4. Design of Discrete-Time Filters

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1 4. Design of Discrete-Time Filters 4.1. Introduction (7.0) 4.2. Frame of Design of IIR Filters (7.1) 4.3. Design of IIR Filters by Impulse Invariance (7.1) 4.4. Design of IIR Filters by Bilinear Transformation (7.1) 4.5. Design of FIR Filters by Windowing (7.2)

2 4.1. Introduction Overview A discrete-time filter is a discrete-time system which passes some frequency components and stops others. An ideal discrete-time filter is best in frequency selectivity. It passes the frequency components in the pass band distortionlessly and stops the frequency components in the stop band completely. Unfortunately, it is noncausal and cannot be implemented in real time. This is undesired in many applications. In such a case, we need to design a causal discrete-time filter which approximates the ideal discrete-time filter functionally. The goal of the design is an impulse response, a frequency response, a system function, a linear constant-coefficient difference equation or another. An ideal filter can be approximated by an IIR or FIR filter. An IIR filter usually needs less cost, i.e., less computation and memory, and an FIR filter usually has a better performance, especially in the phase

3 response. If a generalized linear phase is needed, we usually use an FIR filter Analysis of Ideal Filters Consider an ideal lowpass filter. The analysis can be extended to other types of ideal filters. Over period [, ), the frequency response of an ideal lowpass filter is defined as H( ) exp( j), c. 0, otherwise (4.1) Let x i (n)=a i exp(j i n) be a frequency component of the input signal. Then, the corresponding output signal of this filter is If i c, then y i (n)=a i exp(j i n)h( i ). (4.2)

4 y i (n)=a i exp[j i (n)], (4.3) i.e., x i (n) is passed with a constant delay. If i > c, then y i (n)=0, (4.4) i.e., x i (n) is stopped completely. Thus, the ideal lowpass filter has the best frequency selectivity. The impulse response of the ideal lowpass filter is h(n) c (n ). sin (4.5) (n ) Since h(n)0 for n<0, the ideal lowpass filter is noncausal and cannot be implemented in real time. However, in practical applications, a causal lowpass filter is often required. In such cases, we need to find a causal lowpass filter which approximates the ideal lowpass filter functionally. Depending on a specific application, this causal lowpass filter can be an IIR or FIR filter.

5 4.2. Frame of Design of IIR Filters Conventionally, the design of a discrete-time IIR filter involves a transform from a continuous-time IIR filter into the discrete-time IIR filter. It consists of three steps: 1. Find the specifications on the continuous-time IIR filter from the specifications on the discrete-time IIR filter. 2. Find the system function of the continuous-time IIR filter from the specifications on the continuous-time IIR filter. 3. Find the system function of the discrete-time IIR filter from the system function of the continuous-time IIR filter. There are two typical methods to transform a continuous-time IIR filter into a discrete-time IIR filter. One is the impulse invariance, and the other is the bilinear transformation. Next, we will discuss the two methods.

6 4.3. Design of IIR Filters by Impulse Invariance Definition A typical method to transform a continuous-time IIR filter into a discrete-time IIR filter is the impulse invariance. In this method, the impulse response of the discrete-time IIR filter is obtained by sampling the impulse response of the corresponding continuous-time IIR filter, i.e., h(n)=th c (nt). (4.6) Here, h(n) is the impulse response of the discrete-time IIR filter, h c (t) is the impulse response of the continuous-time IIR filter, and T is the sampling interval Causality and Stability If the continuous-time IIR filter has a rational system function and is causal and stable, the discrete-time IIR filter is causal and stable.

7 Frequency Response If H() and H c () are the frequency responses of the discrete-time IIR filter and the continuous-time IIR filter, respectively, then H 2 T T H m. m c (4.7) h c (t) H c () h(n) t W H(T) n W 2/T Figure 4.1. Relation between H() and H c () when 2/T>W.

