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

2. 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

3. Types of Transfer Functions  In the case of digital transfer functions with frequency-selective frequency responses, there are two types of classifications  (1) Classification based on the shape of the magnitude function |H (e jω ) |  (2) Classification based on the the form of the phase function θ(ω)

4. §7.1 Classification Based on Magnitude Characteristics  One common classification is based on an ideal magnitude response  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

5. §7.1.1 Digital Filters with Ideal Magnitude Responses  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

6. §7.1.1 Digital Filters with Ideal Magnitude Responses  Frequency responses of the four popular types of ideal digital filters with real impulse response coefficients are shown below:

7. §7.1.1 Digital Filters with Ideal Magnitude Responses  Lowpass filter: Passband -0≤ω≤ω c Stopband -ω c ≤ω≤ π  Highpass filter: Passband -ω c ≤ω≤ π Stopband -0≤ω≤ω c  Bandpass filter: Passband -ω c1 ≤ω≤ω c2 Stopband -0≤ω≤ω c1 and ω c2 ≤ω≤ π  Bandstop filter: Stopband -ω c1 ≤ω≤ω c2 Passband -0≤ω≤ω c1 and ω c2 ≤ω≤ π

8. §7.1.1 Digital Filters with Ideal Magnitude Responses  The frequencies ω c, ω c1,and ω 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

9. §7.1.1 Digital Filters with Ideal Magnitude Responses  Earlier in the course we derived the inverse DTFT of the frequency response H LP (e jω ) of the ideal lowpass filter:  We have also shown that the above impulse response is not absolutely summable, and hence, the corresponding transfer function is not BIBO stable

10. §7.1.1 Digital Filters with Ideal Magnitude Responses  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

11. §7.1.1 Digital Filters with Ideal Magnitude Responses  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

12. §7.1.1 Digital Filters with Ideal Magnitude Responses  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 the right

13. §7.1.2 Bounded Real Transfer Functions  A causal stable real-coefficient transfer function H(z) is defined as a bounded real (BR) transfer function if |H LP (e jω )|≤1 for all values of ω  Let x[n] and y[n] denote, respectively, the input and output of a digital filter characterized by a BR transfer function H(z) with X(e jω ) and Y(e jω ) denoting their DTFT s

14. §7.1.2 Bounded Real Transfer Functions  Then the condition |H(e jω )|≤1 implies that  Integrating the above from –π to π, and applying Parseval’s relation we get

15. §7.1.2 Bounded Real Transfer Functions  Thus, for all finite-energy inputs, the output energy is less than or equal to the input energy implying that a digital filter characterized by a BR transfer function can be viewed as a passive structure  If |H(e jω )|=1, then the output energy is equal to the input energy, and such a digital filter is therefore a lossless system

16. §7.1.2 Bounded Real Transfer Functions  A causal stable real-coefficient transfer function H(z) with |H(e jω )|=1 is thus called a lossless bounded real ( LBR ) transfer function  The BR and LBR transfer functions are the keys to the realization of digital filters with low coefficient sensitivity

17. §7.1.2 Bounded Real Transfer Functions  Example – Consider the causal stable IIR transfer function where K is a real constant  Its square-magnitude function is given by

18. §7.1.2 Bounded Real Transfer Functions  The maximum value of |H(e jω )| 2 is obtained when 2αcosω in the denominator is maximum and the minimum value is a obtained when 2αcosω is a minimum  For α > 0, maximum value of 2αcosω is equal to 2α at ω=0, and minimum value is -2α at ω=π

19. §7.1.2 Bounded Real Transfer Functions  Thus, for α>0, the maximum value of |H(e jω )| 2 is equal to K 2 /(1-α) 2 at ω=0 and the minimum value is equal to K 2 /(1+α) 2 at ω=π  On the other hand, for α<0, the maximum value of 2αcosω is equal to - 2α at ω=π and the minimum value is equal to 2α at ω= 0

20. §7.1.2 Bounded Real Transfer Functions  Here, the maximum value of |H(e jω )| 2 is equal to K 2 /(1-α) 2 at ω=π and the minimum value is equal to K 2 /(1-α) 2 at ω=0  Hence, the maximum value can be made equal to 1 by choosing K=±(1-α), in which case the minimum value becomes (1-α) 2 /(1+α) 2

21. §7.1.2 Bounded Real Transfer Functions is a BR function for K=±(1-α)  Plots of the magnitude function for α=±0.5 with values of K chosen to make H(z) a BR function are shown on the next slide  Hence,

22. §7.1.2 Bounded Real Transfer Functions

23. §7.1.3 Allpass Transfer Function Definition  An IIR transfer function A(z) with unity magnitude response for all frequencies, i.e., |A(e jω )| 2 = 1, for all ω is called an allpass transfer function  An M -th order causal real-coefficient allpass transfer function is of the form

24. §7.1.3 Allpass Transfer Function  If we denote the denominator polynomials of A M (z) as D M (z) : D M (z) = 1+d 1 z -1 +···+d M-1 z -M+1 +d M z -M then it follows that A M (z) can be written as:  Note from the above that if z=re jФ is a pole of a real coefficient allpass transfer function, then it has a zero at

25. §7.1.3 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  We shall use the notation to denote the mirror-image polynomial of a degree- M polynomial D M (z), i.e.,

26. §7.1.3 Allpass Transfer Function implies that the poles and zeros of a real- coefficient allpass function exhibit mirror- image symmetry in the z -plane  The expression

27. §7.1.3 Allpass Transfer Function  To show that we observe that  Therefore  Hence

28. §7.1.3 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

29. §7.1.3 Allpass Transfer Function  Figure below shows the principal value of the phase of the 3rd -order allpass function  Note the discontinuity by the amount of 2π in the phase θ(ω)

30. §7.1.3 Allpass Transfer Function  If we unwrap the phase by removing the discontinuity, we arrive at the unwrapped phase function θ c (ω) indicated below  Note: The unwrapped phase function is a continuous function of ω

31. §7.1.3 Allpass Transfer Function  The unwrapped phase function of any arbitrary causal stable allpass function is a continuous function of ω Properties  (1) A causal stable real-coefficient allpass transfer function is a lossless bounded real ( LBR ) function or, equivalently, a causal stable allpass filter is a lossless structure

32. §7.1.3 Allpass Transfer Function  (2) The magnitude function of a stable allpass function A(z) satisfies:  (3) Let τ(ω) denote the group delay function of an allpass filter A(z), i.e.,

