉❡♥❝② f0/2✳ ❚❤✉s✱ ✇❡ t❛❦❡

❛❞✈❛♥t❛❣❡ ♦❢ t❤❡ ❢❛❝t t❤❛t ♠♦st ♦❢ t❤❡ ♣♦✇❡r ♦❢ t❤❡ ✏s❤❛♣❡❞✑ ♦r ✏❝♦❧♦r❡❞✑ ♥♦✐s❡ ✐s ✐♥ t❤❡ ❵❤✐❣❤✬✲❢r❡q✉❡♥❝②

❜❛♥❞s [f0/2, fe/2] ❛♥❞ [−fe/2, −f0/2] ✇✐t❤ ✈❡r② ❧✐tt❧❡ ♥♦✐s❡ ❛t ❢r❡q✉❡♥❝✐❡s ❛r♦✉♥❞ ✵✳

❚♦ ❛ss❡ss t❤❡ ❣❛✐♥ ✇❡ ❝♦♠♣✉t❡ t❤❡ ♣♦✇❡r ♦❢ t❤❡ ♥♦✐s❡ ❛❢t❡r t❤✐s ❧♦✇✲♣❛ss ✜❧t❡r✳ ◆♦t❡ t❤❛t t❤❡ ❧♦✇✲♣❛ss

✜❧t❡r s❡ts N (f) = 0 ❢♦r fe/2 > |f| > f0/2✿ t❤✐s r❡♠♦✈❡s ♠✉❝❤ ♦❢ t❤❡ ♥♦✐s❡ ❛s ❞❡♠♦♥str❛t❡❞ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣

❝♦♠♣✉t❛t✐♦♥❀ ✐t ❛❧s♦ ❣✉❛r❛♥t❡❡s t❤❛t ♣♦✇❡r ✇♦♥✬t ❜❡ ❝❤❛♥❣❡❞ ✇❤❡♥ ❞♦✇♥s❛♠♣❧✐♥❣ ❛❢t❡r ✜❧t❡r✐♥❣ t♦ f0✳ ❯s✐♥❣

❛❣❛✐♥ t❤❛t E (f) = ∆2/12 ✇❡ ❣❡t✿

f

f

P

0 /2

0 /2

shaping

=

1

|N (f ) |2df = 1 2

|E (f ) |22 1 − cos 2πf

fe

−f0/2

fe

0

fe

✭✶✸✳✻✮

=

4∆2

f − fe sin 2πf

|f0/2 = ∆2 4

1 − 1 sin π

12f

0

e

2π

fe

12

2β

2π

β

❯s✐♥❣ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ sin (x)

x − x3/6 + x5/5! − +... ✇❡ ✜♥❞

Pshaping

1

1

π

1

1

π

π3

π2

= 4

−

sin

4

−

−

=

β−3

✭✶✸✳✼✮

Pnoise

2β

2π

β

2β

2π

β

6 · β3

3

◆♦✐s❡ r❡❞✉❝t✐♦♥ ✉♥❞❡r ♦✈❡rs❛♠♣❧✐♥❣ ✇✐t❤ ♥♦✐s❡✲s❤❛♣✐♥❣

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

✺✻

❈❍❆P❚❊❘ ✶✸✳ ◆❖■❙❊✲❙❍❆P■◆●

■♥ ❝♦♥❝❧✉s✐♦♥✱ t❤❡ ❙◆❘ ✐♠♣r♦✈❡s ✭❛s ❝♦♠♣❛r❡❞ t♦ ♥♦ ♦✈❡rs❛♠♣❧✐♥❣✮ ✉♥❞❡r ♥♦✐s❡ s❤❛♣✐♥❣ ❜② t❤❡ ✐♥✈❡rs❡

♦❢ t❤❡ ❢❛❝t♦r ✭✶✸✳✼✮✱ t❤✉s r♦✉❣❤❧② ❜② 3 β3✳ ■♥ ❞❡❝✐❜❡❧✱ t❤✐s ❝♦rr❡s♣♦♥❞s t♦ ❛♣♣r♦①✐♠❛t✐✈❡❧② 10log (3) −

