# Unit impulse response Signal with Echo

Discussion in 'Homework Help' started by jegues, Dec 4, 2012.

1. ### jegues Thread Starter Well-Known Member

Sep 13, 2010
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1. The problem statement, all variables and given/known data

a) Find the unit impulse response h(t) of an audio system that causes two echoes, one occuring at 0.5 seconds and the other one at 1.5 seconds. Please sketch H(w). (Magnitude and phase)

b) Design a system that can eliminate noise coming from the 60Hz power signal (thus noise is an additive 60Hz sinusoid). Derive its time input response h(t).

2. Relevant equations

3. The attempt at a solution

Here is my attempt thus far at the solution,

a)

If I have echoes at time 0.5 and 1.5,

$h(t) = \delta(t) + \delta(t-0.5) + \delta(t-1.5)$

The Fourier transform of this will be,

$H(\omega) = 1 + e^{-j\frac{\omega}{2}} + e^{-j\frac{3\omega}{2}}$

Is this correct? I am having a hard time figuring out how I am suppose to easily sketch the magnitude and phase of such a function.

Thanks again!