33. §7.1.3 Allpass Transfer Function  The unwrapped phase function θ c (ω) of a stable allpass function is a monotonically decreasing function of ω so that τ(ω) is everywhere positive in the range 0<ω<π  The group delay of an M -th order stable real-coefficient allpass transfer function satisfies:

34. §7.1.3 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

35. §7.1.3 Allpass Transfer Function  Since |A(e jω )|=1, we have |G(e jω )A(e jω )|=|G(e jω )|  Overall group delay is the given by the sum of the group delays of G(z) and A(z) G(z)G(z)A(z)A(z)

36. §7.1.3 Allpass Transfer Function  Example – Figure below shows the group delay of a 4 th order elliptic filter with the following specifications: ω p =0.3π,δ p =1dB, δ s =35dB

37. §7.1.3 Allpass Transfer Function  Figure below shows the group delay of the original elliptic filter cascaded with an 8 th order allpass section designed to equalize the group delay in the passband

38. §7.2 Classification Based on Phase Characteristics  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

39. §7.2.1 Zero-Phase Transfer Function  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 not possible to design a causal digital filter with a zero phase

40. §7.2.1 Zero-Phase Transfer Function  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 H(z)H(z) x[n]x[n]v[n]v[n] u[n]=v[-n] H(z)H(z) u[n]u[n]w[n]w[n] y[n]=w[-n]

41. §7.2.1 Zero-Phase Transfer Function  It is easy to verify the above scheme in the frequency domain  Let X(e jω ), V(e jω ), U(e jω ), W(e jω ) and Y(e jω ) denote the DTFT s 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 DTFT s as given on the next slide

42. §7.2.1 Zero-Phase Transfer Function V(e jω )=H(e jω )X(e jω ), W(e jω )=H(e jω )U(e jω ) U(e jω )=V*(e jω ), Y(e jω )=W*(e jω )  Combining the above equations we get Y(e jω )=W*(e jω )=H*(e jω )U*(e jω ) = H*(e jω )V(e jω )=H*(e jω )H(e jω )X(e jω ) = |H(e jω )| 2 X(e jω ) H(z)H(z) x[n]x[n]v[n]v[n] u[n]=v[-n] H(z)H(z) u[n]u[n]w[n]w[n] y[n]=w[-n]

43. §7.2.1 Zero-Phase Transfer Function  The function filtfilt 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

44. §7.2.1 Zero-Phase Transfer Function  The most general type of a filter with a linear phase has a frequency response given by H(e jω )=e -jωD which has a linear phase from ω=0 to ω=2π  Note also |H(e jω )|=1 τ(ω)=D

45. §7.2.2 Linear-Phase Transfer Function  The output y[n] of this filter to an input x[n]=Ae jωn is then given by y[n]=Ae -jωD e jωn =Ae jω(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) isprecisely the group delay of amount D

46. §7.2.2 Linear-Phase Transfer Function  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]  In the latter case, the waveform of the underlying continuous-time output is identical to the waveform of the underlying continuous-time input and delayed D units of time

47. §7.2.2 Linear-Phase Transfer Function  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

48. §7.2.2 Linear-Phase Transfer Function  Figure below shows the frequency response if a lowpass filter with a linear-phase characteristic in the passband

49. §7.2.2 Linear-Phase Transfer Function  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:

50. §7.2.2 Linear-Phase Transfer Function  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

51. §7.2.2 Linear-Phase Transfer Function  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

52. §7.2.2 Linear-Phase Transfer Function  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

53. §7.2.2 Linear-Phase Transfer Function  Figure below shows the filter coefficients obtained using the function sinc for two different values of N

54. §7.2.2 Linear-Phase Transfer Function Zero-Phase Response  Because of the symmetry of the impulse response coefficients as indicated in the two figures, the frequency response of the truncated approximation can be expresse as: where, called the zero-phase response or amplitude response, is a real function of ω

55. §7.2.3 Minimum-Phase and Maximum- Phase Transfer Functions  Consider the two 1st -order transfer function:  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

56. §7.2.3 Minimum-Phase and Maximum- Phase Transfer Functions  Figure below shows the pole-zero plots of the two transfer functions

57. §7.2.3 Minimum-Phase and Maximum- Phase Transfer Functions  However, both transfer functions have an identical magnitude function as  The corresponding phase functions are

58. §7.2.3 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

59. §7.2.3 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)  The excess phase lag property of H 2 (z) with respect to H 1 (z) can also be explained by observing that we can write

60. §7.2.3 Minimum-Phase and Maximum- Phase Transfer Functions where A(z)=(bz+1)/(z+b) is a stable allpass function  The phase function of H 1 (z) and H 2 (z) are thus related through  As the unwrapped phase function of a stable first-order allpass function is a negative function of ω, it follows from the above that H 2 (z) has indeed an excess phase lag with respect to H 1 (z)

61. §7.2.3 Minimum-Phase and Maximum- Phase Transfer Functions  Generalizing the above result, let H m (z) be a causal stable transfer function with all zeros inside the unit circle and let H(z) be another causal stable transfer function satifying | H(e jω ) |=| H m (e jω ) |  These two transfer functions are then related through H(z) = H m (z) A (z) where A (z) is a causal stable allpass function

62. §7.2.3 Minimum-Phase and Maximum- Phase Transfer Functions  The unwrapped phase functions of H m (z) and H(z) are thus related through  H(z) has an excess phase lag with respect to H m (z)  A causal stable transfer function with all zeros inside the unit circle is called a minimum-phase transfer function

63. §7.2.3 Minimum-Phase and Maximum- Phase Transfer Functions  A causal stable transfer function with all zeros outside the unit circle is called a maximum-phase transfer function  A causal stable transfer function with zeros inside and outside the unit circle is called a mixed-phase transfer function

64. §7.2.3 Minimum-Phase and Maximum- Phase Transfer Functions  Example – Consider the mixed-phase transfer function  We can rewrite H(z) as

65. §7.3 Types of Linear-Phase FIR Transfer Functions  It is impossible to design an IIR transfer function with an exact linear-phase  It is always possible to design an FIR transfer function with an exact linear-phase response  We now develop the forms of the linear- phase FIR transfer function H(z) with real impluse response h[n]

66. called the amplitude response, also called the zero-phase response, is a real function of ω §7.3 Types of Linear-Phase FIR Transfer Functions  If H(z) is to have a linear-phase, its frequency response must be of the form where c and β are constants, and,  Let

67. §7.3 Types of Linear-Phase FIR Transfer Functions  For a real impulse response, the magnitude response |H(e jω )| is an even function of ω, i.e.,  Since,the amplitude response is then either an even function or an odd function of ω, i.e.

