数字信号处理教学课件：Chapter10 FIR Digital Filter Design

1、Chapter 10 FIR Digital Filter DesignPreliminary ConsiderationFIR Digital Filter Design MethodsMATLAB Functions in FIR Filter Design10.1 Preliminary ConsiderationDigital Filter SpecificationsSelection of the Filter TypeBasic Approach to FIR Digital Filter DesignFIR Digital Filter Order EstimationScal

2、ing the Digital Transfer Function10.1.1 Basic Approaches to design FIR filtersUnlike IIR digital filter design, FIR filter design does not have any connection with the design of analog filters.The design of FIR filters is based on a direct approximation of the specified magnitude response, with ofte

3、n added requirement that the phase response be linear.10.1.2 Estimation of the Filter OrderThe order (length) of G(z) is directly estimated from the digital filter specification.Several authors have advanced formulas : （1）Kaisers Formula:Estimation of the Filter Order（2）Bellangers Formula:（3） Herman

4、ns Formula:Reference P529 10.6(a-e).Estimation of the Filter OrderA Comparison of FIR Filter Order Formulas:(1)The estimated filter order N of the FIR filter is inversely proportional to the transition band width(p-s) .(2)The order of Kaisers and Bellangers formulas depends on the productpand s.The

5、comparison result is shown in P531.10.2 FIR Filter Design Based on Windowed Fourier SeriesBecause FIR filters are described by a transfer function that is a polynomial in z-1, we will use another way to design them. Methods: (1)Truncating the Fourier series representation of the prescribed frequency

6、 response. (direct and straightforward)(2)Sampling the frequency response to get the DFT of hn, and use IDFT to get hn.10.2.1 Least Integral-Squared Error Design of FIR FiltersLet Hd(ej) denotes the desired frequency response function, hdn denotes the impulse response samples, so :In most applicatio

7、n, for meeting the FR Hd(ej) requirements, hdn is infinite. Least Integral-Squared Error Design of FIR FiltersSo we want find: a Length-N, causal So:The procedure of using htn to replace hdn is called truncating or windowing. Using what approximating criterion: Minimum Integral-Squared Error.Least I

8、ntegral-Squared Error Design of FIR FiltersDefine integral-squared error:Using Parsevals relation, we get:When htn=hdn for -MnM, the R is minimum.10.2.2 Impulse Responses of Ideal FiltersIdeal lowpass filter: Ideal highpass filter:Impulse Responses of Ideal FiltersIdeal bandpass filter: Ideal bandst

9、op filter:Impulse Responses of Ideal FiltersIdeal multi-band filterImpulse Responses of Ideal FiltersHilbert transformer(90-degree phase shifter)Impulse Responses of Ideal FiltersDifferentiator:The N-point linear phase FIR with10.2.3 Gibbs Phenomenon- The Effect of Truncating the hn Gibbs phenomenon

10、 - Oscillatory behavior in the magnitude responses of causal FIR filters obtained by truncating the impulse response coefficients of ideal filterGibbs PhenomenonWiden the discontinuous position of the ideal frequency characteristics to form a transition band. In both sides of ,the H(w) have maximum

11、ripple value. Increasing the truncation-length of window can decrease the width of the transition band, but cant change the relative value of the mainlobe and sidelobe.Gibbs PhenomenonInterpretation:In time-domain:In f-domain:So ,after using a rectangular window, the spectrum of the truncated impuls

12、e response will have oscillatory behavior.Gibbs PhenomenonGibbs PhenomenonSo,we need a window:(1)Its main lobe width is as narrow as possible to ensure a fast transition from passband to stopband.(2)Reducing the relative magnitude of the maximum sidelobe to reduce the passband and stopband ripple.10

