Compute the convolution integral - determine the

Question

Question 1. Consider the two waveforms f(t) and g(t) shown in the figures below.

(a) Characterize both functions by expressing each in a suitable mathematical functional form. Write the resultant equation next to the equal sign for each function.

(b) Compute the convolution integral:

h(t) = _-∞∫^∞ f(t)g(t - τ)dτ

(c) Use a Plotting package of your choice to plot your result for h(t). You may also sketch the plot if desired. What do you observe about the relative widths of f(t), g(t) and h(t)?

Question 2. Discrete-Time Systems in Time Domain

Determine if the following discrete-time systems are linear, time-invariant, BIBO stable, memoryless, causal and/or invertible. Fully justify your answers.

(a) y[n] = x[n] + x[n-2] (b) y[n]=x[n²] (c) y[n] = x²[n] (d) y[n] = exp(x[n])

(e) LTI system with impulse response h[n] = cos(Πn/3)

(f) LTI system with impulse response h[n] = u[n] - u[n-2].

(g) Show that a memoryless discrete-time system is always causal