68. §7.3 Types of Linear-Phase FIR Transfer Functions  The frequency response satisfies the relation  If is an even function, then the above relation leads to or, equivalently, the relation implying that either β=0 or β=π

69. §7.3 Types of Linear-Phase FIR Transfer Functions  Substituting the value of β in the above we get  From we have

70. §7.3 Types of Linear-Phase FIR Transfer Functions  Replacing ω with -ω in the previous equation we get  Making a change of variable l=N-n, we rewrite the above equation as

71. §7.3 Types of Linear-Phase FIR Transfer Functions  The above leads to the condition  As, we have  Thus the FIR filter with an even amplitude ewsponse will have linear phase if it has a symmetric impulse response with c=-N/2

72. §7.3 Types of Linear-Phase FIR Transfer Functions  The above is satisfied if β=π/2 or β=-π/2  Then  If is an odd function of ω, then from we get as reduces to

73. §7.3 Types of Linear-Phase FIR Transfer Functions  The last equation can be rewritten as  As, from the above we get

74. §7.3 Types of Linear-Phase FIR Transfer Functions  Making a change of variable l =N-n we rewrite the last equation as  Equation the above with we arrive at the condition for linear phase as

75. §7.3 Types of Linear-Phase FIR Transfer Functions h[n]=h[n-N], 0≤n≤N with c=-N/2  Therefore, a FIR filter with an odd amplitude response will have linear-phase response if it has an antisymmetric impulse response

76. §7.3 Types of 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

77. §7.3 Types of Linear-Phase FIR Transfer Functions Type 1: N=8Type 2: N=7 Type 3: N=8Type 4: N=7

78. §7.3 Types of 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

79. §7.3 Types of Linear-Phase FIR Transfer Functions  Because of symmetry, we have h[0]=h[8], h[1]=h[7], h[2]=h[6], h[3]=h[5]  Thus we can write H[z]=h[0](1+z -8 )+h[1](z -1 +z -7 )+ h[2](z -2 +z -6 ) +h[3](z -3 +z -5 )+ h[4]z -4 =z -4 {h[0](z 4+ z -4 )+h[1](z 3+ z -3 )+h[2](z 2+ z -2 ) +h[3](z + z -1 )+h[4]}

80. §7.3 Types of Linear-Phase FIR Transfer Functions  The corresponding frequency response is then given by  The quantity inside the braces is a real function of w, and can assume positive or negative values in the range 0  |  | 

81. §7.3 Types of Linear-Phase FIR Transfer Functions  The phase here is given by θ(ω)=-4ω+β where β is either 0 or π,and hence, it is a linear function of ω  The group delay is given by indicating a constant group delay of 4 samples

82. §7.3 Types of 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

83. §7.3 Types of 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

84. §7.3 Types of Linear-Phase FIR Transfer Functions  A plot of the magnitude response of H 0 (z) 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

85. §7.3 Types of 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  Thus, H 0 (z) has a double zero at z=-1, i.e., (ω=π)

86. §7.3 Types of Linear-Phase FIR Transfer Functions Type 2: Symmetric Impulse Response with Even Length  In this case, the degree N is odd  Assume N=7 for simplicity  The transfer function if of the form H(z)=h[0]+h[1]z -1 +h[2]z -2 +h[3]z -3 +h[4]z -4 +h[5]z -5 +h[6]z -6 +h[7]z -7

87. §7.3 Types of Linear-Phase FIR Transfer Functions  Making use of the symmetry of the impulse response coefficients, the transfer function can be written as H(z)=h[0](1+z -7 )+h[1](z -1 +z -6 ) +h[2](z -2 +z -5 )+h[3](z -3 +z -4 ) =z -7/2 {h[0](z 7/2 +z -7/2 )+h[1](z 5/2 +z -5/2 ) +h[2](z 3/2 +z -3/2 )+h[3](z 1/2 +z -1/2 )}

88. §7.3 Types of Linear-Phase FIR Transfer Functions  The corresponding frequency response is given by  As before, the quantity inside the braces is a real function of ω, and can assume positive or negative values in the range 0≤|ω|≤π

89. §7.3 Types of Linear-Phase FIR Transfer Functions  Here the phase function is given by θ(ω)=-7/2ω+β where again is either 0 or π  As a result, the phase is also a linear function of ω  The corresponding group delay is τ(ω)=7/2 indicating a group delay of 7/2 samples

90. §7.3 Types of Linear-Phase FIR Transfer Functions  The expression for the frequency response in the general case for Type 2 FIR filters is of the form where the amplitude response is given by

91. §7.3 Types of Linear-Phase FIR Transfer Functions Type 3:Antiymmetric Impulse Response with Odd Length  In this case, the degree N is even  Assume N=8 for simplicity  Applying the symmetry condition we get H(z)=z -4 {h[0](z 4 -z -4 )+h[1](z 3 -z -3 ) +h[2](z 2 -z -2 )+h[3](z-z -1 )}

92. §7.3 Types of Linear-Phase FIR Transfer Functions  It also exhibits a linear phase response given by  The corresponding frequency response is given by where β is either 0 or π

93. §7.3 Types of Linear-Phase FIR Transfer Functions  The group delay here is τ(ω)=4 indicating a constant group delay of 4 samples  In the general case where the amplitude response is of the form

94. §7.3 Types of Linear-Phase FIR Transfer Functions Type 4:Antiymmetric Impulse Response with Even Length  In this case, the degree N is even  Assume N=7 for simplicity  Applying the symmetry condition we get H(z)=z -7/2 {h[0](z 7/2 -z -7/2 )+h[1](z 5/2 -z -5/2 ) +h[2](z3/ 2 -z -3/2 )+h[3](z 1/2 -z -1/2 )}

95. §7.3 Types of Linear-Phase FIR Transfer Functions  The corresponding frequency response is given by  It again exhibits a linear phase response given by θ(ω)=-7/2ω+π/2+β where β is either 0 or π

96. §7.3 Types of Linear-Phase FIR Transfer Functions  The group delay is constant andis given by τ(ω)=7/2  In the general case we have where now the amplitude response is of the form

97. §7.3 Types of Linear-Phase FIR Transfer Functions General Form of Frequency Response  In each of the four types of linear-phase FIR filters, the frequency response is of the form  The amplitude response for each of the four types of linear-phase FIR filters can become negative over certain frequency ranges, typically in the stopband

