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 T₂. More on this later.