1. Chapter 4 Digital Processing of Continuous-Time Signals

2. §4.1 Introduction  Digital processing of a continuous-time signal involves the following basic steps: (1) Conversion of the continuous-time signal into a discrete-time signal, (2) Processing of the discrete-time signal, (3) Conversion of the processed discrete- time signal back into a continuous-time signal

3. §4.1 Introduction  Conversion of a continuous-time signal into digital form is carried out by an analog-to- digital (A/D) converter  The reverse operation of converting a digital signal into a continuous-time signal is performed by a digital-to-analog (D/A) converter

4. §4.1 Introduction  Since the A/D conversion takes a finite amount of time, a sample-and-hold (S/H) circuit is used to ensure that the analog signal at the input of the A/D converter remains constant in amplitude until the conversion is complete to minimize the error in its representation

5. §4.1 Introduction  To prevent aliasing, an analog anti-aliasing filter is employed before the S/H circuit  To smooth the output signal of the D/A converter, which has a staircase-like waveform, an analog reconstruction filter is used

6. §4.1 Introduction  Since both the anti-aliasing filter and the reconstruction filter are analog lowpass filters, we review first the theory behind the design of such filters  Also, the most widely used IIR digital filter design method is based on the conversion of an analog lowpass prototype Anti- Aliasing filter Digital processor D/A Reconstruction filter A/DS/H  Complete block-diagram

7. §4.2 Sampling of Continuous-Time Signals  As indicated earlier, discrete-time signals in many applications are generated by sampling continuous-time signals  We have seen earlier that identical discrete- time signals may result from the sampling of more than one distinct continuous-time function

8. §4.2 Sampling of Continuous-Time Signals  In fact, there exists an infinite number of continuous-time signals, which when sampled lead to the same discrete-time signal  However, under certain conditions, it is possible to relate a unique continuous-time signal to a given discrete-time signal

9. §4.2 Sampling of Continuous-Time Signals  If these conditions hold, then it is possible to recover the original continuous-time signal from its sampled values  We next develop this correspondence and the associated conditions

10. §4.2.1 Effect of Sampling in the Frequency Domain  Let g a (t) be a continuous-time signal that is sampled uniformly at t = nT, generating the sequence g[n] where g[n]=g a (nT), -∞ <n<∞ with T being the sampling period  The reciprocal of T is called the sampling frequency F T, i.e., F T =1/T

11. §4.2.1 Effect of Sampling in the Frequency Domain  Now, the frequency-domain representation of g a (t) is given by its continuos-time Fourier transform(CTFT):  The frequency-domain representation of g[n] is given by its discrete-time Fourier transform (DTFT):

12. §4.2.1 Effect of Sampling in the Frequency Domain  To establish the relation between G a (jΩ) and G(e jω ),we treat the sampling operation mathematically as a multiplication of ga (t) by a periodic impulse train p(t ):

13. §4.2.1 Effect of Sampling in the Frequency Domain  p(t) consists of a train of ideal impulses with a period T as shown below  The multiplication operation yields an impulse train:

14. §4.2.1 Effect of Sampling in the Frequency Domain  g p (t) is a continuous-time signal consisting of a train of uniformly spaced impulses with the impulse at t = nT weighted by the sampled value g a (nT) of g a (t) at the instant

15. §4.2.1 Effect of Sampling in the Frequency Domain  There are two different forms of G p (jΩ):  One form is given by the weighted sum of the CTFTs of δ(t-nT) : where Ω T =2π/T and Φ( jΩ) is the CTFT of φ(t)  To derive the second form, we make use of the Poisson’s formula:

16. §4.2.1 Effect of Sampling in the Frequency Domain  For t=0  Now, from the frequency shifting property of the CTFT, the CTFT of g a (t) e -jΨt is given by G a (j(Ω+Ψ)) reduces to

17. §4.2.1 Effect of Sampling in the Frequency Domain  Substituting φ(t)=g a (t)e-jΨt in we arrive at  By replacing Ψ with Ω in the above equation we arrive at the alternative form of the CTFT G p (jΩ ) of g p (t)

