Is there a Barcilon-Temes window function that works as well as in harris's classic paper?

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Paul McKenzie 2016-11-23

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编辑： Star Strider 2016-12-19

The batrhewin.m function on the matlab server is not it! harris;s recipe for barcilon-temes doesn't seem to work, but then his recipe for dolph-chebyshev doesn't work either.

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Paul McKenzie 2016-11-28

编辑：Walter Roberson 2016-11-28

在 MATLAB Online 中打开

I was looking for a symmetric Barcilon-Temes; the link you cited seems not to produce a symmetric window.

I went back to the original Barcilon-Temes paper (Optimum Impulse Response and the van der Maas function), which actually is concerned with a continuous window rather than a discrete window, but suggested the following symmetric candidate function, which, despite harris's suggestion, is actually easier to define in the 'time' domain, but it is still seems not quite what harris used. Perhaps some scaling of x would come closer.

function w=bartem(N,a)
%w=bartem(N,a) computes N-pt Barcilon-Temes window (?) with parameter a
% This has slightly better sidelobe performance for a given parameter 'a'
% than the function used in harris's paper:
%       a       harris    this fn
%       3.0   -53 dB   -53 dB
%       3.5    -58 dB  -61 dB
%       4.0    -68 dB   -71 dB
%INPUTS:
%       N       # points of window
%       a       suppression parameter
%OUTPUT:
%       w       N-point window
c=acosh(10^a);
x=(1-N:2:N-1).'/N;y=sqrt(1-x.*x);  %symmetric
w=2*(sinh(c)*cosh(c*y)-cosh(c)*y.*sinh(c*y))./(x.*x*(c+sinh(c)*cosh(c)));
w(x==0)=1;                      %special case for x==0

As for the dolph-chebyshev window, I was looking for a symmetric window; the link you cited seems not to be symmetric.

My own version of symmetric dolph-chebyshev is:

function [w,b]=chebwin(N,a,o)
%w=chebwin(N,A) computes N-pt Chebyshev window with stopband suppression a
%INPUTS:
%       N       # points of window
%       a       dB of stopband suppression [or dB/20]
%OUTPUTS:
%       w       N-point window
%       b       start of stopband in bins
N1=N-1;                         %need (N-1) internally
if abs(a)<10,a=a*20;end         %dB/20 to dB
g=10^abs(.05*a);                %undb
fsf=cosh(acosh(g)/N1);          %freq scale factor
b=acos(1/fsf)*N1/pi;            %beginning of stopband
m=(0:N-2).';                    %compute N-1 spectral points
x=fsf*cos(pi*m/N1);             %input argument to Chebyshev polynomial
W=cos(N1*acos(x));            %small & large x together
w=real(ifft(real(W.*(-1).^m)));  %shift & ifft to time domain
w(1)=.5*w(1);w=[w;w(1)]/max(w);  %fix 1st and Nth points; scale

Victor van der Wolf 2016-12-6

在 MATLAB Online 中打开

Here is my JavaScript solution to the problem. I rearranged the formula until I found a chebychev component. The thing is that beta * cos(PI*k/N) can be>1 and then you can't perform an acos on it (not sure if I explain this correctly, math is not my strong point).

Anyway, this is my version, happy to receive comments:

function cheby(n, x) {
  // returns the n-th chebychev polynomal at x
  var tmp;
  if (Math.abs(x)<=1.0) {
    tmp = Math.cos(n*Math.acos(x));  // between -1 and 1
  }
  else {
    tmp = cosh(n*acosh(x)); // > 1.0
  }
  return tmp;
}
var M = (N-1) * 0.5;
var C = acosh(pow(10, a));      
var A = sinh(C);
var B = cosh(C);
var beta = cosh(C/N);
      
for (var i=0; i<N; i++) {
  var W = 0;
  for (var k=0; k<=(N-1)/2; k++) {
    var mul = 2;
    if (k==0) mul = 1;
    var tt = beta * cos(PI*k/N);
    var yk;
    if ((tt>=-1) && (tt<=1)) {
      yk = N * acos(tt);
    }
    else {
      yk = N * acosh(tt);
    }
    var nom = A*cheby(N, tt)+B*yk/C*sin(yk);
    var denom = ((C+A*B)*(pow(yk/C,2)+1));
    W += mul * nom / denom * cos( (k * 2 * PI * (i-M)) / N);
  }
  z[i] *= W / N;  // because inverse DFT, divide by N
}

I had to copy-paste several snippets, so if something seems missing, please ask.

Victor van der Wolf 2016-12-6

在 MATLAB Online 中打开

Read your reply again, here's some aditional info:

My version does use the formula for the frequency domain and then does an idft to construct the filter in the time domain.

a=3.0 --> -48 dB
a=3.5 --> -58 dB
a=4.0 --> -68 dB

So, I'm getting the Harris numbers, except at 3.0. Will we ever get it right?

Paul McKenzie 2016-12-19

编辑：Star Strider 2016-12-19

在 MATLAB Online 中打开

I finally got a version to work, but I had to add a step to remove the window's pedestal. Then I got the same result that Victor got, namely that harris's result for a=3.0 is optimistic by 5 dB. I think that harris did not oversample the spectrum adequately; if you look at his graph for a=3.0, you can just make out a sidelobe at -48 dB that is mushed into the mainlobe. My Barcilon-Temes function is shown below:

function w=bartemwin(N,a,s)
%w=bartemwin(N,a,s) computes N-pt Barcilon-Temes window with parameter a,
% using procedure from harris's paper, but with pedestal adjustment to 0.
% Comparison of sidelobe levels of this function to harris's results:
%       a  this function   harris
%       3.0  -47.2 dB        -53 dB (perhaps -48 dB?)  
%       3.5  -57.4 dB        -58 dB
%       4.0  -67.6 dB        -68 dB  
% It appears that harris's result for a=3 may be somewhat optimistic,
% perhaps the result of inadequate oversampling of the spectrum. This is
% consistent with the expectation that an increase of a by 0.5 should lower
% the sidelobe level by about 10 dB.
%INPUTS:
%       N       # points of window
%       a       suppression parameter
%       s       symmetry flag (s==0 is asymmetric, s~=0 is symmetric)
%OUTPUT:
%       w       N-point Barcilon-Temes window
%P. McKenzie Dec 2016
if nargin<3||isempty(s),s=1;end
B=10^a;C=acosh(B);A=sinh(C);beta=cosh(C/N);
H=(N-1)/2;k=[0:floor(H),-ceil(H):-1].';  %indices for N spectral points
y=N*acos(beta*cos(pi*k/N));     %y(k)
W=(-1).^k.*(A*cos(y)+(B/C)*(y.*sin(y)))./((C+A*B)*(1+(y/C).^2));
W(1)=W(1)-real(sum(W));     %adjust pedestal to 0
if s;W=W.*exp(1j*k*(pi/N));end;  %make window symmetric
w=real(ifft(W))*N;w=w/max(w);  %ifft to time domain, normalize

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Answer 1

Star Strider 2016-11-25

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Have you seen the File Exchange contribution Window Utilities?

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Answer 2

Paul McKenzie 2016-11-28

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In Windows Utilities, the only function claiming to be a Barcilon-Temes window is the aforementioned barthewin.m function. While it may be a window, it does not result in sidelobe levels comparable to the results shown in Harris's paper.

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Is there a Barcilon-Temes window function that works as well as in harris's classic paper?

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