Question
help?
A real-valued sinusoid is described by three parameters. The mathematical form of the time signal is s[n] = A cos(2 pi fo n + i + phi) for n = 0,1,2,,N-l where N is the signal length, A its amplitude, fo its frequency, and phi the relative phase at n = 0. Compute the 21-point DFT of a sequence representing exactly one cycle of a cosine wave. Determine the frequency of this sinusoid. Make sure to take exactly one cycle, not one cycle plus one point (i.e., don't repeat the first sample). If done correctly, the answer will be extremely simple. Repeat part (a) for a sine, then for a cosine with a 45 degree phase shift. Observe carefully the magnitudes of the DFT coefficients (and compare the phases). Repeat part (a) for three cycles of a sinusoid, still using a 21-point DFT. What is the frequency of this sinusoid (in radians per sample)? Try a vector that is 3.1 cycles of a sinusoid. Why is the DFT so different? Experiment with different frequency sinusoids. Show that choosing the frequency to be fo = k(1/N), when k is an integer, gives an N-point DFT that has only two nonzero values.
Explanation / Answer
first write the cosine term as an complex exponential
cos(2*pi*f*t) = (1/2)*(exp(j2*pi*f*t) + exp(-j2*pi*f*t))
as you need 21 point one cycle DFT. so take f = 1/21 and compute the DFT for each N = 0,1,2,.....20.
for sin(2*pi*f*t) = (1/2j)*(exp(j2*pi*f*t) - exp(-j2*pi*f*t))
apply the same as stated above.
