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