Continuous-Time Fourier Transform - SJTU

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Continuous-Time Fourier Transform

Hongkai Xiong ็†Š็บขๅ‡ฏ

http://min.sjtu.edu.cn

Department of Electronic Engineering Shanghai Jiao Tong University

http://ivm.sjtu.edu.cn/

๐‘ฅ๐‘ฅ๐‘˜๐‘˜[๐‘›๐‘›] โ†’ ๐‘ฆ๐‘ฆ๐‘˜๐‘˜[๐‘›๐‘›] โŸน ๏ฟฝ ๐‘Ž๐‘Ž๐‘˜๐‘˜๐‘ฅ๐‘ฅ๐‘˜๐‘˜[๐‘›๐‘›]๐‘˜๐‘˜ โ†’ โˆ‘ ๐‘Ž๐‘Ž๐‘˜๐‘˜๐‘˜๐‘˜ ๐‘ฆ๐‘ฆ๐‘˜๐‘˜[๐‘›๐‘›]

โ€ข If we can find a set of โ€œbasicโ€ signals, such that โ–ซ a rich class of signals can be represented as linear

combinations of these basic (building block) signals. โ–ซ the response of LTI Systems to these basic signals are both

simple and insightful

โ€ข Candidate sets of โ€œbasicโ€ signals โ–ซ Unit impulse function and its delays: ๐›ฟ๐›ฟ(๐‘ก๐‘ก)/๐›ฟ๐›ฟ[๐‘›๐‘›] โ–ซ Complex exponential/sinusoid signals: ๐‘’๐‘’๐‘ ๐‘ ๐‘ก๐‘ก , ๐‘’๐‘’๐‘—๐‘—๐‘—๐‘—๐‘—/๐‘ง๐‘ง๐‘›๐‘›, ๐‘’๐‘’๐‘—๐‘—๐‘—๐‘—๐‘›๐‘›

LTI Systems

2

Fourier Series of CT Periodic Signals โ€ข Recall a periodic CT signal ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก โ–ซ with fundamental period ๐‘‡๐‘‡, and โ–ซ fundamental frequency ๐œ”๐œ”0 = 2๐œ‹๐œ‹/๐‘‡๐‘‡

โ€ข Fourier series represent ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก in terms of harmonically related complex exponentials โ–ซ Synthesis equation

๐‘ฅ๐‘ฅ ๐‘ก๐‘ก = ๏ฟฝ ๐‘Ž๐‘Ž๐‘˜๐‘˜๐‘’๐‘’๐‘—๐‘—๐‘˜๐‘˜๐‘—๐‘—0๐‘ก๐‘กโˆž

๐‘˜๐‘˜=โˆ’โˆž

= ๏ฟฝ ๐‘Ž๐‘Ž๐‘˜๐‘˜๐‘’๐‘’๐‘—๐‘—๐‘˜๐‘˜(2๐œ‹๐œ‹/๐‘‡๐‘‡)๐‘ก๐‘กโˆž

๐‘˜๐‘˜=โˆ’โˆž

โ–ซ Analysis equation

๐‘Ž๐‘Ž๐‘˜๐‘˜ =1๐‘‡๐‘‡๏ฟฝ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก ๐‘’๐‘’โˆ’๐‘—๐‘—๐‘˜๐‘˜๐‘—๐‘—0๐‘ก๐‘ก ๐‘‘๐‘‘๐‘ก๐‘ก๐‘‡๐‘‡

=1๐‘‡๐‘‡๏ฟฝ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก ๐‘’๐‘’โˆ’๐‘—๐‘—๐‘˜๐‘˜(2๐œ‹๐œ‹/๐‘‡๐‘‡)๐‘ก๐‘ก ๐‘‘๐‘‘๐‘ก๐‘ก๐‘‡๐‘‡

3

โ€ข In one period, ๐‘ฅ๐‘ฅ๐‘‡๐‘‡ ๐‘ก๐‘ก = ๏ฟฝ1, ๐‘ก๐‘ก < ๐‘‡๐‘‡1 0, ๐‘‡๐‘‡1 < ๐‘ก๐‘ก < ๐‘‡๐‘‡/2

โ€ข Fourier coefficients

๐‘Ž๐‘Ž0=1๐‘‡๐‘‡๏ฟฝ 1๐‘‡๐‘‡1

โˆ’๐‘‡๐‘‡1๐‘‘๐‘‘๐‘ก๐‘ก =

2๐‘‡๐‘‡1๐‘‡๐‘‡

, ๐‘Ž๐‘Ž๐‘˜๐‘˜ =1๐‘‡๐‘‡๏ฟฝ ๐‘’๐‘’โˆ’๐‘—๐‘—๐‘˜๐‘˜๐‘—๐‘—0๐‘ก๐‘ก๐‘‘๐‘‘๐‘ก๐‘ก๐‘‡๐‘‡1

โˆ’๐‘‡๐‘‡1=

2sin (๐‘˜๐‘˜๐œ”๐œ”0๐‘‡๐‘‡1)๐‘˜๐‘˜๐œ”๐œ”0๐‘‡๐‘‡

, ๐‘˜๐‘˜ โ‰  0

โ€ข Frequency component at ๐œ”๐œ”๐‘˜๐‘˜ = ๐‘˜๐‘˜๐œ”๐œ”0 satisfies

๐‘‡๐‘‡๐‘Ž๐‘Ž๐‘˜๐‘˜ =2 sin ๐œ”๐œ”๐‘‡๐‘‡1

๐œ”๐œ”๏ฟฝ๐‘—๐‘—=๐‘˜๐‘˜๐‘—๐‘—0

๐‘‡๐‘‡๐‘Ž๐‘Ž๐‘˜๐‘˜ thought as sampled value of envelope ๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” โ‰œ 2 sin(๐œ”๐œ”๐‘‡๐‘‡1)/๐œ”๐œ” at ๐œ”๐œ”๐‘˜๐‘˜ = ๐‘˜๐‘˜๐œ”๐œ”0

Motivating Example: Periodic Square Wave

4

โ€ข ๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ”๐‘˜๐‘˜ โ‰œ 2 sin(๐œ”๐œ”๐‘˜๐‘˜๐‘‡๐‘‡1)/๐œ”๐œ”๐‘˜๐‘˜ for fixed ๐‘‡๐‘‡1 and increasing ๐‘‡๐‘‡

โ€ข Observation: as ๐‘‡๐‘‡ โ†’ โˆž, discrete frequencies sampled more densely

Motivating Example: Periodic Square Wave

5

โ€ข As ๐‘ป๐‘ป โ†’ โˆž, ๐’™๐’™๐‘ป๐‘ป ๐’•๐’• โ†’ ๐’™๐’™ ๐’•๐’• โ‰œ ๐’–๐’– ๐’•๐’• + ๐‘ป๐‘ป๐Ÿ๐Ÿ๐Ÿ๐Ÿ

โˆ’ ๐’–๐’–(๐’•๐’• โˆ’ ๐‘ป๐‘ป๐Ÿ๐Ÿ๐Ÿ๐Ÿ

), a rectangular impulse

๐‘ฅ๐‘ฅ๐‘‡๐‘‡ ๐‘ก๐‘ก = ๏ฟฝ ๐‘Ž๐‘Ž๐‘˜๐‘˜๐‘’๐‘’๐‘—๐‘—๐‘˜๐‘˜๐‘—๐‘—0๐‘ก๐‘กโˆž

๐‘˜๐‘˜=โˆ’โˆž

=1๐‘‡๐‘‡๏ฟฝ ๐‘‹๐‘‹(๐‘—๐‘—๐‘˜๐‘˜๐œ”๐œ”0)๐‘’๐‘’๐‘—๐‘—๐‘˜๐‘˜๐‘—๐‘—0๐‘ก๐‘กโˆž

๐‘˜๐‘˜=โˆ’โˆž

=12๐œ‹๐œ‹

๏ฟฝ ๐‘‹๐‘‹ ๐‘—๐‘—๐‘˜๐‘˜๐œ”๐œ”0 ๐‘’๐‘’๐‘—๐‘—๐‘˜๐‘˜๐‘—๐‘—0๐‘ก๐‘ก๐œ”๐œ”0

โˆž

๐‘˜๐‘˜=โˆ’โˆž

โŸน lim๐‘‡๐‘‡โ†’โˆž

๐‘ฅ๐‘ฅ๐‘‡๐‘‡ ๐‘ก๐‘ก =12๐œ‹๐œ‹

๏ฟฝ ๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” ๐‘’๐‘’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก๐‘‘๐‘‘๐œ”๐œ”โˆž

โˆ’โˆž

โ€ข Thus

๐‘ฅ๐‘ฅ(๐‘ก๐‘ก) =12๐œ‹๐œ‹

๏ฟฝ ๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” ๐‘’๐‘’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก๐‘‘๐‘‘๐œ”๐œ”โˆž

โˆ’โˆž

Motivating Example: Periodic Square Wave

6

โ€ข For envelope ๐‘ฟ๐‘ฟ(๐’‹๐’‹๐’‹๐’‹)

๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ”๐‘˜๐‘˜ = ๐‘‡๐‘‡๐‘Ž๐‘Ž๐‘˜๐‘˜

= ๏ฟฝ ๐‘ฅ๐‘ฅ๐‘‡๐‘‡(๐‘ก๐‘ก)๐‘’๐‘’โˆ’๐‘—๐‘—๐‘˜๐‘˜๐‘—๐‘—๐‘ก๐‘ก๐‘‘๐‘‘๐‘ก๐‘ก๐‘‡๐‘‡2

โˆ’๐‘‡๐‘‡2

= ๏ฟฝ ๐‘ฅ๐‘ฅ(๐‘ก๐‘ก)๐‘’๐‘’โˆ’๐‘—๐‘—๐‘—๐‘—๐‘˜๐‘˜๐‘ก๐‘ก๐‘‘๐‘‘๐‘ก๐‘ก๐‘‡๐‘‡2

โˆ’๐‘‡๐‘‡2

= ๏ฟฝ ๐‘ฅ๐‘ฅ(๐‘ก๐‘ก)๐‘’๐‘’โˆ’๐‘—๐‘—๐‘—๐‘—๐‘˜๐‘˜๐‘ก๐‘ก๐‘‘๐‘‘๐‘ก๐‘กโˆž

โˆ’โˆž

โ€ข Thus

๐‘‹๐‘‹(๐‘—๐‘—๐œ”๐œ”) = ๏ฟฝ ๐‘ฅ๐‘ฅ(๐‘ก๐‘ก)๐‘’๐‘’โˆ’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก๐‘‘๐‘‘๐‘ก๐‘กโˆž

โˆ’โˆž

Motivating Example: Periodic Square Wave

7

Motivating Example: Periodic Square Wave

8

CT Fourier Transform of Aperiodic Signal ๐’™๐’™(๐’•๐’•)

โ€ข General strategy โ–ซ approximate ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก by a periodic signal ๐‘ฅ๐‘ฅ๐‘‡๐‘‡ ๐‘ก๐‘ก with infinite period ๐‘‡๐‘‡

9

๐‘ฅ๐‘ฅ ๐‘ก๐‘ก

โˆ’๐‘‡๐‘‡1 0 ๐‘‡๐‘‡1

โŸน lim๐‘‡๐‘‡โ†’โˆž

๐‘ฅ๐‘ฅ๐‘‡๐‘‡ ๐‘ก๐‘ก = ๐‘ฅ๐‘ฅ(๐‘ก๐‘ก) ๐‘ฅ๐‘ฅ๐‘‡๐‘‡ ๐‘ก๐‘ก

โˆ’๐‘‡๐‘‡ โˆ’ ๐‘‡๐‘‡2

โˆ’ ๐‘‡๐‘‡1 0 ๐‘‡๐‘‡1 ๐‘‡๐‘‡2

๐‘‡๐‘‡

โ‹ฏ โ‹ฏ

CT Fourier Transform of Aperiodic Signal ๐’™๐’™(๐’•๐’•)

โ€ข Step 1: for aperiodic signal ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก with sup ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก โŠ‚ [โˆ’๐‘‡๐‘‡1,๐‘‡๐‘‡1] construct a periodic signal ๐‘ฅ๐‘ฅ๐‘‡๐‘‡ ๐‘ก๐‘ก , for which one period is ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก

๐‘ฅ๐‘ฅ๐‘‡๐‘‡ ๐‘ก๐‘ก = ๏ฟฝ ๐‘ฅ๐‘ฅ(๐‘ก๐‘ก โˆ’ ๐‘˜๐‘˜๐‘‡๐‘‡)โˆž

๐‘˜๐‘˜=โˆ’โˆž

โ–ซ Then ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก = ๐‘ฅ๐‘ฅ๐‘‡๐‘‡ ๐‘ก๐‘ก ,โˆ€ ๐‘ก๐‘ก < ๐‘‡๐‘‡/2

10

๐‘ฅ๐‘ฅ๐‘‡๐‘‡ ๐‘ก๐‘ก

โ‹ฏ โ‹ฏ

๐‘ฅ๐‘ฅ ๐‘ก๐‘ก

โˆ’๐‘‡๐‘‡ โˆ’ ๐‘‡๐‘‡2

โˆ’ ๐‘‡๐‘‡1 0 ๐‘‡๐‘‡1 ๐‘‡๐‘‡2

๐‘‡๐‘‡

CT Fourier Transform of Aperiodic Signal ๐’™๐’™(๐’•๐’•)

โ€ข Step 2: determine the Fourier series representation for ๐‘ฅ๐‘ฅ๐‘‡๐‘‡ ๐‘ก๐‘ก โ–ซ Fourier series coefficients

๐‘Ž๐‘Ž๐‘˜๐‘˜ =1๐‘‡๐‘‡๏ฟฝ ๐‘ฅ๐‘ฅ๐‘‡๐‘‡(๐‘ก๐‘ก)๐‘’๐‘’โˆ’๐‘—๐‘—๐‘˜๐‘˜๐‘—๐‘—0๐‘ก๐‘ก๐‘‘๐‘‘๐‘ก๐‘ก๐‘‡๐‘‡/2

โˆ’๐‘‡๐‘‡/2=

1๐‘‡๐‘‡๏ฟฝ ๐‘ฅ๐‘ฅ(๐‘ก๐‘ก)๐‘’๐‘’โˆ’๐‘—๐‘—๐‘˜๐‘˜๐‘—๐‘—0๐‘ก๐‘ก๐‘‘๐‘‘๐‘ก๐‘กโˆž

โˆ’โˆž

โ–ซ Define the envelope of ๐‘‡๐‘‡๐‘Ž๐‘Ž๐‘˜๐‘˜

๐‘‹๐‘‹(๐‘—๐‘—๐œ”๐œ”) = ๏ฟฝ ๐‘ฅ๐‘ฅ(๐‘ก๐‘ก)๐‘’๐‘’โˆ’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก๐‘‘๐‘‘๐‘ก๐‘กโˆž

โˆ’โˆž

โ–ซ Then

๐‘ฅ๐‘ฅ๐‘‡๐‘‡ ๐‘ก๐‘ก = ๏ฟฝ ๐‘Ž๐‘Ž๐‘˜๐‘˜๐‘’๐‘’๐‘—๐‘—๐‘˜๐‘˜๐‘—๐‘—0๐‘ก๐‘กโˆž

๐‘˜๐‘˜=โˆ’โˆž

=1๐‘‡๐‘‡ ๏ฟฝ ๐‘‹๐‘‹ ๐‘—๐‘—๐‘˜๐‘˜๐œ”๐œ”0 ๐‘’๐‘’๐‘—๐‘—๐‘˜๐‘˜๐‘—๐‘—0๐‘ก๐‘ก =

โˆž

๐‘˜๐‘˜=โˆ’โˆž

12๐œ‹๐œ‹ ๏ฟฝ ๐‘‹๐‘‹ ๐‘—๐‘—๐‘˜๐‘˜๐œ”๐œ”0 ๐‘’๐‘’๐‘—๐‘—๐‘˜๐‘˜๐‘—๐‘—0๐‘ก๐‘ก๐œ”๐œ”0

โˆž

๐‘˜๐‘˜=โˆ’โˆž

11

๐‘ฅ๐‘ฅ๐‘‡๐‘‡ ๐‘ก๐‘ก

โ‹ฏ โ‹ฏ

๐‘ฅ๐‘ฅ ๐‘ก๐‘ก

โˆ’๐‘‡๐‘‡ โˆ’ ๐‘‡๐‘‡2

โˆ’ ๐‘‡๐‘‡1 0 ๐‘‡๐‘‡1 ๐‘‡๐‘‡2

๐‘‡๐‘‡

CT Fourier Transform of Aperiodic Signal ๐’™๐’™(๐’•๐’•)

โ€ข Step 3: approximate ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก with ๐‘ฅ๐‘ฅ๐‘‡๐‘‡ ๐‘ก๐‘ก by increasing ๐‘‡๐‘‡ โ†’ โˆž

โ–ซ As ๐‘‡๐‘‡ โ†’ โˆž โŸน๐œ”๐œ”0 โ†’ 0, ๐‘ฅ๐‘ฅ๐‘‡๐‘‡ ๐‘ก๐‘ก โ†’ ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก ,๐œ”๐œ”0 โ†’ ๐‘‘๐‘‘๐œ”๐œ”,โˆ‘ โ†’ โˆซ

๐‘ฅ๐‘ฅ๐‘‡๐‘‡ ๐‘ก๐‘ก =12๐œ‹๐œ‹

๏ฟฝ ๐‘‹๐‘‹ ๐‘—๐‘—๐‘˜๐‘˜๐œ”๐œ”0 ๐‘’๐‘’๐‘—๐‘—๐‘˜๐‘˜๐‘—๐‘—0๐‘ก๐‘ก๐œ”๐œ”0

โˆž

๐‘˜๐‘˜=โˆ’โˆž

โ‡“

๐‘ฅ๐‘ฅ ๐‘ก๐‘ก = lim๐‘‡๐‘‡โ†’โˆž

๐‘ฅ๐‘ฅ๐‘‡๐‘‡(๐‘ก๐‘ก) = lim๐‘—๐‘—0โ†’0

12๐œ‹๐œ‹

๏ฟฝ ๐‘‹๐‘‹ ๐‘—๐‘—๐‘˜๐‘˜๐œ”๐œ”0 ๐‘’๐‘’๐‘—๐‘—๐‘˜๐‘˜๐‘—๐‘—0๐‘ก๐‘ก๐œ”๐œ”0

โˆž

๐‘˜๐‘˜=โˆ’โˆž

=12๐œ‹๐œ‹

๏ฟฝ ๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” ๐‘’๐‘’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก๐‘‘๐‘‘๐œ”๐œ”โˆž

โˆ’โˆž

12

๐‘ฅ๐‘ฅ๐‘‡๐‘‡ ๐‘ก๐‘ก

โ‹ฏ โ‹ฏ

๐‘ฅ๐‘ฅ ๐‘ก๐‘ก

โˆ’๐‘‡๐‘‡ โˆ’ ๐‘‡๐‘‡2

โˆ’ ๐‘‡๐‘‡1 0 ๐‘‡๐‘‡1 ๐‘‡๐‘‡2

๐‘‡๐‘‡

CT Fourier Transform Pair โ€ข Fourier transform (analysis equation)

๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” = โ„ฑ{๐‘ฅ๐‘ฅ(๐‘ก๐‘ก)} = ๏ฟฝ ๐‘ฅ๐‘ฅ(๐‘ก๐‘ก)๐‘’๐‘’โˆ’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก๐‘‘๐‘‘๐‘ก๐‘กโˆž

โˆ’โˆž

where ๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” called spectrum of ๐‘ฅ๐‘ฅ(๐‘ก๐‘ก)

โ€ข Inverse Fourier transform (synthesis equation)

๐‘ฅ๐‘ฅ ๐‘ก๐‘ก = โ„ฑโˆ’1{๐‘‹๐‘‹(๐‘—๐‘—๐œ”๐œ”)} =12๐œ‹๐œ‹๏ฟฝ ๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” ๐‘’๐‘’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก๐‘‘๐‘‘๐œ”๐œ”

โˆž

โˆ’โˆž

Superposition of complex exponentials at continuum of frequencies, frequency component ๐‘’๐‘’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก has โ€œamplitudeโ€ of ๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” ๐‘‘๐‘‘๐œ”๐œ”/2๐œ‹๐œ‹

13

CT Fourier Series of Periodic Signal ๐’™๐’™๐‘ป๐‘ป(๐’•๐’•)

โ€ข Fourier Transform of a finite duration signal ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก that is equal to ๐‘ฅ๐‘ฅ๐‘‡๐‘‡ ๐‘ก๐‘ก over one period, e.g., for ๐‘ก๐‘ก0 โ‰ค ๐‘ก๐‘ก โ‰ค ๐‘ก๐‘ก0 + ๐‘‡๐‘‡

๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” = โ„ฑ{๐‘ฅ๐‘ฅ(๐‘ก๐‘ก)} = ๏ฟฝ ๐‘ฅ๐‘ฅ(๐‘ก๐‘ก)๐‘’๐‘’โˆ’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก๐‘‘๐‘‘๐‘ก๐‘กโˆž

โˆ’โˆž

โ€ข Fourier coefficients of periodic signal ๐‘ฅ๐‘ฅ๐‘‡๐‘‡ ๐‘ก๐‘ก with period ๐‘‡๐‘‡

๐‘Ž๐‘Ž๐‘˜๐‘˜ =1๐‘‡๐‘‡๏ฟฝ ๐‘ฅ๐‘ฅ๐‘‡๐‘‡ ๐‘ก๐‘ก ๐‘’๐‘’โˆ’๐‘—๐‘—๐‘˜๐‘˜๐‘—๐‘—0๐‘ก๐‘ก๐‘‘๐‘‘๐‘ก๐‘ก =

1๐‘‡๐‘‡๏ฟฝ ๐‘ฅ๐‘ฅ(๐‘ก๐‘ก)๐‘’๐‘’โˆ’๐‘—๐‘—๐‘˜๐‘˜๐‘—๐‘—0๐‘ก๐‘ก๐‘‘๐‘‘๐‘ก๐‘ก

๐‘ก๐‘ก0+๐‘‡๐‘‡

๐‘ก๐‘ก0

๐‘ก๐‘ก0+๐‘‡๐‘‡

๐‘ก๐‘ก0

=1๐‘‡๐‘‡๏ฟฝ ๐‘ฅ๐‘ฅ(๐‘ก๐‘ก)๐‘’๐‘’โˆ’๐‘—๐‘—๐‘˜๐‘˜๐‘—๐‘—0๐‘ก๐‘ก๐‘‘๐‘‘๐‘ก๐‘กโˆž

โˆ’โˆž=

1๐‘‡๐‘‡๐‘‹๐‘‹(๐‘—๐‘—๐œ”๐œ”)๏ฟฝ

๐‘—๐‘—=๐‘˜๐‘˜๐‘—๐‘—0

โŸน proportional to samples of the Fourier transform of ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก , i.e., one period of ๐‘ฅ๐‘ฅ๐‘‡๐‘‡ ๐‘ก๐‘ก

14

Convergence of Fourier Transform โ€ข Suppose from ๐‘ฅ๐‘ฅ(๐‘ก๐‘ก), we have

๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” = ๏ฟฝ ๐‘ฅ๐‘ฅ(๐‘ก๐‘ก)๐‘’๐‘’โˆ’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก๐‘‘๐‘‘๐‘ก๐‘กโˆž

โˆ’โˆž

โ€ข Let ๐‘ฅ๐‘ฅ๏ฟฝ(๐‘ก๐‘ก) be the Fourier transform representation of ๐‘ฅ๐‘ฅ(๐‘ก๐‘ก),

obtained using the synthesis equation

๐‘ฅ๐‘ฅ๏ฟฝ(๐‘ก๐‘ก) =12๐œ‹๐œ‹๏ฟฝ ๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” ๐‘’๐‘’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก๐‘‘๐‘‘๐œ”๐œ”

โˆž

โˆ’โˆž

โ€ข Question: when can ๐‘ฅ๐‘ฅ๏ฟฝ(๐‘ก๐‘ก) be a valid representation of the original

signal ๐‘ฅ๐‘ฅ(๐‘ก๐‘ก)?

15

Finite Energy Condition โ€ข One useful notion for engineers: โ–ซ no energy in approximation error

๐‘’๐‘’ ๐‘ก๐‘ก โ‰œ ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก โˆ’ ๐‘ฅ๐‘ฅ๏ฟฝ ๐‘ก๐‘ก = ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก โˆ’12๐œ‹๐œ‹๏ฟฝ ๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” ๐‘’๐‘’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก๐‘‘๐‘‘๐œ”๐œ”

โˆž

โˆ’โˆž

and ๏ฟฝ ๐‘’๐‘’(๐‘ก๐‘ก) 2๐‘‘๐‘‘๐‘ก๐‘ก = 0โˆž

โˆ’โˆž

โ€ข Finite energy condition (square integrable)

๏ฟฝ ๐‘ฅ๐‘ฅ(๐‘ก๐‘ก) 2๐‘‘๐‘‘๐‘ก๐‘ก < โˆžโˆž

โˆ’โˆž

โ–ซ Note: does not guarantee ๐‘ฅ๐‘ฅ(๐‘ก๐‘ก) equal to its Fourier representation at any time ๐‘ก๐‘ก, but guarantees no energy in approximation error

16

โ€ข Condition 1. โ–ซ ๐‘ฅ๐‘ฅ(๐‘ก๐‘ก) is absolutely integrable, i.e.,

๏ฟฝ |๐‘ฅ๐‘ฅ ๐‘ก๐‘ก |๐‘‘๐‘‘๐‘ก๐‘กโˆž

โˆ’โˆž< โˆž

โ€ข Condition 2. โ–ซ Within any finite interval, ๐‘ฅ๐‘ฅ(๐‘ก๐‘ก) has a finite number of maxima

and minima

โ€ข Condition 3. โ–ซ Within any finite interval, ๐‘ฅ๐‘ฅ(๐‘ก๐‘ก) has only a finite number of

discontinuities with finite values

โ€ข โŸน absolutely integrable signals that are continuous or with a finite number of discontinuities have Fourier transforms

Dirichlet Conditions

17

Example: Right-sided Decaying Exponential

โ€ข ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก = ๐‘’๐‘’โˆ’๐‘Ž๐‘Ž๐‘ก๐‘ก๐‘ข๐‘ข ๐‘ก๐‘ก ,๐‘Ž๐‘Ž > 0 is a real number

๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” = ๏ฟฝ ๐‘ฅ๐‘ฅ(๐‘ก๐‘ก)๐‘’๐‘’โˆ’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก๐‘‘๐‘‘๐‘ก๐‘กโˆž

โˆ’โˆž= ๏ฟฝ ๐‘’๐‘’โˆ’๐‘Ž๐‘Ž๐‘ก๐‘ก๐‘’๐‘’โˆ’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก๐‘‘๐‘‘๐‘ก๐‘ก

โˆž

0

= โˆ’1

๐‘Ž๐‘Ž + ๐‘—๐‘—๐œ”๐œ”๐‘’๐‘’โˆ’(๐‘Ž๐‘Ž+๐‘—๐‘—๐‘—๐‘—)๐‘ก๐‘ก๏ฟฝ

0

โˆž

=1

๐‘Ž๐‘Ž + ๐‘—๐‘—๐œ”๐œ”

๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” =1

๐‘Ž๐‘Ž2 + ๐œ”๐œ”2, arg๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” = โˆ’ arctan

๐œ”๐œ”๐‘Ž๐‘Ž

โ€ข Observation: โ–ซ Magnitude spectrum: even symmetry โ–ซ Phase spectrum: odd symmetry

โ€ข Note: also works for complex ๐‘Ž๐‘Ž with โ„›โ„ฏ ๐‘Ž๐‘Ž > 0

18

Example: Two-sided Decaying Exponential

โ€ข ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก = ๐‘’๐‘’โˆ’|๐‘Ž๐‘Ž|๐‘ก๐‘ก๐‘ข๐‘ข ๐‘ก๐‘ก ,๐‘Ž๐‘Ž > 0 is a real number

๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” = ๏ฟฝ ๐‘ฅ๐‘ฅ(๐‘ก๐‘ก)๐‘’๐‘’โˆ’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก๐‘‘๐‘‘๐‘ก๐‘กโˆž

โˆ’โˆž

= ๏ฟฝ ๐‘’๐‘’โˆ’|๐‘Ž๐‘Ž|๐‘ก๐‘ก๐‘’๐‘’โˆ’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก๐‘‘๐‘‘๐‘ก๐‘กโˆž

โˆ’โˆž

= ๏ฟฝ ๐‘’๐‘’๐‘Ž๐‘Ž๐‘ก๐‘ก๐‘’๐‘’โˆ’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก๐‘‘๐‘‘๐‘ก๐‘ก0

โˆ’โˆž+ ๏ฟฝ ๐‘’๐‘’โˆ’๐‘Ž๐‘Ž๐‘ก๐‘ก๐‘’๐‘’โˆ’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก๐‘‘๐‘‘๐‘ก๐‘ก

โˆž

0

=1

๐‘Ž๐‘Ž โˆ’ ๐‘—๐‘—๐œ”๐œ”๐‘’๐‘’(๐‘Ž๐‘Žโˆ’๐‘—๐‘—๐‘—๐‘—)๐‘ก๐‘ก๏ฟฝ

โˆ’โˆž

0

โˆ’1

๐‘Ž๐‘Ž + ๐‘—๐‘—๐œ”๐œ”๐‘’๐‘’โˆ’(๐‘Ž๐‘Ž+๐‘—๐‘—๐‘—๐‘—)๐‘ก๐‘ก๏ฟฝ

0

โˆž

=1

๐‘Ž๐‘Ž โˆ’ ๐‘—๐‘—๐œ”๐œ”+

1๐‘Ž๐‘Ž + ๐‘—๐‘—๐œ”๐œ”

=2๐‘Ž๐‘Ž

๐‘Ž๐‘Ž2 + ๐œ”๐œ”2

โ€ข Note: also works for complex ๐‘Ž๐‘Ž with โ„›โ„ฏ ๐‘Ž๐‘Ž > 0

19

Example: Rectangular Pulse โ€ข ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก = ๐‘ข๐‘ข ๐‘ก๐‘ก + ๐‘‡๐‘‡1 โˆ’ ๐‘ข๐‘ข(๐‘ก๐‘ก โˆ’ ๐‘‡๐‘‡1)

โ€ข Fourier transform

๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” = ๏ฟฝ ๐‘’๐‘’โˆ’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก๐‘‘๐‘‘๐‘ก๐‘ก๐‘‡๐‘‡1

โˆ’๐‘‡๐‘‡1= 2

sin(๐œ”๐œ”๐‘‡๐‘‡1)๐œ”๐œ”

โ€ข Inverse Fourier transform

๐‘ฅ๐‘ฅ๏ฟฝ ๐‘ก๐‘ก =12๐œ‹๐œ‹

๏ฟฝ 2sin(๐œ”๐œ”๐‘‡๐‘‡1)

๐œ”๐œ”๐‘’๐‘’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก๐‘‘๐‘‘๐œ”๐œ”

โˆž

โˆ’โˆž

โ€ข As ๐‘‡๐‘‡1 โ†’ โˆž, (define sinc ๐œƒ๐œƒ = sin(๐œ‹๐œ‹๐œ‹๐œ‹)๐œ‹๐œ‹๐œ‹๐œ‹

)

โ–ซ Frequency domain: 2๐œ‹๐œ‹ sin(๐‘—๐‘—๐‘‡๐‘‡1)๐œ‹๐œ‹๐‘—๐‘—

= 2๐œ‹๐œ‹ ๐‘‡๐‘‡1๐œ‹๐œ‹

sinc ๐‘—๐‘—๐‘‡๐‘‡1๐œ‹๐œ‹

โ†’ 2๐œ‹๐œ‹๐›ฟ๐›ฟ(๐œ”๐œ”)

โ–ซ Time domain: ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก โ†’ 1, DC

20

1 1

1

1

Example: Ideal Lowpass Filter โ€ข Frequency response ๐ป๐ป(๐‘—๐‘—๐œ”๐œ”) = ๐‘ข๐‘ข ๐œ”๐œ” + ๐œ”๐œ”๐‘๐‘ โˆ’ ๐‘ข๐‘ข(๐œ”๐œ” โˆ’ ๐œ”๐œ”๐‘๐‘)

โ€ข Impulse response

โ„Ž ๐‘ก๐‘ก =12๐œ‹๐œ‹

๏ฟฝ ๐‘’๐‘’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก๐‘‘๐‘‘๐œ”๐œ”๐‘—๐‘—๐‘๐‘

โˆ’๐‘—๐‘—๐‘๐‘

=sin(๐œ”๐œ”๐‘๐‘ ๐‘ก๐‘ก)

๐œ‹๐œ‹๐‘ก๐‘ก=๐œ”๐œ”๐‘๐‘๐œ‹๐œ‹

sinc๐œ”๐œ”๐‘๐‘๐‘ก๐‘ก๐œ‹๐œ‹

โ€ข As ๐œ”๐œ”๐‘๐‘ โ†’ โˆž, โ–ซ Time domain โ„Ž ๐‘ก๐‘ก โ†’ ๐›ฟ๐›ฟ(๐‘ก๐‘ก), becomes identity system

โ–ซ Frequency domain ๐ป๐ป(๐‘—๐‘—๐œ”๐œ”) โ†’ 1, passes all frequencies

21

Duality

22

Examples โ€ข Unit impulse ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก = ๐›ฟ๐›ฟ ๐‘ก๐‘ก

๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” = ๏ฟฝ ๐›ฟ๐›ฟ(๐‘ก๐‘ก)๐‘’๐‘’โˆ’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก๐‘‘๐‘‘๐‘ก๐‘ก+โˆž

โˆ’โˆž= 1

๐‘ฅ๐‘ฅ ๐‘ก๐‘ก =12๐œ‹๐œ‹๏ฟฝ ๐‘’๐‘’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก๐‘‘๐‘‘๐œ”๐œ”

โˆž

โˆ’โˆž= lim

๐‘—๐‘—๐‘๐‘โ†’โˆž

12๐œ‹๐œ‹๏ฟฝ ๐‘’๐‘’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก๐‘‘๐‘‘๐œ”๐œ”

๐‘—๐‘—๐‘๐‘

โˆ’๐‘—๐‘—๐‘๐‘

= lim๐‘—๐‘—๐‘๐‘โ†’โˆž

sin(๐œ”๐œ”๐‘๐‘ ๐‘ก๐‘ก)๐œ‹๐œ‹๐‘ก๐‘ก = ๐›ฟ๐›ฟ(๐‘ก๐‘ก)

๐œน๐œน(๐’•๐’•) has โ€œwhiteโ€ spectrum, equal amount of all frequencies!

โ€ข DC Signal ๐‘ฅ๐‘ฅ(๐‘ก๐‘ก) = 1

๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” = lim๐‘‡๐‘‡1โ†’โˆž

๏ฟฝ ๐‘’๐‘’โˆ’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก๐‘‘๐‘‘๐‘ก๐‘ก๐‘‡๐‘‡1

โˆ’๐‘‡๐‘‡1= lim

๐‘‡๐‘‡1โ†’โˆž2๐œ‹๐œ‹

sin(๐œ”๐œ”๐‘‡๐‘‡1)๐œ‹๐œ‹๐œ”๐œ” = 2๐œ‹๐œ‹๐›ฟ๐›ฟ(๐œ”๐œ”)

๐‘ฅ๐‘ฅ ๐‘ก๐‘ก =12๐œ‹๐œ‹๏ฟฝ ๐‘‹๐‘‹(๐‘—๐‘—๐œ”๐œ”)๐‘’๐‘’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก๐‘‘๐‘‘๐œ”๐œ”

โˆž

โˆ’โˆž=

12๐œ‹๐œ‹๏ฟฝ 2๐œ‹๐œ‹๐›ฟ๐›ฟ(๐œ”๐œ”)๐‘’๐‘’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก๐‘‘๐‘‘๐œ”๐œ” = 1

โˆž

โˆ’โˆž

Spectrum of DC signal is impulse at zero frequency!

23

Fourier Transform of Periodic Signals โ€ข A periodic signal ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก with fundamental frequency ๐œ”๐œ”0 =2๐œ‹๐œ‹/๐‘‡๐‘‡ has a Fourier series representation

๐‘ฅ๐‘ฅ ๐‘ก๐‘ก = ๏ฟฝ ๐‘Ž๐‘Ž๐‘˜๐‘˜๐‘’๐‘’๐‘—๐‘—๐‘˜๐‘˜๐‘—๐‘—0๐‘ก๐‘กโˆž

๐‘˜๐‘˜=โˆ’โˆž

โŸน ๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” = โ„ฑ{๐‘ฅ๐‘ฅ ๐‘ก๐‘ก } = ๏ฟฝ ๐‘Ž๐‘Ž๐‘˜๐‘˜โ„ฑ{๐‘’๐‘’๐‘—๐‘—๐‘˜๐‘˜๐‘—๐‘—0๐‘ก๐‘ก}โˆž

๐‘˜๐‘˜=โˆ’โˆž

โ€ข Question then becomes

โ„ฑ{๐‘’๐‘’๐‘—๐‘—๐‘˜๐‘˜๐‘—๐‘—0๐‘ก๐‘ก} =?