18. §4.2.1 Effect of Sampling in the Frequency Domain  The alternative form of the CTFT of g p (t) is given by  Therefore, G p (jΩ ) is a periodic function of Ω consisting of a sum of shifted and scaled replicas of G a (jΩ ), shifted by integer multiples of Ω T and scaled by 1/T

19. §4.2.1 Effect of Sampling in the Frequency Domain  The term on the RHS of the previous equation for k=0 is the baseband portion of G p (jΩ ), and each of the remaining terms are the frequency translated portions of G p (jΩ )  The frequency range is called the baseband or Nyquist band

20. §4.2.1 Effect of Sampling in the Frequency Domain  Assume g a (t) is a band-limited signal with a CTFT G a (jΩ) as shown below G a (jΩ) ΩmΩm -Ω m Ω 0 ΩTΩT -Ω T 2Ω T 3Ω T 0 P(jΩ) Ω 1 … …  The spectrum P(jΩ) of p(t) having a sampling period T=2π/ Ω T is indicated below

21. §4.2.1 Effect of Sampling in the Frequency Domain  Two possible spectra of G p (jΩ ) are shown below

22. §4.2.1 Effect of Sampling in the Frequency Domain  It is evident from the top figure on the previous slide that if Ω T >2Ω m,there is no overlap between the shifted replicas of G a (jΩ) generating G p (jΩ)  On the other hand, as indicated by the figure on the bottom, if Ω T <2Ω m,there is an overlap of the spectra of the shifted replicas of G a (jΩ) generating G p (jΩ)

23. §4.2.1 Effect of Sampling in the Frequency Domain H r (jΩ) ga(t)ga(t) gp(t)gp(t) p(t)p(t) If Ω T >2Ω m, g a (t) can be recovered exactly from g p (t) by passing it through an ideal lowpass filter H r (jΩ) with a gain T and a cutoff frequency Ω c greater than Ω m and less than Ω T -Ω m as shown below

24. §4.2.1 Effect of Sampling in the Frequency Domain  The spectra of the filter and pertinent signals are shown below

25. §4.2.1 Effect of Sampling in the Frequency Domain  On the other hand, if Ω T <2Ω m,due to the overlap of the shifted replicas of G a (jΩ), the spectrum G a (jΩ) cannot be separated by filtering to recover G a (jΩ) because of the distortion caused by a part of the replicas immediately outside the baseband folded back or aliased into the baseband

26. §4.2.1 Effect of Sampling in the Frequency Domain Sampling theorem – Let g a (t) be a band- limited signal with CTFT G a (jΩ) =0 for |Ω|>Ω m  Then g a (t) is uniquely determined by its samples g a (nT),-∞≤ n ≤∞ if Ω T ≥ 2Ω m where Ω T = 2π/T

27. §4.2.1 Effect of Sampling in the Frequency Domain  The condition Ω T ≥ 2Ω m is often referred to as the Nyquist condition  The frequency Ω T /2 is usually referred to as the folding frequency

28. §4.2.1 Effect of Sampling in the Frequency Domain  Given { g a (nT) }, we can recover exactly g a (t) by generating an impulse train and then passing it through an ideal lowpass filter H r (jΩ) with a gain T and a cutoff frequency Ω c satisfying  m <  c < (  T -  m )

29. §4.2.1 Effect of Sampling in the Frequency Domain  The highest frequency Ω m contained in g a (t) is usually called the Nyquist frequency since it determines the minimum sampling frequency Ω T =2 Ω m that must be used to fully recover g a (t) from its sampled version  The frequency 2 Ω m is called the Nyquist rate

30. §4.2.1 Effect of Sampling in the Frequency Domain  Oversampling – The sampling frequency is higher than the Nyquist rate  Undersampling – The sampling frequency is lower than the Nyquist rate  Critical sampling – The sampling frequency is equal to the Nyquist rate  Note: A pure sinusoid may not be recoverable from its critically sampled version

31. §4.2.1 Effect of Sampling in the Frequency Domain  In digital telephony, a 3.4 kHz signal bandwidth is acceptable for telephone conversation  Here, a sampling rate of 8 kHz, which is greater than twice the signal bandwidth, is used

