6.003: Signals and Systems

Transcription

1 6.003: Signals and Sysems Relaions among Fourier Represenaions November 5, 20

2 Mid-erm Examinaion #3 Wednesday, November 6, 7:30-9:30pm, No reciaions on he day of he exam. Coverage: Lecures 8 Reciaions 6 Homeworks 0 Homework 0 will no be colleced or graded. Soluions are posed. Closed book: 3 pages of noes (8 2 inches; fron and back). No calculaors, compuers, cell phones, music players, or oher aids. Designed as -hour exam; wo hours o complee. Prior erm miderm exams have been posed on he websie. 2

3 Fourier Represenaions We ve seen a variey of Fourier represenaions: CT Fourier series CT Fourier ransform DT Fourier series DT Fourier ransform Today: relaions among he four Fourier represenaions. 3

4 4 Four Fourier Represenaions We have discussed four closely relaed Fourier represenaions. DT Fourier Series a k = a k+n = x[n]e j 2π N kn N n=<n> x[n] = x[n + N] = a k e j 2π N kn k=<n> DT Fourier ransform X(e jω ) = x[n]e jωn n= x[n] = X(e jω )e jωn dω 2π <2π> CT Fourier Series a k = T T x() = x( + T ) = j 2π x()e T k d k= a k e j 2π T k CT Fourier ransform X(jω) = x()e jω d x() = X(jω)e jω dω 2π

5 5 Four Types of Time discree vs. coninuous () and periodic vs aperiodic ( ) DT Fourier Series DT Fourier ransform n n CT Fourier Series CT Fourier ransform

6 6 Four Types of Frequency discree vs. coninuous ( ) and periodic vs aperiodic () DT Fourier Series DT Fourier ransform 2π N k Ω CT Fourier Series 2π T k CT Fourier ransform ω

7 7 Relaion beween Fourier Series and Transform A periodic signal can be represened by a Fourier series or by an equivalen Fourier ransform. Series: represen periodic signal as weighed sum of harmonics 0 jω 0 k 2π x() = x( + T ) = a k e ; ω 0 = T k= The Fourier ransform of a sum is he sum of he Fourier ransforms: 0 X(jω) = 2πa k δ(ω kω 0 ) k= Therefore periodic signals can be equivalenly represened as Fourier ransforms (wih impulses!).

8 Relaion beween Fourier Series and Transform A periodic signal can be represened by a Fourier series or by an equivalen Fourier ransform. Fourier Series a a a 0 a 2 a 2 a 4 a 3 a 3 a4 x() = x( + T ) = a k e jω 0k k= 0 Fourier Transform k 2πa 4 2πa 3 2πa 2 2πa 2πa0 2πa 2πa2 2πa3 2πa4 0 ω 0 ω 8

9 Relaions among Fourier Represenaions Explore oher relaions among Fourier represenaions. Sar wih an aperiodic CT signal. Deermine is Fourier ransform. Conver he signal so ha i can be represened by alernae Fourier represenaions and compare. periodic DT DTFS N periodic exension aperiodic DT DTFT inerpolae sample inerpolae sample periodic CT CTFS T periodic exension aperiodic CT CTFT 9

10 Sar wih he CT Fourier Transform Deermine he Fourier ransform of he following signal. x() 0 Could calculae Fourier ransform from he definiion. X(jω) = x()e jω d Easier o calculae x() by convoluion of wo square pulses: y() y() 2 2

11 Sar wih he CT Fourier Transform The ransform of y() is Y (jω) = sin(ω/2) ω/2 y() 2 2 Y (jω) 2π 2π ω so he ransform of x() = (y y)() is X(jω) = Y (jω) Y (jω). x() X(jω) 2π 2π ω

12 2 Relaion beween Fourier Transform and Series Wha is he effec of making a signal periodic in ime? Find Fourier ransform of periodic exension of x() o period T = 4. z() = x( + 4k) k= 4 4 Could calculae Z(jω) for he definiion... ugly.

