1. Chapter 7 LTI Discrete-Time Systems in the Transform domain

2. §7.1 LTI Discrete-Time Systems in the Transform Domain Such transform-domain representations provide additional insight into the behavior of such systems It is easier to design and implement these systems in the transform-domain for certain applications We consider now the use of the DTFT and the z-transform in developing the transform- domain representations of an LTI system

3. §7.1 LTI Discrete-Time Systems in the Transform Domain In this course we shall be concerned with LTI discrete-time systems characterized by linear constant coefficient difference equations of the form:

4. §7.1 LTI Discrete-Time Systems in the Transform Domain Applying the DTFT to the difference equation and making use of the linearity and the time- invariance properties we arrive at the input- output relation in the transform-domain as where Y(e j  ) and X(e j  ) are the DTFTs of y[n] and x[n], respectively

5. §7.1 LTI Discrete-Time Systems in the Transform Domain In developing the transform-domain representation of the difference equation, it has been tacitly assumed that X(e j  ) and Y(e j  ) exist The previous equation can be alternately written as

6. §7.1 LTI Discrete-Time Systems in the Transform Domain Applying the z-transform to both sides of the difference equation and making use of the linearity and the time-invariance properties we arrive at where Y(z) and X(z) denote the z-transforms of y[n] and x[n] with associated ROCs, respectively

7. §7.1 LTI Discrete-Time Systems in the Transform Domain A more convenient form of the z-domain representation of the difference equation is given by

8. §7.2 The Frequency Response Most discrete-time signals encountered in practice can be represented as a linear combination of a very large, maybe infinite, number of sinusoidal discrete-time signals of different angular frequencies Thus, knowing the response of the LTI system to a single sinusoidal signal, we can determine its response to more complicated signals by making use of the superposition property

9. §7.2 The Frequency Response The quantity H(e j  ) is called the frequency response of the LTI discrete- time system H(e j  ) provides a frequency-domain description of the system H(e j  ) is precisely the DTFT of the impulse response {h[n]} of the system

10. §7.2 The Frequency Response H(e j  ), in general, is a complex function of  with a period 2  It can be expressed in terms of its real and imaginary parts H(e j  )= H re (e j  ) +j H im (e j  ) or, in terms of its magnitude and phase, H(e j  )=|H(e j  )| e  (  ) where  (  )=argH(e j  )

11. §7.2 The Frequency Response The function | H(e j  ) | is called the magnitude response and the function  (  ) is called the phase response of the LTI discrete-time system Design specifications for the LTI discrete-time system, in many applications, are given in terms of the magnitude response or the phase response or both

12. §7.2 The Frequency Response In some cases, the magnitude function is specified in decibels as G(  ) = 20log 10 | H(e j  ) | dB where G (  ) is called the gain function The negative of the gain function A(  ) = - G(  ) is called the attenuation or loss function

13. §7.2 The Frequency Response Note: Magnitude and phase functions are real functions of , whereas the frequency response is a complex function of  If the impulse response h[n] is real then the magnitude function is an even function of  : |H(e j  )| = |H(e - j  )| and the phase function is an odd function of  :  (  ) = -  (-  )

14. §7.3 Frequency Response Computation Using MATLAB The function freqz(h,w) can be used to determine the values of the frequency response vector h at a set of given frequency points w From h, the real and imaginary parts can be computed using the functions real and imag, and the magnitude and phase functions using the functions abs and angle

15. §7.4 The Concept of Filtering One application of an LTI discrete-time system is to pass certain frequency components in an input sequence without any distortion (if possible) and to block other frequency components Such systems are called digital filters and one of the main subjects of discussion in this course

16. §7.4 The Concept of Filtering To understand the mechanism behind the design of frequency-selective filters, consider a real-coefficient LTI discrete- time system characterized by a magnitude function

17. §7.6 Frequency Response of the LTI Discrete-Time System For an LTI system described by a linear constant coefficient difference equation of the form we have It follows from the previous equation

