how to get real frequency components using MUSIC compute frequency?

I used pmusic ( https://au.mathworks.com/help/signal/ref/pmusic.html ). It seems that pmusic cannot get real frequency for the signal.

x = cos(0.257*pi*n) + sin(0.2*pi*n) + 0.01*randn(size(n))

should have frequency components of 0.1258 and 0.1.

1. pmusic with No Sampling Specified example

But, the peaks for x-axis(normalized frequency) are 0.257 and 0.2.

2. Specifying Sampling Frequency and Subspace Dimensions example

But, the peaks for x-axis( frequency) are 812(=0.2*fs/2) and 1258(=0.257*fs/2) .

It seems that pmusic cannot get real frequency for the signal? So the peaks at x-axis is not the real frequency? what is the actually frequency? how could I get them for pmusic

[Walter Roberson]

Please show your function calls.

[evelyn]

I just use the same code as the example in the webpage.

Specifying Sampling Frequency and Subspace Dimensions

rng default
n = 0:199;
x = cos(0.257*pi*n) + sin(0.2*pi*n) + 0.01*randn(size(n));
[P,f] = pmusic(x,[Inf,1.1],[],8000,7); % Window length = 7
plot(f,20*log10(abs(P)))
xlabel 'Frequency (Hz)', ylabel 'Power (dB)'
title 'Pseudospectrum Estimate via MUSIC', grid on

pmusic with No Sampling Specified

n = 0:199;
x = cos(0.257*pi*n) + sin(0.2*pi*n) + 0.01*randn(size(n));
pmusic(x,4)      % Set p to 4 because there are two real inputs

[Walter Roberson]

When you pass fs of 8000 into the first pmusic() call, you are lying completely about the sampling frequency.

[evelyn]

thanks for reply.

how should I set the fs for the first pmusic call?

x = cos(0.257*pi*n) + sin(0.2*pi*n) + 0.01*randn(size(n)) have frequency components of 0.1258 and 0.1.

when I set fs= 2*0.1258=0.257, the result of frequency still seems not right?

rng default
n = 0:199;
x = cos(0.257*pi*n) + sin(0.2*pi*n) + 0.01*randn(size(n));
[P,f] = pmusic(x,[Inf,1.1],[],0.257,7); % Window length = 7
plot(f,20*log10(abs(P)))
xlabel 'Frequency (Hz)', ylabel 'Power (dB)'
title 'Pseudospectrum Estimate via MUSIC', grid on

Answer 1

[Chunru]

rng default
n = 0:199;
f1 = 0.257/2;  f2 = 0.2/2;  
fs= 2;          % fs (fs>2*max(f1, f2)
% generate signal according to f1, f2, fs
t = n/fs;
x = cos(f1*2*pi*t) + sin(f2*2*pi*t) + 0.01*randn(size(t));

nfft=1024;
[P,f] = pmusic(x,4,nfft,fs,7); % Window length = 7

plot(f,20*log10(abs(P)))
xlabel 'Frequency (Hz)', ylabel 'Power (dB)'
title 'Pseudospectrum Estimate via MUSIC', grid on
xline([f1 f2])

[evelyn]

thanks for reply.

but when I choose fs to 3, the frequencies (f1, f2) in the figure below are still not right?

rng default
n = 0:199;
f1 = 0.257/2;  f2 = 0.2/2;  
fs= 3;          % fs (fs>2*max(f1, f2)
% generate signal according to f1, f2, fs
t = n/fs;
x = cos(f1*2*pi*t) + sin(f2*2*pi*t) + 0.01*randn(size(t));
nfft=1024;
[P,f] = pmusic(x,4,nfft,fs,7); % Window length = 7
plot(f,20*log10(abs(P)))
xlim([0 1])
xlabel 'Frequency (Hz)', ylabel 'Power (dB)'
title 'Pseudospectrum Estimate via MUSIC', grid on
xline([f1 f2])

[Chunru]

You have a too small window size, which results in poor resolution. It could not resolve the two freq (when fs increase, the two freq normalized by fs get closer). To increase the resolution, you can increase the windown length (say 16).

rng default
n = 0:199;
f1 = 0.257/2;  f2 = 0.2/2;  
fs= 3;          % fs (fs>2*max(f1, f2)
% generate signal according to f1, f2, fs
t = n/fs;
x = cos(f1*2*pi*t) + sin(f2*2*pi*t) + 0.01*randn(size(t));
nfft=1024;
[P,f] = pmusic(x,4,nfft,fs,16); % Window length = 7
plot(f,20*log10(abs(P)))
xlim([0 1])
xlabel 'Frequency (Hz)', ylabel 'Power (dB)'
title 'Pseudospectrum Estimate via MUSIC', grid on
xline([f1 f2])

[evelyn]

thanks for your reply.

how to choose a proper fs and window length?

when I set fs to 0.4, what is the proper window length?

[Chunru]

Fs should be at least 2x of the max freq of interest to satisfy the sampling requirement.

Window length should be long enough to resolve the frequencies. For conventional method using FT to estimate spectrum, L is proportial to fs/delta_f. For the example above:

f1 = 0.257/2;  f2 = 0.2/2;  
fs= 3;          

% Length of windows for conventional FT
fs/abs(f2-f1)

ans = 105.2632

For MUSIC, the windown length can be significantly smaller. The exact number is subject to SNR, noise properties, signal properties and so on (since signal and noise subspaces are estimated from the data).