32. §4.2.1 Effect of Sampling in the Frequency Domain  In high-quality analog music signal processing, a bandwidth of 20 kHz has been determined to preserve the fidelity  Hence, in compact disc (CD) music systems, a sampling rate of 44.1 kHz, which is slightly higher than twice the signal bandwidth, is used

33. §4.2.1 Effect of Sampling in the Frequency Domain  Example - Consider the three continuous- time sinusoidal signals: g a (t)=cos(6πt) g a (t)=cos(14πt) g a (t)=cos(26πt)  Their corresponding CTFTs are: G 1 (jΩ)=π[δ(Ω-6π)+δ(Ω+6π)] G 2 (jΩ)=π[δ(Ω-14π)+δ(Ω+14π)] G 3 (jΩ)=π[δ(Ω-26π)+δ(Ω+26π)]

34. §4.2.1 Effect of Sampling in the Frequency Domain  These three transforms are plotted below

35. §4.2.1 Effect of Sampling in the Frequency Domain  These continuous-time signals sampled at a rate of T = 0.1 sec, i.e., with a sampling frequency Ω T =20π rad/sec  The sampling process generates the continuous-time impulse trains, g 1p (t), g 2p (t), and g 3p (t)  Their corresponding CTFTs are given by

36. §4.2.1 Effect of Sampling in the Frequency Domain  Plots of the 3 CTFTs are shown below

37. §4.2.1 Effect of Sampling in the Frequency Domain  These figures also indicate by dotted lines the frequency response of an ideal lowpass filter with a cutoff at Ω c =Ω T /2=10π and a gain T=0.1  The CTFTs of the lowpass filter output are also shown in these three figures  In the case of g 1 (t), the sampling rate satisfies the Nyquist condition, hence no aliasing

38. §4.2.1 Effect of Sampling in the Frequency Domain  Moreover, the reconstructed output is precisely the original continuous-time signal  In the other two cases, the sampling rate does not satisfy the Nyquist condition, resulting in aliasing and the filter outputs are all equal to cos( 6 π t )

39. §4.2.1 Effect of Sampling in the Frequency Domain  Note: In the figure below, the impulse appearing at Ω = 6π in the positive frequency passband of the filter results from the aliasing of the impulse in G 2 (jΩ) at Ω = −14π  Likewise, the impulse appearing at Ω = 6π in the positive frequency passband of the filter results from the aliasing of the impulse in G 3 (jΩ) at Ω = 26π

40. §4.2.1 Effect of Sampling in the Frequency Domain  We now derive the relation between the DTFT of g[n] and the CTFT of g p [t]  To this end we compare and make use of g[n]=g a [nT],-∞<n< ∞ with

41. §4.2.1 Effect of Sampling in the Frequency Domain  Observation: We have  From the above observation and or, equivalently,

42. §4.2.1 Effect of Sampling in the Frequency Domain we arrive at the desired result given by

43. §4.2.1 Effect of Sampling in the Frequency Domain  The relation derived on the previous slide can be alternately expressed as it follows that G(e jω ) is obtained from G p (jΩ) by applying the mapping Ω=ω/T or from  From

44. §4.2.1 Effect of Sampling in the Frequency Domain  Now, the CTFT G p (jΩ) is a periodic function of Ω with a period Ω T =2π/T  Because of the mapping, the DTFT G(e jω ) is a periodic function of ω with a period 2π

45. §4.2.2 Recovery of the Analog Signal  We now derive the expression for the output ĝ a (t) of the ideal lowpass reconstruction filter H r (jΩ) as a function of the samples g[n]  The impulse response h r (t) of the lowpass reconstruction filter is obtained by taking the inverse DTFT of H r (jΩ) :

46. §4.2.2 Recovery of the Analog Signal  Thus, the impulse response is given by  The input to the lowpass filter is the impulse train g p (t) :

47. §4.2.2 Recovery of the Analog Signal  Therefore, the output ĝ a (t) of the ideal lowpass filter is given by: ^ *  Substituting h r (t) = sin(Ω c t)/(Ω T t/2) in the above and assuming for simplicity  c =  T /2=  /T, we get

48. §4.2.2 Recovery of the Analog Signal  The ideal bandlimited interpolation process is illustrated below

