Digical signal processing and digigal filter design

Digital Signal Processing and
Digital Filter Design (Draft)

By:
C. Sidney Burrus



Digital Signal Processing and
Digital Filter Design (Draft)

By:
C. Sidney Burrus

Online:
< >

CONNEXIONS
Rice University, Houston, Texas


❚❤✐s s❡❧❡❝t✐♦♥ ❛♥❞ ❛rr❛♥❣❡♠❡♥t ♦❢ ❝♦♥t❡♥t ❛s ❛ ❝♦❧❧❡❝t✐♦♥ ✐s ❝♦♣②r✐❣❤t❡❞ ❜②
❈✳ ❙✐❞♥❡② ❇✉rr✉s✳ ■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❡❞✿ ◆♦✈❡♠❜❡r ✶✼✱ ✷✵✶✷
P❉❋ ❣❡♥❡r❛t❡❞✿ ◆♦✈❡♠❜❡r ✶✼✱ ✷✵✶✷
❋♦r ❝♦♣②r✐❣❤t ❛♥❞ ❛ttr✐❜✉t✐♦♥ ✐♥❢♦r♠❛t✐♦♥ ❢♦r t❤❡ ♠♦❞✉❧❡s ❝♦♥t❛✐♥❡❞ ✐♥ t❤✐s
❝♦❧❧❡❝t✐♦♥✱ s❡❡ ♣✳ ✸✶✵✳


❚❛❜❧❡ ♦❢ ❈♦♥t❡♥ts

Pr❡❢❛❝❡✿ ❉✐❣✐t❛❧ ❙✐❣♥❛❧ Pr♦❝❡ss✐♥❣ ❛♥❞ ❉✐❣✐t❛❧ ❋✐❧t❡r ❉❡s✐❣♥
✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✶

✶ ❙✐❣♥❛❧s ❛♥❞ ❙✐❣♥❛❧ Pr♦❝❡ss✐♥❣ ❙②st❡♠s
✶✳✶ ❈♦♥t✐♥✉♦✉s✲❚✐♠❡ ❙✐❣♥❛❧s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺
✶✳✷ ❉✐s❝r❡t❡✲❚✐♠❡ ❙✐❣♥❛❧s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼
✶✳✸ ❉✐s❝r❡t❡✲❚✐♠❡ ❙②st❡♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼
✶✳✹ ❙❛♠♣❧✐♥❣✱ ❯♣✕❙❛♠♣❧✐♥❣✱ ❉♦✇♥✕❙❛♠♣❧✐♥❣✱
❛♥❞ ▼✉❧t✐✕❘❛t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽

✷ ❋✐♥✐t❡ ■♠♣✉❧s❡ ❘❡s♣♦♥s❡ ❉✐❣✐t❛❧ ❋✐❧t❡rs ❛♥❞ ❚❤❡✐r ❉❡s✐❣♥
✷✳✶ ❋■❘ ❉✐❣✐t❛❧ ❋✐❧t❡rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺
✷✳✷ ❋■❘ ❋✐❧t❡r ❉❡s✐❣♥ ❜② ❋r❡q✉❡♥❝② ❙❛♠♣❧✐♥❣ ♦r
✷✳✸
✷✳✹
✷✳✺
✷✳✻

■♥t❡r♣♦❧❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✷
▲❡❛st ❙q✉❛r❡❞ ❊rr♦r ❉❡s✐❣♥ ♦❢ ❋■❘ ❋✐❧t❡rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✷
❈❤❡❜②s❤❡✈ ♦r ❊q✉❛❧ ❘✐♣♣❧❡ ❊rr♦r ❆♣♣r♦①✐♠❛✲
t✐♦♥ ❋✐❧t❡rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✷
❚❛②❧♦r ❙❡r✐❡s✱ ▼❛①✐♠❛❧❧② ❋❧❛t✱ ❛♥❞ ❩❡r♦ ▼♦✲
♠❡♥t ❉❡s✐❣♥ ❈r✐t❡r✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✺
❈♦♥str❛✐♥❡❞ ❆♣♣r♦①✐♠❛t✐♦♥ ❛♥❞ ▼✐①❡❞ ❈r✐t❡✲
r✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✻

✸ ■♥✜♥✐t❡ ■♠♣✉❧s❡ ❘❡s♣♦♥s❡ ❉✐❣✐t❛❧ ❋✐❧t❡rs ❛♥❞ ❚❤❡✐r ❉❡s✐❣♥
✸✳✶ Pr♦♣❡rt✐❡s ♦❢ ■■❘ ❋✐❧t❡rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽✸
✸✳✷ ❉❡s✐❣♥ ♦❢ ■♥✜♥✐t❡ ■♠♣✉❧s❡ ❘❡s♣♦♥s❡ ✭■■❘✮ ❋✐❧✲

✸✳✸
✸✳✹
✸✳✺
✸✳✻
✸✳✼
✸✳✽
✸✳✾

t❡rs ❜② ❋r❡q✉❡♥❝② ❚r❛♥s❢♦r♠❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽✽
❇✉tt❡r✇♦rt❤ ❋✐❧t❡r Pr♦♣❡rt✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾✷
❈❤❡❜②s❤❡✈ ❋✐❧t❡r Pr♦♣❡rt✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵✵
❊❧❧✐♣t✐❝✲❋✉♥❝t✐♦♥ ❋✐❧t❡r Pr♦♣❡rt✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶✹
❖♣t✐♠❛❧✐t② ♦❢ t❤❡ ❋♦✉r ❈❧❛ss✐❝❛❧ ❋✐❧t❡r ❉❡s✐❣♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸✶
❋r❡q✉❡♥❝② ❚r❛♥s❢♦r♠❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸✷
❈♦♥✈❡rs✐♦♥ ♦❢ ❆♥❛❧♦❣ t♦ ❉✐❣✐t❛❧ ❚r❛♥s❢❡r ❋✉♥❝✲
t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸✽
❉✐r❡❝t ❋r❡q✉❡♥❝② ❉♦♠❛✐♥ ■■❘ ❋✐❧t❡r ❉❡s✐❣♥
▼❡t❤♦❞s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺✷

✹ ❉✐❣✐t❛❧ ❋✐❧t❡r ❙tr✉❝t✉r❡s ❛♥❞ ■♠♣❧❡♠❡♥t❛t✐♦♥
✹✳✶ ❇❧♦❝❦✱ ▼✉❧t✐✲r❛t❡✱ ▼✉❧t✐✲❞✐♠❡♥s✐♦♥❛❧ Pr♦❝❡ss✲
✐♥❣ ❛♥❞ ❉✐str✐❜✉t❡❞ ❆r✐t❤♠❡t✐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻✺

❇✐❜❧✐♦❣r❛♣❤② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼✺
■♥❞❡① ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵✾
❆ttr✐❜✉t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶✵


✐✈

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


Pr❡❢❛❝❡✿ ❉✐❣✐t❛❧ ❙✐❣♥❛❧
Pr♦❝❡ss✐♥❣ ❛♥❞ ❉✐❣✐t❛❧
❋✐❧t❡r ❉❡s✐❣♥
✶

❉✐❣✐t❛❧ s✐❣♥❛❧ ♣r♦❝❡ss✐♥❣ ✭❉❙P✮ ❤❛s ❡①✐st❡❞ ❛s ❧♦♥❣ ❛s q✉❛♥t✐t❛t✐✈❡ ❝❛❧❝✉✲
❧❛t✐♦♥s ❤❛✈❡ ❜❡❡♥ s②st❡♠❛t✐❝❛❧❧② ❛♣♣❧✐❡❞ t♦ ❞❛t❛ ✐♥ ❙❝✐❡♥❝❡✱ ❙♦❝✐❛❧ ❙❝✐❡♥❝❡✱
❛♥❞ ❚❡❝❤♥♦❧♦❣②✳ ❚❤❡ s❡t ♦❢ ❛❝t✐✈✐t✐❡s st❛rt❡❞ ♦✉t ❛s ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ ✐❞❡❛s
❛♥❞ t❡❝❤♥✐q✉❡s ✐♥ ✈❡r② ❞✐✛❡r❡♥t ❛♣♣❧✐❝❛t✐♦♥s✳ ❆r♦✉♥❞ ✶✾✻✺✱ ✇❤❡♥ t❤❡ ❢❛st
❋♦✉r✐❡r tr❛♥s❢♦r♠ ✭❋❋❚✮ ✇❛s r❡❞✐s❝♦✈❡r❡❞✱ ❉❙P ✇❛s ❡①tr❛❝t❡❞ ❢r♦♠ ✐ts
❛♣♣❧✐❝❛t✐♦♥s ❛♥❞ ❜❡❝❛♠❡ ❛ s✐♥❣❧❡ ❛❝❛❞❡♠✐❝ ❛♥❞ ♣r♦❢❡ss✐♦♥❛❧ ❞✐s❝✐♣❧✐♥❡ t♦
❜❡ ❞❡✈❡❧♦♣❡❞ ❛s ❢❛r ❛s ♣♦ss✐❜❧❡✳
❖♥❡ ♦❢ t❤❡ ❡❛r❧✐❡st ❜♦♦❦s ♦♥ ❉❙P ✇❛s ❜② ●♦❧❞ ❛♥❞ ❘❛❞❡r ❬✶✷✺❪✱ ✇r✐tt❡♥
✐♥ ✶✾✻✽✱ ❛❧t❤♦✉❣❤ t❤❡r❡ ❤❛❞ ❜❡❡♥ ❡❛r❧✐❡r ❜♦♦❦s ♦♥ s❛♠♣❧❡❞ ❞❛t❛ ❝♦♥tr♦❧
❛♥❞ t✐♠❡ s❡r✐❡s ❛♥❛❧②s✐s✱ ❛♥❞ ❝❤❛♣t❡rs ✐♥ ❜♦♦❦s ♦♥ ❝♦♠♣✉t❡r ❛♣♣❧✐❝❛t✐♦♥s✳
■♥ t❤❡ ❧❛t❡ ✻✵✬s ❛♥❞ ❡❛r❧② ✼✵✬s t❤❡r❡ ✇❛s ❛♥ ❡①♣❧♦s✐♦♥ ♦❢ ❛❝t✐✈✐t② ✐♥ ❜♦t❤
t❤❡ t❤❡♦r② ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ ❉❙P✳ ❆s t❤❡ ❛r❡❛ ✇❛s ❜❡❣✐♥♥✐♥❣ t♦ ♠❛t✉r❡✱
t✇♦ ✈❡r② ✐♠♣♦rt❛♥t ❜♦♦❦s ♦♥ ❉❙P ✇❡r❡ ♣✉❜❧✐s❤❡❞ ✐♥ ✶✾✼✺✱ ♦♥❡ ❜② ❖♣♣❡♥✲
❤❡✐♠ ❛♥❞ ❙❝❤❛❢❡r ❬✷✷✺❪ ❛♥❞ t❤❡ ♦t❤❡r ❜② ❘❛❜✐♥❡r ❛♥❞ ●♦❧❞ ❬✷✽✹❪✳ ❚❤❡s❡
t❤r❡❡ ❜♦♦❦s ❞♦♠✐♥❛t❡❞ t❤❡ ❡❛r❧② ❝♦✉rs❡s ✐♥ ✉♥✐✈❡rs✐t✐❡s ❛♥❞ s❡❧❢ st✉❞② ✐♥
✐♥❞✉str②✳
❚❤❡ ❡❛r❧② ❛♣♣❧✐❝❛t✐♦♥s ♦❢ ❉❙P ✇❡r❡ ✐♥ t❤❡ ❞❡❢❡♥s❡✱ ♦✐❧✱ ❛♥❞ ♠❡❞✐❝❛❧
✐♥❞✉str✐❡s✳ ❚❤❡② ✇❡r❡ t❤❡ ♦♥❡s ✇❤♦ ♥❡❡❞❡❞ ❛♥❞ ❝♦✉❧❞ ❛✛♦r❞ t❤❡ ❡①♣❡♥s✐✈❡
❜✉t ❤✐❣❤❡r q✉❛❧✐t② ♣r♦❝❡ss✐♥❣ t❤❛t ❞✐❣✐t❛❧ t❡❝❤♥✐q✉❡s ♦✛❡r❡❞ ♦✈❡r ❛♥❛❧♦❣
s✐❣♥❛❧ ♣r♦❝❡ss✐♥❣✳ ❍♦✇❡✈❡r✱ ❛s t❤❡ t❤❡♦r② ❞❡✈❡❧♦♣❡❞ ♠♦r❡ ❡✣❝✐❡♥t ❛❧❣♦✲
r✐t❤♠s✱ ❛s ❝♦♠♣✉t❡rs ❜❡❝❛♠❡ ♠♦r❡ ♣♦✇❡r❢✉❧ ❛♥❞ ❝❤❡❛♣❡r✱ ❛♥❞ ✜♥❛❧❧②✱ ❛s

❉❙P ❝❤✐♣s ❜❡❝❛♠❡ ❝♦♠♠♦❞✐t② ✐t❡♠s ✭❡✳❣✳ t❤❡ ❚❡①❛s ■♥str✉♠❡♥ts ❚▼❙✲
✸✷✵ s❡r✐❡s✮ ❉❙P ♠♦✈❡❞ ✐♥t♦ ❛ ✈❛r✐❡t② ♦❢ ❝♦♠♠❡r❝✐❛❧ ❛♣♣❧✐❝❛t✐♦♥s ❛♥❞ t❤❡
❝✉rr❡♥t ❞✐❣✐t✐③❛t✐♦♥ ♦❢ ❝♦♠♠✉♥✐❝❛t✐♦♥s ❜❡❣❛♥✳ ❚❤❡ ❛♣♣❧✐❝❛t✐♦♥s ❛r❡ ♥♦✇

✶ ❚❤✐s

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

✶


✷

❡✈❡r②✇❤❡r❡✳

❚❤❡② ❛r❡ t❡❧❡✲❝♦♠♠✉♥✐❝❛t✐♦♥s✱ s❡✐s♠✐❝ s✐❣♥❛❧ ♣r♦❝❡ss✐♥❣✱

r❛❞❛r ❛♥❞ s♦♥❛r s✐❣♥❛❧ ♣r♦❝❡ss✐♥❣✱ s♣❡❡❝❤ ❛♥❞ ♠✉s✐❝ s✐❣♥❛❧ ♣r♦❝❡ss✐♥❣✱ ✐♠✲
❛❣❡ ❛♥❞ ♣✐❝t✉r❡ ♣r♦❝❡ss✐♥❣✱ ❡♥t❡rt❛✐♥♠❡♥t s✐❣♥❛❧ ♣r♦❝❡ss✐♥❣✱ ✜♥❛♥❝✐❛❧ ❞❛t❛
s✐❣♥❛❧ ♣r♦❝❡ss✐♥❣✱ ♠❡❞✐❝❛❧ s✐❣♥❛❧ ♣r♦❝❡ss✐♥❣✱ ♥♦♥❞❡str✉❝t✐✈❡ t❡st✐♥❣✱ ❢❛❝✲
t♦r② ✢♦♦r ♠♦♥✐t♦r✐♥❣✱ s✐♠✉❧❛t✐♦♥✱ ✈✐s✉❛❧✐③❛t✐♦♥✱ ✈✐rt✉❛❧ r❡❛❧✐t②✱ r♦❜♦t✐❝s✱
❛♥❞ ❝♦♥tr♦❧✳ ❉❙P ❝❤✐♣s ❛r❡ ❢♦✉♥❞ ✐♥ ✈✐rt✉❛❧❧② ❛❧❧ ❝❡❧❧ ♣❤♦♥❡s✱ ❞✐❣✐t❛❧ ❝❛♠✲
❡r❛s✱ ❤✐❣❤✲❡♥❞ st❡r❡♦ s②st❡♠s✱ ▼P✸ ♣❧❛②❡rs✱ ❉❱❉ ♣❧❛②❡rs✱ ❝❛rs✱ t♦②s✱ t❤❡
✏❙❡❣✇❛②✧✱ ❛♥❞ ♠❛♥② ♦t❤❡r ❞✐❣✐t❛❧ s②st❡♠s✳
■♥ ❛ ♠♦❞❡r♥ ❝✉rr✐❝✉❧✉♠✱ ❉❙P ❤❛s ♠♦✈❡❞ ❢r♦♠ ❛ s♣❡❝✐❛❧✐③❡❞ ❣r❛❞✉❛t❡
❝♦✉rs❡ ❞♦✇♥ t♦ ❛ ❣❡♥❡r❛❧ ✉♥❞❡r❣r❛❞✉❛t❡ ❝♦✉rs❡✱ ❛♥❞✱ ✐♥ s♦♠❡ ❝❛s❡s✱ t♦ t❤❡
✐♥tr♦❞✉❝t♦r② ❢r❡s❤♠❛♥ ♦r s♦♣❤♦♠♦r❡ ❊❊ ❝♦✉rs❡ ❬✶✾✽❪✳ ❆♥ ❡①❝✐t✐♥❣ ♣r♦❥❡❝t
✐s ❡①♣❡r✐♠❡♥t✐♥❣ ✇✐t❤ t❡❛❝❤✐♥❣ ❉❙P ✐♥ ❤✐❣❤ s❝❤♦♦❧s ❛♥❞ ✐♥ ❝♦❧❧❡❣❡s t♦ ♥♦♥✲

