What is the "-20dB" bandwidth of a signal and how to calculate it in MATLAB

I searched to find the answer, I could find the 3dB bandwidth which is half power bandwidth and there is function to find that in MATLAB: bw = powerbw(x)

But I dont know anything about -20dB bandwidth, how to calculate it and the applications of using this one over 3dB.

Any helps appreciated,

Answer 1

[William Rose]

@2NOR_Kh,

@Star Strider has given an excellent answer that explains, with code and figures, how to find the bandwidth of a filter, or transfer function. (I uset those terms interchangeably.)

The bandwidth of a signal is a closely related concept to the bandwidth of a filter or transfer funciton. You compute the power spectrum of a signal. YOU find the peak power in the power spectrum. The -20 dB bandwidth is the range of frequencies, around the frequency of peak power, in which the signal's power is not more than 20 dB less than the peak power.

The explanation and diagrams in powerbw() are good. They mostly show -3 dB bandwidth in the examples, since - 3 dB is the default, but you can use the r argument (r=20) to get the (-)20 dB bandwidth.

[William Rose]

@2NOR_Kh,

[edit: fix chip -> chirp]

Example:

Make a signal wich is a chirp plus some noise. Find the bandwith to -3 dB, -10 dB, -20 dB.

N=2000; fs=1000;            %number of samples, sampling rate (Hz)
t=(1:N)/fs;                 %vector of times
x=chirp(t,50,t(end),150);   %chirp signal: 50 Hz at start to 150 Hz at end
x=x+randn(size(x))/100;     %add noise to x
dB1=3;                      %find the bandwidth to (-)dB1
dB2=10;                     %find the bandwidth to (-)dB1
dB3=20;                     %find the bandwidth to (-)dB1
bw1=powerbw(x,fs,[],dB1);
bw2=powerbw(x,fs,[],dB2);
bw3=powerbw(x,fs,[],dB3);
fprintf('Bandwidth to -(%d,%d,%d) dB = %.1f, %.1f, %.1f Hz.\n',...
    dB1,dB2,dB3,bw1,bw2,bw3)

Bandwidth to -(3,10,20) dB = 94.6, 102.7, 115.9 Hz.

As expected, the bandwidth is wider when you go to -20 dB than when you use -3 dB, and -10 dB bandwidth is in between.

The exact bw values you get may vary slightly, because randn() returns different random values with each call. Try it. Good luck.

[2NOR_Kh]

I tried two different methods:1- using pwelch to find the power spectrum of my signal and then finding -20dB bandwidth based on that, which the result is:

so the -20 bw is around 6Mhz.

in the 2nd method I used: bw1=powerbw(rf(:,220,1),fs,[],dB1);

which rf(:,220,1) is my signal, fs the sampling freq. and dB1=20 here. This one gave me:

Bandwidth to -(20) dB = 91171.5 Hz

Which is way different than the value in the attached image. However, the 6MHz makes more sense, since could see the same bandwidth for the fft of the signal. Now, I am confused why powerbw function is not giving me the correct answer.

[William Rose]

@2NOR_Kh,

Do you want to share a sample signal for which methods 1 and 2 give different answers?

Help for powerbw() says it uses periodogram and a rectangular window to compute the spectrum. I assume that means a rectangular window whose length equals the full length signal. If you are using pwelch() with default options (which means hamming window and multiple segments), then your power spectrum is probaby smoother than the one computed by powerbw(). Perhaps that explains the difference.

In the example below, I compute bandwidth to -20 dB using method1=powerbw() and using method 2 = periodogram() plus @Star Strider's method. I made some tweaks in @Star Strider's method, because they assumed the peak value is 0 dB, which makes sense for a transfer function, but is generally not true for a signal. Also, my noisy signal sometimes has more than two crossings of the -20 dB line. @Star Strider's method does not give a correct answer in this case, but powerbw() gives a reasonable answer. Methods 1 and 2 give very similar answers. The green vertical lines on the plot are at the frequencies found by method 1. The red vertical dashed lines on the plot are at the frequencies found by method 2. The green and red vertical lines are basically superimposed, because the methods are returning the same answers.

N=2000; fs=1000;            %number of samples, sampling rate (Hz)
t=(1:N)/fs;                 %vector of times
x=chirp(t,50,t(end),150);   %chirp signal: 50 Hz at start to 150 Hz at end
x=x+randn(size(x))/50;      %add noise to x
dB1=20;                     %find the bandwidth to (-)dB1
[bw1,flo,fhi]=powerbw(x,fs,[],dB1);     %Method 1 bandwidth
[pxx,f]=periodogram(x,ones(length(x),1),length(x),fs,'power');
pxxdB=10*log10(pxx);        %power spectrum in dB
pxxdBmax=max(pxxdB);
idx = find(diff(sign(pxxdB-pxxdBmax+dB1)));             % Approximate Index (Or Indices) Of -20 dB Point
for k = 1:numel(idx)
    ixr = max(1,idx(k)-1):min(numel(f),idx(k)+1);       % Index Range For Interpolation
    fq(k) = interp1(pxxdB(ixr)-pxxdBmax,f(ixr),-dB1);   % Precise Value Of Frequency For Each Value Of Magnitude
end
bw2=fq(2)-fq(1);            %Method 2 bandwidth

