Chapter 4: Applications of Fourier Representations. Chih-Wei Liu

Chapter 4: Application of Fourier Repreentation Chih-Wei Liu

Outline Introduction Fourier ranform of Periodic Signal Convolution/Multiplication with Non-Periodic Signal Fourier ranform of Dicrete-ime Signal Sampling Recontruction of Continuou-ime Signal Dicrete-ime Proceing of Continuou-ime Signal Fourier Serie of Finite-Duration Nonperiodic Signal 2

Introduction Four clae of ignal in Fourier repreentation Continuou- and dicrete-time ignal Periodic and nonperiodc ignal In order to ue Fourier method to analyze a general ytem involving a mixing of noperiodic ay, impule repone and periodic input ignal, we mut build bridge between Fourier repreentation of different clae of ignal F/DF are mot commonly ued for analyi application We mut develop F/DF repreentation of periodic ignal DFS i the primary repreentation ued for computational application the only one can be evaluated on a computer Ue DFS to repreent the F, FS, and DF 3

F Repreentation of Periodic Signal Recall the FS repreentation of a periodic ignal xt: Note that F 2 Uing the freuency-hift property, we have Let tae the F of xt: F x e xt t F t e 2 e [ ] t 2 t t [ ] e [ ] F [ ] t F xt e [ ] 2 [ ] he F of a periodic ignal i a erie of impule paced by the fundamental freuency. he th impule ha trength 2[] he hape of i identical to that of [] 4

FS and F Repreentation of a Periodic Continuou-ime Signal 6 4 2 2 5

Example 4. F of a Coine Find the F repreentation of xt = co o t <Sol.>, co 2,. FS of xt: FS; t F 2. F of xt: co t co t 2 2 6

Example 4.2 F of a Unit Impule rain Find the F of the impule train <Sol.>. pt = periodic, fundamental period = 2. FS of pt: / t t e dt / P 2 / 2 p t t n n 3. F of pt: 2 P P 2 4 2 2 4 he F of an impule train i alo an impule train. 7

Relating the DF to the DFS 8 Recall the DFS repreentation of a periodic D ignal x[n]: Note that the invere DF of a freuency hifted impule, i.e. -, i a complex inuoid. With one period of n e, we have n DF e, 2 Let contruct an infinite erie of hifted impule eparated by 2, i.e. to obtain the following 2-periodic function 2 Let tae the DF of x[n]: x N n [ n] [ ] e N e [ ] xn n DF e m 2 m DF e 2 Since [] i N periodic and N = 2, we may combine the two um n N [ ] m m 2

Relating the DF to the DFS N [ ] n DF 2 [ ] xn e e he DF of a periodic ignal i a erie of impule paced by the fundamental freuency DFS Weighting factor = 2 DF 9 4 he DFS [] and the correponding DF e have imilar hape. 4 4 2 2 2 4

Example 4.3 DF of a Periodic Signal Determine the invere DF of the freuency-domain repreentation depicted in the following figure, where Ω = π /N. 3 2 <Sol.>. We expre one period of e a 2 2 3 4 e, 2 2 from which we infer that / 4, / 4,, otherwie on N 2 x n e e in n 2 2 2 n n 2 2. he invere DF: x[ n] N [ ] e n

Convolution and Multiplication with Mixture of Periodic and Nonperiodic Signal Example: a periodic ignal fed into a table, nonperiodic impule repone Periodic input xt Stable filter Nonperiodic impule repone ht yt = xt ht 2 F i applied to continuou-time C cae DF i applied to dicrete-time D cae F For C ignal: y t x t h t Y H If xt i C periodic ignal, then x t F 2 y t x t h t Y H * F 2 [ ] y t x t h t Y H F * 2 [ ]

Convolution with Mixture of Periodic and Nonperiodic Signal y t x t h t Y H F * 2 [ ] 4 2 2 [] 2 [2] H 2 2 [] 4 2 4 Y 2 [] H 2 [2] H 2 2 4 Magnitude adutment 2 [] H 4 2 2 4 3