18. §7.7 The Transfer Function A generalization of the frequency response function The convolution sum description of an LTI discrete-time system with an impulse response h[n] is given by

19. §7.7 The Transfer Function Taking the z-transforms of both sides we get

20. §7.8 The Transfer Function Thus, Y(z) = H(z)X(z) Or, Therefore,

21. §7.8 The Transfer Function Hence, H(z) = Y(Z)/X(z) The function H(z), which is the z-transform of the impulse response h[n] of the LTI system, is called the transfer function or the system function The inverse z-transform of the transfer function H(z) yields the impulse response h[n]

22. §7.8 The Transfer Function Its transfer function is obtained by taking the z-transform of both sides of the above equation Thus Consider an LTI discrete-time system characterized by a difference equation

23. §7.8 The Transfer Function Or, equivalently as An alternate form of the transfer function is given by

24. §7.8 The Transfer Function  1,  2,…,  M are the finite zeros, and 1, 2,…, N are the finite poles of H(z) If N > M, there are additional (N-M) zeros at z = 0 If N < M, there are additional (M-N) poles at z = 0 Or, equivalently as

25. §7.8 The Transfer Function For a causal IIR digital filter, the impulse response is a causal sequence The ROC of the causal transfer function Thus the ROC is given by is thus exterior to a circle going through the pole furthest from the origin

26. §7.8 The Transfer Function Example - Consider the M-point moving- average FIR filter with an impulse response Its transfer function is then given by

27. §7.8 The Transfer Function The transfer function has M zeros on the unit circle at z=e j2  k/M, 0  k  M-1 There are M poles at z = 0 and a single pole at z = 1 The pole at z = 1 exactly cancels the zero at z = 1 The ROC is the entire z-plane except z = 0 M = 8

28. §7.8 The Transfer Function Example - A causal LTI IIR digital filter is described by a constant coefficient difference equation given by y[n]=x[n-1]-1.2x[n-2]+x[n-3]+1.3y[n-1] -1.04y[n-2]+0.222y[n-3] Its transfer function is therefore given by

29. §7.8 The Transfer Function Alternate forms: ROC: Note: Poles farthest from z=0 have a magnitude

30. §7.9 Frequency Response from Transfer Function If the ROC of the transfer function H(z) includes the unit circle, then the frequency response H(e j  ) of the LTI digital filter can be obtained simply as follows: For a real coefficient transfer function H(z) it can be shown that

31. §7.9 Frequency Response from Transfer Function For a stable rational transfer function in the form the factored form of the frequency response is given by

32. §7.9 Frequency Response from Transfer Function It is convenient to visualize the contributions of the zero factor (z-  k ) and the pole factor (z- k ) from the factored form of the frequency response The magnitude function is given by

33. §7.10 Types of Transfer Functions The time-domain classification of an LTI digital transfer function sequence is based on the length of its impulse response: - Finite impulse response (FIR) transfer function - Infinite impulse response (IIR) transfer function

34. §7.10 Types of Transfer Functions Several other classifications are also used In the case of digital transfer functions with frequency-selective frequency responses, one classification is based on the shape of the magnitude function H(e i  ) or the form of the phase function  (  ) Based on this four types of ideal filters are usually defined

35. §7.10.1 Ideal Filters A digital filter designed to pass signal components of certain frequencies without distortion should have a frequency response equal to one at these frequencies, and should have a frequency response equal to zero at all other frequencies

36. §7.10.1 Ideal Filters The range of frequencies where the frequency response takes the value of one is called the passband The range of frequencies where the frequency response takes the value of zero is called the stopband

37. §7.10.1 Ideal Filters Frequency responses of the four popular types of ideal digital filters with real impulse response coefficients are shown below:

38. §7.10.1 Ideal Filters The frequencies  c,  c1, and  c2 are called the cutoff frequencies An ideal filter has a magnitude response equal to one in the passband and zero in the stopband, and has a zero phase everywhere

