Use Matlab Function pwelch to Find Power Spectral Density – or Do It Yourself
In my last post, we saw that finding the spectrum of a signal requires several steps beyond computing the discrete Fourier transform (DFT). These include windowing the signal, taking the magnitude-squared of the DFT, and computing the vector of frequencies. The Matlab function pwelch  performs all these steps, and it also has the option to use DFT averaging to compute the so-called Welch power spectral density estimate [3,4].
In this article, I’ll present some examples to show how to use pwelch. You can also “do it yourself”, i.e. compute spectra using the Matlab fft or other fft function. As examples, the appendix provides two demonstration mfiles; one computes the spectrum without DFT averaging, and the other computes the spectrum with DFT averaging. (Note: I have also created a simplified Matlab PSD function based on pwelch: see this post.)
We’ll use this form of the function call for pwelch:
The input arguments are:
x input signal vector
window window vector of length nfft
noverlap number of overlapped samples (used for DFT averaging)
nfft number of points in DFT
fs sample rate in Hz
For this article, we’ll assume x is real. Let length(x) = N. If DFT averaging is not desired, you set nfft equal to N and pwelch takes a single DFT of the windowed vector x. If DFT averaging is desired, you set nfft to a fraction of N and pwelch uses windowed segments of x of length nfft as inputs to the DFT. The |X(k)|2 of the multiple DFT’s are then averaged.
The output arguments are:
pxx power spectral density vector, W/Hz
f vector of frequency values from 0 to fs/2, Hz
The length of the output vectors is nfft/2 + 1 when nfft is even. For example, if nfft= 1024, pxx and f contain 513 samples. pxx has units of W/Hz when x has units of volts and load resistance is one ohm.
The following examples use a signal x consisting of a sine wave plus Gaussian noise. Here is the Matlab code to generate x, where sine frequency is 500 Hz and sample rate is 4000 Hz:
fs= 4000; % Hz sample rate Ts= 1/fs; f0= 500; % Hz sine frequency A= sqrt(2); % V sine amplitude for P= 1 W into 1 ohm. N= 1024; % number of time samples n= 0:N-1; % time index x= A*sin(2*pi*f0*n*Ts) + .1*randn(1,N); % 1 W sinewave + noise
The power of the sine wave into a 1-ohm load is A2/2 = 1 watt. In the following, examples 1 and 2 compute a single DFT, while example 3 computes multiple DFT’s and averages them. Example 4 calculates C/N0 for the spectrum of Example 3.
Example 1. Spectrum in dBW/Hz
We’ll start out with a rectangular window. Note the sinewave frequency of 500 Hz is at the center of a DFT bin to prevent spectral leakage . Here is the code to compute the spectrum. Since we are not using DFT averaging, we set noverlap= 0:
nfft= N; window= rectwin(nfft); [pxx,f]= pwelch(x,window,0,nfft,fs); % W/Hz power spectral density PdB_Hz= 10*log10(pxx); % dBW/Hz
The resulting spectrum is plotted in Figure 1, which shows the component at 500 Hz with amplitude of roughly -6 dBW/Hz. How does this relate to the actual power of the sinewave for a 1-ohm load resistance? To compute the power of the spectral component at 500 Hz, we need to convert the amplitude in dBW/Hz to dBW/bin as follows. First, convert W/Hz to W/bin:
Pbin (W/bin) = pxx (W/Hz) * fs/nfft (Hz/bin) (1)
Then, we have in dB:
PdB_bin = 10*log10(pxx*fs/nfft) dBW/bin (2)
Or PdB_bin = PdB_Hz + 10*log10(fs/nfft) dBW/bin
If we set the noise amplitude to zero, we get PdB_Hz = -5.9176 dB/Hz and
PdB_bin= -5.9176 dBW/Hz + 10log10(4000/1024) = 0 dBW = 1 watt
So we are reassured that the amplitude of the spectrum is correct. This motivates us to try the next example, which displays the spectrum in dBW/bin.
