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Discrete-Time Stochastic Signal Definitions in Proakis

Started by Randy Yates January 28, 2006
Hi Folks,

We're using Proakis' "Digital Communications" (4th ed.) for our (guess
what?) digital communications course at NCSU. In the second chapter,
which is a review of probability and stochastic processes, Proakis
defines the discrete-time autocorrelation function as

  Rxx(n,k) = (1/2) * E[X(n) * X(k)'],

where "'" denotes complex conjugate.

My question: Where does the "1/2" come from? I've never seen it
before, and Papoulis doesn't define it this way, so where does it come
from?

Another oddity is that Proakis uses the "real-valued" definition of
autocorrelation for continuous-time processes,

  Rxx(t1, t2) = E[X(t1) * X(t2)],

but the complex conjugate version for discrete-time processes. Why
the inconsistency? 
-- 
%  Randy Yates                  % "Rollin' and riding and slippin' and
%% Fuquay-Varina, NC            %  sliding, it's magic."
%%% 919-577-9882                %  
%%%% <yates@ieee.org>           % 'Living' Thing', *A New World Record*, ELO
http://home.earthlink.net/~yatescr
Randy Yates wrote:
> Hi Folks, > > We're using Proakis' "Digital Communications" (4th ed.) for our (guess > what?) digital communications course at NCSU. In the second chapter, > which is a review of probability and stochastic processes, Proakis > defines the discrete-time autocorrelation function as > > Rxx(n,k) = (1/2) * E[X(n) * X(k)'], > > where "'" denotes complex conjugate. > > My question: Where does the "1/2" come from? I've never seen it > before, and Papoulis doesn't define it this way, so where does it come > from?
I don't have that book, but I'll guess that this is a "convenience factor". Maybe he defines the time domain correlation function in this way in order to avoid some factor that otherwise would occur in all formulas in the frequency domain? If you compare the formulas in frequency domain in the two books, and find that Proakis' formulas are notationally simpler, I think that's the explanation. Saving ink.
> Another oddity is that Proakis uses the "real-valued" definition of > autocorrelation for continuous-time processes, > > Rxx(t1, t2) = E[X(t1) * X(t2)], > > but the complex conjugate version for discrete-time processes. Why > the inconsistency?
Again, this will only be guesswork, but maybe there is no need to bother with complex-valued continuous-time formulas? Personally, I think the author should be as general as possible, and use the complex-valued form in all developments. I don't know of many books where that is done, though. Maybe there are no applications where the complex-valued continuous-time form is needed. Maybe there is, but the author didn't bother developing the complex-valued forms since no one else have done it. Rune
My guess is to normalize so that:

 (1/2) * E[X(n) * X(n)'] = 1

"Randy Yates" <yates@ieee.org> wrote in message
news:irs59ct7.fsf@ieee.org...
> Hi Folks, > > We're using Proakis' "Digital Communications" (4th ed.) for our (guess > what?) digital communications course at NCSU. In the second chapter, > which is a review of probability and stochastic processes, Proakis > defines the discrete-time autocorrelation function as > > Rxx(n,k) = (1/2) * E[X(n) * X(k)'], > > where "'" denotes complex conjugate. > > My question: Where does the "1/2" come from? I've never seen it > before, and Papoulis doesn't define it this way, so where does it come > from? > > Another oddity is that Proakis uses the "real-valued" definition of > autocorrelation for continuous-time processes, > > Rxx(t1, t2) = E[X(t1) * X(t2)], > > but the complex conjugate version for discrete-time processes. Why > the inconsistency? > -- > % Randy Yates % "Rollin' and riding and slippin' and > %% Fuquay-Varina, NC % sliding, it's magic." > %%% 919-577-9882 % > %%%% <yates@ieee.org> % 'Living' Thing', *A New World Record*,
ELO
> http://home.earthlink.net/~yatescr

Randy Yates wrote:
> Hi Folks, > > We're using Proakis' "Digital Communications" (4th ed.) for our (guess > what?) digital communications course at NCSU.
Good book although may be pretending to cover too wide area.
> In the second chapter, > which is a review of probability and stochastic processes, Proakis > defines the discrete-time autocorrelation function as > > Rxx(n,k) = (1/2) * E[X(n) * X(k)'], > > where "'" denotes complex conjugate. > > My question: Where does the "1/2" come from? I've never seen it > before, and Papoulis doesn't define it this way, so where does it come > from?
Inherited from passband/baseband conversion, perhaps?
> Another oddity is that Proakis uses the "real-valued" definition of > autocorrelation for continuous-time processes, > > Rxx(t1, t2) = E[X(t1) * X(t2)], > > but the complex conjugate version for discrete-time processes. Why > the inconsistency?
Passband vs Baseband. Vladimir Vassilevsky DSP and Mixed Signal Design Consultant http://www.abvolt.com
Randy Yates wrote:
> Hi Folks, > > We're using Proakis' "Digital Communications" (4th ed.) for our (guess > what?) digital communications course at NCSU. In the second chapter, > which is a review of probability and stochastic processes, Proakis > defines the discrete-time autocorrelation function as > > Rxx(n,k) = (1/2) * E[X(n) * X(k)'], > > where "'" denotes complex conjugate. > > My question: Where does the "1/2" come from? I've never seen it > before, and Papoulis doesn't define it this way, so where does it come > from?
I'm kind of guessing, but if you look at Whalen's Detection theory book there is a 1/2 factor that is different for his complex Gaussian than that for a real valued Gaussian. The complex Gaussian is a bit queer because complex numbers don't order i.e. x1 < x2 is ambiguous. You have to treat the complex number as 2 dimensional real object.
> > Another oddity is that Proakis uses the "real-valued" definition of > autocorrelation for continuous-time processes, > > Rxx(t1, t2) = E[X(t1) * X(t2)], > > but the complex conjugate version for discrete-time processes. Why > the inconsistency?