1、Digital Filter Design: Basic ApproachesFIR filter design is based on a direct approximation of the specified magnitude response, with the often added requirement that the phase be linear （how to?）1Digital Filter Design: Basic ApproachesThree commonly used approaches to FIR filter design:(1) Windowed

2、 Fourier series approach(2) Frequency sampling approach(3) Computer-based optimization methods2Impulse Responses of Ideal FiltersIdeal lowpass filter Ideal highpass filter 3Impulse Responses of Ideal FiltersIdeal bandpass filter 4Impulse Responses of Ideal FiltersIdeal bandstop filter 5Gibbs Phenome

3、nonGibbs phenomenon - Oscillatory behavior in the magnitude responses of causal FIR filters obtained by truncating the impulse response coefficients of ideal filters6Gibbs PhenomenonAs can be seen, as the length of the lowpass filter is increased, the number of ripples in both passband and stopband

4、increases, with a corresponding decrease in the ripple widthsHeight of the largest ripples remain the same independent of lengthSimilar oscillatory behavior observed in the magnitude responses of the truncated versions of other types of ideal filters7Gibbs PhenomenonGibbs phenomenon can be explained

5、 by treating the truncation operation as an windowing operation:In the frequency domainwhere and are the DTFTs of and , respectively 8Gibbs PhenomenonThus is obtained by a periodic continuous convolution of with9Gibbs PhenomenonIf is a very narrow pulse centered at (ideally a delta function) compare

6、d to variations in , then will approximate very closelyOn one hand, length 2M+1 of wn should be very largeOn the other hand, length 2M+1 of should be as small as possible to reduce computational complexity10Gibbs PhenomenonA rectangular window is used to achieve simple truncation:Presence of oscilla

7、tory behavior in is basically due to: 1) is infinitely long and not absolutely summable, and hence filter is unstable 2) Rectangular window has an abrupt transition to zero11Gibbs PhenomenonOscillatory behavior can be explained by examining the DTFT of : has a main lobe centered at Other ripples are

8、 called sidelobes12Gibbs PhenomenonMain lobe of characterized by its width defined by first zero crossings on both sides ofAs M increases, width of main lobe decreases as desiredArea under each lobe remains constant while width of each lobe decreases with an increase in MSo Ripples in around the poi

9、nt of discontinuity occur more closely but with no decrease in amplitude as M increases13Gibbs PhenomenonRectangular window has an abrupt transition to zero outside the range , which results in Gibbs phenomenon inGibbs phenomenon can be reduced either:(1) Using a window that tapers smoothly to zero

10、at each end, or(2) Providing a smooth transition from passband to stopband in the magnitude specifications14Fixed Window FunctionsUsing a tapered window causes the height of the sidelobes to diminish, with a corresponding increase in the main lobe width resulting in a wider transition at the discont

11、inuityHanning:Hamming:Blackman:15Fixed Window Functions16Fixed Window FunctionsPlots of magnitudes of the DTFTs of these windows for M = 25 are shown below:17Fixed Window FunctionsMagnitude spectrum of each window characterized by a main lobe centered at w = 0 followed by a series of sidelobes with

12、decreasing amplitudesParameters predicting the performance of a window in filter design are:Main lobe widthRelative sidelobe level18Fixed Window FunctionsMain lobe width - given by the distance between zero crossings on both sides of main lobeRelative sidelobe level - given by the difference in dB b

13、etween amplitudes of largest sidelobe and main lobe19Fixed Window FunctionsObserveThus, Passband and stopband ripples are the same20Fixed Window FunctionsDistance between the locations of the maximum passband deviation and minimum stopband valueWidth of transition band21Fixed Window FunctionsTo ensu

14、re a fast transition from passband to stopband, window should have a very small main lobe widthTo reduce the passband and stopband ripple d, the area under the sidelobes should be very smallUnfortunately, these two requirements are contradictory22Fixed Window FunctionsIn the case of rectangular, Han

15、n, Hamming, and Blackman windows, the value of ripple does not depend on filter length or cutoff frequency , and is essentially constantIn addition,where c is a constant for most practical purposes23Fixed Window FunctionsRectangular window - dB, dB,Hann window - dB, dB,Hamming window - dB, dB,Blackm

16、an window - dB, dB,24Fixed Window FunctionsBasic idea of windows design techniques: choose a proper ideal frequency-selective filter (which has a noncausal, infinite-duration impulse response) and then truncate its impulse response to obtain a linear-phase and causal FIR filter.Filter Design Steps:(

17、1) Set(2) Choose window based on specified(3) Estimate M using(4)(5)FIR: Windowing!25 (6) linear-phase requirement?Note: Windowed Fourier series approach will naturally result in type-1 FIR filter. (why?)26FIR Filter Design ExampleLowpass filter of length 51 and27FIR Filter Design ExampleDesign a FIR lowpass filter with the following specifications: p=0.2, s=0.3,p= s=0.005s=-20log10(s)=46dB, according to Table 10.2, which window can we choose? If we require that the length of FIR filter to be the smallest?We can choose either Hamming