Question: Show your work.

1. The Fourier Transform of the signal x(t) = cos(t) is X(Ω) = (1/2)(δ(Ω - 1) + δ(Ω + 1)). Use the derivative property of Fourier Transforms to derive the Fourier Transform for sin(t).

2. Use the time shifting property of Fourier Transforms to derive the Fourier Transform of the cos(t) from the Fourier Transform of the sin(t).

3. The good human ear can hear frequencies up to 20 kHz. If you build a digital audio application, describe how you would satisfy the two requirements for the Sampling Theorem.

4. The frequency spectrum of human speech filtered to 4 kHz (i.e., frequencies above 4 kHz are cut off) is still easy to understand. If you build a digital speech application and prefilter the speech signal to 4kHz, what is the minimum sampling rate you should use for the speech signal?

5. Your lab partner samples a 10 kHz sinusoid at 12 kHz. If you would like to reconstruct the original signal from your partner's samples, did your partner properly sample the input signal? If you go ahead and reconstruct the signal from the sampled data assuming that it had been properly sampled, what would you get for the reconstructed signal?

6. A 10 Hz square wave signal is sampled at 25 Hz. Does this meet the requirements of the Sampling Theorem? Will you get aliasing in the reconstructed signal if you use the sampled data?