Digital Signal and Image Processing using MATLAB, Volume 1: Fundamentals, 2nd Edition

Chapter 3

Spectral Observation

The purpose of this chapter is to introduce the reader to the two following fundamental concepts:

– the accuracy of the frequency measurement when the DFT is used to evaluate a signal’s DTFT. As we will see, this accuracy depends on the number of points used to calculate the DFT;

– the spectral resolution, which is the ability to discern two distinct frequencies contained in the same signal. It depends on the observation time and on the weighting windows applied to the signal.

3.1 Spectral accuracy and resolution

3.1.1 Observation of a complex exponential

To illustrate the DFT’s use in signal spectrum observation, we will begin with a simple example.

EXAMPLE 3.1 (Sampling a complex exponential)

Consider the sequence resulting from the sampling of a complex exponential e^2jπF₀t at a frequency of F_s = 1/T_s. If we set f₀ = F₀/F_s and assume it to be < 1/2, we get x(n) = e^2jπf₀n.

1. determine the DTFT’s expression for the sequence {x(n) = exp(2jπf₀n)} where f₀ = 7/32 and n ∈ {0, …, 31};

2. using this result, find the DTFT’s values at the points of frequency f = k/32, for k ∈ {0, …, 31};

3. using the fft command, display the modulus of the DFT of {x(n)};

4. now let f₀ = 0.2. Display the modulus of the DFT of {x(n)}. How do you explain the result?

HINTS:

1. Starting off with definition 2.22 of the DTFT, we get:

(3.1)

Because a finite duration sequence is ...