# Homework Help: Signals unit impulse response h(t) ECHO

1. Dec 4, 2012

### jegues

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!

2. Dec 4, 2012

### rude man

Fourier transform looks right (I'm a Laplace man myself, but it's about the same thing).

So, assuming it's right, use the Euler relation on the two exponentials, then use standard method of forming H(ω) = A + jB

magn = √(A2 + B2)
phase = arc tan B/A. Pay attention separately to the signs of A and B (in other words, arc tan (A/-B) ≠ arc tan(-A/B) etc.)

3. Dec 4, 2012

### jegues

I've done this, let's just say it doesn't simplify down to something nice that you can sketch by hand. (You'd need a graphing calculator as far as I can tell)

That's why I'm getting frustrated, it's not turning into something I can simply sketch by hand.

4. Dec 5, 2012

### rude man

What did you get for magn. and phase as functions of ω? Can't be that bad.