Chapter 3 Fourier Representations of Signals and Linear Time-Invariant Systems

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1 Chapter 3 Fourier Representations of Signals and Linear Time-Invariant Systems Introduction Complex Sinusoids and Frequency Response of LTI Systems. Fourier Representations for Four Classes of Signals Discrete-Time Periodic Signals Continuous-Time Periodic Signals Discrete-Time Nonperiodic Signals Continuous-Time Nonperiodic Signals. Properties of Fourier Representation. Linearity and Symmetry Properties.

2 2 Chapter 3 Fourier Representations of Signals and Linear Time-Invariant Systems Convolution Property Differentiation and Integration Properties Time- and Frequency-Shift Properties Finding Inverse Fourier Transforms by Using Partial-Fraction Expansions. Multiplication Property Scaling Properties Parseval Relationships Time-Bandwidth Product Duality

3 Introduction 3 Fourier Analysis Representing Signals a Superposition of Complex Sinusoids Four Fourier Representations DT periodic signals: DT Fourier series (DTFS) CT periodic signals: CT Fourier series (CTFS) DT nonperiodic signals: DT Fourier transform (DTFT) CT nonperiodic signals: CT Fourier transform (CTFT)

4 4 2. Complex Sinusoids and Frequency Response of LTI Systems. The input-output relation of an LTI system is characterized by convolution with the system s impulse response. By representing the input as a sum of complex sinusoids, we obtain a more straightforward understanding of the input-output relation of an LTI system

5 5 2. Complex Sinusoids and Frequency Response of LTI Systems Discrete-Time LTI System The output of a complex sinusoidal input to an LTI system is a complex sinusoid of the same frequency as the input, multiplied by the frequency response of the system. y jnk n hk xn k hk e k k x(t) or x[n] LTI System y(t) or y[n] y n e h k e H e e jn jk j jn k Frequency Response j H( e ) h[ k] e k j

6 6 2. Complex Sinusoids and Frequency Response of LTI Systems Continuous-Time LTI System j ( t ) jt j y( t) h( ) e d e h( ) e d H( j ) e jt Frequency response: j H( j ) h( ) e d Polar form complex number c = a + jb: c j arg{ c} c e where 2 2 and arg{ } tan b c a b c a Polar form for H (j): H( j) H( j) e jarg{ H ( j )} where H j H j ( ) Magnitude response and arg ( ) Phase respnse j t argh j y t H j e

7 7 2. Complex Sinusoids and Frequency Response of LTI Systems RC Circuit: Frequency response Frequency response: H RC j j e u e d RC j RC e RC j RC RC j RC RC j RC RC e j RC d Magnitude response: H j RC 2 Phase response: arg RC h t e u t RC t / RC ( ) ( ) H j arctan RC 2

8 8 2. Complex Sinusoids and Frequency Response of LTI Systems H j arg H 2 j () t [ n] H j t () t H( j ) e jt [ n] e H j n e j H e e j n H

9 2. Complex Sinusoids and Frequency Response of LTI Systems Eigenvalues and Eigenfunctions of a LTI System 9 If e k is an eigenvector of a matrix A with eigenvalue k, then Ae e k k k Eigenfunction () t j t e Eigenvalue H( j) [ n] e jn j He ( ) Arbitrary input = weighted superpositions of eigenfunctions Convolution operation Multiplication Ex. Input: x t j k x[ n] a e Output: M M jkt ake y k k k M M n j k k k k t a H j e j t j k y[ n] a H e e k n

10 3. Fourier Representations for Four Classes of Signals Periodic Signals: Fourier Series Representations. x[n] = discrete-time signal with fundamental period N. DTFS of x[n] is x[ n] A[ k] e jk n o = 2/N Fundamental (3.4) frequency of x[n] k 2. x(t) = continuous-time signal with fundamental period T. FS of x(t) is x t ( ) A[ k] e jk t (3.5) k o = 2/T Fundamental frequency of x(t) ^ denotes approximate value. A[k] = the weight applied to the k th harmonic. jkt jt e is the kth harmonic of e.

11 3. Fourier Representations for Four Classes of Signals 3. The complex sinusoids exp(jk o n) are N-periodics in the frequency index k. <pf.> jnk n jn n jk n e e e j n jn n e 2 e jkn e There are only N distinct complex sinusoids of the form exp(jk n) should be used in Eq. (3.4). xˆ[ n] N k Symmetries in x[k]: k = (N )/2 to (N )/2 4. Continuous-time complex sinusoid exp(jk t) with distinct frequencies k are always distinct. FS of continuous-time periodic signal x(t) becomes x( t) A[ k] e jk t k Mean-square error (MSE) between the signal and its series representation: Discrete-time case: A[ k] e jk n Continuous-time case: N 2 MSE x[ n] x[ n] dt N n T 2 MSE x( t) x( t) dt T

12 2 3. Fourier Representations for Four Classes of Signals Nonperiodic Signals: Fourier-Transform Representations. FT of continuous-time signal: xˆ[ n] 2 X ( j) e j t d X(j)/(2) = the weight applied to a sinusoid of frequency in the FT representation. 2. DTFT of discrete-time signal: xˆ[ n] 2 X ( e j ) e jn d X(e j )/(2) = the weight applied to the sinusoid e j n in the DTFT representation.