13 3 Relaion beween Fourier Transform and Series Easier o calculae z() by convolving x() wih an impulse rain. z() = x( + 4k) k= 4 4 z() = 0 x( + 4k) = (x p)() k= where 0 p() = δ( + 4k) Then k= Z(jω) = X(jω) P (jω)

14 4 Check Yourself Wha s he Fourier ransform of an impulse rain? x() = δ( kt ) k= 0 T

15 Check Yourself Wha s he Fourier ransform of an impulse rain? x() = X(jω) = k= δ( kt ) 0 T a k = T k T k= 2π 2π δ(ω k T T ) 0 2π T 5 2π T k ω

16 6 Relaion beween Fourier Transform and Series Easier o calculae z() by convolving x() wih an impulse rain. z() = x( + 4k) k= 4 4 z() = 0 x( + 4k) = (x p)() k= where 0 p() = δ( + 4k) Then k= Z(jω) = X(jω) P (jω)

17 7 Relaion beween Fourier Transform and Series Convolving in ime corresponds o muliplying in frequency. X(jω) 2π P (jω) 2π ω π 2 π 2 π 2 ω Z(jω) π/2 π 2 π 2 ω

18 8 Relaion beween Fourier Transform and Series The Fourier ransform of a periodically exended funcion is a discree funcion of frequency ω. z() = x( + 4k) k= 4 4 Z(jω) π/2 π 2 π 2 ω

19 9 Relaion beween Fourier Transform and Series The weigh (area) of each impulse in he Fourier ransform of a periodically exended funcion is 2π imes he corresponding Fourier series coefficien. Z(jω) π/2 π 2 π 2 ω a k /4 k

20 Relaion beween Fourier Transform and Series The effec of periodic exension of x() o z() is o sample he frequency represenaion. X(jω) 2π Z(jω) π/2 2π ω π 2 π 2 ω a k /4 20 k

21 Relaion beween Fourier Transform and Series Periodic exension of CT signal discree funcion of frequency. Periodic exension = convolving wih impulse rain in ime = muliplying by impulse rain in frequency sampling in frequency periodic DT DTFS N periodic exension aperiodic DT DTFT inerpolae sample inerpolae sample periodic CT CTFS T periodic exension (sampling in frequency) 2 aperiodic CT CTFT

22 22 Four Types of Time discree vs. coninuous () and periodic vs aperiodic ( ) DT Fourier Series DT Fourier ransform n n CT Fourier Series CT Fourier ransform

23 23 Four Types of Frequency discree vs. coninuous ( ) and periodic vs aperiodic () DT Fourier Series DT Fourier ransform 2π N k Ω CT Fourier Series 2π T k CT Fourier ransform ω

24 Relaion beween Fourier Transform and Series Periodic exension of CT signal discree funcion of frequency. Periodic exension = convolving wih impulse rain in ime = muliplying by impulse rain in frequency sampling in frequency periodic DT DTFS N periodic exension aperiodic DT DTFT inerpolae sample inerpolae sample periodic CT CTFS T periodic exension (sampling in frequency) 24 aperiodic CT CTFT

25 25 Relaions among Fourier Represenaions Compare o sampling in ime. periodic DT DTFS N periodic exension aperiodic DT DTFT inerpolae sample inerpolae sample periodic CT CTFS T periodic exension aperiodic CT CTFT

26 Relaions beween CT and DT ransforms Sampling a CT signal generaes a DT signal. x[n] = x(nt ) x() 0 Take T = 2. x[n] n Wha is he effec on he frequency represenaion? 26

27 27 Relaions beween CT and DT ransforms We can generae a signal wih he same shape by muliplying x() by an impulse rain wih T = 2. 0 x p () = x() p() where p() = δ( + kt ) k= x() 0 x[n] x p () n

28 Relaions beween CT and DT ransforms We can generae a signal wih he same shape by muliplying x() by an impulse rain wih T = 2. 0 x p () = x() p() where p() = δ( + kt ) x() k= 0 x p () 28