13、.2.4 Fixed Window FunctionsRectangular window00.20.40.60.81-100-80-60-40-200w/pGain, dBRectangular windowN11Fixed Window FunctionsHanning window00.20.40.60.81-100-80-60-40-200w/pGain, dBHanning windowFixed Window FunctionsHamming window:00.20.40.60.81-100-80-60-40-200w/pGain, dBHamming windowFixed W

14、indow FunctionsBlackman window:00.20.40.60.81-100-80-60-40-200w/pGain, dBBlackman windowFixed Window FunctionsMagnitude spectrum of each window characterized by a main lobe centered at w = 0 followed by a series of sidelobes with decreasing amplitudes.Parameters predicting the performance of a windo

15、w in filter design are:Main lobe width.Relative sidelobe level.Fixed Window FunctionsMain lobe width ML- given by the distance between zero crossings on both sides of main lobe.Relative sidelobe level Asl - given by the difference in dB between amplitudes of largest sidelobe and main lobe.Fixed Wind

16、ow Functions Passband and stopband ripples are the same.Thus,ObserveFixed Window FunctionsDistance between the locations of the maximum passband deviation and minimum stopband value ML.Width of transition band: w = s - p MLFixed Window FunctionsTo ensure a fast transition from passband to stopband,

17、window should have a very small main lobe width.To reduce the passband and stopband ripple d, the area under the sidelobes should be very small.Unfortunately, these two requirements are contradictory.Fixed Window FunctionsIn the case of rectangular, Hann, Hamming, and Blackman windows, the value of

18、ripple does not depend on filter length or cutoff frequency c , and is essentially constant.In addition, w c / Mwhere c is a constant for most practical purposes.Rectangular and Blackman WindowsFixed Window FunctionsProperties of some fixed window functionsType of windowsMain lobe width MLRelative s

19、idelobe level AslMinimum stopband attenuationTransition bandwidth MRectangular 4/(2M+1)13.3dB20.9dB0.92/MHann 8/(2M+1)31.5dB43.9dB3.11/MHamming8/(2M+1)42.7dB54.5dB3.32/MBlackman 12/(2M+1)75.3dB75.3dB5.56/M10.2.5 Adjustable Window FunctionsDolph-Chebyshev windowWhere: is the relative sidelobe amplitu

20、de. Adjustable Window FunctionsKaiser windowWhere:Its zero-order Bessel function, and is adjustable.So it is called adjustable windowDesign Steps for Windowed Low Pass FIR Filters(1) Choose a pass band edge frequency in Hz for the filter in the middle of the transition width: c =(p +s )/2(2) Substit

21、ute c into h1n, the infinite impulse response for an ideal low pass filter: h1n=sin(nc)/n Design Steps for Windowed Low Pass FIR Filters(3) Choose a window based on the specified, and calculate the number of nonzero window terms.(4) Calculate FIR hn from hn=h1nwn, notice the response is noncausal.(5

22、) Shift the impulse response to the right by (N-1)/2 to make the filter causal.(6) Compute to test.A Example of FIR Filter Design:Design a linear phase FIR LPF:If given the passband edge frequency p=0.2, the passband ripple is 1dB，the stopband edge frequency s=0.3 ,the minimum stopband attenuation-2

23、5dB.FIR Filter Design ExampleSpecifications: p=0.2, s=0.3 , p =1dB, s=25dBThusSo, the ideal LP impulse response is:Select a window based on table 10.2 and specifications.Because of s=25dB, select window Hann.A Example of FIR Filter Design:Calculate order N:N=2M+1=65The impulse response is:Finally, t

24、est the result.Supplement: FIR Filter Design Based on Frequency Sampling ApproachExpand the idea of window function method to frequency domain, we get frequency sampling approach in FIR filters design, that is:Find an approximation of the desired FR H(ej)in frequency domain. Basic Idea of Frequency Sampling ApproachNote:Hk must satisfy constraint that the FIR filter is linear phase.Use to approximate the desired frequency response , then from: we can get the length-N impulse response hn of the FIR filter.Design Steps:(1)From ide