Figure 1. Spectrum in dBW/Hz
Example 2. Spectrum in dBW/bin
To find the spectrum in dBW/bin, the code is the same as Example 1, except we use Equation 2 to find PdB_bin:
nfft= N; window= rectwin(nfft); [pxx,f]= pwelch(x,window,0,nfft,fs); % W/Hz power spectral density PdB_bin= 10*log10(pxx*fs/nfft); % dBW/bin
The spectrum is shown in the top plot of Figure 2. As expected, the component at 500 Hz has power of 0 dBW. Note that the amplitude of a sine’s spectral component in dBW/bin is constant vs nfft, while the amplitude in dBW/Hz varies vs. nfft. If we now replace the rectangular window with a hanning window:
We get the spectrum in the bottom plot of Figure 2. The peak power is now less than 0 dBW, because the power is spread over several frequency samples of the window. Nevertheless, this is a more useful window than the rectangular window, which has bad spectral leakage when f0 is not at the center of a bin.
Figure 2. Spectra in dBW/bin Top: rectangular window
Bottom: hanning windowExample 3. DFT averaging
For this example, fs, Ts, f0, A, and nfft are the same as for the previous examples, but we use N= 8*nfft time samples of x. pwelch takes the DFT of Navg = 15 overlapping segments of x, each of length nfft, then averages the |X(k)|2 of the DFT’s. The segments of x are shown in Figure 3, where we have set the number of overlapping samples to nfft/2 = 512.
In general, if there are an integer number of segments that cover all samples of N, we have :
(Navg – 1)*D + nfft = N (3)
Where D = nfft – noverlap. For our case, with D = nfft/2 and N/nfft = 8, we have
Navg = 2*N/nfft -1 = 15.
For a given number of time samples N, using overlapping segments lets us increase Navg compared with no overlapping. In this case, overlapping of 50% increases Navg from 8 to 15. Here is the Matlab code to compute the spectrum:
nfft= 1024; N= nfft*8; % number of samples in signal n= 0:N-1; x= A*sin(2*pi*f0*n*Ts) + .1*randn(1,N); % 1 W sinewave + noise noverlap= nfft/2; % number of overlapping time samples window= hanning(nfft); [pxx,f]= pwelch(x,window,noverlap,nfft,fs); % W/Hz PSD PdB_bin= 10*log10(pxx*fs/nfft); % dBW/bin
The resulting spectrum is shown in Figure 4. Compared to the no-averaging case in Figure 2, there is less variation in the noise floor. DFT averaging reduces the variance σ2 of the noise spectrum by a factor of Navg, as long as noverlap is not greater than nfft/2 .
Figure 3. Overlapping segments of signal x for N = 8*nfft and noverlap = nfft/2 (50%).
Figure 4. Spectrum with Navg = 15 and noverlap= nfft/2= 512.
Example 4. Calculation of C/N0
Reducing the noise variance by DFT averaging makes calculation of the noise level and thus carrier-to-noise ratio (C/N) more accurate. Let’s calculate C/N0 for the signal of Example 3, where N0 is the noise power in a 1 Hz bandwidth.
First, we’ll find the power of the sinewave (the “C” in C/N0) by performing a peak search on the dBW/bin spectrum, then adding the spectral components near the peak to get the total power. This is necessary because windowing has spread the power over several frequency samples.
y= max(PdB_bin); % peak of spectrum k_peak= find(PdB_bin==y); % index of peak Psum= fs/nfft * sum(pxx(k_peak-2:k_peak+2)); % W sum power of 5 samples C= 10*log10(Psum) % dBW carrier power
The peak power of 0 dBW is shown by the marker in the left plot of Figure 4. Now we’ll find N0 from the dBW/Hz spectrum. Choosing f = 800 Hz, we have:
PdB_Hz= 10*log10(pxx); % dBW/Hz f1= 800; % Hz k1= round(f1/fs *nfft); % index of f1 No= PdB_Hz(k1) % dBW/Hz Noise density at f1
The N0 marker is shown in the right plot of Figure 4. Figure 5 combines the dBW/bin and dBW/Hz plots in one graph. C/N0 is just:
CtoNo= C - No; % dB (1 Hz)
Figure 5. Left: Spectrum in dBW/bin with marker showing the power of the sinewave.
