hilbiir.m

数字通信第四版原书的例程

function [aa, b, c, d, sv] = hilbiir(stp, dly, bandwidth, tol)
%HILBIIR Designs a Hilbert transform digital IIR filter.
%       HILBIIR produces a plot showing the impulse responses of an ideal
%       Hilbert transform filter and a designed fourth order digital Hilbert
%       transform IIR filter. The filter sample time is 2/7 second. The filter
%       time delay is one second. The ideal Hilbert transform filter impulse
%       response of the system is 1/(t-1)/pi. No compensator is applied.
%
%       HILBIIR(ST) produces a plot showing the impulse responses of an ideal
%       theoretical Hilbert transform filter and a designed fourth order
%       digital Hilbert transform IIR filter. The filter sample time is
%       ST (sec). The filter time delay equals to 3.5*ST (a suggested value).
%       The order of the filter is determined by using TOL=0.05. No
%       compensator is added. The filter is assumed to be used for bandwidth
%       1/ST/2 input signal.
%
%       HILBIIR(ST, DLY) produces a plot with impulse response of a designed
%       Hilbert transform filter with sample time as ST, delay as DLY (sec).
%       No compensator is used. For an accurate calculation, the parameters
%       should be chosen such that DLY is at least a few times larger than
%       ST, and rem(DLY, ST) = ST/2.
%
%       HILBIIR(ST, DLY, BANDWIDTH) produces a plot with impulse response of
%       a designed Hilbert transform filter with sample time as ST, delay as
%       DLY. The filter design uses a compensator for the input signal. The
%       input signal has its bandwidth as BANDWIDTH. When BANDWIDTH = 0 or 
%       BANDWIDTH > 1/TS/2, no compensator is added.
%
%       HILBIIR(ST, DLY, BANDWIDTH, TOL) produces a plot with impulse response
%       of a designed Hilbert transform filter with sample time ST, delay as
%       DLY, and specified bandwidth as BANDWIDTH. TOL specifies the tolerance
%       in the filter design. The order of the filter is determined by
%       truncated-singular-value/maximum-singular-value < TOL; when TOL is
%       larger than or equal to one, TOL is the order of the filter. If
%       BANDWIDTH = 0 or BANDWIDTH > 1/TS/2, the filter order is exact TOL,
%       since no compensator is added. Otherwise, the filter order is
%       TOL + max(3,ceil(TOL)/2) with a compensator added.
%
%       [NUM, DEN] = HILBIIR(...) outputs the transfer function NUM(s)/DEN(s).
%
%       [NUM, DEN, SV] = HILBIIR(...) outputs the additional parameter SV, the
%       singular value in the calculation.
%
%       [A, B, C, D] = HILBIIR(...) outputs state-space solution.
%
%       [A, B, C, D, SV] = HILBIIR(...) outputs state-space solution and
%       singular values.
%
%       See also HILBATF

%       Wes Wang 4/4/94, 10/10/95
%       Copyright (c) 1995-96 by The MathWorks, Inc.
%       $Revision: 1.1 $  $Date: 1996/04/01 18:00:48 $

%   Because a idea Hilbert transform filter is a non-causal filter, the
%   specified DLY has to be larger than 0. The idea transfer function of
%   the HILBIIR transfer is H(s) = - j sign(s). The impulse response of
%   the filter is h(t) = 1/t/pi. The time delay is choosen such that the 
%   response before 0 is less important. 
%   SVD method has been used in determining the filter.
%   For an accurate calculation, the parameters should be choosen such that
%   DLY is at leat a few times larger than ST, and rem(DLY, ST) = ST/2.
%   


if nargin < 1
    stp = 2/7;
    bandwidth = 0;
end;
if nargin < 2
    dly = stp*7/2;
    bandwidth = 0;
end;
if nargin < 3
    bandwidth = 0;
end;
if nargin < 4
    tol = 0.05;
end;
if nargin < 1
    bandwidth = 0;
end;
if isempty(stp)
    stp = 2/7;
end;
if isempty(dly)
    dly = stp*7/2;
end;
if isempty(bandwidth)
    badwidth = 0;
end;
if isempty(tol)
    tol = 0.05;
end;
if bandwidth > 1/stp/2
    bandwidth = 0;
end;

% DLY must be positive
if dly <= 0
    error('The time delay factor for designing HILBAFT filter must be larger than 0')
end;

sz = max(2 * dly / stp, tol*1.3 + 2);
if dly / stp < 3
   disp('Warning: the time delay is too small comparing the step size in HILBIIR')
   sz = 6;
end;

% determine the time.
t = [-dly : stp : max(3 * dly, 2*sz*stp)];
ind = find(abs(t) < eps);
if ~isempty(ind)
    t(ind) = eps*ones(1,length(ind));
end

%impulse response of the system
x = 1 ./ t / pi;
if ~isempty(ind)
    x(ind) = zeros(1,length(ind));
    t(ind) = zeros(1,length(ind));
end

% discrete-time model realization.
[a, b, c, d, sv] = imp2sys(x, tol);

% add a compensator to smooth the gain over the entire input signa frequency.
if bandwidth ~= 0
    [a, b] = ss2tf(a,b,c,d,1);
    bandwidth = min(bandwidth, 1/stp/2);
    [h, w] = freqz(a, b, [0:bandwidth*.01:bandwidth]*stp);
    [cn, cd] = yulewalk(ceil(max(3, length(b)/2)), [w(:); 1],...
            [1./abs(h(:)); 1/abs(h(length(h)))]);
    a = conv(a, cn);
    b = conv(b, cd);
    if nargout > 3
        [a, b, c, d] = tf2ss(a, b);
    else
        c = sv;
    end
elseif nargout < 4
    [a, b] = ss2tf(a, b, c, d, 1);
    c = sv;
end;

if nargout < 2
    [hh,t] = impz(a, b);
    h = ishold;
    tt = t * stp - dly;
    x = 1 ./ tt / pi;
    stem(t*stp,hh);
    hold on
    plot(t*stp, x, 'r*')
    ax=get(get(gcf,'child'),'Ylim');
    plot([dly, dly], ax, 'b');
    xlabel('time (second)')
    title('Impulse Response: yellow: IIR; red: ideal; blue symmetric line')
    if ~h
        hold off
    end
    return;
else
    aa = a;
end