Signals unit impulse response h(t) ECHO

[jegues]

Homework Statement

a) Find the unit impulse response h(t) of an audio system that causes two echoes, one occurring 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).

Homework Equations

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!

 

[rude man]

Homework Statement

a) Find the unit impulse response h(t) of an audio system that causes two echoes, one occurring 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).

Homework Equations

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!

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.)

 

[jegues]

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.)

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.

 

[rude man]

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.

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