2-D FFT

Compute 2-D fast Fourier transform (FFT)

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Libraries:
Computer Vision Toolbox / Transforms

Description

The 2-D FFT block computes the discrete Fourier transform (DFT) of a two-dimensional input matrix using the fast Fourier transform (FFT) algorithm. The equation for the two-dimensional DFT F(m, n) of an M-by-N input matrix, f(x, y), is:

$$F(m,n)={\displaystyle \sum _{x=0}^{M-1}{\displaystyle \sum _{y=0}^{N-1}f(x,y)e^{-j\frac{2\pi mx}{M}}}}{e}^{-j\frac{2\pi ny}{N}}$$

where $$0\le m\le M-1$$ and $$0\le n\le N-1$$.

The block supports FFT implementation based on the FFTW library and an implementation based on a collection of Radix-2 algorithms. You can either manually select one of these implementations or let the block select one automatically.

Examples

Filter Image in Frequency Domain

Apply Gaussian lowpass filter to an image using the 2-D FFT block.

• Open Model

Ports

Input

Port_1 — Input data

vector | matrix

Input data, specified as a vector or matrix of intensity values.

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

Output

Port_1 — Output data

vector | matrix

Output data containing the 2-D FFT of the input, returned as a vector or matrix. The size and data type of the output are the same as those of the input.

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

Parameters

Main

FFT implementation — FFT implementation

Auto (default) | Radix-2 | FFTW

Specify the type of implementation to use to compute the FFT as one of these options:

• FFTW — Select this option to support an arbitrary-length input signal. The block restricts generated code with the FFTW implementation to host computers capable of running MATLAB^®.

• Radix-2 — Select this option to support bit-reversed processing, fixed and floating-point data, or portable C-code generation using Simulink Coder. The dimensions of the input matrix, M and N, must be powers of two. To work with other input sizes, use the Image Pad block to pad or truncate these dimensions to powers of two, or, if possible, choose the FFTW implementation. For more information about the algorithms used by the Radix-2 mode, see Radix-2 Implementation.

• Auto — Select this option to let the block choose the FFT implementation. For non-power-of-two transform lengths, the block restricts generated code to MATLAB host computers.

Output in bit-reversed order — Bit-reversed output

off (default) | on

Select this parameter to get the output data in bit-reversed order relative to the order of the input data. If you clear this parameter, the output data appears in linear order relative to the order of the input data. For more information on the bit-reversed order of output, see Bit-Reversed Order.

Dependencies

To enable this parameter, set the FFT implementation parameter to Auto or Radix-2.

Scale result by FFT length — Scale output data

off (default) | on

When you select this parameter, the block divides the output by the FFT length. This option is useful when you want the output of the FFT to stay in the same amplitude range as its input. This is particularly useful when working with fixed-point data types.

Data Types

For details on the fixed-point block parameters, see Specify Fixed-Point Attributes for Blocks (DSP System Toolbox).

Lock data type settings against change by the fixed-point tools — Data type override

off (default) | on

Select this parameter to prevent the fixed-point tools from overriding the data types you specify in this block. For more information, see Lock the Output Data Type Setting (Fixed-Point Designer).

Block Characteristics

Data Types	double | fixed point | integer | single
Multidimensional Signals	no
Variable-Size Signals	yes

More About

Bit-Reversed Order

Two numbers are bit-reversed values of each other when the binary representation of one is the mirror image of the binary representation of the other. For example, in a three-bit system, one and four are bit-reversed values of each other because the three-bit binary representation of one, 001, is the mirror image of the three-bit binary representation of four, 100. The diagram shows the row indices in linear order. To put them in bit-reversed order:

1. Translate the indices into their binary representations with the minimum number of bits. In this example, the minimum number of bits is three because the binary representation of the largest row index, 7, is 111.

2. Find the mirror image of each binary entry, and write it beside the original binary representation.

3. Translate each binary mirror image to its decimal representation.

   The row indices now appear in bit-reversed order.

   

When you select the Output in bit-reversed order parameter of the 2-D FFT block, the block bit-reverses the order of both the rows and columns. All output values remain the same, but they appear in a different order.

Fixed-Point Data Types

These diagrams show the data types used in the 2-D FFT block for fixed-point signals. The block first casts inputs to the output data type and stores them in the output buffer. Each butterfly stage then processes signals in the accumulator data type, with the final butterfly casting its output back into the output data type. The block multiplies by a twiddle factor before each butterfly stage, in a decimation-in-time FFT, and after each butterfly stage in a decimation-in-frequency FFT.

The multiplier output appears in the accumulator data type because both of the inputs to the multiplier are complex. For details on the complex multiplication performed, refer to Multiplication Data Types.

Algorithms

FFTW Implementation

The FFTW implementation provides an optimized FFT calculation, including support for power-of-two and non-power-of-two transform lengths in both simulation and code generation. Generated code using the FFTW implementation can run only on computers capable of running MATLAB. The input must be of a floating-point data type.

Radix-2 Implementation

The Radix-2 implementation supports bit-reversed processing, fixed or floating-point data, and enables the block to provide portable C-code generation using Simulink Coder. The dimensions of the input matrix, M and N, must be powers of two. To work with other input sizes, use the Image Pad block to pad or truncate these dimensions to powers of two.

The block implements one or more of these algorithms for Radix-2 implementation.

• Butterfly operation

• Double-signal algorithm

• Half-length algorithm

• Radix-2 decimation-in-time (DIT) algorithm

• Radix-2 decimation-in-frequency (DIF) algorithm

Radix-2 Algorithms for Real and Complex Signals

Complexity of Input	Output Ordering	Algorithms Used for FFT Computation
Complex	Linear	Bit-reversed operation and Radix-2 DIT
Complex	Bit-reversed	Radix-2 DIF
Real	Linear	Bit-reversed operation and Radix-2 DIT in conjunction with the half-length and double-signal algorithms
Real	Bit-reversed	Radix-2 DIF in conjunction with the half-length and double-signal algorithms

Radix-2 Optimization for Table of Trigonometric Values

In certain situations, the Radix-2 algorithm computes all the possible trigonometric values of the twiddle factor:

$$e^{j\frac{2\pi k}{K}},$$

where K is the greater value of either M or N, and k is an integer in the range [0, K – 1]. The block stores these values in a table and retrieves them during simulation. This table summarizes the number of table entries each for fixed-point and floating-point Radix-2 FFT algorithms:

Number of Table Entries for N-Point FFT
Floating point	3N/4
Fixed point	N

References

[1] “FFTW Home Page.” Accessed February 23, 2022. https://www.fftw.org/.

[2] Frigo, M., and S.G. Johnson. “FFTW: An Adaptive Software Architecture for the FFT.” In Proceedings of the 1998 IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP ’98 (Cat. No.98CH36181), 3:1381–84. Seattle, WA, USA: IEEE, 1998. https://doi.org/10.1109/ICASSP.1998.681704.

Extended Capabilities

C/C++ Code Generation

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

Usage notes and limitations:

• When these conditions apply, the executable generated from this block relies on prebuilt dynamic library files (.dll files) included with MATLAB:

  • FFT implementation is set to FFTW.

  • Inherit FFT length from input dimensions is cleared, and the length of each dimension of the input matrix is a power of two.

  Use the packNGo function to package the code generated from this block and all the relevant files in a compressed ZIP file. Using this ZIP file, you can relocate, unpack, and rebuild your project in another development environment where MATLAB is not installed.

• When the length of each dimension of the input matrix is a power of two, you can generate standalone C and C++ code from this block.

Version History

Introduced before R2006a