29 Relaions beween CT and DT ransforms Muliplying x() by an impulse rain in ime is equivalen o convolving X(jω) by an impulse rain in frequency (hen 2π). X(jω) 2π P (jω) 2π ω 4π 4π 4π ω X p (jω) 2 4π 4π ω 29

30 30 Relaions beween CT and DT ransforms Fourier ransform of sampled signal x p () is periodic in ω, period 4π. x p () X p (jω) 2 4π 4π ω

31 Relaions beween CT and DT ransforms Fourier ransform of sampled signal x p () has same shape as DT Fourier ransform of x[n]. x[n] n X(e jω ) 2 2π 2π Ω 3

32 DT Fourier ransform CT Fourier ransform of sampled signal x p () = DT Fourier ransform of samples x[n] where Ω = ωt, i.e., X(jω) = X(e jω ). Ω=ωT x p () X p (jω) 2 4π ω 4π x[n] X(e jω ) n 2π 2 Ω 2π Ω = ωt = 2 ω 32

33 33 Relaion beween CT and DT Fourier Transforms Compare he definiions: 0 X(e jω ) = x[n]e jωn n X jω p (jω) = x p ()e d 0 = x[n]δ( nt )e jω d n 0 = x[n] δ( nt )e jω d n 0 jωnt = x[n]e n Ω = ωt

34 Relaion Beween DT Fourier Transform and Series Periodic exension of a DT signal is equivalen o convoluion of he signal wih an impulse rain. x[n] n p[n] 8 8 n x p [n] = (x p)[n] n

35 Relaion Beween DT Fourier Transform and Series Convoluion by an impulse rain in ime is equivalen o muliplicaion by an impulse rain in frequency. X(e jω ) 2 2π 2π Ω P (e jω ) π 4 Ω X p (e jω ) π 2 2π π 4 π 4 2π Ω 35

36 Relaion Beween DT Fourier Transform and Series Periodic exension of a discree signal (x[n]) resuls in a signal (x p [n]) ha is boh periodic and discree. Is ransform (X p (e jω )) is also periodic and discree. x p [n] = (x p)[n] 8 8 n X p (e jω ) π 2 2π π 4 π 4 2π Ω 36

37 Relaion Beween DT Fourier Transform and Series The weigh of each impulse in he Fourier ransform of a periodically exended funcion is 2π imes he corresponding Fourier series coefficien. X p (e jω ) π 2 2π π 4 π 4 2π Ω a k Ω 37

38 Relaion beween Fourier Transforms and Series The effec of periodic exension was o sample he frequency represenaion. X(e jω ) 2 2π 2π Ω a k Ω 38

39 39 Four Types of Time discree vs. coninuous () and periodic vs aperiodic ( ) DT Fourier Series DT Fourier ransform n n CT Fourier Series CT Fourier ransform

40 40 Four Types of Frequency discree vs. coninuous ( ) and periodic vs aperiodic () DT Fourier Series DT Fourier ransform 2π N k Ω CT Fourier Series 2π T k CT Fourier ransform ω

41 4 Relaion beween Fourier Transforms and Series Periodic exension of a DT signal produces a discree funcion of frequency. Periodic exension = convolving wih impulse rain in ime = muliplying by impulse rain in frequency sampling in frequency periodic DT DTFS N aperiodic DT DTFT periodic exension inerpolae (sampling in frequency) sample inerpolae sample periodic CT CTFS T periodic exension aperiodic CT CTFT

42 42 Four Fourier Represenaions Underlying srucure view as one ransform, no four. DT Fourier Series a k = a k+n = x[n]e j 2π N kn N n=<n> x[n] = x[n + N] = a k e j 2π N kn k=<n> DT Fourier ransform X(e jω ) = x[n]e jωn n= x[n] = X(e jω )e jωn dω 2π <2π> CT Fourier Series a k = T T x() = x( + T ) = j 2π x()e T k d k= a k e j 2π T k CT Fourier ransform X(jω) = x()e jω d x() = X(jω)e jω dω 2π

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