98. §7.3 Types of Linear-Phase FIR Transfer Functions  The magnitude and phase responses of the linear-phase FIR are given by  The group delay in each case is τ(ω)=N/2

99. §7.3 Types of Linear-Phase FIR Transfer Functions  Note that, even though the group delay is constant, since in general |H(e jω )| is not a constant, the output waveform is not a replica of the input waveform  An FIR filter with a frequency response that is a real function of ω is often called a zero- phase filter  Such a filter must have a noncausal impulse response

100. §7.3.1 Zero Location of Linear- Phase FIR Transfer Functions  Consider first an FIR filter with a symmetric impulse response: h[n]=h[N-n]  Its transfer function can be written as  By making a change of variable m=N-n, we can write

101. §7.3.1 Zero Location of Linear- Phase FIR Transfer Functions  Hence for an FIR filter with a symmetric impulse response of length N+1 we have  But,  A real-coefficient polynomial H(z) satisfying the above condition is called a mirror-image polynomial (MIP)

102. §7.3.1 Zero Location of Linear- Phase FIR Transfer Functions  Now consider first an FIR filter with an antisymmetric impulse response:  Its transfer function can be written as  By making a change of variable m=N-n, we get

103. §7.3.1 Zero Location of Linear- Phase FIR Transfer Functions  Hence, the transfer function H(z) of an FIR filter with an antisymmetric impulse response satisfies the condition  A real-coefficient polynomial H(z) satisfying the above condition is called a antimirror- image polynomial (AIP)

104.  Hence, a zero at z=ξ o is associated with a zero at §7.3.1 Zero Location of Linear- Phase FIR Transfer Functions  It follows from the relation H(z)=±z -N H(z -1 ) that if z=ξ o is a zero of H(z), so is z=1/ξ o  Moreover, for an FIR filter with a real impulse response, the zeros of H(z) occur in complex conjugate pairs

105. §7.3.1 Zero Location of Linear- Phase FIR Transfer Functions  Thus, a complex zero that is not on the unit circle is associated with a set of 4 zeros given by  A zero on the unit circle appear as a pair as its reciprocal is also its complex conjugate

106. §7.3.1 Zero Location of Linear- Phase FIR Transfer Functions  Since a zero at z =±1 is its own reciprocal, it can appear only singly  Now a Type 2 FIR filter satisfies H(z)=z -N H(z -1 ) with degree N odd  Hence, H(-1)=(-1) -N H(-1)=-H(-1) implying H(-1)=0, i.e., H(z) must have a zero at z=-1

107. §7.3.1 Zero Location of Linear- Phase FIR Transfer Functions  Likewise, a Type 3 or 4 FIR filter satisfies H(z)=-z -N H(z -1 )  Thus H(-1)=-(1) -N H(1)=-H(1) implying that H(z) must have a zero at z = 1  On the other hand, only the Type 3 FIR filter is restricted to have a zero at since here the degree N is even and hence, H(-1)=-(-1) -N H(-1)=-H(-1)

108. §7.3.1 Zero Location of Linear- Phase FIR Transfer Functions  Typical zero locations shown below

109. §7.3.1 Zero Location of Linear- Phase FIR Transfer Functions  Summarizing (1) Type 1 FIR filter: Either an even number or no zeros at z =1 and z =-1 (2) Type 2 FIR filter: Either an even number or no zeros at z =1, and an odd number of zeros at z =-1 (3) Type 3 FIR filter: An odd number of zeros at z =1 and z =-1

110. §7.3.1 Zero Location of Linear- Phase FIR Transfer Functions (4) Type 4 FIR filter: An odd number of zeros at z =1, and either an even number of no zeros at z =-1  The presence of zeros at z = ± 1 leads to the following limitations on the use of these linear-phase transfer functions for designing frequency-selective filters

111. §7.3.1 Zero Location of Linear- Phase FIR Transfer Functions  A Type 2 FIR filter cannot be used to design a highpass filter since it always has a zero z =-1  A Type 3 FIR filter has zeros at both z =1 and z =-1, and hence cannot be used to design either a lowpass or a highpass or a bandstop filter

112. §7.3.1 Zero Location of Linear- Phase FIR Transfer Functions  A Type 4 FIR filter is not appropriate to design lowpass and bandstop filters due to the presence of a zero at z =1  Type 1 FIR filter has no such restrictions and can be used to design almost any type of filter

113. §7.4 Simple Digital Filters  Later in the course we shall review various methods of designing frequency-selective filters satisfying prescribed specifications  We now describe several low-order FIR and IIR digital filters with reasonable selective frequency responses that often are satisfactory in a number of applications

114. §7.4.1 Simple FIR Digital Filters  FIR digital filters considered here have integer-valued impulse response coefficients  These filters are employed in a number of practical applications, primarily because of their simplicity, which makes them amenable to inexpensive hardware implementations

115. §7.4.1 Simple FIR Digital Filters Lowpass FIR Digital Filters  The simplest lowpass FIR digital filter is the 2 -point moving-average filter given by  The above transfer function has a zero at z=-1 and a pole at z=0  Note that here the pole vector has a unity magnitude for all values of ω

116. §7.4.1 Simple FIR Digital Filters  On the other hand, as ω increases from 0 to π, the magnitude of the zero vector decreases from a value of 2, the diameter of the unit circle, to 0  Hence, the magnitude response |H 0 (e jω )| is a monotonically decreasing function of ωfrom ω=0 to ω=π

117. §7.4.1 Simple FIR Digital Filters  The maximum value of the magnitude function is 1 at ω = 0, and the minimum value is 0 at ω = π, i.e.,  The frequency response of the above filter is given by

118. §7.4.1 Simple FIR Digital Filters can be seen to be a monotonically decreasing function of ω  The magnitude response

119. §7.4.1 Simple FIR Digital Filters  The frequency ω=ω c at which  since the dc gain G(0)=20log 10 |H(e j0 )|=0 is of practical interest since here the gain G(ω c ) in dB is given by

120. §7.4.1 Simple FIR Digital Filters  Thus, the gain G(ω) at ω=ω c is approximately 3 dB less than the gain at ω=0  As a result, ω c is called the 3-dB cutoff frequency  To determine the value of ω c we set which yields ω c =π/2

121. §7.4.1 Simple FIR Digital Filters  The 3 -dB cutoff frequency ω c can be considered as the passband edge frequency  As a result, for the filter H 0 (z) the passband width is approximately π/2  Note: H 0 (z) has a zero at z=-1 or ω= π, which is in the stopband of the filter

