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## Copyright (C) 2006 Peter V. Lanspeary <pvl@mecheng.adelaide.edu.au>
##
## This program is free software; you can redistribute it and/or modify it under
## the terms of the GNU General Public License as published by the Free Software
## Foundation; either version 3 of the License, or (at your option) any later
## version.
##
## This program is distributed in the hope that it will be useful, but WITHOUT
## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more
## details.
##
## You should have received a copy of the GNU General Public License along with
## this program; if not, see <http://www.gnu.org/licenses/>.
## -*- texinfo -*-
## @deftypefn {Function File} {} ar_psd (@var{a}, @var{v})
## @deftypefnx {Function File} {} ar_psd (@var{a}, @var{v}, @var{freq})
## @deftypefnx {Function File} {} ar_psd (@var{a}, @var{v}, @var{freq}, @var{Fs})
## @deftypefnx {Function File} {} ar_psd (@dots{}, @var{range})
## @deftypefnx {Function File} {} ar_psd (@dots{}, @var{method})
## @deftypefnx {Function File} {} ar_psd (@dots{}, @var{plottype})
## @deftypefnx {Function File} {[@var{psd}, @var{f_out}] =} ar_psd (@dots{})
##
## Calculate the power spectrum of the autoregressive model
##
## @example
## @group
## M
## x(n) = sqrt(v).e(n) + SUM a(k).x(n-k)
## k=1
## @end group
## @end example
##
## where @math{x(n)} is the output of the model and @math{e(n)} is white noise.
## This function is intended for use with
## @code{[a, v, k] = arburg (x, poles, criterion)}
## which use the Burg (1968) method to calculate a "maximum entropy"
## autoregressive model of @var{x}.
##
## If the @var{freq} argument is a vector (of frequencies) the spectrum is
## calculated using the polynomial method and the @var{method} argument is
## ignored. For scalar @var{freq}, an integer power of 2, or @var{method} =
## "FFT", causes the spectrum to be calculated by FFT. Otherwise, the spectrum
## is calculated as a polynomial. It may be computationally more
## efficient to use the FFT method if length of the model is not much
## smaller than the number of frequency values. The spectrum is scaled so
## that spectral energy (area under spectrum) is the same as the time-domain
## energy (mean square of the signal).
##
## ARGUMENTS:
## All but the first two arguments are optional and may be empty.
## @itemize
## @item
## @var{a}
## list of M=(order+1) autoregressive model
## coefficients. The first element of "ar_coeffs" is the
## zero-lag coefficient, which always has a value of 1.
## @item
## @var{v}
## square of the moving-average coefficient of the AR model.
## @item
## @var{freq}
## frequencies at which power spectral density is calculated, or a scalar
## indicating the number of uniformly distributed frequency
## values at which spectral density is calculated.
## (default = 256)
## @item
## @var{Fs}
## sampling frequency (Hertz) (default=1)
## @end itemize
##
## CONTROL-STRING ARGUMENTS -- each of these arguments is a character string.
## Control-string arguments can be in any order after the other arguments.
##
## Range:
##
## 'half', 'onesided' : frequency range of the spectrum is
## from zero up to but not including sample_f/2. Power
## from negative frequencies is added to the positive
## side of the spectrum.
## 'whole', 'twosided' : frequency range of the spectrum is
## -sample_f/2 to sample_f/2, with negative frequencies
## stored in "wrap around" order after the positive
## frequencies; e.g. frequencies for a 10-point 'twosided'
## spectrum are 0 0.1 0.2 0.3 0.4 0.5 -0.4 -0.3 -0.2 -0.1
## 'shift', 'centerdc' : same as 'whole' but with the first half
## of the spectrum swapped with second half to put the
## zero-frequency value in the middle. (See "help
## fftshift". If "freq" is vector, 'shift' is ignored.
