Implementation of Narrow band pass filter ( Butterworth)

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Bharath 2015-3-19

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评论： Star Strider 2021-2-18

Hi all,

I'm struggling with implementation of narrow band pass filter. I understand from few suggestions (feedback) that it's too narrow and impulse is becoming too large. I also tried using the decimation but even it didn't work. I've posted my question in stackoverflow. Here is the link to it.

http://stackoverflow.com/questions/29111768/band-pass-implementation-matlab?noredirect=1#comment46453099_29111768

Could someone help me to solve this. Since this is part of my project. I'm little running short of time.

Thanks in advance for all your help and suggestion.

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Star Strider 2015-3-19

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https://ww2.mathworks.cn/matlabcentral/answers/183972-implementation-of-narrow-band-pass-filter-butterworth#answer_171934

编辑：Star Strider 2015-3-19

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This seems to work, although freqz has problems with it for some reason:

% DEFINE FILTER PARAMETERS 
Fs  = 2E+6;
Fn  = Fs/2;
Fc1 =  630;
Fc2 =  640;
Fs1 = Fc1*0.8;
Fs2 = Fc2/0.8;
Rp  =    1;
Rs  =   10;
% DESIGN FILTER
[n,Wn]  = buttord([Fc1 Fc2]/Fn, [Fs1 Fs2]/Fn, Rp, Rs);
[b,a]   = butter(n,Wn);
[sos,g] = tf2sos(b,a);
% CHECK FILTER PERFORMANCE
% figure(1)
% freqz(sos, 512, Fn)
s = zeros(1, Fs);
s(fix(Fs/2)) = 1;                       % Create Impulse Signal
y = filtfilt(sos, g, s);                % Filter Signal
tfs = fft(y)./fft(s);                   % Calculate Transfer Funciton
Fv = linspace(0,1,length(s)/2+1)*Fn;    % Frequency Vector
ix = 1:length(Fv);                      % Index Vector
figure(2)
plot(Fv, abs(tfs(ix)))
grid
axis([0  1000    ylim])

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Star Strider 2015-3-19

My pleasure.

I leave the orbit plots to you. I’m just sorting out your filter.

There’s a certain ‘art’ to filter design. As a first approximation, I calculate the stopband frequencies (‘Fs1’, ‘Fs2’) for any filter as 0.8 times the low passband and 1.25 times the upper passband. (This usually works; if it doesn’t, I adjust them.) The buttord function uses these to determine the appropriate filter order.

I found significant problems with your filter design, and I gave you code to design and implement it efficiently to get the result you want. I converted your filter to a second-order-section representation (with tf2sos) for stability. This is common practice, and the butter function discusses it in detail.

Since you have the Signal Processing Toolbox, use filtfilt. It does not introduce any phase distortion, while filter does. (All this is explained clearly in the documentation for the various functions, so I won’t go into it in detail here.)

The three ‘tfs’, ‘Fv’, and ‘ix’ assignments calculate the frequency-domain transfer function, the frequency vector for the plot, and the index vector for the plot. (The index vector just makes the plot call easier to write.) Because your passband is very small in relation to your sampling frequency, the freqz function did not show the passband with sufficient resolution, so I created my own version. (Note: I edited my original code to provide a much better ‘s’ and transfer function calcualtion.)

Bharath 2015-3-19

编辑：Bharath 2015-3-19

在 MATLAB Online 中打开

Thanks a ton! "That's true filter design is an art and you are a masterclass artist." Your answer was very informative. I just need to clarify few other points. kindly bear with me.

Once I've [sos,g] values calculated....all I need to do is filter my signal using the

y = filtfilt(sos, g, s); % where s is my raw signal?

Is the impulse response and transfer function only used to plot the filter response right?
How are the Rp and Rs values varied?
I also tried varying the values for 'Fs1' and 'Fs2' sometimes it gives me an error saying they can't have overlapped values.
After applying this filter to my raw signals X and Y. I seem to get distorted shapes. Although the filter impulse response shows its working fine. What do you think the problem could be with? I'm attaching the image for your reference. As you can see I still don't understand if the filter is really doing the job.

Few people commented saying my sampling is a over kill for the frequency I'm filtering into. what is your suggestion about this?

Sorry for many questions. I need to clear my mind when I wanted to learn something in detail.

Thanks for your answers and patience.

Star Strider 2015-3-19

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My pleasure! I very much appreciate the compliment.

1. Correct. My ‘s’ variable is your raw signal.

2. Is the impulse response and transfer function only used to plot the filter response right?

Correct. I always test my filter designs first to be sure the result is close to what I want.

3. How are the Rp and Rs values varied?

They can be any reasonable values. They are really not applicable to a Butterworth design, because it has relatively flat passband and stopband characteristics, but buttord wants them. I always choose 1 dB for the passband ripple, but the stopband ripple can be much higher. I choose 10 dB, but 30 dB is reasonable.

4. I also tried varying the values for 'Fs1' and 'Fs2' sometimes it gives me an error saying they can't have overlapped values.

The ‘Fs1’ value always has to be strictly less than the ‘Fs2’ value. If they are too close together, the filter becomes unstable, and even a second-order-section representation will not produce a usable filter. I would narrow the stopband frequencies instead, if you want a narrower passband. Using a factor of 0.9 produces a stable filter with a much narrower passband:

Fs1 = Fc1*0.9;
Fs2 = Fc2/0.9;

See if that improves your filtered signal.