49. §4.2.2 Recovery of the Analog Signal  It can be shown that when Ω c = Ω T / 2 in for all integer values of r in the range-∞< r <∞ we observe h r (0)=1 and h r (nT)=0 for n ≠ 0  As a result, from

50. §4.2.2 Recovery of the Analog Signal  The relation holds whether or not the condition of the sampling theorem is satisfied  However, ĝ a (rT)=g a (rT) for all values of t only if the sampling frequency Ω T satisfies the condition of the sampling theorem

51. §4.2.3 Implication of the Sampling Process  Consider again the three continuous-time signals: g 1 (t) =cos(6πt),g 2 (t) =cos(14πt), and g 3 (t) =cos(26πt)  The plot of the CTFT G 1p (jΩ) of the sampled version g 1p (t) of g 1 (t) is shown below

52. §4.2.3 Implication of the Sampling Process  From the plot, it is apparent that we can recover any of its frequency-translated versions cos[(20k±6)πt] outside the baseband by passing g 1p (t) through an ideal analog bandpass filter with a passband centered at Ω= ( 20k±6)π

53. §4.2.3 Implication of the Sampling Process  For example, to recover the signal cos(34πt ), it will be necessary to employ a bandpass filter with a frequency response where ∆ is a small number

54. §4.2.3 Implication of the Sampling Process  Likewise, we can recover the aliased baseband component cos(6πt) from the sampled version of either g 2p (t) or g 3p (t) by passing it through an ideal lowpass filter with a frequency response:

55. §4.2.3 Implication of the Sampling Process  There is no aliasing distortion unless the original continuous-time signal also contains the component cos(6πt)  Similarly, from either g 2p (t) or g 3p (t) we can recover any one of the frequency-translated versions, including the parent continuous- time signal g 2 (t) or g 3 (t) as the case may be, by employing suitable filters

56. §4.3 Sampling of Bandpass Signals  The conditions developed earlier for the unique representation of a continuous-time signal by the discrete-time signal obtained by uniform sampling assumed that the continuous-time signal is bandlimited in the frequency range from dc to some frequency Ω m  Such a continuous-time signal is commonly referred to as a lowpass signal

57. §4.3 Sampling of Bandpass Signals  There are applications where the continuous- time signal is bandlimited to a higher frequency range Ω L ≤|Ω|≤Ω H with Ω L > 0  Such a signal is usually referred to as the bandpass signal  To prevent aliasing a bandpass signal can of course be sampled at a rate greater than twice the highest frequency, i.e. by ensuring Ω T ≥2 Ω H

58. §4.3 Sampling of Bandpass Signals  However, due to the bandpass spectrum of the continuous-time signal, the spectrum of the discrete-time signal obtained by sampling will have spectral gaps with no signal components present in these gaps  Moreover, if Ω H is very large, the sampling rate also has to be very large which may not be practical in some situations

59. §4.3 Sampling of Bandpass Signals  A more practical approach is to use under- sampling  Let ΔΩ = Ω H - Ω L define the bandwidth of the bandpass signal  Assume first that the highest frequency Ω H contained in the signal is an integer multiple of the bandwidth, i.e., Ω H =M(ΔΩ )

60. §4.3 Sampling of Bandpass Signals  We choose the sampling frequency Ω T to satisfy the condition  T = 2(  ) = 2  H /M which is smaller than 2Ω H, the Nyquist rate  Substitute the above expression for Ω T in

61. §4.3 Sampling of Bandpass Signals  This leads to  As before, G p (jΩ) consists of a sum of G a (jΩ) and replicas of G a (jΩ) shifted by integer multiples of twice the bandwidth ∆Ω and scaled by 1/T  The amount of shift for each value of k ensures that there will be no overlap between all shifted replicas →no aliasing

62. §4.3 Sampling of Bandpass Signals  Figure below illustrate the idea behind -H-H -L-L LL HH 0 Gp(j)Gp(j)  -H-H -L-L LL HH 0  Gp(j)Gp(j)

63. §4.3 Sampling of Bandpass Signals  As can be seen, g a (t) can be recovered from g p (t) by passing it through an ideal bandpass filter with a passband given by Ω L ≤|Ω|≤Ω H and a gain of T  Note: Any of the replicas in the lower frequency bands can be retained by passing g p (t) through bandpass filters with passbands Ω L ≤ - k( ΔΩ) ≤|Ω|≤Ω H - k( ΔΩ), 1≤k≤ M-1 providing a translation to lower frequency ranges

