PSECH Sampled, periodized hyperbolic secant
Usage: g=psech(L);
g=psech(L,tfr);
g=psech(L,s,'samples);
[g,tfr]=psech( ... );
Input parameters:
L : Length of vector.
tfr : ratio between time and frequency support.
Output parameters:
g : The periodized hyperbolic cosine.
PSECH(L,tfr) computes samples of a periodized hyperbolic secant.
The function returns a regular sampling of the periodization
of the function sech(pi*x)
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');
References:
A. J. E. M. Janssen and T. Strohmer. Hyperbolic secants yield Gabor
frames. Appl. Comput. Harmon. Anal., 12(2):259--267, 2002.
Url: http://ltfat.github.io/doc/fourier/psech.html
See also: pgauss, pbspline, pherm.
Package: ltfat