Power Spectral Density Estimates Using FFT

Even-Length Input with Sample Rate

Obtain the periodogram for an even-length signal sampled at 1 kHz using both fft and periodogram. Compare the results.

Create a signal consisting of a 100 Hz sine wave in N(0,1) additive noise. The sampling frequency is 1 kHz. The signal length is 1000 samples.

fs = 1000;
t = 0:1/fs:1-1/fs;
x = cos(2*pi*100*t) + randn(size(t));

Obtain the periodogram using fft. The signal is real-valued and has even length. Because the signal is real-valued, you only need power estimates for the positive or negative frequencies. In order to conserve the total power, multiply all frequencies that occur in both sets — the positive and negative frequencies — by a factor of 2. Zero frequency (DC) and the Nyquist frequency do not occur twice. Plot the result.

N = length(x);
xdft = fft(x);
xdft = xdft(1:N/2+1);
psdx = (1/(fs*N)) * abs(xdft).^2;
psdx(2:end-1) = 2*psdx(2:end-1);
freq = 0:fs/length(x):fs/2;

plot(freq,pow2db(psdx))
grid on
title("Periodogram Using FFT")
xlabel("Frequency (Hz)")
ylabel("Power/Frequency (dB/Hz)")

Compute and plot the periodogram using periodogram. Show that the two results are identical.

periodogram(x,rectwin(N),N,fs)

mxerr = max(psdx'-periodogram(x,rectwin(N),N,fs))

mxerr = 
3.4694e-18

Input with Normalized Frequency

Use fft to produce a periodogram for an input using normalized frequency. Create a signal consisting of a sine wave in N(0,1) additive noise. The sine wave has an angular frequency of $$\pi /4$$ rad/sample.

N = 1000;
n = 0:N-1;
x = cos(pi/4*n) + randn(size(n));

Obtain the periodogram using fft. The signal is real-valued and has even length. Because the signal is real-valued, you only need power estimates for the positive or negative frequencies. In order to conserve the total power, multiply all frequencies that occur in both sets — the positive and negative frequencies — by a factor of 2. Zero frequency (DC) and the Nyquist frequency do not occur twice. Plot the result.

xdft = fft(x);
xdft = xdft(1:N/2+1);
psdx = (1/(2*pi*N)) * abs(xdft).^2;
psdx(2:end-1) = 2*psdx(2:end-1);
freq = 0:2*pi/N:pi;

plot(freq/pi,pow2db(psdx))
grid on
title("Periodogram Using FFT")
xlabel("Normalized Frequency (\times\pi rad/sample)")
ylabel("Power/Frequency (dB/(rad/sample))")

Compute and plot the periodogram using periodogram. Show that the two results are identical.

periodogram(x,rectwin(N),N)

mxerr = max(psdx'-periodogram(x,rectwin(N),N))

mxerr = 
4.4409e-16

Complex-Valued Input with Normalized Frequency

Use fft to produce a periodogram for a complex-valued input with normalized frequency. The signal is a complex exponential with an angular frequency of $$\pi /4$$ rad/sample in complex-valued N(0,1) noise.

N = 1000;
n = 0:N-1;
x = exp(1j*pi/4*n) + [1 1j]*randn(2,N)/sqrt(2);

Use fft to obtain the periodogram. Because the input is complex-valued, obtain the periodogram from $$[0,2\pi )$$ rad/sample. Plot the result.

xdft = fft(x);
psdx = (1/(2*pi*N)) * abs(xdft).^2;
freq = 0:2*pi/N:2*pi-2*pi/N;

plot(freq/pi,pow2db(psdx))
grid on
title("Periodogram Using FFT")
xlabel("Normalized Frequency (\times\pi rad/sample)")
ylabel("Power/Frequency (dB/(rad/sample))")

Use periodogram to obtain and plot the periodogram. Compare the PSD estimates.

periodogram(x,rectwin(N),N,"twosided")

mxerr = max(psdx'-periodogram(x,rectwin(N),N,"twosided"))

mxerr = 
2.2204e-16