t❡❝❤♥✐❝❛❧ ♠❛❥♦rs ❬✷✸✼❪✳
❖✉r r❡❛s♦♥ ❢♦r ✇r✐t✐♥❣ t❤✐s ❜♦♦❦ ❛♥❞ ❛❞❞✐♥❣ t♦ t❤❡ ❛❧r❡❛❞② ❧♦♥❣ ❧✐st
♦❢ ❉❙P ❜♦♦❦s ✐s t♦ ❝♦✈❡r t❤❡ ♥❡✇ r❡s✉❧ts ✐♥ ❞✐❣✐t❛❧ ✜❧t❡r ❞❡s✐❣♥ t❤❛t ❤❛✈❡
❜❡❝♦♠❡ ❛✈❛✐❧❛❜❧❡ ✐♥ t❤❡ ❧❛st ✶✵ t♦ ✷✵ ②❡❛rs ❛♥❞ t♦ ♠❛❦❡ t❤❡s❡ r❡s✉❧ts ❛✈❛✐❧✲
❛❜❧❡ ♦♥ ❧✐♥❡ ✐♥ ❈♦♥♥❡①✐♦♥s ❛s ✇❡❧❧ ❛s ♣r✐♥t✳ ❉✐❣✐t❛❧ ✜❧t❡rs ❛r❡ ✐♠♣♦rt❛♥t
♣❛rts ♦❢ ❛ ❧❛r❣❡ ♥✉♠❜❡r ♦❢ s②st❡♠s ❛♥❞ ♣r♦❝❡ss❡s✳ ■♥ ♠❛♥② ❝❛s❡s✱ t❤❡ ✉s❡
♦❢ ♠♦❞❡r♥ ♦♣t✐♠❛❧ ❞❡s✐❣♥ ♠❡t❤♦❞s ❛❧❧♦✇s t❤❡ ✉s❡ ♦❢ ❛ ❧❡ss ❡①♣❡♥s✐✈❡ ❉❙P
❝❤✐♣ ❢♦r ❛ ♣❛rt✐❝✉❧❛r ❛♣♣❧✐❝❛t✐♦♥ ♦r ♦❜t❛✐♥✐♥❣ ❤✐❣❤❡r ♣❡r❢♦r♠❛♥❝❡ ✇✐t❤
❡①✐st✐♥❣ ❤❛r❞✇❛r❡✳ ❚❤❡ ❜♦♦❦ s❤♦✉❧❞ ❜❡ ✉s❡❢✉❧ ✐♥ ❛♥ ✐♥tr♦❞✉❝t♦r② ❝♦✉rs❡
✐❢ t❤❡ st✉❞❡♥ts ❤❛✈❡ ❤❛❞ ❛ ❝♦✉rs❡ ♦♥ ❞✐s❝r❡t❡✲t✐♠❡ s②st❡♠s✳ ■t ❝❛♥ ❜❡ ✉s❡❞
✐♥ ❛ s❡❝♦♥❞ ❉❙P ❝♦✉rs❡ ♦♥ ✜❧t❡r ❞❡s✐❣♥ ♦r ✉s❡❞ ❢♦r s❡❧❢✲st✉❞② ♦r r❡❢❡r❡♥❝❡
✐♥ ✐♥❞✉str②✳
❲❡ ✜rst ❝♦✈❡r t❤❡ ♦♣t✐♠❛❧ ❞❡s✐❣♥ ♦❢ ❋✐♥✐t❡ ■♠♣✉❧s❡ ❘❡s♣♦♥s❡ ✭❋■❘✮
✜❧t❡rs ✉s✐♥❣ ❛ ❧❡❛st sq✉❛r❡❞ ❡rr♦r✱ ❛ ♠❛①✐♠❛❧❧② ✢❛t✱ ❛♥❞ ❛ ❈❤❡❜②s❤❡✈ ❝r✐✲
t❡r✐♦♥✳

❆ ❢❡❛t✉r❡ ♦❢ t❤❡ ❜♦♦❦ ✐s ❝♦✈❡r✐♥❣ ✜♥✐t❡ ✐♠♣✉❧s❡ r❡s♣♦♥s❡ ✭❋■❘✮

✜❧t❡r ❞❡s✐❣♥ ❜❡❢♦r❡ ✐♥✜♥✐t❡ ✐♠♣✉❧s❡ r❡s♣♦♥s❡ ✭■■❘✮ ✜❧t❡r ❞❡s✐❣♥✳ ❚❤✐s r❡✲
✢❡❝ts ♠♦❞❡r♥ ♣r❛❝t✐❝❡ ❛♥❞ ♥❡✇ ✜❧t❡r ❞❡s✐❣♥ ❛❧❣♦r✐t❤♠s✳ ❚❤❡ ❋■❘ ✜❧t❡r
❞❡s✐❣♥ ❝❤❛♣t❡r ❝♦♥t❛✐♥s ♥❡✇ ♠❡t❤♦❞s ♦♥ ❝♦♥str❛✐♥❡❞ ♦♣t✐♠✐③❛t✐♦♥✱ ♠✐①❡❞
♦♣t✐♠✐③❛t✐♦♥ ❝r✐t❡r✐❛✱ ❛♥❞ ♠♦❞✐✜❝❛t✐♦♥s t♦ t❤❡ ❜❛s✐❝ P❛r❦s✲▼❝❈❧❡❧❧❛♥ ❛❧✲
❣♦r✐t❤♠ t❤❛t ❛r❡ ✈❡r② ✉s❡❢✉❧✳ ❉❡s✐❣♥ ♣r♦❣r❛♠s ❛r❡ ❣✐✈❡♥ ✐♥ ▼❛t▲❛❜ ❛♥❞
❋❖❘❚❘❆◆✳
❆ ❜r✐❡❢ ❝❤❛♣t❡r ♦♥ str✉❝t✉r❡s ❛♥❞ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♣r❡s❡♥ts ❜❧♦❝❦ ♣r♦✲
❝❡ss✐♥❣ ❢♦r ❜♦t❤ ❋■❘ ❛♥❞ ■■❘ ✜❧t❡rs✱ ❞✐str✐❜✉t❡❞ ❛r✐t❤♠❡t✐❝ str✉❝t✉r❡s
❢♦r ♠✉❧t✐♣❧✐❡r❧❡ss ✐♠♣❧❡♠❡♥t❛t✐♦♥✱ ❛♥❞ ♠✉❧t✐r❛t❡ s②st❡♠s ❢♦r ✜❧t❡r ❜❛♥❦s
❛♥❞ ✇❛✈❡❧❡ts✳

❚❤✐s ✐s ♣r❡s❡♥t❡❞ ❛s ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ t♦ s❛♠♣❧✐♥❣ ❛♥❞ t♦

♣❡r✐♦❞✐❝❛❧❧② t✐♠❡✲✈❛r②✐♥❣ s②st❡♠s✳ ❚❤❡ ❜✐❢r❡q✉❡♥❝② ♠❛♣ ❣✐✈❡s ❛ ❝❧❡❛r❡r
❡①♣❧❛♥❛t✐♦♥ ♦❢ ❛❧✐❛s✐♥❣ ❛♥❞ ❤♦✇ t♦ ❝♦♥tr♦❧ ✐t✳
❚❤❡ ❜❛s✐❝ ♥♦t❡s t❤❛t ✇❡r❡ ❞❡✈❡❧♦♣❡❞ ✐♥t♦ t❤✐s ❜♦♦❦ ❤❛✈❡ ❡✈♦❧✈❡❞ ♦✈❡r
✸✺ ②❡❛rs ♦❢ t❡❛❝❤✐♥❣ ❛♥❞ ❝♦♥❞✉❝t✐♥❣ r❡s❡❛r❝❤ ✐♥ ❉❙P ❛t ❘✐❝❡✱ ❊r❧❛♥❣❡♥✱
❛♥❞ ▼■❚✳ ❚❤❡② ❝♦♥t❛✐♥ t❤❡ r❡s✉❧ts ♦❢ r❡s❡❛r❝❤ ♦♥ ✜❧t❡rs ❛♥❞ ❛❧❣♦r✐t❤♠s
❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s
❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✵✺✾✽✴✶✳✻❃


✸

❞♦♥❡ ❛t t❤♦s❡ ✉♥✐✈❡rs✐t✐❡s ❛♥❞ ♦t❤❡r ✉♥✐✈❡rs✐t✐❡s ❛♥❞ ✐♥❞✉str✐❡s ❛r♦✉♥❞
t❤❡ ✇♦r❧❞✳

❚❤❡ ❜♦♦❦ tr✐❡s t♦ ❣✐✈❡ ♥♦t ♦♥❧② t❤❡ ❞✐✛❡r❡♥t ♠❡t❤♦❞s ❛♥❞

❛♣♣r♦❛❝❤❡s✱ ❜✉t ❛❧s♦ r❡❛s♦♥s ❛♥❞ ✐♥t✉✐t✐♦♥ ❢♦r ❝❤♦♦s✐♥❣ ♦♥❡ ♠❡t❤♦❞ ♦✈❡r
❛♥♦t❤❡r✳ ■t s❤♦✉❧❞ ❜❡ ✐♥t❡r❡st✐♥❣ t♦ ❜♦t❤ t❤❡ ✉♥✐✈❡rs✐t② st✉❞❡♥t ❛♥❞ t❤❡
✐♥❞✉str✐❛❧ ♣r❛❝t✐t✐♦♥❡r✳
❲❡ ✇❛♥t t♦ ❛❝❦♥♦✇❧❡❞❣❡ ✇✐t❤ ❣r❛t✐t✉❞❡ t❤❡ ❧♦♥❣ t✐♠❡ s✉♣♣♦rt ♦❢
❚❡①❛s ■♥str✉♠❡♥ts✱ ■♥❝✳✱ t❤❡ ◆❛t✐♦♥❛❧ ❙❝✐❡♥❝❡ ❋♦✉♥❞❛t✐♦♥✱ ◆❛t✐♦♥❛❧ ■♥✲
str✉♠❡♥ts✱ ■♥❝✳ ❛♥❞ t❤❡ ▼❛t❤❲♦r❦s✱ ■♥❝✳ ❛s ✇❡❧❧ ❛s t❤❡ s✉♣♣♦rt ♦❢ t❤❡
▼❛①✜❡❧❞ ❛♥❞ ❖s❤♠❛♥ ❢❛♠✐❧✐❡s✳

❲❡ ❛❧s♦ ✇❛♥t t♦ t❤❛♥❦ ♦✉r ❧♦♥❣✲t✐♠❡

❝♦❧❧❡❛❣✉❡s ❚♦♠ P❛r❦s✱ ❍❛♥s ❙❝❤✉❡ss❧❡r✱ ❏✐♠ ▼❝❈❧❡❧❧❛♥✱ ❆❧ ❖♣♣❡♥❤❡✐♠✱
❙❛♥❥✐t ▼✐tr❛✱ ■✈❛♥ ❙❡❧❡s♥✐❝❦✱ ❉♦✉❣ ❏♦♥❡s✱ ❉♦♥ ❏♦❤♥s♦♥✱ ▲❡❧❛♥❞ ❏❛❝❦s♦♥✱
❘✐❝❤ ❇❛r❛♥✐✉❦✱ ❛♥❞ ♦✉r ❣r❛❞✉❛t❡ st✉❞❡♥ts ♦✈❡r ✸✵ ②❡❛rs ❢r♦♠ ✇❤♦♠ ✇❡

❤❛✈❡ ❧❡❛r♥❡❞ ♠✉❝❤ ❛♥❞ ✇✐t❤ ✇❤♦♠ ✇❡ ❤❛✈❡ ❛r❣✉❡❞ ♦❢t❡♥✱ ♣❛rt✐❝✉❧❛r❧②✱
❙❡❧❡s♥✐❝❦✱ ●♦♣✐♥❛t❤✱ ❙♦❡✇✐t♦✱ ❛♥❞ ❱❛r❣❛s✳

❲❡ ❛❧s♦ ♦✇❡ ♠✉❝❤ t♦ t❤❡

■❊❊❊ ❙✐❣♥❛❧ Pr♦❝❡ss✐♥❣ ❙♦❝✐❡t② ❛♥❞ t♦ ❘✐❝❡ ❯♥✐✈❡rs✐t② ❢♦r ❡♥✈✐r♦♥♠❡♥ts
t♦ ❧❡❛r♥✱ t❡❛❝❤✱ ❝r❡❛t❡✱ ❛♥❞ ❝♦❧❧❛❜♦r❛t❡✳ ▼✉❝❤ ♦❢ t❤❡ r❡s✉❧ts ✐♥ ❉❙P ✇❛s
s✉♣♣♦rt❡❞ ❞✐r❡❝t❧② ♦r ✐♥❞✐r❡❝t❧② ❜② t❤❡ ◆❙❋✱ ♠♦st r❡❝❡♥t❧② ◆❙❋ ❣r❛♥t
❊❊❈✲✵✺✸✽✾✸✹ ✐♥ t❤❡ P❛rt♥❡rs❤✐♣s ❢♦r ■♥♥♦✈❛t✐♦♥ ♣r♦❣r❛♠ ✇♦r❦✐♥❣ ✇✐t❤
◆❛t✐♦♥❛❧ ■♥str✉♠❡♥ts✱ ■♥❝✳
❲❡ ♣❛rt✐❝✉❧❛r❧② t❤❛♥❦ ❚❡①❛s ■♥str✉♠❡♥ts ❛♥❞ Pr❡♥t✐❝❡ ❍❛❧❧ ❢♦r r❡t✉r♥✲

❉❋❚✴❋❋❚ ❛♥❞
❈♦♥✈♦❧✉t✐♦♥ ❆❧❣♦r✐t❤♠s❬✺✽❪✱ ❉❡s✐❣♥ ♦❢ ❉✐❣✐t❛❧ ❋✐❧t❡rs❬✷✹✺❪✱ ❛♥❞
✏❊✣❝✐❡♥t ❋♦✉r✐❡r ❚r❛♥s❢♦r♠ ❛♥❞ ❈♦♥✈♦❧✉t✐♦♥ ❆❧❣♦r✐t❤♠s✧ ✐♥ ❆❞✈❛♥❝❡❞
❚♦♣✐❝s ✐♥ ❙✐❣♥❛❧ Pr♦❝❡ss✐♥❣❬✹✹❪ ❝♦✉❧❞ ❜❡ ✐♥❝❧✉❞❡❞ ❤❡r❡ ✉♥❞❡r t❤❡ ❈r❡✲
✐♥❣ t❤❡ ❝♦♣②r✐❣❤ts t♦ ♠❡ s♦ t❤❛t ♣❛rt ♦❢ t❤❡ ♠❛t❡r✐❛❧ ✐♥

❛t✐✈❡ ❈♦♠♠♦♥s ❆ttr✐❜✉t✐♦♥ ❝♦♣②r✐❣❤t✳ ■ ❛❧s♦ ❛♣♣r❡❝✐❛t❡ ■❊❊❊ ♣♦❧✐❝② t❤❛t
❛❧❧♦✇s ♣❛rts ♦❢ ♠② ♣❛♣❡rs t♦ ❜❡ ✐♥❝❧✉❞❡❞ ❤❡r❡✳
❆ r❛t❤❡r ❧♦♥❣ ❧✐st ♦❢ r❡❢❡r❡♥❝❡s ✐s ✐♥❝❧✉❞❡❞ t♦ ♣♦✐♥t t♦ ♠♦r❡ ❜❛❝❦❣r♦✉♥❞✱
t♦ ♠♦r❡ ❛❞✈❛♥❝❡❞ t❤❡♦r②✱ ❛♥❞ t♦ ❛♣♣❧✐❝❛t✐♦♥s✳

❆ ❜♦♦❦ ♦❢ ▼❛t❧❛❜ ❉❙P

❡①❡r❝✐s❡s t❤❛t ❝♦✉❧❞ ❜❡ ✉s❡❞ ✇✐t❤ t❤✐s ❜♦♦❦ ❤❛s ❜❡❡♥ ♣✉❜❧✐s❤❡❞ t❤r♦✉❣❤
Pr❡♥t✐❝❡ ❍❛❧❧ ❬✺✻❪✱ ❬✶✾✾❪✳ ❙♦♠❡ ▼❛t❧❛❜ ♣r♦❣r❛♠s ❛r❡ ✐♥❝❧✉❞❡❞ t♦ ❛✐❞ ✐♥
✉♥❞❡rst❛♥❞✐♥❣ t❤❡ ❞❡s✐❣♥ ❛❧❣♦r✐t❤♠s ❛♥❞ t♦ ❛❝t✉❛❧❧② ❞❡s✐❣♥ ✜❧t❡rs✳ ▲❛❜✲
❱✐❡✇ ❢r♦♠ ◆❛t✐♦♥❛❧ ■♥str✉♠❡♥ts ✐s ❛ ✈❡r② ✉s❡❢✉❧ t♦♦❧ t♦ ❜♦t❤ ❧❡❛r♥ ✇✐t❤
❛♥❞ ✉s❡ ✐♥ ❛♣♣❧✐❝❛t✐♦♥✳ ❆❧❧ ♦❢ t❤❡ ♠❛t❡r✐❛❧ ✐♥ t❤❡s❡ ♥♦t❡s ✐s ❜❡✐♥❣ ♣✉t ✐♥t♦

