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Diffstat (limited to 'macros')
| -rw-r--r-- | macros/ar_psd.sci | 301 |
1 files changed, 262 insertions, 39 deletions
diff --git a/macros/ar_psd.sci b/macros/ar_psd.sci index 540dfec..1953043 100644 --- a/macros/ar_psd.sci +++ b/macros/ar_psd.sci @@ -1,40 +1,263 @@ -function [P, F]= ar_psd(A, varargin) -//Calculate the power spectrum of the autoregressive model -//Calling Sequence -// [PSD,F_OUT]=ar_psd (A, V) -// [PSD,F_OUT]=ar_psd (A, V, FREQ) -// [PSD,F_OUT]=ar_psd (A, V, FREQ, FS) -// [PSD,F_OUT]=ar_psd (..., RANGE) -// [PSD,F_OUT]=ar_psd (..., METHOD) -// [PSD,F_OUT]=ar_psd (..., PLOTTYPE) -//Parameters -//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. -//V:Square of the moving-average coefficient of the AR model. -//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) -//FS:Sampling frequency (Hertz) (default=1) -//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. -//PSD: estimate of power-spectral density -//F_OUT: frequency values -//Description -//If the FREQ argument is a vector (of frequencies) the spectrum is calculated using the polynomial method and the METHOD argument is ignored. For scalar FREQ, an integer power of 2, or 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). -//Examples -//[a,b]= ar_psd([1,2,3],2) - - funcprot(0); - rhs= argn(2); - if(rhs <2 | rhs>5) - error("Wrong number of input arguments"); - end - select(rhs) - case 2 then - [P,F]= callOctave("ar_psd", A, varargin(1)); - case 3 then - [P,F]= callOctave("ar_psd", A, varargin(1), varargin(2)); - case 4 then - [P,F]= callOctave("ar_psd", A, varargin(1), varargin(2), varargin(3)); - case 5 then - [P,F]= callOctave("ar_psd", A, varargin(1), varargin(2), varargin(3), varargin(4)); - end +function varargout = ar_psd(a, v, varargin) +//Calculate the power spectrum of the autoregressive model. + +//Calling Sequence: +// [psd, f_out] = ar_psd(a, v) +// [psd, f_out] = ar_psd (a, v, freq) +// [psd, f_out] = ar_psd (a, v, freq, fs) +// [psd, f_out] = ar_psd (..., range) +// [psd, f_out] = ar_psd (..., method) +// [psd, f_out] = ar_psd (..., plottype) + +//Parameters: +//Every parameter except for the first two is optional. +// +//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. +//v- Square of the moving-average coefficient of the AR model. +//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) +//fs- Sampling frequency (Hertz) (default=1) +//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. 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. +//psd: estimate of power-spectral density. +//f_out: frequency values. + +//Description: +//If the 'freq' argument is a vector (of frequencies) the spectrum is calculated using the polynomial method and the METHOD argument is ignored. For scalar 'freq', an integer power of 2, or 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 methodif 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). + +//Examples: +//[a,b]= ar_psd([1,2,3], 2) + + funcprot(0); + // Check fixed arguments + if nargin < 2 then + error("ar_psd: needs at least 2 args. Use help ar_psd."); + elseif ~isvector(a) | length(a) < 2 then + error("ar_psd: arg 1 (a) must be vector, length >= 2."); + elseif ~isscalar(v) then + error("ar_psd: arg 2 (v) must be real scalar >0."); + else + real_model = isreal(a); + // Default values for optional arguments + 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 = 0; + for iarg = 1:length(varargin) + arg = varargin(iarg); + end_numeric_args = end_numeric_args | (type(arg) == 10); + // Skip empty arguments + if isempty(arg) then + // Do nothing + elseif (type(arg) ~= 10) then + if end_numeric_args then + error("ar_psd: control arg must be string."); + // First optional numeric arg is "freq" + elseif iarg == 1 then + user_freqs = isvector(arg) & length(arg) > 1; + if ~isscalar(arg) & ~user_freqs then + error("ar_psd: arg 3 (freq) must be vector or scalar."); + elseif ~user_freqs & (~isreal(arg) | fix(arg) ~= arg | arg <= 2 | arg >= 1048576) then + error("ar_psd: arg 3 (freq) must be integer >=2, <=1048576"); + elseif user_freqs & ~isreal(arg) then + error("ar_psd: arg 3 (freq) vector must be real."); + end + freq = arg(:); // -> column vector + // Second optional numeric arg is "Fs" - sampling frequency + elseif iarg == 2 then + if ~isscalar(arg) | ~isreal(arg) | arg <= 0 then + error("ar_psd: arg 4 (Fs) must be real positive scalar."); + end + Fs = arg; + else + error("ar_psd: control arg must be string."); + end + // Decode control-string arguments + elseif ~strcmp(arg, "plot") | ~strcmp(arg, "squared") then + plot_type = 1; + elseif ~strcmp(arg, "semilogx") then + plot_type = 2; + elseif ~strcmp(arg, "semilogy") then + plot_type = 3; + elseif ~strcmp(arg, "loglog") then + plot_type = 4; + elseif ~strcmp(arg, "dB") then + plot_type = 5; + elseif ~strcmp(arg, "fft") then + force_FFT = 1; + force_poly = 0; + elseif ~strcmp(arg, "poly") then + force_FFT = 0; + force_poly = 1; + elseif ~strcmp(arg, "half") | ~strcmp(arg, "onesided") then + pad_fact = 2; // FFT zero-padding factor (pad FFT to double length) + do_shift = 0; + elseif ~strcmp(arg, "whole") | ~strcmp(arg, "twosided") then + pad_fact = 1; // FFT zero-padding factor (do not pad) + do_shift = 0; + elseif ~strcmp(arg, "shift") | ~strcmp(arg, "centerdc") then + pad_fact = 1; + do_shift = 1; + else + error("ar_psd: string arg: illegal value: %s", arg); + end + end + // End of decoding and checking args + if user_freqs then + // User provides (column) vector of frequencies + if or(abs(freq) > Fs/2) then + error("ar_psd: arg 3 (freq) cannot exceed half sampling frequency."); + elseif pad_fact == 2 & or(freq < 0) then + error("ar_psd: arg 3 (freq) must be positive in onesided spectrum"); + end + 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 = modulo(log(freq_len), log(2)) < 10 * %eps; + use_FFT = (~force_poly & is_power_of_2) | force_FFT; + fft_len = freq_len * pad_fact; + end + // Calculate denominator of Equation 2.28, Kay and Marple, ref [1] Jr.: + len_coeffs = length(a); + if use_FFT then + // FFT method + x = [a(:); zeros(fft_len - len_coeffs, 1)]; + fft_out = fft(x); + else + // Polynomial method + // Complex data on "half" frequency range needs -ve frequency values + if pad_fact == 2 & ~real_model then + freq = [freq; -freq(freq_len:-1:1)]; + fft_len = 2 * freq_len; + end + fft_out = horner(a($:-1:1), exp((-%i * 2 * %pi / Fs) * freq)); + end + // 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 then + freq = freq(1:freq_len); + if real_model then + // Real data, double the psd + psd = 2 * psd(1:freq_len); + elseif use_FFT then + // 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); + end + // Range='shift' + // Disabled for user-supplied frequencies + // Shift zero-frequency to the middle (pad_fact == 1) + elseif do_shift then + 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)]; + end + // Plot the spectrum if there are no return variables. + if nargout() >= 2 then + varargout(1) = psd; + varargout(2) = freq; + elseif nargout() == 1 then + varargout(1) = psd; + else + if plot_type == 1 then + plot(freq, psd); + elseif plot_type == 2 then + semilogx(freq, psd); + elseif plot_type == 3 then + semilogy(freq, psd); + elseif plot_type == 4 then + loglog(freq, psd); + elseif plot_type == 5 then + plot(freq, 10 * log10(psd)); + end + end + end endfunction + +//tests: + +//a = [1, -0.5]; +////v = 1; +//[psd, freq] = ar_psd(a, v); +//plot(freq, psd); +//title('Power Spectral Density of the AR Model'); +//xlabel('Frequency'); +//ylabel('Power/Frequency'); + +//a = [1, -1.5, 0.7]; +//v = 2; +//Fs = 2.0; +//[psd, freq] = ar_psd(a, v, 512, Fs); +//plot(freq, psd); +//title('Power Spectral Density with Different Sampling Frequency'); +//xlabel('Frequency (Hz)'); +//ylabel('Power/Frequency'); + +//a = [1, -0.9, 0.4]; +//v = 0.8; +//Fs = 1.0; +//ar_psd(a, v, 512, Fs, 'semilogx'); +//title('Power Spectral Density (Semilogx)'); +//xlabel('Frequency (Hz)'); +//ylabel('Power/Frequency'); +// +//figure(); +//ar_psd(a, v, 512, Fs, 'loglog'); +//title('Power Spectral Density (Loglog)'); +//xlabel('Frequency (Hz)'); +//ylabel('Power/Frequency'); + +//a = [1, -0.7, 0.2]; +//v = 1.5; +//Fs = 1.0; +//[psd_fft, freq] = ar_psd(a, v, 512, Fs, 'fft'); +//[psd_poly, freq] = ar_psd(a, v, 512, Fs, 'poly'); +//plot(freq, psd_fft, 'r', freq, psd_poly, 'b'); +//title('Power Spectral Density (FFT vs Polynomial)'); +//xlabel('Frequency (Hz)'); +//ylabel('Power/Frequency'); +//legend('FFT Method', 'Polynomial Method'); + +//a = [1, -1.2, 0.5]; +//v = 1; +//[psd_half, freq_half] = ar_psd(a, v, 512, 1, 'half'); +//[psd_whole, freq_whole] = ar_psd(a, v, 512, 1, 'whole'); +//subplot(2, 1, 1); +//plot(freq_half, psd_half); +//title('Power Spectral Density (Half Spectrum)'); +//xlabel('Frequency (Hz)'); +//ylabel('Power/Frequency'); +// +//subplot(2, 1, 2); +//plot(freq_whole, psd_whole); +//title('Power Spectral Density (Whole Spectrum)'); +//xlabel('Frequency (Hz)'); +//ylabel('Power/Frequency'); |