24

Fourier Transform of Periodic Signals โ€ข Using the basic Fourier transform pair

๐‘’๐‘’๐‘—๐‘—๐‘—๐‘—0๐‘ก๐‘ก โ„ฑ

2๐œ‹๐œ‹๐›ฟ๐›ฟ ๐œ”๐œ” โˆ’ ๐œ”๐œ”0

โ–ซ โ„ฑ{๐‘’๐‘’๐‘—๐‘—๐‘—๐‘—0๐‘ก๐‘ก} = lim๐‘‡๐‘‡1โ†’โˆž

โˆซ ๐‘’๐‘’โˆ’๐‘—๐‘— ๐‘—๐‘—โˆ’๐‘—๐‘—0 ๐‘ก๐‘ก๐‘‘๐‘‘๐‘ก๐‘ก๐‘‡๐‘‡1โˆ’๐‘‡๐‘‡1

= lim๐‘‡๐‘‡1โ†’โˆž

2๐œ‹๐œ‹ sin[(๐‘—๐‘—โˆ’๐‘—๐‘—0)๐‘‡๐‘‡1]๐œ‹๐œ‹ ๐‘—๐‘—โˆ’๐‘—๐‘—0

= 2๐œ‹๐œ‹๐›ฟ๐›ฟ(๐œ”๐œ” โˆ’ ๐œ”๐œ”0)

โ–ซ โ„ฑโˆ’1{2๐œ‹๐œ‹๐›ฟ๐›ฟ(๐œ”๐œ” โˆ’ ๐œ”๐œ”0)} = 12๐œ‹๐œ‹ โˆซ 2๐œ‹๐œ‹๐›ฟ๐›ฟ ๐œ”๐œ” โˆ’ ๐œ”๐œ”0 ๐‘’๐‘’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก๐‘‘๐‘‘๐œ”๐œ”

โˆžโˆ’โˆž = ๐‘’๐‘’๐‘—๐‘—๐‘—๐‘—0๐‘ก๐‘ก

โ€ข Thus, the Fourier transform of periodic signal ๐‘ฅ๐‘ฅ(๐‘ก๐‘ก) is

๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” = โ„ฑ{๐‘ฅ๐‘ฅ ๐‘ก๐‘ก } = ๏ฟฝ ๐‘Ž๐‘Ž๐‘˜๐‘˜โ„ฑ{๐‘’๐‘’๐‘—๐‘—๐‘˜๐‘˜๐‘—๐‘—0๐‘ก๐‘ก}โˆž

๐‘˜๐‘˜=โˆ’โˆž

= ๏ฟฝ 2๐œ‹๐œ‹๐‘Ž๐‘Ž๐‘˜๐‘˜๐›ฟ๐›ฟ(๐œ”๐œ” โˆ’ ๐‘˜๐‘˜๐œ”๐œ”0)โˆž

๐‘˜๐‘˜=โˆ’โˆž

25

Fourier Transform of Periodic Signals โ€ข A periodic signal ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก with fundamental frequency ๐œ”๐œ”0 = 2๐œ‹๐œ‹/๐‘‡๐‘‡

โ€ข Fourier series

๐‘ฅ๐‘ฅ ๐‘ก๐‘ก = ๏ฟฝ ๐‘Ž๐‘Ž๐‘˜๐‘˜๐‘’๐‘’๐‘—๐‘—๐‘˜๐‘˜๐‘—๐‘—0๐‘ก๐‘กโˆž

๐‘˜๐‘˜=โˆ’โˆž

๐‘Ž๐‘Ž๐‘˜๐‘˜ =1๐‘‡๐‘‡๏ฟฝ ๐‘ฅ๐‘ฅ(๐‘ก๐‘ก)๐‘’๐‘’โˆ’๐‘—๐‘—๐‘˜๐‘˜๐‘—๐‘—0๐‘ก๐‘ก๐‘‘๐‘‘๐‘ก๐‘ก๐‘‡๐‘‡2

โˆ’๐‘‡๐‘‡2

โ€ข Fourier transform

๐‘ฅ๐‘ฅ(๐‘ก๐‘ก) โ„ฑ

๐‘‹๐‘‹(๐‘—๐‘—๐œ”๐œ”๏ผ‰ = ๏ฟฝ 2๐œ‹๐œ‹๐‘Ž๐‘Ž๐‘˜๐‘˜๐›ฟ๐›ฟ(๐œ”๐œ” โˆ’ ๐‘˜๐‘˜๐œ”๐œ”0)โˆž

๐‘˜๐‘˜=โˆ’โˆž

โ–ซ Spectrum of periodic signal consists of a impulse train occurring at harmonically related frequencies!

โ–ซ Areas of impulses are ๐Ÿ๐Ÿ๐Ÿ๐Ÿ times Fourier series coefficients

26

Example: Cosine and Sine โ€ข ๐‘ฅ๐‘ฅ๐‘๐‘(๐‘ก๐‘ก) = cos(๐œ”๐œ”0๐‘ก๐‘ก) = 1

2๐‘’๐‘’๐‘—๐‘—๐‘—๐‘—0๐‘ก๐‘ก + 1

2๐‘’๐‘’โˆ’๐‘—๐‘—๐‘—๐‘—0๐‘ก๐‘ก

โŸน ๐‘Ž๐‘Ž1= 1,๐‘Ž๐‘Žโˆ’1 = 1, ๐‘Ž๐‘Ž๐‘˜๐‘˜ = 0๏ผŒ๐‘˜๐‘˜ โ‰  ยฑ1

โŸน ๐‘‹๐‘‹๐‘๐‘(๐‘—๐‘—๐œ”๐œ”) = ๐œ‹๐œ‹[๐›ฟ๐›ฟ(๐œ”๐œ” โˆ’ ๐œ”๐œ”0) + ๐›ฟ๐›ฟ(๐œ”๐œ” + ๐œ”๐œ”0)]

โ€ข ๐‘ฅ๐‘ฅ๐‘ ๐‘ (๐‘ก๐‘ก) = sin(๐œ”๐œ”0๐‘ก๐‘ก) = 12๐‘—๐‘—๐‘’๐‘’๐‘—๐‘—๐‘—๐‘—0๐‘ก๐‘ก โˆ’ 1

2๐‘—๐‘—๐‘’๐‘’โˆ’๐‘—๐‘—๐‘—๐‘—0๐‘ก๐‘ก

โŸน ๐‘Ž๐‘Ž1= 12๐‘—๐‘—

,๐‘Ž๐‘Žโˆ’1 = โˆ’ 12๐‘—๐‘—

, ๐‘Ž๐‘Ž๐‘˜๐‘˜ = 0๏ผŒ๐‘˜๐‘˜ โ‰  ยฑ1

โŸน ๐‘‹๐‘‹๐‘ ๐‘  ๐‘—๐‘—๐œ”๐œ” =๐œ‹๐œ‹๐‘—๐‘—๐›ฟ๐›ฟ ๐œ”๐œ” โˆ’ ๐œ”๐œ”0 โˆ’

๐œ‹๐œ‹๐‘—๐‘—๐›ฟ๐›ฟ(๐œ”๐œ” + ๐œ”๐œ”0)

27

Example: Periodic Square Wave

๐‘Ž๐‘Ž0=1๐‘‡๐‘‡๏ฟฝ 1๐‘‡๐‘‡1

โˆ’๐‘‡๐‘‡1๐‘‘๐‘‘๐‘ก๐‘ก =

2๐‘‡๐‘‡1๐‘‡๐‘‡

, ๐‘Ž๐‘Ž๐‘˜๐‘˜ =1๐‘‡๐‘‡๏ฟฝ ๐‘’๐‘’โˆ’๐‘—๐‘—๐‘˜๐‘˜๐‘—๐‘—0๐‘ก๐‘ก๐‘‘๐‘‘๐‘ก๐‘ก๐‘‡๐‘‡1

โˆ’๐‘‡๐‘‡1=

2sin (๐‘˜๐‘˜๐œ”๐œ”0๐‘‡๐‘‡1)๐‘˜๐‘˜๐œ”๐œ”0๐‘‡๐‘‡

, ๐‘˜๐‘˜ โ‰  0

๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” = ๏ฟฝ 2๐œ‹๐œ‹2 sin ๐‘˜๐‘˜๐œ”๐œ”0๐‘‡๐‘‡1

๐‘˜๐‘˜๐œ”๐œ”0๐‘‡๐‘‡

โˆž

๐‘˜๐‘˜=โˆ’โˆž

๐›ฟ๐›ฟ ๐œ”๐œ” โˆ’ ๐‘˜๐‘˜๐œ”๐œ”0 = ๏ฟฝ2 sin ๐‘˜๐‘˜๐œ”๐œ”0๐‘‡๐‘‡1

๐‘˜๐‘˜

โˆž

๐‘˜๐‘˜=โˆ’โˆž

๐›ฟ๐›ฟ ๐œ”๐œ” โˆ’ ๐‘˜๐‘˜๐œ”๐œ”0

28

Example: Periodic Impulse Train โ€ข ๐‘ฅ๐‘ฅ(๐‘ก๐‘ก) = โˆ‘ ๐›ฟ๐›ฟ(๐‘ก๐‘ก โˆ’ ๐‘˜๐‘˜๐‘‡๐‘‡)โˆž

๐‘˜๐‘˜=โˆ’โˆž

โŸน ๐‘Ž๐‘Ž๐‘˜๐‘˜=1๐‘‡๐‘‡๏ฟฝ ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก๐‘‡๐‘‡2

โˆ’๐‘‡๐‘‡2

๐‘’๐‘’โˆ’๐‘—๐‘—๐‘˜๐‘˜๐‘—๐‘—0๐‘ก๐‘ก๐‘‘๐‘‘๐‘ก๐‘ก =1๐‘‡๐‘‡

โŸน ๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” = ๏ฟฝ2๐œ‹๐œ‹๐‘‡๐‘‡๐›ฟ๐›ฟ(๐œ”๐œ” โˆ’ ๐‘˜๐‘˜

2๐œ‹๐œ‹๐‘‡๐‘‡

)โˆž

๐‘˜๐‘˜=โˆ’โˆž

29

Properties of CT Fourier Transform

๐‘ฅ๐‘ฅ ๐‘ก๐‘ก โ„ฑ

๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” ๐‘ฆ๐‘ฆ(๐‘ก๐‘ก) โ„ฑ

๐‘Œ๐‘Œ(๐‘—๐‘—๐œ”๐œ”)

โ€ข Linearity

๐‘Ž๐‘Ž๐‘ฅ๐‘ฅ(๐‘ก๐‘ก) + ๐‘๐‘๐‘ฆ๐‘ฆ(๐‘ก๐‘ก) โ„ฑ

๐‘Ž๐‘Ž๐‘‹๐‘‹(๐‘—๐‘—๐œ”๐œ”) + ๐‘๐‘๐‘Œ๐‘Œ(๐‘—๐‘—๐œ”๐œ”)

โ€ข Time shifting

๐‘ฅ๐‘ฅ(๐‘ก๐‘ก โˆ’ ๐‘ก๐‘ก0) โ„ฑ

๐‘’๐‘’โˆ’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก0๐‘‹๐‘‹(๐‘—๐‘—๐œ”๐œ”) โ€ข Frequency shifting

๐‘’๐‘’๐‘—๐‘—๐‘—๐‘—0๐‘ก๐‘ก๐‘ฅ๐‘ฅ(๐‘ก๐‘ก) โ„ฑ

๐‘‹๐‘‹(๐‘—๐‘— ๐œ”๐œ” โˆ’ ๐œ”๐œ”0 )

30

Example: Multipath Fading Effect Transmitter: ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก Receiver: ๐‘ฆ๐‘ฆ ๐‘ก๐‘ก = ๐‘Ž๐‘Ž0๐‘ฅ๐‘ฅ ๐‘ก๐‘ก + ๐‘Ž๐‘Ž1๐‘ฅ๐‘ฅ(๐‘ก๐‘ก โˆ’ ๐œ๐œ)

โ€ข ๐‘Œ๐‘Œ ๐‘—๐‘—๐œ”๐œ” = ๐‘Ž๐‘Ž0๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” + ๐‘’๐‘’โˆ’๐‘—๐‘—๐‘—๐‘—๐‘—๐‘—๐‘Ž๐‘Ž1๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” = (๐‘Ž๐‘Ž0 + ๐‘’๐‘’โˆ’๐‘—๐‘—๐‘—๐‘—๐‘—๐‘—๐‘Ž๐‘Ž1)๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” โ€ข Let ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก = ๐‘ข๐‘ข ๐‘ก๐‘ก + ๐‘‡๐‘‡ โˆ’ ๐‘ข๐‘ข ๐‘ก๐‘ก โˆ’ ๐‘‡๐‘‡

๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” =2 sin(๐œ”๐œ”๐‘‡๐‘‡)

๐œ”๐œ”

๐‘Œ๐‘Œ ๐‘—๐‘—๐œ”๐œ” = (๐‘Ž๐‘Ž0 + ๐‘’๐‘’โˆ’๐‘—๐‘—๐‘—๐‘—๐‘—๐‘—๐‘Ž๐‘Ž1)2 sin(๐œ”๐œ”๐‘‡๐‘‡)

๐œ”๐œ”

โ€ข When ๐‘Ž๐‘Ž0 = ๐‘Ž๐‘Ž1 = 1

๐‘Œ๐‘Œ ๐‘—๐‘—๐œ”๐œ” = 4๐‘’๐‘’โˆ’๐‘—๐‘—๐‘—๐‘—๐‘—๐‘—/2 cos ๐œ”๐œ”๐œ๐œ/22 sin(๐œ”๐œ”๐‘‡๐‘‡)

๐œ”๐œ”

31

Example: Modulation Baseband signal: ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก Modulated signal: ๐‘ฆ๐‘ฆ ๐‘ก๐‘ก = ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก cos(๐œ”๐œ”๐‘๐‘๐‘ก๐‘ก)

โ€ข ๐‘ฆ๐‘ฆ ๐‘ก๐‘ก = 12๐‘ฅ๐‘ฅ(๐‘ก๐‘ก)(๐‘’๐‘’๐‘—๐‘—๐‘—๐‘—๐‘๐‘๐‘ก๐‘ก + ๐‘’๐‘’โˆ’๐‘—๐‘—๐‘—๐‘—๐‘๐‘๐‘ก๐‘ก )

๐‘Œ๐‘Œ ๐‘—๐‘—๐œ”๐œ” =12๐‘‹๐‘‹ ๐‘—๐‘—(๐œ”๐œ” โˆ’ ๐œ”๐œ”๐‘๐‘ +

12๐‘‹๐‘‹ ๐‘—๐‘—(๐œ”๐œ” + ๐œ”๐œ”๐‘๐‘

โ€ข Let ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก = ๐‘ข๐‘ข ๐‘ก๐‘ก + ๐‘‡๐‘‡ โˆ’ ๐‘ข๐‘ข ๐‘ก๐‘ก โˆ’ ๐‘‡๐‘‡

๐‘Œ๐‘Œ ๐‘—๐‘—๐œ”๐œ” =sin((๐œ”๐œ” โˆ’ ๐œ”๐œ”๐‘๐‘)๐‘‡๐‘‡)

๐œ”๐œ” โˆ’ ๐œ”๐œ”๐‘๐‘+

sin((๐œ”๐œ” + ๐œ”๐œ”๐‘๐‘)๐‘‡๐‘‡)๐œ”๐œ” + ๐œ”๐œ”๐‘๐‘

32

Properties of CT Fourier Transform โ€ข Time reversal

๐‘ฅ๐‘ฅ(โˆ’๐‘ก๐‘ก) โ„ฑ

๐‘‹๐‘‹(โˆ’๐‘—๐‘—๐œ”๐œ”)

โ€ข Conjugation

๐‘ฅ๐‘ฅโˆ— ๐‘ก๐‘ก โ„ฑ

๐‘‹๐‘‹โˆ— โˆ’๐‘—๐‘—๐œ”๐œ”

Proof: โ„ฑ ๐‘ฅ๐‘ฅโˆ— ๐‘ก๐‘ก = โˆซ ๐‘ฅ๐‘ฅโˆ— ๐‘ก๐‘ก ๐‘’๐‘’โˆ’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก๐‘‘๐‘‘๐‘ก๐‘ก =+โˆžโˆ’โˆž โˆซ ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก ๐‘’๐‘’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก๐‘‘๐‘‘๐‘ก๐‘ก+โˆž

โˆ’โˆž

โˆ—= ๐‘‹๐‘‹โˆ— (โˆ’๐‘—๐‘—๐œ”๐œ”)

โ€ข Conjugate Symmetry โ–ซ ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก real โŸบ ๐‘‹๐‘‹(โˆ’๐‘—๐‘—๐œ”๐œ”) = ๐‘‹๐‘‹โˆ—(๐‘—๐‘—๐œ”๐œ”) โ–ซ ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก even โŸบ ๐‘‹๐‘‹(๐‘—๐‘—๐œ”๐œ”) even, ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก odd โŸบ ๐‘‹๐‘‹(๐‘—๐‘—๐œ”๐œ”) odd โ–ซ ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก real and even โŸบ ๐‘‹๐‘‹(๐‘—๐‘—๐œ”๐œ”) real and even โ–ซ ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก real and odd โŸบ ๐‘‹๐‘‹(๐‘—๐‘—๐œ”๐œ”) purely imaginary and odd

33

โ€ข For a real signal: ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก โ„ฑ

๐‘‹๐‘‹(๐‘—๐‘—๐œ”๐œ”)

๐‘ฅ๐‘ฅ๐‘’๐‘’ ๐‘ก๐‘ก =12 ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก + ๐‘ฅ๐‘ฅ โˆ’๐‘ก๐‘ก

โ„ฑ โ„›โ„ฏ{๐‘‹๐‘‹(๐‘—๐‘—๐œ”๐œ”)}

๐‘ฅ๐‘ฅ๐‘œ๐‘œ ๐‘ก๐‘ก =12 ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก โˆ’ ๐‘ฅ๐‘ฅ โˆ’๐‘ก๐‘ก

โ„ฑ ๐‘—๐‘— โ‹… โ„๐“‚๐“‚{๐‘‹๐‘‹(๐‘—๐‘—๐œ”๐œ”)}

โ€ข Proof: โ–ซ Since ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก is real, then ๐‘‹๐‘‹(๐‘—๐‘—๐œ”๐œ”) is conjugate symmetric

โ„ฑ ๐‘ฅ๐‘ฅ๐‘’๐‘’ ๐‘ก๐‘ก =12๐‘‹๐‘‹(๐‘—๐‘—๐œ”๐œ”) + ๐‘‹๐‘‹(โˆ’๐‘—๐‘—๐œ”๐œ”) =

12๐‘‹๐‘‹(๐‘—๐‘—๐œ”๐œ”) + ๐‘‹๐‘‹โˆ—(๐‘—๐‘—๐œ”๐œ”) = โ„›โ„ฏ ๐‘‹๐‘‹(๐‘—๐‘—๐œ”๐œ”)

โ„ฑ ๐‘ฅ๐‘ฅ๐‘œ๐‘œ ๐‘ก๐‘ก =12๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” โˆ’ ๐‘‹๐‘‹(โˆ’๐‘—๐‘—๐œ”๐œ”) =

12๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” โˆ’ ๐‘‹๐‘‹โˆ—(๐‘—๐‘—๐œ”๐œ”) = ๐‘—๐‘— โ‹… โ„๐“‚๐“‚{๐‘‹๐‘‹(๐‘—๐‘—๐œ”๐œ”)}

Even-Odd Decomposition

34

Example For ๐‘Ž๐‘Ž > 0, let

๐‘ฅ๐‘ฅ ๐‘ก๐‘ก = ๏ฟฝ๐‘’๐‘’โˆ’๐‘Ž๐‘Ž๐‘ก๐‘ก, ๐‘ก๐‘ก > 0โˆ’๐‘’๐‘’๐‘Ž๐‘Ž๐‘ก๐‘ก ๐‘ก๐‘ก < 0

โ€ข Let ๐‘ฆ๐‘ฆ ๐‘ก๐‘ก = ๐‘’๐‘’โˆ’๐‘Ž๐‘Ž๐‘ก๐‘ก๐‘ข๐‘ข(๐‘ก๐‘ก), with ๐‘Œ๐‘Œ ๐‘—๐‘—๐œ”๐œ” =

