May 4, 2004
Problem Set
12: Discrete (Fast) Fourier Transform
1. Suppose that we are studying the spectral properties
of a soprano singing a tone of middle C (f = 256 Hz). We want to see overtones up to 2 octaves
above the fundamental. Let's say that we
decide to measure frequency components up to a frequency of 2 kHz.
Determine the sampling frequency required to
observe this maximum frequency. (This is
the Nyquist frequency.) Also calculate the time Dt between samples at the Nyquist
frequency.
2. Suppose we digitize for a time T = 10 seconds
at the Nyquist frequency calculated above, and
calculate the discrete Fourier transform.
Calculate the number of points N
recorded, and the frequency interval Dw between points in the transform.
3. Now suppose that we shorten the length T of the recording interval, with a
corresponding loss in frequency resolution, manifested by an increase in Dw. How long must
the recording be to determine (by looking at the highest value in the frequency
spectrum) whether the soprano is singing C (256 Hz) or C# (271 Hz)?