Example 4.4 A periodic uare wave i applied to a ytem with impule repone ht=/tint. Ue the convolution property to find the output of thi ytem. <Sol.>. he freuency repone of the LI ytem i by taing the F of the impule repone ht:, F H h t H, 2. F of the periodic uare wave: 2in / 2 2 3. he F of the ytem output i Y = H: Y 2 2 2 2 H act a a low-pa filter, paing harmonic at /2,, and /2, while uppreing all other. 4. Invere 4 F of Y: yt / 2 2 / cot / 2 2 2 2 2

Convolution with Mixture of Periodic and Nonperiodic Signal Periodic input x[n] Stable filter Nonperiodic impule repone h[n] y[n] = x[n] h[n] For D ignal: If x[n] i D periodic ignal, then N [ ] n DF 2 [ ] xn e e DF yn xn* bn Y e 2 H e [ ]. he form ofye indicate that y[n] i alo periodic with the ame period a x[n]. 5

Multiplication of Periodic and Nonperiodic Signal Continuou-time cae: Original multiplication property: F y t g t x t Y G 2 If xt i periodic. he F of xt i [ ] xt e t F 2 [ ] y t g t x t Y G [ ] F y t g t x t Y G F [ ] 6 Multiplication of gt with the periodic function xt give an F coniting of a weighted um of hifted verion of G

y t g t x t Y G F [ ] G G G Y 2 2 2 2 yt become nonperiodic ignal! he product of periodic and nonperiodic ignal i nonperiodic 7

Example 4.4 Conider a ytem with output yt = gtxt. Let xt be the uare wave and gt = cot/2, Find Y in term of G. <Sol.> FS repreentation of the uare wave: x t FS; /2 in / 2 And, we have G / 2 / 2 in / 2 Y 2 / 2 / 2 / 2 / 8

Example 4.6 AM Radio A implified AM tranmitter and receiver are depicted in Fig. 4.3a. he effect of propagation and channel noie are ignored in thi ytem. he ignal at the receiver antenna, rt, i aumed eual to the tranmitted ignal. he paband of the low-pa filter in the receiver i eual to the meage bandwidth, W < < W. Analyze thi ytem in the freuency domain. <Sol.>. he tranmitted ignal i expreed a F 9 r t m t co t R / 2 M / 2 M c c c

G 2 C 2 C R Y C C F g t r t co t G / 2 R / 2 R c c c c c G /4 M 2 /2 M /4 M 2 he original meage i recovered by low-pa filtering to remove the meage replicate centered at twice the carrier freuency. 2

Multiplication of Periodic and Nonperiodic Signal 2 Dicrete-time cae: Original multiplication property: If x[n] i periodic. he DFS of x[n] and it DF i 2 ] [ ] [ ] [ DF e Z e e Y n z n x n y [ ] 2 [ ] N n DF xn e e d e Z e Z e e Y ] [ 2 In any 2 interval of, there are exactly N multiple of the form, ince =2/N. ] [ ] [ N N e Z d e Z e Y [ ] N DF yn xnzn Y e Z e

Example 4.7 Windowing Effect Windowing or truncating: one common data-proceing application, which acce only to a portion of a data record. 7 9 Conider the ignal x n co n co n 6 6 Uing only the 2M+ value of x[n], n M, evaluate the effect of computing the DF. <Sol.>. Since hen [ ] n DF 2 [ ] N xn e e 9 7 7 9 e 6 6 which conit of impule at ± 7/6 and ± 9/6., n M 2. Define a ignal y[n] = x[n]w[n], where w n, n M 6 DF y[ n] x[ n] z[ n] Y e e Z e 2 22 Y e W e W e W e W e 2 6 9 /6 7 /6 7 /6 9 /6