64. §4.4.1 Analog Lowpass Filter Specifications  Typical magnitude response | H a (jΩ)| of an analog lowpass filter may be given as indicated below

65. §4.4.1 Analog Lowpass Filter Specifications  In the passband, defined by 0≤Ω≤Ω p, we require 1-  p  |H a (j  )|  1+  p, |  |   p  i.e., | H a (jΩ)| approximates unity within an error of ± δ p  In the stopband, defined by Ω s ≤Ω<∞, we require |H a (j  )|   s,  s     i.e., | H a (jΩ)| approximates zero within an error of δ s

66. §4.4.1 Analog Lowpass Filter Specifications  Ω p – passband edge frequency  Ω s – stopband edge frequency  δ p – peak ripple value in the passband  δ s – peak ripple value in the stopband  Peak passband ripple  Minimum stopband attenuation

67. §4.4.1 Analog Lowpass Filter Specifications  Magnitude specifications may alternately be given in a normalized form as indicated below

68. §4.4.1 Analog Lowpass Filter Specifications  Here, the maximum value of the magnitude in the passband assumed to be untiy  – Maximum passband deviation, given by the minimum value of the magnitude in the passband  1/A – Maximum stopband magnitude

69. §4.4.1 Analog Lowpass Filter Specifications  Two additional parameters are defined – (1) Transition ratio k =  p /  s For a lowpass filter k<1 (2) Discrimination parameter Usually k<<1

70. §4.4.2 Butterworth Approximation  The magnitude-square response of an N -th order analog lowpass Butterworth filter is given by  First 2N-1 derivatives of | H a (jΩ)| 2 at Ω=0 are equal to zero  The Butterworth lowpass filter thus is said to have a maximally-flat magnitude at Ω=0

71. §4.4.2 Butterworth Approximation  Gain in dB is G (Ω)=10 log 10 | H a (jΩ)| 2  As G(0)=0 and G (Ω c )= 10 log 10 (0.5)= −3.0103 ≅ - 3dB Ω c is called the 3-dB cutoff frequency

72. §4.4.2 Butterworth Approximation  Typical magnitude responses with Ω c =1

73. §4.4.2 Butterworth Approximation  Two parameters completely characterizing a Butterworth lowpass filter are Ω c and N  These are determined from the specified bandedges Ω p and Ω s, and minimum passband magnitude, and maximum stopband ripple 1/A

74. §4.4.2 Butterworth Approximation  Ω c and N are thus determined from  Solving the above we get

75. §4.4.2 Butterworth Approximation  Since order N must be an integer, value obtained is rounded up to the next highest integer  This value of N is used next to determine Ω c by satisfying either the stopband edge or the passband edge specification exactly  If the stopband edge specification is satisfied, then the passband edge specification is exceeded providing a safety margin

76. §4.4.2 Butterworth Approximation  Transfer function of an analog Butterworth lowpass filter is given by  Denominator D N (s) is known as the Butterworth polynomial of order N where

77. §4.4.2 Butterworth Approximation  Example – Determine the lowest order of a Butterworth lowpass filter with a 1-dB cutoff frequency at 1kHz and a minimum attenuation of 40 dB at 5kHz  Now which yields A 2 =10,000 which yields ε 2 =0.25895 and

78. §4.4.2 Butterworth Approximation  Therefore  We choose N=4  Hence and

79. §4.4.3 Chebyshev Approximation  The magnitude-square response of an N -th order analog lowpass Type 1 Chebyshev filter is given by where T N (Ω) is the Chebyshev polynomial of order N :

80. §4.4.3 Chebyshev Approximation  Typical magnitude response plots of the analog lowpass Type 1 Chebyshev filter are shown below

81. §4.4.3 Chebyshev Approximation  If at Ω=Ω s the magnitude is equal to 1/A, then  Order N is chosen as the nearest integer greater than or equal to the above value  Solving the above we get

82. §4.4.3 Chebyshev Approximation  The magnitude-square response of an N -th order analog lowpass Type 2 Chebyshev (also called inverse Chebyshev) filter is given by where T N (Ω) is the Chebyshev polynomial of order N

