why 50hz can't be removed compeletely?

Dear, I am try to remove the 50 hz. It works very well if I plot the result using amplitude. However when when I plot the result using 10*log10(amp), the 50hz will be still there with large amplitude.

%% build filter
N     = 20;    % Order
F3dB1 = 49;  % First
F3dB2 = 51;  % Second
Fs    = 1000;  % Sampling Frequency
d = designfilt('bandstopiir','FilterOrder',2, ...
               'HalfPowerFrequency1',49,'HalfPowerFrequency2',51, ...
               'DesignMethod','butter','SampleRate',Fs);

%% test data
fs = 1000;
t = 0:1/fs:5-1/fs;
x = cos(2*pi*50*t)+cos(2*pi*100*t);
%plot(t,x)
%plotChannelSpectral(x);
[psd,f] = pwelch(x,500,200,500,1000);
plot(f,10*log10(psd));
plot(f,(psd));
fdata=filtfilt(d,double(x));
[psd2,f2] = pwelch(fdata,500,200,500,1000);
figure;
plot(f2,psd2); % 50hz almost gone in this plot
plot(f,10*log10(psd2)); % however I still can see large 50hz in this plot, even I re-run the filtfilt 20 times, I still can see relative large amplitude around 50hz, though there is a groove at 50hz.

Does it means that I should be satisfied as long as I can't see the 50hz in amplitude? And there is no way to totally eliminate the large value under 10*log10(amplitude)?

Thanks in advance.

Answer 1

[Mathieu NOE]

hello again

I like simple solutions - see demo code below adapted to your case

I hope 200 dB attenuation can be considered as "complete washed out" !!

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% FFT parameters
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
NFFT = 1024*4;    % 
NOVERLAP = round(0.75*NFFT);
w = hanning(NFFT);     % Hanning window / Use the HANN function to get a Hanning window which has the first and last zero-weighted samples.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% load signal
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%% test data
Fs = 1000;
t = 0:1/Fs:10-1/Fs;
signal = cos(2*pi*50*t)+cos(2*pi*100*t);

 
%% notch filter section %%%%%%
% H(s) = (s^2 + 1) / (s^2 + s/Q + 1)

fc = 50;       % notch freq
wc = 2*pi*fc;
Q = 2;         % adjust Q factor for wider (low Q) / narrower (high Q) notch
                % at f = fc the filter has gain = 0

w0 = 2*pi*fc/Fs;
alpha = sin(w0)/(2*Q);

            b0 =   1;
            b1 =  -2*cos(w0);
            b2 =   1;
            a0 =   1 + alpha;
            a1 =  -2*cos(w0);
            a2 =   1 - alpha;
            
% analog notch (for info)
num1=[1/wc^2 0 1];
den1=[1/wc^2 1/(wc*Q) 1];

% digital notch (for info)
num1z=[b0 b1 b2];
den1z=[a0 a1 a2];

freq = linspace(fc-1,fc+1,200);
[g1,p1] = bode(num1,den1,2*pi*freq);
g1db = 20*log10(g1);

[gd1,pd1] = dbode(num1z,den1z,1/Fs,2*pi*freq);
gd1db = 20*log10(gd1);

figure(1);
plot(freq,g1db,'b',freq,gd1db,'+r');
title(' Notch: H(s) = (s^2 + 1) / (s^2 + s/Q + 1)');
legend('analog','digital');
xlabel('Fréquence (Hz)');
ylabel(' dB')
 
% now let's filter the signal 
signal_filtered = filtfilt(num1z,den1z,signal);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% display : averaged FFT spectrum before / after notch filter
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

[sensor_spectrum, freq] = pwelch(signal,w,NOVERLAP,NFFT,Fs);
sensor_spectrum_dB = 20*log10(sensor_spectrum);% convert to dB scale (ref = 1)


[sensor_spectrum_filtered, freq] = pwelch(signal_filtered,w,NOVERLAP,NFFT,Fs);
sensor_spectrum_filtered_dB = 20*log10(sensor_spectrum_filtered);% convert to dB scale (ref = 1)


figure(2),semilogx(freq,sensor_spectrum_dB,'b',freq,sensor_spectrum_filtered_dB,'r');grid
title(['Averaged FFT Spectrum  / Fs = ' num2str(Fs) ' Hz / Delta f = ' num2str(freq(2)-freq(1)) ' Hz ']);
legend('before notch filter','after notch filter');
xlabel('Frequency (Hz)');ylabel(' dB')

Answer 2

[Mathieu NOE]

hello

2 gifts for you in the same day (you lucky ! )

• a code for spectral analysis and time / frequency analysis
• a the end , some code for a better notch filter digital implementation ; your notch is not effective enough

hope it helps

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% FFT parameters
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
NFFT = 1024;    % 
NOVERLAP = round(0.75*NFFT);
w = hanning(NFFT);     % Hanning window / Use the HANN function to get a Hanning window which has the first and last zero-weighted samples.

