Exercises

1. If 0 dB corresponds to an amplitude of 1, how many dB corresponds to amplitudes of 1.5, 2, 3, and 5? (Answer: about 3, 6, 10, and 14.)
2. Two uncorrelated signals of RMS amplitude 3 and 4 are added; what's the RMS amplitude of the sum?
3. How many uncorrelated signals, all of equal amplitude, would you have to add to get a signal that is 9 dB hotter?
4. What is the angular frequency of middle C at 44100 samples per second?
5. If $x[n]$ is an audio signal, show that:
   
   $${A_{\mathrm{RMS}}} \{x[n]\} \le {A_{\mathrm{peak}}} \{x[n]\}$$
   
   and
   
   $${A_{\mathrm{RMS}}} \{x[n]\} \ge {A_{\mathrm{peak}}} \{x[n]\} / {\sqrt N} ,$$
   
   where $N$ is the window size. Under what conditions does equality hold for each one?
6. If $x[n]$ is the SINUSOID of Section 1.1, 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.