Periodic Convolution and Auto Correlation - Electrical Engineering (EE) PDF Download

Periodic Convolution and Auto-Correlation

Periodic Convolution 

We have applied the convolution theorem to convolutions involving:
(i) two aperiodic signals
(ii) one aperiodic and one periodic signal.

But, convolutions between periodic signals diverge, and hence the convolution theorem cannot be applied in this context. However a modified definition of convolution for periodic signals whose periods are rationally related is found useful. We look at this definition now. Later, we will prove a result similar to the Convolution theorem in the context of periodic signals.

Consider the following signals
 x(t) periodic with period T₁ and h(t) periodic with period T₂ where T₁ and T₂ are rationally related.
Let T₁/T₂ = m / n (where m and n are integers) Hence, m T₂ = n T₁ = T is a common period for both x(t) and h(t).
Periodic convolution or circular convolution of x(.) with h(.) is denoted by  and is defined as :

Note the definition holds even if T is not the smallest common period for x(t) and h(t) due to the division by T. Thus we don't need m and n to be the smallest possible integers satisfying T₁ / T₂ = m / n in the process of finding T.
Also, show for yourself that the periodic convolution is commutative, i.e: . Also, notice that the convolution is periodic with period T₁ as well as T_2. More on this later.
Fourier Transform of 
Say x(t) is periodic with period T₁ and h(t) is periodic with period T₂ with T_{1 /} T₂ = m / n (where m and n are integers).
Thus m T₂ = n T₁ = T is a common period for the two.
We can expand x(t) and h(t) into Fourier Series with fundamental frequency   

If one compares the Fourier co-efficients in these expansions with those in the expansions with the original fundamental frequencies, i.e:

Now,

But then, we have seen that C_k can be non-zero only when k is a multiple of n, and d_k can be non-zero only when k is a multiple of m.
Their product can clearly be non-zero only when k is a multiple of m and n. Thus if p is the LCM (least common multiple) of m and n, we have:

What can we make out of this?
The Fourier Transform of the circular convolution has impulses at all (common) frequencies where the Fourier transforms of x(t) and h(t) have impulses. The circular convolution therefore "picks out" common frequencies, at which the spectra of x(t) and h(t) are non-zero and the strength of the impulse at that frequency is the product of the strengths of the impulses at that frequency in the original two spectra.

This result is the equivalent of the Convolution theorem in the context of periodic convolution.

Parseval's Theorem 

We now obtain the result equivalent to the Parseval's theorem we have already seen in the context of periodic signals.
Let x(t) and y(t) be periodic with a common period T.

Applying the Convolution theorem equivalent we have just proved on   we get:

 Put t = 0, to get: 

Compare this equation with the Parseval's theorem we had proved earlier.

If we take x = y, then T becomes the fundamental period of x and:

Note the left-hand side of the above equation is the power of x(t).
Note also that the periodic convolution of  yields a periodic signal with Fourier coefficients that are the modulus square of the coefficients of x(t).

Another important result

If   

Then  represents the power of y(t), where T is a period common to x(t) and h(t).

If 

Applying the Parseval's theorem to y,
 

The Auto-correlation and the Cross-correlation.
Proceeding with our work on the Fourier transform, let us define two important functions, the Auto-correlation and the Cross-correlation.
Auto Correlation

You have seen that for a Periodic signal y(t),   has Fourier series coefficients that the modulus square of the Fourier series coefficients of y(t).

Lets look at an equivalent situation with aperiodic signals, i.e:

Assume that:

then

Notice that

Since

We have, 

Using the dual of the convolution theorem,

The auto-correlation of x(t), denoted by   is defined as:
 

ts Spectrum is the modulus square of the spectrum of x(t).

It can also be interpreted as the projection of x(t) on its own shifted version, shifted back by an interval ‘t'.

It can be shown that   ( note that is nothing but the energy in the signal x(t) )

 Cross Correlation
The cross correlation between two signals x(t) and y(t) is defined as : 

Note that the cross-correlation R_xy  is the convolution of .

If  then using the fact that the auto-correlation integral peaks at 0 , the cross correlation peaks at .
It may be said that cross-correlation function gives a measure of resemblance between the shifted versions of signal x(t) and y(t). Hence it is used to in Radar and Sonar applications to measure distances . In these systems, a transmitter transmits signals which on reflection from a target are received by a receiver. Thus the received signal is a time shifted version of the transmitted signal . By seeing where the cross-correlation of these two signals peaks, one can determine the time shift and hence the distance of the target.
The Fourier transform of R_xy (t)  is of-course 

Conclusion: 

In this lecture you have learnt:
Periodic convolution or circular convolution of x(.) with h(.) is denoted by  and is defined as :
  
Fourier Transform of 
Parseval's theorem in the context of periodic signals is  

Auto correlation is defined as  

Cross correlation is defined as