122. §7.4.1 Simple FIR Digital Filters  A cascade of the simple FIR filter  results in an improved lowpass frequency response as illustrated below for a cascade of 3 sections

123. §7.4.1 Simple FIR Digital Filters  The 3 -dB cutoff frequency of a cascade of  M sections is given by  For M = 3, the above yields ω c =0.302 π  Thus, the cascade of first-order sections yields a sharper magnitude response but at the expense of a decrease in the width of the passband

124. §7.4.1 Simple FIR Digital Filters  A better approximation to the ideal lowpass filter is given by a higher-order moving- average filter  Signals with rapid fluctuations in sample values are generally associated with high- frequency components  These high-frequency components are essentially removed by an moving-average filter resulting in a smoother output

125. §7.4.1 Simple FIR Digital Filters Highpass FIR Digital Filters  The simplest highpass FIR filter is obtained from the simplest lowpass FIR filter by replacing z with - z  This results in

126. §7.4.1 Simple FIR Digital Filters  Corresponding frequency response is given by whose magnitude response is plotted below

127. §7.4.1 Simple FIR Digital Filters  The monotonically increasing behavior of the magnitude function can again be demonstrated by examining the pole-zero pattern of the transfer function H 1 (z)  The highpass transfer function H 1 (z) has a zero at z =1 or ω=0 which is in the

128. §7.4.1 Simple FIR Digital Filters  Improved highpass magnitude response can again be obtained by cascading several sections of the first-order highpass filter  Alternately, a higher-order highpass filter of the form is obtained by replacing z with -z in the transfer in function of a moving average filter

129. §7.4.1 Simple FIR Digital Filters  An application of the FIR highpass filters is in moving-target-indicator (MTI) radars  In these radars, interfering signals, called clutters, are generated from fixed objects in the path of the radar beam  The clutter, generated mainly from ground echoes and weather returns, has frequency components near zero frequency (dc)

130. §7.4.1 Simple FIR Digital Filters  The clutter can be removed by filtering the radar return signal through a two-pulse canceler, which is the first-order FIR highpass filter H 1 (z)=1/2(1-z -1 )  For a more effective removal it may be necessary to use a three-pulse canceler obtained by cascading two two-pulse cancelers

131. §7.4.2 Simple IIR Digital Filters Lowpass IIR Digital Filters  We have shown earlier that the first-order causal IIR transfer function has a lowpass magnitude response for α>0

132. §7.4.2 Simple IIR Digital Filters  An improved lowpass magnitude response is obtained by adding a factor (1+z -1 ) to the numerator of transfer function  This forces the magnitude response to have a zero at ω=π in the stopband of the

133. §7.4.2 Simple IIR Digital Filters  On the other hand, the first-order causal IIR transfer function has a highpass magnitude response for α< 0

134. §7.4.2 Simple IIR Digital Filters  However, the modified transfer function obtained with the addition of a factor (1+z -1 ) to the numerator exhibits a lowpass magnitude response

135. §7.4.2 Simple IIR Digital Filters  The modified first-order lowpass transfer function for both positive and negative values of α is then given by  As ω increases from 0 to π, the magnitude of the zero vector decreases from a value of 2 to 0

136. §7.4.2 Simple IIR Digital Filters  The maximum values of the magnitude function is 2K/(1-α) at ω=0 and the minimum value is 0 at ω=π, i.e.,  Therefore, |H LP (e jω )| is a monotonically decreasing function of ω from ω=0 to ω=π

137. §7.4.2 Simple IIR Digital Filters  For most applications, it is usual to have a dc gain of 0 dB, that is to have |H LP (e j0 )| =1  To this end, we choose K=(1-α)/2 resulting in the first-order IIR lowpass transfer function  The above transfer function has a zero at i.e., at ω=π which is in the stopband

138. §7.4.2 Simple IIR Digital Filters Lowpass IIR Digital Filters  A first-order causal lowpass IIR digital filter has a transfer function given by where |α|<1 for stability  The above transfer function has a zero at z =-1 i.e., at ω=π which is in the stopband

139. §7.4.2 Simple IIR Digital Filters  H LP (z) has a real pole at z =α  As ω increases from 0 to π, the magnitude of the zero vector decreases from a value of 2 to 0, whereas, for a positive value of α, the magnitude of the pole vector increases from a value of 1-α to 1+α  The maximum value of the magnitude function is 1 at ω= 0, and the minimum value is 0 at ω=π

140. §7.4.2 Simple IIR Digital Filters  i.e., |H LP (e j0 )| =1, |H LP (e jπ )| =0  Therefore, |H LP (e jω )| is a monotonically decreasing function of ω from ω=0 to ω=π as indicated below

141. §7.4.2 Simple IIR Digital Filters  The squared magnitude function is given by  The derivative of |H LP (e jω )| 2 with respect to ω is given by

142. §7.4.2 Simple IIR Digital Filters d|H LP (e jω )| 2 /dω≤0 in the range 0≤ω≤π verifying again the monotonically decreasing behavior of the magnitude function  To determine the 3 -dB cutoff frequency we set in the expression for the square magnitude function resulting in

143. §7.4.2 Simple IIR Digital Filters The above quadratic equation can be solved for α yielding two solutions or which when solved yields

144. §7.4.2 Simple IIR Digital Filters  The solution resulting in a stable transfer function H LP (z) is given by  It follows from that H LP (z) is a BR function for |α|<1

145. §7.4.2 Simple IIR Digital Filters Highpass IIR Digital Filters  A first-order causal highpass IIR digital filter has a transfer function given by where |α|<1 for stability  The above transfer function has a zero at z=1 i.e., at ω=0 which is in the stopband

146. §7.4.2 Simple IIR Digital Filters  Its 3 -dB cutoff frequency ω c is given by which is the same as that of H LP (z)  Magnitude and gain responses of H HP (z) are shown below

147. §7.4.2 Simple IIR Digital Filters  H HP (z) is a BR function for |α|<1  Example – Design a first-order highpass digital filter with a 3 -dB cutoff frequency of 0.8π  Now, sin( ω c )=sin(0.8π)=0.587785 and cos(0.8π)=-0.80902  Therefore α=(1-sin ω c )/cos ω c =-0.5095245

148. §7.4.2 Simple IIR Digital Filters  Therefore

149. §7.4.2 Simple IIR Digital Filters Bandpass IIR Digital Filters  A 2nd -order bandpass digital transfer function is given by  Its squared magnitude function is