Right: Spectrum in dBW/Hz with marker at 800 Hz showing N0.
Figure 6. Spectrum in dBW/bin (blue) and dBW/Hz (grey).
Appendix Matlab Code to compute spectra without using pwelch
These examples use the Malab fft function to compute spectra. See my earlier post  for derivations of the formulas for power/bin and normalized window function.
Demonstrate Spectrum Computation/ No Averaging
%spectrum_demo.m 1/3/19 Neil Robertson % Use FFT to find spectrum of sine + noise in units of dBW/bin and dBW/Hz. fs= 4000; % Hz sample frequency Ts= 1/fs; f0= 500; % Hz sine frequency A= sqrt(2 % V sine amplitude for P= 1 W into 1 ohm. N= 1024; % number of time samples nfft= N; % number of frequency samples n= 0:N-1; % time index % x= A*sin(2*pi*f0*n*Ts) + .1*randn(1,N); % 1 W sinewave + noise w= hanning(N); % window function window= w.*sqrt(N/sum(w.^2)); % normalize window for P= 1 xw= x.*window'; % apply window to x % X= fft(xw,nfft); % DFT X= X(1:nfft/2); % retain samples from 0 to fs/2 magsq= real(X).^2 + imag(X).^2; % DFT magnitude squared P_bin= 2/nfft.^2 * magsq; % W/bin power spectrum into 1 ohm P_Hz= P_bin*nfft/fs; % W/Hz power spectrum % PdB_bin= 10*log10(P_bin); % dBW/bin PdB_Hz= 10*log10(P_Hz); % dBW/Hz % k= 0:nfft/2-1; % frequency index f= k*fs/nfft; % Hz frequency vector % % plot(f,PdB_bin),grid axis([0 fs/2 -60 10]) xlabel('Hz'),ylabel('dBW/bin') title('Spectrum with amplitude units of dBW/bin') figure y= max(PdB_Hz); plot(f,PdB_Hz),grid axis([0 fs/2 y-60 y+10]) xlabel('Hz'),ylabel('dBW/Hz') title('Spectrum with amplitude units of dBW/Hz')
Demonstrate Spectrum Computation with Averaging
Suppose we have segmented a signal x into four segments of length nfft, that could be overlapped. If we window each segment and take its DFT, we have four DFT’s as shown in Table A.1. Here, we have defined M= nfft and we have numbered the FFT bins from 1 to M (instead of 0 to M - 1).
Table A.1 DFT’s of four segments of the signal x.
Now we form the magnitude-squared of each bin of the four DFT’s, as shown in Table A.2 If we then sum the magnitude-square values of each bin over the four DFT’s and divide by 4, we obtain the average magnitude-squared value, as shown in the right column. Finally, we scale the magnitude-squared value to obtain the averaged power per bin  or averaged power per Hz:
Pbin(k) = 2/M2 * sum(|Xk|2)/4 W/bin (A.1)
PHz(k)= Pbin(k)*M/fs W/Hz (A.2)
PHz(k)= 2/(M*fs) * sum(|Xk|2)/4 W/Hz (A.3)
PHz(k) is the Welch power spectral density estimate.
Table A.2 Magnitude-square values of Four DFT’s, and their average (right column).
The following example m-file uses Navg = 8, summing 8 values of each |X(k)|2 and dividing by 8. For simplicity, noverlap = 0 samples. Another option for averaging, useful for the case where DFT’s are taken continuously, is exponential averaging, which basically runs the |X(k)|2 through a first-order IIR filter .
