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. Use the default settings of the random number generator for reproducible results.

rng default
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,10*log10(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(length(x)),length(x),Fs)

mxerr = max(psdx'-periodogram(x,rectwin(length(x)),length(x),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. Use the default settings of the random number generator for reproducible results.

rng default
n = 0:999;
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.

N = length(x);
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,10*log10(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(length(x)),length(x))

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

mxerr = 1.4211e-14

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. Set the random number generator to the default settings for reproducible results.

rng default
n = 0:999;
x = exp(1j*pi/4*n) + [1 1j]*randn(2,length(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.

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

plot(freq/pi,10*log10(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(length(x)),length(x),'twosided')

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

mxerr = 2.8422e-14