39. §7.10.1 Ideal Filters Earlier in the course we derived the inverse DTFT of the frequency response H LP (e j  ) of the ideal lowpass filter: h LP [n]=sin  c n/n , -  <n<  We have also shown that the above impulse response is not absolutely summable, and hence, the corresponding transfer function is not BIBO stable

40. §7.10.1 Ideal Filters Also, h LP [n] is not causal and is of doubly infinite length The remaining three ideal filters are also characterized by doubly infinite, noncausal impulse responses and are not absolutely summable Thus, the ideal filters with the ideal “brick wall” frequency responses cannot be realized with finite dimensional LTI filter

41. §7.10.1 Ideal Filters To develop stable and realizable transfer functions, the ideal frequency response specifications are relaxed by including a transition band between the passband and the stopband This permits the magnitude response to decay slowly from its maximum value in the passband to the zero value in the stopband

42. §7.10.1 Ideal Filters Moreover, the magnitude response is allowed to vary by a small amount both in the passband and the stopband Typical magnitude response specifications of a lowpass filter are shown as:

43. §7.10.2 Zero-Phase and Linear- Phase Transfer Functions A second classification of a transfer function is with respect to its phase characteristics In many applications, it is necessary that the digital filter designed does not distort the phase of the input signal components with frequencies in the passband

44. §7.10.2 Zero-Phase and Linear- Phase Transfer Functions One way to avoid any phase distortion is to make the frequency response of the filter real and nonnegative, i.e., to design the filter with a zero phase characteristic However, it is impossible to design a causal digital filter with a zero phase

45. §7.10.2 Zero-Phase and Linear- Phase Transfer Functions For non-real-time processing of real-valued input signals of finite length, zero-phase filtering can be very simply implemented by relaxing the causality requirement One zero-phase filtering scheme is sketched below x[n] v[n] u[n] w[n] H(z) u[n]=v[-n], y[n]=w[-n]

46. §7.10.2 Zero-Phase and Linear- Phase Transfer Functions It is easy to verify the above scheme in the frequency domain Let X(e j  ),V(e j  ),U(e j  ),W(e j  ), and Y(e j  ) denote the DTFTs of x[n], v[n], u[n], w[n], and y[n], respectively From the figure shown earlier and making use of the symmetry relations we arrive at the relations between various DTFTs as given on the next slide

47. §7.10.2 Zero-Phase and Linear- Phase Transfer Functions V(e j  )= H(e j  )X(e j  ), W(e j  )=H(e j  )U(e j  ) U(e j  )= V*(e j  ), Y(e j  )= W*(e j  ) Combining the above equations we get Y(e j  ) = W*(e j  ) = H*(e j  )U*(e j  ) = H*(e j  )V(e j  ) = H*(e j  )H(e j  )X(e j  ) = |H(e j  )| 2 X(e j  ) This is a zero-phase filter with a frequency response |H(e j  )| 2 x[n] v[n] u[n] w[n] H(z) u[n]=v[-n], y[n]=w[-n]

48. §7.10.2 Zero-Phase and Linear- Phase Transfer Functions The function fftfilt implements the above zero- phase filtering scheme In the case of a causal transfer function with a nonzero phase response, the phase distortion can be avoided by ensuring that the transfer function has a unity magnitude and a linear- phase characteristic in the frequency band of interest

49. §7.10.2 Zero-Phase and Linear- Phase Transfer Functions The most general type of a filter with a linear phase has a frequency response given by H(e j  )= e -j  D which has a linear phase from  = 0 to  = 2  Note also |H(e j  )|=1  (  )=D

50. §7.10.2 Zero-Phase and Linear- Phase Transfer Functions The output y[n] of this filter to an input x[n]=Ae j  n is then given by(P156) y[n]= Ae -j  D e j  n = Ae j  (n-D) If x a (t) and y a (t) represent the continuous-time signals whose sampled versions, sampled at t = nT, are x[n] and y[n] given above, then the delay between x a (t) and y a (t) is precisely the group delay of amount D