1๐‘Ž๐‘Ž + ๐‘—๐‘—๐œ”๐œ”

โ€ข Since ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก = ๐‘ฆ๐‘ฆ ๐‘ก๐‘ก โˆ’ ๐‘ฆ๐‘ฆ(โˆ’๐‘ก๐‘ก)

๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” = ๐‘Œ๐‘Œ ๐‘—๐‘—๐œ”๐œ” โˆ’ ๐‘Œ๐‘Œ(โˆ’๐‘—๐‘—๐œ”๐œ”)

=1

๐‘Ž๐‘Ž + ๐‘—๐‘—๐œ”๐œ”โˆ’

1๐‘Ž๐‘Ž โˆ’ ๐‘—๐‘—๐œ”๐œ”

=โˆ’2๐‘—๐‘—๐œ”๐œ”๐‘Ž๐‘Ž2 + ๐œ”๐œ”2

โ€ข ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก real and odd โŸน ๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” purely imaginary and odd

35

Time and Frequency Scaling

๐‘ฅ๐‘ฅ ๐‘ก๐‘ก โ„ฑ

๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ”

๐‘ฅ๐‘ฅ ๐‘Ž๐‘Ž๐‘ก๐‘ก โ„ฑ 1

๐‘Ž๐‘Ž ๐‘‹๐‘‹๐‘—๐‘—๐œ”๐œ”๐‘Ž๐‘Ž ,๐‘Ž๐‘Ž โ‰  0

โ€ข Proof: change variables by ๐œ๐œ = ๐‘Ž๐‘Ž๐‘ก๐‘ก โ–ซ For ๐‘Ž๐‘Ž > 0,

๏ฟฝ ๐‘ฅ๐‘ฅ ๐‘Ž๐‘Ž๐‘ก๐‘ก ๐‘’๐‘’โˆ’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก๐‘‘๐‘‘๐‘ก๐‘ก = ๏ฟฝ ๐‘ฅ๐‘ฅ ๐œ๐œ ๐‘’๐‘’โˆ’๐‘—๐‘—๐‘—๐‘—๐‘Ž๐‘Ž๐‘—๐‘—๐‘‘๐‘‘(

๐œ๐œ๐‘Ž๐‘Ž

) =1๐‘Ž๐‘Ž๐‘‹๐‘‹(๐‘—๐‘—๐œ”๐œ”๐‘Ž๐‘Ž

) +โˆž

โˆ’โˆž

+โˆž

โˆ’โˆž

โ–ซ For ๐‘Ž๐‘Ž < 0,

๏ฟฝ ๐‘ฅ๐‘ฅ ๐‘Ž๐‘Ž๐‘ก๐‘ก ๐‘’๐‘’โˆ’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก๐‘‘๐‘‘๐‘ก๐‘ก = ๏ฟฝ ๐‘ฅ๐‘ฅ ๐œ๐œ ๐‘’๐‘’โˆ’๐‘—๐‘—๐‘—๐‘—๐‘Ž๐‘Ž๐‘—๐‘—๐‘‘๐‘‘

๐œ๐œ๐‘Ž๐‘Ž

= โˆ’1๐‘Ž๐‘Ž๐‘‹๐‘‹(๐‘—๐‘—๐œ”๐œ”๐‘Ž๐‘Ž

) โˆ’โˆž

โˆž

+โˆž

โˆ’โˆž

36

Time and Frequency Scaling

๐‘ฅ๐‘ฅ ๐‘ก๐‘ก โ„ฑ

๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” ๐‘ฅ๐‘ฅ ๐‘Ž๐‘Ž๐‘ก๐‘ก โ„ฑ 1

๐‘Ž๐‘Ž ๐‘‹๐‘‹๐‘—๐‘—๐œ”๐œ”๐‘Ž๐‘Ž ,๐‘Ž๐‘Ž โ‰  0

compression in time โŸบ stretching in frequency stretching in time โŸบ compression in frequency

โ€ข Example: faster playback an audio clip โŸน higher pitch

๐‘ฅ๐‘ฅ ๐‘ก๐‘ก = ๐‘’๐‘’โˆ’ ๐‘Ž๐‘Ž๐‘ก๐‘ก 2 โ„ฑ

๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” =๐œ‹๐œ‹๐‘Ž๐‘Ž๐‘’๐‘’โˆ’

14๐‘—๐‘—๐‘Ž๐‘Ž

2

37

Exercise โ€ข Problem: If we have the Fourier transform pair

๐‘ฅ๐‘ฅ ๐‘ก๐‘ก โ†” ๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” ,

determine the Fourier transform for the signal ๐‘ฅ๐‘ฅ(๐‘Ž๐‘Ž๐‘ก๐‘ก + ๐‘๐‘).

โ€ข Answer:

โ„ฑ ๐‘ฅ๐‘ฅ ๐‘Ž๐‘Ž๐‘ก๐‘ก + ๐‘๐‘ =1

|๐‘Ž๐‘Ž|๐‘‹๐‘‹(๐‘—๐‘—๐œ”๐œ”๐‘Ž๐‘Ž )๐‘’๐‘’๐‘—๐‘—

๐‘—๐‘—๐œ”๐œ”๐‘Ž๐‘Ž

38

๐‘ฅ๐‘ฅโ€ฒ ๐‘ก๐‘ก โ„ฑ

๐‘—๐‘—๐œ”๐œ”๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” , ๐‘ฅ๐‘ฅ ๐‘˜๐‘˜ ๐‘ก๐‘ก โ„ฑ

๐‘—๐‘—๐œ”๐œ” ๐‘˜๐‘˜๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ”

โ€ข Proof: differentiate both sides of FT synthesis equation

๐‘ฅ๐‘ฅ ๐‘ก๐‘ก =12๐œ‹๐œ‹

๏ฟฝ ๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” ๐‘’๐‘’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก๐‘‘๐‘‘๐œ”๐œ” โŸน โˆž

โˆ’โˆž๐‘ฅ๐‘ฅโ€ฒ ๐‘ก๐‘ก =

12๐œ‹๐œ‹

๏ฟฝ ๐‘—๐‘—๐œ”๐œ”๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” ๐‘’๐‘’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก๐‘‘๐‘‘๐œ”๐œ”โˆž

โˆ’โˆž

โ€ข Example: differentiator ๐‘ฆ๐‘ฆ ๐‘ก๐‘ก = ๐‘ฅ๐‘ฅโ€ฒ ๐‘ก๐‘ก , with frequency response

๐ป๐ป ๐‘—๐‘—๐œ”๐œ” = โ„ฑ ๐›ฟ๐›ฟโ€ฒ ๐‘ก๐‘ก = ๏ฟฝ ๐›ฟ๐›ฟโ€ฒ ๐‘ก๐‘ก ๐‘’๐‘’โˆ’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก๐‘‘๐‘‘๐‘ก๐‘ก+โˆž

โˆ’โˆž

= ๐›ฟ๐›ฟ(๐‘ก๐‘ก)๐‘’๐‘’โˆ’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก๏ฟฝโˆ’โˆž

+โˆž+ ๐‘—๐‘—๐œ”๐œ”๏ฟฝ ๐›ฟ๐›ฟ(๐‘ก๐‘ก)๐‘’๐‘’โˆ’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก๐‘‘๐‘‘๐‘ก๐‘ก

+โˆž

โˆ’โˆž= ๐‘—๐‘—๐œ”๐œ”

โ–ซ ๐‘Œ๐‘Œ ๐‘—๐‘—๐œ”๐œ” = ๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” ๐ป๐ป(๐‘—๐‘—๐œ”๐œ”) โ–ซ amplifies high frequencies โ–ซ suppresses low frequencies โ–ซ DC completely eliminated, cannot be completely recovered from ๐‘ฅ๐‘ฅโ€ฒ ๐‘ก๐‘ก

Differentiation

39

๐‘ฆ๐‘ฆ ๐‘ก๐‘ก = ๏ฟฝ ๐‘ฅ๐‘ฅ ๐œ๐œ ๐‘‘๐‘‘๐œ๐œ๐‘ก๐‘ก

โˆ’โˆž

โ„ฑ ๐‘Œ๐‘Œ ๐‘—๐‘—๐œ”๐œ” =

1๐‘—๐‘—๐œ”๐œ”

๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” + ๐œ‹๐œ‹๐‘‹๐‘‹ 0 ๐›ฟ๐›ฟ(๐œ”๐œ”)

โ€ข Since ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก = ๐‘ฆ๐‘ฆโ€ฒ ๐‘ก๐‘ก , by differentiation property

๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” = ๐‘—๐‘—๐œ”๐œ”๐‘Œ๐‘Œ ๐‘—๐‘—๐œ”๐œ” โŸน ๐‘Œ๐‘Œ ๐‘—๐‘—๐œ”๐œ” =1๐‘—๐‘—๐œ”๐œ”

๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” ,โˆ€๐œ”๐œ” โ‰  0

โ€ข Intuitively, ๐‘ฆ๐‘ฆ(๐‘ก๐‘ก) has a DC component

๐‘ฆ๐‘ฆ๏ฟฝ = lim๐‘‡๐‘‡โ†’โˆž

12๐‘‡๐‘‡

๏ฟฝ ๐‘ฆ๐‘ฆ ๐‘ก๐‘ก ๐‘‘๐‘‘๐‘ก๐‘ก๐‘‡๐‘‡

โˆ’๐‘‡๐‘‡= lim

๐‘‡๐‘‡โ†’โˆž

12๐‘‡๐‘‡

๏ฟฝ ๏ฟฝ ๐‘ฅ๐‘ฅ ๐œ๐œ ๐‘ข๐‘ข ๐‘ก๐‘ก โˆ’ ๐œ๐œ ๐‘‘๐‘‘๐œ๐œโˆž

โˆ’โˆž๐‘‘๐‘‘๐‘ก๐‘ก

๐‘‡๐‘‡

โˆ’๐‘‡๐‘‡

= ๏ฟฝ ๐‘ฅ๐‘ฅ ๐œ๐œ lim๐‘‡๐‘‡โ†’โˆž

12๐‘‡๐‘‡

๏ฟฝ ๐‘ข๐‘ข(๐‘ก๐‘ก โˆ’ ๐œ๐œ)๐‘‘๐‘‘๐‘ก๐‘ก๐‘‡๐‘‡

โˆ’๐‘‡๐‘‡๐‘‘๐‘‘๐œ๐œ

โˆž

โˆ’โˆž=

12๏ฟฝ ๐‘ฅ๐‘ฅ ๐œ๐œ ๐‘‘๐‘‘๐œ๐œโˆž

โˆ’โˆž=

12๐‘‹๐‘‹(0)

โ€ข Therefore, at ๐œ”๐œ” = 0, ๐‘Œ๐‘Œ ๐‘—๐‘—๐œ”๐œ” has a component 2๐œ‹๐œ‹๐‘ฆ๐‘ฆ๏ฟฝ๐›ฟ๐›ฟ ๐œ”๐œ” = ๐œ‹๐œ‹๐‘‹๐‘‹ 0 ๐›ฟ๐›ฟ(๐œ”๐œ”)

Integration

40

โ€ข Sign function:

sgn ๐‘ก๐‘ก = ๏ฟฝ 1, ๐‘ก๐‘ก > 0โˆ’1, ๐‘ก๐‘ก < 0

โ„ฑ

2๐‘—๐‘—๐œ”๐œ”

โ€ข Constant function:

1 โ„ฑ

2๐œ‹๐œ‹๐›ฟ๐›ฟ(๐œ”๐œ”) โ€ข Unit step function:

๐‘ข๐‘ข ๐‘ก๐‘ก =sgn ๐‘ก๐‘ก + 1

2 โ„ฑ

๐‘ˆ๐‘ˆ ๐‘—๐‘—๐œ”๐œ” =1๐‘—๐‘—๐œ”๐œ”

+ ๐œ‹๐œ‹๐›ฟ๐›ฟ ๐œ”๐œ”

โ€ข Integrator:

๐‘ฆ๐‘ฆ ๐‘ก๐‘ก = ๏ฟฝ ๐‘ฅ๐‘ฅ ๐œ๐œ ๐‘‘๐‘‘๐œ๐œ๐‘ก๐‘ก

โˆ’โˆž= ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก โˆ— ๐‘ข๐‘ข ๐‘ก๐‘ก

โ„ฑ ๐‘Œ๐‘Œ ๐‘—๐‘—๐œ”๐œ” = ๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” ๐‘ˆ๐‘ˆ ๐‘—๐‘—๐œ”๐œ” =

1๐‘—๐‘—๐œ”๐œ” ๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” + ๐œ‹๐œ‹๐‘‹๐‘‹ 0 ๐›ฟ๐›ฟ(๐œ”๐œ”)

Integration

41

โ€ข Sign function:

๐‘ฅ๐‘ฅ(๐‘ก๐‘ก) = ๏ฟฝ1 โˆ’2|๐‘ก๐‘ก|๐‘‡๐‘‡

, ๐‘ก๐‘ก โ‰ค๐‘‡๐‘‡2

0, otherwise

๐‘‹๐‘‹2 ๐‘—๐‘—๐œ”๐œ” =2๐‘‡๐‘‡

๐‘’๐‘’๐‘—๐‘—๐‘—๐‘—๐‘‡๐‘‡2 โˆ’ 2 + ๐‘’๐‘’๐‘—๐‘—

๐‘—๐‘—๐‘‡๐‘‡2 = โˆ’

8๐‘‡๐‘‡

sin2๐œ”๐œ”๐‘‡๐‘‡4

๐‘‹๐‘‹1 ๐‘—๐‘—๐œ”๐œ” =๐‘‹๐‘‹2 ๐‘—๐‘—๐œ”๐œ”๐‘—๐‘—๐œ”๐œ”

+ ๐œ‹๐œ‹๐‘‹๐‘‹2 0 ๐›ฟ๐›ฟ(๐œ”๐œ”) = โˆ’8๐‘—๐‘—๐œ”๐œ”๐‘‡๐‘‡

sin2๐œ”๐œ”๐‘‡๐‘‡4

๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” =๐‘‹๐‘‹1 ๐‘—๐‘—๐œ”๐œ”๐‘—๐‘—๐œ”๐œ”

+ ๐œ‹๐œ‹๐‘‹๐‘‹1 0 ๐›ฟ๐›ฟ(๐œ”๐œ”) =8

๐œ”๐œ”2๐‘‡๐‘‡sin2

๐œ”๐œ”๐‘‡๐‘‡4

Example: Triangular Pulse

42

Calculation of ๐‘ฟ๐‘ฟ(๐ŸŽ๐ŸŽ) โ€ข Method 1:

๐‘‹๐‘‹ 0 = ๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” ๏ฟฝ๐‘—๐‘—=0

โ€ข Method 2:

๐‘‹๐‘‹ 0 = ๏ฟฝ ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก ๐‘‘๐‘‘๐‘ก๐‘ก+โˆž

โˆ’โˆž

43

Example: ๐ฌ๐ฌ๐ฌ๐ฌ๐ฌ๐ฌ ๐’•๐’• Function โ€ข In terms of exponential function in the limit (๐‘Ž๐‘Ž > 0)

๐‘ฅ๐‘ฅ ๐‘ก๐‘ก = ๏ฟฝ๐‘’๐‘’โˆ’๐‘Ž๐‘Ž๐‘ก๐‘ก, ๐‘ก๐‘ก > 0โˆ’๐‘’๐‘’๐‘Ž๐‘Ž๐‘ก๐‘ก ๐‘ก๐‘ก < 0

sgn ๐‘ก๐‘ก = lim๐‘Ž๐‘Žโ†’0

๐‘ฅ๐‘ฅ (๐‘ก๐‘ก)

sgn ๐‘ก๐‘ก โ„ฑ

lim๐’‚๐’‚โ†’0

๐‘‹๐‘‹ (๐‘—๐‘—๐œ”๐œ”)๏ผ2๐‘—๐‘—๐œ”๐œ”

โ€ข In terms of unit step functions

sgn ๐‘ก๐‘ก = ๐‘ข๐‘ข ๐‘ก๐‘ก โˆ’ ๐‘ข๐‘ข(โˆ’๐‘ก๐‘ก)

sgn ๐‘ก๐‘ก โ„ฑ

[๐œ‹๐œ‹๐›ฟ๐›ฟ(๐œ”๐œ”) +1๐‘—๐‘—๐œ”๐œ”

] โˆ’ [๐œ‹๐œ‹๐›ฟ๐›ฟ(โˆ’๐œ”๐œ”) +1

โˆ’๐‘—๐‘—๐œ”๐œ”] =

2๐‘—๐‘—๐œ”๐œ”

44

Example: ๐ฌ๐ฌ๐ฌ๐ฌ๐ฌ๐ฌ ๐’•๐’• Function โ€ข Exploiting integration property

sgn ๐‘ก๐‘ก = 2๐‘ข๐‘ข ๐‘ก๐‘ก โˆ’ 1 โ‰œ ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก

sgnโ€ฒ ๐‘ก๐‘ก = 2๐›ฟ๐›ฟ(๐‘ก๐‘ก) โ‰œ ๐‘ฅ๐‘ฅ1 ๐‘ก๐‘ก

๐‘ฅ๐‘ฅ1 ๐‘ก๐‘ก = 2๐›ฟ๐›ฟ ๐‘ก๐‘ก โ„ฑ

๐‘‹๐‘‹1 ๐‘—๐‘—๐œ”๐œ” = 2

sgn ๐‘ก๐‘ก = ๐‘ฅ๐‘ฅ(๐‘ก๐‘ก) = ๐‘ฅ๐‘ฅ โˆ’โˆž + ๏ฟฝ ๐‘ฅ๐‘ฅ1(๐‘ก๐‘ก)๐‘‘๐‘‘๐‘ก๐‘ก๐‘ก๐‘ก

โˆ’โˆž

โŸนโ„ฑ sgn ๐‘ก๐‘ก = ๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” = 2๐œ‹๐œ‹๐‘ฅ๐‘ฅ โˆ’โˆž ๐›ฟ๐›ฟ ๐œ”๐œ” +๐‘‹๐‘‹1(๐‘—๐‘—๐œ”๐œ”)๐‘—๐‘—๐œ”๐œ”

+ ๐œ‹๐œ‹๐‘‹๐‘‹1(0)๐›ฟ๐›ฟ(๐œ”๐œ”)

= 2๐œ‹๐œ‹ โ‹… โˆ’1 โ‹… ๐›ฟ๐›ฟ ๐œ”๐œ” +2๐‘—๐‘—๐œ”๐œ”

+ ๐œ‹๐œ‹ โ‹… 2๐›ฟ๐›ฟ(๐œ”๐œ”) =2๐‘—๐‘—๐œ”๐œ”

45

Duality โ€ข Both time and frequency functions are continuous and aperiodic

in general, identical form except for โ–ซ different signs in exponent of complex exponential โ–ซ constant factor 1/2๐œ‹๐œ‹

๐‘ฅ๐‘ฅ ๐‘ก๐‘ก =1

2๐œ‹๐œ‹๏ฟฝ ๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” ๐‘’๐‘’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘กโˆž

โˆ’โˆž๐‘‘๐‘‘๐œ”๐œ”

