dsp.Autocorrelator

Autocorrelation sequence

Description

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:

1. Create the dsp.Autocorrelator object and set its properties.

2. Call the object with arguments, as if it were a function.

To learn more about how System objects work, see What Are System Objects?.

Creation

Syntax

ac = dsp.Autocorrelator

ac = dsp.Autocorrelator(Name,Value)

Description

ac = dsp.Autocorrelator returns an autocorrelator, ac, 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.

ac = dsp.Autocorrelator(Name,Value) returns an autocorrelator, ac, with each property set to the specified value.

Properties

Unless otherwise indicated, properties are nontunable, which means you cannot change their values after calling the object. Objects lock when you call them, and the release function unlocks them.

If a property is tunable, you can change its value at any time.

For more information on changing property values, see System Design in MATLAB Using System Objects.

MaximumLagSource — Source of maximum lag

Auto (default) | Property

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].

MaximumLag — Maximum positive lag

1 (default) | 0 | positive integer

Specify the maximum lag as an integer greater than or equal to 0. The MaximumLag must be less than the length of the input data.

Dependencies

This property applies only when the MaximumLagSource property is Property.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64

Scaling — Autocorrelation function scaling

None (default) | Biased | Unbiased | Unity at zero-lag

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. The default is None.

Method — Domain for computing autocorrelations

Time Domain (default) | Frequency Domain

Specify the domain for computing autocorrelations as Time Domain or Frequency Domain. You must set this property to Time Domain for fixed-point signals.

Fixed-Point Properties

FullPrecisionOverride — Full precision override for fixed-point arithmetic

true (default) | false

Specify whether to use full precision rules. If you set FullPrecisionOverride to true, which is the default, the object computes all internal arithmetic and output data types using full precision rules. These rules provide the most accurate fixed-point numerics. It also turns off the display of other fixed-point properties because they do not apply individually. These rules guarantee that no quantization occurs within the object. Bits are added, as needed, to ensure that no roundoff or overflow occurs. If you set FullPrecisionOverride to false, fixed-point data types are controlled through individual fixed-point property settings. For more information, see Full Precision for Fixed-Point System Objects.

RoundingMethod — Rounding method for fixed-point operations

Floor (default) | Ceiling | Convergent | Nearest | Round | Simplest | Zero

Specify the rounding method as Ceiling, Convergent, Floor, Nearest, Round, Simplest, or Zero.

Dependencies

This property applies only when you set the Method property to Time Domain and the object is not in full precision mode.

OverflowAction — Overflow action for fixed-point operations

Wrap (default) | Saturate

Specify the overflow action as Wrap or Saturate.

Dependencies

This property applies only when you set the Method property to Time Domain and the object is not in full precision mode.

ProductDataType — Product word and fraction lengths

Full precision (default) | Same as input | Custom

Specify the product fixed-point data type as one of Full precision, Same as input, or Custom.

Dependencies

This property applies only when you set the Method property to Time Domain.

CustomProductDataType — Product word and fraction lengths

numerictype([],32,30) (default) | numerictype

Specify the product fixed-point type as a scaled numerictype (Fixed-Point Designer) object with a Signedness of Auto.

Dependencies

This property applies only when you set the Method property to Time Domain and the ProductDataType property to Custom.

AccumulatorDataType — Accumulator word and fraction lengths

Full precision (default) | Same as product | Same as input | Custom

Specify the accumulator fixed-point data type as one of Full precision, Same as product, Same as input, or Custom.

Dependencies

This property applies only when the Method property is Time Domain.

CustomAccumulatorDataType — Accumulator word and fraction lengths

numerictype([],32,30) (default) | numerictype

Specify the accumulator fixed-point type as a scaled numerictype (Fixed-Point Designer) object with a Signedness of Auto.

Dependencies

This property applies only when you set the Method property to Time Domain and the AccumulatorDataType property to Custom.

