离散时间信号处理DSP第章.ppt
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1、Digital Signal Processing,Chapter 7 Filter Design Techniques,2,7.1 Design of Discrete-Time IIR Filters from Continuous-Time Filters,In the chapter, we deal with the digital filter design methods in which a desired frequency response of the system is approximated by a system function consisting of a
2、ratio of polynomials. Generally, the design of an IIR digital filter is carried out in three steps as follows. Specifications Approximations Realization,3,7.1 Design of Discrete-Time IIR Filters from Continuous-Time Filters,4,Figure 1 Ideal magnitude specifications for digital lowpass filter,理想滤波器的幅
3、度特性有理想、陡截止的通带和无穷大衰减的阻带两个范围,如图1所示,这显然是无法实现的,因为它们的单位取样响应均是非因果和无限长的。 实践中只能用一种因果可实现的滤波器去与之逼近,使其满足给定的误差容限。一个实际滤波器的幅度特性在通带中允许有一定的波动,阻带衰减则应大于给定的衰减要求,且在通带与阻带之间允许有一定宽度的过渡带,如图2所示。,5,Figure 2 Magnitude specifications for digital lowpass filter,6,Lowpass filter specification,Passband(通带) The frequency range of
4、0 pc is called the passband; pc is the passband cutoff frequency(通带截止频率); p is the passband tolerance, that is Stopband(阻带) The frequency range of sc is called the stopband; sc is the stopband cutoff frequency(阻带截止频率); s is the stopband tolerance, that is Transition band The frequency range of pc sc
5、 is called the transition band;,7,Lowpass filter specification,Maximum passband attenuation p Minimum stopband attenuation p Commonly, the maximum magnitude response is assumed to be normalized to unity. For a lowpass filter, we have,8,Design Stages for Digital Filter,Design stages Analog filter app
6、roximations, including Butterworth, Chebyshev and elliptic. Continuous-time to discrete-time transformation, including impulse invariance and bilinear transformation. Frequency transformations, that is, transforming a lowpass filter into a highpass or bandpass or bandstop filter.,9,1 Analog Butterwo
7、rth(巴特沃思)Lowpass Filters,The Butterworth lowpass filter has several properties: All poles, and no zero. No ripples(波纹) in the passband and stopband. The Butterworth lowpass filter is defined in a magnitude-squared function(幅度平方函数): where N is a positive integer and is called the order of the filter,
8、 c is the passband cutoff frequency,10,1 Analog Butterworth Lowpass Filters,Because The cutoff frequency c is called the half-power frequency point of the filter. The maximum passband attenuation is the frequency c is also called 3-dB cutoff frequency or 3-dB bandwidth of the filter.,11,1 Analog But
9、terworth Lowpass Filters,In practical applications, the analog lowpass filter is specified by the specifications as follows: pc: the passband cutoff frequency in rad/s; p: the maximum passband attenuation in dB; sc: the stopband cutoff frequency in rad/s; s: the minimum stopband attenuation in dB; T
10、o use the Butterworth lowpass filter to approximate a lowpass filter, we should obtain the order N and the 3-dB cutoff frequency c.,12,1 Analog Butterworth Lowpass Filters,The magnitude response of the filter is The maximum passband attenuation p arrives at =pc, which is represented as The minimum s
11、topband attenuation s arrives at =sc, which is represented as,(7.141),(7.142),(7.140),13,1 Analog Butterworth Lowpass Filters,Solving the above two equations, we have,14,1 Analog Butterworth Lowpass Filters,There are two choices to determine c Substituting N in Eq.(7.141), we have Using this formula
12、 to determine c, p is exactly met at =pc and s is exceeded for the stopband, provided that p3dB. Substituting N in Eq.(7.142), we have Using this formula to determine c, s is exactly met at =sc and p is exceeded for the passband, provided that s3dB.,15,1 Analog Butterworth Lowpass Filters,To determi
