# pyulear

Autoregressive power spectral density estimate — Yule-Walker method

## Syntax

• `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)`

## Description

````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`. ```

example

````pyulear(___)` with no output arguments plots the AR PSD estimate in dB per unit frequency in the current figure window.```

## Examples

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### Yule-Walker PSD Estimate of an AR(4) Process

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 Signal

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) ```

## Input Arguments

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### `x` — Input signalvector | matrix

Input signal, specified as a row or column vector, or as a matrix. If `x` is a matrix, then its columns are treated as independent channels.

Example: `cos(pi/4*(0:159))+randn(1,160)` is a single-channel row-vector signal.

Example: `cos(pi./[4;2]*(0:159))'+randn(160,2)` is a two-channel signal.

Data Types: `single` | `double`
Complex Number Support: Yes

### `order` — Order of autoregressive modelpositive integer

Order of the autoregressive model, specified as a positive integer.

Data Types: `double`

### `nfft` — Number of DFT points`256` (default) | integer | `[]`

Number of DFT points, specified as a positive integer. For a real-valued input signal, `x`, the PSD estimate, `pxx` has length (`nfft`/2+1) if `nfft` is even, and (`nfft`+1)/2 if `nfft` is odd. For a complex-valued input signal,`x`, the PSD estimate always has length `nfft`. If `nfft` is specified as empty, the default `nfft` is used.

Data Types: `single` | `double`

### `fs` — Sampling frequencypositive scalar

Sampling frequency, specified as a positive scalar. The sampling frequency is the number of samples per unit time. If the unit of time is seconds, the sampling frequency has units of hertz.

### `w` — Normalized frequenciesvector

Normalized frequencies, specified as a row or column vector with at least 2 elements. Normalized frequencies are in rad/sample.

Example: `w = [pi/4 pi/2]`

Data Types: `double`

### `f` — Cyclical frequenciesvector

Cyclical frequencies, specified as a row or column vector with at least 2 elements. The frequencies are in cycles per unit time. The unit time is specified by the sampling frequency, `fs`. If `fs` has units of samples/second, then `f` has units of Hz.

Example: `fs = 1000; f= [100 200]`

Data Types: `double`

### `freqrange` — Frequency range for PSD estimate`'onesided'` | `'twosided'` | `'centered'`

Frequency range for the PSD estimate, specified as a one of `'onesided'`, `'twosided'`, or `'centered'`. The default is `'onesided'` for real-valued signals and `'twosided'` for complex-valued signals. The frequency ranges corresponding to each option are

• `'onesided'` — returns the one-sided PSD estimate of a real-valued input signal, `x`. If `nfft` is even, `pxx` has length `nfft`/2 + 1 and is computed over the interval [0,π] rad/sample. If `nfft` is odd, the length of `pxx` is (`nfft` + 1)/2 and the interval is [0,π) rad/sample. When `fs` is optionally specified, the corresponding intervals are [0,`fs`/2] cycles/unit time and [0,`fs`/2) cycles/unit time for even and odd length `nfft` respectively.

• `'twosided'` — returns the two-sided PSD estimate for either the real-valued or complex-valued input, `x`. In this case, `pxx` has length `nfft` and is computed over the interval [0,2π) rad/sample. When `fs` is optionally specified, the interval is [0,`fs`) cycles/unit time.

• `'centered'` — returns the centered two-sided PSD estimate for either the real-valued or complex-valued input, `x`. In this case, `pxx` has length `nfft` and is computed over the interval (–π,π] rad/sample for even length `nfft` and (–π,π) rad/sample for odd length `nfft`. When `fs` is optionally specified, the corresponding intervals are (–`fs`/2, `fs`/2] cycles/unit time and (–`fs`/2, `fs`/2) cycles/unit time for even and odd length `nfft` respectively.

Data Types: `char`

### `probability` — Confidence interval for PSD estimate0.95 (default) | scalar in the range (0,1)

Coverage probability for the true PSD, specified as a scalar in the range (0,1). The output, `pxxc`, contains the lower and upper bounds of the `probability` × 100% interval estimate for the true PSD.

## Output Arguments

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### `pxx` — PSD estimatevector | matrix

PSD estimate, returned as a real-valued, nonnegative column vector or matrix. Each column of `pxx` is the PSD estimate of the corresponding column of `x`. The units of the PSD estimate are in squared magnitude units of the time series data per unit frequency. For example, if the input data is in volts, the PSD estimate is in units of squared volts per unit frequency. For a time series in volts, if you assume a resistance of 1 Ω and specify the sampling frequency in hertz, the PSD estimate is in watts per hertz.

Data Types: `single` | `double`

### `w` — Normalized frequenciesvector

Normalized frequencies, returned as a real-valued column vector. If `pxx` is a one-sided PSD estimate, `w` spans the interval [0,π] if `nfft` is even and [0,π) if `nfft` is odd. If `pxx` is a two-sided PSD estimate, `w` spans the interval [0,2π). For a DC-centered PSD estimate, `f` spans the interval (–π,π] rad/sample for even length `nfft` and (–π,π) radians/sample for odd length `nfft`.

Data Types: `double`

### `f` — Cyclical frequenciesvector

Cyclical frequencies, returned as a real-valued column vector. For a one-sided PSD estimate, `f` spans the interval [0,`fs`/2] when `nfft` is even and [0,`fs`/2) when `nfft` is odd. For a two-sided PSD estimate, `f` spans the interval [0,`fs`). For a DC-centered PSD estimate, `f` spans the interval (–`fs`/2, `fs`/2] cycles/unit time for even length `nfft` and (–`fs`/2, `fs`/2) cycles/unit time for odd length `nfft`.

Data Types: `double`

### `pxxc` — Confidence boundsmatrix

Confidence bounds, returned as a matrix with real-valued elements. The row size of the matrix is equal to the length of the PSD estimate, `pxx`. `pxxc` has twice as many columns as `pxx`. Odd-numbered columns contain the lower bounds of the confidence intervals, and even-numbered columns contain the upper bounds. Thus, `pxxc(m,2*n-1)` is the lower confidence bound and `pxxc(m,2*n)` is the upper confidence bound corresponding to the estimate `pxx(m,n)`. The coverage probability of the confidence intervals is determined by the value of the `probability` input.

Data Types: `single` | `double`