FIR filter design

Catalog

Linear phase filter

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         With infinite duration impulse response (IIR) Compared with the filter , Digital filter with finite duration impulse response （ All zero or FIR filter ） There are both advantages and disadvantages .

FIR The filter has the following main advantages ：

• They can have precise linear phase
• They are always stable
• The design method is usually linear
• They can be implemented efficiently in hardware

The filter start transient has a finite duration .

        FIR The main disadvantage of the filter is , To achieve the same level of performance , The required order is much higher than IIR filter . Accordingly , The delay of these filters is usually better than that of the same performance IIR The filter is much larger .

FIR filter

Filter design method	explain	Filter function
Add windows	Apply windowing to the truncated inverse Fourier transform of the specified rectangular filter	fir1, fir2, kaiserord
Multiband （ Including transition zone ）	Use equal ripple or least square methods for subbands in the frequency range	firls、firpm、firpmord
Constrained least squares	According to the maximum error constraint , Minimize the square integral error over the entire frequency range	fircls, fircls1
Any response	Any response , Including nonlinear phase and complex filter	cfirpm
rrc	Low pass response with smooth sinusoidal transition	rcosdesign

Linear phase filter

         except cfirpm Outside , all FIR Filter design functions only design linear phase filters . These filter coefficients or “ Takeout ” Follow even or odd symmetry . According to this symmetry and the order of the filter n Is it even or odd , Linear phase filter （ Stored in... Length n +1 Vector b in ） There are certain inherent limitations on its frequency response .

Linear phase filter type	The order of the filter	Symmetry of coefficients	Respond to H(f)、f = 0	Respond to H(f)、f = 1 (Nyquist)
I class	even numbers	even numbers ： b(k)=b(n+2−k), k=1,...,n+1	unlimited	unlimited
II class	Odd number	even numbers ： b(k)=b(n+2−k), k=1,...,n+1	unlimited	H(1) = 0
III class	even numbers	Odd number ： b(k)=−b(n+2−k), k=1,...,n+1	H(0) = 0	H(1) = 0
IV class	Odd number	Odd number ： b(k)=−b(n+2−k), k=1,...,n+1	H(0) = 0	unlimited

         Linear phase FIR The phase delay and group delay of the filter are equal and constant in the whole frequency band . about n Order linear phase FIR filter , The group delay is n/2, Filtered signal delay n/2 Time steps （ The amplitude of its Fourier transform is scaled according to the amplitude response of the filter ）. This attribute maintains the waveform of the signal in the passband ; in other words , No phase distortion .

         By default , function fir1、fir2、firls、firpm、fircls and fircls1 Can be used for design I Classes and II Class linear phase FIR filter .rcosdesign Only for design I Class filter . In the given 'hilbert' or 'differentiator' In the case of signs ,firls and firpm Can be used for design III and IV Class linear phase FIR filter .cfirpm It can be used to design any type of linear phase filter and nonlinear phase filter .

         Be careful ： because II Class filter in Nyquist frequency （“ high ” frequency ） The frequency response is zero at ,fir1 Not for design II Quasi high pass and band stop filters . In these cases , If n Is an odd number ,fir1 Add steps 1, And back to I Class filter .

Windowing method
         Suppose a cut-off frequency is ω0 radian / Second ideal rectangular digital low-pass filter . The filter has an amplitude less than ω0 Has amplitude at all frequencies 1, The amplitude is between ω0 and π There is an amplitude in the frequency between 0. Its impulse response sequence h(n) by ：

         This filter cannot be realized , Because its impulse response is infinite and non causal . To create a finite duration impulse response , Please truncate it by applying windowing . By keeping the central part of the impulse response in this truncation , Linear phase can be obtained FIR filter . for example , A low-pass cut-off frequency ω0  by 0.4 π radian / Of a second 、 The length is 51 The filter is ：

b = 0.4*sinc(0.4*(-25:25));

         The windowing applied here is a simple rectangular window . according to Parseval Theorem , The length is 51 The filter of is closest to the ideal low-pass filter in the sense of integral least square . The following order is in FVTool The frequency response of the filter is displayed in ：

fvtool(b,1)

         Please note that , The following figure shows y The shaft adopts the quadratic power of amplitude . You can right-click the axis tab and select Amplitude squared To set this . As shown in the figure ：

         Ringing and ripple will appear in the response , Especially near the edge of the band . such “ Gibbs effect ” It will not disappear with the increase of filter length , But non rectangular windows will reduce their amplitude . Multiplying the signal by a window function in the time domain will make the signal convolute or smooth in the frequency domain . The length will be 51 Of Hamming Windows should be used for filters , And use FVTool Show results ：

b = 0.4*sinc(0.4*(-25:25));
b = b.*hamming(51)';
fvtool(b,1)

         Please note that , The following figure shows y The shaft adopts the quadratic power of amplitude . You can right-click the axis tab and select Amplitude squared To set this . As shown in the figure ：

         Use Hamming Windows can greatly reduce ringing . This improvement comes at the cost of transition width and optimality ： Windowed filters take longer to fall from passband to stopband , And the square error integral cannot be minimized .

