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.

 

[rezeew]

Hello,

Your attempt for part (a) is correct. The unit impulse response for an audio system with two echoes at 0.5 seconds and 1.5 seconds can be represented by the sum of three delta functions, as you have shown.

To sketch the magnitude and phase of the Fourier transform, you can use a graphing calculator or software such as MATLAB to plot the function H(w). Alternatively, you can use the properties of the Fourier transform to simplify the expression and then sketch the magnitude and phase based on the simplified expression.

For part (b), to design a system that can eliminate noise coming from the 60Hz power signal, you can use a notch filter. The transfer function of a notch filter is given by H(\omega) = 1 - e^{-j\omega t}, where t is the time delay. You can set t = \frac{1}{60} seconds to eliminate the 60Hz noise.

Using the inverse Fourier transform, you can derive the time input response h(t) as h(t) = \delta(t) - \delta(t - \frac{1}{60}). This impulse response will eliminate the 60Hz noise in the input signal.

I hope this helps. Let me know if you have any further questions.

Best regards,