13 3 3. Fourier Representations for Four Classes of Signals

14 4 3. Fourier Representations for Four Classes of Signals Periodic Discrete

15 5 4. Discrete-Time Periodic Signals: The Discrete-Time Fourier Series. DTFS pair of periodic signal x[n]: N x[ n] [ ] jkn X k e (3.) k N X[ k] [ ] N Notation: jkn x n e (3.) n x n DTFS; Fundamental period = N; Fundamental frequency = o = 2/N Fourier coefficients; Frequency domain representation X k

16 6 4. Discrete-Time Periodic Signals: The Discrete-Time Fourier Series Example 3.6 x n, M n M, M n N M. Period of DTFS coefficients = N o = 2/N 2. It is convenient to evaluate Eq.(3.) over the interval n = M to n = N M. NM X[ k] x[ n] e N N nm M nm e jk n jk n 3. For k =, N, 2N,, we have e o e jk jk o 2M 2M m X[ k] N 2M m e N X k, k, N, 2 N, N N e jk ( mm ) 2M jk M jk m m e

17 7 4. Discrete-Time Periodic Signals: The Discrete-Time Fourier Series 3. For k, N, 2N,, we may sum the geometric series in Eq. (3.5) to obtain jkm jkm (2M ) e e X[ k], k, N, 2 N,... jk N e jk 2M / 2 jk 2M jk 2M / 2 jk 2M / 2 e e e e Xk jk / 2 jk jk / 2 jk / 2 N e e N e e X k sin k 2M / N, k, N, 2 N, Xk N sin k / N 2M / N, k, N, 2 N, N sin k sin 2M k 4. Substituting o = 2/N, yields X k N sin k 2M sin k / N / N / 2 / 2, lim k, N, 2 N, L Hô pital s Rule sin k 2M / N 2M N sin k / N N The value of X[k] for k =, N, 2N,, is obtained from the limit as k.

18 L'Hôpital's rule 8 In its simplest form, l'hôpital's rule states that for functions ƒ and g:

19 9 4. Discrete-Time Periodic Signals: The Discrete-Time Fourier Series Figure 3.2 (p. 2) The DTFS coefficients for the square wave shown in Fig. 3., assuming a period N = 5: (a) M = 4. (b) M = 2.

20 2 4. Discrete-Time Periodic Signals: The Discrete-Time Fourier Series Symmetry property of DTFS coefficient: X[k] = X[ k].. DTFS of Eq. (3.) can be written as a series involving harmonically related cosines. 2. Assume that N is even and let k range from ( N/2 ) + to N/2. Eq. (3.) becomes x n N / 2 X kn / 2 x k e jk n / 2 jk n m e jkn 2 jkn n X X N / 2e X me X N / m 3. Use X[m] = X[ m] and the identity N o = 2 to obtain N / 2 jmn jmn j n e e xn X X N / 2e 2X m m 2 N / 2 / 2cos 2 cos X X N n X m m n 4. Define the new set of coefficients m e jn = cos(n) since sin(n) = for integer n.

21 2 4. Discrete-Time Periodic Signals-- The Discrete-Time Fourier Series 4. Define the new set of coefficients Bk X k, k, N / 2 2 X k, k, 2,, N / 2 and write the DTFS for the square wave in terms of a series of harmonically related cosines as N /2 (3.7) k x[ n] B[ k]cos( k n) A similar expression may be derived for N odd. Example 3.7 Building a Square Wave From DTFS Coefficients J xj[ n] B[ k]cos( k n) k

22 22 4. Discrete-Time Periodic Signals-- The Discrete-Time Fourier Series (a) J =. (b) J = 3. (c) J = 5. (d) J = 23. (e) J = 25.

23 Example Figure 3.5 Electrocardiograms for two different heartbeats and the first 6 coefficients of their magnitude spectra. (a) Normal heartbeat. (b) Ventricular tachycardia. 心動過速 (c) Magnitude spectrum for the normal heartbeat. (d) Magnitude spectrum for ventricular tachycardia.

24 24 5. Continuous-Time Periodic Signals The Fourier Series. Continuous-time periodic signals are represented by the Fourier series (FS). 2. A signal with fundamental period T and fundamental frequency o = 2/T, the FS pair of x(t) is x t ( ) X[ k] e jk t (3.9) k T jkt X[ k] x( t) e dt T (3.2) 3. Notation: x t FS; X k Frequency domain representation of x(t) The variable k determines the frequency of the complex sinusoid associated with X[k] in Eq. (3.9).

25 25 5. Continuous-Time Periodic Signals The Fourier Series Mean-square error (MSE). Suppose we define xˆ t k jk ot X k e 2. If x(t) is square integrable, T T x t then the MSE between x(t) and ˆx t is zero, or, mathematically, 2 T x() t xˆ t at all values of t; it simply implies x t xt dt T that there is zero power in their difference. 3. Pointwise convergence of x() t xˆ t is guaranteed at all values of t except those corresponding to discontinuities if the Dirichlet conditions are satisfied: x(t) is bounded. x(t) has a finite number of maximum and minima in one period. x(t) has a finite number of discontinuities in one period. If a signal x(t) satisfies the Dirichlet conditions and is not continuous, then ˆx t converges to the mipoint of the left and right limits of x(t) at each discontinuity. 2 dt