π2

10

20log

(π) + 30log

(β)✱ t❤✉s✱ r♦✉❣❤❧② ✸ t✐♠❡s ♠♦r❡ t❤❛♥ ✇✐t❤ ♦✈❡rs❛♠♣❧✐♥❣ ❛❧♦♥❡✳

10

10

◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡s✿

❲✐t❤ ❛♥ ❖❙❘ ♦❢ β = 128 t❤❡ ❙◆❘ ✐♠♣r♦✈❡s

• ✉♥❞❡r ♦✈❡rs❛♠♣❧✐♥❣ ❛❧♦♥❡ ❜② 10log (128)

21❞❇

10

• ✉♥❞❡r ♦✈❡rs❛♠♣❧✐♥❣ ❝♦✉♣❧❡❞ ✇✐t❤ ❜❛s✐❝ ♥♦✐s❡✲s❤❛♣✐♥❣ ❛s ❛❜♦✈❡ ❜② ❛♣♣r♦①✐♠❛t✐✈❡❧② 10log (3) −

10

20log

(π) + 30log

(128) = 58.0445❞❇✱ ♦r r♦✉❣❤❧② ✸ t✐♠❡s ♠♦r❡✳

10

10

❲❤❡♥ ❞♦✉❜❧✐♥❣ t❤❡ ❖❙❘ t❤❡ ❙◆❘ ✐♠♣r♦✈❡s

• ✉♥❞❡r ♦✈❡rs❛♠♣❧✐♥❣ ❛❧♦♥❡ ❜② 10log (2)

3.01❞❇

10

• ✉♥❞❡r ♦✈❡rs❛♠♣❧✐♥❣ ✇✐t❤ ♥♦✐s❡✲s❤❛♣✐♥❣ ❜② ❛♣♣r♦①✐♠❛t✐✈❡❧② 30log (2) = 9.03❞❇✱ ♦r ❛♣♣r♦①✐♠❛t✐✈❡❧② ✸