✏❈♦♥♥❡①✐♦♥s✧ ❬✷✷❪ ✇❤✐❝❤ ✐s ❛ ♠♦❞❡r♥ ✇❡❜✲❜❛s❡❞ ♦♣❡♥✲❝♦♥t❡♥t ✐♥❢♦r♠❛t✐♦♥
s②st❡♠ ✇✇✇✳❝♥①✳♦r❣✳ ❋✉rt❤❡r ✐♥❢♦r♠❛t✐♦♥ ✐s ❛✈❛✐❧❛❜❧❡ ♦♥ ♦✉r ✇❡❜ s✐t❡ ❛t
✇✇✇✳❞s♣✳r✐❝❡✳❡❞✉ ✇✐t❤ ❧✐♥❦s t♦ ♦t❤❡r r❡❧❛t❡❞ ✇♦r❦✳

❲❡ t❤❛♥❦ ❘✐❝❤❛r❞

❇❛r❛♥✐✉❦✱ ❉♦♥ ❏♦❤♥s♦♥✱ ❘❛② ❲❛❣♥❡r✱ ❉❛♥✐❡❧ ❲✐❧❧✐❛♠s♦♥✱ ❛♥❞ ▼❛r❝✐❛
❍♦rt♦♥ ❢♦r t❤❡✐r ❤❡❧♣✳
❚❤✐s ✈❡rs✐♦♥ ♦❢ t❤❡ ❜♦♦❦ ✐s ❛ ❞r❛❢t ❛♥❞ ✇✐❧❧ ❝♦♥t✐♥✉❡ t♦ ❡✈♦❧✈❡ ✉♥❞❡r
❈♦♥♥❡①✐♦♥s✳ ❆ ❝♦♠♣❛♥✐♦♥ ❋❋❚ ❜♦♦❦ ✐s ❜❡✐♥❣ ✇r✐tt❡♥ ❛♥❞ ✐s ❛❧s♦ ❛✈❛✐❧✲
❛❜❧❡ ✐♥ ❈♦♥♥❡①✐♦♥s ❛♥❞ ♣r✐♥t ❢♦r♠✳

❆❧❧ ♦❢ t❤❡s❡ t✇♦ ❜♦♦❦s ❛r❡ ✐♥ t❤❡

r❡♣♦s✐t♦r② ♦❢ ❈♦♥♥❡①✐♦♥s ❛♥❞✱ t❤❡r❡❢♦r❡✱ ❛✈❛✐❧❛❜❧❡ t♦ ❛♥②♦♥❡ ❢r❡❡ t♦ ✉s❡✱
❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s
❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✵✺✾✽✴✶✳✻❃


✹

r❡✉s❡✱ ♠♦❞✐❢②✱ ❡t❝✳ ❛s ❧♦♥❣ ❛s ❛ttr✐❜✉t✐♦♥ ✐s ❣✐✈❡♥✳
❈✳ ❙✐❞♥❡② ❇✉rr✉s
❍♦✉st♦♥✱ ❚❡①❛s
❏✉♥❡ ✷✵✵✽

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

❈❤❛♣t❡r ✶

❙✐❣♥❛❧s ❛♥❞ ❙✐❣♥❛❧
Pr♦❝❡ss✐♥❣ ❙②st❡♠s
✶✳✶ ❈♦♥t✐♥✉♦✉s✲❚✐♠❡ ❙✐❣♥❛❧s✶
❙✐❣♥❛❧s ♦❝❝✉r ✐♥ ❛ ✇✐❞❡ r❛♥❣❡ ♦❢ ♣❤②s✐❝❛❧ ♣❤❡♥♦♠❡♥♦♥✳

❚❤❡② ♠✐❣❤t ❜❡

❤✉♠❛♥ s♣❡❡❝❤✱ ❜❧♦♦❞ ♣r❡ss✉r❡ ✈❛r✐❛t✐♦♥s ✇✐t❤ t✐♠❡✱ s❡✐s♠✐❝ ✇❛✈❡s✱ r❛❞❛r
❛♥❞ s♦♥❛r s✐❣♥❛❧s✱ ♣✐❝t✉r❡s ♦r ✐♠❛❣❡s✱ str❡ss ❛♥❞ str❛✐♥ s✐❣♥❛❧s ✐♥ ❛ ❜✉✐❧❞✐♥❣
str✉❝t✉r❡✱ st♦❝❦ ♠❛r❦❡t ♣r✐❝❡s✱ ❛ ❝✐t②✬s ♣♦♣✉❧❛t✐♦♥✱ ♦r t❡♠♣❡r❛t✉r❡ ❛❝r♦ss ❛
♣❧❛t❡✳ ❚❤❡s❡ s✐❣♥❛❧s ❛r❡ ♦❢t❡♥ ♠♦❞❡❧❡❞ ♦r r❡♣r❡s❡♥t❡❞ ❜② ❛ r❡❛❧ ♦r ❝♦♠♣❧❡①
✈❛❧✉❡❞ ♠❛t❤❡♠❛t✐❝❛❧ ❢✉♥❝t✐♦♥ ♦❢ ♦♥❡ ♦r ♠♦r❡ ✈❛r✐❛❜❧❡s✳

❋♦r ❡①❛♠♣❧❡✱

s♣❡❡❝❤ ✐s ♠♦❞❡❧❡❞ ❜② ❛ ❢✉♥❝t✐♦♥ r❡♣r❡s❡♥t✐♥❣ ❛✐r ♣r❡ss✉r❡ ✈❛r②✐♥❣ ✇✐t❤
t✐♠❡✳

❚❤❡ ❢✉♥❝t✐♦♥ ✐s ❛❝t✐♥❣ ❛s ❛ ♠❛t❤❡♠❛t✐❝❛❧ ❛♥❛❧♦❣② t♦ t❤❡ s♣❡❡❝❤

s✐❣♥❛❧ ❛♥❞✱ t❤❡r❡❢♦r❡✱ ✐s ❝❛❧❧❡❞ ❛♥

❛♥❛❧♦❣

s✐❣♥❛❧✳ ❋♦r t❤❡s❡ s✐❣♥❛❧s✱ t❤❡

✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡ ✐s t✐♠❡ ❛♥❞ ✐t ❝❤❛♥❣❡s ❝♦♥t✐♥✉♦✉s❧② s♦ t❤❛t t❤❡ t❡r♠

❝♦♥t✐♥✉♦✉s✲t✐♠❡

s✐❣♥❛❧ ✐s ❛❧s♦ ✉s❡❞✳ ■♥ ♦✉r ❞✐s❝✉ss✐♦♥✱ ✇❡ t❛❧❦ ♦❢ t❤❡

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

❋♦r t❤❛t r❡❛✲

s♦♥✱ ✇❡ ✇✐❧❧ st❛rt t❤❡ ❞✐s❝✉ss✐♦♥ ♦❢ s✐❣♥❛❧s ✇✐t❤ ❛ ❞❡✈❡❧♦♣♠❡♥t ♦❢ ❋♦✉r✐❡r
tr❛♥s❢♦r♠ ♠❡t❤♦❞s✳ ❲❡ ✇✐❧❧ ✜rst r❡✈✐❡✇ t❤❡ ❝♦♥t✐♥✉♦✉s✲t✐♠❡ ♠❡t❤♦❞s ♦❢
t❤❡ ❋♦✉r✐❡r s❡r✐❡s ✭❋❙✮✱ t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ♦r ✐♥t❡❣r❛❧ ✭❋❚✮✱ ❛♥❞ t❤❡
▲❛♣❧❛❝❡ tr❛♥s❢♦r♠ ✭▲❚✮✳ ◆❡①t t❤❡ ❞✐s❝r❡t❡✲t✐♠❡ ♠❡t❤♦❞s ✇✐❧❧ ❜❡ ❞❡✈❡❧♦♣❡❞
✐♥ ♠♦r❡ ❞❡t❛✐❧ ✇✐t❤ t❤❡ ❞✐s❝r❡t❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ✭❉❋❚✮ ❛♣♣❧✐❡❞ t♦ ✜♥✐t❡

✶ ❚❤✐s

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

✺


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

✻

❧❡♥❣t❤ s✐❣♥❛❧s ❢♦❧❧♦✇❡❞ ❜② t❤❡ ❞✐s❝r❡t❡✲t✐♠❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ✭❉❚❋❚✮ ❢♦r
✐♥✜♥✐t❡❧② ❧♦♥❣ s✐❣♥❛❧s ❛♥❞ ❡♥❞✐♥❣ ✇✐t❤ t❤❡ ❩✲tr❛♥s❢♦r♠ ✇❤✐❝❤ ❛❧❧♦✇s t❤❡
♣♦✇❡r❢✉❧ t♦♦❧s ♦❢ ❝♦♠♣❧❡① ✈❛r✐❛❜❧❡ t❤❡♦r② t♦ ❜❡ ❛♣♣❧✐❡❞✳
▼♦r❡ r❡❝❡♥t❧②✱ ❛ ♥❡✇ t♦♦❧ ❤❛s ❜❡❡♥ ❞❡✈❡❧♦♣❡❞ ❢♦r t❤❡ ❛♥❛❧②s✐s ♦❢ s✐❣✲
♥❛❧s✳

❲❛✈❡❧❡ts ❛♥❞ ✇❛✈❡❧❡t tr❛♥s❢♦r♠s ❬✶✺✵❪✱ ❬✻✸❪✱ ❬✾✷❪✱ ❬✸✽✵❪✱ ❬✸✹✼❪ ❛r❡

❛♥♦t❤❡r ♠♦r❡ ✢❡①✐❜❧❡ ❡①♣❛♥s✐♦♥ s②st❡♠ t❤❛t ❛❧s♦ ❝❛♥ ❞❡s❝r✐❜❡ ❝♦♥t✐♥✉♦✉s
❛♥❞ ❞✐s❝r❡t❡✲t✐♠❡✱ ✜♥✐t❡ ♦r ✐♥✜♥✐t❡ ❞✉r❛t✐♦♥ s✐❣♥❛❧s✳ ❲❡ ✇✐❧❧ ✈❡r② ❜r✐❡✢②
✐♥tr♦❞✉❝❡ t❤❡ ✐❞❡❛s ❜❡❤✐♥❞ ✇❛✈❡❧❡t✲❜❛s❡❞ s✐❣♥❛❧ ❛♥❛❧②s✐s✳

✶✳✶✳✶ ❚❤❡ ❋♦✉r✐❡r ❙❡r✐❡s
❚❤❡ ♣r♦❜❧❡♠ ♦❢ ❡①♣❛♥❞✐♥❣ ❛ ✜♥✐t❡ ❧❡♥❣t❤ s✐❣♥❛❧ ✐♥ ❛ tr✐❣♦♥♦♠❡tr✐❝ s❡r✐❡s
✇❛s ♣♦s❡❞ ❛♥❞ st✉❞✐❡❞ ✐♥ t❤❡ ❧❛t❡ ✶✼✵✵✬s ❜② r❡♥♦✇♥❡❞ ♠❛t❤❡♠❛t✐❝✐❛♥s
s✉❝❤ ❛s ❇❡r♥♦✉❧❧✐✱ ❞✬❆❧❡♠❜❡rt✱ ❊✉❧❡r✱ ▲❛❣r❛♥❣❡✱ ❛♥❞ ●❛✉ss✳ ■♥❞❡❡❞✱ ✇❤❛t
✇❡ ♥♦✇ ❝❛❧❧ t❤❡ ❋♦✉r✐❡r s❡r✐❡s ❛♥❞ t❤❡ ❢♦r♠✉❧❛s ❢♦r t❤❡ ❝♦❡✣❝✐❡♥ts ✇❡r❡
✉s❡❞ ❜② ❊✉❧❡r ✐♥ ✶✼✽✵✳

❍♦✇❡✈❡r✱ ✐t ✇❛s t❤❡ ♣r❡s❡♥t❛t✐♦♥ ✐♥ ✶✽✵✼ ❛♥❞

t❤❡ ♣❛♣❡r ✐♥ ✶✽✷✷ ❜② ❋♦✉r✐❡r st❛t✐♥❣ t❤❛t ❛♥ ❛r❜✐tr❛r② ❢✉♥❝t✐♦♥ ❝♦✉❧❞
❜❡ r❡♣r❡s❡♥t❡❞ ❜② ❛ s❡r✐❡s ♦❢ s✐♥❡s ❛♥❞ ❝♦s✐♥❡s t❤❛t ❜r♦✉❣❤t t❤❡ ♣r♦❜❧❡♠
t♦ ❡✈❡r②♦♥❡✬s ❛tt❡♥t✐♦♥ ❛♥❞ st❛rt❡❞ s❡r✐♦✉s t❤❡♦r❡t✐❝❛❧ ✐♥✈❡st✐❣❛t✐♦♥s ❛♥❞
♣r❛❝t✐❝❛❧ ❛♣♣❧✐❝❛t✐♦♥s t❤❛t ❝♦♥t✐♥✉❡ t♦ t❤✐s ❞❛② ❬✶✹✼❪✱ ❬✻✾❪✱ ❬✶✻✺❪✱ ❬✶✻✹❪✱
❬✶✶✻❪✱ ❬✷✷✸❪✳ ❚❤❡ t❤❡♦r❡t✐❝❛❧ ✇♦r❦ ❤❛s ❜❡❡♥ ❛t t❤❡ ❝❡♥t❡r ♦❢ ❛♥❛❧②s✐s ❛♥❞
t❤❡ ♣r❛❝t✐❝❛❧ ❛♣♣❧✐❝❛t✐♦♥s ❤❛✈❡ ❜❡❡♥ ♦❢ ♠❛❥♦r s✐❣♥✐✜❝❛♥❝❡ ✐♥ ✈✐rt✉❛❧❧② ❡✈✲
❡r② ✜❡❧❞ ♦❢ q✉❛♥t✐t❛t✐✈❡ s❝✐❡♥❝❡ ❛♥❞ t❡❝❤♥♦❧♦❣②✳

❋♦r t❤❡s❡ r❡❛s♦♥s ❛♥❞

♦t❤❡rs✱ t❤❡ ❋♦✉r✐❡r s❡r✐❡s ✐s ✇♦rt❤ ♦✉r s❡r✐♦✉s ❛tt❡♥t✐♦♥ ✐♥ ❛ st✉❞② ♦❢
s✐❣♥❛❧ ♣r♦❝❡ss✐♥❣✳

✶✳✶✳✶✳✶ ❉❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❋♦✉r✐❡r ❙❡r✐❡s
❲❡ ❛ss✉♠❡ t❤❛t t❤❡ s✐❣♥❛❧

x (t)

t♦ ❜❡ ❛♥❛❧②③❡❞ ✐s ✇❡❧❧ ❞❡s❝r✐❜❡❞ ❜② ❛ r❡❛❧

♦r ❝♦♠♣❧❡① ✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ ♦❢ ❛ r❡❛❧ ✈❛r✐❛❜❧❡

{0 ≤ t ≤ T }✳
x (t) =
✇❤❡r❡

t ❞❡✜♥❡❞ ♦✈❡r ❛ ✜♥✐t❡ ✐♥t❡r✈❛❧
x (t) ✐s ❣✐✈❡♥ ❜②

❚❤❡ tr✐❣♦♥♦♠❡tr✐❝ s❡r✐❡s ❡①♣❛♥s✐♦♥ ♦❢

a (0)
+
2

∞

a (k) cos
k=1

2π
kt + b (k) sin
T

2π
kt .
T

✭✶✳✶✮

xk (t) = cos (2πkt/T ) ❛♥❞ yk (t) = sin (2πkt/T ) ❛r❡ t❤❡ ❜❛s✐s ❢✉♥❝✲

t✐♦♥s ❢♦r t❤❡ ❡①♣❛♥s✐♦♥✳ ❚❤❡ ❡♥❡r❣② ♦r ♣♦✇❡r ✐♥ ❛♥ ❡❧❡❝tr✐❝❛❧✱ ♠❡❝❤❛♥✐❝❛❧✱
❡t❝✳ s②st❡♠ ✐s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ sq✉❛r❡ ♦❢ ✈♦❧t❛❣❡✱ ❝✉rr❡♥t✱ ✈❡❧♦❝✐t②✱ ♣r❡s✲
s✉r❡✱ ❡t❝✳

❋♦r t❤✐s r❡❛s♦♥✱ t❤❡ ♥❛t✉r❛❧ s❡tt✐♥❣ ❢♦r ❛ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢

s✐❣♥❛❧s ✐s t❤❡ ❍✐❧❜❡rt s♣❛❝❡ ♦❢

L2 [0, T ]✳

❚❤✐s ♠♦❞❡r♥ ❢♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡

♣r♦❜❧❡♠ ✐s ❞❡✈❡❧♦♣❡❞ ✐♥ ❬✶✵✹❪✱ ❬✶✻✺❪✳ ❚❤❡ s✐♥✉s♦✐❞❛❧ ❜❛s✐s ❢✉♥❝t✐♦♥s ✐♥ t❤❡

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s

❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✵✺✾✽✴✶✳✻❃


✼

tr✐❣♦♥♦♠❡tr✐❝ ❡①♣❛♥s✐♦♥ ❢♦r♠ ❛ ❝♦♠♣❧❡t❡ ♦rt❤♦❣♦♥❛❧ s❡t ✐♥

L2 [0, T ]✳

❚❤❡

♦rt❤♦❣♦♥❛❧✐t② ✐s ❡❛s✐❧② s❡❡♥ ❢r♦♠ ✐♥♥❡r ♣r♦❞✉❝ts

cos
T
0

2π
kt ,
T
cos 2π
kt
T

t
cos 2π
T
2π
cos T t

=

✭✶✳✷✮

2π
t
T

dt = 0

dt = δ (k − )

