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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.


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absAbsolute value and complex magnitude
anglePhase angle
fftFast Fourier transform
ifftInverse fast Fourier transform
fftshiftShift zero-frequency component to center of spectrum
ifftshiftInverse zero-frequency shift
dftmtxDiscrete Fourier transform matrix
fft22-D fast Fourier transform
ifft22-D inverse fast Fourier transform
instfreqEstimate instantaneous frequency
cztChirp Z-transform
goertzelDiscrete Fourier transform with second-order Goertzel algorithm
dctDiscrete cosine transform
idctInverse discrete cosine transform
envelopeSignal envelope
fwhtFast Walsh-Hadamard transform
ifwhtInverse Fast Walsh-Hadamard transform
hilbertDiscrete-time analytic signal using Hilbert transform
emdEmpirical mode decomposition
fsstFourier synchrosqueezed transform
ifsstInverse Fourier synchrosqueezed transform
hhtHilbert-Huang transform
pspectrumAnalyze signals in the frequency and time-frequency domains
spectrogramSpectrogram using short-time Fourier transform
xspectrogramCross-spectrogram using short-time Fourier transforms
stftShort-time Fourier transform
dlstftDeep learning short-time Fourier transform
stftLayerShort-time Fourier transform layer
stftmag2sigSignal reconstruction from STFT magnitude
istftInverse short-time Fourier transform
vmdVariational mode decomposition
wvdWigner-Ville distribution and smoothed pseudo Wigner-Ville distribution
xwvdCross Wigner-Ville distribution and cross smoothed pseudo Wigner-Ville distribution
ccepsComplex cepstral analysis
iccepsInverse complex cepstrum
rcepsReal cepstrum and minimum-phase reconstruction
bitrevorderPermute data into bit-reversed order
digitrevorderPermute input into digit-reversed order


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.

Featured Examples