Correlation

Cross-correlation of two inputs

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Libraries:
DSP System Toolbox / Statistics

Description

The Correlation block computes the cross-correlation of two N-D input arrays along the first-dimension. 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 block convolves the first input signal, u, with the time-reversed complex conjugate of the second input signal, v. In the frequency domain, to compute the cross-correlation, the block:

1. Takes the Fourier transform of both input signals, U and V.

2. Multiplies U and V^*, where * denotes the complex conjugate.

3. Computes the inverse Fourier transform of the product.

If you set Computation domain to Fastest, the block chooses the domain that minimizes the number of computations. For information on these computation methods, see Algorithms.

Ports

Input

Port_1 — First data input signal

vector | matrix | N-D array

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. When one or both of the input signals are complex, the output signal is also complex.

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

Port_2 — Second data input signal

vector | matrix | N-D array

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. When one or both of the input signals are complex, the output signal is also complex.

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

Output

Port_1 — Cross-correlated output

vector | matrix | N-D array

Cross-correlated output of the two input signals.

When the inputs are N-D arrays, the object outputs an N-D array, where all the dimensions, except for the first dimension, match with the input array. For example,

• When the inputs u and v have dimensions M_u-by-N-by-P and M_v-by-N-by-P, respectively, the Correlation block outputs an (M_u + M_v – 1)-by-N-by-P array.

• When the inputs u and v have the dimensions M_u-by-N and M_v-by-N, the block outputs an (M_u + M_v – 1)-by-N matrix.

If one input is a column vector and the other input is an N-D array, the Correlation block computes the cross-correlation of the vector with each column in the N-D array. For example,

• When the input u is an M_u-by-1 column vector and v is an M_v-by-N matrix, the block outputs an (M_u + M_v – 1)-by-N matrix.

• Similarly, when u and v are column vectors with lengths M_u and M_v, respectively, the block performs the vector cross-correlation.

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

Parameters

Main Tab

Computation domain — Domain in which the block computes the cross-correlation

Time (default) | Frequency | Fastest

• Time — Computes the cross-correlation in the time domain, which minimizes the memory usage.

• Frequency — Computes the cross-correlation in the frequency domain. For more information, see Algorithms.

• Fastest — Computes the cross-correlation in the domain that minimizes the number of computations.

To cross-correlate 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 Correlation 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 output of an accumulation operation in the Correlation 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 Correlation 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 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 the 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

Cross-Correlation

Cross-correlation is the measure of similarity of two discrete-time sequences as a function of the lag of one relative to the other.

For two length-N deterministic inputs or realizations of jointly wide-sense stationary (WSS) random processes, x and y, the cross-correlation is computed using the following relationship:

$$\begin{array}{l}{r}_{xy}(h)=\{\begin{array}{ll}{\displaystyle \sum _{n=0}^{N-h-1}x}(n+h)y^{*}(n)\hfill & 0\le h\le N-1\hfill \\ r_{yx}^{*}(h)\hfill & -(N-1)\le h\le 0\hfill \end{array}\\ \end{array}$$

where h is the lag and * denotes the complex conjugate. If the inputs are realizations of jointly WSS stationary random processes, r_xy(h) is an unnormalized estimate of the theoretical cross-correlation:

$${\rho }_{xy}(h)=E\{x(n+h)y^{*}(n)\}$$

where E{ } is the expectation operator.

Fixed-Point Data Types

The following diagram shows the data types the Correlation 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 multiplier is in the accumulator data type. For details on the complex multiplication performed, see Multiplication Data Types.

Note

When one or both of the inputs are signed fixed-point signals, all internal block data types are signed fixed point. The internal block data types are unsigned fixed point only when both inputs are unsigned fixed-point signals.

Algorithms

Time-Domain Computation

When you set the computation domain to time, the algorithm computes the cross-correlation of two signals in the time domain. The input signals can be fixed-point signals in this domain.

Correlate Two 2-D Arrays

When the inputs are two 2-D arrays, the jth column of the output, y_uv, has these elements:

$$\begin{array}{l}{y}_{uv(i,j)}={\displaystyle \sum _{k=0}^{\max (M_u,M_v)-1}{u}_{k,j}^{*}{v}_{(k+i),j}^{}}0\le i<M_v\\ \\ y_{uv(i,j)}=y_{vu(-i,j)}^{*}-M_u<i<0\end{array}$$

where:

• * denotes the complex conjugate.

• u is an M_u-by-N input matrix.

• v is an M_v-by-N input matrix.

• y_u,v is an (M_u + M_v – 1)-by-N matrix.

Inputs u and v are zero when indexed outside their valid ranges.

Correlate a Column Vector with a 2-D Array

When one input is a column vector and the other input is a 2-D array, the algorithm independently cross-correlates the input vector with each column of the 2-D array. The jth column of the output, y_u,v, has these elements:

$$\begin{array}{l}{y}_{uv(i,j)}={\displaystyle \sum _{k=0}^{\max (M_u,M_v)-1}{u}_k^{*}{v}_{(k+i),j}^{}}0\le i<M_v\\ \\ y_{uv(i,j)}=y_{vu(-i,j)}^{*}-M_u<i<0\end{array}$$

where:

• * denotes the complex conjugate.

• u is an M_u-by-1 column vector.

• v is an M_v-by-N matrix.

• y_uv is an (M_u + M_v – 1)-by-N matrix.

Inputs u and v are zero when indexed outside their valid ranges.

Correlate Two Column Vectors

When the inputs are two column vectors, the jth column of the output, y_uv, has these elements:

$$\begin{array}{l}{y}_{uv(i)}={\displaystyle \sum _{k=0}^{\max (M_u,M_v)-1}{u}_k^{*}{v}_{(k+i)}^{}}0\le i<M_v\\ \\ y_{uv(i)}=y_{vu(-i)}^{*}-M_u<i<0\end{array}$$

where:

• * denotes the complex conjugate.

• u is an M_u-by-1 column vector.

• v is an M_v-by-1 column vector.

• y_uv is an (M_u + M_v – 1)-by-1 column vector.

Inputs u and v are zero when indexed outside their valid ranges.

Frequency-Domain Computation

When you set the computation domain to frequency, the algorithm computes the cross-correlation in the frequency domain.

To compute the cross-correlation, the algorithm:

1. Takes the Fourier transform of both input signals, U and V.

2. Multiplies U and V^*, where * denotes the complex conjugate.

3. Computes 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™.

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

Version History

Introduced before R2006a