PSECH - Sampled, periodized hyperbolic secant

Usage

g=psech(L);
g=psech(L,tfr);
g=psech(L,s,'samples);
[g,tfr]=psech( ... );

Input parameters

Output parameters

Description

psech(L,tfr) computes samples of a periodized hyperbolic secant. The function returns a regular sampling of the periodization of the function

The returned function has norm equal to 1.

The parameter tfr determines the ratio between the effective support of g and the effective support of the DFT of g. If \(tfr>1\) then g has a wider support than the DFT of g.

psech(L) does the same setting \(tfr=1\).

psech(L,s,'samples') returns a hyperbolic secant with an effective support of s samples. This means that approx. 96% of the energy or 74% or the area under the graph is contained within s samples. This is equivalent to psech(L,s^2/L).

[g,tfr] = psech( ... ) additionally returns the time-to-frequency support ratio. This is useful if you did not specify it (i.e. used the 'samples' input format).

The function is whole-point even. This implies that fft(psech(L,tfr)) is real for any L and tfr.

If this function is used to generate a window for a Gabor frame, then the window giving the smallest frame bound ratio is generated by psech(L,a*M/L).

Examples:

This example creates a psech function, and demonstrates that it is its own Discrete Fourier Transform:

g=psech(128);

% Test of DFT invariance: Should be close to zero.
norm(g-dft(g))

The next plot shows the psech in the time domain compared to the Gaussian:

plot((1:128)',fftshift(pgauss(128)),...
     (1:128)',fftshift(psech(128)));
legend('pgauss','psech');

The next plot shows the psech in the frequency domain on a log scale compared to the Gaussian:

hold all;
magresp(pgauss(128),'dynrange',100);
magresp(psech(128),'dynrange',100);
legend('pgauss','psech');

The next plot shows psech in the time-frequency plane:

sgram(psech(128),'tc','nf','lin');