26 26 5. Continuous-Time Periodic Signals The Fourier Series. Fundamental frequency: o = 2/T. 2. X[k]: by integrating over T/2 t T/2 T /2 jk t jkt X k x t e dt e dt T /2 T T e Tjk 2sin 2 e Tk jk t k T Tk For k =, we have X T T dt T T T e 2 j, k jk T jk T 2T T, k T T, k 3. By means of L Hôpital s rule, we have

27 27 5. Continuous-Time Periodic Signals The Fourier Series lim k 2sin k T Tk X k 2sin 2T T k T Tk 4. X[k] is real valued. Using o = 2/T gives X[k] as a function of the ratio T o /T: Xk [ ] 2sin( k2 T / T) k2 (3.23) 5. Fig (a)-(c) depict X[k], 5 k 5, for T o /T =/4, T o /T =/6, and T o /T = /64, respectively. Definition of Sinc function: sin( u) sinc( u) (3.24) Fig u ) Maximum of sinc function is unity at u =, the zero crossing occur at integer values of u, and the amplitude dies off as /u. 2) The portion of this function between the zero crossings at u = is manilobe of the sinc function.

28 28 5. Continuous-Time Periodic Signals The Fourier Series The FS coefficients, X[k], 5 k 5, for three square waves. (see Fig. 3.2.) (a) T o /T = /4. (b) T o /T = /6. (c) T o /T = /64.

29 29 5. Continuous-Time Periodic Signals The Fourier Series Example 3.5 x J J t Bk k t k cos Gibbs phenomeno Ripple near discontinuiitiies Figure 3.25b-3 (p. 226) (b) J = 3. (c) J = 7. (d) J = 29.

30 3 5. Continuous-Time Periodic Signals The Fourier Series Trigonometric FS for Real Signal (3.25) k x( t) B[] B[ k]cos( k t) A[ k]sin( k t) FS coefficients: B[] = X[] represents the timeaveraged value of the signal. T B[] x( t) dt T 2 T B[ k] x( t)cos( k t) dt T (3.26) 2 T A[ k] x( t)sin( k t) dt T Relation between exponential FS and trigonometric FS coefficients B[ k] X[ k] X[ k] for k A[ k] j( X[ k] X[ k])

31 3 6. Discrete-Time Nonperiodic Signals The Discrete-Time Fourier Transform. The DTFT is used to represent a discrete-time nonperiodic signal as a superposition of complex sinusoids. 2. DTFT representation of time-domain signal x[n]: j jn x[ n] X ( e ) e d 2 j jn X ( e ) x[ n] e (3.32) 3. Notation: n (3.3) DTFT j X e x n DTFT pair Frequency-domain representation x[n]

32 32 6. Discrete-Time Nonperiodic Signals The Discrete-Time Fourier Transform Condition for convergence of DTFT: If x[n] is of infinite duration, then the sum converges only for certain classes of signals. If x[n] is absolutely summable, i.e., n xn The sum in Eq. (3.32) converges uniformly to a continuous function of. If x[n] is not absolutely summable, but does satisfy (i.e., if x[n] has finite energy), n x n 2 It can be shown that the sum in Eq. (3.32) converges in a mean-square error sense, but does not converge pointwise.

33 33 6. Discrete-Time Nonperiodic Signals The Discrete-Time Fourier Transform Let xn,, n n M M 2M 2M j j( mm ) jm jm X e e e e m m j jn. DTFT of x[n]: X e e. j2( M) jm e e,, 2, 4, j e 2M, =, 2, 4, M nm Change of variable m = n + M The expression for X(e j ) when X ( e ) e j jm e e e e j 2M / 2 j 2M / 2 j 2M / 2 e e e e j / 2 j / 2 j / 2 j 2M / 2 j 2M / 2 e e j / 2 j / 2 sin 2M / 2 j Xe ( ) sin 2, 2, 4, L Hôptital s Rule sin2m / 2 lim 2M ;, 2, 4,, sin 2 With understanding that X(e j ) for, 2, 4, is obtained as limit.

34 34 6. Discrete-Time Nonperiodic Signals The Discrete-Time Fourier Transform Example 3.9 Inverse DTFT of A Rectangular Pulse X e j,, W, W n W jn xn e d 2 W W sin /, cwn x n

35 35 6. Discrete-Time Nonperiodic Signals The Discrete-Time Fourier Transform Example 3.2 DTFT of The Unit Impulse Find the DTFT of x[n] = [n]. <Sol.>. DTFT of x[n]: j jn ne X e n DTFT n. 2. This DTFT pair is depicted in Fig Figure 3.32 (p. 235) Example 3.2. (a) Unit impulse in the time domain. (b) DTFT of unit impulse in the frequency domain.

36 36 6. Discrete-Time Nonperiodic Signals The Discrete-Time Fourier Transform Example 3.2 Inverse DTFT of A Unit Impulse Spectrum. Inverse DTFT of X(e j j n ): xn e d. 2 DTFT, 2. We can define X(e j ) over all by writing it as an infinite sum of delta functions shifted by integer multiples of This DTFT pair is depicted in Fig j k X e k 2. Dilemma: The DTFT of x[n] = /(2) does not converge, since it is not a square summable signal, yet x[n] is a valid inverse DTFT!

