Autocorrelation

Autocorrelation of N-D array

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Libraries:
DSP System Toolbox / Statistics

Description

The Autocorrelation block computes the autocorrelation along the first dimension of an N-D input array. The computation can be done in the time domain or frequency domain. You can specify the domain through the Computation domain parameter. In the time domain, the input signal is convolved with its time-reversed complex conjugate. In the frequency domain, the block computes the autocorrelation by taking the Fourier transform of the input signal, multiplying the Fourier transform with its conjugate, and computing the inverse Fourier transform of the product. In this domain, depending on the input length, the block can require fewer computations. For information on these two computation methods, see Algorithms.

You can specify the maximum lag for autocorrelation using the Compute all non-negative lags and Maximum non-negative lag (less than input length) parameters.

The block accepts fixed-point signals when you set the Computation domain to Time.

Ports

Input

Port_1 — Data input

vector | matrix | N-D array

Data input. The block accepts real-valued or complex-valued multichannel and multidimensional inputs. The input can be a fixed-point signal when you set the Computation domain to Time.

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

Output

Port_1 — Autocorrelated output

vector | matrix | N-D array

Autocorrelated output of the data input.

• When the input is an M-by-N matrix, u, the output, y, is an (l+1)-by-N matrix. l is the maximum positive lag for autocorrelation.

• When the input is an N-D array, the block outputs an N-D array. The size of the first dimension is l+1, and the sizes of all other dimensions match those of the input array. For example, when the input is an M-by-N-by-P array, the block outputs an (l+1)-by-N-by-P array.

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

Parameters

Main Tab

Compute all non-negative lags — Compute autocorrelation over all nonnegative lags

on (default) | off

When you select this parameter, the Autocorrelation block computes the autocorrelation over all nonnegative lags in the range [0, length(input) – 1]. When you clear this parameter, the block computes the autocorrelation using the lags in the range [0, l], where l is the value you specify in Maximum non-negative lag (less than input length).

Maximum non-negative lag (less than input length) — Maximum positive lag

1 (default) | integer greater than or equal to 0 and less than input length

Maximum positive lag for autocorrelation, specified as an integer that is greater than or equal to 0 and less than the input length.

Dependencies

To enable this parameter, clear the Compute all non-negative lags parameter.

Scaling — Scaling of the output

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

Scaling applied to the output.

• None — Generates the raw autocorrelation y_i,j without normalization.

• Biased — Generates the biased estimate of the autocorrelation.

  $$y_{i,j}^{biased}=\frac{y_{i,j}}{M}$$

• Unbiased — Generates the unbiased estimate of the autocorrelation.

  $$y_{i,j}^{unbiased}=\frac{y_{i,j}}{M-i}$$

• Unity at zero-lag — Normalizes the estimate of the autocorrelation for each channel so that the zero-lag sum, the first element in each column, is identically 1.

  $$y_{0,j}=1$$

Computation domain — Domain in which the block computes the autocorrelation

Time (default) | Frequency

• Time — Computes the convolutions in the time domain, which minimizes the memory usage.

• Frequency — Computes the autocorrelation in frequency domain. For more information, see Algorithms.

To autocorrelate fixed-point signals, set this parameter to Time.

Data Types Tab

Note

Fixed-point signals are supported for the time domain only. To use these parameters, on the Main tab, set Computation domain to Time.

Rounding mode — Method of rounding operation

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

Specify the rounding mode for fixed-point operations as one of the following:

• Floor

• Ceiling

• Convergent

• Nearest

• Round

• Simplest

• Zero

For more details, see rounding mode.

Note

The Rounding mode and Saturate on integer overflow parameters have no effect on numerical results when all these conditions are met:

• Product output data type is Inherit: Inherit via internal rule.

• Accumulator data type is Inherit: Inherit via internal rule.

• Output data type is Inherit: Same as accumulator.

With these data type settings, the block operates in full-precision mode.

Saturate on integer overflow — Method of overflow action

off (default) | on

When you select this parameter, the block saturates the result of its fixed-point operation. When you clear this parameter, the block wraps the result of its fixed-point operation. For details on saturate and wrap, see overflow mode for fixed-point operations.

Note

The Rounding mode and Saturate on integer overflow parameters have no effect on numeric results when all these conditions are met:

• Product output data type is Inherit: Inherit via internal rule.

• Accumulator data type is Inherit: Inherit via internal rule.

