# Chirp-Z Transform (G Dataflow)

Version:

Computes the Chirp-Z transform of a sequence. The Chirp-Z transform algorithm is also known as Bluestein's FFT algorithm.

## x

A real vector.

This input can be a 1D array of double-precision, floating-point numbers or a 1D array of complex double-precision, floating-point numbers.

## number of bins

The length of the sequence after the Chirp-Z transform.

If number of bins is less than or equal to 0, this node sets number of bins to the length of 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.

Default: 1 + 0i

## 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

## algorithm

Transform method to use.

Name Value Description
direct 0 Computes the Chirp-Z transform using the direct form method. Use this method if the size of x or the number of bins is small.
frequency domain 1 Computes the Chirp-Z transform using an FFT-based technique. Use this method if the size of x or the number of bins is large.

Computing the Chirp-Z Transform Using the Direct Form Method

The direct form method computes the Chirp-Z transform as follows:

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

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

where N is the length of X.

Computing the Chirp-Z Transform Using the Frequency Domain Method

The direct form can be reformulated with the convolution between gi and ${W}^{-{i}^{2}/2}$ as follows:

${Y}_{k}={W}^{{k}^{2}/2}\underset{i=0}{\overset{N-1}{\sum }}{g}_{i}{W}^{-{\left(k-i\right)}^{2}/2}$

where ${g}_{i}={x}_{i}{A}^{-i}{W}^{-{i}^{2}/2}$. You can perform the convolution operation using an FFT-based technique.

Default: frequency domain

## chirp-z{x}

The Chirp-Z transform of the 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.

## Evaluating the Chirp-Z Transform Along a Spiral

This node evaluates the z transform along a spiral in the z-plane at the following points:

${z}_{k}=A{W}^{-k}$

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

where

• M is number of bins
• A is starting point
• W is increment

The following illustration shows samples in the z-plane.

You can use the Chirp-Z transform to calculate the FFT result. Set starting point and increment as follows:

$\mathrm{starting}\text{\hspace{0.17em}}\mathrm{point}=1$
$\mathrm{increment}={e}^{-j\frac{2\pi }{N}}$

where N is the length of X. Let number of bins equal N. When the samples are evenly distributed on the unit circle, as shown in the following image, the Chirp-Z transform is the same as the fast Fourier transform (FFT).

You also can use the Chirp-Z transform to calculate the partial FFT result. Set starting point and increment as follows:

$\mathrm{starting}\text{\hspace{0.17em}}\mathrm{point}={e}^{j\frac{2\pi }{N}s}$
$\mathrm{increment}={e}^{-j\frac{2\pi }{N}}$

where s is the start bin and N is the length of X. This is useful when you are interested in only a small portion of a spectrum of a very long signal, as shown in the following image.

Where This Node Can Run:

Desktop OS: Windows

FPGA: Not supported

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