# Convolution and Correlation

Started by 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.
--
%% 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

vimal_bhatia2@yahoo.com (Vimal) wrote in message
> 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