โ„ฑ ๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” = ๏ฟฝ ๐‘ฅ๐‘ฅ(๐‘ก๐‘ก)๐‘’๐‘’โˆ’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก

โˆž

โˆ’โˆž๐‘‘๐‘‘๐‘ก๐‘ก

โ€ข Suppose two functions are related by

๐‘“๐‘“ ๐‘Ÿ๐‘Ÿ = ๏ฟฝ ๐‘”๐‘”(๐œ๐œ)๐‘’๐‘’โˆ’๐‘—๐‘—๐‘—๐‘—๐‘—๐‘—โˆž

โˆ’โˆž๐‘‘๐‘‘๐œ๐œ

โ–ซ Let ๐œ๐œ = ๐‘ก๐‘ก, and ๐‘Ÿ๐‘Ÿ = ๐œ”๐œ”, โŸน ๐‘”๐‘” ๐‘ก๐‘ก โ„ฑ

๐‘“๐‘“ ๐œ”๐œ”

โ–ซ Let ๐œ๐œ = โˆ’๐œ”๐œ”, and ๐‘Ÿ๐‘Ÿ = ๐‘ก๐‘ก,โŸน ๐‘“๐‘“ ๐‘ก๐‘ก โ„ฑ

2๐œ‹๐œ‹๐‘”๐‘” โˆ’๐œ”๐œ”

โ€ข Duality: ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก โ„ฑ

๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” โŸบ ๐‘‹๐‘‹(๐‘ก๐‘ก) โ„ฑ

2๐œ‹๐œ‹๐‘ฅ๐‘ฅ(โˆ’๐‘—๐‘—๐œ”๐œ”)

46

Duality

๐‘ข๐‘ข ๐‘ก๐‘ก + ๐‘Ž๐‘Ž โˆ’ ๐‘ข๐‘ข ๐‘ก๐‘ก โˆ’ ๐‘Ž๐‘Ž โ„ฑ

2 sin(๐‘Ž๐‘Ž๐œ”๐œ”)

๐œ”๐œ”

2 sin(๐‘Ž๐‘Ž๐‘ก๐‘ก)

๐‘ก๐‘ก โ„ฑ

2๐œ‹๐œ‹[๐‘ข๐‘ข ๐œ”๐œ” + ๐‘Ž๐‘Ž โˆ’ ๐‘ข๐‘ข ๐œ”๐œ” โˆ’ ๐‘Ž๐‘Ž ]

47

Duality Example: ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก = 1

๐œ‹๐œ‹๐‘ก๐‘ก

โ€ข Known

sgn ๐‘ก๐‘ก โ„ฑ 2

๐‘—๐‘—๐œ”๐œ”

โ€ข By duality โ–ซ switch LHS and RHS, switch ๐œ”๐œ” and ๐‘ก๐‘ก โ–ซ substitute ๐œ”๐œ” โ†’ โˆ’๐œ”๐œ” or ๐‘ก๐‘ก โ†’ โˆ’๐‘ก๐‘ก (but not both) โ–ซ multiply RHS by 2๐œ‹๐œ‹

โˆ’2๐‘—๐‘—๐‘ก๐‘ก

โ„ฑ 2๐œ‹๐œ‹ โ‹… sgn ๐œ”๐œ”

โ€ข By linearity 1๐œ‹๐œ‹๐‘ก๐‘ก

โ„ฑ โˆ’ ๐‘—๐‘— โ‹… sgn ๐œ”๐œ”

48

๐‘ฅ๐‘ฅ ๐‘ก๐‘ก โ„ฑ

๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” โŸบ ๐‘‹๐‘‹(๐‘ก๐‘ก) โ„ฑ

2๐œ‹๐œ‹๐‘ฅ๐‘ฅ(โˆ’๐‘—๐‘—๐œ”๐œ”)

Duality Example: ๐‘ฅ๐‘ฅ(๐‘ก๐‘ก) = 1

1+๐‘ก๐‘ก2

โ€ข Known

๐‘’๐‘’โˆ’|๐‘ก๐‘ก| โ„ฑ 21 + ๐œ”๐œ”2

โ€ข By duality โ–ซ switch LHS and RHS, switch ๐œ”๐œ” and ๐‘ก๐‘ก โ–ซ substitute ๐œ”๐œ” โ†’ โˆ’๐œ”๐œ” or ๐‘ก๐‘ก โ†’ โˆ’๐‘ก๐‘ก (but not both) โ–ซ multiply RHS by 2๐œ‹๐œ‹

21 + ๐‘ก๐‘ก2

โ„ฑ

2๐œ‹๐œ‹ โ‹… ๐‘’๐‘’โˆ’|๐‘—๐‘—|

โ€ข By linearity 1

1 + ๐‘ก๐‘ก2 โ„ฑ

๐œ‹๐œ‹ โ‹… ๐‘’๐‘’โˆ’|๐‘—๐‘—|

49

๐‘ฅ๐‘ฅ ๐‘ก๐‘ก โ„ฑ

๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” โŸบ ๐‘‹๐‘‹(๐‘ก๐‘ก) โ„ฑ

2๐œ‹๐œ‹๐‘ฅ๐‘ฅ(โˆ’๐‘—๐‘—๐œ”๐œ”)

Duality โ€ข Differentiation in frequency

โˆ’๐‘—๐‘—๐‘ก๐‘ก๐‘ฅ๐‘ฅ ๐‘ก๐‘ก โ„ฑ ๐‘‘๐‘‘๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ”

๐‘‘๐‘‘๐œ”๐œ”, ๐‘ก๐‘ก๐‘ฅ๐‘ฅ ๐‘ก๐‘ก

โ„ฑ ๐‘—๐‘—๐‘‘๐‘‘๐‘‹๐‘‹(๐‘—๐‘—๐œ”๐œ”)๐‘‘๐‘‘๐œ”๐œ”

โ€ข Proof:

๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” = ๏ฟฝ ๐‘ฅ๐‘ฅ(๐‘ก๐‘ก)๐‘’๐‘’โˆ’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘กโˆž

โˆ’โˆž๐‘‘๐‘‘๐‘ก๐‘ก โŸน

๐‘‘๐‘‘๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ”๐‘‘๐‘‘๐œ”๐œ”

= ๏ฟฝ (โˆ’๐‘—๐‘—๐‘ก๐‘ก)๐‘ฅ๐‘ฅ(๐‘ก๐‘ก)๐‘’๐‘’โˆ’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘กโˆž

โˆ’โˆž๐‘‘๐‘‘๐‘ก๐‘ก

โ€ข Example: Unit ramp function ๐‘ข๐‘ขโˆ’2 ๐‘ก๐‘ก = โˆซ ๐‘ข๐‘ข ๐œ๐œ ๐‘‘๐‘‘๐œ๐œ๐‘ก๐‘กโˆ’โˆž = ๐‘ก๐‘ก๐‘ข๐‘ข(๐‘ก๐‘ก)

โ„ฑ ๐‘ข๐‘ขโˆ’2 ๐‘ก๐‘ก = ๐‘—๐‘—โ„ฑ โˆ’๐‘—๐‘—๐‘ก๐‘ก๐‘ข๐‘ข(๐‘ก๐‘ก) = ๐‘—๐‘—๐‘‘๐‘‘๐‘ˆ๐‘ˆ(๐‘—๐‘—๐œ”๐œ”)๐‘‘๐‘‘๐œ”๐œ”

= โˆ’1๐œ”๐œ”2 + ๐‘—๐‘—๐œ‹๐œ‹๐›ฟ๐›ฟโ€ฒ(๐œ”๐œ”)

โ€ข Integration in frequency

โˆ’1๐‘—๐‘—๐‘ก๐‘ก๐‘ฅ๐‘ฅ ๐‘ก๐‘ก + ๐œ‹๐œ‹๐‘ฅ๐‘ฅ 0 ๐›ฟ๐›ฟ ๐‘ก๐‘ก

โ„ฑ ๏ฟฝ ๐‘‹๐‘‹ ๐œ‚๐œ‚ ๐‘‘๐‘‘๐œ‚๐œ‚

๐‘—๐‘—

โˆ’โˆž

50

Example

โ€ข Problem: ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก = ๐‘ก๐‘ก๐‘’๐‘’โˆ’2๐‘ก๐‘ก๐‘ข๐‘ข ๐‘ก๐‘ก โ„ฑ

?

โ€ข Solution:

๐‘’๐‘’โˆ’2๐‘ก๐‘ก๐‘ข๐‘ข ๐‘ก๐‘ก โ„ฑ 1

2 + ๐‘—๐‘—๐œ”๐œ”

๐‘ก๐‘ก๐‘’๐‘’โˆ’2๐‘ก๐‘ก๐‘ข๐‘ข ๐‘ก๐‘ก โ„ฑ

๐‘—๐‘—๐‘‘๐‘‘๐‘‘๐‘‘๐œ”๐œ”

12 + ๐‘—๐‘—๐œ”๐œ” =

1(2 + ๐‘—๐‘—๐œ”๐œ”)2

โ€ข Exercise: ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก = ๐‘ก๐‘ก๐‘’๐‘’โˆ’2๐‘ก๐‘ก๐‘ข๐‘ข ๐‘ก๐‘ก โˆ’ 1 โ†”?

51

Exercise โ€ข Determine ๐‘ฅ๐‘ฅ(๐‘ก๐‘ก) according to ๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ”

โ€ข Hints: exploit integration property in frequency domain

๐‘ฅ๐‘ฅ1 ๐‘ก๐‘ก โ„ฑ

๐‘‹๐‘‹โ€ฒ ๐‘—๐‘—๐œ”๐œ” โŸน ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก = ๐‘ฅ๐‘ฅ1 ๐‘ก๐‘กโˆ’๐‘—๐‘—๐‘ก๐‘ก

+ ๐œ‹๐œ‹๐‘ฅ๐‘ฅ1 0 ๐›ฟ๐›ฟ ๐‘ก๐‘ก

where ๐‘ฅ๐‘ฅ1 0 = โˆซ๐‘‹๐‘‹โ€ฒ(๐‘—๐‘—๐œ”๐œ”)๐‘‘๐‘‘๐œ”๐œ” = 0

52

โˆ’๐œ”๐œ”1 ๐œ”๐œ”1

๐‘‹๐‘‹(๐‘—๐‘—๐œ”๐œ”)

1

๐‘‹๐‘‹โ€ฒ(๐‘—๐‘—๐œ”๐œ”)

๐œ”๐œ”1 โˆ’๐œ”๐œ”1

1/๐œ”๐œ”1

-1/๐œ”๐œ”1

โ€ข For a Fourier transform pair ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก โ„ฑ

๐‘‹๐‘‹(๐‘—๐‘—๐œ”๐œ”)

๏ฟฝ ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก 2โˆž

โˆ’โˆž๐‘‘๐‘‘๐‘ก๐‘ก =

12๐œ‹๐œ‹

๏ฟฝ ๐‘‹๐‘‹(๐‘—๐‘—๐œ”๐œ”) 2โˆž

โˆ’โˆž๐‘‘๐‘‘๐œ”๐œ” = ๏ฟฝ ๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” 2

โˆž

โˆ’โˆž

๐‘‘๐‘‘๐œ”๐œ”2๐œ‹๐œ‹

โ€ข Note: ๐œ”๐œ” is angular frequency and ๐‘“๐‘“ = ๐œ”๐œ”/2๐œ‹๐œ‹ is frequency

โ€ข Interpretation: Energy conservation โ–ซ ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก 2: power, or energy per unit time (in second) โ–ซ ๐‘‹๐‘‹(๐‘—๐‘—๐œ”๐œ”) 2: energy per unit frequency (in Hertz), called energy-

density spectrum

โ€ข lim๐‘‡๐‘‡โ†’โˆž

1๐‘‡๐‘‡ โˆซ |๐‘ฅ๐‘ฅ(๐‘ก๐‘ก)|2๐‘‘๐‘‘๐‘ก๐‘ก = 1

2๐œ‹๐œ‹๐‘‡๐‘‡ โˆซ lim๐‘‡๐‘‡โ†’โˆž

|๐‘‹๐‘‹(๐‘—๐‘—๐‘—๐‘—)|2

๐‘‡๐‘‡โˆžโˆ’โˆž ๐‘‘๐‘‘๐œ”๐œ”

โ–ซ lim๐‘‡๐‘‡โ†’โˆž

|๐‘‹๐‘‹(๐‘—๐‘—๐‘—๐‘—)|2

๐‘‡๐‘‡ : called power-density spectrum

Parsevalโ€™s Relation

53

โ€ข For a Fourier transform pair ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก โ„ฑ

๐‘‹๐‘‹(๐‘—๐‘—๐œ”๐œ”)

๏ฟฝ ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก 2โˆž

โˆ’โˆž๐‘‘๐‘‘๐‘ก๐‘ก =

12๐œ‹๐œ‹

๏ฟฝ ๐‘‹๐‘‹(๐‘—๐‘—๐œ”๐œ”) 2โˆž

โˆ’โˆž๐‘‘๐‘‘๐œ”๐œ” = ๏ฟฝ ๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” 2

โˆž

โˆ’โˆž

๐‘‘๐‘‘๐œ”๐œ”2๐œ‹๐œ‹

โ€ข Proof:

๏ฟฝ ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก 2โˆž

โˆ’โˆž๐‘‘๐‘‘๐‘ก๐‘ก = ๏ฟฝ ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก ๐‘ฅ๐‘ฅโˆ— ๐‘ก๐‘ก

โˆž

โˆ’โˆž๐‘‘๐‘‘๐‘ก๐‘ก = ๏ฟฝ ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก

12๐œ‹๐œ‹

๏ฟฝ ๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ”โˆž

โˆ’โˆž๐‘’๐‘’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก๐‘‘๐‘‘๐œ”๐œ”

โˆ—โˆž

โˆ’โˆž๐‘‘๐‘‘๐‘ก๐‘ก

= ๏ฟฝ ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก12๐œ‹๐œ‹

๏ฟฝ ๐‘‹๐‘‹โˆ— ๐‘—๐‘—๐œ”๐œ”โˆž

โˆ’โˆž๐‘’๐‘’โˆ’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก๐‘‘๐‘‘๐œ”๐œ”

โˆž

โˆ’โˆž๐‘‘๐‘‘๐‘ก๐‘ก

=12๐œ‹๐œ‹

๏ฟฝ ๐‘‹๐‘‹โˆ— ๐‘—๐‘—๐œ”๐œ” ๏ฟฝ ๐‘ฅ๐‘ฅ ๐‘ก๐‘กโˆž

โˆ’โˆž๐‘’๐‘’โˆ’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก๐‘‘๐‘‘๐‘ก๐‘ก

โˆž

โˆ’โˆž๐‘‘๐‘‘๐œ”๐œ”

=12๐œ‹๐œ‹

๏ฟฝ ๐‘‹๐‘‹โˆ— ๐‘—๐‘—๐œ”๐œ” ๐‘‹๐‘‹(๐‘—๐‘—๐œ”๐œ”)โˆž

โˆ’โˆž๐‘‘๐‘‘๐œ”๐œ” =

12๐œ‹๐œ‹

๏ฟฝ ๐‘‹๐‘‹(๐‘—๐‘—๐œ”๐œ”) 2โˆž

โˆ’โˆž๐‘‘๐‘‘๐œ”๐œ”

Parsevalโ€™s Relation

54

โ€ข Example: determine the following time domain expressions:

๐ธ๐ธ = ๏ฟฝ ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก |2๐‘‘๐‘‘๐‘ก๐‘ก, and ๐ท๐ท = ๐‘ฅ๐‘ฅโ€ฒ ๐‘ก๐‘ก ๐‘ก๐‘ก=0

โˆž

โˆ’โˆž

โ€ข Solution:

๐ธ๐ธ = ๏ฟฝ |๐‘ฅ๐‘ฅ ๐‘ก๐‘ก |2๐‘‘๐‘‘๐‘ก๐‘กโˆž

โˆ’โˆž

=12๐œ‹๐œ‹

๏ฟฝ ๐‘‹๐‘‹(๐‘—๐‘—๐œ”๐œ”) 2โˆž

โˆ’โˆž๐‘‘๐‘‘๐œ”๐œ” =

12๐œ‹๐œ‹

โ‹…5๐œ‹๐œ‹4

=58

๐‘”๐‘” ๐‘ก๐‘ก = ๐‘ฅ๐‘ฅโ€ฒ ๐‘ก๐‘ก โ„ฑ

๐บ๐บ ๐‘—๐‘—๐œ”๐œ” = ๐‘—๐‘—๐œ”๐œ”๐‘‹๐‘‹(๐‘—๐‘—๐œ”๐œ”)

โŸน ๐ท๐ท = ๐‘”๐‘” 0 =12๐œ‹๐œ‹

๏ฟฝ ๐บ๐บ ๐‘—๐‘—๐œ”๐œ”โˆž

โˆ’โˆž๐‘‘๐‘‘๐œ”๐œ” =

12๐œ‹๐œ‹

๏ฟฝ ๐‘—๐‘—๐œ”๐œ”๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” โˆž

โˆ’โˆž๐‘‘๐‘‘๐œ”๐œ” = 0

Parsevalโ€™s Relation

55

๐‘‹๐‘‹(๐‘—๐‘—๐œ”๐œ”) ๐œ‹๐œ‹

๐œ‹๐œ‹/2

-1 -0.5 1 0.5

โ€ข Example: sin ๐‘Š๐‘Š๐‘ก๐‘ก๐œ‹๐œ‹๐‘ก๐‘ก

โ„ฑ

๐‘ข๐‘ข ๐œ”๐œ” + ๐‘Š๐‘Š โˆ’ ๐‘ข๐‘ข(๐œ”๐œ” โˆ’๐‘Š๐‘Š)

โ€ข By Parsevalโ€™s relation

๏ฟฝsin2 ๐‘Š๐‘Š๐‘ก๐‘ก๐œ‹๐œ‹2๐‘ก๐‘ก2

๐‘‘๐‘‘๐‘ก๐‘ก =12๐œ‹๐œ‹

โˆž

โˆ’โˆž๏ฟฝ ๐‘ข๐‘ข ๐œ”๐œ” + ๐‘Š๐‘Š โˆ’ ๐‘ข๐‘ข(๐œ”๐œ” โˆ’๐‘Š๐‘Š) 2โˆž

โˆ’โˆž๐‘‘๐‘‘๐œ”๐œ”

=12๐œ‹๐œ‹

๏ฟฝ 1๐‘Š๐‘Š

โˆ’๐‘Š๐‘Š๐‘‘๐‘‘๐œ”๐œ”

=๐‘Š๐‘Š๐œ‹๐œ‹

Parsevalโ€™s Relation

56

Example โ€ข Use Fourier transform of typical signals and Fourier

transform properties to determine the Fourier transform of the following signal

57

-1 -2 1 2

1

2

๐‘ฅ๐‘ฅ(๐‘ก๐‘ก)

Example

โ€ข Solution 1: ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก = 2๐‘”๐‘”2 ๐‘ก๐‘ก + ๐‘ข๐‘ข โˆ’๐‘ก๐‘ก โˆ’ 2 + ๐‘ข๐‘ข(๐‘ก๐‘ก โˆ’ 2)

โ–ซ ๐‘”๐‘”๐œ๐œ(๐‘ก๐‘ก) is the rectangular pulse with width of ๐œ๐œ and unit magnitude