150. §7.4.2 Simple IIR Digital Filters  |H BP (e jω )| 2 goes to zero at ω=0 and ω=π  It assumes a maximum value of 1 at ω=ω 0, called the center frequency of the bandpass filter, where  The frequencies ω c1 and ω c2 where |H BP (e jω )| 2 becomes 1/2 are called the 3 -dB cutoff frequencies

151. §7.4.2 Simple IIR Digital Filters  The difference between the two cutoff frequencies, assuming ω c2 >ω c1 is called the 3-dB bandwidth and is given by  The transfer function H BP (z) is a BR function if |α|<1 and |β|<1

152. §7.4.2 Simple IIR Digital Filters  Plots of |H BP (e jω )| are shown below

153. §7.4.2 Simple IIR Digital Filters  Example – Design a 2nd order bandpass digital filter with center frequency at 0.4π and a 3 -dB bandwidth of 0.1π  Here β=cos(ω 0 )=cos(0.4π)=0.309017 and  The solution of the above equation yields: α=1.376382 and α=0.72654253

154. §7.4.2 Simple IIR Digital Filters  The corresponding transfer functions are and  The poles of H’ BP (z) are at z=0.3671712± j1.11425636 and have a magnitude >1

155. §7.4.2 Simple IIR Digital Filters  Thus, the poles of H’ BP (z) are outside the unit circle making the transfer function unstable  On the other hand, the poles of H” BP (z) are at z= 0.2667655±j0.85095546 and have a magnitude of 0.8523746  Hence H” BP (z) is BIBOstable  Later we outline a simpler stability test

156. §7.4.2 Simple IIR Digital Filters  Figures below show the plots of the magnitude function and the group delay of H” BP (z)

157. §7.4.2 Simple IIR Digital Filters Bandstop IIR Digital Filters  A 2nd -order bandstop digital filter has a transfer function given by  The transfer function H BS (z) is a BR function if |α|<1 and |β|<1

158. §7.4.2 Simple IIR Digital Filters  Its magnitude response is plotted below

159. §7.4.2 Simple IIR Digital Filters  Here, the magnitude function takes the maximum value of 1 at ω=0 and ω=π  It goes to 0 at ω=ω 0, where ω 0, called the notch frequency, is given by ω 0 =cos -1 (β)  The digital transfer function H BS (z) is more commonly called a notch filter

160. §7.4.2 Simple IIR Digital Filters  The frequencies ω c2 and ω c1 where |H BS (e jω )| 2 becomes 1/2 are called the 3-dB cutoff frequencies  The difference between the two cutoff frequencies, assuming ω c2 >ω c1 is called the 3-dB notch bandwidth and is given by

161. §7.4.2 Simple IIR Digital Filters Higher-Order IIR Digital Filters  By cascading the simple digital filters discussed so far, we can implement digital filters with sharper magnitude responses  Consider a cascade of K first-order lowpass sections characterized by the transfer

162. §7.4.2 Simple IIR Digital Filters  The overall structure has a transfer function given by  The corresponding squared-magnitude function is given by

163. §7.4.2 Simple IIR Digital Filters  To determine the relation between its 3-dB cutoff frequency ω c and the parameter α, we set which when solved for α, yields for a stable G LP (z):

164. §7.4.2 Simple IIR Digital Filters for K=1 where  It should be noted that the expression for αgiven earilier reduces to

165. §7.4.2 Simple IIR Digital Filters  Example – Design a lowpass filter with a 3- dB cutoff frequency at ω c =0.4π using a single first-order section and a cascade of 4 first- order sections, and compare their gain responses  For the single first-order lowpass filter we have

166. §7.4.2 Simple IIR Digital Filters  For the cascade of 4 first-order sections, we substitute K=4 and get  Next we compute

167. §7.4.2 Simple IIR Digital Filters  The gain responses of the two filters are shown below  As can be seen, cascading has resulted in a sharper roll-off in the gain response

168. §7.4.3 Comb Filters  The simple filters discussed so far are characterized either by a single passband and/or a single stopband  There are applications where filters with multiple passbands and stopbands are required  The comb filter is an example of such filters

169. §7.4.3 Comb Filters  In its most general form, a comb filter has a frequency response that is a periodic function of ω with a period 2π/L, where L is a positive integer  If H(z) is a filter with a single passband and/or a single stopband, a comb filter can be easily generated from it by replacing each delay in its realization with L delays resulting in a structure with a transfer function given by G(z)=H(z L )

170. §7.4.3 Comb Filters  If |H(e jω )| exhibits a peak at ω p,then |G(e jω )| will exhibit L peaks at ω p k/L, 0≤k≤L-1 in the frequency range 0≤ω<2π  Likewise, if |H(e jω )| has a notch atω 0, then |G(e jω )| will have L notches at ω 0 k/L, 0≤k≤L-1 in the frequency range 0≤ω<2π  A comb filter can be generated from either an FIR or an IIR prototype filter

171. §7.4.3 Comb Filters  For example, the comb filter generated from the prototype lowpass FIR filter H 0 (z)=1/2(1+z -1 ) has a transfer function  |G 0 (e jω )| has L notches at ω =(2k+1)π/L and L peaks at ω =2πk/L, 0≤k≤L-1, in the frequency range 0≤ω<2π

172. §7.4.3 Comb Filters  For example, the comb filter generated from the prototype highpass FIR filter H 1 (z)=1/2(1-z -1 ) has a transfer function  |G 1 (e jω )| has L peaks atω =(2k+1)π/L and L notches at ω =2πk/L, 0≤k≤L-1, in the frequency range 0≤ω<2π

173. §7.4.3 Comb Filters  Depending on applications, comb filters with other types of periodic magnitude responses can be easily generated by appropriately choosing the prototype filter  For example, the M -point moving average filter has been used as a prototype

174. §7.4.3 Comb Filters  This filter has a peak magnitude at ω=0, and M-1 notches at ω=2πl/M, 1≤l≤M-1  The corresponding comb filter has a transfer function whose magnitude has L peaks at ω=2πk/L, 0≤k≤L-1 and L(M-1) notches at ω=2πk/LM, 1≤k≤L(M-1)

175. §7.5 Complementary Transfer Functions  A set of digital transfer functions with complementary characteristics often finds useful applications in practice  Four useful complementary relations are described next along with some applications

176. §7.5.1 Delay-Complementary Transfer Functions  A set of L transfer functions, {H i (z)}, 0≤i≤L-1, is defined to be delay-complementary of each other if the sum of their transfer functions is equal to some integer multiple of unit delays, i.e., where n 0 is a nonnegative integer