10

t✐♠❡s ♠♦r❡✳

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

❆❚❚❘■❇❯❚■❖◆❙

✺✼

❆ttr✐❜✉t✐♦♥s

❈♦❧❧❡❝t✐♦♥✿ ❙❛♠♣❧✐♥❣ ❘❛t❡ ❈♦♥✈❡rs✐♦♥

❊❞✐t❡❞ ❜②✿ ❉❡♥✈❡r ●r❡❡♥❡

❯❘▲✿ ❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷✴

▲✐❝❡♥s❡✿ ❤tt♣✿✴✴❝r❡❛t✐✈❡❝♦♠♠♦♥s✳♦r❣✴❧✐❝❡♥s❡s✴❜②✴✸✳✵✴

▼♦❞✉❧❡✿ ✧❋♦✉r✐❡r ❙❡r✐❡s✿ ❘❡✈✐❡✇✧

❇②✿ ❘✉❞♦❧❢ ❘✐❡❞✐

❯❘▲✿ ❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✹✻✽✶✼✴✶✳✺✴

P❛❣❡s✿ ✶✲✸

❈♦♣②r✐❣❤t✿ ❘✉❞♦❧❢ ❘✐❡❞✐

▲✐❝❡♥s❡✿ ❤tt♣✿✴✴❝r❡❛t✐✈❡❝♦♠♠♦♥s✳♦r❣✴❧✐❝❡♥s❡s✴❜②✴✸✳✵✴

▼♦❞✉❧❡✿ ✧❉✐s❝r❡t❡ ❋♦✉r✐❡r ❚r❛♥s❢♦r♠✧

❇②✿ ❘✉❞♦❧❢ ❘✐❡❞✐

❯❘▲✿ ❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✹✻✽✵✹✴✶✳✷✴

P❛❣❡s✿ ✺✲✻

❈♦♣②r✐❣❤t✿ ❘✉❞♦❧❢ ❘✐❡❞✐

▲✐❝❡♥s❡✿ ❤tt♣✿✴✴❝r❡❛t✐✈❡❝♦♠♠♦♥s✳♦r❣✴❧✐❝❡♥s❡s✴❜②✴✸✳✵✴

▼♦❞✉❧❡✿ ✧❋♦✉r✐❡r ■♥t❡❣r❛❧✧

❇②✿ ❘✉❞♦❧❢ ❘✐❡❞✐

❯❘▲✿ ❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✹✻✽✶✾✴✶✳✸✴

P❛❣❡s✿ ✼✲✽

❈♦♣②r✐❣❤t✿ ❘✉❞♦❧❢ ❘✐❡❞✐

▲✐❝❡♥s❡✿ ❤tt♣✿✴✴❝r❡❛t✐✈❡❝♦♠♠♦♥s✳♦r❣✴❧✐❝❡♥s❡s✴❜②✴✸✳✵✴

▼♦❞✉❧❡✿ ✧❊♥❡r❣② ❛♥❞ P♦✇❡r✧

❇②✿ ❘✉❞♦❧❢ ❘✐❡❞✐

❯❘▲✿ ❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✹✻✽✶✶✴✶✳✹✴

P❛❣❡s✿ ✾✲✶✷

❈♦♣②r✐❣❤t✿ ❘✉❞♦❧❢ ❘✐❡❞✐

▲✐❝❡♥s❡✿ ❤tt♣✿✴✴❝r❡❛t✐✈❡❝♦♠♠♦♥s✳♦r❣✴❧✐❝❡♥s❡s✴❜②✴✸✳✵✴

▼♦❞✉❧❡✿ ✧❊①❛♠♣❧❡s✧

❇②✿ ❘✉❞♦❧❢ ❘✐❡❞✐

❯❘▲✿ ❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✹✻✽✶✺✴✶✳✸✴

P❛❣❡s✿ ✶✸✲✶✽

❈♦♣②r✐❣❤t✿ ❘✉❞♦❧❢ ❘✐❡❞✐

▲✐❝❡♥s❡✿ ❤tt♣✿✴✴❝r❡❛t✐✈❡❝♦♠♠♦♥s✳♦r❣✴❧✐❝❡♥s❡s✴❜②✴✸✳✵✴

▼♦❞✉❧❡✿ ✧❊st✐♠❛t✐♦♥ ♦❢ ❙♣❡❝tr✉♠ ❛♥❞ P♦✇❡r ✈✐❛ ❉❋❚✧

❇②✿ ❘✉❞♦❧❢ ❘✐❡❞✐

❯❘▲✿ ❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✹✻✽✶✻✴✶✳✸✴

P❛❣❡s✿ ✶✾✲✷✷

❈♦♣②r✐❣❤t✿ ❘✉❞♦❧❢ ❘✐❡❞✐

▲✐❝❡♥s❡✿ ❤tt♣✿✴✴❝r❡❛t✐✈❡❝♦♠♠♦♥s✳♦r❣✴❧✐❝❡♥s❡s✴❜②✴✸✳✵✴

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

✺✽

❆❚❚❘■❇❯❚■❖◆❙

▼♦❞✉❧❡✿ ✧❙❛♠♣❧✐♥❣✿ ❘❡✈✐❡✇✧

