Question

Consider the random process n(u, t) with t ∈ R, defined by

n(u, t) = ∑ p(t − nT − θ(u))

where ∑ is n=−∞ to ∞

Using the shifting property of delta functions, indicate how you would generate y(u, t) from n(u, t) and p(t).

where θ(u) is a random variable that uniformly distributed on (−T/2, T/2).

Answer 1

To generate y(u, t) from n(u, t) and p(t) using the shifting property of delta functions, we can express y(u, t) as follows:

y(u, t) = ∑ p(t - nT - θ(u)) = ∑ p(t - nT) * δ(t - nT - θ(u))

In the above equation, δ(t) is the Dirac delta function.

The shifting property of the delta function states that if δ(t - a) is the delta function shifted by a, then δ(t - a) = 0 when t ≠ a and the integral of δ(t - a) over an interval containing a is equal to 1.

Applying the shifting property, we can rewrite the equation as:

y(u, t) = ∑ p(t - nT) * δ(t - nT - θ(u)) = ∑ p(t - nT) * δ(t - nT + θ(u))

Now, we have y(u, t) expressed as a summation of p(t - nT) multiplied by shifted delta functions.

Note that the shifting property of the delta function ensures that the delta functions are non-zero only when their arguments satisfy the condition t = nT - θ(u).

This expression allows us to generate y(u, t) by evaluating the function p(t - nT) at the points t = nT - θ(u) for all integer values of n. The resulting values can then be summed to obtain y(u, t).

Please keep in mind that this is a general explanation of how to generate y(u, t) using the shifting property of delta functions. The specific implementation and calculations may vary depending on the context and properties of p(t) and θ(u).