Consider computing the 1000point DFT of a 1000 point long si

Consider computing the 1000-point DFT of a 1,000 point long sine wave having a frequency of 200 Hz sampled at a rate of 10,000 samples per second. X(k) = DFT[x(n)] a) What is the digital frequency resolution (frequency spacing between adjacent values of k, in radians/sample)? What is the equivalent analog frequency resolution (in Hertz)? b) At what value of k would we expect to see evidence of the 200 Hz sinusoid in the DFT?

Solution

freq Resolution(Analog)=F_s/N

F_s=Sampling freq=10,000Hz

N=1000;

freq Resolution(Analog)R=10000/1000=10Hz

For Digital=2*pi*R=20pi radian/samples

b)

sine wave freq=200H

res freq=10,

so one full cycles take samples K=200/10=20

when k=20 it will give evidance ofsinewave at 200hz