# Chirp-Z Transform (G Dataflow)

Version:
Last Modified: January 9, 2017

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.

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.

## 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