❛♥❞

2π
kt
T

cos

, sin

T

2π
t
T

=

cos
0

2π
kt
T

sin

✭✶✳✸✮

δ (t) ✐s t❤❡ ❑r♦♥❡❝❦❡r ❞❡❧t❛ ❢✉♥❝t✐♦♥ ✇✐t❤ δ (0) = 1 ❛♥❞ δ (k = 0) =
0✳ ❇❡❝❛✉s❡ ♦❢ t❤✐s✱ t❤❡ k t❤ ❝♦❡✣❝✐❡♥ts ✐♥ t❤❡ s❡r✐❡s ❝❛♥ ❜❡ ❢♦✉♥❞ ❜② t❛❦✐♥❣
t❤❡ ✐♥♥❡r ♣r♦❞✉❝t ♦❢ x (t) ✇✐t❤ t❤❡ k t❤ ❜❛s✐s ❢✉♥❝t✐♦♥s✳ ❚❤✐s ❣✐✈❡s ❢♦r t❤❡
✇❤❡r❡

❝♦❡✣❝✐❡♥ts

a (k) =

2
T

T

x (t) cos

2π
kt dt
T

✭✶✳✹✮

x (t) sin

2π
kt dt
T

✭✶✳✺✮

0

❛♥❞

b (k) =
✇❤❡r❡

T

2
T

T
0

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

❇❡❝❛✉s❡ ♦❢ t❤❡ ♦rt❤♦❣♦♥❛❧✐t② ♦❢ t❤❡ ❜❛s✐s ❢✉♥❝t✐♦♥s✱ ❛ ✜♥✐t❡ ❋♦✉r✐❡r s❡r✐❡s
❢♦r♠❡❞ ❜② tr✉♥❝❛t✐♥❣ t❤❡ ✐♥✜♥✐t❡ s❡r✐❡s ✐s ❛♥ ♦♣t✐♠❛❧ ❧❡❛st sq✉❛r❡❞ ❡rr♦r

❛♣♣r♦①✐♠❛t✐♦♥ t♦
❫

x (t) =

x (t)✳

a (0)
+
2

■❢ t❤❡ ✜♥✐t❡ s❡r✐❡s ✐s ❞❡✜♥❡❞ ❜②

N

a (k) cos
k=1

2π
kt + b (k) sin
T

2π
kt ,
T

✭✶✳✻✮

t❤❡ sq✉❛r❡❞ ❡rr♦r ✐s

=

1
T

✇❤✐❝❤ ✐s ♠✐♥✐♠✐③❡❞ ♦✈❡r ❛❧❧

T

❫

2

|x (t) − x (t) | dt

✭✶✳✼✮

0

a (k)

❛♥❞

b (k)

❜② ✭✶✳✹✮ ❛♥❞ ✭✶✳✺✮✳ ❚❤✐s ✐s ❛♥

❡①tr❛♦r❞✐♥❛r✐❧② ✐♠♣♦rt❛♥t ♣r♦♣❡rt②✳

x (t) ∈ L2 [0, T ]✱ t❤❡♥ t❤❡ s❡r✐❡s ❝♦♥✈❡r❣❡s t♦ x (t) ✐♥

→ 0 ❛s N → ∞❬✶✵✹❪✱ ❬✶✻✺❪✳ ❚❤❡ q✉❡st✐♦♥ ♦❢ ♣♦✐♥t✲✇✐s❡

■t ❢♦❧❧♦✇s t❤❛t ✐❢
t❤❡ s❡♥s❡ t❤❛t

❝♦♥✈❡r❣❡♥❝❡ ✐s ♠♦r❡ ❞✐✣❝✉❧t✳ ❆ s✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥ t❤❛t ✐s ❛❞❡q✉❛t❡ ❢♦r
♠♦st ❛♣♣❧✐❝❛t✐♦♥ st❛t❡s✿ ■❢

f (x) ✐s ❜♦✉♥❞❡❞✱ ✐s ♣✐❡❝❡✲✇✐s❡ ❝♦♥t✐♥✉♦✉s✱ ❛♥❞

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


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

✽

❤❛s ♥♦ ♠♦r❡ t❤❛♥ ❛ ✜♥✐t❡ ♥✉♠❜❡r ♦❢ ♠❛①✐♠❛ ♦✈❡r ❛♥ ✐♥t❡r✈❛❧✱ t❤❡ ❋♦✉r✐❡r

f (x)

s❡r✐❡s ❝♦♥✈❡r❣❡s ♣♦✐♥t✲✇✐s❡ t♦

❛t ❛❧❧ ♣♦✐♥ts ♦❢ ❝♦♥t✐♥✉✐t② ❛♥❞ t♦ t❤❡

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

f (x)

✐s ❝♦♥t✐♥✉♦✉s✱ t❤❡

s❡r✐❡s ❝♦♥✈❡r❣❡s ✉♥✐❢♦r♠❧② ❛t ❛❧❧ ♣♦✐♥ts ❬✶✻✺❪✱ ❬✶✹✼❪✱ ❬✻✾❪✳
❆ ✉s❡❢✉❧ ❝♦♥❞✐t✐♦♥ ❬✶✵✹❪✱ ❬✶✻✺❪ st❛t❡s t❤❛t ✐❢
t❤r♦✉❣❤ t❤❡

q t❤

❋♦✉r✐❡r ❝♦❡✣❝✐❡♥ts
❛s

1
kq+1

❛s

x (t)

❛♥❞ ✐ts ❞❡r✐✈❛t✐✈❡s

❞❡r✐✈❛t✐✈❡ ❛r❡ ❞❡✜♥❡❞ ❛♥❞ ❤❛✈❡ ❜♦✉♥❞❡❞ ✈❛r✐❛t✐♦♥✱ t❤❡

k → ∞✳

a (k)

❛♥❞

b (k)

❛s②♠♣t♦t✐❝❛❧❧② ❞r♦♣ ♦✛ ❛t ❧❡❛st ❛s ❢❛st

❚❤✐s t✐❡s ❣❧♦❜❛❧ r❛t❡s ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ❝♦❡✣❝✐❡♥ts

t♦ ❧♦❝❛❧ s♠♦♦t❤♥❡ss ❝♦♥❞✐t✐♦♥s ♦❢ t❤❡ ❢✉♥❝t✐♦♥✳
❚❤❡ ❢♦r♠ ♦❢ t❤❡ ❋♦✉r✐❡r s❡r✐❡s ✉s✐♥❣ ❜♦t❤ s✐♥❡s ❛♥❞ ❝♦s✐♥❡s ♠❛❦❡s
❞❡t❡r♠✐♥❛t✐♦♥ ♦❢ t❤❡ ♣❡❛❦ ✈❛❧✉❡ ♦r ♦❢ t❤❡ ❧♦❝❛t✐♦♥ ♦❢ ❛ ♣❛rt✐❝✉❧❛r ❢r❡q✉❡♥❝②
t❡r♠ ❞✐✣❝✉❧t✳ ❆ ❞✐✛❡r❡♥t ❢♦r♠ t❤❛t ❡①♣❧✐❝✐t❧② ❣✐✈❡s t❤❡ ♣❡❛❦ ✈❛❧✉❡ ♦❢ t❤❡
s✐♥✉s♦✐❞ ♦❢ t❤❛t ❢r❡q✉❡♥❝② ❛♥❞ t❤❡ ❧♦❝❛t✐♦♥ ♦r ♣❤❛s❡ s❤✐❢t ♦❢ t❤❛t s✐♥✉s♦✐❞
✐s ❣✐✈❡♥ ❜②

x (t) =

∞

d (0)
+
2

d (k) cos
k=1

2π
kt + θ (k)
T

✭✶✳✽✮

❛♥❞✱ ✉s✐♥❣ ❊✉❧❡r✬s r❡❧❛t✐♦♥ ❛♥❞ t❤❡ ✉s✉❛❧ ❡❧❡❝tr✐❝❛❧ ❡♥❣✐♥❡❡r✐♥❣ ♥♦t❛t✐♦♥

♦❢

j=

√

−1✱
ejx = cos (x) + jsin (x) ,

✭✶✳✾✮

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

∞
2π

c (k) ej T

x (t) =

kt

✭✶✳✶✵✮

k=−∞
✇❤❡r❡

c (k) = a (k) + j b (k) .

✭✶✳✶✶✮

❚❤❡ ❝♦❡✣❝✐❡♥t ❡q✉❛t✐♦♥ ✐s

c (k) =

1
T

T

2π

x (t) e−j T

kt

dt

✭✶✳✶✷✮

0

❚❤❡ ❝♦❡✣❝✐❡♥ts ✐♥ t❤❡s❡ t❤r❡❡ ❢♦r♠s ❛r❡ r❡❧❛t❡❞ ❜②

2

2

|d| = |c| = a2 + b2

✭✶✳✶✸✮

❛♥❞

θ = arg{c} = tan−1

b
a

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

✭✶✳✶✹✮


✾

■t ✐s ❡❛s✐❡r t♦ ❡✈❛❧✉❛t❡ ❛ s✐❣♥❛❧ ✐♥ t❡r♠s ♦❢
✐♥ t❡r♠s ♦❢

a (k)

❛♥❞

b (k)✳

c (k)

d (k)

♦r

❛♥❞

θ (k)

t❤❛♥

❚❤❡ ✜rst t✇♦ ❛r❡ ♣♦❧❛r r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛

❝♦♠♣❧❡① ✈❛❧✉❡ ❛♥❞ t❤❡ ❧❛st ✐s r❡❝t❛♥❣✉❧❛r✳ ❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢♦r♠ ✐s ❡❛s✐❡r
t♦ ✇♦r❦ ✇✐t❤ ♠❛t❤❡♠❛t✐❝❛❧❧②✳
❆❧t❤♦✉❣❤ t❤❡ ❢✉♥❝t✐♦♥ t♦ ❜❡ ❡①♣❛♥❞❡❞ ✐s ❞❡✜♥❡❞ ♦♥❧② ♦✈❡r ❛ s♣❡❝✐✜❝
✜♥✐t❡ r❡❣✐♦♥✱ t❤❡ s❡r✐❡s ❝♦♥✈❡r❣❡s t♦ ❛ ❢✉♥❝t✐♦♥ t❤❛t ✐s ❞❡✜♥❡❞ ♦✈❡r t❤❡ r❡❛❧
❧✐♥❡ ❛♥❞ ✐s ♣❡r✐♦❞✐❝✳ ■t ✐s ❡q✉❛❧ t♦ t❤❡ ♦r✐❣✐♥❛❧ ❢✉♥❝t✐♦♥ ♦✈❡r t❤❡ r❡❣✐♦♥
♦❢ ❞❡✜♥✐t✐♦♥ ❛♥❞ ✐s ❛ ♣❡r✐♦❞✐❝ ❡①t❡♥s✐♦♥ ♦✉ts✐❞❡ ♦❢ t❤❡ r❡❣✐♦♥✳

■♥❞❡❡❞✱

♦♥❡ ❝♦✉❧❞ ❛rt✐✜❝✐❛❧❧② ❡①t❡♥❞ t❤❡ ❣✐✈❡♥ ❢✉♥❝t✐♦♥ ❛t t❤❡ ♦✉ts❡t ❛♥❞ t❤❡♥ t❤❡
❡①♣❛♥s✐♦♥ ✇♦✉❧❞ ❝♦♥✈❡r❣❡ ❡✈❡r②✇❤❡r❡✳

✶✳✶✳✶✳✷ ❆ ●❡♦♠❡tr✐❝ ❱✐❡✇
■t ❝❛♥ ❜❡ ✈❡r② ❤❡❧♣❢✉❧ t♦ ❞❡✈❡❧♦♣ ❛ ❣❡♦♠❡tr✐❝ ✈✐❡✇ ♦❢ t❤❡ ❋♦✉r✐❡r s❡r✐❡s
✇❤❡r❡

x (t)

✐s ❝♦♥s✐❞❡r❡❞ t♦ ❜❡ ❛ ✈❡❝t♦r ❛♥❞ t❤❡ ❜❛s✐s ❢✉♥❝t✐♦♥s ❛r❡ t❤❡

❝♦♦r❞✐♥❛t❡ ♦r ❜❛s✐s ✈❡❝t♦rs✳

x (t)

♦♥ t❤❡ ❝♦♦r❞✐♥❛t❡s✳

❚❤❡ ❝♦❡✣❝✐❡♥ts ❜❡❝♦♠❡ t❤❡ ♣r♦❥❡❝t✐♦♥s ♦❢

❚❤❡ ✐❞❡❛s ♦❢ ❛ ♠❡❛s✉r❡ ♦❢ ❞✐st❛♥❝❡✱ s✐③❡✱ ❛♥❞

♦rt❤♦❣♦♥❛❧✐t② ❛r❡ ✐♠♣♦rt❛♥t ❛♥❞ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❡rr♦r ✐s ❡❛s② t♦ ♣✐❝t✉r❡✳
❚❤✐s ✐s ❞♦♥❡ ✐♥ ❬✶✵✹❪✱ ❬✶✻✺❪✱ ❬✸✾✵❪ ✉s✐♥❣ ❍✐❧❜❡rt s♣❛❝❡ ♠❡t❤♦❞s✳

✶✳✶✳✶✳✸ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❋♦✉r✐❡r ❙❡r✐❡s
❚❤❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❋♦✉r✐❡r s❡r✐❡s ❛r❡ ✐♠♣♦rt❛♥t ✐♥ ❛♣♣❧②✐♥❣ ✐t t♦ s✐❣♥❛❧
❛♥❛❧②s✐s ❛♥❞ t♦ ✐♥t❡r♣r❡t✐♥❣ ✐t✳ ❚❤❡ ♠❛✐♥ ♣r♦♣❡rt✐❡s ❛r❡ ❣✐✈❡♥ ❤❡r❡ ✉s✐♥❣
t❤❡ ♥♦t❛t✐♦♥ t❤❛t t❤❡ ❋♦✉r✐❡r s❡r✐❡s ♦❢ ❛ r❡❛❧ ✈❛❧✉❡❞ ❢✉♥❝t✐♦♥

{0 ≤ t ≤ T }

✐s ❣✐✈❡♥ ❜②

❡①t❡♥s✐♦♥s ♦❢

x (t)✳

✶✳ ▲✐♥❡❛r✿

F{x (t)} = c (k)

❛♥❞

x (t)

♦✈❡r

❞❡♥♦t❡s t❤❡ ♣❡r✐♦❞✐❝

F{x + y} = F{x} + F{y}

■❞❡❛ ♦❢ s✉♣❡r♣♦s✐t✐♦♥✳ ❆❧s♦ s❝❛❧❛❜✐❧✐t②✿
✷✳ ❊①t❡♥s✐♦♥s ♦❢

x
˜ (t)

x
˜ (t)

F{ax} = aF{x}

x (t)✿ x
˜ (t) = x
˜ (t + T )

✐s ♣❡r✐♦❞✐❝✳

✸✳ ❊✈❡♥ ❛♥❞ ❖❞❞ P❛rts✿

x (t) = u (t) + jv (t)

❛♥❞

C (k) = A (k) +

jB (k) = |C (k) | ejθ(k)
u

v

A

B

|C|

θ

❡✈❡♥

✵

❡✈❡♥

✵

❡✈❡♥

✵

♦❞❞

✵

✵

♦❞❞

❡✈❡♥

✵

✵

❡✈❡♥

✵

❡✈❡♥

❡✈❡♥

π/2

✵

♦❞❞

♦❞❞

✵

❡✈❡♥

π/2

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


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

✶✵

❚❛❜❧❡ ✶✳✶

✹✳ ❈♦♥✈♦❧✉t✐♦♥✿ ■❢ ❝♦♥t✐♥✉♦✉s ❝②❝❧✐❝ ❝♦♥✈♦❧✉t✐♦♥ ✐s ❞❡✜♥❡❞ ❜②

T

˜ (t − τ ) x
h
˜ (τ ) dτ

y (t) = h (t) ◦ x (t) =

✭✶✳✶✺✮

0

t❤❡♥

F{h (t) ◦ x (t)} = F{h (t)} F{x (t)}

✺✳ ▼✉❧t✐♣❧✐❝❛t✐♦♥✿ ■❢ ❞✐s❝r❡t❡ ❝♦♥✈♦❧✉t✐♦♥ ✐s ❞❡✜♥❡❞ ❜②

∞

e (n) = d (n) ∗ c (n) =

d (m) c (n − m)

✭✶✳✶✻✮

m=−∞
t❤❡♥

F{h (t) x (t)} = F{h (t)} ∗ F {x (t)}

❚❤✐s ♣r♦♣❡rt② ✐s t❤❡ ✐♥✈❡rs❡ ♦❢ ♣r♦♣❡rt② ✹ ✭❧✐st✱ ♣✳ ✾✮ ❛♥❞ ✈✐❝❡ ✈❡rs❛✳

2
∞
2
1 T
k=−∞ |C (k) |
T 0 |x (t) | dt =
❚❤✐s ♣r♦♣❡rt② s❛②s t❤❡ ❡♥❡r❣② ❝❛❧❝✉❧❛t❡❞ ✐♥ t❤❡ t✐♠❡ ❞♦♠❛✐♥ ✐s t❤❡