❇②✿ ❘✉❞♦❧❢ ❘✐❡❞✐

❯❘▲✿ ❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✹✻✽✸✸✴✶✳✹✴

P❛❣❡s✿ ✷✸✲✷✾

❈♦♣②r✐❣❤t✿ ❘✉❞♦❧❢ ❘✐❡❞✐

▲✐❝❡♥s❡✿ ❤tt♣✿✴✴❝r❡❛t✐✈❡❝♦♠♠♦♥s✳♦r❣✴❧✐❝❡♥s❡s✴❜②✴✸✳✵✴

▼♦❞✉❧❡✿ ✧❉❡❝✐♠❛t✐♦♥ ❛♥❞ ❉♦✇♥s❛♠♣❧✐♥❣✧

❇②✿ ❘✉❞♦❧❢ ❘✐❡❞✐

❯❘▲✿ ❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✹✻✽✵✾✴✶✳✹✴

P❛❣❡s✿ ✸✶✲✸✸

❈♦♣②r✐❣❤t✿ ❘✉❞♦❧❢ ❘✐❡❞✐

▲✐❝❡♥s❡✿ ❤tt♣✿✴✴❝r❡❛t✐✈❡❝♦♠♠♦♥s✳♦r❣✴❧✐❝❡♥s❡s✴❜②✴✸✳✵✴

▼♦❞✉❧❡✿ ✧■♥t❡r♣♦❧❛t✐♦♥ ❛♥❞ ❯♣s❛♠♣❧✐♥❣✧

❇②✿ ❘✉❞♦❧❢ ❘✐❡❞✐

❯❘▲✿ ❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✹✻✽✷✶✴✶✳✹✴

P❛❣❡s✿ ✸✺✲✹✶

❈♦♣②r✐❣❤t✿ ❘✉❞♦❧❢ ❘✐❡❞✐

▲✐❝❡♥s❡✿ ❤tt♣✿✴✴❝r❡❛t✐✈❡❝♦♠♠♦♥s✳♦r❣✴❧✐❝❡♥s❡s✴❜②✴✸✳✵✴

▼♦❞✉❧❡✿ ✧❙❛♠♣❧✐♥❣ ❘❛t❡ ❈♦♥✈❡rs✐♦♥✧

❇②✿ ❘✉❞♦❧❢ ❘✐❡❞✐

❯❘▲✿ ❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✹✻✽✷✽✴✶✳✷✴

P❛❣❡✿ ✹✶

❈♦♣②r✐❣❤t✿ ❘✉❞♦❧❢ ❘✐❡❞✐

▲✐❝❡♥s❡✿ ❤tt♣✿✴✴❝r❡❛t✐✈❡❝♦♠♠♦♥s✳♦r❣✴❧✐❝❡♥s❡s✴❜②✴✸✳✵✴

▼♦❞✉❧❡✿ ✧▼♦❞❡❧s ♦❢ ◆♦✐s❡✧

❇②✿ ❘✉❞♦❧❢ ❘✐❡❞✐

❯❘▲✿ ❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✹✻✽✷✷✴✶✳✹✴

P❛❣❡s✿ ✹✸✲✹✻

❈♦♣②r✐❣❤t✿ ❘✉❞♦❧❢ ❘✐❡❞✐

▲✐❝❡♥s❡✿ ❤tt♣✿✴✴❝r❡❛t✐✈❡❝♦♠♠♦♥s✳♦r❣✴❧✐❝❡♥s❡s✴❜②✴✸✳✵✴

▼♦❞✉❧❡✿ ✧❖✈❡rs❛♠♣❧✐♥❣✧

❇②✿ ❘✉❞♦❧❢ ❘✐❡❞✐

❯❘▲✿ ❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✹✻✽✸✹✴✶✳✺✴

P❛❣❡s✿ ✹✼✲✺✷

❈♦♣②r✐❣❤t✿ ❘✉❞♦❧❢ ❘✐❡❞✐

▲✐❝❡♥s❡✿ ❤tt♣✿✴✴❝r❡❛t✐✈❡❝♦♠♠♦♥s✳♦r❣✴❧✐❝❡♥s❡s✴❜②✴✸✳✵✴

▼♦❞✉❧❡✿ ✧◆♦✐s❡✲❙❤❛♣✐♥❣✧

❇②✿ ❘✉❞♦❧❢ ❘✐❡❞✐

❯❘▲✿ ❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✹✻✽✷✸✴✶✳✷✴

P❛❣❡s✿ ✺✸✲✺✻

❈♦♣②r✐❣❤t✿ ❘✉❞♦❧❢ ❘✐❡❞✐

▲✐❝❡♥s❡✿ ❤tt♣✿✴✴❝r❡❛t✐✈❡❝♦♠♠♦♥s✳♦r❣✴❧✐❝❡♥s❡s✴❜②✴✸✳✵✴

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✺✷✾✴✶✳✷❃

❆❜♦✉t ❈♦♥♥❡①✐♦♥s

❙✐♥❝❡ ✶✾✾✾✱ ❈♦♥♥❡①✐♦♥s ❤❛s ❜❡❡♥ ♣✐♦♥❡❡r✐♥❣ ❛ ❣❧♦❜❛❧ s②st❡♠ ✇❤❡r❡ ❛♥②♦♥❡ ❝❛♥ ❝r❡❛t❡ ❝♦✉rs❡ ♠❛t❡r✐❛❧s ❛♥❞