## If model coefficients "ar_coeffs" are real, the default
## range is 'half', otherwise default range is 'whole'.
##
## Method:
##
## 'fft': use FFT to calculate power spectrum.
## 'poly': calculate power spectrum as a polynomial of 1/z
## N.B. this argument is ignored if the "freq" argument is a
## vector. The default is 'poly' unless the "freq"
## argument is an integer power of 2.
##
## Plot type:
##
## 'plot', 'semilogx', 'semilogy', 'loglog', 'squared' or 'db':
## specifies the type of plot. The default is 'plot', which
## means linear-linear axes. 'squared' is the same as 'plot'.
## 'dB' plots "10*log10(psd)". This argument is ignored and a
## spectrum is not plotted if the caller requires a returned
## value.
##
## RETURNED VALUES:
## If returned values are not required by the caller, the spectrum
## is plotted and nothing is returned.
## @itemize
## @item
## @var{psd}
## estimate of power-spectral density
## @item
## @var{f_out}
## frequency values
## @end itemize
##
## REFERENCE
## [1] Equation 2.28 from Steven M. Kay and Stanley Lawrence Marple Jr.:
## "Spectrum analysis -- a modern perspective",
## Proceedings of the IEEE, Vol 69, pp 1380-1419, Nov., 1981
##
## @end deftypefn
function varargout = ar_psd(a,v,varargin)
##
## Check fixed arguments
if ( nargin < 2 )
error( 'ar_psd: needs at least 2 args. Use help ar_psd.' );
elseif ( ~isvector(a) || length(a)<2 )
error( 'ar_psd: arg 1 (a) must be vector, length>=2.' );
elseif ( ~isscalar(v) )
error( 'ar_psd: arg 2 (v) must be real scalar >0.' );
else
real_model = isreal(a);
##
## default values for optional areguments
freq = 256;
user_freqs = 0; ## boolean: true for user-specified frequencies
Fs = 1.0;
## FFT padding factor (is also frequency range divisor): 1=whole, 2=half.
pad_fact = 1 + real_model;
do_shift = 0;
force_FFT = 0;
force_poly = 0;
plot_type = 1;
##
## decode and check optional arguments
## end_numeric_args is boolean; becomes true at 1st string arg
end_numeric_args = 0;
for iarg = 1:length(varargin)
arg = varargin{iarg};
end_numeric_args = end_numeric_args || ischar(arg);
## skip empty arguments
if ( isempty(arg) )
1;
## numeric optional arguments must be first, cannot follow string args
## N.B. older versions of matlab may not have the function "error" so
## the user writes "function error(msg); disp(msg); end" and we need
## a "return" here.
elseif ( ~ischar(arg) )
if ( end_numeric_args )
error( 'ar_psd: control arg must be string.' );
##
## first optional numeric arg is "freq"
elseif ( iarg == 1 )
user_freqs = isvector(arg) && length(arg)>1;
if ( ~isscalar(arg) && ~user_freqs )
error( 'ar_psd: arg 3 (freq) must be vector or scalar.' );
elseif ( ~user_freqs && ( ~isreal(arg) || ...
fix(arg)~=arg || arg <= 2 || arg >= 1048576 ) )
error('ar_psd: arg 3 (freq) must be integer >=2, <=1048576' );
elseif ( user_freqs && ~isreal(arg) )
error( 'ar_psd: arg 3 (freq) vector must be real.' );
endif
freq = arg(:); # -> column vector
##
## second optional numeric arg is "Fs" - sampling frequency
elseif ( iarg == 2 )
if ( ~isscalar(arg) || ~isreal(arg) || arg<=0 )
error( 'ar_psd: arg 4 (Fs) must be real positive scalar.' );
endif
Fs = arg;
##
else
error( 'ar_psd: control arg must be string.' );
endif
##
## decode control-string arguments
elseif ( strcmp(arg,'plot') || strcmp(arg,'squared') )
