# Problem 2.28, HW3, ECE301, Summer 2008

Determine if each system is causal and stable.

A

h[n] = (1/5)$^n$ u[n]

For n < 0 h[n] = 0 therefore h[n] is causal.

$\Sigma_{n=0}^\infty$ (1/5)$^n$ < $\infty$ since lim$_{n->\infty}$ = 0

The system is both causal and stable.

B

h[n] = $(0.8)^n$ u[n+2]

Since u[n+2] = 1 for n >= -2 and 0 for n < -2 the system is not causal because h[n] $\neq$ 0 for t < 0.

$\Sigma_{n = -2}^\infty$ $(0.8)^n$ < $\infty$ since $lim_{n->\infty} (0.8)^n = 0$, the system is stable.

The system is not causal and stable.

D

h[n] = 5$^n$u[3-n]

Since u[3-n] = 1 for n <= 3 and 0 for n > 3, h[n] $\neq$ 0 for t < 0.

$\Sigma_{-\infty}^\infty 5^n u[3-n] = \Sigma_{-\infty}^3 5^n < \infty$, therefore the system is stable.

This system is stable but not causal.

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