13、ne the transfer function Ha(s) of the filter, we substitute s = j, therefore The poles of the magnitude-squared function Ha(s)Ha(-s) are given by Figure 3 shows all the poles when N = 3.,16,Figure 3 Pole plot for a third-order Butterworth filter,k = 0,k = 1,k = 2,k = 3,k = 4,k = 5,j,17,1 Analog Butt
14、erworth Lowpass Filters,The 2N poles equally spaced in angle on a circle in the s-plane. They are symmetric about the imaginary axis. In order to obtain a stable system, we choose the n poles on the left-side of the s-plane. Then, we get the transfer function as follows where,18,1 Analog Butterworth
15、 Lowpass Filters,The attenuation of the Butterworth approximation increases monotonically(单调地)with frequency. And it increases very slowly in the passband and quickly in the stopband. If one wants to increase the attenuation one has to increase the filter order. The 3-dB bandwidth is unrelated to th
16、e filter order.,19,Figure 4 The magnitude-frequency response of Butterworth Lowpass Filters,20,2 Analog Chebyshev(切贝雪夫)Lowpass Filters,The Chebyshev-I lowpass filters have equiripple(等波纹的) magnitude response in the passband and monotonic(单调的) magnitude response in the stopband. The magnitude-squared
17、 response of an analog Chebyshev-I lowpass filter is given by where N is the order of the filter, is a positive and less than unity number which is the passband ripple factor, c is the passband cutoff frequency at which the attenuation of the magnitude response is not necessary to be 3-dB. The funct
18、ion TN(x) is an Nth-order Chebyshev polynomial defined by,(7.147),21,2 Analog Chebyshev Lowpass Filters,The characteristics of the Chebyshev-I filters are as follows: At = 0, |Ha(j0)| = 1 for N odd and |Ha(j0)| = for N even; At =c, |Ha(jc)| = for all N; Within the passband of 0c, |Ha(j)| oscillates
19、between 1 and ; For c, |Ha(j)| approaches zero monotonically and rapidly; At =s (the stopband cutoff frequency), |Ha(js)| = 1/A.,22,Figure 5 Analog Chebyshev Lowpass Filters,23,2 Analog Chebyshev Lowpass Filters,In the design of Chebyshev-I lowpass filter, the specifications are given by: c: the pas
20、sband cutoff frequency in rad/s; p: the passband ripple in dB; s: the stopband cutoff frequency in rad/s; s: the minimum stopband attenuation in dB; To design the filter, the order N and ripple factor should be determined.,24,2 Analog Chebyshev Lowpass Filters,Since Then And Therefore,25,2 Analog Ch
21、ebyshev Lowpass Filters,To obtain the transfer function Ha(s) of the Chebyshev-I filter, we substitute =s/j into Eq.(7.147) (p19), and then we get There are 2N poles of the magnitude-squared function Ha(s)Ha(-s), They are spaced on a ellipse in the s-plane and symmetric about the imaginary axis. In
22、order to obtain a stable system, we choose the N poles on the left-side of the s-plane and get the transfer function as follows.,26,2 Analog Chebyshev Lowpass Filters,where and,27,两种典型模拟滤波器,两种典型模拟滤波器: Butterworth(巴特沃思)滤波器:幅频特性单调下降,但衰减特性较差; Chebyshev(切贝雪夫)滤波器:在通带(或阻带)中幅频特性单调下降,在阻带(或通带)中有波纹,衰减特性好于巴特沃思
23、滤波器;,28,Analog-to-Digital Filter Transformations,The continuous-time to discrete-time transformations include three steps: Transformations of specifications in discrete-time domain into ones in continuous-time domain. Designing the analog filter according to the specifications in continuous-time dom
24、ain. Transform the filter in s domain into the one in z domain. The main methods of transformations have two kinds: Impulse-invariance method Bilinear transformation method,29,z transform and Laplace transform,Laplace transform z transform The Laplace transform of x(nT) is Therefore the relationship
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