Note: This page has been translated by MathWorks. Click here to see

To view all translated materials including this page, select Country from the country navigator on the bottom of this page.

To view all translated materials including this page, select Country from the country navigator on the bottom of this page.

This example shows how to obtain nonparametric power spectral density (PSD) estimates equivalent to the periodogram using `fft`

. The examples show you how to properly scale the output of `fft`

for even-length inputs, for normalized frequency and hertz, and for one- and two-sided PSD estimates.

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

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

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 $$(-\pi ,\pi ]$$ 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

`fft`

|`periodogram`

|`pspectrum`