%spectrum_demo_avg 1/3/19 Neil Robertson % Perform DFT averaging on signal = sine + noise % noverlap = 0 % Given signal x(1:8192), % % |x(1) x(1025) . . . . x(7169)| % |x(2) x(1026) . . . . x(7170)| % xMatrix= | . . . | % | . . . | % |x(1024) x(2048) . . . . x(8192)| % % fs= 4000; % Hz sample frequency Ts= 1/fs; f0= 500; % Hz sine frequency A= sqrt(2 % V sine amplitude for P= 1 W into 1 ohm nfft= 1024; % number of frequency samples Navg= 8; % number of DFT's to be averaged N= nfft*Navg; % number of time samples n= 0:N-1; % time index x= A*sin(2*pi*f0*n*Ts) + .1*randn(1,N); % 1 W sinewave + noise w= hanning(nfft); % window function window= w.*sqrt(nfft/sum(w.^2)); % normalize window for P= 1 W % % Create matrix xMatrix with Navg columns, % each column a segment of x of length nfft xMatrix= reshape(x,nfft,Navg); magsq_sum= zeros(nfft/2); for i= 1:Navg x_column= xMatrix(:,i); xw= x_column.*window; % apply window of length nfft X= fft(xw); % DFT X= X(1:nfft/2); % retain samples from 0 to fs/2 magsq= real(X).^2 + imag(X).^2; % DFT magnitude squared magsq_sum= magsq_sum + magsq; % sum of DFT mag squared end mag_sq_avg= magsq_sum/Navg; % average of DFT mag squared P_bin= 2/nfft.^2 *mag_sq_avg; % W/bin power spectrum P_Hz= P_bin*nfft/fs; % W/Hz power spectrum PdB_bin= 10*log10(P_bin); % dBW/bin PdB_Hz= 10*log10(P_Hz); % dBW/Hz k= 0:nfft/2 -1; % frequency index f= k*fs/nfft; % Hz frequency vector % plot(f,PdB_bin),grid axis([0 fs/2 -60 10]) xlabel('Hz'),ylabel('dBW/bin') title('Averaged Spectrum with amplitude units of dBW/bin') figure plot(f,PdB_Hz),grid axis([0 fs/2 -60 10]) xlabel('Hz'),ylabel('dBW/Hz') title('Averaged Spectrum with amplitude units of dBW/Hz')
- Robertson, Neil, “Evaluate Window Functions for the Discrete Fourier Transform” https://www.dsprelated.com/showarticle/1211.php
- Mathworks website https://www.mathworks.com/help/signal/ref/pwelch.html
- Oppenheim, Alan V. and Shafer, Ronald W., Discrete-Time Signal Processing, Prentice Hall, 1989, pp. 737—742.
- Welch, Peter D., “The Use of Fast Fourier Transform for the Estimation of Power Spectra: A Method Based on Time Averaging Over Short, Modified Periodograms”, IEEE Trans. Audio and Electroacoustics, vol AU-15, pp. 70 – 73, June 1967. https://www.utd.edu/~cpb021000/EE%204361/Great%20DSP%20Papers/Welchs%20Periodogram.pdf
- Lyons, Richard G., Understanding Digital Signal Processing, 2nd Ed., Prentice Hall, 2004, section 3.8.
- Oppenheim and Shafer, p. 738.
- Robertson, Neil, “The Power Spectrum”, https://www.dsprelated.com/showarticle/1004.php
- Lyons, section 11.5.
Neil Robertson January, 2019
Previous post by Neil Robertson:
Digital PLL's -- Part 2
Thanks Neil for the very informative article. I'm just wondering at the end where you calculate the SNR, should you have taken the processing gain into account. The noise floor as shown in figure 6 is lowered by the processing gain?
In figure 6, the left plot has amplitude units of dBW/bin, and the right plot has amplitude units of dBW/Hz. This is the reason for the difference in noise floor (and difference in carrier level as well).
Regarding processing gain due to DFT averaging: If you look at the un-averaged spectrum in Figure 2, and you try to put a marker on the noise, you can get a 20 dB variation of the noise reading depending on exactly where you place the marker. In the averaged spectrum of figure 5 and 6, we have less variation in the noise floor, so maybe only 6 dB variation. You can say processing gain is inherent in the plotted spectrum.
