Documentation |
Package: dsp
Autocorrelation sequence
The Autocorrelator object returns the autocorrelation sequence for a discrete-time, deterministic input, or the autocorrelation sequence estimate for a discrete-time, wide-sense stationary (WSS) random process at positive lags.
To obtain the autocorrelation sequence:
Define and set up your autocorrelator. See Construction.
Call step to compute the autocorrelation sequence according to the properties of dsp.Autocorrelator. The behavior of step is specific to each object in the toolbox.
H = dsp.Autocorrelator returns an autocorrelator, H, that computes the autocorrelation along the first dimension of an N-D array. By default, the autocorrelator computes the autocorrelation at lags from zero to N – 1, where N is the length of the input vector or the row dimension of the input matrix. Inputting a row vector results in a row of zero-lag autocorrelation sequence values, one for each column of the row vector. The default autocorrelator returns the unscaled autocorrelation and performs the computation in the time domain.
H = dsp.Autocorrelator('PropertyName',PropertyValue, ...) returns an autocorrelator, H, with each property set to the specified value.
MaximumLagSource |
Source of maximum lag Specify how to determine the range of lags for the autocorrelation as Auto or Property. If the value of MaximumLagSource is Auto, the autocorrelator computes the autocorrelation over all nonnegative lags in the interval [0, N-1], where N is the length of the first dimension of the input. Otherwise, the object computes the autocorrelation using lags in the range [0,MaximumLag]. The default is Auto. |
MaximumLag |
Maximum positive lag Specify the maximum lag as a positive integer. This property applies only when the MaximumLagSource property is Property. The MaximumLag must be less than the length of the input data. The default is 1. |
Scaling |
Autocorrelation function scaling Specify the scaling to apply to the output as None, Biased, Unbiased, or Unity at zero-lag. Set this property to None to generate the autocorrelation function without scaling. This option is appropriate if you are computing the autocorrelation of a nonrandom (deterministic) input. The Biased option scales the autocorrelation by 1/N, where N is the length of the input data. Scaling by 1/N yields a biased, finite-sample approximation to the theoretical autocorrelation of a WSS random process. In spite of the bias, scaling by 1/N has the desirable property that the sample autocorrelation matrix is nonnegative definite, a property possessed by the theoretical autocorrelation matrices of all wide-sense stationary random processes. The Fourier transform of the biased autocorrelation estimate is the periodogram, a widely used estimate of the power spectral density of a WSS process. The Unbiased option scales the estimate of the autocorrelation by 1/N-1. Scaling by N – 1 produces an unbiased estimate of the theoretical autocorrelation. However, using the unbiased option, you can obtain an estimate of the autocorrelation function that fails to have the nonnegative definite property. Use the Unity at zero-lag option to normalize the autocorrelation estimate as identically one at lag zero. See Definitions for more detail on scaling. The default is None. |
Method |
Domain for computing autocorrelations Specify the domain for computing autocorrelations as Time Domain or Frequency Domain. You must set this property to Time Domain for fixed-point signals. The default is Time Domain. |
clone | Create autocorrelator object with same property values |
getNumInputs | Number of expected inputs to step method |
getNumOutputs | Number of outputs of step method |
isLocked | Locked status for input attributes and nontunable properties |
release | Allow property value and input characteristics changes |
step | Autocorrelation sequence |
The autocorrelation of a deterministic discrete-time sequence, x(n) is:
$${r}_{x}(h)={\displaystyle \sum _{n=0}^{N-h-1}{x}^{*}}(n)x(n+h)\text{\hspace{1em}}h=0,1,\dots ,N-1$$
where h is the lag and * denotes the complex conjugate. If the input is a length N realization of a WSS stationary random process, r_{x}(h) is an estimate of the theoretical autocorrelation:
$${\rho}_{x}(h)=E\{{x}^{*}(n)x(n+h)\}$$
where E{ } is the expectation operator. The Unity at zero-lag normalization divides each sequence value by the autocorrelation or autocorrelation estimate at zero lag.
$$\frac{{\rho}_{x}(h)}{{\rho}_{x}(0)}=\frac{E\{{x}^{*}(n)x(n+h)\}}{E\left\{\right|x(0){|}^{2}\}}$$
The most commonly used estimate of the theoretical autocorrelation of a WSS random process is the biased estimate:
$${\widehat{\rho}}_{x}(h)=\frac{1}{N}{\displaystyle \sum _{k=0}^{N-h-1}{x}^{*}}(n)x(n+h)$$
Compute autocorrelation for all positive lags:
hac1 = dsp.Autocorrelator; % x is a column vector x = [1:100]'; y = step(hac1, x);
Compute the autocorrelation of a sine wave in white Gaussian noise with approximate 95% upper and lower confidence limits:
S = rng('default'); % Sine wave with period N=4 x = 1.4*cos(pi/2*(1:100))'+randn(100,1); MaxLag = 20; H = dsp.Autocorrelator('MaximumLagSource',... 'Property','MaximumLag',MaxLag,'Scaling','Unity at zero-lag'); SigAutocorr = step(H,x); stem(SigAutocorr,'b','markerfacecolor',[0 0 1]); line(1:MaxLag+1,1.96/sqrt(100)*ones(MaxLag+1,1),... 'linestyle','-.','linewidth',2); line(1:MaxLag+1,-1.96/sqrt(100)*ones(MaxLag+1,1),... 'linestyle','-.','linewidth',2); axis([1 20 -1 1]); title('Sine Wave + Noise Autocorrelation'); xlabel('Lag');
As this figure shows, the autocorrelation estimate demonstrates the four sample periodic sine wave with excursions outside the 95% white Gaussian noise confidence limits every two samples.
This object implements the algorithm, inputs, and outputs described on the Autocorrelation block reference page. The object properties correspond to the block parameters.