# Inverse FFT (Inverse Real FFT) (G Dataflow)

Computes the inverse discrete Fourier transform (IDFT) of a sequence. You can use this node when the input sequence is the Fourier transform of a real time-domain signal.

## reset

A Boolean that specifies whether to reset the internal state of the node.

 True Resets the internal state of the node. False Does not reset the internal state of the node.

This input is available only if you wire a complex double-precision, floating-point number to FFT{x}.

Default: False

## FFT{x}

Complex valued input sequence, which should be conjugated centrosymmetric except for the first element.

This node uses only the anterior half of FFT{x}.

This input accepts the following data types:

• Complex double-precision, floating-point number
• 1D array of complex double-precision, floating-point numbers
• 2D array of complex double-precision, floating-point numbers

## sample length

Length of each set of data. The node performs computation for each set of data.

sample length must be greater than zero.

This input is available only if you wire a complex double-precision, floating-point number to FFT{x}.

Default: 100

## shift?

A Boolean that determines whether the DC component is at the center of the FFT of the input sequence.

 True The DC component is at the center of the FFT{x}. False The DC component is not at the center of the FFT{x}.

This input is available only if you wire a 1D array of complex double-precision, floating-point numbers or a 2D array of complex double-precision, floating-point numbers to FFT{x}.

How This Input Affects 1D FFT

The following table illustrates the pattern of the elements of FFT{x} with various length of the FFT, when shift? is False. Y is FFT{x} and n is the length of the FFT:

n is even (k = n/2) n is odd (k = (n-1)/2)
Array Element Corresponding Frequency Array Element Corresponding Frequency
Y 0 DC component Y 0 DC component
Y 1 $\mathrm{\Delta }f$ Y 1 $\mathrm{\Delta }f$
Y 2 $2\mathrm{\Delta }f$ Y 2 $2\mathrm{\Delta }f$
Y 3 $3\mathrm{\Delta }f$ Y 3 $3\mathrm{\Delta }f$
$\begin{array}{c}\cdot \\ \cdot \\ \cdot \end{array}$ $\begin{array}{c}\cdot \\ \cdot \\ \cdot \end{array}$ $\begin{array}{c}\cdot \\ \cdot \\ \cdot \end{array}$ $\begin{array}{c}\cdot \\ \cdot \\ \cdot \end{array}$
Y k-2 $\left(k-2\right)\mathrm{\Delta }f$ Y k-2 $\left(k-2\right)\mathrm{\Delta }f$
Y k-1 $\left(k-1\right)\mathrm{\Delta }f$ Y k-1 $\left(k-1\right)\mathrm{\Delta }f$
Y k Nyquist Frequency Y k $k\mathrm{\Delta }f$
Y k+1 $-\left(k-1\right)\mathrm{\Delta }f$ Y k+1 $-k\mathrm{\Delta }f$
Y k+2 $-\left(k-2\right)\mathrm{\Delta }f$ Y k+2 $-\left(k-1\right)\mathrm{\Delta }f$
$\begin{array}{c}\cdot \\ \cdot \\ \cdot \end{array}$ $\begin{array}{c}\cdot \\ \cdot \\ \cdot \end{array}$ $\begin{array}{c}\cdot \\ \cdot \\ \cdot \end{array}$ $\begin{array}{c}\cdot \\ \cdot \\ \cdot \end{array}$
Y n-3 $-3\mathrm{\Delta }f$ Y n-3 $-3\mathrm{\Delta }f$
Y n-2 $-2\mathrm{\Delta }f$ Y n-2 $-2\mathrm{\Delta }f$
Y n-1 $-\mathrm{\Delta }f$ Y n-1 $-\mathrm{\Delta }f$

The following table illustrates the pattern of the elements of FFT{x} with various length of the FFT, when shift? is True. Y is FFT{x} and n is the length of the FFT:

n is even (k = n/2) n is odd (k = (n-1)/2)
Array Element Corresponding Frequency Array Element Corresponding Frequency
Y 0 -(Nyquist Frequency) Y 0 $-k\mathrm{\Delta }f$
Y 1 $-\left(k-1\right)\mathrm{\Delta }f$ Y 1 $-\left(k-1\right)\mathrm{\Delta }f$
Y 2 $-\left(k-2\right)\mathrm{\Delta }f$ Y 2 $-\left(k-2\right)\mathrm{\Delta }f$
Y 3 $-\left(k-3\right)\mathrm{\Delta }f$ Y 3 $-\left(k-3\right)\mathrm{\Delta }f$
$\begin{array}{c}\cdot \\ \cdot \\ \cdot \end{array}$ $\begin{array}{c}\cdot \\ \cdot \\ \cdot \end{array}$ $\begin{array}{c}\cdot \\ \cdot \\ \cdot \end{array}$ $\begin{array}{c}\cdot \\ \cdot \\ \cdot \end{array}$
Y k-2 $-2\mathrm{\Delta }f$ Y k-2 $-2\mathrm{\Delta }f$
Y k-1 $-\mathrm{\Delta }f$ Y k-1 $-\mathrm{\Delta }f$
Y k DC component Y k DC component
Y k+1 $\mathrm{\Delta }f$ Y k+1 $\mathrm{\Delta }f$
Y k+2 $2\mathrm{\Delta }f$ Y k+2 $2\mathrm{\Delta }f$
$\begin{array}{c}\cdot \\ \cdot \\ \cdot \end{array}$ $\begin{array}{c}\cdot \\ \cdot \\ \cdot \end{array}$ $\begin{array}{c}\cdot \\ \cdot \\ \cdot \end{array}$ $\begin{array}{c}\cdot \\ \cdot \\ \cdot \end{array}$
Y n-3 $\left(k-3\right)\mathrm{\Delta }f$ Y n-3 $\left(k-2\right)\mathrm{\Delta }f$
Y n-2 $\left(k-2\right)\mathrm{\Delta }f$ Y n-2 $\left(k-1\right)\mathrm{\Delta }f$
Y n-1 $\left(k-1\right)\mathrm{\Delta }f$ Y n-1 $k\mathrm{\Delta }f$

How This Input Affects 2D FFT

The illustration below shows the effect of shift? on the 2D FFT result:

2D input signals FFT without shift FFT with shift

Default: False

## error in

Error conditions that occur before this node runs.

The node responds to this input according to standard error behavior.

Standard Error Behavior

Many nodes provide an error in input and an error out output so that the node can respond to and communicate errors that occur while code is running. The value of error in specifies whether an error occurred before the node runs. Most nodes respond to values of error in in a standard, predictable way.

error in does not contain an error error in contains an error
If no error occurred before the node runs, the node begins execution normally.

If no error occurs while the node runs, it returns no error. If an error does occur while the node runs, it returns that error information as error out.

If an error occurred before the node runs, the node does not execute. Instead, it returns the error in value as error out.

Default: No error

## x

Inverse real FFT of the complex valued input sequence.

This output can return a 1D array of double-precision, floating-point numbers or a 2D array of double-precision, floating-point numbers.

## error out

Error information.

The node produces this output according to standard error behavior.

Standard Error Behavior

Many nodes provide an error in input and an error out output so that the node can respond to and communicate errors that occur while code is running. The value of error in specifies whether an error occurred before the node runs. Most nodes respond to values of error in in a standard, predictable way.

error in does not contain an error error in contains an error
If no error occurred before the node runs, the node begins execution normally.

If no error occurs while the node runs, it returns no error. If an error does occur while the node runs, it returns that error information as error out.

If an error occurred before the node runs, the node does not execute. Instead, it returns the error in value as error out.

## Algorithm Definition for 1D Inverse FFT

For a 1D, N-sample, frequency domain sequence Y, the inverse discrete Fourier transform (IDFT) is defined as:

${X}_{n}=\frac{1}{N}\sum _{k=0}^{N-1}Yk{e}^{j2\pi kn/N}$

for n = 0, 1, 2, ..., N-1.

## Algorithm Definition for 2D Inverse FFT

For a 2D, M-by-N frequency domain array Y, the inverse discrete Fourier transform (IDFT) is defined as:

$X\left(m,n\right)=\frac{1}{MN}\sum _{u=0}^{M-1}\sum _{v=0}^{N-1}Y\left(u,v\right){e}^{j2\pi mu/M}{e}^{j2\pi nv/N}$

for m = 0, 1, ..., M-1, n=0, 1, ..., M-1.

## Conjugated Centrosymmetric Property of 1D Inverse Real FFT

When shift? is False and FFT{x} is the Fourier transform of a 1D real time-domain signal with length N, the posterior half part of FFT{x} can be constructed by the anterior half part. The centrosymmetric relationship between the anterior and posterior half part of FFT{x} can be written as

${f}_{N-1}={{f}_{i}}^{*},\text{}i=1,\text{}2,\text{}...,\text{}⌊\frac{N}{2}⌋$

where f i is the element in FFT{x}.

This node uses only the anterior half part, from f 0 to $f_⌊\frac{N}{2}⌋$ to perform the inverse real FFT, where $⌊•⌋$ means the floor operation.

## Conjugated Centrosymmetric Property of 2D Inverse Real FFT

When shift? is False and FFT{x} is the Fourier transform of a 2D real time-domain signal with M rows and N columns, the lower half part of FFT{x} can be constructed by the upper half part. The centrosymmetric relationship between the upper and lower half part of FFT{x} can be written as

$\left\{\begin{array}{c}{f}_{M-i,j}={f}_{i,N-j}^{*}\text{}i=1,\text{}2,\text{}...,\text{}⌊\frac{M}{2}⌋,\text{}j=1,\text{}2,\text{}...,\text{}N-1\\ {f}_{M-i,j}={f}_{i,j}^{*}\text{}i=1,\text{}2,\text{}...,\text{}⌊\frac{M}{2}⌋,\text{}j=0\end{array}$

where f i,j is the element in FFT{x}.

This node uses only the upper half part, from f 0,0 to $f_⌊\frac{M}{2}⌋,N-1$ to perform the inverse real FFT, where $⌊•⌋$ means the floor operation.

Where This Node Can Run:

Desktop OS: Windows

FPGA: Not supported

Web Server: Not supported in VIs that run in a web application