数字信号处理课件群discrete

1、7. Design of Discrete-Time Filters7.1. Introduction (7.0)7.2. Frame of Design of IIR Filters (7.1)7.3. Design of IIR Filters by Impulse Invariance (7.1)7.4. Design of IIR Filters by Bilinear Transformation (7.1)7.5. Design of FIR Filters by Windowing (7.2)7.1. Introduction7.1.1. Overview A discrete-

2、time filter is a discrete-time system that passes certain frequency components and stops others. In general, an ideal discrete-time filter is noncausal and thus cannot be implemented in real time. In practical applications, we often need design a causal discrete-time filter which approximates the id

3、eal discrete-time filter functionally. The result of the design may be an impulse response, a frequency response, a system function, a linear constant-coefficient difference equation or others. An ideal filter can be approximated by an IIR or FIR filter. An IIR filter usually requires less cost, i.e

4、., less computation and memory. However, an FIR filter usually has better performance, especially in the phase response. If a generalized linear phase is required, we need to use an FIR filter usually.7.1.2. Analysis of Ideal Filters Consider an ideal lowpass filter. The analysis can be extended to

5、other types of ideal filters. Over period -, ), the frequency response of an ideal lowpass filter is defined as(7.1)Let xi(n)=Aiexp(jin) be a frequency component of the input signal. Then, the corresponding output signal of this filter isyi(n)=Aiexp(jin)H(i). (7.2)If |i|c, then yi(n)=Aiexpji(n-). (7

6、.3)(7.5)That is, xi(n) is passed with a constant delay . If |i|c, thenyi(n)=0. (7.4)That is, xi(n) is stopped. The impulse response of the ideal lowpass filter isSince h(n)0 for nW.2/TWW(7.7)(7.7) shows that H(T) equals Hc() extended with period 2/T. Let W be the bandwidth of Hc(). If 2/TW, then H(T

7、) equals Hc() over the interval -/T,/T) (figure 7.1). Otherwise, aliasing is caused(figure 7.2). The above analysis also implies that the impulse invariance can be used to design band-limited filters, like low-pass filters and band-pass filters but cannot be used to design other types of filters, su

8、ch as high-pass filters and band-stop filters.thc(t)nh(n)Hc()H(T)2/TWWFigure 7.2. Relation between H() and Hc() when 2/TW.7.3.3. Causality and Stability In general, the discrete-time IIR filter will be causal and stable if the continuous-time IIR filter is causal and stable. Let the continuous-time

9、IIR filter have a rational system function. If it is causal and stable, hc(t) can be expressed as(7.8)where (m)0. According to (7.6), h(n) can be expressed as(7.9)where 0e(m)T0, by the impulse invariance.7.4. Design of IIR Filters by Bilinear Transformation7.4.1. Definition Another typical method to

10、 transform a continuous-time IIR filter into a discrete-time IIR filter is the bilinear transformation. Assume that H(z) and Hc(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 Hc(s) by

11、 letting(7.13)that is,(7.14)where T is a positive constant.7.4.2. Frequency Response Letting z=exp(j) in (7.14), we obtain-00Hexp(j)0Hc(j)(7.15)Figure 7.3. Relation between Hexp(j) and Hc(j).(7.15) shows that Hexp(j) equals Hc(j) when(7.16)This mapping is nonlinear, and thus a nonlinear distortion i

12、s caused. However, since the interval (-, ) of corresponds to the interval (-, ) of , no aliasing arises. The relation between Hexp(j) and Hc(j) is illustrated in figure 7.3.7.4.3. Causality and Stability The discrete-time IIR filter is causal and stable if the continuous-time IIR filter is causal a

13、nd stable. From (7.13), we obtain(7.17)If the continuous-time IIR filter is causal and stable, then Hc(s) will be convergent over regionRe(s)0. (7.18)According to (7.17), H(z) will be convergent over region |z|1. (7.19)Thus, the discrete-time IIR filter is also causal and stable.7.4.4. Procedure The

14、 bilinear transformation is carried out in the following steps: 1. Find the specifications for the continuous-time IIR filter from the specifications for the discrete-time IIR filter according to (7.16). 2. Find the system function of the continuous-time IIR filter from the specifications obtained i

15、n step 1. 3. Find the system function of the discrete-time IIR filter from the system function of the continuous-time IIR filter according to (7.14). Example. Design a causal lowpass discrete-time IIR filter which satisfies(7.20)where H(z) is the system function of the filter. It is required that H(

16、z) should be obtained from a causal lowpass continuous-time IIR filter(7.21)where a0, by the bilinear transformation.7.5. Design of FIR Filters by Windowing7.5.1. Principle In the windowing method, the impulse response of the causal FIR filter is obtained by windowing that of the ideal filter. Let u

17、s consider the design of a lowpass filter. The addressed ideas and methods, however, also apply to other cases. Over the period -, ), the frequency response of an ideal lowpass filter is defined as(7.22)(7.23)Taking the inverse Fourier transform of Hd(), we obtain the impulse response of the ideal l

18、owpass filter, i.e.,Since hd(n)0 for n0, the ideal lowpass filter is noncausal. However, windowing hd(n), we can obtain the impulse response of a causal FIR lowpass filter. That is, h(n)=hd(n)w(n), (7.24)where w(n) is a window function and is equal to 0 for n0. Generally, w(n) is real and symmetric,

19、 and has a maximum value of unit. is chosen as =(N-1)/2, (7.25)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.7.5.2. Frequency Response Let W() be the spectrum of w(n). Then the frequency response of the causal FIR lo

20、wpass filter equals the periodic convolution of Hd()(7.26)Hd() can be expressed asHd()=Ad()exp(-j), (7.27)where Ad() is the amplitude of Hd(). W() can be expressed as W()=E()exp(-j), (7.28)where E() is a real function. From (7.26)-(7.28), we obtain H()=A()exp(-j), (7.29)where A() is a real function

21、and equals the periodic convolution of Ad() and E() divided by 2, i.e.,(7.30)and W() divided by 2, i.e.,Ad()c-nhd(n)w(n)E()nh(n)nFigure 7.4. Differences between Ad() and A().-A()-c From (7.27) and (7.29), we see that the difference between Hd() and H() is actually the difference between Ad() and A()

22、. Based on (7.30), there exist two significant differences between Ad() and A() (figure 7.4). 1. At each cutoff frequency, Ad() 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(). A narrower main lobe results in a

23、narrower transition band. In order that A() has a narrow transition band, E() should have a narrow main lobe. (2) The width of the main lobe is determined by the type and the length of w(n). A longer w(n) results in a narrower main lobe. 2. Over the passband and the stopbands, Ad() is equal to 1 and

24、 0, respectively, but A() has ripples. (1) The amplitudes of the ripples are determined by the areas of the side lobes of E(). Smaller side lobes result in smaller ripples. In order that A() has small ripples, E() should have small side lobes. (2) The areas of the side lobes aredetermined by the typ

25、e of w(n) only. From (7.29), 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.7.5.3. Windows The rectangular windo

26、w is defined as(7.31) The Bartlett window is defined as(7.32)(7.33) The Hamming window is defined as(7.34) The Blackman window is defined as(7.35) Figure 7.4 shows the shapes of these commonly used windows. The Hann window is defined asFigure 7.4. Shapes of Commonly Used Windows. Table 7.1 gives the features of these commonly used windows. It can be seen that when the length of the window is fixed, a narrow transition band corresponds to large ripples and thus there exists a tradeoff between the decrease of the width of the transition band and the decrease of t