37 37 7. Continuous-Time Nonperiodic Signals The Fourier Transform. The Fourier transform (FT) is used to represent a continuous-time nonperiodic signal as a superposition of complex sinusoids. 2. FT representation of time-domain signal x(t): jt x( t) X ( j) e d 2 (3.35) jt X ( j) x( t) e dt (3.36) 3. Notation for FT pair: FT. x t X j Frequency-domain representation of the signal x(t)

38 38 7. Continuous-Time Nonperiodic Signals The Fourier Transform Convergence condition for FT: jt xˆ t X j e d, 2 ) Approximation: Squared error between x(t) and 2 x t xˆ t dt, ˆx t is zero if x(t) is square integrable, i.e., if : the error energy, given by 2 dt. Zero squared error does not imply pointwise convergence [i.e., x t xˆ t ] at all values of t. There is zero energy in the difference of x( t) and xˆ t Pointwise convergence of x() t xˆ t is guaranteed at all values of t except those corresponding to discontinuities if x(t) satisfies the Dirichlet conditions for nonperiodic signals: x t dt x(t) is absolutely integrable:. x(t) has a finite number of maximum, minima, and discontinuities in any finite interval. The size of each discontinuity is finite. x t

39 39 7. Continuous-Time Nonperiodic Signals The Fourier Transform Example 3.25 FT of A Rectangular Pulse Consider the rectangular pulse depicted in Fig. 3.4 (a) and defined as, T t T xt Find the FT of x(t)., t T. The rectangular pulse x(t) is absolutely integrable, provided that T o <. 2. FT of x(t): jt jt X j x t e dt e dt T jt T 2 e T sin T, j 3. For =, the integral simplifies to 2T o. 2 lim sin T 2. T 2 sin, T X j T With understanding that the value at = is obtained by evaluating a limit.

40 4 7. Continuous-Time Nonperiodic Signals The Fourier Transform 4. Magnitude spectrum: X j sin T 2, 5. Phase spectrum:, argx j, sin T sin T Fig (b). 6. X(j) in terms of sinc function: X j 2T sinc T. As T o increases, the nonzero time extent of x(t) increases, while X(j) becomes more concentrated about the frequency origin. v ; p

41 4 7. Continuous-Time Nonperiodic Signals The Fourier Transform Example 3.26 Inverse FT of A Rectangular Pulse Find the inverse FT of the rectangular spectrum depicted in Fig (a) and given by Figure 3.42 (p. 246), W W Example (a) Rectangular spectrum in X j <Sol.>, W the frequency domain. (b) Inverse FT in the time domain. v ; p. Inverse FT of x(t): W jt jt W xt e d e W sin( Wt ), t 2 W j t t

42 42 7. Continuous-Time Nonperiodic Signals The Fourier Transform 2. When t =, the integral simplifies to W/, i.e., lim sin Wt W, t t 3. Inverse FT is usually written as sin, t Wt or xt sinc, x t W Wt With understanding that the value at t = is obtained as a limit. Fig (b). As W increases, the frequency-domain reprsentation becomes less concentrated about =, while the time-domain representation X(j) becomes more concentrated about t =.

43 43 7. Continuous-Time Nonperiodic Signals The Fourier Transform <Sol.> Example 3.27 FT of The Unit Impulse Find the FT of x(t) = (t).. x(t) does not satisfy the Dirichlet conditions, since the discontinuity at the origin is infinite. 2. FT of x(t): jt X j te dt FT Notation for FT pair: t, Example 3.28 Inverse FT of An Impulse Spectrum Find the inverse FT of X(j) = 2(). <Sol.>. Inverse FT of X(j): jt xt 2 e d 2 2. Notation of FT pair: FT 2 Duality between Example 3.27 and 3.28

44 44 8. Properties of Fourier Representation

45 45 8. Properties of Fourier Representation

46 8. Properties of Fourier Representation 46 Periodic Discrete

47 9. Linearity and Symmetry Properties 47 Linearity property for four Fourier representations FT FS; o z t ax t by t Z j ax j by j z t ax t by t Z k ax k by k DTFT j j j z n ax n by n Z e ax e by e DTFS ; o z n ax n by n Z k ax k by k

48 9. Linearity and Symmetry Properties 48. Signal z(t): 3 z t x t y t 2 2 where x(t) and y(t) 2. X[k] and Y[k]: FS;2 FS;2 sin 4 sin 2 x t X k k k y t Y k k k 3. FS of z(t): FS;2 3 2 sin 2 2 sin 2 z t Z k k k k k

49 9. Linearity and Symmetry Properties 49 Symmetry Properties: Real and Imaginary Signals Symmetry property for real-valued signal x(t):. FT of x(t): * * j t * jt X ( j ) x( t) e dt x ( t) e dt (3.37) 2. Since x(t) is real valued, x(t) = x (t). Eq. (3.37) becomes * j ( ) t X j x t e dt X * ( j) X ( j) (3.38) X(j) is complex-conjugate symmetric X j X j X j X j Re ( ) Re ( ) and Im ( ) Im ( )

50 9. Linearity and Symmetry Properties 5 The conjugate symmetry property for DTFS: X * [ k] X[ N k] Because the DTFS coefficients are N periodic, and thus X[ k] X[ N k] The symmetry conditions in all four Fourier representations of real-valued signals are indicated in Table 3.4.