plot_type = 1;
elseif ( strcmp(arg,'semilogx') )
plot_type = 2;
elseif ( strcmp(arg,'semilogy') )
plot_type = 3;
elseif ( strcmp(arg,'loglog') )
plot_type = 4;
elseif ( strcmp(arg,'dB') )
plot_type = 5;
elseif ( strcmp(arg,'fft') )
force_FFT = 1;
force_poly = 0;
elseif ( strcmp(arg,'poly') )
force_FFT = 0;
force_poly = 1;
elseif ( strcmp(arg,'half') || strcmp(arg,'onesided') )
pad_fact = 2; # FFT zero-padding factor (pad FFT to double length)
do_shift = 0;
elseif ( strcmp(arg,'whole') || strcmp(arg,'twosided') )
pad_fact = 1; # FFT zero-padding factor (do not pad)
do_shift = 0;
elseif ( strcmp(arg,'shift') || strcmp(arg,'centerdc') )
pad_fact = 1;
do_shift = 1;
else
error( 'ar_psd: string arg: illegal value: %s', arg );
endif
endfor
## end of decoding and checking args
##
if ( user_freqs )
## user provides (column) vector of frequencies
if ( any(abs(freq)>Fs/2) )
error( 'ar_psd: arg 3 (freq) cannot exceed half sampling frequency.' );
elseif ( pad_fact==2 && any(freq<0) )
error( 'ar_psd: arg 3 (freq) must be positive in onesided spectrum' );
endif
freq_len = length(freq);
fft_len = freq_len;
use_FFT = 0;
do_shift = 0;
else
## internally generated frequencies
freq_len = freq;
freq = (Fs/pad_fact/freq_len) * [0:freq_len-1]';
## decide which method to use (poly or FFT)
is_power_of_2 = rem(log(freq_len),log(2))<10.*eps;
use_FFT = ( ~ force_poly && is_power_of_2 ) || force_FFT;
fft_len = freq_len * pad_fact;
endif
##
## calculate denominator of Equation 2.28, Kay and Marple, ref [1]Jr.:
len_coeffs = length(a);
if ( use_FFT )
## FFT method
fft_out = fft( [ a(:); zeros(fft_len-len_coeffs,1) ] );
else
## polynomial method
## complex data on "half" frequency range needs -ve frequency values
if ( pad_fact==2 && ~real_model )
freq = [freq; -freq(freq_len:-1:1)];
fft_len = 2*freq_len;
endif
fft_out = polyval( a(len_coeffs:-1:1), exp( (-i*2*pi/Fs) * freq ) );
endif
##
## The power spectrum (PSD) is the scaled squared reciprocal of amplitude
## of the FFT/polynomial. This is NOT the reciprocal of the periodogram.
## The PSD is a continuous function of frequency. For uniformly
## distributed frequency values, the FFT algorithm might be the most
## efficient way of calculating it.
##
psd = ( v / Fs ) ./ ( fft_out .* conj(fft_out) );
##
## range='half' or 'onesided',
## add PSD at -ve frequencies to PSD at +ve frequencies
## N.B. unlike periodogram, PSD at zero frequency _is_ doubled.
if ( pad_fact==2 )
freq = freq(1:freq_len);
if ( real_model )
## real data, double the psd
psd = 2 * psd(1:freq_len);
elseif ( use_FFT )
## complex data, FFT method, internally-generated frequencies
psd = psd(1:freq_len)+[psd(1); psd(fft_len:-1:freq_len+2)];
else
## complex data, polynomial method
## user-defined and internally-generated frequencies
psd = psd(1:freq_len)+psd(fft_len:-1:freq_len+1);
endif
##
## range='shift'
## disabled for user-supplied frequencies
## Shift zero-frequency to the middle (pad_fact==1)
elseif ( do_shift )
len2 = fix((fft_len+1)/2);
psd = [psd(len2+1:fft_len); psd(1:len2)];
freq = [freq(len2+1:fft_len)-Fs; freq(1:len2)];
endif
##
## Plot the spectrum if there are no return variables.
if ( nargout >= 2 )
varargout{1} = psd;
varargout{2} = freq;
elseif ( nargout == 1 )
varargout{1} = psd;
else
if ( plot_type == 1 )
plot(freq,psd);
elseif ( plot_type == 2 )
semilogx(freq,psd);
elseif ( plot_type == 3 )
semilogy(freq,psd);
elseif ( plot_type == 4 )
loglog(freq,psd);
elseif ( plot_type == 5 )
plot(freq,10*log10(psd));
endif
endif
endif
endfunction

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