% spectrogram dB scale
spectrogram_dB_scale = 80;  % dB range scale (means , the lowest displayed level is XX dB below the max level)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% options 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% if you are dealing with acoustics, you may wish to have A weighted
% spectrums 

% option_w = 0 : linear spectrum (no weighting dB (L) )
% option_w = 1 : A weighted spectrum (dB (A) )
option_w = 0;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% load signal
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%[data,Fs]=wavread('Approach_Gear_Drop_Aft Ctr.wav ');  %(older matlab)
% or 
[data,Fs]=audioread('myWAVaudiofile.wav'); %(newer matlab)

channel = 1;

signal = data(:,channel);
samples = length(signal);
dt = 1/Fs;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% display 1 : averaged FFT spectrum
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

[sensor_spectrum, freq] = pwelch(signal,w,NOVERLAP,NFFT,Fs);

% convert to dB scale (ref = 1)
sensor_spectrum_dB = 20*log10(sensor_spectrum);

% apply A weigthing if needed
if option_w == 1
    pondA_dB = pondA_function(freq);
    sensor_spectrum_dB = sensor_spectrum_dB+pondA_dB;
    my_ylabel = ('Amplitude (dB (A))');
else
    my_ylabel = ('Amplitude (dB (L))');
end


figure(1),semilogx(freq,sensor_spectrum_dB);grid
title(['Averaged FFT Spectrum  / Fs = ' num2str(Fs) ' Hz / Delta f = ' num2str(freq(2)-freq(1)) ' Hz ']);
xlabel('Frequency (Hz)');ylabel(my_ylabel);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% display 2 : time / frequency analysis : spectrogram demo
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

[sg,fsg,tsg] = specgram(signal,NFFT,Fs,w,NOVERLAP);  

% FFT normalisation and conversion amplitude from linear to dB (peak)
sg_dBpeak = 20*log10(abs(sg))+20*log10(2/length(fsg));     % NB : X=fft(x.*hanning(N))*4/N; % hanning only

% apply A weigthing if needed
if option_w == 1
    pondA_dB = pondA_function(fsg);
    sg_dBpeak = sg_dBpeak+(pondA_dB*ones(1,size(sg_dBpeak,2)));
    my_title = ('Spectrogram (dB (A))');
else
    my_title = ('Spectrogram (dB (L))');
end

% saturation of the dB range : 
% saturation_dB = 60;  % dB range scale (means , the lowest displayed level is XX dB below the max level)
min_disp_dB = round(max(max(sg_dBpeak))) - spectrogram_dB_scale;
sg_dBpeak(sg_dBpeak<min_disp_dB) = min_disp_dB;

% plots spectrogram
figure(2);
imagesc(tsg,fsg,sg_dBpeak);colormap('jet');
axis('xy');colorbar('vert');grid
title([my_title ' / Fs = ' num2str(Fs) ' Hz / Delta f = ' num2str(fsg(2)-fsg(1)) ' Hz ']);
xlabel('Time (s)');ylabel('Frequency (Hz)');


function pondA_dB = pondA_function(f)
	% dB (A) weighting curve
	n = ((12200^2*f.^4)./((f.^2+20.6^2).*(f.^2+12200^2).*sqrt(f.^2+107.7^2).*sqrt(f.^2+737.9^2)));
	r = ((12200^2*1000.^4)./((1000.^2+20.6^2).*(1000.^2+12200^2).*sqrt(1000.^2+107.7^2).*sqrt(1000.^2+737.9^2))) * ones(size(f));
	pondA = n./r;
	pondA_dB = 20*log10(pondA(:));
end

% %%%%%%%%%%% other infos %%%%%%%%%%%%%%%%%%%%%%%%
% 
% 
% %% notch filter section %%%%%%
% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % notch :      H(s) = (s^2 + 1) / (s^2 + s/Q + 1)
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% fc = 50;       % notch freq
% 
% Q = 100;         % the higher the Q factor, the deeper the notch
% 
% %%%%%%
% w0 = 2*pi*fc;
% 
% alpha = sin(w0)/(2*Q);
% 
% 
%             b0 =   1;
%             b1 =  -2*cos(w0);
%             b2 =   1;
%             a0 =   1 + alpha;
%             a1 =  -2*cos(w0);
%             a2 =   1 - alpha;
% % 
% num1z=[b0 b1 b2];
% den1z=[a0 a1 a2];
% 
% % now let's filter the signal 
% signal_filtered = filter(num1z,den1z,signal);

[Xiaolong wu]

I think so as well, my filter just not powerful enough. I am not sure how to design a powerful notch filter with designfilt.