58

-1 -2 1 2

1

2

๐‘ฅ๐‘ฅ(๐‘ก๐‘ก)

1 -1

2

+

-2

1 +

2

1

Example

โ€ข Solution 2: โ–ซ Assume

๐‘ฅ๐‘ฅ1 ๐‘ก๐‘ก = ๐‘ฅ๐‘ฅโ€ฒ ๐‘ก๐‘ก โ„ฑ

๐‘‹๐‘‹1(๐‘—๐‘—๐œ”๐œ”)

โ–ซ Then

๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” = 2๐œ‹๐œ‹๐‘ฅ๐‘ฅ โˆ’โˆž ๐›ฟ๐›ฟ ๐œ”๐œ” +๐‘‹๐‘‹1(๐‘—๐‘—๐œ”๐œ”)๐‘—๐‘—๐œ”๐œ”

+ ๐œ‹๐œ‹๐‘‹๐‘‹1 0 ๐›ฟ๐›ฟ ๐œ”๐œ”

59

-1 -2 1 2

1

2

๐‘ฅ๐‘ฅ(๐‘ก๐‘ก)

-1

1

2

-2

1 2

๐‘ฅ๐‘ฅโ€ฒ(๐‘ก๐‘ก)

-1 -2

Exercise โ€ข Determine the inverse Fourier transform

โ–ซ Plot 1:

โ–ซ Plot 2:

60

0ฯ‰โˆ’ 0ฯ‰

A

|)(| ฯ‰jX )( ฯ‰jXโˆ 

0ฯ‰โˆ’ 0ฯ‰ w

A

|)(| ฯ‰jX )( ฯ‰jXโˆ 

0ฯ‰0ฯ‰โˆ’

2/ฯ€

2/ฯ€โˆ’

Note: different

phase spectrum

Exercise โ€ข Determine the Fourier transform of the following signals

โ–ซ ๐‘ฅ๐‘ฅ(๐‘ก๐‘ก) = ๐‘’๐‘’โˆ’2๐‘ก๐‘ก๐‘ข๐‘ข(๐‘ก๐‘ก)

โ–ซ ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก = ๐‘’๐‘’โˆ’2 ๐‘ก๐‘กโˆ’1 ๐‘ข๐‘ข ๐‘ก๐‘ก

โ–ซ ๐‘ฅ๐‘ฅ(๐‘ก๐‘ก) = ๐‘’๐‘’โˆ’2๐‘ก๐‘ก๐‘ข๐‘ข(๐‘ก๐‘ก โˆ’ 1)

61

Convolution Property

๐‘ฆ๐‘ฆ ๐‘ก๐‘ก = ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก โˆ— โ„Ž ๐‘ก๐‘ก โ„ฑ

๐‘Œ๐‘Œ ๐‘—๐‘—๐œ”๐œ” = ๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” ๐ป๐ป(๐‘—๐‘—๐œ”๐œ”)

convolution in time โŸบ multiplication in frequency

โ€ข Proof:

๐‘Œ๐‘Œ ๐‘—๐‘—๐œ”๐œ” = ๏ฟฝ ๐‘ฆ๐‘ฆ ๐‘ก๐‘ก ๐‘’๐‘’โˆ’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก๐‘‘๐‘‘๐‘ก๐‘ก = ๏ฟฝ ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก โˆ— โ„Ž(๐‘ก๐‘ก)๐‘’๐‘’โˆ’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก๐‘‘๐‘‘๐‘ก๐‘ก+โˆž

โˆ’โˆž

+โˆž

โˆ’โˆž

= ๏ฟฝ ๏ฟฝ ๐‘ฅ๐‘ฅ ๐œ๐œ โ„Ž(๐‘ก๐‘ก โˆ’ ๐œ๐œ)+โˆž

โˆ’โˆž๐‘‘๐‘‘๐œ๐œ ๐‘’๐‘’โˆ’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก๐‘‘๐‘‘๐‘ก๐‘ก

+โˆž

โˆ’โˆž

= ๏ฟฝ ๐‘ฅ๐‘ฅ ๐œ๐œ ๏ฟฝ โ„Ž(๐‘ก๐‘ก โˆ’ ๐œ๐œ)+โˆž

โˆ’โˆž๐‘’๐‘’โˆ’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก๐‘‘๐‘‘๐‘ก๐‘ก ๐‘‘๐‘‘๐œ๐œ

+โˆž

โˆ’โˆž

= ๏ฟฝ ๐‘ฅ๐‘ฅ ๐œ๐œ ๐‘’๐‘’โˆ’๐‘—๐‘—๐‘—๐‘—๐‘—๐‘—๐ป๐ป ๐‘—๐‘—๐œ”๐œ” ๐‘‘๐‘‘๐œ๐œ+โˆž

โˆ’โˆž= ๐ป๐ป ๐‘—๐‘—๐œ”๐œ” ๏ฟฝ ๐‘ฅ๐‘ฅ ๐œ๐œ ๐‘’๐‘’โˆ’๐‘—๐‘—๐‘—๐‘—๐‘—๐‘—๐‘‘๐‘‘๐œ๐œ

+โˆž

โˆ’โˆž= ๐ป๐ป(๐‘—๐‘—๐œ”๐œ”)๐‘‹๐‘‹(๐‘—๐‘—๐œ”๐œ”)

62

Convolution Property โ€ข For an LTI system with โ–ซ impulse response โ„Ž(๐‘ก๐‘ก) โ–ซ frequency response ๐ป๐ป ๐‘—๐‘—๐œ”๐œ” = โˆซ โ„Ž ๐‘ก๐‘กโˆž

โˆ’โˆž ๐‘’๐‘’โˆ’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก๐‘‘๐‘‘๐‘ก๐‘ก = โ„ฑ{โ„Ž ๐‘ก๐‘ก }

โ–ซ ๐‘’๐‘’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก is eigenfunction associated with eigenvalue ๐ป๐ป ๐‘—๐‘—๐œ”๐œ”

๐‘ฅ๐‘ฅ ๐‘ก๐‘ก = ๐‘’๐‘’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก โ†’ ๐‘ฆ๐‘ฆ ๐‘ก๐‘ก = ๏ฟฝ โ„Ž ๐œ๐œ ๐‘’๐‘’๐‘—๐‘—๐‘—๐‘—(๐‘ก๐‘กโˆ’๐‘—๐‘—)๐‘‘๐‘‘๐œ๐œโˆž

โˆ’โˆž= ๐ป๐ป ๐‘—๐‘—๐œ”๐œ” ๐‘’๐‘’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก

โ€ข Input ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก as a linear combination of complex exponentials

๐‘ฅ๐‘ฅ ๐‘ก๐‘ก =12๐œ‹๐œ‹

๏ฟฝ ๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” ๐‘’๐‘’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก๐‘‘๐‘‘๐œ”๐œ”โˆž

โˆ’โˆž= lim

๐‘—๐‘—0โ†’0

12๐œ‹๐œ‹

๏ฟฝ ๐‘‹๐‘‹ ๐‘—๐‘—๐‘˜๐‘˜๐œ”๐œ”0 ๐‘’๐‘’๐‘—๐‘—๐‘˜๐‘˜๐‘—๐‘—0๐‘ก๐‘ก๐œ”๐œ”0

โˆž

๐‘˜๐‘˜=โˆ’โˆž

โ‡’ ๐‘ฆ๐‘ฆ ๐‘ก๐‘ก = lim๐‘—๐‘—0โ†’0

12๐œ‹๐œ‹

๏ฟฝ ๐‘‹๐‘‹ ๐‘—๐‘—๐‘˜๐‘˜๐œ”๐œ”0 ๐ป๐ป ๐‘—๐‘—๐‘˜๐‘˜๐œ”๐œ”0 ๐‘’๐‘’๐‘—๐‘—๐‘˜๐‘˜๐‘—๐‘—0๐‘ก๐‘ก๐œ”๐œ”0

โˆž

๐‘˜๐‘˜=โˆ’โˆž

=12๐œ‹๐œ‹

๏ฟฝ ๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” ๐ป๐ป ๐‘—๐‘—๐œ”๐œ” ๐‘’๐‘’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก๐‘‘๐‘‘๐œ”๐œ”โˆž

โˆ’โˆž=

12๐œ‹๐œ‹

๏ฟฝ ๐‘Œ๐‘Œ ๐‘—๐‘—๐œ”๐œ” ๐‘’๐‘’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก๐‘‘๐‘‘๐œ”๐œ”โˆž

โˆ’โˆž

63

โ€ข Rectangular pulse:

๐‘ฅ๐‘ฅ1 ๐‘ก๐‘ก = 2/๐‘‡๐‘‡ ๐‘ข๐‘ข ๐‘ก๐‘ก +๐‘‡๐‘‡4

โˆ’ ๐‘ข๐‘ข(๐‘ก๐‘ก โˆ’๐‘‡๐‘‡4

)

๐‘‹๐‘‹1 ๐‘—๐‘—๐œ”๐œ” = 2/๐‘‡๐‘‡ โ‹…2 sin(๐œ”๐œ”๐‘‡๐‘‡/4)

๐œ”๐œ”

โ€ข Triangular pulse:

๐‘ฅ๐‘ฅ(๐‘ก๐‘ก) = 1 โˆ’2|๐‘ก๐‘ก|๐‘‡๐‘‡

๐‘ข๐‘ข ๐‘ก๐‘ก +๐‘‡๐‘‡2

โˆ’ ๐‘ข๐‘ข(๐‘ก๐‘ก โˆ’๐‘‡๐‘‡2

)

๐‘‹๐‘‹1 ๐‘—๐‘—๐œ”๐œ” = ๐‘‹๐‘‹12 ๐‘—๐‘—๐œ”๐œ” =8 sin2(๐œ”๐œ”๐‘‡๐‘‡/4)

๐œ”๐œ”2๐‘‡๐‘‡

Example

64

Frequency Response of LTI Systems โ€ข Fully characterized by impulse response โ„Ž(๐‘ก๐‘ก)

๐‘ฆ๐‘ฆ ๐‘ก๐‘ก = ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก โˆ— โ„Ž ๐‘ก๐‘ก

โ€ข Also fully characterized by frequency response ๐ป๐ป ๐‘—๐‘—๐œ”๐œ” =โ„ฑ{โ„Ž ๐‘ก๐‘ก }, if ๐ป๐ป ๐‘—๐‘—๐œ”๐œ” is well defined โ–ซ BIBO stable system: โˆซ โ„Ž ๐‘ก๐‘ก ๐‘‘๐‘‘๐‘ก๐‘ก < โˆž+โˆž

โˆ’โˆž โ–ซ other systems: e.g., identity โ„Ž ๐‘ก๐‘ก = ๐›ฟ๐›ฟ(๐‘ก๐‘ก), differentiator โ„Ž ๐‘ก๐‘ก = ๐›ฟ๐›ฟโ€ฒ(๐‘ก๐‘ก)

โ€ข Convolution property implies โ–ซ instead of computing ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก โˆ— โ„Ž ๐‘ก๐‘ก in time domain, we can analyze

a system in frequency domain ๐‘ฆ๐‘ฆ ๐‘ก๐‘ก = ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก โˆ— โ„Ž(๐‘ก๐‘ก)

โ‡“

๐‘Œ๐‘Œ ๐‘—๐‘—๐œ”๐œ” = ๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” โ‹… ๐ป๐ป ๐‘—๐‘—๐œ”๐œ” , ๐ป๐ป ๐‘—๐‘—๐œ”๐œ” =๐‘Œ๐‘Œ ๐‘—๐‘—๐œ”๐œ”๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ”

= ๐ป๐ป ๐‘—๐‘—๐œ”๐œ” ๐‘’๐‘’๐‘—๐‘—๐‘—๐‘—๐‘— ๐‘—๐‘—๐‘—๐‘—

65

Frequency Response of LTI Systems โ€ข Some typical systems:

66

โ€ข Example: Differentiation property

๐‘ฆ๐‘ฆ ๐‘ก๐‘ก = ๐‘ฅ๐‘ฅโ€ฒ ๐‘ก๐‘ก = ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก โˆ— ๐›ฟ๐›ฟโ€ฒ ๐‘ก๐‘ก

๐‘Œ๐‘Œ ๐‘—๐‘—๐œ”๐œ” = ๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” โ‹… โ„ฑ ๐›ฟ๐›ฟโ€ฒ ๐‘ก๐‘ก = ๐‘—๐‘—๐œ”๐œ” โ‹… ๐‘‹๐‘‹(๐‘—๐‘—๐œ”๐œ”)

โ€ข Example. Integration property

๐‘ฆ๐‘ฆ ๐‘ก๐‘ก = ๏ฟฝ ๐‘ฅ๐‘ฅ ๐œ๐œ ๐‘‘๐‘‘๐œ๐œ๐‘ก๐‘ก

โˆ’โˆž= ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก โˆ— ๐‘ข๐‘ข ๐‘ก๐‘ก

๐‘Œ๐‘Œ ๐‘—๐‘—๐œ”๐œ” = ๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” โ‹… ๐‘ˆ๐‘ˆ(๐‘—๐‘—๐œ”๐œ”) = ๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ”1๐‘—๐‘—๐œ”๐œ”

+ ๐œ‹๐œ‹๐›ฟ๐›ฟ ๐œ”๐œ”

=1๐‘—๐‘—๐œ”๐œ”

๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” + ๐œ‹๐œ‹๐‘‹๐‘‹ 0 ๐›ฟ๐›ฟ ๐œ”๐œ”

Examples

67

โ€ข Unit ramp function: ๐‘ข๐‘ขโˆ’2 ๐‘ก๐‘ก = ๐‘ข๐‘ข ๐‘ก๐‘ก โˆ— ๐‘ข๐‘ข(๐‘ก๐‘ก) = ๐‘ก๐‘ก๐‘ข๐‘ข(๐‘ก๐‘ก)

โ€ข Convolution property suggests

โ„ฑ ๐‘ข๐‘ขโˆ’2 ๐‘ก๐‘ก = ๐‘ˆ๐‘ˆ2 ๐‘—๐‘—๐œ”๐œ” =1๐‘—๐‘—๐œ”๐œ”

+ ๐œ‹๐œ‹๐›ฟ๐›ฟ ๐œ”๐œ”2

= โˆ’1๐œ”๐œ”2 +

2๐œ‹๐œ‹๐‘—๐‘—๐œ”๐œ”

๐›ฟ๐›ฟ ๐œ”๐œ” + ๐œ‹๐œ‹2๐›ฟ๐›ฟ ๐œ”๐œ” ๐›ฟ๐›ฟ(๐œ”๐œ”)

โ–ซ But ๐›ฟ๐›ฟ(๐‘—๐‘—)๐‘—๐‘—๐‘—๐‘—

and ๐›ฟ๐›ฟ ๐œ”๐œ” ๐›ฟ๐›ฟ(๐œ”๐œ”) not well-defined!

โ–ซ Convolution property not applicable here โ–ซ Rule of thumb: applicable when formula is well-defined

โ€ข Known from the differentiation in frequency property

โ„ฑ ๐‘ข๐‘ขโˆ’2 ๐‘ก๐‘ก = ๐‘—๐‘—โ„ฑ โˆ’๐‘—๐‘—๐‘ก๐‘ก๐‘ข๐‘ข(๐‘ก๐‘ก) = ๐‘—๐‘—๐‘‘๐‘‘๐‘ˆ๐‘ˆ(๐‘—๐‘—๐œ”๐œ”)๐‘‘๐‘‘๐œ”๐œ”

= โˆ’1๐œ”๐œ”2 + ๐‘—๐‘—๐œ‹๐œ‹๐›ฟ๐›ฟโ€ฒ(๐œ”๐œ”)

Example

68

Example โ€ข Determine the Response of an LTI system with impulse response โ„Ž ๐‘ก๐‘ก = ๐‘’๐‘’โˆ’๐‘Ž๐‘Ž๐‘ก๐‘ก๐‘ข๐‘ข(๐‘ก๐‘ก), to the input ๐‘ฅ๐‘ฅ(๐‘ก๐‘ก) = ๐‘’๐‘’โˆ’๐œ”๐œ”๐‘ก๐‘ก๐‘ข๐‘ข(๐‘ก๐‘ก), where ๐‘Ž๐‘Ž, ๐‘๐‘ > 0

โ€ข Method 1: convolution in time domain ๐‘ฆ๐‘ฆ ๐‘ก๐‘ก = ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก โˆ— โ„Ž ๐‘ก๐‘ก

โ€ข Method 2: solving the differential equation with initial rest condition ๐‘ฆ๐‘ฆโ€ฒ ๐‘ก๐‘ก + ๐‘Ž๐‘Ž๐‘ฆ๐‘ฆ ๐‘ก๐‘ก = ๐‘’๐‘’โˆ’๐œ”๐œ”๐‘ก๐‘ก๐‘ข๐‘ข(๐‘ก๐‘ก)

โ€ข Method 3: Fourier transform

๐ป๐ป ๐‘—๐‘—๐œ”๐œ” =1

๐‘Ž๐‘Ž + ๐‘—๐‘—๐œ”๐œ”,๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” =

1๐‘๐‘ + ๐‘—๐‘—๐œ”๐œ”

โŸน ๐‘Œ๐‘Œ ๐‘—๐‘—๐œ”๐œ” =1

(๐‘Ž๐‘Ž + ๐‘—๐‘—๐œ”๐œ”)(๐‘๐‘ + ๐‘—๐‘—๐œ”๐œ”)

โ–ซ If ๐‘Ž๐‘Ž โ‰  ๐‘๐‘,๐‘Œ๐‘Œ ๐‘—๐‘—๐œ”๐œ” = 1๐œ”๐œ”โˆ’๐‘Ž๐‘Ž

1๐‘Ž๐‘Ž+๐‘—๐‘—๐‘—๐‘—

โˆ’ 1๐œ”๐œ”+๐‘—๐‘—๐‘—๐‘—

โŸน ๐‘ฆ๐‘ฆ ๐‘ก๐‘ก = 1๐œ”๐œ”โˆ’๐‘Ž๐‘Ž

๐‘’๐‘’โˆ’๐‘Ž๐‘Ž๐‘ก๐‘ก โˆ’ ๐‘’๐‘’โˆ’๐œ”๐œ”๐‘ก๐‘ก ๐‘ข๐‘ข(๐‘ก๐‘ก)

โ–ซ If ๐‘Ž๐‘Ž = ๐‘๐‘,๐‘Œ๐‘Œ ๐‘—๐‘—๐œ”๐œ” = ๐‘—๐‘— ๐‘‘๐‘‘๐‘‘๐‘‘๐‘—๐‘—

1๐‘Ž๐‘Ž+๐‘—๐‘—๐‘—๐‘—

โŸน ๐‘ฆ๐‘ฆ ๐‘ก๐‘ก = ๐‘ก๐‘ก โ‹… โ„ฑโˆ’1{ 1๐‘Ž๐‘Ž+๐‘—๐‘—๐‘—๐‘—

} = ๐‘ก๐‘ก๐‘’๐‘’โˆ’๐‘Ž๐‘Ž๐‘ก๐‘ก๐‘ข๐‘ข(๐‘ก๐‘ก)