8 h c (t) H c () h(n) t W H(T) n Figure 4.2. Relation between H() and H c () when 2/TW. (4.7) shows that H(T) equals H c () extended with period 2/T. Let W be the bandwidth of H c (). If 2/T>W, then H(T) equals H c () over the interval [/T, /T) (figure 4.1). Aliasing happens otherwise (figure 4.2). The above analysis implies that the impulse invariance can be used W 2/T

9 to design band-limited filters, such as lowpass filters and bandpass filters but cannot be used to design other filters, like highpass filters and bandstop filters Procedure The impulse invariance is carried out in three steps: 1. Find the specifications on the continuous-time IIR filter from the specifications on the discrete-time IIR filter according to =/T, (4.8) where is the physical frequency of the continuous-time IIR filter, is the normalized frequency of the discrete-time IIR filter, and T is the sampling interval. 2. Find the system function of the continuous-time IIR filter from the specifications on the continuous-time IIR filter. 3. Find the system function of the discrete-time IIR filter from the

10 system function of the continuous-time IIR filter. This consists of three steps: (1) Find the impulse response of the continuous-time IIR filter from its system function. (2) Find the impulse response of the discrete-time IIR filter from the impulse response of the continuoustime IIR filter. (3) Find the system function of the discrete-time IIR filter from its impulse response. Example. Design a causal lowpass discrete-time IIR filter with the cutoff frequency 0.2, i.e., H[exp(j0.2)] H[exp(j0)] 1 2, (4.9) where H(z) is the system function of the filter. A requirement is that the filter should be generated from a causal lowpass continuous-time IIR filter H c a (s), Re(s) a, (4.10) s a

11 where a>0, by the impulse invariance Design of IIR Filters by Bilinear Transformation Definition Another typical method to transform a continuous-time IIR filter into a discrete-time IIR filter is the bilinear transformation. Assume that H(z) and H c (s) are the system functions of the discrete-time IIR filter and the continuous-time IIR filter, respectively. In the bilinear transformation, H(z) is obtained from H c (s) by letting that is, 2 T 1 z 1 z 1 s 1, (4.11) z H(z) Hc 1 T 1 z, (4.12)

12 where T is a positive constant Causality and Stability The discrete-time IIR filter is causal and stable if the continuoustime IIR filter is causal and stable Frequency Response Letting z=exp(j) in (4.12), we obtain 2 H c T 2 exp(j) H j tan. (4.13) shows that H[exp(j)] equals H c (j) when (figure 4.3). 2 T tan 2 (4.13) (4.14)

13 0 H c (j) Aliasing is avoided. However, a nonlinear distortion is caused due to the nonlinear mapping from to. 0 3 H[exp(j)] 0 3 Figure 4.3. Relation between H[exp(j)] and H c (j).

14 Procedure The bilinear transformation is carried out in the following steps: 1. Find the specifications on the continuous-time IIR filter from the specifications on the discrete-time IIR filter according to (4.14). 2. Find the system function of the continuous-time IIR filter from the specifications on the continuous-time IIR filter. 3. Find the system function of the discrete-time IIR filter from the system function of the continuous-time IIR filter according to (4.12). Example. Design a causal lowpass discrete-time IIR filter which has the cutoff frequency 0.2, i.e., H[exp(j0.2)] H[exp(j0)] 1 2, (4.15) where H(z) is the system function of the filter. A requirement is that

15 the filter should be obtained from a causal lowpass continuous-time IIR filter H c where a>0, by the bilinear transformation Design of FIR Filters by Windowing Principle a (s), Re(s) a, (4.16) s a An ideal filter can be approximated by a causal FIR filter as well. This can be carried out by different methods. A widely used method is the windowing method. In this method, the impulse response of the causal FIR filter is obtained by windowing the impulse response of the ideal filter. Let us consider the design of a lowpass filter. The addressed ideas and methods, however, also apply to other cases.

16 In period [, ), the frequency response of an ideal lowpass filter is defined as H d ( ) h d exp( j), c. 0, otherwise c (n ). (4.17) Taking the inverse Fourier transform of H d (), we obtain the impulse response of the ideal lowpass filter, i.e., sin (n) (4.18) (n ) Since h d (n)0 for n<0, the ideal lowpass filter is noncausal. However, windowing h d (n), we can obtain the impulse response of a causal FIR lowpass filter. That is, h(n)=h d (n)w(n), (4.19) where w(n) is a window function and is equal to 0 for n<0. Generally,