✻✳ P❛rs❡✈❛❧✿

s❛♠❡ ❛s t❤❛t ❝❛❧❝✉❧❛t❡❞ ✐♥ t❤❡ ❢r❡q✉❡♥❝② ✭♦r ❋♦✉r✐❡r✮ ❞♦♠❛✐♥✳
✼✳ ❙❤✐❢t✿

F{˜
x (t − t0 )} = C (k) e−j2πt0 k/T

❆ s❤✐❢t ✐♥ t❤❡ t✐♠❡ ❞♦♠❛✐♥ r❡s✉❧ts ✐♥ ❛ ❧✐♥❡❛r ♣❤❛s❡ s❤✐❢t ✐♥ t❤❡
❢r❡q✉❡♥❝② ❞♦♠❛✐♥✳
✽✳ ▼♦❞✉❧❛t❡✿

F{x (t) ej2πKt/T } = C (k − K)

▼♦❞✉❧❛t✐♦♥ ✐♥ t❤❡ t✐♠❡ ❞♦♠❛✐♥ r❡s✉❧ts ✐♥ ❛ s❤✐❢t ✐♥ t❤❡ ❢r❡q✉❡♥❝②
❞♦♠❛✐♥✳ ❚❤✐s ♣r♦♣❡rt② ✐s t❤❡ ✐♥✈❡rs❡ ♦❢ ♣r♦♣❡rt② ✼✳
✾✳ ❖rt❤♦❣♦♥❛❧✐t② ♦❢ ❜❛s✐s ❢✉♥❝t✐♦♥s✿

T

e−j2πmt/T ej2πnt/T dt = T δ (n − m) = {
0

T

✐❢

n=m

0

✐❢

n = m.
✭✶✳✶✼✮

❖rt❤♦❣♦♥❛❧✐t② ❛❧❧♦✇s t❤❡ ❝❛❧❝✉❧❛t✐♦♥ ♦❢ ❝♦❡✣❝✐❡♥ts ✉s✐♥❣ ✐♥♥❡r ♣r♦❞✲
✉❝ts ✐♥ ✭✶✳✹✮ ❛♥❞ ✭✶✳✺✮✳ ■t ❛❧s♦ ❛❧❧♦✇s P❛rs❡✈❛❧✬s ❚❤❡♦r❡♠ ✐♥ ♣r♦♣✲
❡rt② ✻ ✭❧✐st✱ ♣✳

✶✵✮✳

❆ r❡❧❛①❡❞ ✈❡rs✐♦♥ ♦❢ ♦rt❤♦❣♦♥❛❧✐t② ✐s ❝❛❧❧❡❞

✏t✐❣❤t ❢r❛♠❡s✧ ❛♥❞ ✐s ✐♠♣♦rt❛♥t ✐♥ ♦✈❡r✲s♣❡❝✐✜❡❞ s②st❡♠s✱ ❡s♣❡❝✐❛❧❧②
✐♥ ✇❛✈❡❧❡ts✳

✶✳✶✳✶✳✹ ❊①❛♠♣❧❡s
•

❆♥ ❡①❛♠♣❧❡ ♦❢ t❤❡ ❋♦✉r✐❡r s❡r✐❡s ✐s t❤❡ ❡①♣❛♥s✐♦♥ ♦❢ ❛ sq✉❛r❡ ✇❛✈❡
s✐❣♥❛❧ ✇✐t❤ ♣❡r✐♦❞

x (t) =

2π ✳

❚❤❡ ❡①♣❛♥s✐♦♥ ✐s

4
1

1
sin (t) + sin (3t) + sin (5t) · · · .
π
3
5
❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s

❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✵✺✾✽✴✶✳✻❃

✭✶✳✶✽✮


✶✶

❇❡❝❛✉s❡

x (t)

✐s ♦❞❞✱ t❤❡r❡ ❛r❡ ♥♦ ❝♦s✐♥❡ t❡r♠s ✭❛❧❧

a (k) = 0✮ ❛♥❞✱
k t❡r♠s

❜❡❝❛✉s❡ ♦❢ ✐ts s②♠♠❡tr✐❡s✱ t❤❡r❡ ❛r❡ ♥♦ ❡✈❡♥ ❤❛r♠♦♥✐❝s ✭❡✈❡♥

❛r❡ ③❡r♦✮✳ ❚❤❡ ❢✉♥❝t✐♦♥ ✐s ✇❡❧❧ ❞❡✜♥❡❞ ❛♥❞ ❜♦✉♥❞❡❞❀ ✐ts ❞❡r✐✈❛t✐✈❡

1
k✳
❆ s❡❝♦♥❞ ❡①❛♠♣❧❡ ✐s ❛ tr✐❛♥❣❧❡ ✇❛✈❡ ♦❢ ♣❡r✐♦❞ 2π ✳ ❚❤✐s ✐s ❛ ❝♦♥t✐♥✲

✐s ♥♦t✱ t❤❡r❡❢♦r❡✱ t❤❡ ❝♦❡✣❝✐❡♥ts ❞r♦♣ ♦✛ ❛s

•

✉♦✉s ❢✉♥❝t✐♦♥ ✇❤❡r❡ t❤❡ sq✉❛r❡ ✇❛✈❡ ✇❛s ♥♦t✳ ❚❤❡ ❡①♣❛♥s✐♦♥ ♦❢ t❤❡
tr✐❛♥❣❧❡ ✇❛✈❡ ✐s

x (t) =

4
1
1
sin (t) − 2 sin (3t) + 2 sin (5t) + · · · .
π
3
5

❍❡r❡ t❤❡ ❝♦❡✣❝✐❡♥ts ❞r♦♣ ♦✛ ❛s

✭✶✳✶✾✮

1
k2 s✐♥❝❡ t❤❡ ❢✉♥❝t✐♦♥ ❛♥❞ ✐ts ✜rst

❞❡r✐✈❛t✐✈❡ ❡①✐st ❛♥❞ ❛r❡ ❜♦✉♥❞❡❞✳
◆♦t❡ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ tr✐❛♥❣❧❡ ✇❛✈❡ ✐s ❛ sq✉❛r❡ ✇❛✈❡✳

❊①❛♠✐♥❡ t❤❡

s❡r✐❡s ❝♦❡✣❝✐❡♥ts t♦ s❡❡ t❤✐s✳ ❚❤❡r❡ ❛r❡ ♠❛♥② ❜♦♦❦s ❛♥❞ ✇❡❜ s✐t❡s ♦♥ t❤❡

❋♦✉r✐❡r s❡r✐❡s t❤❛t ❣✐✈❡ ✐♥s✐❣❤t t❤r♦✉❣❤ ❡①❛♠♣❧❡s ❛♥❞ ❞❡♠♦s✳

✶✳✶✳✶✳✺ ❚❤❡♦r❡♠s ♦♥ t❤❡ ❋♦✉r✐❡r ❙❡r✐❡s
❋♦✉r ♦❢ t❤❡ ♠♦st ✐♠♣♦rt❛♥t t❤❡♦r❡♠s ✐♥ t❤❡ t❤❡♦r② ♦❢ ❋♦✉r✐❡r ❛♥❛❧②s✐s
❛r❡ t❤❡ ✐♥✈❡rs✐♦♥ t❤❡♦r❡♠✱ t❤❡ ❝♦♥✈♦❧✉t✐♦♥ t❤❡♦r❡♠✱ t❤❡ ❞✐✛❡r❡♥t✐❛t✐♦♥
t❤❡♦r❡♠✱ ❛♥❞ P❛rs❡✈❛❧✬s t❤❡♦r❡♠ ❬✼✶❪✳

•

❚❤❡ ✐♥✈❡rs✐♦♥ t❤❡♦r❡♠ ✐s t❤❡ tr✉t❤ ♦❢ t❤❡ tr❛♥s❢♦r♠ ♣❛✐r ❣✐✈❡♥ ✐♥
✭✶✳✶✮✱ ✭✶✳✹✮✱ ❛♥❞ ✭✶✳✺✮✳✳

•
•

❚❤❡ ❝♦♥✈♦❧✉t✐♦♥ t❤❡♦r❡♠ ✐s ♣r♦♣❡rt② ✹ ✭❧✐st✱ ♣✳ ✾✮✳

•

P❛rs❡✈❛❧✬s t❤❡♦r❡♠ ✐s ❣✐✈❡♥ ✐♥ ♣r♦♣❡rt② ✻ ✭❧✐st✱ ♣✳ ✶✵✮✳

❚❤❡ ❞✐✛❡r❡♥t✐❛t✐♦♥ t❤❡♦r❡♠ s❛②s t❤❛t t❤❡ tr❛♥s❢♦r♠ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡
♦❢ ❛ ❢✉♥❝t✐♦♥ ✐s

jω

t✐♠❡s t❤❡ tr❛♥s❢♦r♠ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✳

❆❧❧ ♦❢ t❤❡s❡ ❛r❡ ❜❛s❡❞ ♦♥ t❤❡ ♦rt❤♦❣♦♥❛❧✐t② ♦❢ t❤❡ ❜❛s✐s ❢✉♥❝t✐♦♥ ♦❢ t❤❡
❋♦✉r✐❡r s❡r✐❡s ❛♥❞ ✐♥t❡❣r❛❧ ❛♥❞ ❛❧❧ r❡q✉✐r❡ ❦♥♦✇❧❡❞❣❡ ♦❢ t❤❡ ❝♦♥✈❡r❣❡♥❝❡

♦❢ t❤❡ s✉♠s ❛♥❞ ✐♥t❡❣r❛❧s✳

❚❤❡ ♣r❛❝t✐❝❛❧ ❛♥❞ t❤❡♦r❡t✐❝❛❧ ✉s❡ ♦❢ ❋♦✉r✐❡r

❛♥❛❧②s✐s ✐s ❣r❡❛t❧② ❡①♣❛♥❞❡❞ ✐❢ ✉s❡ ✐s ♠❛❞❡ ♦❢ ❞✐str✐❜✉t✐♦♥s ♦r ❣❡♥❡r❛❧✐③❡❞
❢✉♥❝t✐♦♥s ✭❡✳❣✳ ❉✐r❛❝ ❞❡❧t❛ ❢✉♥❝t✐♦♥s✱

δ (t)✮

❬✷✸✾❪✱ ❬✸✷❪✳ ❇❡❝❛✉s❡ ❡♥❡r❣② ✐s

❛♥ ✐♠♣♦rt❛♥t ♠❡❛s✉r❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✐♥ s✐❣♥❛❧ ♣r♦❝❡ss✐♥❣ ❛♣♣❧✐❝❛t✐♦♥s✱ t❤❡
❍✐❧❜❡rt s♣❛❝❡ ♦❢

L2

❢✉♥❝t✐♦♥s ✐s ❛ ♣r♦♣❡r s❡tt✐♥❣ ❢♦r t❤❡ ❜❛s✐❝ t❤❡♦r② ❛♥❞

❛ ❣❡♦♠❡tr✐❝ ✈✐❡✇ ❝❛♥ ❜❡ ❡s♣❡❝✐❛❧❧② ✉s❡❢✉❧ ❬✶✵✹❪✱ ❬✼✶❪✳
❚❤❡ ❢♦❧❧♦✇✐♥❣ t❤❡♦r❡♠s ❛♥❞ r❡s✉❧ts ❝♦♥❝❡r♥ t❤❡ ❡①✐st❡♥❝❡ ❛♥❞ ❝♦♥✈❡r✲
❣❡♥❝❡ ♦❢ t❤❡ ❋♦✉r✐❡r s❡r✐❡s ❛♥❞ t❤❡ ❞✐s❝r❡t❡✲t✐♠❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ❬✷✷✻❪✳
❉❡t❛✐❧s✱ ❞✐s❝✉ss✐♦♥s ❛♥❞ ♣r♦♦❢s ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ t❤❡ ❝✐t❡❞ r❡❢❡r❡♥❝❡s✳

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


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

✶✷

•

■❢

f (x)

❤❛s ❜♦✉♥❞❡❞ ✈❛r✐❛t✐♦♥ ✐♥ t❤❡ ✐♥t❡r✈❛❧

s❡r✐❡s ❝♦rr❡s♣♦♥❞✐♥❣ t♦

f (x)

(−π, π)✱ t❤❡ ❋♦✉r✐❡r
f (x) ❛t ❛♥②

❝♦♥✈❡r❣❡s t♦ t❤❡ ✈❛❧✉❡

♣♦✐♥t ✇✐t❤✐♥ t❤❡ ✐♥t❡r✈❛❧✱ ❛t ✇❤✐❝❤ t❤❡ ❢✉♥❝t✐♦♥ ✐s ❝♦♥t✐♥✉♦✉s❀ ✐t

1
2 [f (x + 0) + f (x − 0)] ❛t ❛♥② s✉❝❤ ♣♦✐♥t ❛t
✇❤✐❝❤ t❤❡ ❢✉♥❝t✐♦♥ ✐s ❞✐s❝♦♥t✐♥✉♦✉s✳ ❆t t❤❡ ♣♦✐♥ts π, −π ✐t ❝♦♥✈❡r❣❡s
1
t♦ t❤❡ ✈❛❧✉❡
2 [f (−π + 0) + f (π − 0)]✳ ❬✶✹✼❪
■❢ f (x) ✐s ♦❢ ❜♦✉♥❞❡❞ ✈❛r✐❛t✐♦♥ ✐♥ (−π, π)✱ t❤❡ ❋♦✉r✐❡r s❡r✐❡s ❝♦♥✲
❝♦♥✈❡r❣❡s t♦ t❤❡ ✈❛❧✉❡

•

✈❡r❣❡s t♦

f (x)✱

•

■❢

f (x)

1
2 [f (x
(−π, π)✳ ❬✶✹✼❪

•

■❢

f (x)

a

❛♥❞

✐s ♦❢ ❜♦✉♥❞❡❞ ✈❛r✐❛t✐♦♥ ✐♥

✈❡r❣❡s t♦

(a, b) ✐♥ ✇❤✐❝❤ f (x) ✐s
b ❜❡✐♥❣ ♦♥ ❜♦t❤ s✐❞❡s✳ ❬✶✹✼❪
(−π, π)✱ t❤❡ ❋♦✉r✐❡r s❡r✐❡s ❝♦♥✲

✉♥✐❢♦r♠❧② ✐♥ ❛♥② ✐♥t❡r✈❛❧

❝♦♥t✐♥✉♦✉s✱ t❤❡ ❝♦♥t✐♥✉✐t② ❛t

+ 0) + f (x − 0)]✱

❜♦✉♥❞❡❞ t❤r♦✉❣❤♦✉t t❤❡ ✐♥t❡r✈❛❧

✐s ❜♦✉♥❞❡❞ ❛♥❞ ✐❢ ✐t ✐s ❝♦♥t✐♥✉♦✉s ✐♥ ✐ts ❞♦♠❛✐♥ ❛t ❡✈❡r②

♣♦✐♥t✱ ✇✐t❤ t❤❡ ❡①❝❡♣t✐♦♥ ♦❢ ❛ ✜♥✐t❡ ♥✉♠❜❡r ♦❢ ♣♦✐♥ts ❛t ✇❤✐❝❤ ✐t ♠❛②
❤❛✈❡ ♦r❞✐♥❛r② ❞✐s❝♦♥t✐♥✉✐t✐❡s✱ ❛♥❞ ✐❢ t❤❡ ❞♦♠❛✐♥ ♠❛② ❜❡ ❞✐✈✐❞❡❞ ✐♥t♦
❛ ✜♥✐t❡ ♥✉♠❜❡r ♦❢ ♣❛rts✱ s✉❝❤ t❤❛t ✐♥ ❛♥② ♦♥❡ ♦❢ t❤❡♠ t❤❡ ❢✉♥❝t✐♦♥ ✐s
♠♦♥♦t♦♥❡❀ ♦r✱ ✐♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❢✉♥❝t✐♦♥ ❤❛s ♦♥❧② ❛ ✜♥✐t❡ ♥✉♠❜❡r

f (x) ❝♦♥✲
[f (x + 0) + f (x − 0)]

♦❢ ♠❛①✐♠❛ ❛♥❞ ♠✐♥✐♠❛ ✐♥ ✐ts ❞♦♠❛✐♥✱ t❤❡ ❋♦✉r✐❡r s❡r✐❡s ♦❢
✈❡r❣❡s t♦

f (x) ❛t ♣♦✐♥ts ♦❢ ❝♦♥t✐♥✉✐t② ❛♥❞ t♦

1
2

❛t ♣♦✐♥ts ♦❢ ❞✐s❝♦♥t✐♥✉✐t②✳ ❬✶✹✼❪✱ ❬✻✾❪

•

■❢

f (x)

