# 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 *f*_{0} = *F*_{0}/*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(2*j**πf*_{0}*n*)} where *f*_{0} = 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

*f*_{0} = 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 ...