1. Impulse response examples for each of the following systems : linear and non-linear, causal and non-causal, with and without memory, invertible/non-invertible, stable/non-stable, time variant and time invariant.
Linear: y[n] = 2x[3n − 4] + ( − 1)n * x[n]
Nonlinear: y(t) = x2[t]
Causal: h(t) = (t − 1) * u(t − 1)
Noncausal: h(t) = ln( − t)
With memory: h(t) = 1 − u(t + 1)
Without memory: h[n] = u[n] − u[n − 1]
Invertible: h(t) = 2u(t − 5)
Noninvertible: y[n] = cos(x[n])
Stable: h(t) = [e-t]u(t)
Nonstable: y(t) = d/dt x(t)
Time variant: y[n] = n * x[n − 1]
Time invariant: y[n] = ( − j)n * x[n]
2. Example of graphical convolution.
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3. Example question related to fundamental period.
x[n] = ( − 1)n * c''o's(p'i * n − pi / 2)) + c'o's[p'i * n] * si'n[pi * n]
The first term is always zero because of the cosine. The second term uses trigonometric properties to convert it to sin(2pi*n)/2 whose period is 1.
Fundamental period = 1
