Obtain the convolution of the pairs of signals in Figure 2 S

Obtain the convolution of the pairs of signals in Figure 2.

Solution

A convolution is a mathematical operation that represents a signal passing through a LTI (Linear and Time-Invariant) system or filter. If we have a signal s(t) passing through a system with impulse response h(t), the output is the convolution of s(t) with h(t). The convolution is simply the integral of the product of the two functions (in this example the functions are s(t) and h(t)), where one is reversed.

(sh)(t)=0s()h(t)d

So why reverse one function? Consider the response of the system to an impulse at time tn: it is h(ttn), right? At t=tnyou will have h(0) and all other points of h(t) will be shifted in the same way. At this point, we are convolving the systems\'s function with a dirac delta function:

(ttn)h(t)=0(tn)h(t)d=h(ttn)

Only for =tn the multiplication inside the integral is non-zero and it is equal to h(t)=h(ttn). Now imagine a sequence of impulses at times t0, t1, and so on until tn that form a discrete signal. The impulse responses accumulate, and you will have the accumulation of all the impulse responses shifted by each impulse.

i0(ti)h(t)d

we can put the sum inside the integral:

0(i(ti))h(t)d

In the limit of a continuous signal (the combination of very close together impulses creates a signal), the summation will lead to signal s(t):

0s()h(t)d

R11()=R()=x(t)x(t)dt[+ve shift]R11()=R()=x(t)x(t)dt[+ve shift]

=x(t)x(t+)dt[-ve shift]=x(t)x(t+)dt[-ve shift]

Where = searching or scanning or delay parameter.

If the signal is complex then auto correlation function is given by

R11()=R()=x(t)x(t)dt[+ve shift]R11()=R()=x(t)x(t)dt[+ve shift]

=x(t+)x(t)dt