Hi Neil, thank you for the brilliant article. I am new to DSP and Matlab and will be using both to analyse long term acoustic data. Can you explain/point me to any resources for why I would/wouldn't use DFT averaging? I want to apply pwelch using a Hanning window with 50% overlap, in 1Hz frequency bins and ultimately transform the results (using 10log10?) to get values in db re 1 micro Pascal. How does this impact the nfft if my sampling rate is 144kHz? Do the two even relate? Thanks for any help you can offer,
What is the maximum frequency of interest of your signal?
Thank you for sharing knowledge in DSP, it is very helpful. I'm learning a lot, by I still need your help.
I'm using a GNSS receiver for some measurements and some times the results are
not what I inspected. Researches in the Internet tell me that by plotting the power spetrum (density) of the signal I can see if there are some interferences or any others signals that can disturb the receiver.
For GNSS receivers, a Interference threshold mask is defined by ICAO in the GNSS SARPs (Section 3.7 of appendix B of
ICAO Annex 10 to the Chicago Convention, amendment 76, Nov. 2001), and helps to identify interferences, or others unwanted signals.
To fullfill this requirements I assume that I need to plot the power spectrum density from noise floor of the signal that is approx. -164.5 dBW.
Number of Samples 2000
Sampling frequency 60 MHz
Frequency(Center) 1584 MHz
1) Is my thinking correct?
2) can I plot this power spectrum density with I/Q Samples?
3) How or references?
If more data are needed, please let me know, I will try provide these data.
Thank you against for your help.
You can look at the spectrum of the I or Q channel using pwelch. You should use 2^N samples, for example 2048. With 2048 samples, the frequency resolution is 60E6/2048 = 29.3 kHz.
The receiver converts the signal centered at 1584 MHz to baseband I/Q. The baseband frequency range is 0 to fs/2 = 0 to 30 MHz.
You can also find the spectrum of the complex baseband signal I + jQ, although I did not discuss that in the post.
I was not clear enough.
What are the steps to estimate self noise (noise floor), if there is one?
There is not a short answer to your question. Here are a couple of references that may apply:
Hi, thanks for the great article. I'm trying to calculate the SNR of my ECG signal. But I do not have a characteristic peak that I'm looking for. Instead, I'm using a frequency range (everything less than 40 Hz) and summing over that frequency range, just as you did with the additional 5 samples around your peak. My sampling rate is 2000.
I'm having an issue in determining how I should calculate the noise. I thought that I should use a range of of frequencies from the dB/Hz spectrum, instead of choosing a single point, as you showed. But when I do use that range of frequencies (800-900 Hz), the SNR values are unrealistically high. However, if I use the same range from the dB/bin spectrum (which is not what you mentioned in the article), then the SNR value is reasonable. Below is my code:
npoints = 2000;
nfft = npoints;
window = hanning(nfft);
noverlap = floor(npoints*0.50);
crit_f = 40;
noise_f1 = 800;
noise_f2 = 900;
[pxx,f] = pwelch(ecgnostretch,window,noverlap,nfft,Fs);
PdB_bin = 10*log10(pxx*Fs/nfft);
ind = find(f==crit_f);
Psum = Fs/nfft * sum(pxx(1:ind));
S = 10*log10(Psum);
PdB_Hz = 10*log10(pxx);
% k1 = round(noise_f/Fs*nfft);
ind1 = find(f==noise_f1);
ind2 = find(f==noise_f2);
No = Fs/nfft * mean(pxx(ind1:ind2)); %for noise, we should average, that's why use mean instead of sum
SNR_ecgnostretch = S-No;
Could you offer any suggestions as to if this is an accurate way to calculate the SNR?
I think you have a mistake in computing No: it would need to be a dB quantity. I am not familiar with ECG; however, here is an example of SNR calculation. Note that SNR and S/No are two different things. Since you say you are interested in SNR, that's what I calculate here. The result is SNR_dB = 40.
%snr1.m 2/17/20 nr
% compute SNR of sine + gaussian noise
% generate signal
% signal with SNR of 40 dB (20 log10(.01) = -40 dBW)
x= A*sin(2*pi*f0*n*Ts) + .01*randn(1,N);
% sum noise, excluding carrier neighborhood
sump= sum(p(1:252)) + sum(p(262:end));
pnoise_bin= sump*fs/nfft; % total noise power in W
N= 10*log10(pnoise_bin); % dBW
SNR_dB= 0 - N % note: signal power is 0 dBW
axis([0 fs/2 -90 10])
Thank you very much for your reply. For ECG, I don't have any characteristic peak, so I'm just using the frequency range between 0-40 Hz as that is the relevant frequency range for this signal.