Standard frequency band FIR Filter design ：fir1

        fir1 Use least square approximation to calculate filter coefficients , Then the impulse response is smoothed by windowing . An overview of windowing and its properties , You can refer to Windowing .fir1 Be similar to IIR Design function of filter , Because it is used to design standard band configuration （ Lowpass 、 Bandpass 、 High pass and band stop ） Filter under . The following sentence ：

n = 50;
Wn = 0.4;
b = fir1(n,Wn);

         Create a row vector b, It includes n rank Hamming Window filter coefficients . This is a low-pass linear phase FIR filter , The cut-off frequency is Wn.Wn Is between 0 and 1 Number between , among 1 Corresponding to Nyquist frequency , That is, half the sampling frequency .（ Different from other methods , here Wn Corresponding to 6 dB spot .） To get a high pass filter , Just put the 'high' Add to the parameter list of the function . To obtain a bandpass or bandstop filter , take Wn Specified as a two element vector containing the passband edge frequency . Add 'stop'.

        b = fir1(n,Wn,window) Use column vectors window Design the window specified in . vector window The length of must be n+1 Elements . If no window is specified ,fir1 Will be applied Hamming window .

        Kaiser Window order estimation .  kaiserord Function estimation filter order 、 Cut off frequency and Kaiser window β Parameters , Make it meet a given set of filter settings . Given the frequency band edge vector, the corresponding amplitude vector and the maximum allowable ripple ,kaiserord by fir1 Function returns the appropriate input parameters .

Multiband FIR Filter design ：fir2

        fir2 Function can also be used to design windowed FIR filter , But it has piecewise linear frequency response of arbitrary shape . This is related to fir1 Different , The latter is only designed with standard low-pass 、 qualcomm 、 Bandpass and bandstop configured filters . The following command ：

n = 50;
f = [0 .4 .5 1];
m = [1  1  0 0];
b = fir2(n,f,m);

         Returns the row vector b, It includes n rank FIR The filter n+1 A coefficient of , Its frequency amplitude characteristic and vector f and m The given frequency amplitude characteristics match .f Is the vector of the frequency point , Range from 0 To 1, among 1 representative Nyquist frequency .m It's a vector , contain f The specified amplitude response of the specified point in .（ The corresponding of this function IIR The function is yulewalk, The latter can also design filters based on arbitrary piecewise linear amplitude response .）

         Reference code ：

 author ： Master fat cat 
 link ：https://zhuanlan.zhihu.com/p/337397832
 source ： You know 
 The copyright belongs to the author . Commercial reprint please contact the author for authorization , Non-commercial reprint please indicate the source .

%%  The original signal 
fs = 1000;          %  sampling frequency 
t = 0:1/fs:2-1/fs;  %  Time 
singal0 = sin(2*pi*1*t);      %  A signal without noise 
SNR = 15;           %  Signal to noise ratio 
singal = awgn(singal0,SNR);    %  After adding Gaussian white noise 
%% IIR wave filtering  （butterworth filter ）
fstop = 20;         %  Low pass cut-off frequency 
wn = fstop/(fs/2);  %  Normalized cut-off frequency 
N_IIR = 5;          % IIR Order 
[b_iir,a_iir] = butter(N_IIR,wn) % IIR Filter parameters 
singal_iir = filter(b_iir,a_iir,singal); %IIR wave filtering 
%% FIR wave filtering  （hamming window ）
N_FIR = 15;            % FIR Order 
b_fir = fir1(N_IIR,wn) % FIR Filter parameters 
singal_fir = filter(b_fir,1,singal); %FIR wave filtering 
%%  moving average  （ Smoothing constant M=10;）
wsize = 10;            %  Smoothing constant 
b_avg = ones(1,wsize)/wsize;   %  A weight 
singal_avg = filter(b_avg,1,singal);  % Moving average filtering 
%%  Output comparison 
figure
subplot(2,2,1)
plot(t,singal)
title('\fontname{ Song style }\fontsize{12} The original signal ');
xlabel('\fontname{Arial}\fontsize{12}t');       
ylabel('\fontname{Arial}\fontsize{12}singal');
grid on;
subplot(2,2,2)
plot(t,singal_iir)
title('\fontname{ Song style }\fontsize{12}5 rank IIR wave filtering ');
xlabel('\fontname{Arial}\fontsize{12}t');       
ylabel('\fontname{Arial}\fontsize{12}singal(iir)');
grid on;
subplot(2,2,3)
plot(t,singal_fir)
title('\fontname{ Song style }\fontsize{12}15 rank FIR wave filtering ');
xlabel('\fontname{Arial}\fontsize{12}t');       
ylabel('\fontname{Arial}\fontsize{12}singal(fir)');
grid on;
subplot(2,2,4)
plot(t,singal_avg)
title('\fontname{ Song style }\fontsize{12} moving average ( constant 10)');
xlabel('\fontname{Arial}\fontsize{12}t');       
ylabel('\fontname{Arial}\fontsize{12}singal(avg)');
grid on;

         As shown in the figure ：