Exercises

1. A sinusoid (page ) has initial phase $\phi=0$ and angluar frequency $\omega = \pi/10$. What is its period in samples? What is the phase at sample number $n=10$?

2. Two sinusoids have periods of 20 and 30 sample, respectively. What is the period of the sum of the two?

3. If 0 dB corresponds to an amplitude of 1, how many dB corresponds to amplitudes of 1.5, 2, 3, and 5?

4. Two uncorrelated signals of RMS amplitude 3 and 4 are added; what's the RMS amplitude of the sum?

5. How many uncorrelated signals, all of equal amplitude, would you have to add to get a signal that is 9 dB greater in amplitude?

6. What is the angular frequency of middle C at 44100 samples per second?

7. Two sinusoids play at middle C (MIDI 60) and the neighboring C sharp (MIDI 61). What is the difference, in Hertz, between their frequuencies?

8. How many cents is the interval between the seventh and the eighth harmonic of a periodic signal?

9. If an audio signal $x[n], n = 0, ..., M-1$ has peak amplitude 1, what is the minimum possible RMS amplitude? What is the maximum possible?

10. If $x[n]$ is the sinusoid of page , and making the assumptions of section 1.2, show that its RMS amplitude is approximately $a / {\sqrt 2}$. Hint: use an integral to approximate the sum. Since the window contains many periods, you can assume that the integral covers a whole number of periods.