# Chapter 3

# Spectral Observation

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

*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;

*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*. If we set

_{s}*f*

_{0}=

*F*

_{0}/

*F*and assume it to be < 1/2, we get

_{s}*x*(

*n*) =

*e*

^{2jπf0n}.

*x*(

*n*) = exp(2

*j*

*πf*

_{0}

*n*)} where

*f*

_{0}= 7/32 and

*n*∈ {0, …, 31};

*f*=

*k*/32, for

*k*∈ {0, …, 31};

**fft**command, display the modulus of the DFT of {

*x*(

*n*)};

*f*

_{0}= 0.2. Display the modulus of the DFT of {

*x*(

*n*)}. How do you explain the result?

(3.1)

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