51 9. Linearity and Symmetry Properties 5 Continuous-time case:. Input signal of LTI system: cos x t A t x t A/ 2 e A/ 2 e j t j t 2. Real-valued impulse response of LTI system is denoted by h(t). 3. Output signal of LTI system: Euler s formula cos arg y t H j A t H j y( t) H( j) ( A/ 2) e H( j) ( A/ 2) e Exploiting the symmetry conditions: H( j) H( j) H j H j arg ( ) arg ( ) j( t arg H ( j ) ) j( t arg H ( j ) ) Applying Eq. (3.2) and linearity

52 9. Linearity and Symmetry Properties 52 Discrete-time case: x[n] = Acos( n ) 2. Real-valued impulse response of LTI system is denoted by h[n]. 3. Output signal of LTI system: j j cos arg y n H e A n H e The LTI system modifies the amplitude of the input sinusoid by H(e j )and the phase by arg{h(e j )}. x(t) is purely imaginary:. x (t) = x(t). 2. Eq.(3.37) becomes * j ( ) t X j x t e dt X * ( j) X ( j) (3.39) X j X j X j X j Re ( ) Re ( ) and Im ( ) Im ( )

53 9. Linearity and Symmetry Properties 53 Symmetry Properties: Even and Odd Signals. x(t) is real valued and has even symmetry. x (t) = x(t) and x ( t) = x(t) x (t) = x( t) 2. Eq.(3.37) becomes * j ( t) X j x t e dt * j X j x e d X j 3. Conclusion: ) The imaginary part of X(j) = : X ( j) X ( j) Change of variable = t If x(t) is real and even, then X(j) is real. 2) If x(t) is real and odd, then X (j) = X(j) and X(j) is imaginary.

54 . Convolution Property 54 Nonperiodic Signals. Convolution of two nonperiodic continuous-time signals x(t) and h(t): y t h t x t h x t d 2. FT of x(t ): j( t) xt X j e d 2 3. Substituting this expression into the convolution integral yields jt j y t h X j e e d d 2 j jt h e d X j e d 2 jt yt H j X j e d 2 and we identify H(j)X(j) as the FT of y(t). FT y( t) h( t)* x( t) Y( j) X ( j) H( j) (3.4)

55 . Convolution Property 55 Example FT 4 2 xt X j 2 sin, t z t Z j, t FT x t z t z t

56 . Convolution Property 56 Convolution of Nonperiodic Signals Discrete-time case DTFT j If xn X e DTFT j and hn H e DTFT j j j y[ n] x[ n]* h[ n] Y( e ) X ( e ) H( e ) (3.4)

57 .2 Filtering 57. Multiplication in frequency domain Filtering. 2. The terms filtering implies that some frequency components of the input are eliminated while others are passed by the system unchanged. 3. System Types of filtering: ) Low-pass filter 2) High-pass filter 3) Band-pass filter 4. Realistic filter: ) Gradual transition band 2) Nonzero gain of stop band 5. Magnitude response of filter: Fig (a), (b), and (c) The characterization of DT filter is based on its behavior in the frequency range < because its frequency response is 2 -periodic. j 2log H j or 2log He [db]

58 58.2 Filtering

59 . Convolution Property 59 Example 3.33 RC Circuit: Filtering For the RC circuit depicted in Fig. 3.54, the impulse response for the case where y C (t) is the output is given by t RC hc t e u t RC Since y R (t) = x(t) y C (t), the impulse response for the case where y R (t) is the output is given by t RC hr t t e u t RC Figure 3.54 (p. 264) RC circuit with input x(t) and outputs y c(t) and y R (t). HC j irc H R j irc irc

60 6

61 . Convolution Property 6 Frequency response of a system The ratio of the FT or DTFT of the output to the input. For CT system: For DT system: Y( j ) H( j ) X( j ) (3.42) j j Ye ( ) He ( ) j Xe ( ) (3.43) Frequency response

62 . Convolution Property 62 Recover the input of the system from the output inv where Equalizer CT case inv X j H j Y j DT case and j inv j X e H e Y e j H j / H( j) inv j and j H e / H( e ) The process of recovering the input from the output is known as equalization

63 . Convolution Property 63 Convolution of Periodic Signals Define the periodic convolution of two CT signals x(t) and z(t), each having period T, as T # y t x t z t x z t d where the symbol denotes that integration is performed over a single period of the signals involved. 2 FS; T y( t) x( t) # z( t) Y[ k] TX[ k] Z[ k]

64 . Convolution Property 64 Discrete-Time Convolution of Periodic Sequences Define the discrete-time convolution of two N-periodic sequences x[n] and z[n] N # y n x n z n x k z n k k DTFS of y[n] 2 DTFS ; N y[ n] x[ n] # z[ n] Y[ k] NX [ k] Z[ k] Table 3.5 Convolution Properties FT x( t) z( t) X ( j) Z( j) FS; x t z t TX k Z k ( )# ( ) [ ] [ ] FT j j x[ n] z[ n] X ( e ) Z( e ) x n z n NX k Z k DTFS ; [ ]# [ ] [ ] [ ]