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

❛ ✜♥✐t❡ ♥✉♠❜❡r ♦❢ ♣♦✐♥ts ✐♥ ✇❤♦s❡ ♥❡✐❣❤❜♦r❤♦♦❞
✉♣♣❡r ❜♦✉♥❞ ❤❛✈❡ ❜❡❡♥ ❡①❝❧✉❞❡❞✱

f (x)

|f (x) |

❤❛s ♥♦

❜❡❝♦♠❡s ❛ ❢✉♥❝t✐♦♥ ✇✐t❤

❜♦✉♥❞❡❞ ✈❛r✐❛t✐♦♥✱ t❤❡♥ t❤❡ ❋♦✉r✐❡r s❡r✐❡s ❝♦♥✈❡r❣❡s t♦ t❤❡ ✈❛❧✉❡

•

1
2 [f (x + 0) + f (x − 0)]✱ ❛t ❡✈❡r② ♣♦✐♥t ✐♥ (−π, π)✱ ❡①❝❡♣t t❤❡ ♣♦✐♥ts
♦❢ ✐♥✜♥✐t❡ ❞✐s❝♦♥t✐♥✉✐t② ♦❢ t❤❡ ❢✉♥❝t✐♦♥✱ ♣r♦✈✐❞❡❞ t❤❡ ✐♠♣r♦♣❡r ✐♥✲
π
f (x) dx ❡①✐st✱ ❛♥❞ ✐s ❛❜s♦❧✉t❡❧② ❝♦♥✈❡r❣❡♥t✳ ❬✶✹✼❪

t❡❣r❛❧
−π
■❢ ❢ ✐s ♦❢ ❜♦✉♥❞❡❞ ✈❛r✐❛t✐♦♥✱ t❤❡ ❋♦✉r✐❡r s❡r✐❡s ♦❢ ❢ ❝♦♥✈❡r❣❡s ❛t ❡✈❡r②

[f (x + 0) + f (x − 0)] /2✳ ■❢ ❢ ✐s✱ ✐♥ ❛❞❞✐t✐♦♥✱
I = (a, b)✱ ✐ts ❋♦✉r✐❡r s❡r✐❡s
✐s ✉♥✐❢♦r♠❧② ❝♦♥✈❡r❣❡♥t ✐♥ I ✳ ❬✸✾✼❪
• ■❢ a (k) ❛♥❞ b (k) ❛r❡ ❛❜s♦❧✉t❡❧② s✉♠♠❛❜❧❡✱ t❤❡ ❋♦✉r✐❡r s❡r✐❡s ❝♦♥✲
✈❡r❣❡s ✉♥✐❢♦r♠❧② t♦ f (x) ✇❤✐❝❤ ✐s ❝♦♥t✐♥✉♦✉s✳ ❬✷✷✻❪
• ■❢ a (k) ❛♥❞ b (k) ❛r❡ sq✉❛r❡ s✉♠♠❛❜❧❡✱ t❤❡ ❋♦✉r✐❡r s❡r✐❡s ❝♦♥✈❡r❣❡s
t♦ f (x) ✇❤❡r❡ ✐t ✐s ❝♦♥t✐♥✉♦✉s✱ ❜✉t ♥♦t ♥❡❝❡ss❛r✐❧② ✉♥✐❢♦r♠❧②✳ ❬✷✷✻❪
• ❙✉♣♣♦s❡ t❤❛t f (x) ✐s ♣❡r✐♦❞✐❝✱ ♦❢ ♣❡r✐♦❞ X ✱ ✐s ❞❡✜♥❡❞ ❛♥❞ ❜♦✉♥❞❡❞
♦♥ [0, X] ❛♥❞ t❤❛t ❛t ❧❡❛st ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦✉r ❝♦♥❞✐t✐♦♥s ✐s
s❛t✐s✜❡❞✿ ✭✐✮ f ✐s ♣✐❡❝❡✇✐s❡ ♠♦♥♦t♦♥✐❝ ♦♥ [0, X]✱ ✭✐✐✮ f ❤❛s ❛ ✜♥✐t❡
♥✉♠❜❡r ♦❢ ♠❛①✐♠❛ ❛♥❞ ♠✐♥✐♠❛ ♦♥ [0, X] ❛♥❞ ❛ ✜♥✐t❡ ♥✉♠❜❡r ♦❢
❞✐s❝♦♥t✐♥✉✐t✐❡s ♦♥ [0, X]✱ ✭✐✐✐✮ f ✐s ♦❢ ❜♦✉♥❞❡❞ ✈❛r✐❛t✐♦♥ ♦♥ [0, X]✱ ✭✐✈✮
f ✐s ♣✐❡❝❡✇✐s❡ s♠♦♦t❤ ♦♥ [0, X]✿ t❤❡♥ ✐t ✇✐❧❧ ❢♦❧❧♦✇ t❤❛t t❤❡ ❋♦✉r✐❡r
♣♦✐♥t

x

t♦ t❤❡ ✈❛❧✉❡

❝♦♥t✐♥✉♦✉s ❛t ❡✈❡r② ♣♦✐♥t ♦❢ ❛♥ ✐♥t❡r✈❛❧

s❡r✐❡s ❝♦❡✣❝✐❡♥ts ♠❛② ❜❡ ❞❡✜♥❡❞ t❤r♦✉❣❤ t❤❡ ❞❡✜♥✐♥❣ ✐♥t❡❣r❛❧✱ ✉s✐♥❣
❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s
❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✵✺✾✽✴✶✳✻❃

✶✸

♣r♦♣❡r ❘✐❡♠❛♥♥ ✐♥t❡❣r❛❧s✱ ❛♥❞ t❤❛t t❤❡ ❋♦✉r✐❡r s❡r✐❡s ❝♦♥✈❡r❣❡s t♦

f (x)

❛t ❛✳❛✳x✱ t♦

1
2 [f
❋♦r ❛♥② 1
✈❛❧✉❡

•

f (x) ❛t ❡❛❝❤ ♣♦✐♥t ♦❢ ❝♦♥t✐♥✉✐t② ♦❢ f ✱ ❛♥❞
(x− ) + f (x+ )] ❛t ❛❧❧ x✳ ❬✼✶❪
≤ p < ∞ ❛♥❞ ❛♥② f ∈ C p S 1 ✱ t❤❡ ♣❛rt✐❛❧ s✉♠s

t♦ t❤❡

❫

Sn = Sn (f ) =

f (k) ek

✭✶✳✷✵✮

|k|≤n

❝♦♥✈❡r❣❡ t♦

f✱

n → ∞❀ ✐♥ ❢❛❝t✱ ||Sn − f ||∞
n−p+1/2 ✳ ❬✶✵✹❪

✉♥✐❢♦r♠❧② ❛s

❜② ❛ ❝♦♥st❛♥t ♠✉❧t✐♣❧❡ ♦❢

✐s ❜♦✉♥❞❡❞

❚❤❡ ❋♦✉r✐❡r s❡r✐❡s ❡①♣❛♥s✐♦♥ r❡s✉❧ts ✐♥ tr❛♥s❢♦r♠✐♥❣ ❛ ♣❡r✐♦❞✐❝✱ ❝♦♥t✐♥✉♦✉s
t✐♠❡ ❢✉♥❝t✐♦♥✱

b (k)

x
˜ (t)✱ t♦ t✇♦ ❞✐s❝r❡t❡ ✐♥❞❡①❡❞ ❢r❡q✉❡♥❝② ❢✉♥❝t✐♦♥s✱ a (k) ❛♥❞

t❤❛t ❛r❡ ♥♦t ♣❡r✐♦❞✐❝✳

✶✳✶✳✷ ❚❤❡ ❋♦✉r✐❡r ❚r❛♥s❢♦r♠
▼❛♥② ♣r❛❝t✐❝❛❧ ♣r♦❜❧❡♠s ✐♥ s✐❣♥❛❧ ❛♥❛❧②s✐s ✐♥✈♦❧✈❡ ❡✐t❤❡r ✐♥✜♥✐t❡❧② ❧♦♥❣
♦r ✈❡r② ❧♦♥❣ s✐❣♥❛❧s ✇❤❡r❡ t❤❡ ❋♦✉r✐❡r s❡r✐❡s ✐s ♥♦t ❛♣♣r♦♣r✐❛t❡✳ ❋♦r t❤❡s❡
❝❛s❡s✱ t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ✭❋❚✮ ❛♥❞ ✐ts ✐♥✈❡rs❡ ✭■❋❚✮ ❤❛✈❡ ❜❡❡♥ ❞❡✲
✈❡❧♦♣❡❞✳

❚❤✐s tr❛♥s❢♦r♠ ❤❛s ❜❡❡♥ ✉s❡❞ ✇✐t❤ ❣r❡❛t s✉❝❝❡ss ✐♥ ✈✐rt✉❛❧❧②

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

❲❤✐❧❡ t❤❡ ❋♦✉r✐❡r s❡r✐❡s ✇❛s ✉s❡❞ ❜❡❢♦r❡ ❋♦✉r✐❡r

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

T

✐♥❝r❡❛s❡ t♦ ✐♥✜♥✐t② ♦r t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ❝❛♥ ❜❡ ✐♥❞❡♣❡♥✲

❞❡♥t❧② ❞❡✜♥❡❞ ❛♥❞ t❤❡♥ t❤❡ ❋♦✉r✐❡r s❡r✐❡s s❤♦✇♥ t♦ ❜❡ ❛ s♣❡❝✐❛❧ ❝❛s❡ ♦❢
✐t✳ ❚❤❡ ❧❛tt❡r ❛♣♣r♦❛❝❤ ✐s t❤❡ ♠♦r❡ ❣❡♥❡r❛❧ ♦❢ t❤❡ t✇♦✱ ❜✉t t❤❡ ❢♦r♠❡r ✐s
♠♦r❡ ✐♥t✉✐t✐✈❡ ❬✷✸✾❪✱ ❬✸✷❪✳

✶✳✶✳✷✳✶ ❉❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❋♦✉r✐❡r ❚r❛♥s❢♦r♠
❚❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ✭❋❚✮ ♦❢ ❛ r❡❛❧✲✈❛❧✉❡❞ ✭♦r ❝♦♠♣❧❡①✮ ❢✉♥❝t✐♦♥ ♦❢ t❤❡
r❡❛❧✲✈❛r✐❛❜❧❡

t

✐s ❞❡✜♥❡❞ ❜②

∞

x (t) e−jωt dt

X (ω) =

✭✶✳✷✶✮

−∞
❣✐✈✐♥❣ ❛ ❝♦♠♣❧❡① ✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ r❡❛❧ ✈❛r✐❛❜❧❡

ω

r❡♣r❡s❡♥t✐♥❣

❢r❡q✉❡♥❝②✳ ❚❤❡ ✐♥✈❡rs❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ✭■❋❚✮ ✐s ❣✐✈❡♥ ❜②

x (t) =

1
2π

∞

X (ω) ejωt dω.
−∞

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

✭✶✳✷✷✮


❈❍❆P❚❊❘ ✶✳ ❙■●◆❆▲❙ ❆◆❉ ❙■●◆❆▲

P❘❖❈❊❙❙■◆● ❙❨❙❚❊▼❙

✶✹

❇❡❝❛✉s❡ ♦❢ t❤❡ ✐♥✜♥✐t❡ ❧✐♠✐ts ♦♥ ❜♦t❤ ✐♥t❡❣r❛❧s✱ t❤❡ q✉❡st✐♦♥ ♦❢ ❝♦♥✈❡r✲
❣❡♥❝❡ ✐s ✐♠♣♦rt❛♥t✳ ❚❤❡r❡ ❛r❡ ✉s❡❢✉❧ ♣r❛❝t✐❝❛❧ s✐❣♥❛❧s t❤❛t ❞♦ ♥♦t ❤❛✈❡
❋♦✉r✐❡r tr❛♥s❢♦r♠s ✐❢ ♦♥❧② ❝❧❛ss✐❝❛❧ ❢✉♥❝t✐♦♥s ❛r❡ ❛❧❧♦✇❡❞ ❜❡❝❛✉s❡ ♦❢ ♣r♦❜✲
❧❡♠s ✇✐t❤ ❝♦♥✈❡r❣❡♥❝❡✳ ❚❤❡ ✉s❡ ♦❢ ❞❡❧t❛ ❢✉♥❝t✐♦♥s ✭❞✐str✐❜✉t✐♦♥s✮ ✐♥ ❜♦t❤
t❤❡ t✐♠❡ ❛♥❞ ❢r❡q✉❡♥❝② ❞♦♠❛✐♥s ❛❧❧♦✇s ❛ ♠✉❝❤ ❧❛r❣❡r ❝❧❛ss ♦❢ s✐❣♥❛❧s t♦
❜❡ r❡♣r❡s❡♥t❡❞ ❬✷✸✾❪✳

✶✳✶✳✷✳✷ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❋♦✉r✐❡r ❚r❛♥s❢♦r♠
❚❤❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ❛r❡ s♦♠❡✇❤❛t ♣❛r❛❧❧❡❧ t♦ t❤♦s❡ ♦❢
t❤❡ ❋♦✉r✐❡r s❡r✐❡s ❛♥❞ ❛r❡ ✐♠♣♦rt❛♥t ✐♥ ❛♣♣❧②✐♥❣ ✐t t♦ s✐❣♥❛❧ ❛♥❛❧②s✐s ❛♥❞
✐♥t❡r♣r❡t✐♥❣ ✐t✳

❚❤❡ ♠❛✐♥ ♣r♦♣❡rt✐❡s ❛r❡ ❣✐✈❡♥ ❤❡r❡ ✉s✐♥❣ t❤❡ ♥♦t❛t✐♦♥

t❤❛t t❤❡ ❋❚ ♦❢ ❛ r❡❛❧ ✈❛❧✉❡❞ ❢✉♥❝t✐♦♥

x (t)

♦✈❡r ❛❧❧ t✐♠❡

t

✐s ❣✐✈❡♥ ❜②

F{x} = X (ω)✳
✶✳ ▲✐♥❡❛r✿

F{x + y} = F{x} + F{y}
x (t) = u (t) + jv (t)

✷✳ ❊✈❡♥ ❛♥❞ ❖❞❞♥❡ss✿ ✐❢

jB (ω)

❛♥❞

X (ω) = A (ω) +

t❤❡♥

u

v

A

B

|X|

θ

❡✈❡♥

✵

❡✈❡♥

✵

❡✈❡♥

✵

♦❞❞

✵

✵

♦❞❞

❡✈❡♥

✵

✵

❡✈❡♥

✵

❡✈❡♥

❡✈❡♥

π/2

✵

♦❞❞

♦❞❞

✵

❡✈❡♥

π/2

❚❛❜❧❡ ✶✳✷

✸✳ ❈♦♥✈♦❧✉t✐♦♥✿ ■❢ ❝♦♥t✐♥✉♦✉s ❝♦♥✈♦❧✉t✐♦♥ ✐s ❞❡✜♥❡❞ ❜②✿

y (t) = h (t) ∗ x (t) =
∞
h (λ) x (t − λ) dλ
−∞

∞
−∞

h (t − τ ) x (τ ) dτ =

F{h (t) ∗ x (t)} = F{h (t)}F{x (t)}
1

F{h (t) x (t)} = 2π
F{h (t)} ∗ F {x (t)}
∞
∞
2
2
1
P❛rs❡✈❛❧✿
|x
(t)
|
dt
=
2π −∞ |X (ω) | dω
−∞
−jωT
❙❤✐❢t✿ F{x (t − T )} = X (ω) e
j2πKt
▼♦❞✉❧❛t❡✿ F{x (t) e
} = X (ω − 2πK)
dx
❉❡r✐✈❛t✐✈❡✿ F{
}
=
jωX
(ω)
dt
1
X
(ω/a)

❙tr❡t❝❤✿ F{x (at)} =
|a|
∞
❖rt❤♦❣♦♥❛❧✐t②✿
e−jω1 t ejω2 t = 2πδ (ω1 − ω2 )
−∞

t❤❡♥

✹✳ ▼✉❧t✐♣❧✐❝❛t✐♦♥✿
✺✳
✻✳
✼✳
✽✳
✾✳
✶✵✳

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

✭✶✳✷✸✮


✶✺

✶✳✶✳✷✳✸ ❊①❛♠♣❧❡s ♦❢ t❤❡ ❋♦✉r✐❡r ❚r❛♥s❢♦r♠
❉❡r✐✈✐♥❣ ❛ ❢❡✇ ❜❛s✐❝ tr❛♥s❢♦r♠s ❛♥❞ ✉s✐♥❣ t❤❡ ♣r♦♣❡rt✐❡s ❛❧❧♦✇s ❛ ❧❛r❣❡
❝❧❛ss ♦❢ s✐❣♥❛❧s t♦ ❜❡ ❡❛s✐❧② st✉❞✐❡❞✳ ❊①❛♠♣❧❡s ♦❢ ♠♦❞✉❧❛t✐♦♥✱ s❛♠♣❧✐♥❣✱
❛♥❞ ♦t❤❡rs ✇✐❧❧ ❜❡ ❣✐✈❡♥✳

