数字信号处理-希尔伯特变换

Hilbert变换时信号分析的一个重要工具，在信号处理系统和通信系统中是非常有用的。主要作用有以下三点：

1、用来构建解析信号，使信号频谱仅含有正频率成分，从而降低信号的抽样率；

2、可以用来表示带通信号，从而为无线电通信中的信号调制提供了一种方法；

3、与其他变换及分解结合在一起，进行非平稳信号的频谱分析。

希尔伯特变换定义如下：

希尔伯特变换matlab代码

function x = hilbert(xr,n)
%HILBERT  Discrete-time analytic signal via Hilbert transform.
%   X = HILBERT(Xr) computes the so-called discrete-time analytic signal
%   X = Xr + i*Xi such that Xi is the Hilbert transform of real vector Xr.
%   If the input Xr is complex, then only the real part is used: Xr=real(Xr).
%   If Xr is a matrix, then HILBERT operates along the columns of Xr.
%
%   HILBERT(Xr,N) computes the N-point Hilbert transform.  Xr is padded with
%   zeros if it has less than N points, and truncated if it has more.
%
%   For a discrete-time analytic signal X, the last half of fft(X) is zero,
%   and the first (DC) and center (Nyquist) elements of fft(X) are purely real.
%
%   EXAMPLE:
%          Xr = [1 2 3 4];
%          X = hilbert(Xr)
%          % produces X=[1+1i 2-1i 3-1i 4+1i] such that Xi=imag(X)=[1 -1 -1 1] is the
%          % Hilbert transform of Xr, and Xr=real(X)=[1 2 3 4].  Note that the last half
%          % of fft(X)=[10 -4+4i -2 0] is zero (in this example, the last half is just
%          % the last element).  Also note that the DC and Nyquist elements of fft(X)
%          % (10 and -2) are purely real.
%
%   See also FFT, IFFT, ENVELOPE.%   Copyright 1988-2008 The MathWorks, Inc.%   References:
%     [1] Alan V. Oppenheim and Ronald W. Schafer, Discrete-Time
%     Signal Processing, 2nd ed., Prentice-Hall, Upper Saddle River,
%     New Jersey, 1998.
%
%     [2] S. Lawrence Marple, Jr., Computing the discrete-time analytic
%     signal via FFT, IEEE Transactions on Signal Processing, Vol. 47,
%     No. 9, September 1999, pp.2600--2603.%#codegennarginchk(1,2);
if isempty(coder.target)if nargin == 1x = hilbert_ml(xr);elsex = hilbert_ml(xr,n);end
elseif nargin == 1x = hilbert_cg(xr);elsecoder.internal.prefer_const(n);x = hilbert_cg(xr,n);end
end%--------------------------------------------------------------------------function x = hilbert_ml(xr,n)
if nargin<2, n=[]; end
if ~isreal(xr)warning(message('signal:hilbert:Ignore'))xr = real(xr);
end
% Work along the first nonsingleton dimension
[xr,nshifts] = shiftdim(xr);
if isempty(n)n = size(xr,1);
end
x = fft(xr,n,1); % n-point FFT over columns.
h  = zeros(n,~isempty(x)); % nx1 for nonempty. 0x0 for empty.
if n > 0 && 2*fix(n/2) == n% even and nonemptyh([1 n/2+1]) = 1;h(2:n/2) = 2;
elseif n>0% odd and nonemptyh(1) = 1;h(2:(n+1)/2) = 2;
end
x = ifft(x.*h(:,ones(1,size(x,2))),[],1);% Convert back to the original shape.
x = shiftdim(x,-nshifts);%--------------------------------------------------------------------------function x = hilbert_cg(xr,n)
narginchk(1,2);
if nargin == 2coder.internal.prefer_const(n);
end
if ~isreal(xr)coder.internal.warning('signal:hilbert:Ignore');if nargin == 2x = hilbert_cg(real(xr),n);elsex = hilbert_cg(real(xr));endreturn
end
% Operate over first non-singleton dimension.
dim = HilbertNonSingletonDim(xr);
if nargin == 1x = coder.nullcopy(complex(xr));if isvector(xr)x(:) = HilbertColumnwise(xr(:));elseif dim == 1x(:) = HilbertColumnwise(xr);else% Use reshape rather than shiftdim because the code generation% shiftdim ultimately requires a fixed number of shifts.ncols = coder.internal.prodsize(xr,'above',dim);xrshifted = reshape(xr,size(xr,dim),ncols);x(:) = HilbertColumnwise(xrshifted);end
elsesz = size(xr);sz(dim) = n;x = coder.nullcopy(zeros(sz,'like',complex(xr)));if isvector(xr)x(:) = HilbertColumnwise(xr(:),n);elseif dim == 1x(:) = HilbertColumnwise(xr,n);else% Use reshape rather than shiftdim because the code generation% shiftdim ultimately requires a fixed number of shifts.ncols = coder.internal.prodsize(xr,'above',dim);xrshifted = reshape(xr,size(xr,dim),ncols);x(:) = HilbertColumnwise(xrshifted,n);end
end%--------------------------------------------------------------------------function x = HilbertColumnwise(xr,n)
if nargin == 2coder.internal.prefer_const(n);x = fft(xr,n,1); % n-point FFT over columns.
elsen = size(xr,1);x = fft(xr,[],1); % n-point FFT over columns.
end
nrows = coder.internal.indexInt(max(0,n));
ncols = coder.internal.prodsize(x,'above',1);
ONE = coder.internal.indexInt(1);
halfn = eml_rshift(nrows,ONE);
if eml_bitand(nrows,ONE) == 0lastIndexToDouble = halfn;
elselastIndexToDouble = halfn + 1;
end
firstIndexToZero = halfn + 2;
for j = 1:ncolsfor i = 2:lastIndexToDoublex(i,j) = x(i,j)*2;endfor i = firstIndexToZero:nrowsx(i,j) = 0;end
end
x = ifft(x,[],1);%--------------------------------------------------------------------------function dim = HilbertNonSingletonDim(x)
% Like MATLAB's first nonsingleton dim except for scalars returns 1 instead
% of 2.
ONE = coder.internal.indexInt(1);
dim = ONE;
for k = ONE:eml_ndims(x)if size(x,k) ~= 1dim = k;breakend
end%--------------------------------------------------------------------------

希尔伯特变换matlab实战代码--提取信号的瞬时振幅、瞬时相位和瞬时频率

%希尔伯特变换demo
clc
clear
Fs = 2e3;
ts = 0:1/Fs:5;
sig = sin(ts*2*pi*50);
sig_ht = hilbert(sig);
sig_r = real(sig_ht);
sig_i = imag(sig_ht);
%%瞬时振幅
IA = sqrt(sig_r.^2+sig_i.^2);
subplot(4,1,1)
plot(ts,sig)
subplot(4,1,2)
plot(ts,IA)
%%瞬时相位
IP = angle(sig_ht);
subplot(4,1,3)
plot(ts,IP)
%%瞬时频率
d_t = diff(ts);
d_IP = diff(IP);
IF = (d_IP./d_t)/2/pi;
subplot(4,1,4)
plot(ts(2:end),IF)

结果展示

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