65 65. Differentiation and Integration Properties Differentiation in Time. A nonperiodic signal x(t) and its FT, X(j), is related by x Differentiating both j e t d t X j 2 sides with respect to t X j d t x ( t FT ) j X ( j ) dt Differentiation of x(t) in Time-Domain (j) X(j) in Frequency-Domain d F x ( t ) ( j ) X ( j ) dt d dt x 2 je j t d

66 66. Differentiation and Integration Properties Differentiation in Frequency. FT of signal x(t): X jt j xt e dt Differentiating both sides with respect to 2. Differentiation property in frequency domain: d jtx t X j d FT d d X j jtxt e jt dt Differentiation of X(j) in Frequency-Domain ( jt) x(t) in Time-Domain

67 67. Differentiation and Integration Properties Integration Integrating nonperiodic signals. Define y t t d x d y ( t ) x ( t ) (3.5) dt 2. By differentiation property, we have Indeterminate at = Y( j) X ( j) (3.52) j 3. When the average value of x(t) is not zero, then it is possible that y(t) is no square integrable. FT of y(t) may not converge! W can get around this problem by including impulse in the transform, i.e. Y ( j) X ( j) c ( ) c can be determined by j the average value of x(t) C = X(j)

68 68. Differentiation and Integration Properties Table 3.6 Commonly Used Differentiation and Integration Properties. d x ( t FT ) j X ( j ) dt d x FS; t jk X k dt FT d jtx t X j d DTFT d j jnx n X e d t FT x( ) d X ( j) X ( j) ( ) j 68

69 2. Time- and Frequency-Shift Properties 69 Time-Shift Property. Let z(t) = x(t t ) be a time-shifted version of x(t). 2. FT of z(t): Z jt jt j zt e dt xt t e dt 3. Change variable by = t t : Z j x e j t jt d j jt e x e d e X j Time-shifting of x(t) by t in Time-Domain (e j t ) X(j) in Frequency-Domain Z( j) ( ) X j and arg{ Z( j)} arg{ X ( j)} t

70 2. Time- and Frequency-Shift Properties 7 x t t e X j FT jt x t t e X k FT ; jkt DTFT jn j x n n e X e DTFS ; jkn x n n e X k

71 2. Time- and Frequency-Shift Properties 7 Frequency Shift. Suppose that: FT FT x( t) X ( j) and z( t) Z( j) 2. By the definition of the inverse FT, we have z 2 t Z j e j t d 2 X j e j t d Substituting variables = into above Eq. gives j t jt z t X je d 2 jt e X j e d 2 Frequency-shifting of X(j) by in Frequency-Domain [i.e. X(j( ))] (e t ) x(t) in Time-Domain e jt x t

72 2. Time- and Frequency-Shift Properties 72 j t FT e x t X j ; jk t FS e x t x k k e jn x[ n] DTFT X e j jk n FS; e x[ n] X k k

73 73 3. Finding Inverse Fourier Transforms by Using Partial-Fraction Expansions.. FT X(j) in terms of a ratio of polynomials in j: X j M bm j b j b N N j a j a j a N Assume that M < N. If M N, then we may use long division to express X(j) in the form Partial-fraction expansion M N k B j is applied to this term X j f j k k A j B A j j Applying the differentiation property and the pair ( t) FT to these terms 2. Let the roots of the denominator A(j) be d k, k =, 2,, N. These roots are found by replacing j with a generic variable v and determining the roots of the polynomial v N a N Nv av a

74 74 3. Finding Inverse Fourier Transforms by Using Partial-Fraction Expansions 3. For M < N, we may the write X j M k N j d k k b k j k Assuming that all the roots d k, k =, 2,, N, are distinct, we may write N Ck X j Appendix B j d k k In Example 3.24, we derived the transform pair dt e u t FT j d C k, k =, 2,, N are determined by the method of residues This pair is valid even if d is complex, provided that Re{d} <. Assuming that the real part of each d k, k =, 2,, N, is negative, we use linearity to write N N dt FT k k k k j dk k x t C e u t X j C

75 3. Finding Inverse Fourier Transforms by Using Partial-Fraction Expansions Inverse Discrete-Time Fourier Transform 75. Suppose X(e j ) is given by a ratio of polynomial in e j, i.e. j N j N N j N j M j M j e e e e e e X Normalized to 2. Factor the denominator polynomial as N k j k j N j N N j N e d e e e Replace e j with the generic variable v: 2 2 N N N N N v v v v Find d k, the roots of this polynomial 3. Partial-fraction expansion: Assuming that M < N and all the d k are distinct, we may express X(e j )as N k j k k j e d C e X

76 76 3. Finding Inverse Fourier Transforms by Using Partial-Fraction Expansions Since n d un k DTFT j de k Expansions for repeated roots are treated in Appendix B. The linearity property implies that x N n n C d un k k k

77 4. Multiplication Property 77 Non-periodic continuous-time signals. Non-periodic signals: x(t), z(t), and y(t) = x(t)z(t). Find the FT of y(t). 2. FT of x(t) and z(t): jvt x t X jve dv 2 j t and z t Z j e d 2 j vt yt X jv Z j e ddv 2 2 Change variable: = v Inner Part: Z(j) X(j) jt y t X jv Z j vdv e d FT of y(t): Outer Part: FT of y(t) y( t) x( t) z( t) FT Y ( j) X ( j)* Z( j) 2 (3.56) Scaled by /2