83. §4.4.3 Chebyshev Approximation  Typical magnitude response plots of the analog lowpass Type 2 Chebyshev filter are shown below

84. §4.4.3 Chebyshev Approximation  The order N of the Type 2 Chebyshev is determined from given ε, Ω s, and A using  Example – Determine the lowest order of a Chebyshev lowpass filter with a 1 -dB cutoff frequency at 1 kHz and a minimum attenuation of 40 dB at 5 kHz -

85. §4.4.4 Elliptic Approximation  The square-magnitude response of an elliptic lowpass filter is given by where R N (Ω) is a rational function of order N satisfying R N (1/Ω)=1/R N (Ω), with the roots of its numerator lying in the interval 0< Ω<1 and the roots of its denominator lying in the interval 1< Ω<∞

86. §4.4.4 Elliptic Approximation  For given Ω p, Ω s,ε, and A, the filter order can be estimated using where

87. §4.4.4 Elliptic Approximation  Example - Determine the lowest order of a elliptic lowpass filter with a 1-dB cutoff frequency at 1 kHz and a minimum attenuation of 40 dB at 5 kHz Note: k = 0.2 and 1/k=196.5134  Substituting these values we get k’ = 0.979796, ρ 0 =0 00255135, ρ= 0 0025513525  and hence N = 2.23308  Choose N = 3

88. §4.4.4 Elliptic Approximation  Typical magnitude response plots with Ω p =1 are shown below

89. §4.4.6 Analog Lowpass Filter Design  Example – Design an elliptic lowpass filter of lowest order with a 1-dB cutoff frequency at 1 kHz and a minimum attenuation of 40 dB at 5 kHz  Code fragments used [N, Wn] = ellipord(Wp, Ws, Rp, Rs, ‘s’); [b, a] = ellip(N, Rp, Rs, Wn, ‘s’); with Wp = 2*pi*1000; Ws = 2*pi*5000; Rp = 1; Rs = 40;

90. §4.4.6 Analog Lowpass Filter Design  Gain plot

91. §4.5 Design of Analog Highpass Bandpass and Bandstop Filters  Steps involved in the design process: Step 1 – Develop of specifications of a prototype analog lowpass filter H LP (s) from specifications of desired analog filter H D (s) using a frequency transformation  Step 2 – Design the prototype analog lowpass filter  Step 3 – Determine the transfer function H D (s) of desired analog filter by applying the inverse frequency transformation to H LP (s)

92. §4.5 Design of Analog Highpass Bandpass and Bandstop Filters  Let s denote the Laplace transform variable of prototype analog lowpass filter H LP (s) and ŝ denote the Laplace transform variable of desired analog filter H D ( ŝ )  The mapping from s -domain to ŝ-domain is given by the invertible transform s=F(ŝ)  Then

93. H LP (s) and is the passband edge §4.5.1 Analog Highpass Filter Design  Spectral Transformation: frequency of H HP ( ŝ )  On the imaginary axis the transformation is where Ω p is the passband edge frequency of

94. §4.5.1 Analog Highpass Filter Design

95.  Example - Design an analog Butterworth highpass filter with the specifications: F p =4 kHz, F s =1 kHz,α p =0.1 dB,α s =40 dB  Choose Ω p =1  Then  Analog lowpass filter specifications: Ω p =1, Ω s =4, α p =0.1 dB α s =40 dB

96. §4.5.1 Analog Highpass Filter Design  Code fragments used [ N, Wn ] = buttord (1, 4, 0.1, 40, ‘s’) ; [ B, A ] = butter (N, Wn, ‘s’) ; [num, den] = lp2hp (B, A, 2*pi*4000 );  Gain plots

97. §4.5.2 Analog Bandpass Filter Design  Spectral Transformation upper passband edge frequencies of desired bandpass filter H BP (ŝ) H LP (s), and and are the lower and where Ω p is the passband edge frequency of

98. §4.5.2 Analog Bandpass Filter Design  On the imaginary axis the transformation is where is the width of passband and is the passband center frequency of the bandpass filter  passband edge frequency ±Ω p is mapped into and, lower and upper passband edge frequencies

