Transforms

Fourier, chirp Z, DCT, Hilbert, cepstrum, Walsh-Hadamard

Signal Processing Toolbox™ provides functions that let you compute widely used forward and inverse transforms, including the fast Fourier transform (FFT), the discrete cosine transform (DCT), and the Walsh-Hadamard transform. Extract signal envelopes and estimate instantaneous frequencies using the analytic signal. Analyze signals in the time-frequency domain. Investigate magnitude-phase relationships, estimate fundamental frequencies, and detect spectral periodicity using the cepstrum. Compute discrete Fourier transforms using the second-order Goertzel algorithm.

Functions

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Discrete Fourier and Cosine Transforms

`abs`	Absolute value and complex magnitude
`angle`	Phase angle
`fft`	Fast Fourier transform
`ifft`	Inverse fast Fourier transform
`fftshift`	Shift zero-frequency component to center of spectrum
`ifftshift`	Inverse zero-frequency shift
`dftmtx`	Discrete Fourier transform matrix
`fft2`	2-D fast Fourier transform
`ifft2`	2-D inverse fast Fourier transform
`instfreq`	Estimate instantaneous frequency
`czt`	Chirp Z-transform
`goertzel`	Discrete-Time Fourier transform with second-order Goertzel algorithm
`dct`	Discrete cosine transform
`idct`	Inverse discrete cosine transform

Hilbert and Walsh-Hadamard Transforms

`envelope`	Signal envelope
`fwht`	Fast Walsh-Hadamard transform
`ifwht`	Inverse Fast Walsh-Hadamard transform
`hilbert`	Discrete-time analytic signal using Hilbert transform

Time-Frequency Analysis

`emd`	Empirical mode decomposition
`fsst`	Fourier synchrosqueezed transform
`ifsst`	Inverse Fourier synchrosqueezed transform
`hht`	Hilbert-Huang transform
`pspectrum`	Analyze signals in the frequency and time-frequency domains
`spectrogram`	Spectrogram using short-time Fourier transform
`xspectrogram`	Cross-spectrogram using short-time Fourier transforms
`stft`	Short-time Fourier transform
`dlstft`	Deep learning short-time Fourier transform (Since R2021a)
`stftLayer`	Short-time Fourier transform layer (Since R2021b)
`istftLayer`	Inverse short-time Fourier transform layer (Since R2024a)
`stftmag2sig`	Signal reconstruction from STFT magnitude (Since R2020b)
`istft`	Inverse short-time Fourier transform
`dlistft`	Deep learning inverse short-time Fourier transform (Since R2024a)
`vmd`	Variational mode decomposition
`wvd`	Wigner-Ville distribution and smoothed pseudo Wigner-Ville distribution
`xwvd`	Cross Wigner-Ville distribution and cross smoothed pseudo Wigner-Ville distribution

Cepstral Analysis

`cceps`	Complex cepstral analysis
`icceps`	Inverse complex cepstrum
`rceps`	Real cepstrum and minimum-phase reconstruction

Bit Reversal

`bitrevorder`	Permute data into bit-reversed order
`digitrevorder`	Permute input into digit-reversed order

Topics

Discrete Fourier and Cosine Transforms

Discrete Fourier Transform
Explore the primary tool of digital signal processing.
Chirp Z-Transform
Use the CZT to evaluate the Z-transform outside of the unit circle and to compute transforms of prime length.
Discrete Cosine Transform
Compute discrete cosine transforms and learn about their energy compaction properties.
DCT for Speech Signal Compression
Use the discrete cosine transform to compress speech signals.

Hilbert and Walsh-Hadamard Transforms

Hilbert Transform
The Hilbert transform helps form the analytic signal.
Analytic Signal for Cosine
Determine the analytic signal for a cosine and verify its properties.
Envelope Extraction
Extract the envelope of a signal using the hilbert and envelope functions.
Analytic Signal and Hilbert Transform
Generate the analytic signal for a finite block of data using the hilbert function and an FIR Hilbert transformer.
Hilbert Transform and Instantaneous Frequency
Estimate the instantaneous frequency of a monocomponent signal using the Hilbert transform. Show that the procedure does not work for multicomponent signals.
Single-Sideband Amplitude Modulation
Perform single-sideband amplitude modulation of a signal using the Hilbert transform. Single-sideband AM signals have less bandwidth than normal AM signals.
Walsh-Hadamard Transform
Learn about the Walsh-Hadamard transform, a non-sinusoidal, orthogonal transformation technique.
Walsh-Hadamard Transform for Spectral Analysis and Compression of ECG Signals
Use an electrocardiogram signal to illustrate the Walsh-Hadamard transform.

Cepstral Analysis

Complex Cepstrum — Fundamental Frequency Estimation
Use the complex cepstrum to estimate a speaker’s fundamental frequency. Compare the result with the estimate obtained with a zero-crossing method.
Cepstrum Analysis
Apply the complex cepstrum to detect echo in a signal.

Related Information

Fourier Analysis (MathWorks Teaching Resources)

Featured Examples

DFT Estimation with the Goertzel Algorithm

Use the Goertzel method to implement a DFT-based Dual-Tone Multi-Frequency detection algorithm.

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Discrete Walsh-Hadamard Transform

Apply the Walsh-Hadamard transform and its properties to spread-spectrum communications and ECG signal processing.

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Single Sideband Modulation via the Hilbert Transform

Use the discrete Hilbert transform to implement single sideband modulation, an efficient form of AM.

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