Function: psech
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