17 w(n) is real and symmetric, and has a maximum of 1. is chosen as =(N1)/2, (4.20) where N is the length of w(n). Thus, h(n) is real and symmetric, and the causal FIR lowpass filter has a generalized linear phase Frequency Response Let W() be the spectrum of w(n). Then the frequency response of the causal FIR lowpass filter equals the periodic convolution of H d () and W() divided by 2, i.e., H( ) H d () can be expressed as H d ( )W( )d. (4.21) H d ()=A d ()exp(j), (4.22)

18 where A d () is the amplitude of H d (). W() can be expressed as W()=E()exp(j), (4.23) where E() is a real function. Substituting (4.22)-(4.23) into (4.21), we obtain H()=A()exp(j), (4.24) where A() is the periodic convolution of A d () and E() divided by 2, i.e., Evidently, A() is also a real function. 1 A( ) Ad ( )E( )d. 2 (4.25) 2 From (4.22) and (4.24), we see that the difference between H d () and H() is actually the difference between A d () and A(). Based on (4.25), there exist two significant differences between A d () and A() (figure 4.4).

19 h d (n) A d () n c w(n) E() n h(n) A() n c Figure 4.4. Differences between A d () and A().

20 1. At each cutoff frequency, A d () is discontinuous, but A() has a transition band. (1) The width of the transition band is determined by the width of the main lobe of E(). In order for A() to have a narrow transition band, E() should have a narrow main lobe. (2) The width of the main lobe of E() is determined by both the shape and the length of w(n). A longer w(n) results in a narrower main lobe. 2. Over the passband and the stopbands, A d () is equal to 1 and 0 respectively, but A() has ripples. (1) The amplitudes of the ripples are determined by the areas of the side lobes of E(). In order for A() to have small-amplitude ripples, E() should have small-area side lobes. (2) The areas of the side lobes of E() are determined by the shape of w(n) only. From (4.24), we have a better understanding about the generalized linear phase of the causal FIR lowpass filter. It has a strict linear phase in the passband, which is desired. Its phase is not strictly linear in the stopbands, but this is trivial.

21 Windows The rectangular window is defined as 1, 0 n N 1 w(n). (4.26) 0, otherwise The Bartlett window is defined as 2n /(N 1), 0 n (N 1) / 2 w(n) 2 2n /(N 1), (N 1)/2 n N 1. 0, otherwise The Hann window is defined as (4.27) w(n) cos 0, 2n, N 1 0 n N 1. otherwise (4.28)

22 The Hamming window is defined as w(n) cos 0, 2n, N 1 0 n N 1. otherwise (4.29) The Blackman window is defined as w(n) cos 0, 2n 0.08cos N 1 4n, 0 n N 1 N 1. otherwise The above windows are illustrated in figure 4.5. (4.30) Table 4.1 gives the features of these windows. When the length of the window is fixed, a narrow transition band always corresponds to large-amplitude ripples, and therefore there exists a tradeoff between the width of the transition band and the amplitudes of the ripples.

23 Figure 4.5. Shapes of Commonly Used Windows.

24 Shape of Window Width of Transition Band ( in Figure 4.4) Peak Amplitude of Ripples Rectangular 4/N 21 db Bartlett 8/(N1) 25 db Hann 8/(N1) 44 db Hamming 8/(N1) 53 db Blackman 12/(N1) 74 db Table 4.1. Features of Commonly Used Windows. The Kaiser window is defined as w(n) I 0 0, 1 I 0 2n N 1 ( ) 1 2, 0 n N 1. otherwise (4.31)

25 Here I 0 ( ) is the zero-order modified Bessel function of the first kind and is a shape parameter Procedure The windowing method is carried out in the following steps: 1. Determine the shape of the window based on the specifications on the amplitudes of the ripples. 2. Determine the length of the window based on the specifications on the width of the transition band. 3. Determine based on (4.20). 4. Determine the impulse response of the causal FIR lowpass filter based on (4.19). Example. Design a causal FIR lowpass filter. The cutoff frequency is 0.2. The width of the transition band is less than or equal to 0.01.

26 The amplitudes of the ripples are less than or equal to 0.01.

(i) Understanding of the characteristics of linear-phase finite impulse response (FIR) filters

FIR Filter Design Chapter Intended Learning Outcomes: (i) Understanding of the characteristics of linear-phase finite impulse response (FIR) filters (ii) Ability to design linear-phase FIR filters according