%% Plot power spectrum with lines at max power and at max power-20dB
plot(f,pxxdB,'-b.')
xlabel('Frequency (Hz)'); ylabel('Power (dB)'); grid on; hold on
yline(max(pxxdB),'--r'); yline(max(pxxdB)-20,'--r')
xline(flo,'g'); xline(fhi,'g')
for i=1:numel(fq),xline(fq(i),'--r');end

%% Display results on console
fprintf('Bandwidth to -%d dB = %.1f (method 1), %.1f (method 2) Hz.\n',...
    dB1,bw1,bw2)

Bandwidth to -20 dB = 116.5 (method 1), 116.5 (method 2) Hz.

I hope that helps.

[2NOR_Kh]

Thank you @William Rose, and @Star Strider I appreciate your time and help.

I should use pwelch and finding the -20dB based on that but I also wanyed to learn how to work with powerbw function and then realized the answer is so much different than what I know about the bandwidth (I also find the bandwidth by using fft).

I attached my code and data, I couldn't attach my whole data cause it's more than 5MB. I just attached one line, that is enough.

fs=40e6;
dB1=20;%find the bandwidth to (-20)dB1
figure,pwelch(rf(:,220,1),128,[],[],fs);%128 is the window length, 40e6 is the sampling frequency of the signal
%FE=fftshift((fft(rf(:,220,1))));
bw1=powerbw(rf(:,220,1),fs,[],dB1);%why this is not giving me the right answer
fprintf('Bandwidth to -(%d) dB = %.1f Hz.\n',dB1,bw1);

[William Rose]

@2NOR_Kh,

The code below shows why powerbw gives an unsatisfactory answer for some signals. As I said earlier, powerbw() computes the spectrum with periodogram(), a rectangular window, and no segment averaging. This means the spectrum is not smoothed. Therefore powerbw() finds frequencies that are 20dB down, very close to the frequency of max power. Thus the bandwith estimate by powerbw() is very narrow when the spectrum is noisy.

x=importdata('rf_line220_f1.mat');  %read data file
fs=40e6;                            %sampling rate
dB1=20;                             %bandwidth limit
%% Compute & plot spectrum with pwelch & winlen=128
figure;
pwelch(x,128,[],[],fs);             %plot pwelch spectrum

%% Compute bandwidth with powerbw()
[bw1,flo,fhi]=powerbw(x,fs,[],dB1);
%% Compute spectrum with periodogram
[pxx,f]=periodogram(x,ones(length(x),1),length(x),fs,'power');
pxxdB=10*log10(pxx);        %power spectrum in dB
%% Plot power spectrum 
%% Add lines at max power, max power-20dB, flo, fhi
figure; subplot(211)
plot(f,pxxdB,'-b.')
xlabel('Frequency (Hz)'); ylabel('Power (dB)'); grid on; hold on
yline(max(pxxdB),'--r'); yline(max(pxxdB)-20,'--r')
xline(flo,'r'); xline(fhi,'r')
%% Plot same spectrum with zoomed in frequency scale
subplot(212)
plot(f,pxxdB,'-b.')
xlabel('Frequency (Hz)'); ylabel('Power (dB)'); grid on; hold on
yline(max(pxxdB),'--r'); yline(max(pxxdB)-20,'--r')
xline(flo,'r'); xline(fhi,'r')
xlim([flo-bw1,fhi+bw1])     %specify narrow frequency range for plot

%% Display bandwith from powerbw() on console
fprintf('powerbw: bandwidth %.2e to %.2e = %.2e Hz.\n',...
    flo,fhi,bw1);

powerbw: bandwidth 4.02e+06 to 4.11e+06 = 9.12e+04 Hz.

Note the difference between the smooth spectrum computed with pwelch (top figure) and a window width of 128 (and, by default, a Hamming window) and the noisy spectrum computed by periodogram with no segment averaging and a rectangular window.

[William Rose]

@2NOR_Kh,

Another suggestion for getting reasonable results from powerbw() is to use the spectrum computed the way you want to compute it:

x=importdata('rf_line220_f1.mat');  %read data file
fs=40e6;                            %sampling rate
dB1=20;                             %bandwidth limit
N=128;                              %window width for spectrum computation
[pxx,f]=pwelch(x,N,[],[],fs);       %compute power spectrum
[bw1,flo,fhi]=powerbw(pxx,f,[],dB1); %compute bandwidth with powerbw()
pxxdB=pow2db(pxx);
plot(f,pxxdB,'-b.')                 %plot power spectrum 
xlabel('Frequency (Hz)'); ylabel('Power (dB)'); grid on; hold on
yline(max(pxxdB),'--r'); yline(max(pxxdB)-20,'--r')
xline(flo,'--r'); xline(fhi,'--r')

The bandwidth range, when we use powerbw() this way, looks reasonable.