Convolution and Correlation

Started by Vimal December 25, 2003
Hello all,

When the two sequances are symmetrical then the correlation and the
convolution are same from the classical definitions.

My query is that, the correlation (if not ergodic) has joint pdf term
in its definition which is not there in convolution. So can I assume
that the above definition as limited to ergodicity or there is
something more to it.

Thanks,

Merry Christmas and a Happy New Year

Best regards,
Vimal
PS: One of the classical confussion 'conv and corr' has again caught
me!!
Vimal wrote:
> Hello all, > > When the two sequances are symmetrical then the correlation and the > convolution are same from the classical definitions.
Actually this is true if only one sequence is (even) symmetric.
> My query is that, the correlation (if not ergodic) has joint pdf term > in its definition which is not there in convolution. So can I assume > that the above definition as limited to ergodicity or there is > something more to it.
There are two definitions for autocorrelation: deterministic and probabilistic. The deterministic autocorrelation can use either a deterministic function or a random process (at a particular realization), x(t), as its argument, and is defined as R_xx(\tau) = \int_{-\infty}^{+\infty} x(\tau)x(t-\tau) dtau The probabilistic definition of autocorrelation requires a random process {X(t)} as its argument and is defined as R_XX(t, s) = E[X(t)X(s)], where E[Y] is the probabilistic expectation operator on the random variable Y, i.e., E[Y] = \int_{-\infty}^{+\infty} y*f_Y(y) dy, where f_Y(y) is the pdf of the random variable Y. If the random process is wide-sense stationary, its autocorrelation is a function only of time difference \tau = t - s, i.e., R_XX(t, s) = R_XX(0, t - s). If the random process is ergodic, then the deterministic autocorrelation approximates the probabilistic autocorrelation. -- % Randy Yates % "...the answer lies within your soul %% Fuquay-Varina, NC % 'cause no one knows which side %%% 919-577-9882 % the coin will fall." %%%% <yates@ieee.org> % 'Big Wheels', *Out of the Blue*, ELO http://home.earthlink.net/~yatescr
vimal_bhatia2@yahoo.com (Vimal) wrote in message
news:<b6fc6dda.0312250716.7a1504f2@posting.google.com>...
> Hello all, > > When the two sequances are symmetrical then the correlation and the > convolution are same from the classical definitions. > > My query is that, the correlation (if not ergodic) has joint pdf term > in its definition which is not there in convolution. So can I assume > that the above definition as limited to ergodicity or there is > something more to it. > > Thanks, > > Merry Christmas and a Happy New Year > > Best regards, > Vimal > PS: One of the classical confussion 'conv and corr' has again caught > me!!
The formal definition of correlation includes the absolute time t as well as the lag tau, Rxy(t,tau) = E[x(t)y^*(t+tau)]. [1] Only for stationary (and ergodig) processes does this simplify to the familiar lag-only expression where Rxy(t,tau) = Rxy(tau). The reason for the joint PDF to appear with the correlation is that the correlation function is defined in terms of the statistical Expectation operator E[x] = integral xp(x) dx [2] for some PDF p(x). This PDF does not appear in any definitions of the convolution operator and the familiar convolution operator for Linear Time Invariant (LTI) systems holds for any t when applied to a non- stationary input signal. In the case of a Linear Time Varying (LTV) system, the impulse response of the system changes with time, and the convolution operator must be specified as x(t) (*) h(t,tau) = integral x(t)*h(t,t-tau) dt [3] even for stationary x(t). The non-stationary Rxy(t,tau) finds its counterpart in the LTV system convolution [3]. Rune