♠❛❦❡ t❤❡♠ ❢✉❧❧② ❛❝❝❡ss✐❜❧❡ ❛♥❞ ❡❛s✐❧② r❡✉s❛❜❧❡ ❢r❡❡ ♦❢ ❝❤❛r❣❡✳ ❲❡ ❛r❡ ❛ ❲❡❜✲❜❛s❡❞ ❛✉t❤♦r✐♥❣✱ t❡❛❝❤✐♥❣ ❛♥❞

❧❡❛r♥✐♥❣ ❡♥✈✐r♦♥♠❡♥t ♦♣❡♥ t♦ ❛♥②♦♥❡ ✐♥t❡r❡st❡❞ ✐♥ ❡❞✉❝❛t✐♦♥✱ ✐♥❝❧✉❞✐♥❣ st✉❞❡♥ts✱ t❡❛❝❤❡rs✱ ♣r♦❢❡ss♦rs ❛♥❞

❧✐❢❡❧♦♥❣ ❧❡❛r♥❡rs✳ ❲❡ ❝♦♥♥❡❝t ✐❞❡❛s ❛♥❞ ❢❛❝✐❧✐t❛t❡ ❡❞✉❝❛t✐♦♥❛❧ ❝♦♠♠✉♥✐t✐❡s✳

❈♦♥♥❡①✐♦♥s✬s ♠♦❞✉❧❛r✱ ✐♥t❡r❛❝t✐✈❡ ❝♦✉rs❡s ❛r❡ ✐♥ ✉s❡ ✇♦r❧❞✇✐❞❡ ❜② ✉♥✐✈❡rs✐t✐❡s✱ ❝♦♠♠✉♥✐t② ❝♦❧❧❡❣❡s✱ ❑✲✶✷

s❝❤♦♦❧s✱ ❞✐st❛♥❝❡ ❧❡❛r♥❡rs✱ ❛♥❞ ❧✐❢❡❧♦♥❣ ❧❡❛r♥❡rs✳ ❈♦♥♥❡①✐♦♥s ♠❛t❡r✐❛❧s ❛r❡ ✐♥ ♠❛♥② ❧❛♥❣✉❛❣❡s✱ ✐♥❝❧✉❞✐♥❣

❊♥❣❧✐s❤✱ ❙♣❛♥✐s❤✱ ❈❤✐♥❡s❡✱ ❏❛♣❛♥❡s❡✱ ■t❛❧✐❛♥✱ ❱✐❡t♥❛♠❡s❡✱ ❋r❡♥❝❤✱ P♦rt✉❣✉❡s❡✱ ❛♥❞ ❚❤❛✐✳ ❈♦♥♥❡①✐♦♥s ✐s ♣❛rt

♦❢ ❛♥ ❡①❝✐t✐♥❣ ♥❡✇ ✐♥❢♦r♠❛t✐♦♥ ❞✐str✐❜✉t✐♦♥ s②st❡♠ t❤❛t ❛❧❧♦✇s ❢♦r Pr✐♥t ♦♥ ❉❡♠❛♥❞ ❇♦♦❦s✳ ❈♦♥♥❡①✐♦♥s

❤❛s ♣❛rt♥❡r❡❞ ✇✐t❤ ✐♥♥♦✈❛t✐✈❡ ♦♥✲❞❡♠❛♥❞ ♣✉❜❧✐s❤❡r ◗❖❖P t♦ ❛❝❝❡❧❡r❛t❡ t❤❡ ❞❡❧✐✈❡r② ♦❢ ♣r✐♥t❡❞ ❝♦✉rs❡

♠❛t❡r✐❛❧s ❛♥❞ t❡①t❜♦♦❦s ✐♥t♦ ❝❧❛ssr♦♦♠s ✇♦r❧❞✇✐❞❡ ❛t ❧♦✇❡r ♣r✐❝❡s t❤❛♥ tr❛❞✐t✐♦♥❛❧ ❛❝❛❞❡♠✐❝ ♣✉❜❧✐s❤❡rs✳

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