数字信号处理与DSP器件：Fourier transform and Filter Shape

1、Fourier transform and Filter Shape数字信号处理与DSP器件 Fourier transform and Filter Shapethe frequency response H(W) is the DTFT of the impulse response hnThe discrete time Fourier transform (DTFT) is a frequency domain tool used to find the spectrum X(W) of a signal xn (next chapter) or the frequency respo

2、nse of a filter (this chapter). This property also makes conversion from a difference equation to a frequency response H(W) straightforward. yn + a1yn-1 + a2yn-2 + = b0 xn + b1xn-1 + b2xn-2 + Fourier transform and Filter ShapeSampling frequency = 12 kHz|H(-W)| = |H(W)|q(-W) = -q(W)Fourier transform

3、and Filter ShapeComb Filter ShapeFourier transform and Filter ShapeFourier transform and Filter ShapeThe closer the zeros and poles are to the unit circle, the more selective the filter, that is, the steeper the transition between the pass band and stop band regionsFinite Impulse Response Filters Fi

4、nite impulse response filters（FIR）are non-recursive filters, meaning that a new filter output relies only on inputs past and present, and not on past outputs. The difference equation for this type of filter is yn = b0 xn + b1xn-1 + + bMxn-MThe transfer function (Z transform) is The frequency respons

5、e is H(Z) = b0 + b1Z-1 + + bMZ-MFinite Impulse Response Filters A moving average filter is a simple example of an FIR filter. This tends to smooth the input, confirmed by the low pass nature of the moving average filters magnitude response. Impulse Response of 5-term Moving Average FilterFilter Shap

6、e of 5-term Moving Average FilterFinite Impulse Response FiltersPole-Zero Plot of 11-term Moving Average FilterFIR Filter MeritCompany LogoPhase distortion affectionCompany LogoElimination of phase distortionCompany LogoFinite Impulse Response FiltersSymmetrical Impulse ResponseMagnitude Response Im

7、pulse Response of 11-term Moving Average FilterFinite Impulse Response FiltersPhase Response (Note that phase response is linear in pass band.) Finite Impulse Response Filtersideal low pass filter, which has an infinite impulse responseIdeal Low Pass Filter ShapeImpulse Response of Ideal Low Pass Fi

8、lterFinite Impulse Response FiltersNon-Ideal Low Pass Filter Shape, After WindowingFinite Impulse Response Filters The degree of acceptable non-ideality is measured by parameters like the pass band edge frequency, the stop band edge frequency, the transition width, the pass band ripple dp, and the s

9、top band ripple ds (or the stop band attenuation20logds).the transition width= the stop band edge frequency- the pass band edge frequencyFinite Impulse Response Filters The problems caused by the rectangular window can be solved to a large extent by using a window with smoother edges. The Hanning, H

10、amming, Blackman, and Kaiser windows give better stop band attenuations.Rectangular WindowFinite Impulse Response FiltersHanning WindowHamming WindowBlackman WindowKaiser Window (b = 8)Impulse Response of Ideal Low Pass FilterRectangular WindowFinite Impulse Response Filters A low pass FIR filter is

11、 designed from a set of specifications as follows: 1. Choose a pass band edge frequency for design that is midway through the transition width, to account for smearing. 2. Find the impulse response h1n for the ideal low pass filter with this pass band edge frequency.Finite Impulse Response Filters 3

12、. Choose a window wn that gives adequate stop band attenuation.Window Stop Band Attenuation Number of TermsRectangular 210.91fS/TW Hanning 443.32fS/TW Hamming 55 3.44fS/TW Blackman 75 5.98fS/TW Kaiser (b = 8) 81 5.25fS/TW 4. Find the impulse response hn = h1nwn for the FIR filter. 5. Shift the impul

13、se response to the right by (N-1)/2 samples.（设计中用的通带边缘频率=所要求的通带频率+（过渡带宽度）/2）Matlab Programfs=50.96*103; fpass=1*103; fstop=10*103;window_type=2; % 1: blackman 75dB 2: hanning 44dB 3 hamming 55dB;TW=fstop-fpass;omw1=2*pi*(fpass+TW/2)/fs);if window_type=1 N=5.98*fs/TW; end; % blackman 75dBN=floor(N/2)

14、*2+1;if window_type=1 x=sinc(N,omw1).*blackman(N); endxN % type the N on the screenFinite Impulse Response FiltersLow Pass Filter Impulse ResponseLow Pass Filter ShapeFinite Impulse Response FiltersWindowed band pass and high pass FIR filters can be obtained by shifting a low pass filter with the de

15、sired shape to a new location in the frequency domain.Low Pass Filter Magnitude ResponseBand Pass Filter Magnitude ResponseBand Pass Filter Company LogoFinite Impulse Response Filters In the case of a band pass filter, the size of the shift in the frequency domain is equal to the center frequency. I

16、n the case of a high pass filter, the size of the shift is equal to half the sampling frequency.Band Pass Filter Impulse ResponseBand Pass Filter ShapeBand Pass Filter Company LogoHow to obtain a High Pass Filter ?Finite Impulse Response FiltersHigh Pass Filter Impulse ResponseHigh Pass Filter Shape

17、Finite Impulse Response Filters Band stop filters may be created by summing together the outputs of a low pass filter and a high pass filter with suitably separated pass band edge frequencies.Band stop filtersFinite Impulse Response FiltersWindowed FIR Filter DesignEquiripple FIR Filter DesignFinite

18、 Impulse Response Filters The equiripple method centers around minimizing the maximum error between and ideal filter shape and the true filter shape.Finite Impulse Response Filters The simplest window is the rectangular window. Unfortunately, it produces a filter with poor stop band attenuation, only about 21 dB. This is not sufficient to remove elements outside the pass band decisively.Reference SoundReference Sound Attenuated by 21 dBFinite Impulse Response FiltersExamplesA. Low Pass Filtering Example