littlewoodPaleySum

Littlewood-Paley sum

Syntax

lpsum = littlewoodPaleySum(sf)

lpsum = littlewoodPaleySum(sf,fb)

[lpsum,f] = littlewoodPaleySum(___)

Description

lpsum = littlewoodPaleySum(sf) returns the Littlewood-Paley sum for the scattering filter banks in sf, the scattering decomposition framework. lpsum is an M-by-L matrix, where M is the number of elements in the Fourier transform of the scattering filters, and L is the number of scattering filter banks. The columns of lpsum are ordered by the position of the filter bank in the scattering decomposition. For example, the first column of lpsum corresponds to the filter bank used for the first-order scattering coefficients.

Since the scattering transform is contractive, the Littlewood-Paley sums will not exceed one.

lpsum = littlewoodPaleySum(sf,fb) returns the Littlewood-Paley sum for the specified filter bank fb in sf. The argument fb is a positive integer between 1 and the number of filter banks in sf inclusive. The number of filter banks in sf is equal to the number of specified QualityFactors in sf.

example

[lpsum,f] = littlewoodPaleySum(___) returns the frequencies for the Littlewood-Paley sum. If you specify a sampling frequency in sf, f is in hertz. If you do not specify a sampling frequency, f is in cycles/sample. You can use these output arguments with any of the input syntaxes shown previously.

Examples

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Plot Littlewood-Paley Sum

Open Live Script

Create a scattering decomposition framework with three filter banks for data sampled at 25 Hz.

sf = waveletScattering('QualityFactors',[8 4 1],...
    'SamplingFrequency',25)

sf = 
  waveletScattering with properties:

          SignalLength: 1024
       InvarianceScale: 20.4800
        QualityFactors: [8 4 1]
              Boundary: "periodic"
     SamplingFrequency: 25
             Precision: "double"
    OversamplingFactor: 0

Plot the Littlewood-Paley sums for the second and third filter banks. Note that the sums do not exceed 1. This shows the filters have been normalized so that the scattering transform is contractive.

[lpsum,f] = littlewoodPaleySum(sf);
plot(f,lpsum(:,2:3))
grid on
legend('Filter Bank 2','Filter Bank 3')
xlabel('Hz')

Input Arguments

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`sf` — Scattering decomposition framework
`waveletScattering` object

Scattering decomposition framework, specified as a waveletScattering object.

`fb` — Filter bank index
positive integer

Filter bank index in the scattering decomposition framework, specified as a positive integer between 1 and the number of filter banks in sf inclusive. The number of filter banks in sf is equal to the number of specified QualityFactors in sf.

Data Types: double

Output Arguments

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`lpsum` — Littlewood-Paley sum
real-valued matrix

Littlewood-Paley sum for the filter banks in the scattering decomposition framework sf, returned as a real-valued matrix. lpsum is an M-by-L matrix, where M is the number of elements in the Fourier transform of the scattering filters and L is the number of scattering filter banks. For example, the first column of lpsum corresponds to the filter bank used for the first-order scattering coefficients.

`f` — Frequencies
real-valued vector

Frequencies for the Littlewood-Paley sum, returned as a real-valued vector. If you specify a sampling frequency in sf, f is in hertz. If you do not specify a sampling frequency, f is in cycles/sample.

Data Types: double