•
•
•

■❢

•

❖t❤❡r ✐♥t❡r❡st✐♥❣ ❛♥❞ ✐❧❧✉str❛t✐✈❡ ❡①❛♠♣❧❡s ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❬✷✸✾❪✱

x (t) = δ (t) t❤❡♥ X (ω) = 1
x (t) = 1 t❤❡♥ X (ω) = 2πδ (ω)
■❢ x (t) ✐s ❛♥ ✐♥✜♥✐t❡ s❡q✉❡♥❝❡ ♦❢ ❞❡❧t❛ ❢✉♥❝t✐♦♥s s♣❛❝❡❞ T ❛♣❛rt✱
∞
x (t) = n=−∞ δ (t − nT )✱ ✐ts tr❛♥s❢♦r♠ ✐s ❛❧s♦ ❛♥ ✐♥✜♥✐t❡ s❡q✉❡♥❝❡
♦❢ ❞❡❧t❛ ❢✉♥❝t✐♦♥s ♦❢ ✇❡✐❣❤t 2π/T s♣❛❝❡❞ 2π/T ❛♣❛rt✱ X (ω) =
∞
2π k=−∞ δ (ω − 2πk/T )✳

■❢

❬✸✷❪✳
◆♦t❡ t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ t❛❦❡s ❛ ❢✉♥❝t✐♦♥ ♦❢ ❝♦♥t✐♥✉♦✉s t✐♠❡ ✐♥t♦ ❛
❢✉♥❝t✐♦♥ ♦❢ ❝♦♥t✐♥✉♦✉s ❢r❡q✉❡♥❝②✱ ♥❡✐t❤❡r ❢✉♥❝t✐♦♥ ❜❡✐♥❣ ♣❡r✐♦❞✐❝✳ ■❢ ✏❞✐s✲
tr✐❜✉t✐♦♥✧ ♦r ✏❞❡❧t❛ ❢✉♥❝t✐♦♥s✧ ❛r❡ ❛❧❧♦✇❡❞✱ t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ♦❢ ❛
♣❡r✐♦❞✐❝ ❢✉♥❝t✐♦♥ ✇✐❧❧ ❜❡ ❛ ✐♥✜♥✐t❡❧② ❧♦♥❣ str✐♥❣ ♦❢ ❞❡❧t❛ ❢✉♥❝t✐♦♥s ✇✐t❤
✇❡✐❣❤ts t❤❛t ❛r❡ t❤❡ ❋♦✉r✐❡r s❡r✐❡s ❝♦❡✣❝✐❡♥ts✳

✶✳✶✳✸ ❚❤❡ ▲❛♣❧❛❝❡ ❚r❛♥s❢♦r♠
❚❤❡ ▲❛♣❧❛❝❡ tr❛♥s❢♦r♠ ❝❛♥ ❜❡ t❤♦✉❣❤t ♦❢ ❛s ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❋♦✉r✐❡r

tr❛♥s❢♦r♠ ✐♥ ♦r❞❡r t♦ ✐♥❝❧✉❞❡ ❛ ❧❛r❣❡r ❝❧❛ss ♦❢ ❢✉♥❝t✐♦♥s✱ t♦ ❛❧❧♦✇ t❤❡ ✉s❡ ♦❢
❝♦♠♣❧❡① ✈❛r✐❛❜❧❡ t❤❡♦r②✱ t♦ s♦❧✈❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✱ ❛♥❞
t♦ ❣✐✈❡ ❛ t♦♦❧ ❢♦r ✐♥♣✉t✲♦✉t♣✉t ❞❡s❝r✐♣t✐♦♥ ♦❢ ❧✐♥❡❛r s②st❡♠s✳ ■ts ✉s❡ ✐♥
s②st❡♠ ❛♥❞ s✐❣♥❛❧ ❛♥❛❧②s✐s ❜❡❝❛♠❡ ♣♦♣✉❧❛r ✐♥ t❤❡ ✶✾✺✵✬s ❛♥❞ r❡♠❛✐♥s ❛s
t❤❡ ❝❡♥tr❛❧ t♦♦❧ ❢♦r ♠✉❝❤ ♦❢ ❝♦♥t✐♥✉♦✉s t✐♠❡ s②st❡♠ t❤❡♦r②✳ ❚❤❡ q✉❡st✐♦♥
♦❢ ❝♦♥✈❡r❣❡♥❝❡ ❜❡❝♦♠❡s st✐❧❧ ♠♦r❡ ❝♦♠♣❧✐❝❛t❡❞ ❛♥❞ ❞❡♣❡♥❞s ♦♥ ❝♦♠♣❧❡①
✈❛❧✉❡s ♦❢

s

✉s❡❞ ✐♥ t❤❡ ✐♥✈❡rs❡ tr❛♥s❢♦r♠ ✇❤✐❝❤ ♠✉st ❜❡ ✐♥ ❛ ✏r❡❣✐♦♥ ♦❢

❝♦♥✈❡r❣❡♥❝❡✧ ✭❘❖❈✮✳

✶✳✶✳✸✳✶ ❉❡✜♥✐t✐♦♥ ♦❢ t❤❡ ▲❛♣❧❛❝❡ ❚r❛♥s❢♦r♠
❚❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ▲❛♣❧❛❝❡ tr❛♥s❢♦r♠ ✭▲❚✮ ♦❢ ❛ r❡❛❧ ✈❛❧✉❡❞ ❢✉♥❝t✐♦♥
❞❡✜♥❡❞ ♦✈❡r ❛❧❧ ♣♦s✐t✐✈❡ t✐♠❡

t

✐s

∞

f (t) e−st dt

F (s) =

✭✶✳✷✹✮

−∞
❛♥❞ t❤❡ ✐♥✈❡rs❡ tr❛♥s❢♦r♠ ✭■▲❚✮ ✐s ❣✐✈❡♥ ❜② t❤❡ ❝♦♠♣❧❡① ❝♦♥t♦✉r ✐♥t❡❣r❛❧

f (t) =

1
2πj

c+j∞

F (s) est ds
c−j∞

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

✭✶✳✷✺✮


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

✶✻

✇❤❡r❡

s = σ + jω

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

t❤❡ ■▲❚ ♠✉st ❜❡ ✐♥ t❤❡ r❡❣✐♦♥ ♦❢ t❤❡
✐♥t❡❣r❛❧ ❝♦♥✈❡r❣❡s✳

s ♣❧❛♥❡ ✇❤❡r❡ t❤❡ ▲❛♣❧❛❝❡ tr❛♥s❢♦r♠

❚❤✐s ❞❡✜♥✐t✐♦♥ ✐s ♦❢t❡♥ ❝❛❧❧❡❞ t❤❡ ❜✐❧❛t❡r❛❧ ▲❛♣❧❛❝❡

tr❛♥s❢♦r♠ t♦ ❞✐st✐♥❣✉✐s❤ ✐t ❢r♦♠ t❤❡ ✉♥✐❧❛t❡r❛❧ tr❛♥s❢♦r♠ ✭❯▲❚✮ ✇❤✐❝❤
✐s ❞❡✜♥❡❞ ✇✐t❤ ③❡r♦ ❛s t❤❡ ❧♦✇❡r ❧✐♠✐t ♦❢ t❤❡ ❢♦r✇❛r❞ tr❛♥s❢♦r♠ ✐♥t❡❣r❛❧
✭✶✳✷✹✮✳ ❯♥❧❡ss st❛t❡❞ ♦t❤❡r✇✐s❡✱ ✇❡ ✇✐❧❧ ❜❡ ✉s✐♥❣ t❤❡ ❜✐❧❛t❡r❛❧ tr❛♥s❢♦r♠✳
◆♦t✐❝❡ t❤❛t t❤❡ ▲❛♣❧❛❝❡ tr❛♥s❢♦r♠ ❜❡❝♦♠❡s t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ♦♥
t❤❡ ✐♠❛❣✐♥❛r② ❛①✐s✱ ❢♦r

s = jω ✳

■❢ t❤❡ ❘❖❈ ✐♥❝❧✉❞❡s t❤❡

jω

❛①✐s✱ t❤❡

❋♦✉r✐❡r tr❛♥s❢♦r♠ ❡①✐sts ❜✉t ✐❢ ✐t ❞♦❡s ♥♦t✱ ♦♥❧② t❤❡ ▲❛♣❧❛❝❡ tr❛♥s❢♦r♠ ♦❢
t❤❡ ❢✉♥❝t✐♦♥ ❡①✐sts✳
❚❤❡r❡ ✐s ❛ ❝♦♥s✐❞❡r❛❜❧❡ ❧✐t❡r❛t✉r❡ ♦♥ t❤❡ ▲❛♣❧❛❝❡ tr❛♥s❢♦r♠ ❛♥❞ ✐ts ✉s❡
✐♥ ❝♦♥t✐♥✉♦✉s✲t✐♠❡ s②st❡♠ t❤❡♦r②✳ ❲❡ ✇✐❧❧ ❞❡✈❡❧♦♣ ♠♦st ♦❢ t❤❡s❡ ✐❞❡❛s ❢♦r
t❤❡ ❞✐s❝r❡t❡✲t✐♠❡ s②st❡♠ ✐♥ t❡r♠s ♦❢ t❤❡ ③✲tr❛♥s❢♦r♠ ❧❛t❡r ✐♥ t❤✐s ❝❤❛♣t❡r
❛♥❞ ✇✐❧❧ ♦♥❧② ❜r✐❡✢② ❝♦♥s✐❞❡r ♦♥❧② t❤❡ ♠♦r❡ ✐♠♣♦rt❛♥t ♣r♦♣❡rt✐❡s ❤❡r❡✳
❚❤❡ ✉♥✐❧❛t❡r❛❧ ▲❛♣❧❛❝❡ tr❛♥s❢♦r♠ ❝❛♥♥♦t ❜❡ ✉s❡❞ ✐❢ ✉s❡❢✉❧ ♣❛rts ♦❢ t❤❡
s✐❣♥❛❧ ❡①✐sts ❢♦r ♥❡❣❛t✐✈❡ t✐♠❡✳ ■t ❞♦❡s ♥♦t r❡❞✉❝❡ t♦ t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠
❢♦r s✐❣♥❛❧s t❤❛t ❡①✐st ❢♦r ♥❡❣❛t✐✈❡ t✐♠❡✱ ❜✉t ✐❢ t❤❡ ♥❡❣❛t✐✈❡ t✐♠❡ ♣❛rt ♦❢ ❛

s✐❣♥❛❧ ❝❛♥ ❜❡ ♥❡❣❧❡❝t❡❞✱ t❤❡ ✉♥✐❧❛t❡r❛❧ tr❛♥s❢♦r♠ ✇✐❧❧ ❝♦♥✈❡r❣❡ ❢♦r ❛ ♠✉❝❤
❧❛r❣❡r ❝❧❛ss ♦❢ ❢✉♥❝t✐♦♥ t❤❛t t❤❡ ❜✐❧❛t❡r❛❧ tr❛♥s❢♦r♠ ✇✐❧❧✳ ■t ❛❧s♦ ♠❛❦❡s t❤❡
s♦❧✉t✐♦♥ ♦❢ ❧✐♥❡❛r✱ ❝♦♥st❛♥t ❝♦❡✣❝✐❡♥t ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✇✐t❤ ✐♥✐t✐❛❧
❝♦♥❞✐t✐♦♥s ♠✉❝❤ ❡❛s✐❡r✳

✶✳✶✳✸✳✷ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ▲❛♣❧❛❝❡ ❚r❛♥s❢♦r♠
▼❛♥② ♦❢ t❤❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ▲❛♣❧❛❝❡ tr❛♥s❢♦r♠ ❛r❡ s✐♠✐❧❛r t♦ t❤♦s❡ ❢♦r
❋♦✉r✐❡r tr❛♥s❢♦r♠ ❬✸✷❪✱ ❬✷✸✾❪✱ ❤♦✇❡✈❡r✱ t❤❡ ❜❛s✐s ❢✉♥❝t✐♦♥s ❢♦r t❤❡ ▲❛♣❧❛❝❡
tr❛♥s❢♦r♠ ❛r❡ ♥♦t ♦rt❤♦❣♦♥❛❧✳ ❙♦♠❡ ♦❢ t❤❡ ♠♦r❡ ✐♠♣♦rt❛♥t ♦♥❡s ❛r❡✿

L{x + y} = L{x} + L{y}
y (t) = h (t) ∗ x (t) = h (t − τ ) x (τ ) dτ
t❤❡♥ L{h (t) ∗ x (t)} = L{h (t)} L{x (t)}
dx
❉❡r✐✈❛t✐✈❡✿ L{
dt } = sL{x (t)}
dx
❉❡r✐✈❛t✐✈❡ ✭❯▲❚✮✿ L{
dt } = sL{x (t)} − x (0)
1
■♥t❡❣r❛❧✿ L{ x (t) dt} = L{x (t)}
s
−T s
❙❤✐❢t✿ L{x (t − T )} = C (k) e
jω t
▼♦❞✉❧❛t❡✿ L{x (t) e 0 } = X (s − jω0 )

✶✳ ▲✐♥❡❛r✿

✷✳ ❈♦♥✈♦❧✉t✐♦♥✿ ■❢

✸✳
✹✳
✺✳
✻✳
✼✳

❊①❛♠♣❧❡s ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❬✷✸✾❪✱ ❬✸✷❪ ❛♥❞ ❛r❡ s✐♠✐❧❛r t♦ t❤♦s❡ ♦❢ t❤❡ ③✲
tr❛♥s❢♦r♠ ♣r❡s❡♥t❡❞ ❧❛t❡r ✐♥ t❤❡s❡ ♥♦t❡s✳ ■♥❞❡❡❞✱ ♥♦t❡ t❤❡ ♣❛r❛❧❧❛❧s ❛♥❞
❞✐✛❡r❡♥❝❡s ✐♥ t❤❡ ❋♦✉r✐❡r s❡r✐❡s✱ ❋♦✉r✐❡r tr❛♥s❢♦r♠✱ ❛♥❞ ❩✲tr❛♥s❢♦r♠✳

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


✶✼

✶✳✷ ❉✐s❝r❡t❡✲❚✐♠❡ ❙✐❣♥❛❧s✷
❆❧t❤♦✉❣❤ t❤❡ ❞✐s❝r❡t❡✲t✐♠❡ s✐❣♥❛❧

x (n)

❝♦✉❧❞ ❜❡ ❛♥② ♦r❞❡r❡❞ s❡q✉❡♥❝❡

♦❢ ♥✉♠❜❡rs✱ t❤❡② ❛r❡ ✉s✉❛❧❧② s❛♠♣❧❡s ♦❢ ❛ ❝♦♥t✐♥✉♦✉s✲t✐♠❡ s✐❣♥❛❧✳
t❤✐s ❝❛s❡✱ t❤❡ r❡❛❧ ♦r ✐♠❛❣✐♥❛r② ✈❛❧✉❡❞ ♠❛t❤❡♠❛t✐❝❛❧ ❢✉♥❝t✐♦♥
✐♥t❡❣❡r

■♥

x (n) ♦❢ t❤❡

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

s❡♥t❛t✐♦♥ ♦❢ ✐t ✭s✉❝❤ ❛s s❛♠♣❧❡s✮✳ ■♥ s♦♠❡ ❝❛s❡s✱ t❤❡ t❡r♠

❞✐❣✐t❛❧ s✐❣♥❛❧ ✐s

✉s❡❞ ✐♥t❡r❝❤❛♥❣❡❛❜❧② ✇✐t❤ ❞✐s❝r❡t❡✲t✐♠❡ s✐❣♥❛❧✱ ♦r t❤❡ ❧❛❜❡❧ ❞✐❣✐t❛❧ s✐❣♥❛❧
♠❛② ❜❡ ✉s❡ ✐❢ t❤❡ ❢✉♥❝t✐♦♥ ✐s ♥♦t r❡❛❧ ✈❛❧✉❡❞ ❜✉t t❛❦❡s ✈❛❧✉❡s ❝♦♥s✐st❡♥t
✇✐t❤ s♦♠❡ ❤❛r❞✇❛r❡ s②st❡♠✳
■♥❞❡❡❞✱ ♦✉r ✈❡r② ✉s❡ ♦❢ t❤❡ t❡r♠ ✏❞✐s❝r❡t❡✲t✐♠❡✧ ✐♥❞✐❝❛t❡s t❤❡ ♣r♦❜✲
❛❜❧❡ ♦r✐❣✐♥ ♦❢ t❤❡ s✐❣♥❛❧s ✇❤❡♥✱ ✐♥ ❢❛❝t✱ t❤❡ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡ ❝♦✉❧❞
❜❡ ❧❡♥❣t❤ ♦r ❛♥② ♦t❤❡r ✈❛r✐❛❜❧❡ ♦r s✐♠♣❧② ❛♥ ♦r❞❡r✐♥❣ ✐♥❞❡①✳ ❚❤❡ t❡r♠
✏❞✐❣✐t❛❧✧ ✐♥❞✐❝❛t❡s t❤❡ s✐❣♥❛❧ ✐s ♣r♦❜❛❜❧② ❣♦✐♥❣ t♦ ❜❡ ❝r❡❛t❡❞✱ ♣r♦❝❡ss❡❞✱ ♦r
st♦r❡❞ ✉s✐♥❣ ❞✐❣✐t❛❧ ❤❛r❞✇❛r❡✳ ❆s ✐♥ t❤❡ ❝♦♥t✐♥✉♦✉s✲t✐♠❡ ❝❛s❡✱ t❤❡ ❋♦✉r✐❡r
tr❛♥s❢♦r♠ ✇✐❧❧ ❛❣❛✐♥ ❜❡ ♦✉r ♣r✐♠❛r② t♦♦❧ ❬✷✷✼❪✱ ❬✷✹✵❪✱ ❬✸✸❪✳
◆♦t❛t✐♦♥ ❤❛s ❜❡❡♥ ❛♥ ✐♠♣♦rt❛♥t ❡❧❡♠❡♥t ✐♥ ♠❛t❤❡♠❛t✐❝s✳