It seems that, for the noise, I need to select my entire noise range as that outside my frequency range of interest, i.e. everything above 40 Hz. If I do that, I still get a negative SNR, which is probably due to some of the noise values being above 0 dB. I think this is due to my data acquisition system as all my PSD plots show similar behavior. If I pick a frequency range which is for sure in the noise floor (i.e. 800 to 1000 Hz), I can get an SNR of about 1, which doesn't make sense just by looking at the raw signal.
Attached is an example of the PSD plot (dbW/bin on the y axis, frequency on x) I'm getting along with the code. Below that is a picture of one of the signals.
Also, why is 0 used as the power of the signal?
The power of the signal in my example is 1 watt = 0 dBW.
Looking at your signal spectrum, I guess it makes sense to sum the power/bin from 0 to 40 Hz. One way to find the noise power would be to use a signal with the transducer not attached to a person (or attached to a dead person), then just sum the noise spectrum.
If that is not possible, you could assume that the noise floor is flat. Then you could find No near 800 Hz or so, where No = 10*log10(pxx) at a single point (note pxx is in W/Hz). You could average No over a few nearby points if desired. Then total noise would be No + 10*log10(fs/2) dBW.
Again, I don't know anything about ECG. I am not responsible for the use or misuse of this advice.
Thank you again for the response. I think what you've mentioned is definitely logical and applies to my case actually. I've reviewed some literature and this seems to be a plausible strategy. Thank you for the comments.
thank you for this article.
I would like to ask you what are the best way to calculate the parameters used in pwelch function.
I have a signal length of N= 5678000 and a sampling rate of fs= 1000.
Then I divide this signal in 3 depending on the onset and offset of a stimulus. For example N1= 1126535, N2=951465, N3=3600000.
I'm using the pwelch function in matlab but I don know what are the best parameters to choose for the window, the noverlap and the nfft.
I'm using now :
[pxx1,f1] = pwelch(BigMatPre(chs(e),:),2000,1000,2000,fs);
(BigMatPre is my signal)
All my best,
In my post, I use 5 inputs to pwelch, but you are showing 6 inputs, so I'm a little confused.
First of all, the computation will be faster if you choose nfft as a power of 2: you could use 2048 instead of 2000, although then the frequency step would not be a simple fraction.
Frequency resolution depends on nfft: For fs= 1000 and nfft = 2000, you have a resolution of fs/nfft = 0.5 Hz per bin. Higher values of nfft give better resolution at the expense of less spectrum averaging. So you may want to try different values of nfft to get a feeling for the trade-off. For example, you could try nfft = 4000 (or 4096).
The choice of noverlap = nfft/2 is probably OK.
For a window, you could start with a Hanning window. See my post discussing window functions:
Thank you for your answer
BigMatPre(chs(e),:) is because is a loop analysing the pwelch for e row.
I have an other question if you can help me.
I then want to compare the PSD of different groups, for example between the frequency range [40-80] but I m confuised. Is it good to compare the mean of a normalize PSD (mean of PSD value between 40-80 / mean of the PSD along [0.5-80] or the area under the curve?
Your article is really useful
Sorry, I don't understand the question.
I mean how can you compare the PSD of two groups? By a numeric way not only the shape of the PSD.
I was thinking to take the PSD values for a particular frequency range and compare statistically. Ex: the mean PSD power of 3 individual from a group 1 and the mean PSD power of 3 individual from group 2
Now you are getting beyond my knowledge base! This link might conceivably be of some help:
In your code, I think there may be a mistake in your call to pwelch. The 2nd input is "2000". However, the 2nd input should be a window function, for example hanning(2000).
Yes I change the values to a choose the nfft as a power of 2 and I tried hanning and hamming windows
Thank you !!!
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