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1、1,Introduction FIR filter: direct design of DT filter with the often added linear-phase requirement (1) Windowed Fourier series approach (10.2) (2) Frequency sampling approach (Problem 10.31,10.32) (3) Computer-based optimization method (10.3),Chap.10 FIR Digital Filter Design,2,10.1 Preliminary Con
2、siderations,For FIR system: real polynomial approximation,if a linear phase is desired,10.1.1 Basic Approaches to FIR Digital Filter Design,3,10.1.2 Estimation of the Filter Order,Kaisers Formula,For lowpass FIR filter design: P397-398,Bellangers Formula,Hermanns Formula,Parameters see P398.,4,10.2
3、Design of FIR Filters by Windowing(P400),10.2.1 Least Integral-Squared Error Design of FIR Filters,5,10.2.2 Impulse Responses of Ideal Filters,Ideal linear phase lowpass filter,6,Impulse Responses of Ideal Filters (II),Ideal linear phase bandpass filter,7,Impulse Responses of Ideal Filters (III),Ide
4、al multiband filter,Ideal discrete-time Hilbert transformer,Ideal discrete-time differentiator,8,10.2.3 Gibbs Phenomenon,9,mainlobe,sidelobe,Mainlobe width-,truncation,=N/2,10.2.3 Gibbs Phenomenon (II),10,11,10.2.3 Gibbs Phenomenon (III),12,N oscillate more rapidly, but the amplitudes of the largest
5、 ripples = constant,For , N m , sidelobe ,10.2.3 Gibbs Phenomenon(IV),(2) For the integral , oscillation will occur at each sidelobe of moves past the discontinuity,(3) The methods to reduce Gibbs phenomenon: -tapering the window smoothly to zero at each end , but m -a smooth transition in magnitude
6、 specifications,13,10.2.4 Fixed Window Functions,(1) Hanning window: A= B=1/2, C=0; Hamming window: A=0.54, B = 0.46, C=0 Blackman window: A=0.42, B = 0.5, C = 0.08., Rectangular window: wn= un un N 1, Hanning, Hamming, Blackman:,Bartlett window: triangular,14,P406 Fig. 10.6 Commonly used fixed wind
7、ows,10.2.4 Fixed Window Functions (II),N/2,N,Rectangular,Hamming,Hanning,Bartlett,Blackman,n,wn,1,15,10.2.4 Fixed Window Functions (III),P407 Fig. 10.7,16,10.2.4 Fixed Window Functions (IV),Parameters predicting the performance of a window main lobe width relative sidelobe level (dB),Same ripples in
8、 passband and stopband,width of transition band,17,10.2.4 Fixed Window Functions (V),P408 Table 10.2,18,10.2.4 Fixed Window Functions (VI),Example to illustrate the effect of windows N=50 P409,19,10.2.4 Fixed Window Functions (VII),Compute impulse response of the desired filter (according to the inv
9、erse Fourier equation),(2) Determine the suitable window by the minimum stopband attenuation and (3) Determine the length of FIR by the transition width (4) Obtain the designed FIR filter:,Steps for FIR filter design:,20,Example 10.6 Page 410,Design an FIR lowpass digital filter with specifications
10、: the attenuation of the stopband should more than 40dB; .,2) According to Table 10.2, we could select Hanning,hamming, Blackman window, then the bandwidth of the transition band should satisfy (for Hanning),Type I: N = 32;,Type II: N = 33,10.2.4 Fixed Window Functions (VIII),1),Please select a suit
11、able window function and determine the smallest length of the window .,21,10.2.4 Fixed Window Functions,Example,Show that the ideal highpass transformer with a frequency response defined by,(1) Determine the impulse response hn ， the relation of and N?,(2) What type of linear-phase FIR filter?,(3) W
12、rite the impulse response hn using the Hann windows-base method.,Solution:,22,10.2.4 Fixed Window Functions,23,10.2.4 Fixed Window Functions,（2） If N is even,is integer，hdn is anti-symmetries ， and hn=-hN-n, the filter is type III.,If N is odd,isnt integer，hdn is symmetries ， and hn=hN-n, the filter
13、 is type II.,24,10.2.4 Fixed Window Functions,（3）,25,with = N/2. controls the side-lobe amplitudes (attenuation) controls the main lobe width Prediction formula: attenuation s = 20 log10s transition region width = sp together with attenuation s N,(10.39),10.2.5 Adjustable Window Functions (P410),Kai
14、ser window,N,26,10.2.5 Adjustable Window Functions (II),27,Kaiser window design example,(1) Determine the window function Kaiser window : , N,i.e., s=0.01, Assume:,Question: Is it suitable for N to be 23?,28,Kaiser window design example (II),(2) The desired impulse response,29,Kaiser window design e
15、xample (III),(3) The FIR filter designed,Where N=24, =3.395 Type I linear phase FIR,30,10.3 CAD of Equiripple Linear-Phase FIR Filters,Approximation methods:,(2) Least Integral-Squared approximation,Windowed Fourier Series approach,31,10.3 CAD of Equiripple Linear-Phase FIR Filters (II),Weighted err
16、or function:,(10.47),or,(10.62),(10.68),32,10.3 CAD of Equiripple Linear-Phase FIR Filters (III),Chebyshev or Minimax criterion:,equiripple FIR filter,Minimize the peak absolute value of,33,10.3 CAD of Equiripple Linear-Phase FIR Filters (IV),Alternation Theorem:,Let R be a union of disjoint closed subse