78 4. Multiplication Property 78 where X j * Z j X jvz j v Multiplication of two signals in Time-Domain Convolution in Frequency-Domain (/2) Non-periodic continuous-time signals. Non-periodic DT signals: x[n], z[n], and y[n] = x[n]z[n]. 2. DTFT of y[n]: DTFT j j j y[ n] x[ n] z[ n] Y ( e ) X ( e ) # Z( e ) (3.57) 2 dv Find the DTFT of y[n]. where the symbol denotes periodic convolution. Here, X(e j ) and X(e j ) are 2-periodic, so we evaluate the convolution over a 2 interval: j j # X j Z j X e Z e d Multiplication of two signals in Time-Domain Convolution in Frequency-Domain (/2)

79 4. Multiplication Property 79 Multiplication property can be used to study the effects of truncating a timedomain Windowing! signal on its frequencydomain. Truncate signal x(t) by a window function w(t) is represented by y(t) = x(t)w(t) Fig (a) Time Interval T t T

80 4. Multiplication Property 8. FT of y(t): y t Y j X j W j 2 FT * 2. If w(t) is the rectangular window depicted in Fig (b), we have 2 W j sin T Smoothing the details in X(j) and introducing oscillation near discontinuities in X(j). Figure 3.65b (p. 293) (b) Convolution of the signal and window FT s resulting from truncation in time. p ; v

81 4. Multiplication Property 8 Example 3.46 Truncating the Impulse Response The frequency response H(e j ) of an ideal discrete-time system is depicted in Fig (a). Describe the frequency response of a system whose impulse response is the ideal system impulse response truncated to the interval M n M. <Sol.>. The ideal impulse response is the inverse DTFT of H(e j ). Using the result of Example 3.9, we write hn sin n 2n 2. Let h t [n] be the truncated impulse response: hn, n M ht n t, otherwise h [ n] h[ n] w[ n] wn, n M, otherwise

82 4. Multiplication Property 82 DTFT j 3. Let h [ n] H ( e ) t t Using the multiplication property in Eq. (3.57), we have j j j H t e H e W e d 2 4. Since and H e W e H t j j,, / 2 / 2 M sin / 2 sin 2 / 2 / 2 e j F 2 where we have defined / 2 d On the basis of Example 3.8 Discussion: refer to p. 295 in textbook. j, / 2 W e j j F H e W e, / 2

83 4. Multiplication Property 83 v, u Figure 3.66 (p. 294) The effect of truncating the impulse response of a discrete-time system. (a) Frequency response of ideal system. (b) F () for near zero.

84 4. Multiplication Property 84 v, u H t (e j ) is the area under F ( ) between = /2 and = /2. Figure 3.66 (p. 294) The effect of truncating the impulse response of a discretetime system. (c) F () for slightly greater than /2.(d) Frequency response of system with truncated impulse

85 4. Multiplication Property 85 Table 3.9 Multiplication Properties of Fourier Representations FT x( t) z( t) X ( j) Z( j) 2 FS; o x( t) z( t) X[ k] Z[ k] DTFT j j x[ n] z[ n] X ( e ) # Z( e ) 2 DTFT ; o x[ n] z[ n] X[ k] # Z[ k]

86 5. Scaling Properties 86. Let z(t) = x(at). 2. FT of z(t) : ( ) ( ) j t jt Z j z t e dt x( at) e dt Z( j) j( / a) (/ a) x( ) e d, a j( / a) (/ a) x( ) e d, a Changing variable: = at j( / a) ( ) (/ ) ( ), Z j a x e d FT z( t) x( at) (/ a ) X ( j / a). (3.6) Scaling in Time-Domain Inverse Scaling in Frequency-Domain Signal expansion or compression! Refer to Fig. 3.7.

87 5. Scaling Properties 87 v Figure 3.7 (p. 3) The FT scaling property. The figure assumes that < a <.

88 5. Scaling Properties 88 Example 3.48 Scaling a Rectangular Pulse Let the rectangular pulse Use the FT of x(t) and the scaling property to find the FT of the scaled rectangular pulse, t xt (), t, t 2 yt (), t 2 <Sol.>. Substituting T = into the result of Example gives 2 X( j) sin( ) Scaling property and a = /2 2. Note that y(t) = x(t/2). 2 2 Y ( j) 2 X ( j2 ) 2 sin(2 ) sin(2 ) Fig Substituting T = 2 into the result of Example 3.25 can also give the answer!

89 89 5. Scaling Properties Figure 3.7 (p. 3) Application of the FT scaling property in Example (a) Original time signal. (b) Original FT. (c) Scaled time signal y(t) = x(t/2). (d) Scaled FT Y(j) = 2X(j2). v

90 6. Parseval Relationships 9 The energy or power in the time-domain representation of a signal is equal to the energy or power in the frequency-domain representation. Case for CT nonperiodic signal: x(t). Energy in x(t): Wx x() t dt 2. Note that 2 2 x( t) x( t) x ( t) Express x*(t) in terms of its FT X(j): jt Wx x( t) X ( j) e d dt 2 j t W x X ( j) x( t) e dt d 2 Wx X ( j) X ( j) d 2 jt x ( t) X ( j) e d 2 The integral inside the braces is the FT of x(t). 2 2 x( t) dt X ( j) d 2

91 6. Parseval Relationships 9 Representations FT FS DTFT DTFS Parseval Relationships 2 2 x( t) dt X ( j) d 2 T T 2 2 x( t) dt X[ k] k 2 n N j x[ n] X ( e ) d 2 N 2 N 2 x[ n] X[ k] n k 2

92 6. Parseval Relationships 92 Example 3.5 Calculating Energy in a Signal Let sin( Wn) xn [ ] n Use Parseval s theorem to evaluate <Sol.> n 2 xn [ ] 2 n j X ( e ) 2 2 sin ( Wn) d n 2 2. Using the DTFT Parseval relationship in Table 3., we have 2. Since, x[ n] DTFT X ( e j ) W, W 2 W d W / W Direct calculation in timedomain is very difficult!