OutputDataType — Output word and fraction lengths

Same as accumulator (default) | Same as product | Same as input | Custom

Specify the output fixed-point data type as Same as accumulator, Same as product, Same as input, or Custom.

Dependencies

This property applies only when the Method property is Time Domain.

CustomOutputDataType — Output word and fraction lengths

numerictype([],16,15) (default) | numerictype

Specify the output fixed-point type as a scaled numerictype (Fixed-Point Designer) object with a Signedness of Auto.

Dependencies

This property applies only when you set the Method property to Time Domain and the OutputDataType property to Custom.

Usage

Syntax

y = ac(x)

Description

y = ac(x) computes the autocorrelation sequence y for the columns of the input x.

Input Arguments

x — Data input

vector | matrix | N-D array

Data input, specified as a vector, matrix, or an N-D array. The object accepts real-valued or complex-valued multichannel and multidimensional inputs. The input can be a fixed-point signal when you set the Method property to 'Time Domain'. When the input signal is complex, the output signal is complex.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | fi
Complex Number Support: Yes

Output Arguments

y — Autocorrelated output

vector | matrix | N-D array

Autocorrelated output of the two input signals. The size, data type, and complexity of the output matches that of the input signal.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | fi
Complex Number Support: Yes

Object Functions

To use an object function, specify the System object™ as the first input argument. For example, to release system resources of a System object named obj, use this syntax:

release(obj)

Common to All System Objects

step	Run System object algorithm
release	Release resources and allow changes to System object property values and input characteristics
reset	Reset internal states of System object

Examples

Autocorrelation of Number Sequence

Note: If you are using R2016a or an earlier release, replace each call to the object with the equivalent step syntax. For example, obj(x) becomes step(obj,x).

ac1 = dsp.Autocorrelator;
% x is a column vector
x = (1:100)';
y = ac1(x);

Autocorrelation of Noisy Sine Wave

Note: If you are using R2016a or an earlier release, replace each call to the object with the equivalent step syntax. For example, obj(x) becomes step(obj,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;
ac = dsp.Autocorrelator('MaximumLagSource',...
'Property','MaximumLag',MaxLag,'Scaling','Unity at zero-lag');
SigAutocorr = ac(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.

More About

Autocorrelation

Autocorrelation is the correlation of a signal with itself at different points in time.

For a deterministic discrete-time sequence, x(n), the autocorrelation is computed using the following relationship:

$$r_x(h)={\displaystyle \sum _{n=0}^{N-h-1}{x}^{*}}(n)x(n+h)h=0,1,\dots ,N-1$$

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:

$${\widehat{\rho }}_x(h)=\frac{1}{N}{\displaystyle \sum _{k=0}^{N-h-1}{x}^{*}}(n)x(n+h)$$

Algorithms

Time-Domain Computation

When you set the computation domain to time, the algorithm computes the autocorrelation of the input signal in the time domain. The input signal can be a fixed-point signal in this domain.

The autocorrelation sequence, y, is computed using this equation:

$$y_{i,j}={\displaystyle \sum _{k=0}^{M-l-1}{u}_{k,j}^{*}{u}_{(k+i),j}^{}}0\le i\le l$$

• y_0,j is the zero-lag element in the jth column of the input.

• i is the index of the lag.

• j is the index of the input data column.

• * denotes the complex conjugate.

• M is the number of elements in each column.

• l is the maximum positive lag for autocorrelation. When you choose to compute the autocorrelation with all nonnegative lags, l=M–1. Otherwise, l is the maximum nonnegative integer lag value specified.

• u is an M-by-N input matrix.

Frequency-Domain Computation

When you set the computation domain to frequency, the algorithm computes the autocorrelation in the frequency domain.

In this domain, the algorithm computes the autocorrelation sequence by taking the Fourier transform of the input signal, multiplying the Fourier transform with its complex conjugate, and taking the inverse Fourier transform of the product. In this domain, depending on the input length, the algorithm can require fewer computations.

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

See System Objects in MATLAB Code Generation (MATLAB Coder).