3. he windowing introduce replica of We centered at the freuencie 7/6 and 9/6, intead of the impule that are preent in e. 4. We may view thi tate of affair a a mearing of broadening of the original impule 5. he energy in Ye i now meared over a band centered on the freuencie of the coine Effect of windowing a data record. Ye for different value of M, auming that = 7/6 and 2 = 9/6. a M = 8, b M = 2, c M = 8. If the number of available data point i mall relative to the freuency eparation, the DF i unable to ditinguih the preence of two ditinct 23 inuoid e.g. M=8

Fourier ranform of Dicrete-ime Signal A mixture of dicrete-time and continuou-time ignal By incorporating dicrete-time impule into the decription of the ignal in the appropriate manner. Baic concept: Conider the following ignal: xt e and gn [ ] e 24 t n Let force g[n] to be eual to the ample of xt with ample period ; i.e., g[n] = xn. e n n e i.e. = n Now, conider the DF of a D ignal x[n]: e n e xne n n x n e Define the continuou time ignal x t with the Fourier tranform : F n tn e aing the invere F of, uing the F pair. x t x n t n n

Relating the F to the DF F n, x t x n t n x n e n n x t a C ignal correpond to x[n]; Fourier tranform correpond to the C Fourier tranform e he DF e i periodic in while the F i 2/ periodic in. 2 2 2 d 25 2 2

Example 4.8 Determine the F pair aociated with the DF pair n DF xn aun e ae hi pair i derived in Example 3.7 auming that a < o the DF converge. <Sol.> n. We firt define the continuou-time ignal x t a t n 2. Uing = give F x t ae n 26

Relating the F to the DFS 27 Suppoe that x[n] i an N-periodic ignal, then Now, define x t a C ignal correpond to x[n], then the Fourier tranform of the continuou time ignal x t i [ ] 2 [ ] N n DF xn e e where [] = DFS coefficient e ] [ 2 ] [ 2 Ue the caling property of the impule, a = /a, to rewrite a 2

F 2 t x[ n] t n [ ] n. Recall that [] i N-periodic, which implie that i periodic with periodic N / = 2/ 2. Recall that x[n] i N-periodic, which implie that x t i periodic with periodic N x d 2 2 2 28 [] and x[n] are N-periodic function x t and are periodic impule train with period N and 2/. N

Sampling We ue F repreentation of dicrete-time ignal to analyze the effect of uniformly ampling a ignal. Sampling the continuou-time ignal i often performed in order to manipulate the ignal on a computer. C ignal Sampling D ignal xn [ ] xn Let xt be a C ignal and x[n], a D ignal, i the ample of xt at integer multiple of a ampling interval. pt tn. n 3 = x t x n t n n

Sampling Proce 3 We note that hen ] [ t p t x n t t x n t t x n t n x n t n x t x n n n n 2 P t p t x t x F 2 2 = 2/ Example 4.2 An infinite um of hifted verion of the original ignal,

Sampling heorem x F t x[ n] t n n he hifted verion of may overlap with each other if i not large enough, compared with the bandwidth of, i.e. W. d d W > W 2 d 2 W < W aliaing 2 2 3 2 W 2 3 A increae or decreae, the hifted replica of move cloer together, finally 32 overlapping one another when < 2W.

Example 4.9 Conider the effect of ampling the inuoidal ignal xt co t Determine the F of the ampled ignal for = / 4. <Sol.> Recall that x F t x[ n] t n n. Firt find that 2. hen, we have which conit of pair of impule eparated by 2, centered on integer multiple of = 2/ = ¼. 33

d 2 = 4 2 2 4 d 2 /3 for = 3/2 3 2 3 2 3 3 34

Down-Sampling: Sampling Dicrete-ime Signal 35 Recall Let y[n] = x[n] be a ubampled verion of x[n], i poitive integer. Let, then F n n t n x t x ] [ ] [ ' ] [ n n n t n x n t n y t y F Y ', m m l m m l m m l Y

Down-Sampling: Sampling Dicrete-ime Signal 36 DF Y Y e Y n y ] [ ' 2 2 m m m m m m m m e Y e 2 m m e e Y F n Y n t n x t y ] [ 2 ] [ ] [ m m DF e e Y n x n y Since, therefore Ue, we have Summary

Down-Sampling by 2 e 2 2 2 2 2 2 2 4 2 2 4 2 d e 2 2 W 2 W 2 2 4 3 2 2 3 4 Figure 4.29 Effect of ubampling on the DF. a Original ignal pectrum. b m = term in E. 4.27 c m = term in E. 4.27. d m = term in E. 4.27. e Ye, auming that W < /. f Ye, auming that W > /. If the highet freuency component of e, W, i le than /, the 37 aliaing can be prevented.

