2.28 (a,b,c)
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)<math>^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.
