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# IFFT

Inverse fast Fourier transform (IFFT) of input

## Library

Transforms

`dspxfrm3`

## Description

The IFFT block computes the inverse fast Fourier transform (IFFT) across the first dimension of an N-D input array.

When you specify an FFT length not equal to the length of the input vector, (or first dimension of the input array), the block implements zero-padding, truncating, or modulo-M, (FFT length) data wrapping. This occurs before the IFFT operation, as per Orfanidis [1].

`y = ifft(u,M) % P ≤ M`

Wrapping:

`y(:,l) = ifft(datawrap(u(:,l),M)) % P > M; l = 1,...,N`

Truncating:

`y (:,l) = ifft(u,M) % P > M; l = 1,...,N`

When the input length, P, is greater than the FFT length, M, you may see magnitude increases in your IFFT output. These magnitude increases occur because the IFFT block uses modulo-M data wrapping to preserve all available input samples.

To avoid such magnitude increases, you can truncate the length of your input sample, P, to the FFT length, M. To do so, place a Pad block before the IFFT block in your model.

The kth entry of the lth output channel, y(k, l), is equal to the kth point of the M-point inverse discrete Fourier transform (IDFT) of the lth input channel:

`$y\left(k,l\right)=\frac{1}{M}\sum _{p=1}^{P}u\left(p,l\right){e}^{j2\pi \left(p-1\right)\left(k-1\right)/M}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}k=1,\dots ,M$`

The output of this block has the same dimensions as the input. If the input signal has a floating-point data type, the data type of the output signal uses the same floating-point data type. Otherwise, the output can be any fixed-point data type. The block computes scaled and unscaled versions of the IFFT.

The input to this block can be floating-point or fixed-point, real or complex, and conjugate symmetric. The block uses one of two possible FFT implementations. You can select an implementation based on the FFTW library [1], [2], or an implementation based on a collection of Radix-2 algorithms. You can select `Auto` to allow the block to choose the implementation.

### 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 will be restricted to MATLAB® host computers. The data type must be floating-point. Refer to Simulink® Coder™ for more details on generating code.

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

With Radix-2 selected, the block implements one or more of the following algorithms:

• Butterfly operation

• Double-signal algorithm

• Half-length algorithm

#### Radix-2 Algorithms for Real or Complex Input Complexity

Parameter SettingsAlgorithms Used for IFFT Computation

Bit-reversal operation and radix-2 DIT in conjunction with the half-length and double-signal algorithms

Radix-2 DIT in conjunction with the half-length and double-signal algorithms

#### Radix-2 Optimization for the Table of Trigonometric Values

In certain situations, the block's 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 $k=0,\cdots ,K-1$. The block stores these values in a table and retrieves them during simulation. The number of table entries for fixed-point and floating-point is summarized in the following table:

Number of Table Entries for N-Point FFT

floating-point

3 N/4

fixed-point

N

#### Fixed-Point Data Types

The following diagrams show the data types used in the IFFT block for fixed-point signals. You can set the sine table, accumulator, product output, and output data types displayed in the diagrams in the IFFT dialog box as discussed in Dialog Box.

Inputs to the IFFT block are first cast to the output data type and stored in the output buffer. Each butterfly stage then processes signals in the accumulator data type, with the final output of the butterfly being cast back into the output data type. The block multiplies in a twiddle factor before each butterfly stage in a decimation-in-time IFFT and after each butterfly stage in a decimation-in-frequency IFFT.

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 in the DSP System Toolbox™ documentation.

### Fixed-Point Data Types

The following diagrams show the data types used within the IFFT block for fixed-point signals. You can set the sine table, accumulator, product output, and output data types displayed in the diagrams in the IFFT block dialog, as discussed in Dialog Box.

The IFFT block first casts input to the output data type and then stores it in the output buffer. Each butterfly stage then processes signals in the accumulator data type, with the final output of the butterfly being cast back into the output data type. The block multiplies in a twiddle factor before each butterfly stage in a decimation-in-time IFFT, and after each butterfly stage in a decimation-in-frequency IFFT.

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

 Note:   When the block input is fixed point, all internal data types are signed fixed point.

## Dialog Box

The Main pane of the IFFT block dialog appears as follows.