■♥ s♦♠❡

❝❛s❡s✱ ❞✐s❝r❡t❡✲t✐♠❡ s✐❣♥❛❧s ❛r❡ ❜❡st ❞❡♥♦t❡❞ ❛s ❛ s❡q✉❡♥❝❡ ♦❢ ✈❛❧✉❡s✱ ✐♥
♦t❤❡r ❝❛s❡s✱ ❛ ✈❡❝t♦r ✐s ❝r❡❛t❡❞ ✇✐t❤ ❡❧❡♠❡♥ts ✇❤✐❝❤ ❛r❡ t❤❡ s❡q✉❡♥❝❡
✈❛❧✉❡s✳

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

✈❛❧✉❡s ❛s ❝♦❡✣❝✐❡♥ts ❢♦r ❛ ❝♦♠♣❧❡① ✈❛r✐❛❜❧❡✳ ❚❤❡ ✈❡❝t♦r ❢♦r♠✉❧❛t✐♦♥ ❛❧❧♦✇s
t❤❡ ✉s❡ ♦❢ ❧✐♥❡❛r ❛❧❣❡❜r❛ ❛♥❞ t❤❡ ♣♦❧②♥♦♠✐❛❧ ❢♦r♠✉❧❛t✐♦♥ ❛❧❧♦✇s t❤❡ ✉s❡
♦❢ ❝♦♠♣❧❡① ✈❛r✐❛❜❧❡ t❤❡♦r②✳

✶✳✷✳✶ ❚❤❡ ❉✐s❝r❡t❡ ❋♦✉r✐❡r ❚r❛♥s❢♦r♠
❚❤❡ ❞❡s❝r✐♣t✐♦♥ ♦❢ s✐❣♥❛❧s ✐♥ t❡r♠s ♦❢ t❤❡✐r s✐♥✉s♦✐❞❛❧ ❢r❡q✉❡♥❝② ❝♦♥t❡♥t
❤❛s ♣r♦✈❡♥ t♦ ❜❡ ❛s ♣♦✇❡r❢✉❧ ❛♥❞ ✐♥❢♦r♠❛t✐✈❡ ❢♦r ❞✐s❝r❡t❡✲t✐♠❡ s✐❣♥❛❧s ❛s ✐t
❤❛s ❢♦r ❝♦♥t✐♥✉♦✉s✲t✐♠❡ s✐❣♥❛❧s✳ ■t ✐s ❛❧s♦ ♣r♦❜❛❜❧② t❤❡ ♠♦st ♣♦✇❡r❢✉❧ ❝♦♠✲
♣✉t❛t✐♦♥❛❧ t♦♦❧ ✇❡ ✇✐❧❧ ✉s❡✳ ❲❡ ♥♦✇ ❞❡✈❡❧♦♣ t❤❡ ❜❛s✐❝ ❞✐s❝r❡t❡✲t✐♠❡ ♠❡t❤✲
♦❞s st❛rt✐♥❣ ✇✐t❤ t❤❡ ❞✐s❝r❡t❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ✭❉❋❚✮ ❛♣♣❧✐❡❞ t♦ ✜♥✐t❡
❧❡♥❣t❤ s✐❣♥❛❧s✱ ❢♦❧❧♦✇❡❞ ❜② t❤❡ ❞✐s❝r❡t❡✲t✐♠❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ✭❉❚❋❚✮
❢♦r ✐♥✜♥✐t❡❧② ❧♦♥❣ s✐❣♥❛❧s✱ ❛♥❞ ❡♥❞✐♥❣ ✇✐t❤ t❤❡ ③✲tr❛♥s❢♦r♠ ✇❤✐❝❤ ✉s❡s t❤❡
♣♦✇❡r❢✉❧ t♦♦❧s ♦❢ ❝♦♠♣❧❡① ✈❛r✐❛❜❧❡ t❤❡♦r②✳

✶✳✷✳✶✳✶ ❉❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❉❋❚
■t ✐s ❛ss✉♠❡❞ t❤❛t t❤❡ s✐❣♥❛❧

x (n)

❉❋❚ ♦❢

✷ ❚❤✐s

x (n)✱

N
n✳

t♦ ❜❡ ❛♥❛❧②③❡❞ ✐s ❛ s❡q✉❡♥❝❡ ♦❢

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

x (n)✱

✐s ❛ ❧❡♥❣t❤

N

s❡q✉❡♥❝❡ ♦❢

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

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

r❡❛❧
❚❤❡


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

✶✽

❝♦♠♣❧❡① ♥✉♠❜❡rs ❞❡♥♦t❡❞

C (k)

❛♥❞ ❞❡✜♥❡❞ ❜②

N −1
2π

x (n) e−j N nk

C (k) =

✭✶✳✷✻✮

n=0

√
j = −1✳ ❚❤❡
C (k) ✐s ❣✐✈❡♥ ❜②

✉s✐♥❣ t❤❡ ✉s✉❛❧ ❡♥❣✐♥❡❡r✐♥❣ ♥♦t❛t✐♦♥✿
✭■❉❋❚✮ ✇❤✐❝❤ r❡tr✐❡✈❡s

x (n)

❢r♦♠

x (n) =

1
N

✐♥✈❡rs❡ tr❛♥s❢♦r♠

N −1
2π

C (k) ej N nk

✭✶✳✷✼✮

k=0

✇❤✐❝❤ ✐s ❡❛s✐❧② ✈❡r✐✜❡❞ ❜② s✉❜st✐t✉t✐♦♥ ✐♥t♦ ✭✶✳✷✻✮✳ ■♥❞❡❡❞✱ t❤✐s ✈❡r✐✜❝❛✲
t✐♦♥ ✇✐❧❧ r❡q✉✐r❡ ✉s✐♥❣ t❤❡ ♦rt❤♦❣♦♥❛❧✐t② ♦❢ t❤❡ ❜❛s✐s ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❉❋❚
✇❤✐❝❤ ✐s

N −1
2π

2π

e−j N mk ej N nk = {

N

✐❢

n=m

0

✐❢

n = m.

k=0
❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ❜❛s✐s ❢✉♥❝t✐♦♥s✱

✈❛❧✉❡s ♦❢ t❤❡

2π

e−j N k ✱

❢♦r

✭✶✳✷✽✮

k ∈ {0, N − 1}✱

❛r❡ t❤❡

N

N

N t❤ r♦♦ts ♦❢ ✉♥✐t② ✭t❤❡ ◆ ③❡r♦s ♦❢ t❤❡ ♣♦❧②♥♦♠✐❛❧ (s − 1)

✮✳

❚❤✐s ♣r♦♣❡rt② ✐s ✇❤❛t ❝♦♥♥❡❝ts t❤❡ ❉❋❚ t♦ ❝♦♥✈♦❧✉t✐♦♥ ❛♥❞ ❛❧❧♦✇s ❡✣❝✐❡♥t
❛❧❣♦r✐t❤♠s ❢♦r ❝❛❧❝✉❧❛t✐♦♥ t♦ ❜❡ ❞❡✈❡❧♦♣❡❞ ❬✺✾❪✳ ❚❤❡② ❛r❡ ✉s❡❞ s♦ ♦❢t❡♥
t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ♥♦t❛t✐♦♥ ✐s ❞❡✜♥❡❞ ❜②
2π

WN = e−j N

✭✶✳✷✾✮

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

N −1

x (n) WNnk

C (k) =

✭✶✳✸✵✮

n=0
❖♥❡ s❤♦✉❧❞ ♥♦t✐❝❡ t❤❛t ✇✐t❤ t❤❡ ✜♥✐t❡ s✉♠♠❛t✐♦♥ ♦❢ t❤❡ ❉❋❚✱ t❤❡r❡ ✐s
♥♦ q✉❡st✐♦♥ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦r ♦❢ t❤❡ ❛❜✐❧✐t② t♦ ✐♥t❡r❝❤❛♥❣❡ t❤❡ ♦r❞❡r ♦❢
s✉♠♠❛t✐♦♥✳ ◆♦ ✏❞❡❧t❛ ❢✉♥❝t✐♦♥s✑ ❛r❡ ♥❡❡❞❡❞ ❛♥❞ t❤❡

N

tr❛♥s❢♦r♠ ✈❛❧✉❡s

❝❛♥ ❜❡ ❝❛❧❝✉❧❛t❡❞ ❡①❛❝t❧② ✭✇✐t❤✐♥ t❤❡ ❛❝❝✉r❛❝② ♦❢ t❤❡ ❝♦♠♣✉t❡r ♦r ❝❛❧❝✉✲
❧❛t♦r ✉s❡❞✮ ❢r♦♠ t❤❡

N

s✐❣♥❛❧ ✈❛❧✉❡s ✇✐t❤ ❛ ✜♥✐t❡ ♥✉♠❜❡r ♦❢ ❛r✐t❤♠❡t✐❝

♦♣❡r❛t✐♦♥s✳

✶✳✷✳✶✳✷ ▼❛tr✐① ❋♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ❉❋❚
❚❤❡r❡ ❛r❡ s❡✈❡r❛❧ ❛❞✈❛♥t❛❣❡s t♦ ✉s✐♥❣ ❛ ♠❛tr✐① ❢♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ❉❋❚✳
❚❤✐s ✐s ❣✐✈❡♥ ❜② ✇r✐t✐♥❣ ✭✶✳✷✻✮ ♦r ✭✶✳✸✵✮ ✐♥ ♠❛tr✐① ♦♣❡r❛t♦r ❢♦r♠ ❛s

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


✶✾












C0
C1
C2
✳
✳
✳

CN −1





W0 W0 W0 ···




 W0 W1 W2




 =  W0 W2 W4



 ✳✳

 ✳


W0
···

W0

✳
✳

✳

W (N −1)(N −1)












x0





✭✶✳✸✶✮
x2 


✳
✳

✳


xN −1
x1

♦r

C = Fx.

✭✶✳✸✷✮

❚❤❡ ♦rt❤♦❣♦♥❛❧✐t② ♦❢ t❤❡ ❜❛s✐s ❢✉♥❝t✐♦♥ ✐♥ ✭✶✳✷✻✮ s❤♦✇s ✉♣ ✐♥ t❤✐s ♠❛tr✐①

F ❜❡✐♥❣ ♦rt❤♦❣♦♥❛❧ t♦ ❡❛❝❤ ♦t❤❡r ❛s ❛r❡
FT F = kI✱ ✇❤❡r❡ k ✐s ❛ s❝❛❧❛r ❝♦♥st❛♥t✱ ❛♥❞✱

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

FT = kF−1 ✳

❚❤✐s ✐s ❝❛❧❧❡❞ ❛ ✉♥✐t❛r② ♦♣❡r❛t♦r✳

❚❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❉❋❚ ✐♥ ✭✶✳✷✻✮ ❡♠♣❤❛s✐③❡s t❤❡ ❢❛❝t t❤❛t ❡❛❝❤ ♦❢
t❤❡

N

❉❋❚ ✈❛❧✉❡s ❛r❡ t❤❡ s✉♠ ♦❢

N

♣r♦❞✉❝ts✳ ❚❤❡ ♠❛tr✐① ❢♦r♠✉❧❛t✐♦♥ ✐♥

k ✲t❤ ❉❋❚ t❡r♠ ✐s t❤❡ ✐♥♥❡r ♣r♦❞✉❝t
F ❛♥❞ x❀ ♦r✱ t❤❡ ❉❋❚ ✈❡❝t♦r✱ C ✐s ❛ ✇❡✐❣❤t❡❞
F ✇✐t❤ ✇❡✐❣❤ts ❜❡✐♥❣ t❤❡ ❡❧❡♠❡♥ts ♦❢ t❤❡ s✐❣♥❛❧

✭✶✳✸✶✮ ❤❛s t✇♦ ✐♥t❡r♣r❡t❛t✐♦♥s✳ ❊❛❝❤
♦❢ t✇♦ ✈❡❝t♦rs✱
s✉♠ ♦❢ t❤❡
✈❡❝t♦r

x✳

N

k ✲t❤

r♦✇ ♦❢

❝♦❧✉♠♥s ♦❢

❆ t❤✐r❞ ✈✐❡✇ ♦❢ t❤❡ ❉❋❚ ✐s t❤❡ ♦♣❡r❛t♦r ✈✐❡✇ ✇❤✐❝❤ ✐s s✐♠♣❧②

t❤❡ s✐♥❣❧❡ ♠❛tr✐① ❡q✉❛t✐♦♥ ✭✶✳✸✷✮✳
■t ✐s ✐♥str✉❝t✐✈❡ ❛t t❤✐s ♣♦✐♥t t♦ ✇r✐t❡ ❛ ❝♦♠♣✉t❡r ♣r♦❣r❛♠ t♦ ❝❛❧❝✉❧❛t❡
t❤❡ ❉❋❚ ♦❢ ❛ s✐❣♥❛❧✳ ■♥ ▼❛t❧❛❜ ❬✷✶✼❪✱ t❤❡r❡ ✐s ❛ ♣r❡✲♣r♦❣r❛♠♠❡❞ ❢✉♥❝t✐♦♥
t♦ ❝❛❧❝✉❧❛t❡ t❤❡ ❉❋❚✱ ❜✉t t❤❛t ❤✐❞❡s t❤❡ s❝❛❧❛r ♦♣❡r❛t✐♦♥s✳ ❖♥❡ s❤♦✉❧❞
♣r♦❣r❛♠ t❤❡ tr❛♥s❢♦r♠ ✐♥ t❤❡ s❝❛❧❛r ✐♥t❡r♣r❡t✐✈❡ ❧❛♥❣✉❛❣❡ ♦❢ ▼❛t❧❛❜ ♦r
s♦♠❡ ♦t❤❡r ❧♦✇❡r ❧❡✈❡❧ ❧❛♥❣✉❛❣❡ s✉❝❤ ❛s ❋❖❘❚❘❆◆✱ ❈✱ ❇❆❙■❈✱ P❛s✲

❝❛❧✱ ❡t❝✳ ❚❤✐s ✇✐❧❧ ✐❧❧✉str❛t❡ ❤♦✇ ♠❛♥② ♠✉❧t✐♣❧✐❝❛t✐♦♥s ❛♥❞ ❛❞❞✐t✐♦♥s ❛♥❞
tr✐❣♦♥♦♠❡tr✐❝ ❡✈❛❧✉❛t✐♦♥s ❛r❡ r❡q✉✐r❡❞ ❛♥❞ ❤♦✇ ♠✉❝❤ ♠❡♠♦r② ✐s ♥❡❡❞❡❞✳
❉♦ ♥♦t ✉s❡ ❛ ❝♦♠♣❧❡① ❞❛t❛ t②♣❡ ✇❤✐❝❤ ❛❧s♦ ❤✐❞❡s ❛r✐t❤♠❡t✐❝✱ ❜✉t ✉s❡
❊✉❧❡r✬s r❡❧❛t✐♦♥s

ejx = cos (x) + jsin (x)
t♦ ❡①♣❧✐❝✐t❧② ❝❛❧❝✉❧❛t❡ t❤❡ r❡❛❧ ❛♥❞ ✐♠❛❣✐♥❛r② ♣❛rt ♦❢

✭✶✳✸✸✮

C (k)✳

■❢ ▼❛t❧❛❜ ✐s ❛✈❛✐❧❛❜❧❡✱ ✜rst ♣r♦❣r❛♠ t❤❡ ❉❋❚ ✉s✐♥❣ ♦♥❧② s❝❛❧❛r ♦♣❡r❛✲
t✐♦♥s✳ ■t ✇✐❧❧ r❡q✉✐r❡ t✇♦ ♥❡st❡❞ ❧♦♦♣s ❛♥❞ ✇✐❧❧ r✉♥ r❛t❤❡r s❧♦✇❧② ❜❡❝❛✉s❡
t❤❡ ❡①❡❝✉t✐♦♥ ♦❢ ❧♦♦♣s ✐s ✐♥t❡r♣r❡t❡❞✳ ◆❡①t✱ ♣r♦❣r❛♠ ✐t ✉s✐♥❣ ✈❡❝t♦r ✐♥♥❡r
♣r♦❞✉❝ts t♦ ❝❛❧❝✉❧❛t❡ ❡❛❝❤

C (k) ✇❤✐❝❤ ✇✐❧❧ r❡q✉✐r❡ ♦♥❧② ♦♥❡ ❧♦♦♣ ❛♥❞ ✇✐❧❧

r✉♥ ❢❛st❡r✳ ❋✐♥❛❧❧②✱ ♣r♦❣r❛♠ ✐t ✉s✐♥❣ ❛ s✐♥❣❧❡ ♠❛tr✐① ♠✉❧t✐♣❧✐❝❛t✐♦♥ r❡q✉✐r✲
✐♥❣ ♥♦ ❧♦♦♣s ❛♥❞ r✉♥♥✐♥❣ ♠✉❝❤ ❢❛st❡r✳ ❈❤❡❝❦ t❤❡ ♠❡♠♦r② r❡q✉✐r❡♠❡♥ts
❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s
❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✵✺✾✽✴✶✳✻❃