99. §4.5.2 Analog Bandpass Filter Design

100.  If bandedge frequencies do not satisfy the above condition, then one of the frequencies needs to be changed to a new value so that the condition is satisfied  Stopband edge frequency ±Ω s is mapped into and, lower and upper stopband edge frequencies  Also,

101. §4.5.2 Analog Bandpass Filter Design increase any one of the stopband edges or decrease any one of the passband edges as shown below  Case 1: to make we can either

102. §4.5.2 Analog Bandpass Filter Design (1) Decrease to larger passband and shorter leftmost transition band (2) Increase to No change in passband and shorter leftmost transition band

103. §4.5.2 Analog Bandpass Filter Design  Note: The condition can also be satisfied by decreasing which is not acceptable as the passband is reduced from the desired value  Alternately, the condition can be satisfied by increasing which is not acceptable as the upper stopband is reduced from the desired value

104. §4.5.2 Analog Bandpass Filter Design decrease any one of the stopband edges or increase any one of the passband edges as shown below  Case 2: to make we can either

105. §4.5.2 Analog Bandpass Filter Design (1) Increase to larger passband and shorter rightmost transition band (2) Decrease to No change in passband and shorter rightmost transition band

106. §4.5.2 Analog Bandpass Filter Design  Note: The condition can also be satisfied by increasing which is not acceptable as the passband is reduced from the desired value  Alternately, the condition can be satisfied by decreasing which is not acceptable as the lower stopband is reduced from the desired value

107. §4.5.2 Analog Bandpass Filter Design  Example – Design an analog elliptic bandpass filter with the specifications:  Now and  Since we choose

108. §4.5.2 Analog Bandpass Filter Design  We choose Ω p =1  Hence  Analog lowpass filter specifications: Ω p =1, Ω s =1.4, α p =1dB,α s =22dB

109. §4.5.2 Analog Bandpass Filter Design  Code fragments used [N, Wn] = ellipord(1, 1.4, 1, 22, ‘s’); [B, A] = ellip(N, 1, 22, Wn, ‘s’); [num, den]= lp2bp(B, A, 2*pi*4.8989795, 2*pi*25/7);  Gain plot

110. §4.5.3 Analog Bandstop Filter Design  Spectral Transformation where Ω s is the stopband edge frequency of H LP (s), and and are the lower and upper stopband edge frequencies of the desired bandstop filter H BS (ŝ)

111. §4.5.3 Analog Bandstop Filter Design  On the imaginary axis the transformation where is the widrth of stopband and is the stopband center frequency of the bandstop filter  Stopband edge frequency is mapped into and, lower and upper stopband edge frequencies

112. §4.5.3 Analog Bandstop Filter Design  Passband edge frequency ±Ω p is mapped into and, lower and upper passband edge frequencies

113. §4.5.3 Analog Bandstop Filter Design  If bandedge frequencies do not satisfy the above condition, then one of the frequencies needs to be changed to a new value so that the condition is satisfied  Also,

114. §4.5.3 Analog Bandstop Filter Design  Case 1:  To make we can either increase any one of the stopband edges or decrease any one of the passband edges as shown below

115. §4.5.3 Analog Bandstop Filter Design (1) Decrease to larger high-frequency passband and shorter rightmost transition band (2) Increase to No change in passband and shorter rightmost transition band

116. §4.5.3 Analog Bandstop Filter Design  Note: The condition can also be satisfied by decreasing which is not acceptable as the low – frequency passband is reduced from the desired value  Alternately, the condition can be satisfied by increasing which is not acceptable as the stopband is reduced from the desired value

117. §4.5.3 Analog Bandstop Filter Design  Case 1:  To make we can either decrease any one of the stopband edges or increase any one of the passband edges as shown below

118. §4.5.3 Analog Bandstop Filter Design (1) Increase to larger passband and shorter leftmost transition band (2) Decrease to No change in passband and shorter leftmost transition band

119. §4.5.3 Analog Bandstop Filter Design  Note: The condition can also be satisfied by increasing which is not acceptable as the high – frequency passband is decreasd from the desired value  Alternately, the condition can be satisfied by decreasing which is not acceptable as the stopband is decreased