FFT implementation

Set this parameter to `FFTW` [1], [2] to support an arbitrary length input signal. The block restricts generated code with FFTW implementation to MATLAB host computers.

Set this parameter to `Radix-2` for bit-reversed processing, fixed or floating-point data, or for portable C-code generation using the Simulink Coder. The dimension M of the M-by-N input matrix, must be a power of two. To work with other input sizes, use the Pad block to pad or truncate these dimensions to powers of two, or if possible choose the FFTW implementation. See Radix-2 Implementation.

Set this parameter to `Auto` to let the block choose the FFT implementation. For non-power-of-two transform lengths, the block restricts generated code to MATLAB host computers.

Input is in bit-reversed order

Select or clear this check box to designate the order of the input channel elements. Select this check box when the input should appear in reversed order, and clear it when the input should appear in linear order. The block yields invalid outputs when you do not set this parameter correctly. This check box only appears when you set the FFT implementation parameter to `Radix-2` or `Auto`.

You cannot select this check box if you have cleared the Inherit FFT length from input dimensions check box, and you are specifying the FFT length using the FFT length parameter. Also, it cannot be selected when you set the FFT implementation parameter to `FFTW`.

For more information on ordering of the output, see Linear and Bit-Reversed Output Order.

Input is conjugate symmetric

Select this option when the block inputs conjugate symmetric data and you want real-valued outputs. Selecting this check box optimizes the block's computation method.

The FFT block yields conjugate symmetric output when you input real-valued data. Taking the IFFT of a conjugate symmetric input matrix produces real-valued output. Therefore, if the input to the block is both floating point and conjugate symmetric, and you select this check box, the block produces real-valued outputs.

You cannot select this check box if you have cleared the Inherit FFT length from input dimensions check box, and you are specifying the FFT length using the FFT length parameter.

If you input conjugate symmetric data to the IFFT block and do not select this check box, the IFFT block outputs a complex-valued signal with small imaginary parts. The block outputs invalid data if you select this option with non conjugate symmetric input data.

Divide output by FFT length

When you select this check box, the block computes its output according to the IDFT equation, discussed in the Description section.

When you clear this check box, the block computes the output using a modified version of the IDFT: $M\cdot y\left(k,l\right)$, which is defined by the following equation:

`$\begin{array}{cc}M\cdot y\left(k,l\right)=\sum _{p=1}^{P}u\left(p,l\right){e}^{j2\pi \left(p-1\right)\left(k-1\right)/M}& k=1,...,M\end{array}$`

Notice, the modified IDFT equation does not include the multiplication factor of 1/M.

Inherit FFT length from input dimensions

Select to inherit the FFT length from the input dimensions. If you do not select this parameter, the FFT length parameter becomes available to specify the length. You cannot clear this parameter when you select either the Input is in bit-reversed order or the Input is conjugate symmetric parameter.

FFT length

Specify FFT length. This parameter only becomes available if you do not select the Inherit FFT length from input dimensions parameter.

When you set the FFT implementation parameter to `Radix-2`, or when you check the Output in bit-reversed order check box, this value must be a power of two.

Wrap input data when FFT length is shorter than input length

Choose to wrap or truncate the input, depending on the FFT length. If this parameter is checked, modulo-length data wrapping occurs before the FFT operation, given FFT length is shorter than the input length. If this property is unchecked, truncation of the input data to the FFT length occurs before the FFT operation. The default is checked.

The Data Types pane of the IFFT block dialog appears as follows.

Rounding mode

Select the rounding mode for fixed-point operations. The sine table values do not obey this parameter; instead, they always round to `Nearest`.

Saturate on integer overflow

When you select this check box, the block saturates the result of its fixed-point operation. When you clear this check box, the block wraps the result of its fixed-point operation. By default, this check box is cleared. For details on `saturate` and `wrap`, see overflow mode for fixed-point operations. The sine table values do not obey this parameter; instead, they are always saturated.

Sine table data type

Choose how you specify the word length of the values of the sine table. The fraction length of the sine table values always equals the word length minus one. You can set this parameter to:

• A rule that inherits a data type, for example, ```Inherit: Same word length as input```

• An expression that evaluates to a valid data type, for example, `fixdt(1,16)`

The sine table values do not obey the Rounding mode and Saturate on integer overflow parameters; instead, they are always saturated and rounded to `Nearest`.

Product output data type

Specify the product output data type. See Fixed-Point Data Types and Multiplication Data Types for illustrations depicting the use of the product output data type in this block. You can set this parameter to:

• A rule that inherits a data type, for example, ```Inherit: Inherit via internal rule```. For more information on this rule, see Inherit via Internal Rule.

• An expression that evaluates to a valid data type, for example, `fixdt(1,16,0)`

Click the Show data type assistant button to display the Data Type Assistant, which helps you set the Product output data type parameter.

Accumulator data type

Specify the accumulator data type. See Fixed-Point Data Types for illustrations depicting the use of the accumulator data type in this block. You can set this parameter to:

• A rule that inherits a data type, for example, ```Inherit: Inherit via internal rule```. For more information on this rule, see Inherit via Internal Rule.

• An expression that evaluates to a valid data type, for example, `fixdt(1,16,0)`

Click the Show data type assistant button to display the Data Type Assistant, which helps you set the Accumulator data type parameter.

Output data type

Specify the output data type. See Fixed-Point Data Types for illustrations depicting the use of the output data type in this block. You can set this parameter to:

• A rule that inherits a data type, for example, ```Inherit: Inherit via internal rule```.

When you select `Inherit: Inherit via internal rule`, the block calculates the output word length and fraction length automatically. The equations that the block uses to calculate the ideal output word length and fraction length depend on the setting of the Divide output by FFT length check box.

• When you select the Divide output by FFT length check box, the ideal output word and fraction lengths are the same as the input word and fraction lengths.

• When you clear the Divide output by FFT length check box, the block computes the ideal output word and fraction lengths according to the following equations:

`$W{L}_{idealoutput}=W{L}_{input}+floor\left({\mathrm{log}}_{2}\left(FFTlength-1\right)\right)+1$`
`$F{L}_{idealoutput}=F{L}_{input}$`

Using these ideal results, the internal rule then selects word lengths and fraction lengths that are appropriate for your hardware. For more information, see Inherit via Internal Rule.

• An expression that evaluates to a valid data type, for example, `fixdt(1,16,0)`

Click the Show data type assistant button to display the Data Type Assistant, which helps you set the Output data type parameter.

Output Minimum

Specify the minimum value that the block should output. The default value is `[]` (unspecified). Simulink software uses this value to perform:

• Simulation range checking (see Signal Ranges)

• Automatic scaling of fixed-point data types

Output Maximum

Specify the maximum value that the block should output. The default value is `[]` (unspecified). Simulink software uses this value to perform:

• Simulation range checking (see Signal Ranges)

• Automatic scaling of fixed-point data types

Lock data type settings against changes by the fixed-point tools

Select this parameter to prevent the fixed-point tools from overriding the data types you specify on the block mask.

## Examples

See Transform Frequency-Domain Data into Time Domain in the DSP System Toolbox User's Guide.

## References

[1] Orfanidis, S. J. Introduction to Signal Processing. Upper Saddle River, NJ: Prentice Hall, 1996, p. 497.

[2] Proakis, John G. and Dimitris G. Manolakis. Digital Signal Processing, 3rd ed. Upper Saddle River, NJ: Prentice Hall, 1996.

[4] Frigo, M. and S. G. Johnson, "FFTW: An Adaptive Software Architecture for the FFT,"Proceedings of the International Conference on Acoustics, Speech, and Signal Processing, Vol. 3, 1998, pp. 1381-1384.

## Supported Data Types

PortSupported Data Types

Input

• Double-precision floating point

• Single-precision floating point

• Fixed point

• 8-, 16-, and 32-bit signed integers

• 8-, 16-, and 32-bit unsigned integers

Output

• Double-precision floating point

• Single-precision floating point

• Fixed point (signed only)

• 8-, 16-, and 32-bit signed integers

 FFT DSP System Toolbox IDCT DSP System Toolbox Pad DSP System Toolbox `bitrevorder` Signal Processing Toolbox `fft` MATLAB `ifft` MATLAB