177. §7.5.1 Delay-Complementary Transfer Functions  A delay-complementary pair {H 0 (z), {H 1 (z)}, can be readily designed if one of the pairs is a known Type 1 FIR transfer function of odd length  Let H 0 (z) be a Type 1 FIR transfer function of length M=2K+1  Then its delay-complementary transfer function is given by

178. §7.5.1 Delay-Complementary Transfer Functions  Let the magnitude response of H 0 (z) be equal to and 1±δ p in the passband and less than or equal to δ s in the stopband where δ p and δ s are very small numbers  Now the frequency response of H 0 (z) can be expressed as where is the amplitude response

179. §7.5.1 Delay-Complementary Transfer Functions  Its delay-complementary transfer function H 1 (z) has a frequency response given by  Now, in the passband, and in the stopband,  It follows from the above equation that in the stopband, and in the passband,

180. §7.5.1 Delay-Complementary Transfer Functions  As a result, H 1 (z) has a complementary magnitude response characteristic to that of H 0 (z) with a stopband exactly identical to the passband of H 0 (z), and a passband that is exactly identical to the stopband of H 0 (z)  Thus, if H 0 (z) is a lowpass filter, H 1 (z) will be a highpass filter, and vice versa

181. §7.5.1 Delay-Complementary Transfer Functions  The frequency ω 0 at which the gain responses of both filters are 6 dB below their maximum values  The frequency ω 0 is thus called the 6 dB crossover frequency

182. §7.5.1 Delay-Complementary Transfer Functions  Example – Consider the Type 1 bandstop transfer function  Its delay-complementary Type 1 bandpass transfer function is given by

183. §7.5.1 Delay-Complementary Transfer Functions  Plots of the magnitude responses of H BS (z) and H BP (z) are shown below

184. §7.5.2 Allpass Complementary Transfer Functions  A set of M digital transfer functions, {H i (z)}, 0≤i≤M-1, is defined to be allpass- complementary of each other, if the sum of their transfer functions is equal to an allpass function, i.e.,

185. §7.5.3 Power-Complementary Transfer Functions  A set of M digital transfer functions, {H i (z)}, 0≤i≤M-1, is defined to be power- complementary of each other, if the sum of their square-magnitude responses is equal to a constant K for all values of ω, i.e.,

186. §7.5.3 Power-Complementary Transfer Functions  By analytic continuation, the above property is equal to for real coefficient H i (z)  Usually, by scaling the transfer functions, the power-complementary property is defined for K=1

187. §7.5.3 Power-Complementary Transfer Functions  For a pair of power-complementary transfer functions, H 0 (z) and H 1 (z), the frequency ω 0 where |H 0 (e jω 0 )| 2 = |H 1 (e jω 0 )| 2 =0.5, is called the cross-over frequency  At this frequency the gain responses of both filters are 3-dB below their maximum values  As a result, ω 0 is called the 3-dB cross-over frequency

188. §7.5.3 Power-Complementary Transfer Functions  Example – Consider the two transfer functions H 0 (z) and H 1 (z) given by where A 0 (z) ana A 1 (z) are stable allpass transfer functions  Note that H 0 (z)+H 1 (z)=A 0 (z)  Hence, H 0 (z) and H 1 (z) are allpass complementary

189. §7.5.3 Power-Complementary Transfer Functions  It can be shown that H 0 (z) and H 1 (z) are also power-complementary  Moreover, H 0 (z) and H 1 (z) are bounded-real transfer functions

190. §7.5.4 Double-Complementary Transfer Functions  A set of M transfer functions satisfying both the allpass complementary and the power- complementary properties is known as a doubly-complementary set

191.  A pair of doubly-complementary IIR transfer functions, H 0 (z) and H 1 (z), with a sum of allpass decomposition can be simply realized as indicated below §7.5.4 Double-Complementary Transfer Functions A0(z)A0(z) A1(z)A1(z) ● ● ● X(z)X(z) Y0(z)Y0(z) Y1(z)Y1(z) 1/2

192. §7.5.4 Double-Complementary Transfer Functions  Example – The first-order lowpass transfer function can be expressed as where

193. §7.5.4 Double-Complementary Transfer Functions  Its power-complementary highpass transfer function is thus given by  The above expression is precisely the first- order highpass transfer function described earlier

194. Complementary Transfer Functions  Figure below demonstrates the allpass complementary property and the power complementary property of H LP (z) and H HP (z)

195. §7.5.5 Power-Symmetric Filters and Conjugate Quadrature Filters  A real-coefficient causal digital filter with a transfer function H(z) is said to be a power- symmetric filter if it satisfies the condition where K>0 is aconstant

196. §7.5.5 Power-Symmetric Filters and Conjugate Quadrature Filters  It can be shown that the gain function G(ω) of a power-symmetric transfer function at ω=π is given by  If we defined G(z)= H(-z), then it follows from the definition of the power-symmetric filter that H(z) and G(z) are power- complementary as

197. §7.5.5 Power-Symmetric Filters and Conjugate Quadrature Filters Conjugate Quadratic Filter  If a power-symmetric filter has an FIR transfer function H(z) of order N, then the FIR digital filter with a transfer function is called a conjugate quadratic filter of H(z) and vice-versa

198. §7.5.5 Power-Symmetric Filters and Conjugate Quadrature Filters  It follows from the definition that G(z) is also a power-symmetric causal filter  It also can be seen that a pair of conjugate quadratic filters H(z) and G(z) are also power-complementary

199. §7.5.5 Power-Symmetric Filters and Conjugate Quadrature Filters  Example – Let H(z)=1-2z -1 +6z -2 +3z -3  We form  H(z) is a power-symmetric transfer function

200. §7.8 Digital Two-Pairs  The LTI discrete-time systems considered so far are single-input, single-output structures characterized by a transfer function  Often, such a system can be efficiently realized by interconnecting two-input, two- output structures, more commonly called two-pairs

201. §7.8 Digital Two-Pairs  Figures below show two commonly used block diagram representations of a two-pair  Here Y 1 and Y 2 denote the two outputs X 1 and X 2 denote the two inputs,where the dependencies on the variable z has been omitted for simplicity X1X1 Y1Y1 Y2Y2 X2X2 X1X1 X2X2 Y1Y1 Y2Y2

202. §7.8.1 Characterization  The input-output relation of a digital two- pair is given by  In the above relation the matrix τ given by is called the transfer matrix of the two-pair

