AR, Burg methods worse resolution than FFT
in [Matlab]
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From: Dean on 7 Aug 2010 18:56 Hi, I've been told Autoregressive Process (AR) offers better frequency resolution than classic Periodogram methods. So I used an example available in the MATLAB manual and compared with Periodogram. Results showed that Periodogram had better frequency resolution than AR via Burg method. I also studied pyulear, pcov, pmcov, their spectral looks almost identical. I've two questions. 1. Why my Periodogram has better frequency resolution than AR-based methods? 2. Have you experienced very similar spectrum in your data analysis? Thank you all, Cheers ----------Code---------------------- randn('state',0) fs = 1000; % Sampling frequency t = (0:fs)/fs; % One second worth of samples A = [1 2]; % Sinusoid amplitudes f = [150;140]; % Sinusoid frequencies xn = A*sin(2*pi*f*t) + 0.1*randn(size(t)); % [P1,f] = pwelch(xn,hamming(256),128,1024,fs); [P1,f] = periodogram(xn,rectwin(length(xn)),1024,fs); [P2,f] = pburg(xn,14,1024,fs); plot(f,10*log10(P1),'r-',f,10*log10(P2),'b--'); grid legend('Periodogram','AR-Burg') ylabel('PSD Estimates (dB/Hz)'); xlabel('Frequency (Hz)');
From: dbd on 7 Aug 2010 20:37 On Aug 7, 3:56 pm, "Dean " <jiangwei0...(a)gmail.com> wrote: > Hi, I've been told Autoregressive Process (AR) offers better frequency resolution than classic Periodogram methods. > ... Perhaps you have rediscovered a quaint historical artifact that a number of algorithms commonly labeled "high resolution" should have been labeled "high SNR only". Dale B. Dalrymple
From: Dean on 8 Aug 2010 11:14 dbd <dbd(a)ieee.org> wrote in message <c855b29e-c1e0-447d-a6d4-8a6ac14dbffe(a)m35g2000prn.googlegroups.com>... > On Aug 7, 3:56 pm, "Dean " <jiangwei0...(a)gmail.com> wrote: > > Hi, I've been told Autoregressive Process (AR) offers better frequency resolution than classic Periodogram methods. > > ... > > Perhaps you have rediscovered a quaint historical artifact that a > number of algorithms commonly labeled "high resolution" should have > been labeled "high SNR only". > > Dale B. Dalrymple Hi Dale, I also found all these AR methods, such as Burg, Covariance, YW, Modified Covariance actually producing almost visually identical spectrum (even with noise added). Is that something textbook won't tell me? Cheers, Dean
From: Wayne King on 8 Aug 2010 11:32 "Dean " <jiangwei0515(a)gmail.com> wrote in message <i3koa2$2nv$1(a)fred.mathworks.com>... > Hi, I've been told Autoregressive Process (AR) offers better frequency resolution than classic Periodogram methods. > > So I used an example available in the MATLAB manual and compared with Periodogram. Results showed that Periodogram had better frequency resolution than AR via Burg method. > > I also studied pyulear, pcov, pmcov, their spectral looks almost identical. I've two questions. > 1. Why my Periodogram has better frequency resolution than AR-based methods? > 2. Have you experienced very similar spectrum in your data analysis? > > Thank you all, > Cheers > ----------Code---------------------- > randn('state',0) > fs = 1000; % Sampling frequency > t = (0:fs)/fs; % One second worth of samples > A = [1 2]; % Sinusoid amplitudes > f = [150;140]; % Sinusoid frequencies > xn = A*sin(2*pi*f*t) + 0.1*randn(size(t)); > % [P1,f] = pwelch(xn,hamming(256),128,1024,fs); > [P1,f] = periodogram(xn,rectwin(length(xn)),1024,fs); > [P2,f] = pburg(xn,14,1024,fs); > plot(f,10*log10(P1),'r-',f,10*log10(P2),'b--'); grid > legend('Periodogram','AR-Burg') > ylabel('PSD Estimates (dB/Hz)'); > xlabel('Frequency (Hz)'); Hi Dean, Typically resolution improvements with AR methods for sinusoids are seen with short data records. So for example: reset(RandStream.getDefaultStream); fs = 1000; t = (0:fs/15)/fs; A = [2 1]; f = [140;150]; xn = A*sin(2*pi*f*t) + 0.1*randn(size(t)); hburg = spectrum.burg(24); psd(hburg,xn,'Fs',fs); hper = spectrum.periodogram; figure; psd(hper,xn,'Fs',fs); Note how the AR estimate resolves the two peaks, but the periodogram smears them together because they are closer than 14.92 hertz resolution of the periodogram (in this case). Don't get me wrong, I'm a big fan of nonparametric methods and have used them much more frequently than parametric methods (which have their own issues), but the resolution superiority is usually brought up in the context of short data records. Of course, if you overmodel the data using parametric methods, you can get line splitting where one true sinusoid gets split into two separate ones. Kinda the opposite problem :) Also, there are the so-called superresolution methods, which begin with a signal model that is a superposition of complex exponentials plus noise. These methods decompose the autocorrelation matrix into a signal subspace and a noise subspace. They don't yield a PSD estimate, so they're used when you want to estimate frequencies, see spectrum.music and spectrum.eigenvector for examples in MATLAB. Wayne
From: Rune Allnor on 8 Aug 2010 13:29
On 8 Aug, 00:56, "Dean " <jiangwei0...(a)gmail.com> wrote: > Hi, I've been told Autoregressive Process (AR) offers better frequency resolution than classic Periodogram methods. Don't believe what people tell you. There are plenty people around who tell you all sort of stuff, that they - when challenged - turn out to know nothing whatsoever about. The AR methods don't provide frequency resolution. They characterize the covariance structure of the signal in terms of an AR model. The good part about this is that you can get good characteristics with small ampunts of data (a novice or amateur might interpret this as 'better frequency resolution'). The downside is that the charactersitics might not be easily interpreted. The really bad side is that if the the data do not comply to the assumed model (AR in this case), the computed parameters might not make sense at all. Rune |