51. §7.10.2 Zero-Phase and Linear- Phase Transfer Functions If D is an integer, then y[n] is identical to x[n], but delayed by D samples If D is not an integer, y[n], being delayed by a fractional part, is not identical to x[n]

52. §7.10.2 Zero-Phase and Linear- Phase Transfer Functions If it is desired to pass input signal components in a certain frequency range undistorted in both magnitude and phase, then the transfer function should exhibit a unity magnitude response and a linear-phase response in the band of interest

53. §7.10.2 Zero-Phase and Linear- Phase Transfer Functions Figure right shows the frequency response if a lowpass filter with a linear-phase characteristic in the passband

54. §7.10.2 Zero-Phase and Linear- Phase Transfer Functions Since the signal components in the stopband are blocked, the phase response in the stopband can be of any shape Example - Determine the impulse response of an ideal lowpass filter with a linear phase response:

55. §7.10.2 Zero-Phase and Linear- Phase Transfer Functions Applying the frequency-shifting property of the DTFT to the impulse response of an ideal zero-phase lowpass filter we arrive at As before, the above filter is noncausal and of doubly infinite length, and hence, unrealizable

56. §7.10.2 Zero-Phase and Linear- Phase Transfer Functions By truncating the impulse response to a finite number of terms, a realizable FIR approximation to the ideal lowpass filter can be developed The truncated approximation may or may not exhibit linear phase, depending on the value of n 0 chosen

57. §7.10.2 Zero-Phase and Linear- Phase Transfer Functions If we choose n 0 = N/2 with N a positive integer, the truncated and shifted approximation ^ will be a length N+1 causal linear- phase FIR filter

58. §7.10.2 Zero-Phase and Linear- Phase Transfer Functions Figure below shows the filter coefficients obtained using the function sinc for two different values of N N=12N=13

59. §7.10.2 Zero-Phase and Linear- Phase Transfer Functions Because of the symmetry of the impulse response coefficients as indicated in the two figures, the frequency response of the truncated approximation can be expressed as: ^ ^  where, called the zero-phase response or amplitude response, is a real function of  

60. §7.11 Linear-Phase FIR Transfer Functions It is nearly impossible to design a linear- phase IIR transfer function It is always possible to design an FIR transfer function with an exact linear- phase response Consider a causal FIR transfer function H(z) of length N+1, i.e., of order N:

61. §7.11 Linear-Phase FIR Transfer Functions The above transfer function has a linear phase, if its impulse response h[n] is either symmetric, i.e., h[n]=h[N-n], 0  n  N or is antisymmetric, i.e., h[n]=-h[N-n], 0  n  N

62. §7.11 Linear-Phase FIR Transfer Functions Since the length of the impulse response can be either even or odd, we can define four types of linear-phase FIR transfer functions For an antisymmetric FIR filter of odd length, i.e., N even h[N/2] = 0 We examine next the each of the 4 cases

63. §7.11 Linear-Phase FIR Transfer Functions Type 1: N = 8 Type 2: N = 7 Type 3: N = 8 Type 4: N = 7

64. §7.11 Linear-Phase FIR Transfer Functions Type 1: Symmetric Impulse Response with Odd Length In this case, the degree N is even Assume N = 8 for simplicity The transfer function H(z) is given by

65. §7.11 Linear-Phase FIR Transfer Functions Because of symmetry, we have h[0]=h[8], h[1] = h[7], h[2] = h[6], and h[3] = h[5] Thus, we can write

66. §7.11 Linear-Phase FIR Transfer Functions The corresponding frequency response is then given by The quantity inside the braces is a real function of , and can assume positive or negative values in the range 0  |  | 

67. §7.11 Linear-Phase FIR Transfer Functions where  is either 0 or , and hence, it is a linear function of  in the generalized sense The group delay is given by indicating a constant group delay of 4 samples The phase function here is given by

68. §7.11 Linear-Phase FIR Transfer Functions In the general case for Type 1 FIR filters, the frequency response is of the form ~ ~ where the amplitude response, also called the zero-phase response, is of the form

