Hello, I think I am missing something very basic here. I am confused as to which definition of autocorrelation to use. From Wikipedia (http://en.wikipedia.org/wiki/Autocorrelation), there are two definitions. 1) Based on convolution 2) Based on 'expectation' - E[f(t).f*(t-t')] I am not sure which version to use. I am working on OFDM so I will be mostly interested in calculation the autocrrelation on a Rayleigh channel or the autocorrelation of the datainput sequence. All of the Filtering/Estimation texts use the definition of expectation: R(t')=E[f(t).f*(t-t')] But according to CNX (http://cnx.org/content/m10676/latest/), for discrete-time real signal sequences, the 'convolution' based method is given. Can somebody please clearup this confusion. Cheers.

# Which version of Autocorrelation function to use?

Started by ●September 3, 2008

Reply by ●September 3, 20082008-09-03

On 3 Sep, 08:46, "m26k9" <maduranga.liyan...@gmail.com> wrote:> Hello, > > I think I am missing something very basic here. I am confused as to which > definition of autocorrelation to use. > > From Wikipedia (http://en.wikipedia.org/wiki/Autocorrelation), there are > two definitions. > 1) Based on convolutionNope. The estimator might look like convolution, but it isn't.> 2) Based on �'expectation' - E[f(t).f*(t-t')]This is the proper definition. The difference between the two is that the proper definition is based on the probablilty demsithy function p(t) which is nevber known outside pure academic settings. The best one can do with data is to *estimate* the correlation function, and there are several *estimators* which can be used. This is a fairly basic argument but unfortunately one that isn't emphasized in texts on statistics. Rune

Reply by ●September 3, 20082008-09-03

> But according to CNX (http://cnx.org/content/m10676/latest/), for > discrete-time real signal sequences, the 'convolution' based method is > given. > > Can somebody please clearup this confusion.They are equivalent if the process is ergodic. If the process is not ergodic, than 1) will *not* work.

Reply by ●September 3, 20082008-09-03

On Sep 3, 11:54 am, Ikaro <ikarosi...@hotmail.com> wrote:> > But according to CNX (http://cnx.org/content/m10676/latest/), for > > discrete-time real signal sequences, the 'convolution' based method is > > given. > > > Can somebody please clearup this confusion. > > They are equivalent if the process is ergodic. If the process is not > ergodic, than 1) will *not* work.this is true, if they are ergodic, any average that you can express in the time domain is equal to the average you get in the "probability domain", the latter we usually call the "expectation value". anyway, for finite-energy signals, the convolution definition (where one copy of the input function to the autocorrelation is time-reversed and sent to the convolver) might be the only one to use. that's how we compute the autocorrelation for a given (usually real-time) signal. we copy a piece of it (maybe windowed) and we compute some kinda inner product of that piece against a corresponding seqment that is delayed by some given time. r b-j

Reply by ●September 4, 20082008-09-04

>From Wikipedia (http://en.wikipedia.org/wiki/Autocorrelation), there are >two definitions. >1) Based on convolution >2) Based on 'expectation' - E[f(t).f*(t-t')]The fundamental difference between (1) and (2) is that of a deterministic and a random function. (2) is the definition of autocorrelation of a random function, you are likely interested in that one. Given a realization, which is deterministic, you can estimate the autocorrelation using (1). In some instances, though, you can find the autocorrelation parametrically, in which case you would not need to use (1).>I am not sure which version to use. I am working on OFDM so I will be >mostly interested in calculation the autocrrelation on a Rayleighchannel>or the autocorrelation of the datainput sequence.If you are interested in studying the channel model per se, and not estimating it from, say, a training sequence, you are more likely to use parametric analysis (2). The following might be helpful: http://en.wikipedia.org/wiki/Rayleigh_fading Emre