69

Example โ€ข Determine the Response of an LTI system with impulse response โ„Ž ๐‘ก๐‘ก = ๐‘’๐‘’โˆ’๐‘Ž๐‘Ž๐‘ก๐‘ก๐‘ข๐‘ข(๐‘ก๐‘ก), to the input ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก = cos(๐œ”๐œ”0๐‘ก๐‘ก), where ๐‘Ž๐‘Ž > 0

โ€ข Frequency response:

๐ป๐ป ๐‘—๐‘—๐œ”๐œ” =1

๐‘Ž๐‘Ž + ๐‘—๐‘—๐œ”๐œ”=

๐‘Ž๐‘Ž โˆ’ ๐‘—๐‘—๐œ”๐œ”๐‘Ž๐‘Ž2 + ๐œ”๐œ”2 =

1๐‘Ž๐‘Ž2 + ๐œ”๐œ”2

๐‘’๐‘’โˆ’๐‘—๐‘—โ‹…arc๐‘—an(๐‘—๐‘—/๐‘Ž๐‘Ž)

โ€ข Method 1: using eigenfunction property

๐‘ฅ๐‘ฅ ๐‘ก๐‘ก =12๐‘’๐‘’๐‘—๐‘—๐‘—๐‘—0๐‘ก๐‘ก +

12๐‘’๐‘’โˆ’๐‘—๐‘—๐‘—๐‘—0๐‘ก๐‘ก

๐‘ฆ๐‘ฆ ๐‘ก๐‘ก =12๐ป๐ป ๐‘—๐‘—๐œ”๐œ”0 ๐‘’๐‘’๐‘—๐‘—๐‘—๐‘—0๐‘ก๐‘ก +

12๐ป๐ป โˆ’๐‘—๐‘—๐œ”๐œ”0 ๐‘’๐‘’โˆ’๐‘—๐‘—๐‘—๐‘—0๐‘ก๐‘ก = โ„›โ„ฏ ๐ป๐ป ๐‘—๐‘—๐œ”๐œ”0 ๐‘’๐‘’๐‘—๐‘—๐‘—๐‘—0๐‘ก๐‘ก

=๐‘Ž๐‘Ž cos(๐œ”๐œ”0๐‘ก๐‘ก) + ๐œ”๐œ”0sin(๐œ”๐œ”0๐‘ก๐‘ก)

๐‘Ž๐‘Ž2 + ๐œ”๐œ”02= 1/ ๐‘Ž๐‘Ž2 + ๐œ”๐œ”02 cos ๐œ”๐œ”0๐‘ก๐‘ก โˆ’ arctan

๐œ”๐œ”๐‘Ž๐‘Ž

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Example โ€ข Determine the Response of an LTI system with impulse response โ„Ž ๐‘ก๐‘ก = ๐‘’๐‘’โˆ’๐‘Ž๐‘Ž๐‘ก๐‘ก๐‘ข๐‘ข(๐‘ก๐‘ก), to the input ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก = cos(๐œ”๐œ”0๐‘ก๐‘ก), where ๐‘Ž๐‘Ž > 0

โ€ข Frequency response:

๐ป๐ป ๐‘—๐‘—๐œ”๐œ” =1

๐‘Ž๐‘Ž + ๐‘—๐‘—๐œ”๐œ”=

๐‘Ž๐‘Ž โˆ’ ๐‘—๐‘—๐œ”๐œ”๐‘Ž๐‘Ž2 + ๐œ”๐œ”2 =

1๐‘Ž๐‘Ž2 + ๐œ”๐œ”2

๐‘’๐‘’โˆ’๐‘—๐‘—โ‹…arc๐‘—an(๐‘—๐‘—/๐‘Ž๐‘Ž)

โ€ข Method 2: using Fourier transform

๐‘ฅ๐‘ฅ ๐‘ก๐‘ก =12๐‘’๐‘’๐‘—๐‘—๐‘—๐‘—0๐‘ก๐‘ก +

12๐‘’๐‘’โˆ’๐‘—๐‘—๐‘—๐‘—0๐‘ก๐‘ก โŸน ๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” = ๐œ‹๐œ‹๐›ฟ๐›ฟ ๐œ”๐œ” โˆ’ ๐œ”๐œ”0 + ๐œ‹๐œ‹๐›ฟ๐›ฟ(๐œ”๐œ” + ๐œ”๐œ”0)

๐‘Œ๐‘Œ ๐‘—๐‘—๐œ”๐œ” = ๐‘‹๐‘‹(๐‘—๐‘—๐œ”๐œ”)๐ป๐ป ๐‘—๐‘—๐œ”๐œ” = ๐œ‹๐œ‹๐ป๐ป ๐‘—๐‘—๐œ”๐œ”0 ๐›ฟ๐›ฟ ๐œ”๐œ” โˆ’ ๐œ”๐œ”0 + ๐œ‹๐œ‹๐ป๐ป โˆ’๐‘—๐‘—๐œ”๐œ”0 ๐›ฟ๐›ฟ(๐œ”๐œ” + ๐œ”๐œ”0)

โŸน ๐‘ฆ๐‘ฆ ๐‘ก๐‘ก =12๐ป๐ป ๐‘—๐‘—๐œ”๐œ”0 ๐‘’๐‘’๐‘—๐‘—๐‘—๐‘—0๐‘ก๐‘ก +

12๐ป๐ป โˆ’๐‘—๐‘—๐œ”๐œ”0 ๐‘’๐‘’โˆ’๐‘—๐‘—๐‘—๐‘—0๐‘ก๐‘ก

71

Example โ€ข Ideal lowpass filter with impulse response โ„Ž ๐‘ก๐‘ก to input ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก

โ„Ž ๐‘ก๐‘ก =sin(๐œ”๐œ”๐‘๐‘๐‘ก๐‘ก)

๐œ‹๐œ‹๐‘ก๐‘ก, ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก =

sin(๐œ”๐œ”๐‘–๐‘–๐‘ก๐‘ก)๐œ‹๐œ‹๐‘ก๐‘ก

โ€ข Fourier transform:

๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” = ๐‘ข๐‘ข ๐œ”๐œ” + ๐œ”๐œ”๐‘–๐‘– โˆ’ ๐‘ข๐‘ข(๐œ”๐œ” โˆ’ ๐œ”๐œ”๐‘–๐‘–)

๐ป๐ป ๐‘—๐‘—๐œ”๐œ” = ๐‘ข๐‘ข ๐œ”๐œ” + ๐œ”๐œ”๐‘๐‘ โˆ’ ๐‘ข๐‘ข(๐œ”๐œ” โˆ’ ๐œ”๐œ”๐‘๐‘)

โ€ข Use ๐‘Œ๐‘Œ ๐‘—๐‘—๐œ”๐œ” = ๐‘‹๐‘‹(๐‘—๐‘—๐œ”๐œ”)๐ป๐ป ๐‘—๐‘—๐œ”๐œ”

๐‘Œ๐‘Œ ๐‘—๐‘—๐œ”๐œ” = ๏ฟฝ๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” , if ๐œ”๐œ”๐‘–๐‘– โ‰ค ๐œ”๐œ”๐‘๐‘๐ป๐ป ๐‘—๐‘—๐œ”๐œ” , if ๐œ”๐œ”๐‘–๐‘– > ๐œ”๐œ”๐‘๐‘

โŸน ๐‘ฆ๐‘ฆ ๐‘ก๐‘ก = ๏ฟฝ๐‘ฅ๐‘ฅ(๐‘ก๐‘ก), if ๐œ”๐œ”๐‘–๐‘– โ‰ค ๐œ”๐œ”๐‘๐‘โ„Ž(๐‘ก๐‘ก), if ๐œ”๐œ”๐‘–๐‘– > ๐œ”๐œ”๐‘๐‘

โ€ข Note: Convolution of two ๐‘ ๐‘ ๐‘ ๐‘ ๐‘›๐‘›๐‘ ๐‘ () functions is another ๐‘ ๐‘ ๐‘ ๐‘ ๐‘›๐‘›๐‘ ๐‘ () function

72

๐‘ฆ๐‘ฆ = (๐‘ฅ๐‘ฅ โˆ— โ„Ž1) โˆ— โ„Ž2 = ๐‘ฅ๐‘ฅ โˆ— (โ„Ž1 โˆ— โ„Ž2) = (๐‘ฅ๐‘ฅ โˆ— โ„Ž2) โˆ— โ„Ž1 โ„Ž(๐‘ก๐‘ก) = โ„Ž1(๐‘ก๐‘ก) โˆ— โ„Ž2(๐‘ก๐‘ก) โ„Ž(๐‘ก๐‘ก) = โ„Ž2(๐‘ก๐‘ก) โˆ— โ„Ž1(๐‘ก๐‘ก) Note: Order of processing usually not important for LTI systems

System Connections: Cascade

73

โ„Ž1(๐‘ก๐‘ก) ๐‘ฅ๐‘ฅ(๐‘ก๐‘ก) ๐‘ฆ๐‘ฆ(๐‘ก๐‘ก) โ„Ž2(๐‘ก๐‘ก) h(๐‘ก๐‘ก)

โ„Ž2(๐‘ก๐‘ก) ๐‘ฅ๐‘ฅ(๐‘ก๐‘ก) ๐‘ฆ๐‘ฆ(๐‘ก๐‘ก) โ„Ž1(๐‘ก๐‘ก) h(๐‘ก๐‘ก)

commutative

๐‘Œ๐‘Œ = ๐‘‹๐‘‹ โ‹… ๐ป๐ป1 โ‹… ๐ป๐ป2= ๐‘‹๐‘‹ โ‹… ๐ป๐ป1 โ‹… ๐ป๐ป2 = ๐‘‹๐‘‹ โ‹… ๐ป๐ป2 โ‹… ๐ป๐ป1 ๐ป๐ป(๐‘—๐‘—๐œ”๐œ”) = ๐ป๐ป1(๐‘—๐‘—๐œ”๐œ”)๐ป๐ป2(๐‘—๐‘—๐œ”๐œ”)

๐ป๐ป(๐‘—๐‘—๐œ”๐œ”) = ๐ป๐ป2(๐‘—๐‘—๐œ”๐œ”)๐ป๐ป1(๐‘—๐‘—๐œ”๐œ”) Note: Order of processing usually not important for LTI systems

System Connections: Cascade

74

๐ป๐ป1(๐‘—๐‘—๐œ”๐œ”) ๐‘ฅ๐‘ฅ(๐‘ก๐‘ก) ๐‘ฆ๐‘ฆ(๐‘ก๐‘ก) ๐ป๐ป2(๐‘—๐‘—๐œ”๐œ”) ๐ป๐ป(๐‘—๐‘—๐œ”๐œ”)

๐ป๐ป2(๐‘—๐‘—๐œ”๐œ”) ๐‘ฅ๐‘ฅ(๐‘ก๐‘ก) ๐‘ฆ๐‘ฆ(๐‘ก๐‘ก) ๐ป๐ป1(๐‘—๐‘—๐œ”๐œ”)

commutative

๐ป๐ป(๐‘—๐‘—๐œ”๐œ”)

System Connections: Parallel โ€ข Time domain

โ„Ž ๐‘ก๐‘ก = โ„Ž1 ๐‘ก๐‘ก + โ„Ž2(๐‘ก๐‘ก)

โ€ข Frequency domain

๐ป๐ป ๐‘—๐‘—๐œ”๐œ” = ๐ป๐ป1 ๐‘—๐‘—๐œ”๐œ” + ๐ป๐ป2(๐‘—๐‘—๐œ”๐œ”)

75

โ„Ž1(๐‘ก๐‘ก) ๐‘ฆ๐‘ฆ(๐‘ก๐‘ก)

โ„Ž2(๐‘ก๐‘ก) ๐‘ฅ๐‘ฅ(๐‘ก๐‘ก) +

โ„Ž ๐‘ก๐‘ก

๐ป๐ป1(๐‘—๐‘—๐œ”๐œ”) ๐‘ฆ๐‘ฆ(๐‘ก๐‘ก)

๐ป๐ป2(๐‘—๐‘—๐œ”๐œ”) ๐‘ฅ๐‘ฅ(๐‘ก๐‘ก) +

๐ป๐ป ๐‘—๐‘—๐œ”๐œ”

System Connections: Feedback โ€ข Time domain ๐‘ฆ๐‘ฆ = โ„Ž1 โˆ— ๐‘ฅ๐‘ฅ โˆ’ โ„Ž2 โˆ— ๐‘ฆ๐‘ฆ โ„Ž ๐‘ก๐‘ก = ?

โ€ข Frequency domain

๐‘Œ๐‘Œ = ๐ป๐ป1๐‘‹๐‘‹ โˆ’ ๐ป๐ป1๐ป๐ป2๐‘Œ๐‘Œ

๐ป๐ป =๐‘Œ๐‘Œ๐‘‹๐‘‹ =

๐ป๐ป11 + ๐ป๐ป1๐ป๐ป2

76

โ„Ž1(๐‘ก๐‘ก) ๐‘ฆ๐‘ฆ(๐‘ก๐‘ก)

โ„Ž2(๐‘ก๐‘ก)

๐‘ฅ๐‘ฅ(๐‘ก๐‘ก)

โ„Ž ๐‘ก๐‘ก

+ +

-

๐ป๐ป1(๐‘—๐‘—๐œ”๐œ”) ๐‘ฆ๐‘ฆ(๐‘ก๐‘ก)

๐ป๐ป2(๐‘—๐‘—๐œ”๐œ”)

๐‘ฅ๐‘ฅ(๐‘ก๐‘ก)

๐ป๐ป ๐‘—๐‘—๐œ”๐œ”

+ +

-

Filtering โ€ข changes relative amplitudes of frequency components, or eliminates

some frequency components entirely

โ€ข LPS:

โ€ข HPS:

โ€ข BPS:

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๐ป๐ป(๐‘—๐‘—๐œ”๐œ”)

๐œ”๐œ”๐‘๐‘ โˆ’๐œ”๐œ”๐‘๐‘

1

๐œ”๐œ”

passband stopband cutoff frequency

๐ป๐ป(๐‘—๐‘—๐œ”๐œ”)

๐ป๐ป(๐‘—๐‘—๐œ”๐œ”)

๐ป๐ป(๐‘—๐‘—๐œ”๐œ”) = ๏ฟฝ1, ๐œ”๐œ” โ‰ค ๐œ”๐œ”๐‘๐‘0, ๐œ”๐œ” > ๐œ”๐œ”๐‘๐‘

๐ป๐ป(๐‘—๐‘—๐œ”๐œ”) = ๏ฟฝ1, ๐œ”๐œ” โ‰ฅ ๐œ”๐œ”๐‘๐‘0, ๐œ”๐œ” < ๐œ”๐œ”๐‘๐‘

๐ป๐ป(๐‘—๐‘—๐œ”๐œ”) = ๏ฟฝ1, ๐œ”๐œ”๐‘๐‘1 < ๐œ”๐œ” < ๐œ”๐œ”๐‘๐‘20, otherwise

๐œ”๐œ”๐‘๐‘ โˆ’๐œ”๐œ”๐‘๐‘ ๐œ”๐œ”

๐œ”๐œ”๐‘๐‘1 โˆ’๐œ”๐œ”๐‘๐‘1 ๐œ”๐œ” โˆ’๐œ”๐œ”๐‘๐‘2

lower cutoff frequency

๐œ”๐œ”๐‘๐‘2

Ideal Zero Phase LPF โ€ข Ideal lowpass filter (unit magnitude, zero phase)

โ„Ž ๐‘ก๐‘ก =sin(๐œ”๐œ”๐‘๐‘๐‘ก๐‘ก)

๐œ‹๐œ‹๐‘ก๐‘ก, ๐ป๐ป ๐‘—๐‘—๐œ”๐œ” = ๏ฟฝ1, ๐œ”๐œ” โ‰ค ๐œ”๐œ”๐‘๐‘

0, otherwise

โ€ข ๐’‹๐’‹๐’„๐’„: cutoff frequency

78

Simple RC Lowpass Filter โ€ข Differential Equation

๐‘…๐‘…๐‘…๐‘…๐‘‘๐‘‘๐‘ฃ๐‘ฃ๐‘๐‘ ๐‘ก๐‘ก๐‘‘๐‘‘๐‘ก๐‘ก + ๐‘ฃ๐‘ฃ๐‘๐‘ ๐‘ก๐‘ก = ๐‘ฃ๐‘ฃ๐‘ ๐‘ (๐‘ก๐‘ก)

โ€ข Input ๐‘ฃ๐‘ฃ๐‘ ๐‘  ๐‘ก๐‘ก , output ๐‘ฃ๐‘ฃ๐‘๐‘ ๐‘ก๐‘ก

โ€ข Taking Fourier transform ๐‘—๐‘—๐œ”๐œ”๐‘…๐‘…๐‘…๐‘… โ‹… ๐‘‰๐‘‰๐‘๐‘ ๐‘—๐‘—๐œ”๐œ” + ๐‘‰๐‘‰๐‘๐‘ ๐‘—๐‘—๐œ”๐œ” = ๐‘‰๐‘‰๐‘ ๐‘ (๐‘—๐‘—๐œ”๐œ”) โ€ข Frequency response:

๐ป๐ป ๐‘—๐‘—๐œ”๐œ” =๐‘‰๐‘‰๐‘๐‘ ๐‘—๐‘—๐œ”๐œ”๐‘‰๐‘‰๐‘ ๐‘  ๐‘—๐‘—๐œ”๐œ”

=1

1 + ๐‘—๐‘—๐œ”๐œ”๐‘…๐‘…๐‘…๐‘…

79

Simple RC Lowpass Filter โ€ข Frequency response

๐ป๐ป ๐‘—๐‘—๐œ”๐œ” =1

1 + ๐‘…๐‘…๐‘…๐‘…๐‘—๐‘—๐œ”๐œ”

|๐ป๐ป ๐‘—๐‘—๐œ”๐œ” | =1

1 + ๐‘…๐‘…๐‘…๐‘…๐œ”๐œ” 2

arg๐ป๐ป(๐‘—๐‘—๐œ”๐œ”) = โˆ’ arctan ๐‘…๐‘…๐‘…๐‘…๐œ”๐œ”

โ€ข Impulse response

โ„Ž ๐‘ก๐‘ก =1๐‘…๐‘…๐‘…๐‘…

๐‘’๐‘’โˆ’๐‘ก๐‘ก/๐‘…๐‘…๐‘…๐‘…๐‘ข๐‘ข(๐‘ก๐‘ก)

โ€ข Nonideal lowpass filter โ–ซ passes lower frequencies โ–ซ attenuates higher frequencies

โ€ข Larger RC โŸน passes smaller range of lower frequencies

80

โ€ข Ideal lowpass filter (unit magnitude, linear phase)

๐ป๐ป ๐‘—๐‘—๐œ”๐œ” = ๏ฟฝ๐‘’๐‘’โˆ’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก0 , ๐œ”๐œ” โ‰ค ๐œ”๐œ”๐‘๐‘

0, otherwise

โŸน |๐ป๐ป(๐‘—๐‘—๐œ”๐œ”)| = ๏ฟฝ1,โ€ƒ|๐œ”๐œ”| โ‰ค ๐œ”๐œ”๐‘๐‘0,โ€ƒ|๐œ”๐œ”| > ๐œ”๐œ”๐‘๐‘