Recontruction of Sampled Signal Recontruction? Not o eay!! he ample of a ignal do not alway uniuely determine the correponding continuou-time ignal 38 Nyuit Sampling heorem F Let xt be a band-limited ignal, i.e. = for > m. If > 2 m, where = 2/ i the ampling freuency, then xt i uniuely determined by it ample xn, n =,, 2,.

Ideal/Perfect Recontruction If the ignal i not band-limited, an antialiaing filter i neceary Hereafter, we conider a band-limited ignal F Suppoe that xt, then the F of the ampled ignal i 39 m a m Ideal recontruction: H r, otherwie m Ideal low-pa filter 2 : Nyuit freuency / 2 H r m / 2 Non-caual!! the perfect-recontruction cannot be implemented in any practical ytem H r

H xt x t h t / 2 H r F r / 2 h t H r r n r h t x t x t hr t x[ n] t n n r r t in 2 t in c t h x[ n] h t n r n t t n x[ n]in c t Infinite length of h r t Non-caual filter = Perfect recontruction 4

A Practical Recontruction Zero-Order Hold Filter he zero-order hold filter, which hold the input value for econd. a tair-tep approximation to the original ignal Impule repone rectangular pule:, t ho t, t, t in / 2 F /2 2 h t H e 4. A linear phae hift correponding to a time delay of /2 econd 2. A ditortion of the portion of between m and m. [he ditortion i produced by the curvature of the mainlobe of H o.] 3. Ditorted and attenuated verion of the image of, centered at nonzero multiple of.

H O m m 2 arg H O m m 2 O m m 2 econd holding for x[n] ime hift of /2 in x o t for ditortion- Stair-tep approximation main reaon for ditortion-2 & 3. Uing Eualizer for non-flattened mag. in paband 2. Uing anti-image filter Compenation filter 42 Both ditortion and 2 are reduced by increaing or, euivalently, decreaing, e.g. overampling!!

Compenation Filter he freuency repone: Hc Eualizer m m m m Anti-imaging H c, m 2in /2, m 43

Example 4.3 In thi example, we explore the benefit of overampling in recontructing a continuou-time audio ignal uing an audio compact dic player. Aume that the maximum ignal freuency i f m = 2 Hz. Conider two cae: a recontruction uing the tandard digital audio rate of / = 44. Hz, and b recontruction uing eight-time overampling, for an effective ampling rate of / 2 = 352.8 Hz. In each cae, determine the contraint on the magnitude repone of an anti-imaging filter o that the overall magnitude repone of the zero-order hold recontruction ytem i between.99 and. in the ignal paband and the image of the original ignal pectrum centered at multiple of the ampling freuency are attenuated by a factor of 3 or more. <Sol.>.99. H' c f, 2Hz f 2Hz H' f H' f o o 44

Dicrete-ime Signal Proceing Several advantage for DSP algorithm Signal manipulation are more eaily performed by uing arithmetic operation of a computer than through the ue of analog component DSP ytem are eaily modified in real-time Direct dependence of the dynamic range and ignal-to-noie ratio on the number of bit ued to repreent the dicrete-time ignal Eaily implement the decimation and the interpolation H a H O Hc G 47