5. The problem is likely that the filter passband is too wide. See #4 for my suggestion on narrowing it. Experiment with higher numbers as well, perhaps up to 0.95, until you get a filter that is stable and that also gives you a filtered signal that is much closer to what you want. If you have a signal that is very close to the one you want and you cannot successfully filter it with the passband filter, consider notching it out with cascaded narrowband bandstop (‘notch’) filter. You would design it the same way as your bandpass filter, except to specify 'stop' as your filter type. (See the documentation for Bandstop Butterworth Filter for details.) The two-filter implementation for this would be to filter your raw signal with your bandpass filter first, then filter the output of that filter procedure with your bandstop filter to get your final filtered signal. You can also experiment with a Chebyshev Type II filter design. It has much steeper stopband attenuation than a Butterworth filter.

Don’t apologise for asking questions! That is what MATLAB Answers is for. I just have to have the time to answer them in as much detail as I feel they need. Right now, I do.

Star Strider 2015-3-19

My pleasure.

Your first two items don’t seem to require a reply.

3. Unfortunately, freqz has problems with your filters because of the high sampling frequency and low passband frequency. Otherwise I would suggest it. Using my version is probably better for your application. (In that instance, ‘y’ is the last output of all your filter operations. The rest of my transfer function calculation and plotting code is unchanged.)

4. It could, but that is simply a problem in any signal processing application. If you want a ‘clean’ signal with an extremely narrow passband, you have to live with such tradeoffs.

5. Generally, using the fewest filters possible is the best solution. Usually you can do everything you need to with at most two or three cascaded filters. (I generally don’t use windows with filters because they usually don’t add anything.) One possible approach to produce an extremely narrowband filter is to cascade two Chebyshev Type II filters — one lowpass and one highpass — and adjust the overlap. This is not the ideal solution, but it may be the only workable option if you have an extremely narrow band of frequencies you want to filter. It should be stable, where an extremely narrow bandpass filter might not be.

Bharath 2015-3-19

在 MATLAB Online 中打开

I forgot to ask about this error which I get when using filtfilt command.

Warning: The first two inputs are interpreted as an SOS matrix and G vector. Use column vectors if you meant to specify a
transfer function. 
> In filtfilt>getCoeffsAndInitialConditions (line 126)
  In filtfilt (line 97)
  In Test_run_chin_half_spectrum_2 (line 61) 
Warning: The first two inputs are interpreted as an SOS matrix and G vector. Use column vectors if you meant to specify
a transfer function. 
> In filtfilt>getCoeffsAndInitialConditions (line 126)
  In filtfilt (line 97)
  In Test_run_chin_half_spectrum_2 (line 62) 
Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND =  5.552883e-17. 
> In filtfilt>getCoeffsAndInitialConditions (line 200)
  In filtfilt (line 97)
  In Test_run_chin_half_spectrum_2 (line 67) 
Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND =  5.552883e-17. 
> In filtfilt>getCoeffsAndInitialConditions (line 200)
  In filtfilt (line 97)
  In Test_run_chin_half_spectrum_2 (line 69) 
>>

Is this something to be ignored? Also in other context like butterworth pass filter can we use filtfilt instead of filter command?

Star Strider 2015-5-6

‘...same data set to extract another freqeuncy.’

That’s likely the problem. You have to design and test an entirely new filter to extract the second frequency. Filter design is heuristic and can be challenging, especially with the narrow passband filters you seem to need. You can use the essence of my code to help you with your design, but you have to determine the passband and stability of the second-order-section (SOS) implementation before you use it.

It is important to note that your wanting to design a filter to do a particular task does not mean that design is feasible. It might actually be easier to design and implement your filters in hardware and recording those filtered signals separately (along with your unfiltered signal) than to do the processing digitally. Narrowband hardware filters are easier to design and implement (at least in my experience) than narrowband digital filters.

Star Strider 2021-2-18

在 MATLAB Online 中打开

Harry — Were I to design that same filter today, I would do it differently, and use an elliptic filter instead of a Butterworth design:

% DEFINE FILTER PARAMETERS 
Fs  = 2E+6;
Fn  = Fs/2;
Fc1 =  630;
Fc2 =  640;
Fs1 = Fc1*0.8;
Fs2 = Fc2/0.8;
Rp  =    1;
Rs  =   10;
% DESIGN FILTER
[n,Wp]  = ellipord([Fc1 Fc2]/Fn, [Fs1 Fs2]/Fn, Rp, Rs);
[z,p,k] = ellip(n,Rp,Rs,Wp);
[sos,g] = zp2sos(z,p,k);
figure
freqz(sos, 2^14, Fs) 

The design procedure apparently appears to be important in producing the correct filter result, since even the Butterworth design is significantly impproved:

% DESIGN FILTER
[n,Wn]  = buttord([Fc1 Fc2]/Fn, [Fs1 Fs2]/Fn, Rp, Rs);
[z,p,k] = butter(n,Wn);
[sos,g] = zp2sos(z,p,k);
figure
freqz(sos, 2^14, Fs)

I have built a number of hardware filters, however they were all op-amp active filters with discrete components (and none recently), not FPGA filters.

See if this approach produces a better result.

Please post back with a Comment reporting the result you get, and how it works in the FPGA implementation.

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