With these data type settings, the block operates in full-precision mode.

Product output — Product output data type

Inherit: Inherit via internal rule (default) | Inherit: Same as input | fixdt([],16,0)

Product output specifies the data type of the output of a product operation in the Autocorrelation block. For more information on the product output data type, see Multiplication Data Types and Fixed-Point Data Types.

• Inherit: Inherit via internal rule — The block inherits the product output data type based on an internal rule. For more information on this rule, see Inherit via Internal Rule.

• Inherit: Same as input — The block specifies the product output data type to be the same as the input data type.

• fixdt([],16,0) — The block specifies an autosigned, binary-point, scaled, fixed-point data type with a word length of 16 bits and a fraction length of 0.

Alternatively, you can set the Product output data type by using the Data Type Assistant. To use the assistant, click the Show data type assistant button.

For more information on the data type assistant, see Specify Data Types Using Data Type Assistant (Simulink).

Accumulator — Accumulator data type

Inherit: Inherit via internal rule (default) | Inherit: Same as input | Inherit: Same as product output | fixdt([],16,0)

Accumulator specifies the data type of the output of an accumulation operation in the Autocorrelation block. For illustrations on how to use the accumulator data type in this block, see Fixed-Point Data Types.

• Inherit: Inherit via internal rule — The block inherits the accumulator data type based on an internal rule. For more information on this rule, see Inherit via Internal Rule.

• Inherit: Same as input — The block specifies the accumulator data type to be the same as the input data type.

• Inherit: Same as product output — The block specifies the accumulator data type to be the same as the product output data type.

• fixdt([],16,0) — The block specifies an autosigned, binary-point, scaled, fixed-point data type with a word length of 16 bits and a fraction length of 0.

Alternatively, you can set the Accumulator data type by using the Data Type Assistant. To use the assistant, click the Show data type assistant button.

For more information on the data type assistant, see Specify Data Types Using Data Type Assistant (Simulink).

Output — Output data type

Inherit: Same as accumulator (default) | Inherit: Same as input | Inherit: Same as product output | fixdt([],16,0)

Output specifies the data type of the output of the Autocorrelation block. For more information on the output data type, see Fixed-Point Data Types.

• Inherit: Same as input — The block specifies the output data type to be the same as the input data type.

• Inherit: Same as product output — The block specifies the output data type to be the same as the product output data type.

• Inherit: Same as accumulator — The block specifies the output data type to be the same as the accumulator data type.

• fixdt([],16,0) — The block specifies an autosigned, binary-point, scaled, fixed-point data type with a word length of 16 bits and a fraction length of 0.

Alternatively, you can set the Output data type by using the Data Type Assistant. To use the assistant, click the Show data type assistant button.

For more information on the data type assistant, see Specify Data Types Using Data Type Assistant (Simulink).

Output Minimum — Minimum value the block can output

[] (default) | scalar

Specify the minimum value the block can output. Simulink^® software uses this minimum value to perform:

• Simulation range checking. See Specify Signal Ranges (Simulink).

• Automatic scaling of fixed-point data types.

Output Maximum — Maximum value block can output

[] (default) | scalar

Specify the maximum value the block can output. Simulink software uses this maximum value to perform:

• Simulation range checking. See Specify Signal Ranges (Simulink).

• Automatic scaling of fixed-point data types.

Lock data type settings against changes by the fixed-point tools — Prevent fixed-point tools from overriding data types

off (default) | on

Select this parameter to prevent the fixed-point tools from overriding the data types you specify in the block dialog box.

Block Characteristics

Data Types	double | fixed point | integer | single
Direct Feedthrough	no
Multidimensional Signals	no
Variable-Size Signals	no
Zero-Crossing Detection	no

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:

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

Fixed-Point Data Types

These diagrams show the data types that the Autocorrelation block uses for fixed-point signals (time domain only).

You can set the product output, accumulator, and output data types on the Data Types tab of the block.

When the input is real, the output of the multiplier is in the product output data type. When the input is complex, the output of the multiplication is in the accumulator data type. For details on the complex multiplication performed, see Multiplication Data Types.

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 Simulink® Coder™.

Generated code relies on the memcpy or memset function (string.h) under certain conditions.

Fixed-Point Conversion
Design and simulate fixed-point systems using Fixed-Point Designer™.

Version History

Introduced before R2006a