69. §7.11 Linear-Phase FIR Transfer Functions which is seen to be a slightly modified version of a length-7 moving-average FIR filter The above transfer function has a symmetric impulse response and therefore a linear phase response Example - Consider

70. §7.11 Linear-Phase FIR Transfer Functions A plot of the magnitude response of along with that of the 7-point moving- average filter is shown below 00.20.40.60.81 0 0.2 0.4 0.6 0.8 1  /  Magnitude modified filter moving-average

71. §7.11 Linear-Phase FIR Transfer Functions Note the improved magnitude response obtained by simply changing the first and the last impulse response coefficients of a moving- average (MA) filter It can be shown that we can express which is seen to be a cascade of a 2-point MA filter with a 6-point MA filter

72. §7.12 Allpass Transfer Function is called an allpass transfer function An M-th order causal real-coefficient allpass transfer function is of the form Definition: An IIR transfer function A(z) with unity magnitude response for all frequencies, i.e.,

73. §7.12 Allpass Transfer Function If we denote the denominator polynomials of A M (z) as D M (z) : Note from the above that if z=re j  is a pole of a real coefficient allpass transfer function, then it has a zero at z=(1/r)e -j  then it follows that A M (z) can be written as:

74. §7.12 Allpass Transfer Function The numerator of a real-coefficient allpass transfer function is said to be the mirror-image polynomial of the denominator, and vice versa

75. §7.12 Allpass Transfer Function The expression implies that the poles and zeros of a real- coefficient allpass function exhibit mirror- image symmetry in the z-plane

76. §7.12 Allpass Transfer Function Therefore Hence To show that we observe that

77. §7.12 Allpass Transfer Function Now, the poles of a causal stable transfer function must lie inside the unit circle in the z-plane Hence, all zeros of a causal stable allpass transfer function must lie outside the unit circle in a mirror-image symmetry with its poles situated inside the unit circle

78. §7.12 Allpass Transfer Function A Simple Application A simple but often used application of an allpass filter is as a delay equalizer Let G(z) be the transfer function of a digital filter designed to meet a prescribed magnitude response The nonlinear phase response of G(z) can be corrected by cascading it with an allpass filter A(z) so that the overall cascade has a constant group delay in the band of interest

79. §7.12 Allpass Transfer Function G(z) A(z) Overall group delay is the given by the sum of the group delays of G(z) and A(z) Since, we have

80. §7.13 Minimum-Phase and Maximum-Phase Transfer Functions Both transfer functions have a pole inside the unit circle at the same location z=-a and are stable But the zero of H 1 (z) is inside the unit circle at z=-b, whereas, the zero of H 2 (z) is at z=-1/b situated in a mirror-image symmetry Consider the two 1st-order transfer functions:

81. §7.13 Minimum-Phase and Maximum-Phase Transfer Functions Figure below shows the pole-zero plots of the two transfer functions H 1 (z) H 2 (z)

82. §7.13 Minimum-Phase and Maximum-Phase Transfer Functions However, both transfer functions have an identical magnitude function as The corresponding phase functions are

83. §7.13 Minimum-Phase and Maximum-Phase Transfer Functions Figure below shows the unwrapped phase responses of the two transfer functions for a = 0.8 and b =-0.5

84. §7.13 Minimum-Phase and Maximum-Phase Transfer Functions From this figure it follows that H 2 (z) has an excess phase lag with respect to H 1 (z) Generalizing the above result, we can show that a causal stable transfer function with all zeros outside the unit circle has an excess phase compared to a causal transfer function with identical magnitude but having all zeros inside the unit circle

85. §7.13 Minimum-Phase and Maximum-Phase Transfer Functions A causal stable transfer function with all zeros inside the unit circle is called a minimum- phase transfer function A causal stable transfer function with all zeros outside the unit circle is called a maximum- phase transfer function Any nonminimum-phase transfer function can be expressed as the product of a minimum- phase transfer function and a stable allpass transfer function

86. Homework Read the textbook from p.353 to 401 Problems 7.24(a)