Chirp Z-Transform (CZT)

From Wikipedia:

The chirp Z-transform (CZT) is a generalization of the discrete Fourier transform (DFT). While the DFT samples the Z plane at uniformly-spaced points along the unit circle, the chirp Z-transform samples along spiral arcs in the Z-plane, corresponding to straight lines in the S plane. The DFT, real DFT, and zoom DFT can be calculated as special cases of the CZT.

Getting Started

You can install the CZT package using pip:

# to install the latest release (from PyPI)
pip install czt

# to install the latest commit (from GitHub)
git clone https://github.com/garrettj403/CZT.git
cd CZT
pip install -e .

# to install dependencies for examples
pip install -e .[examples]

# to install dependencies for testing
pip install -e .[testing]

Example

Consider the following time-domain signal:

<p align="center"> <img src="https://raw.githubusercontent.com/garrettj403/CZT/main/examples/results/signal.png" width="500"> </p>

This is an exponentially decaying sine wave with some distortion from higher-order frequencies. We can convert this signal to the frequency-domain to investigate the frequency content using the Chirp Z-Transform (CZT):

<p align="center"> <img src="https://raw.githubusercontent.com/garrettj403/CZT/main/examples/results/freq-domain.png" width="500"> </p>

Note that the CZT also allows us to calculate the frequency response over an arbitrary frequency range:

<p align="center"> <img src="https://raw.githubusercontent.com/garrettj403/CZT/main/examples/results/zoom-czt.png" width="500"> </p>

We can see that the signal has frequency components at 1 kHz, 2.5 kHz and 3.5 kHz. To remove the distortion and isolate the 1 kHz signal, we can apply a simple window in the frequency-domain:

<p align="center"> <img src="https://raw.githubusercontent.com/garrettj403/CZT/main/examples/results/windowed-freq-domain.png" width="500"> </p>

Finally, we can use the Inverse Chirp Z-Transform (ICZT) to transform back to the time domain:

<p align="center"> <img src="https://raw.githubusercontent.com/garrettj403/CZT/main/examples/results/windowed-time-domain.png" width="500"> </p>

As we can see, we were able to remove the higher-order frequencies that were distorting our 1 kHz signal.

You can find this example and others in the examples/ directory.

References

• L. Rabiner, R. Schafer and C. Rader, "The chirp z-transform algorithm," IEEE Transactions on Audio and Electroacoustics, vol. 17, no. 2, pp. 86-92, Jun. 1969, doi: 10.1109/TAU.1969.1162034.

• V. Sukhoy and A. Stoytchev, "Generalizing the inverse FFT off the unit circle," Scientific Reports, vol. 9, no. 14443, Oct. 2019, doi: 10.1038/s41598-019-50234-9.

• Chirp Z-Transform (Wikipedia)

• Discrete Fourier Transform (Wikipedia)