Comments and Further Reading

4.11 Comments and Further Reading

Spectral analysis is the topic of several good books, such as [SM05]. As discussed in these books dedicated to spectral analysis, there are involved issues on whether or not random signals can be analyzed with Fourier transforms in the same way done for deterministic power signals. Because it turns out the answer is yes, and the result is elegant and even intuitive, the proof and corresponding discussion was omitted here. Another strategy adopted here to avoid going deeper in the theory of spectral analysis was to focus primarily on discrete-time signals.

Two classic works about windows for FFT-based analysis are [Har78, Nut81]. Scalloping loss and its compensation are well described in [Lyo11]. The windows described here are the so-called “symmetric” windows in Matlab’s documentation, which also discusses “periodic” windows. Another variation is to center the window at the origin $n = 0$ as discussed in [Har78], while here a window with $N$ samples starts at $n = 0$ and ends at $n = N - 1$ . The issue of correcting the amplitude and power of a FFT-based spectral analysis is discussed in [Bra11].

There are distinct definitions in the literature for the periodogram. For example, it can be defined as e. g. in [ url4spe]:

Ŝ (e^{j Ω}) \overset{def}{=} \frac{1}{N} | DTFT {x_{N} [n]} |^{2}

(4.72)

to be an approximation of the discrete-time PSD $S (e^{j Ω})$ . However, this definition is not directly compatible with the periodogram function in Matlab/Octave.

More information about linear prediction can be obtained in specialized textbooks such as The Theory of Linear Prediction, by P. P. Vaidyanathan, 2008, available on the Web [ url4ppv].

In [Hua90], it is discussed the minimum description length (MDL) criteria applied to autoregressive models. In spite of the relative simplicity of MDL, its application to autoregressive modeling is involved and [Hua90] addresses several important issues.

The PSD and, equivalently, the autocorrelation, cannot convey all the information of a signal. Similarly, spectrograms are just one of the many techniques for time-frequency analysis. For example, wavelets and Wigner-Ville distributions [VK95, Mal08] have been applied in several problems.

In many applications, higher-order spectra analysis [NP93] can bring additional insight when compared to the techniques discussed in this chapter. Another very powerful tool is cyclostationary analysis [Gar94, Gia99], which is important in applications such as blind signal identification [GG98] and spectrum sensing for cognitive radios [DIM10].