Dr. Mallard (known as Ducky) needs the response of a causal LTI system with impu

Question

Dr. Mallard (known as Ducky) needs the response of a causal LTI system with impulse response h(t), to a specific input signal, x(t), given by h(t) = t[u(t - 1) - u(t - 2)] and x(t) = e_t[u(t)]. He wants to find the closed form result and prepare a plot of his output using two techniques, namely, using time domain convolution in the following form that is, by flipping and shifting ONLY x(t) and leaving h{f) stationary (there will be three regions, one of which will be zero), |HINT: betaezbeta d beta = e a beta (alpha beta - 1) / a2],|,

Explanation / Answer

code using matlab :

Suppose we have two signals:

u(t) : unit step function and h(t) = exp(-t) * u(t)

Let us calculate their convolution. Doing that on paper is pretty easy, the result will be y(t) = (1-exp(-t)) * u(t). i.e the function will increase till it reaches the value of 1 and then it becomes constant = 1.

The big question is that why the following code produces wrong answer after time of 10s (this time is the length of the original signals)? In other words, why the result starts decaying after this time instant and reaches zero ?!

Code:

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