Sytem Repone Analyi Aume that the dicrete-time proceing operation i repreented by a D ytem with freuency repone He =, ampling interval H a HO Hc H a a a H a Y He Ha 48 Y Ho Hc He Ha

he anti-imaging filter H c eliminate freuency component above /2, hence eliminating all the term in the infinite um except for the = term. herefore, we have Y H o Hc H e Ha he overall ytem i euivalent to a continuou-time LI ytem having the repone: G H o Hc H e Ha If we chooe the anti-aliaing and anti-imaging filter uch that / H H H G H e o c a hat i, we may implement a C ytem in D by chooing ampling parameter appropriately and deigning a correponding D ytem. 49

Overampling A high ampling rate lead to high computation cot Overampling can relax the reuirement of the anti-aliaing filter a well a the anti-image filter a wide tranition band Anti-aliaing filter i ued to prevent aliaing by limiting the bandwidth of the ignal prior to ampling. W topband of filter, W t = W W width of tranition band. 5 -W -W -W > W W t = W W < -2W

Decimation wo ampled ignal x [n] and x 2 [n] with different ampling interval and 2 Sampling!! 4 2 2 4 4 2 2 4 x 2 [n] g[n] -fold pread out 5 4 2 2 4 he maximum freuency component of 2 e atifie W 2 < /

Decimation Filter Decimation filter h d [n] : a low-pa filter prevent from aliaing problem when downampling by. H d, e, otherwie 52

Interpolation Interpolation increae the ampling rate and reuire that we omehow produce value between two ample of the ignal 2 2 -fold ueeze 53 2 4 2 2 4 W W 2 Image occur!!

Interpolation Filter Decimation filter h i [n] : a low-pa filter prevent from image problem when upampling by. x[n] Upampling by Dicrete-time low pa H i e x i [n] H i, e, otherwie 2 2 54 2 2

Why DFS for Finite Nonperiodic Signal? he primary motivation i for the numerical computation of Fourier tranform DFS i only Fourier repreentation that can be evaluated numerically!! Let x[n] be a finite-duration ignal of length M Introduce a periodic D ignal xn [ ] with period NM zero padding e M n xne n 56 2 2

Relating the DFS to the DF Conider a periodic D ignal xn [ ] with period NM, then N ~ ~ [ n ~ [ ] x n] e Apply x[n], we have [ ] N n M o = 2/N n Recall that e xne, we conclude that ~ [ ] N e n xn [ ] N M n x[ n] e he DFS coefficient of are ample of the DF of x[n], divided by N and evaluated at interval of 2/N. n 57 he effect of ampling the DF of a finite-duration nonperiodic ignal i to periodically extend the ignal in the time domain Dual to ampling in time domain In order to prevent overlap in time domain, we reuire NM, or the ampling freuency 2/M

Example 4.4 3 co n, n 3 Conider the ignal xn 8, otherwie Derive both the DF, e, and the DFS, [], of x[n], auming a period N > 3. Evaluate and plot e and N [] for N = 32, 6, and 2. <Sol.>. We rewrite the ignal x[n] a g[n]w[n], where g[n] = co3n/8 and a rectangular window wn 2. hen, n 3, otherwie DF W e e G e 2 W e e 3 / 2 in 6 in 2 3 3 /8 / 2 3 3 /8 / 2 in 6 3 8 in 6 3 8 e e e 2 in 3 8 2 2 in 3 8 2 3. Sample at = and divide by N, we have 58

e e e e e 2Ne 3 /86 3 /86 3 /86 o o o 3 /82 3 /82 3 /8 o o o 2 e e e o 2Ne e e e 3 /86 3 /86 3 /86 o o o 3 /8/2 3 /82 3 /82 3 3 /8 / 2 o in 6 o 3 8 2N in o 3 8 2 3 o 3 /8 / 2 e in 6 o 3 8 2N in o 3 8 2 o 59

Relating the FS to the F he relationhip between the FS and the F of a finite-duration nonperiodic continuou-time ignal i analogou to that of dicrete-time cae Let xt have duration, o that xt, t or t Contruct a periodic ignal xt xt m with o m Conider the FS of x t Recall that o ot o [ ] x t e dt t x t e dt D x t t x t e dt x t e t dt [ ] o he FS coefficient are ample of the F, normalized by 6