93 7. Time-Bandwidth Product 93. Preview:, t To FT x( t) X ( j ) 2sin( To ) /, t To Fig v, p Figure 3.72 (p. 35) Rectangular pulse illustrating the inverse relationship between the time and frequency extent of a signal. 2. x(t) has time extent 2T. X(j) is actually of infinite extent frequency. 3. The mainlobe of sinc function is fallen in the interval of < /T, in which contains the majority of its energy.

94 7. Time-Bandwidth Product The product of the time extent T and mainlobe width 2/T is a constant. 5. Compressing a signal in time leads expansion in the frequency domain and vice versa. Signla s time-bandwidth product! 6. Bandwidth: The extent of the signal s significant frequency content. ) A mainlobe bounded by nulls. Ex. Lowpass filter One half the width of mainlobe. 2) The frequency at which the magnitude spectrum is /2 times its peak values. 7. Effective duration of signal x(t): T d 2 2 t x() t dt 2 x() t dt / 2 (3.63) 8. Effective bandwidth of signal x(t): B w / X ( j ) d 2 X ( j) d (3.64)

95 7. Time-Bandwidth Product The time-bandwidth product for any signal is lower bounded according to the relationship TB / 2 (3.65) d w We cannot simultaneously decrease the duration and bandwidth of a signal. Uncertainty principle! Example 3.5 Bounding the Bandwidth of a Rectangular Pulse Let, xt (), t t T <Sol.>. T d of x(t): T / 2 o 2 t dt To T d To dt T o o T o Use the uncertainty principle to place a lower bound on the effective bandwidth of x(t).

96 7. Time-Bandwidth Product 96 T / 2 o 2 t dt To 3 To / 2 d [(/(2 o))(/ 3) T ] d / 3 T T T t T o o dt T o 2. The uncertainty principle given by Eq. (3.65) states that B w /(2 T ) B 3 /(2 T ) d w o

97 8. Duality 97 Duality of the FT. FT pair: jt jt x( t) X ( j) e d 2 and X ( j) x( t) e dt 2. General equation: jv y( v) z( ) e dv 2 (3.66) Choose = t and = the Eq. (3.66) implies that jt FT y( t) z( ) e d 2 y( t) z( ) (3.67) 3. Interchange the role of time and frequency by letting = and = t, then Eq. (3.66) implies that jt y( ) z( t) e dt 2 FT z( t) 2 y( ) (3.68)

98 98 8. Duality

99 Example 99 Example 3.52 Applying Duality Find the FT of <Sol.>. Note that: xt () 2. Replacing by t, we obtain jt t FT f ( t) e u( t) F( j) j F( jt) x(t) has been expressed as F(jt). jt 3. Duality property: X ( j) 2 f ( ) 2 e u( )

100 Duality v ; p Figure 3.74 (p. 39) The FT duality property.

101 8. Duality. DTFS Pair for x[n]: N jko x[ n] X[ k] e k n and N jko X[ k] x[ n] e N n n 2. The DTFS duality is stated as follows: if 2 DTFS ; N x[ n] X[ k] (3.7) 2 DTFS ; N X[ n] x[ k] (3.72) N N = time index; k = frequency index

102 8. Duality 2 DTFT jk ot z( t) Z[ k] e X ( e ) x[ n] e k 3. Duality relationship between z(t) and X(e j ) j jn k Require T = 2 In the DTFT t in the FS Assumption: = n In the DTFT k in the FS DTFS j x[ n] X ( e ) X e x k jt FS; ( ) [ ] FT FT FT f ( t) F( j) F( jt) 2 f ( ) DTFS x n DTFS ; 2 / N [ ] X[ k] DTFS j FS-DTFT x[ n] X ( e ) X n N x k DTFS ; 2 / N [ ] (/ ) [ ] X e x k jt FS; ( ) [ ] Table 3. Duality Properties of Fourier Representations

103 A. Wind Instruments Fipples, pressure-excited reeds, or buzzing lips are coupled with resonating air column.

104 A.2 String Instruments One or more stretched stringed strings are made to vibrate, typically by bowing, plunking or striking, in conjunction with resonating boxes or sound boards.

105 A.3 Percussion Instruments Include virtually any object that may be struck, typically consisting of bars, plates, or stretched membrances and associated resonators

106 A.4 Cello & Clarinet

107 A.5 Trumpet & Bass Drum

108 A.6 Speech Models p y ( n ) a y ( n k ) G b x ( n r ) k k Pitch period P q r r Voiced V U voiced/unvoiced switch All-Pole Filter y(n) signal speech Unvoiced Gain estimate ( z)