4. 
Here is a short audio signal which continuously repeats:
0.6, 0.14, 0.2, 0.14, 0.2, 0.14, 0.2, 0.14
This signal only contains cosine sinusoids, so you will not have to
measure the signal with complex sinusoids.
There are five distinct possible frequencies in this signal: 0 Hz, f_{s}/8,
f_{s}/4, 3 f_{s}/8, and f_{s}/2. Here are the
timeseries for each frequency:
0 Hz = 1, 1, 1, 1, 1, 1, 1, 1
fs/8 = 1, 0.707, 0, 0.707, 1, 0.707, 0, 0.707
fs/4 = 1, 0, 1, 0, 1, 0, 1, 0
3 fs/8 = 1, 0.707, 0, 0.707, 1, 0.707, 0, 0.707
fs/2 = 1, 1, 1, 1, 1, 1, 1, 1
Here is a plot of each possible sinewave:
(a) Draw the 5point spectrum of the signal by extracting the amplitudes
of the 5 sinusoids present in the signal. Here are some of the
amplitudes:
0 Hz = 0
fs/8 = ???
fs/4 = 0.2
3 fs/8 = ???
fs/2 = 0
(b) What are the amplitudes of the of the fs/8 and 3fs/8 frequencies
present in the signal? The normalization factor is 4 (divide by 4
after summing the multiplied signals) to get the correct initial
amplitudes. Check your answers by calculating the fs/4 frequency and
see if you get 0.2 as the initial amplitude.
(c) Verify that either the frequency 0 Hz or f_{s}/2
have an amplitude of 0 in the signal.
(d) For extra credit, demonstrate that the amplitudes of the frequencies
in the spectrum you calculated in part (b) can be used to generate the
original sequence:
0.6, 0.14, 0.2, 0.14, 0.2, 0.14, 0.2, 0.14
In other words, what are the sequences of sampled values from the
three (co)sinewaves that add up to generate the final signal?
