4.2  Introduction

This chapter presents a brief introduction to spectral analysis. The main goal of spectral analysis is to estimate how the signal power is distributed over frequency. It is a very broad area with many applications. Beware that definitions and jargon vary throughout the literature! The discussion here will focus on the important task of estimating the power spectrum (also called power spectral density).

Spectral analysis differs from frequency estimation. The latter, also called spectral line analysis, assumes there are few frequencies that need to be estimated while, in contrast, spectral analysis aims at estimating the whole spectrum. When comparing the two, it is intuitive that having only few frequencies of interest is beneficial to the estimation process.

Frequency estimation could be eventually performed by obtaining the signal spectrum via an FFT and picking its magnitude peaks. However, there are many specialized frequency estimation algorithms that can outperform this FFT-based strategy. There are frequency estimation methods for a single tone, multi-tone and multi-harmonic signals, for example. In the latter case, the signal of interest is composed by a sum of harmonically related sinusoids. This text concentrates in spectral analysis and the reader is encouraged to get information elsewhere about frequency estimation methods if he/she faces the task of finding a small set of frequencies that compose a given signal of interest.

The spectral analysis algorithms can be broadly divided in parametric and nonparametric. The former category encompasses algorithms that assume a model for the signal under analysis and estimate the parameters of this model. Examples of such models are autoregressive (AR), moving-average (MA) and ARMA (combination of the two), which can be seen as stochastic processes with realizations obtained by passing white noise through digital filters of types all-poles IIR $1∕A(z)$, FIR $B(z)$, and an IIR with finite zeros $B(z)∕A(z)$, respectively.

For these parametric models, the “parameters” are the coefficients of the corresponding filter and the noise power. The nonparametric algorithms try to estimate the spectrum without imposing that the signal of interest was generated by a model. This text discusses the following spectral analysis methods:

• Nonparametric: PSD estimation with the periodogram and Welch’s algorithm
• Parametric: estimation of an AR model via linear predictive coding (LPC)

We will start by studying nonparametric methods based on the FFT. Assuming we want to perform spectral analysis on a discrete-time signal $x[n]$, we will go over two steps:

• Extract $N$ samples of $x[n]$ to enable using a FFT, and mathematically model this process as multiplying $x[n]$ by a window function $w[n]$ to create the windowed signal with DTFT $X_w(e^{j\mathrm{\Omega }})$.
• Use the FFT to obtain $X_w[k]$, which consists of discrete values of the DTFT $X_w(e^{j\mathrm{\Omega }})$ over the discrete-time angular frequency $\mathrm{\Omega }$, as discussed in Section 2.5.5.0.

These two processes bring artifacts, which will be discussed in the next sections: spectrum leakage and picket-fence effect, respectively. If one starts from an analog signal $x(t)$ instead of $x[n]$, he/she needs to also consider the C/D process that relates $x(t)$ and $x[n]$ and may be impaired by other artifacts, besides leakage and picket-fence.

To understand spectrum leakage, a brief review of windows for spectrum estimation is presented in the sequel.