, โ‰ฎ ๐ป๐ป(๐‘—๐‘—๐œ”๐œ”) = ๏ฟฝโˆ’๐œ”๐œ”๐‘ก๐‘ก0 ๐œ”๐œ” โ‰ค ๐œ”๐œ”๐‘๐‘ 0, otherwise

โ€ข ๐‘Œ๐‘Œ ๐‘—๐‘—๐œ”๐œ” = ๐‘’๐‘’โˆ’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก0๐‘‹๐‘‹๐ฟ๐ฟ๐ฟ๐ฟ ๐‘—๐‘—๐œ”๐œ” โŸน ๐‘ฆ๐‘ฆ ๐‘ก๐‘ก = ๐‘ฅ๐‘ฅ๐ฟ๐ฟ๐ฟ๐ฟ(๐‘ก๐‘ก โˆ’ ๐‘ก๐‘ก0) โ–ซ Linear phase โŸบ simply a shift in time, no distortion โ–ซ Nonlinear phase โŸบ distortion in addition to time shift

Ideal Linear Phase LPF

81

๐œ”๐œ”๐‘๐‘ โˆ’๐œ”๐œ”๐‘๐‘

โ‰ฎ ๐ป๐ป(๐‘—๐‘—๐œ”๐œ”) ๐œ”๐œ”๐‘๐‘๐‘ก๐‘ก0

โˆ’๐œ”๐œ”๐‘๐‘๐‘ก๐‘ก0

Example โ€ข Frequency response of an LTI system is

๐ป๐ป ๐‘—๐‘—๐œ”๐œ” = ๏ฟฝ๐‘—๐‘—๐œ”๐œ”3๐œ‹๐œ‹

, ๐œ”๐œ” โ‰ค 3๐œ‹๐œ‹

0, otherwise

โ€ข Considered as a cascade of an LPF and a differentiator

82

๐‘ฅ๐‘ฅ(๐‘ก๐‘ก) ๐‘ฆ๐‘ฆ(๐‘ก๐‘ก) ๐ป๐ป2(๐‘—๐‘—๐œ”๐œ”) = ๐‘—๐‘—๐œ”๐œ”

๐ป๐ป(๐‘—๐‘—๐œ”๐œ”)

๐ป๐ป1(๐‘—๐‘—๐œ”๐œ”)

1/3๐œ‹๐œ‹

3๐œ‹๐œ‹ โˆ’3๐œ‹๐œ‹

โ€ข Dual of convolution property:

๐‘ฅ๐‘ฅ ๐‘ก๐‘ก โ‹… ๐‘ฆ๐‘ฆ ๐‘ก๐‘ก โ„ฑ

12๐œ‹๐œ‹

[๐‘‹๐‘‹(๐‘—๐‘—๐œ”๐œ”) โˆ— ๐‘Œ๐‘Œ(๐‘—๐‘—๐œ”๐œ”)]

multiplication in time โŸบ convolution in frequency

โ€ข Proof: let ๐‘๐‘ ๐‘—๐‘—๐œ”๐œ” = 12๐œ‹๐œ‹ โˆซ ๐‘‹๐‘‹ ๐‘—๐‘—๐œƒ๐œƒ ๐‘Œ๐‘Œ(๐‘—๐‘—(๐œ”๐œ” โˆ’ ๐œƒ๐œƒ))๐‘‘๐‘‘๐œƒ๐œƒ+โˆž

โˆ’โˆž

โ„ฑโˆ’1 ๐‘๐‘ ๐‘—๐‘—๐œ”๐œ” =12๐œ‹๐œ‹

๏ฟฝ12๐œ‹๐œ‹

๏ฟฝ ๐‘‹๐‘‹ ๐‘—๐‘—๐œƒ๐œƒ ๐‘Œ๐‘Œ(๐‘—๐‘—(๐œ”๐œ” โˆ’ ๐œƒ๐œƒ))๐‘‘๐‘‘๐œƒ๐œƒ+โˆž

โˆ’โˆž๐‘’๐‘’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก๐‘‘๐‘‘๐œ”๐œ”

+โˆž

โˆ’โˆž

=12๐œ‹๐œ‹

๏ฟฝ ๐‘‹๐‘‹ ๐‘—๐‘—๐œƒ๐œƒ12๐œ‹๐œ‹

๏ฟฝ ๐‘Œ๐‘Œ(๐‘—๐‘—(๐œ”๐œ” โˆ’ ๐œƒ๐œƒ))๐‘’๐‘’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก๐‘‘๐‘‘๐œ”๐œ”+โˆž

โˆ’โˆž๐‘‘๐‘‘๐œƒ๐œƒ

+โˆž

โˆ’โˆž

=12๐œ‹๐œ‹

๏ฟฝ ๐‘‹๐‘‹ ๐‘—๐‘—๐œƒ๐œƒ ๐‘ฆ๐‘ฆ ๐‘ก๐‘ก ๐‘’๐‘’๐‘—๐‘—๐œ‹๐œ‹๐‘ก๐‘ก๐‘‘๐‘‘๐œƒ๐œƒ = ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก ๐‘ฆ๐‘ฆ(๐‘ก๐‘ก)+โˆž

โˆ’โˆž

Multiplication Property

83

Example โ€ข Define a signal as

๐‘ฅ๐‘ฅ ๐‘ก๐‘ก =sin ๐‘ก๐‘ก โ‹… sin(๐‘ก๐‘ก/2)

๐œ‹๐œ‹๐‘ก๐‘ก2โ‰œ ๐œ‹๐œ‹๐‘ฅ๐‘ฅ1 ๐‘ก๐‘ก ๐‘ฅ๐‘ฅ2 ๐‘ก๐‘ก

where,

๐‘ฅ๐‘ฅ1 ๐‘ก๐‘ก =sin ๐‘ก๐‘ก๐œ‹๐œ‹๐‘ก๐‘ก

๐‘ฅ๐‘ฅ2 ๐‘ก๐‘ก =sin(๐‘ก๐‘ก/2)

๐œ‹๐œ‹๐‘ก๐‘ก

โ€ข Fourier transform

๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” =12๐‘‹๐‘‹1 ๐‘—๐‘—๐œ”๐œ” โˆ— ๐‘‹๐‘‹2(๐‘—๐‘—๐œ”๐œ”)

84

Example: Modulation Baseband signal: ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก Carrier:

๐‘๐‘ ๐‘ก๐‘ก = cos(๐œ”๐œ”๐‘๐‘๐‘ก๐‘ก) =12

(๐‘’๐‘’๐‘—๐‘—๐‘—๐‘—๐‘๐‘๐‘ก๐‘ก + ๐‘’๐‘’โˆ’๐‘—๐‘—๐‘—๐‘—๐‘๐‘๐‘ก๐‘ก )

๐‘ƒ๐‘ƒ ๐‘—๐‘—๐œ”๐œ” = ๐œ‹๐œ‹๐›ฟ๐›ฟ ๐œ”๐œ” โˆ’ ๐œ”๐œ”๐‘๐‘ + ๐œ‹๐œ‹๐›ฟ๐›ฟ ๐œ”๐œ” + ๐œ”๐œ”๐‘๐‘

โ€ข Modulated signal: ๐‘ฆ๐‘ฆ ๐‘ก๐‘ก = ๐‘ฅ๐‘ฅ(๐‘ก๐‘ก)๐‘๐‘ ๐‘ก๐‘ก

๐‘Œ๐‘Œ ๐‘—๐‘—๐œ”๐œ” =12๐‘‹๐‘‹ ๐‘—๐‘—(๐œ”๐œ” โˆ’ ๐œ”๐œ”๐‘๐‘) +

12๐‘‹๐‘‹ ๐‘—๐‘—(๐œ”๐œ” + ๐œ”๐œ”๐‘๐‘)

85

Example: Demodulation โ€ข Modulated signal: ๐‘ฆ๐‘ฆ ๐‘ก๐‘ก = ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก ๐‘๐‘ ๐‘ก๐‘ก

โ€ข Carrier: ๐‘๐‘ ๐‘ก๐‘ก = cos(๐œ”๐œ”๐‘๐‘๐‘ก๐‘ก)

โ€ข Demodulated signal: ๐‘ง๐‘ง ๐‘ก๐‘ก = ๐‘ฆ๐‘ฆ ๐‘ก๐‘ก ๐‘๐‘ ๐‘ก๐‘ก

๐‘๐‘ ๐‘—๐‘—๐œ”๐œ” =12๐‘Œ๐‘Œ ๐‘—๐‘—(๐œ”๐œ” โˆ’ ๐œ”๐œ”๐‘๐‘) +

12๐‘Œ๐‘Œ ๐‘—๐‘—(๐œ”๐œ” + ๐œ”๐œ”๐‘๐‘)

=12๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” +

14

[๐‘‹๐‘‹(๐‘—๐‘— ๐œ”๐œ” โˆ’ 2๐œ”๐œ”๐‘๐‘ ) + ๐‘‹๐‘‹(๐‘—๐‘— ๐œ”๐œ” + 2๐œ”๐œ”๐‘๐‘ )] ๐‘…๐‘… ๐‘—๐‘—๐œ”๐œ” = ๐‘๐‘ ๐‘—๐‘—๐œ”๐œ” ๐ป๐ปlowpass(๐‘—๐‘—๐œ”๐œ”) ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก = ๐‘Ÿ๐‘Ÿ ๐‘ก๐‘ก

86

Ideal AM Communication System โ€ข Amplitude modulation

87

LCCDEs โ€ข Determine the frequency response of an LTI system described by

๏ฟฝ๐‘Ž๐‘Ž๐‘˜๐‘˜๐‘‘๐‘‘๐‘˜๐‘˜๐‘ฆ๐‘ฆ(๐‘ก๐‘ก)๐‘‘๐‘‘๐‘ก๐‘ก๐‘˜๐‘˜

๐‘๐‘

๐‘˜๐‘˜=0

= ๏ฟฝ๐‘๐‘๐‘˜๐‘˜๐‘‘๐‘‘๐‘˜๐‘˜๐‘ฅ๐‘ฅ(๐‘ก๐‘ก)๐‘‘๐‘‘๐‘ก๐‘ก๐‘˜๐‘˜

๐‘€๐‘€

๐‘˜๐‘˜=0

โ€ข Method 1: using the eigenfunction property

let ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก = ๐‘’๐‘’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก โ†’ ๐‘ฆ๐‘ฆ ๐‘ก๐‘ก = ๐ป๐ป ๐‘—๐‘—๐œ”๐œ” ๐‘’๐‘’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก

substitution in to the differential equation yields

๏ฟฝ๐‘Ž๐‘Ž๐‘˜๐‘˜๐ป๐ป ๐‘—๐‘—๐œ”๐œ” ๐‘—๐‘—๐œ”๐œ” ๐‘˜๐‘˜๐‘’๐‘’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก๐‘๐‘

๐‘˜๐‘˜=0

= ๏ฟฝ๐‘๐‘๐‘˜๐‘˜ ๐‘—๐‘—๐œ”๐œ” ๐‘˜๐‘˜๐‘’๐‘’๐‘—๐‘—๐‘—๐‘—๐‘ก๐‘ก๐‘€๐‘€

๐‘˜๐‘˜=0

๐ป๐ป ๐‘—๐‘—๐œ”๐œ” =โˆ‘ ๐‘๐‘๐‘˜๐‘˜ ๐‘—๐‘—๐œ”๐œ” ๐‘˜๐‘˜๐‘€๐‘€๐‘˜๐‘˜=0

โˆ‘ ๐‘Ž๐‘Ž๐‘˜๐‘˜ ๐‘—๐‘—๐œ”๐œ” ๐‘˜๐‘˜๐‘๐‘๐‘˜๐‘˜=0

88

LCCDEs โ€ข Method 2: taking the Fourier transform of both sides

โ„ฑ ๏ฟฝ๐‘Ž๐‘Ž๐‘˜๐‘˜๐‘‘๐‘‘๐‘˜๐‘˜๐‘ฆ๐‘ฆ(๐‘ก๐‘ก)๐‘‘๐‘‘๐‘ก๐‘ก๐‘˜๐‘˜

๐‘๐‘

๐‘˜๐‘˜=0

= โ„ฑ ๏ฟฝ๐‘๐‘๐‘˜๐‘˜๐‘‘๐‘‘๐‘˜๐‘˜๐‘ฅ๐‘ฅ(๐‘ก๐‘ก)๐‘‘๐‘‘๐‘ก๐‘ก๐‘˜๐‘˜

๐‘€๐‘€

๐‘˜๐‘˜=0

โ€ข By linearity and differentiation in time property

๏ฟฝ๐‘Ž๐‘Ž๐‘˜๐‘˜ ๐‘—๐‘—๐œ”๐œ” ๐‘˜๐‘˜๐‘Œ๐‘Œ(๐‘—๐‘—๐œ”๐œ”) =๐‘๐‘

๐‘˜๐‘˜=0

๏ฟฝ๐‘๐‘๐‘˜๐‘˜ ๐‘—๐‘—๐œ”๐œ” ๐‘˜๐‘˜๐‘‹๐‘‹(๐‘—๐‘—๐œ”๐œ”) ๐‘€๐‘€

๐‘˜๐‘˜=0

๐ป๐ป ๐‘—๐‘—๐œ”๐œ” =๐‘Œ๐‘Œ ๐‘—๐‘—๐œ”๐œ”๐‘‹๐‘‹(๐‘—๐‘—๐œ”๐œ”)

=โˆ‘ ๐‘๐‘๐‘˜๐‘˜ ๐‘—๐‘—๐œ”๐œ” ๐‘˜๐‘˜๐‘€๐‘€๐‘˜๐‘˜=0

โˆ‘ ๐‘Ž๐‘Ž๐‘˜๐‘˜ ๐‘—๐‘—๐œ”๐œ” ๐‘˜๐‘˜๐‘๐‘๐‘˜๐‘˜=0

โ€ข ๐ป๐ป ๐‘—๐‘—๐œ”๐œ” is a rational function of (๐‘—๐‘—๐œ”๐œ”), i.e., ratio of polynomials

89

Example ๐‘ฆ๐‘ฆโ€ฒโ€ฒ ๐‘ก๐‘ก + 4๐‘ฆ๐‘ฆโ€ฒโ€ฒ ๐‘ก๐‘ก + 3๐‘ฆ๐‘ฆ ๐‘ก๐‘ก = ๐‘ฅ๐‘ฅโ€ฒ ๐‘ก๐‘ก + 2๐‘ฅ๐‘ฅ(๐‘ก๐‘ก)

โ€ข Frequency response

๐ป๐ป ๐‘—๐‘—๐œ”๐œ” =๐‘—๐‘—๐œ”๐œ” + 2

๐‘—๐‘—๐œ”๐œ” 2 + 4 ๐‘—๐‘—๐œ”๐œ” + 3

โ–ซ Using partial fraction expansion

๐ป๐ป ๐‘—๐‘—๐œ”๐œ” =๐‘—๐‘—๐œ”๐œ” + 2

๐‘—๐‘—๐œ”๐œ” + 1 (๐‘—๐‘—๐œ”๐œ” + 3)=

๐ด๐ด1๐‘—๐‘—๐œ”๐œ” + 1

+๐ด๐ด2

๐‘—๐‘—๐œ”๐œ” + 3

โ–ซ Let ๐‘ฃ๐‘ฃ = ๐‘—๐‘—๐œ”๐œ”

๐ด๐ด1= ๐‘ฃ๐‘ฃ + 1 ๐ป๐ป ๐‘ฃ๐‘ฃ ๏ฟฝ๐‘ฃ๐‘ฃ=โˆ’1

=12

,๐ด๐ด2 = ๐‘ฃ๐‘ฃ + 3 ๐ป๐ป ๐‘ฃ๐‘ฃ ๏ฟฝ๐‘ฃ๐‘ฃ=โˆ’3

=12

โ–ซ Taking inverse Fourier transform to find impulse response

โ„Ž ๐‘ก๐‘ก =12๐‘’๐‘’โˆ’๐‘ก๐‘ก +

12๐‘’๐‘’โˆ’3๐‘ก๐‘ก ๐‘ข๐‘ข ๐‘ก๐‘ก

90

Example ๐‘ฆ๐‘ฆโ€ฒโ€ฒ ๐‘ก๐‘ก + 4๐‘ฆ๐‘ฆโ€ฒโ€ฒ ๐‘ก๐‘ก + 3๐‘ฆ๐‘ฆ ๐‘ก๐‘ก = ๐‘ฅ๐‘ฅโ€ฒ ๐‘ก๐‘ก + 2๐‘ฅ๐‘ฅ(๐‘ก๐‘ก)

โ€ข Determine zero-state response to ๐‘ฅ๐‘ฅ ๐‘ก๐‘ก = ๐‘’๐‘’โˆ’๐‘ก๐‘ก๐‘ข๐‘ข ๐‘ก๐‘ก

๐‘Œ๐‘Œ ๐‘—๐‘—๐œ”๐œ” = ๐ป๐ป ๐‘—๐‘—๐œ”๐œ” ๐‘‹๐‘‹ ๐‘—๐‘—๐œ”๐œ” =๐‘—๐‘—๐œ”๐œ” + 2

๐‘—๐‘—๐œ”๐œ” + 1 ๐‘—๐‘—๐œ”๐œ” + 3 โ‹…1

๐‘—๐‘—๐œ”๐œ” + 1=

๐ด๐ด1,1๐‘—๐‘—๐œ”๐œ” + 1 +

๐ด๐ด1,2๐‘—๐‘—๐œ”๐œ” + 1 2 โˆ’

๐ด๐ด2๐‘—๐‘—๐œ”๐œ” + 3

โ–ซ Using partial fraction expansion, let ๐‘ฃ๐‘ฃ = ๐‘—๐‘—๐œ”๐œ”

๐ด๐ด1,1=1

2 โˆ’ 1 !๐‘‘๐‘‘๐‘‘๐‘‘๐‘ฃ๐‘ฃ

๐‘ฃ๐‘ฃ + 1 2๐‘Œ๐‘Œ ๐‘ฃ๐‘ฃ ๏ฟฝ๐‘ฃ๐‘ฃ=โˆ’1

=๐‘‘๐‘‘๐‘‘๐‘‘๐‘ฃ๐‘ฃ

๐‘ฃ๐‘ฃ + 2๐‘ฃ๐‘ฃ + 3

๏ฟฝ๐‘ฃ๐‘ฃ=โˆ’1

=14

๐ด๐ด1,2= ๐‘ฃ๐‘ฃ + 1 2๐‘Œ๐‘Œ ๐‘ฃ๐‘ฃ ๏ฟฝ๐‘ฃ๐‘ฃ=โˆ’1

=12

,๐ด๐ด2 = ๐‘ฃ๐‘ฃ + 3 ๐‘Œ๐‘Œ ๐‘ฃ๐‘ฃ ๏ฟฝ๐‘ฃ๐‘ฃ=โˆ’3

= โˆ’14

โ–ซ Taking inverse Fourier transform to find output

๐‘ฆ๐‘ฆ ๐‘ก๐‘ก =14๐‘’๐‘’โˆ’๐‘ก๐‘ก +

12๐‘ก๐‘ก๐‘’๐‘’โˆ’๐‘ก๐‘ก โˆ’

14๐‘’๐‘’โˆ’3๐‘ก๐‘ก ๐‘ข๐‘ข ๐‘ก๐‘ก

91

92

Many Thanks

Q & A

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