203. §7.8.1 Characterization  It follows from the input-output relation that the transfer parameters can be found as follows:

204. §7.8.1 Characterization  An alternate characterization of the two-pair is in terms of its chain parameters where the matrix Γ given by is called the chain matrix of the two-pair

205. §7.8.1 Characterization  The relation between the transfer parameters and the chain parameters are given by

206. §7.8.2 Two-Pair Interconnection Schemes  Cascade Connection – Γ-cascade  Here

207. §7.8.2 Two-Pair Interconnection Schemes  But from figure, X ” 1 =Y ’ 2 and Y ” 1 =X ’ 2  Substituting the above relations in the first equation on the previous slide and combining the two equations we get  Hence

208. §7.8.2 Two-Pair Interconnection Schemes  Cascade Connection –τ-cascade  Here

209. §7.8.2 Two-Pair Interconnection Schemes  But from figure, X ” 1 =Y ’ 1 and X ’ 2 = Y ” 2  Substituting the above relations in the first equation on the previous slide and combining the two equations we get  Hence,

210. §7.8.2 Two-Pair Interconnection Schemes Constrained Two-Pair  It can be shown that Y2Y2 G(z)G(z) X1X1 X2X2 Y1Y1 H(z)H(z)

211. §7.9 Algebraic Stability Test  We have shown that the BIBO stability of a causal rational transfer function requires that all its poles be inside the unit circle  For very high-order transfer functions, it is very difficult to determine the pole locations analytically  Root locations can of course be determined on a computer by some type of root finding algorithms

212. §7.9.1 The Stability Triangle  We now outline a simple algebraic test that does not require the determination of pole locations The Stability Triangle  For a 2nd-order transfer function the stability can be easily checked by examining its denominator coefficients

213. §7.9.1 The Stability Triangle denote the denominator of the transfer function  In terms of its poles, D(z) can be expressed as  Let  Comparing the last two equations we get

214. §7.9.1 The Stability Triangle  Now the coefficient d 2 is given by the product of the poles  Hence we must have  The poles are inside the unit circle if  It can be shown that the second coefficient condition is given by

215. §7.9.1 The Stability Triangle  The region in the (d 1, d 2 )- plane where the two coefficient condition are satisfied, called the stability triangle, is shown below

216. §7.9.1 The Stability Triangle  Example – Consider the two 2nd -order bandpass transfer functions designed earlier:

217. §7.9.1 The Stability Triangle  In the case of H ’ BP (z), we observe that d 1 =-0.7343424, d 2 =1.3763819  Since here |d 2 |>1, H ’ BP (z) is unstable  On the other hand, in the case of H ” BP (z), we observe that d 1 =-0.53353098, d 2 =0.726542528  Here, |d 2 |<1 and |d 1 |<1+ d 2, and hence H ” BP (z) is BIBO stable

218. §7.9.2 A Stability Test Procedure  Let D M (z) denote the denominator of an M -th order causal IIR transfer function H(z): where we assume d 2 =1 for simplicity  Defined an M -th order allpass transfer function:

219. §7.9.2 A Stability Test Procedure then it follows that  Or, equivalently  If we express

220. §7.9.2 A Stability Test Procedure  Now for stability we must have, |λ i |<1, which implies the condition |d M |<1  Define k M = A M (∞)<1= d M  Then a necessary condition for stability of A M (z), and hence,the transfer function H(z) is given by

221. §7.9.2 A Stability Test Procedure  Assume the above condition holds  We now form a new function  Substituting the rational form of A M (z) in the above equation we get

222. §7.9.2 A Stability Test Procedure Hence, A M-1 (z) is an allpass function of order M-1  Now the poles λ o of A M-1 (z) are given by the roots of the equation where

223. §7.9.2 A Stability Test Procedure  If A M (z) is a stable allpass function, then  Hence  By assumption  Thus, if A M (z) is a stable allpass function, then the condition holds only if |λ o | <1

224. §7.9.2 A Stability Test Procedure  Thus, if A M (z) is a stable allpass function and, then A M-1 (z) is also a stable allpass function of one order lower  We now prove the converse, i.e., if A M-1 (z) is a stable allpass function and, then A M (z) is also a stable allpass function  Or, in other words A M-1 (z) is a stable allpass function

225. §7.9.2 A Stability Test Procedure  To this end, we express A M (z) in terms of A M-1 (z) arriving at  By assumption holds  If ζ 0 is a pole of A M (z), then

226. §7.9.2 A Stability Test Procedure |A M-1 (ζ 0 )|>|ζ 0 |  The above condition implies |A M-1 (ζ 0 )|>1 if |ζ 0 |≥1  Assume A M-1 (z) is a stable allpass function  Then A M-1 (z) ≤1 for |z|≥1  Thus, for |ζ 0 |≥1, we should have |A M-1 (ζ 0 )| ≤1  Therefore, i.e.,

227. §7.9.2 A Stability Test Procedure  Thus there is a contradiction  On the other hand, if |ζ 0 |<1 then from A M-1 (z) >1 for |z|<1 we have |A M-1 (ζ 0 )|>1  The above condition does not violate the condition |A M-1 (ζ 0 )|>|ζ 0 |

228. §7.9.2 A Stability Test Procedure  Summarizing, a necessary and sufficient set of conditions for the causal allpass function A M-1 (z) to be stable is therefore:  Thus, if and if A M-1 (z) is a stable allpass function, then A M (z) is also a stable allpass function (1), and (2) The allpass function A M-1 (z) is stable

229. §7.9.2 A Stability Test Procedure  Thus, once we have checked the condition, we test next for the stability of the lower-order allpass function A M-1 (z)  The process is then repeated, generating a set of coefficients: and a set of allpass functions of decreasing order:

230. §7.9.2 A Stability Test Procedure  The allpass function A M (z) is atable if and only if for i  Note:  Example – Test the stability of  From H(z) we generate a 4-th order allpass function

231. §7.9.2 A Stability Test Procedure we determine the coefficients {d ’ i } of the third-order allpass function A 3 (z) from the dcoefficients { d ’ i } of A 4 (z):  Using

232. §7.9.2 A Stability Test Procedure  Following the above procedure, we derive the next two lower-order allpass functions:  Note:

233. §7.9.2 A Stability Test Procedure  Since all of the stability conditions are satisfied, A 4 (z) and hence H(z) are stable  Note: It is not necessary to derive A 1 (z) since A 2 (z) can be tested for stability using the coefficient conditions  Note: