# Inverse Chirp-Z Transform (G Dataflow)

Computes the inverse Chirp-Z transform of a sequence.

## chirp-z{x}

The complex valued input sequence.

The length of chirp-z{x} must be greater than or equal to number of samples.

## number of samples

Length of the sequence after the inverse Chirp-Z transform.

number of samples must be less than or equal to the length of chirp-z{x}. If number of samples is less than or equal to 0, the node sets number of samples to the length of chirp-z{x}.

Default: -1

## starting point

The point at which this node begins evaluating the Chirp-Z transform.

If starting point is 0, the node returns an error.

Default: 1 + 0i

## increment

The increment between each point to evaluate for the Chirp-Z transform.

increment cannot be 0.

Avoiding Singular Cases of the Inverse Chirp-Z Transform

To avoid singular cases of the inverse Chirp-Z transform, increment must be different from where and N is number of samples.

Default: 1 + 0.1i

## 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

The inverse Chirp-Z transform of the complex valued input sequence.

## 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 for Computing the Inverse Chirp-Z Transform

If Y represents the input sequence chirp-z{x}, the following equation shows how this node performs the Chirp-Z transform to obtain the output sequence x:

${y}_{k}=\underset{n=0}{\overset{N-1}{\sum }}{x}_{n}{\left(A{W}^{-k}\right)}^{-n}$

for k=0, 1, ..., M-1

where

• N is the length of x (number of samples)
• M is the length of chirp-z{x}
• A is the starting point
• W is the increment
• Xn is the nth element of x
• Yk is the kth element of chirp-z{x}

## Implementing the Inverse Chirp-Z Transform Using a Convolution-Based Method

This node employs a convolution-based method to implement the inverse Chirp-Z transform according to the following equations.

${x}_{n}={h}_{n}×{m}_{n}$

with

${h}_{n}=\left\{\begin{array}{c}\begin{array}{cc}\underset{k=0}{\overset{N-1}{\sum }}{C}_{k}{y}_{k}{z}_{k}^{n}& n\ge 0\end{array}\\ \begin{array}{c}\begin{array}{cc}0& n<0\end{array}\end{array}\end{array}$

where

${C}_{0}=\underset{p=1}{\overset{N-1}{\prod }}1/\left(1-{W}^{-P}\right)$
${C}_{k}/{C}_{k-1}=\left(1-{W}^{k}{W}^{-N}\right)/\left(1-{W}^{k}\right)$
$k=1,\text{\hspace{0.17em}}2,...,\text{\hspace{0.17em}}N-1\text{\hspace{0.17em}}$
${z}_{k}=0,\text{\hspace{0.17em}}1,\text{\hspace{0.17em}}...,\text{\hspace{0.17em}}N-1$

mn can be obtained from its z-transform M(z):

$M\left(z\right)=\underset{p=0}{\overset{N-1}{\prod }}\left(1-zp{z}^{-1}\right)$

Where This Node Can Run:

Desktop OS: Windows

FPGA: This product does not support FPGA devices

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