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# pyulear() - Signal Processing

### Syntax

### Example

### Output / Return Value

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### Reference

pxx = pyulear(x,order) returns the power spectral density estimate, pxx, of a discrete-time signal, x, found using the Yule-Walker method. When x is a vector, it is treated as a single channel. When x is a matrix, the PSD is computed independently for each column and stored in the corresponding column of pxx. pxx is the distribution of power per unit frequency. The frequency is expressed in units of rad/sample. order is the order of the autoregressive (AR) model used to produce the PSD estimate. pxx = pyulear(x,order,nfft) uses nfft points in the discrete Fourier transform (DFT). For real x, pxx has length (nfft/2 + 1) if nfft is even, and (nfft + 1)/2 if nfft is odd. For complex-valued x, pxx always has length nfft. pyulear uses a default DFT length of 256. [pxx,w] = pyulear(___) returns the vector of normalized angular frequencies, w, at which the PSD is estimated. w has units of rad/sample. For real-valued signals, w spans the interval [0,π] when nfft is even and [0,π) when nfft is odd. For complex-valued signals, w always spans the interval [0,2π). example[pxx,f] = pyulear(___,fs) returns a frequency vector, f, in cycles per unit time. The sampling frequency, fs, is the number of samples per unit time. If the unit of time is seconds, then f is in cycles/second (Hz). For real–valued signals, f spans the interval [0,fs/2] when nfft is even and [0,fs/2) when nfft is odd. For complex-valued signals, f spans the interval [0,fs). [pxx,w] = pyulear(x,order,w) returns the two-sided AR PSD estimates at the normalized frequencies specified in the vector, w. The vector, w, must contain at least two elements. [pxx,f] = pyulear(x,order,f,fs) returns the two-sided AR PSD estimates at the frequencies specified in the vector, f. The vector, f, must contain at least two elements. The frequencies in f are in cycles per unit time. The sampling frequency, fs, is the number of samples per unit time. If the unit of time is seconds, then f is in cycles/second (Hz). [___] = pyulear(x,order,___,freqrange) returns the AR PSD estimate over the frequency range specified by freqrange. Valid options for freqrange are: 'onesided', 'twosided', or 'centered'. [___,pxxc] = pyulear(___,'ConfidenceLevel',probability) returns the probability × 100% confidence intervals for the PSD estimate in pxxc. examplepyulear(___) with no output arguments plots the AR PSD estimate in dB per unit frequency in the current figure window.

pxx = pyulear(x,order)pxx = pyulear(x,order,nfft)[pxx,w] = pyulear(___)[pxx,f] = pyulear(___,fs) example[pxx,w] = pyulear(x,order,w)[pxx,f] = pyulear(x,order,f,fs)[___] = pyulear(x,order,___,freqrange)[___,pxxc] = pyulear(___,'ConfidenceLevel',probability)pyulear(___) example

Yule-Walker PSD Estimate of an AR(4) ProcessOpen This Example Create a realization of an AR(4) wide-sense stationary random process. Estimate the PSD using the Yule-Walker method. Compare the PSD estimate based on a single realization to the true PSD of the random process. Create an AR(4) system function. Obtain the frequency response and plot the PSD of the system.A = [1 -2.7607 3.8106 -2.6535 0.9238]; [H,F] = freqz(1,A,[],1); plot(F,20*log10(abs(H))) xlabel('Frequency (Hz)') ylabel('PSD (dB/Hz)') Create a realization of the AR(4) random process. Set the random number generator to the default settings for reproducible results. The realization is 1000 samples in length. Assume a sampling frequency of 1 Hz. Use pyulear to estimate the PSD for a 4th-order process. Compare the PSD estimate with the true PSD.rng default x = randn(1000,1); y = filter(1,A,x); [Pxx,F] = pyulear(y,4,1024,1); hold on plot(F,10*log10(Pxx)) legend('True Power Spectral Density','pyulear PSD Estimate') Yule-Walker PSD Estimate of a Multichannel SignalOpen This Example Create a multichannel signal consisting of three sinusoids in additive white Gaussian noise. The sinusoids' frequencies are 100 Hz, 200 Hz, and 300 Hz. The sampling frequency is 1 kHz, and the signal has a duration of 1 s. Fs = 1000; t = 0:1/Fs:1-1/Fs; f = [100;200;300]; x = cos(2*pi*f*t)'+randn(length(t),3); Estimate the PSD of the signal using the Yule-Walker method with a 12th-order autoregressive model. Use the default DFT length. Plot the estimate.morder = 12; pyulear(x,morder,[],Fs)