13. Non-stationary Spectral Analysis

When it is necessary to capture audio covering the entire 20–20,000 Hz range of human hearing, such as when recording music or many types of acoustic events, audio waveforms are typically sampled at 44.1 (Compact Disks), 48, 88.2, or 96 kHz.

A wavelet is, in general, a finite-energy, zero-mean waveform that in the wavelet transform is stretched or compressed in an auto-similar way, exactly like the Gaussian function in Fig. 4.​14 (which has an unbounded time support) and the boxy function of Fig. 4.​15 (which has a compact temporal support). For example, we will introduce, on one hand, the historical real Haar wavelet, a waveform with compact time support, and the analytic Morlet wavelet (shown in its real and imaginary parts in Fig. 13.15) that, being a complex exponential oscillation modulated by a Gaussian envelope, never vanishes identically at finite time values, and therefore has an unbounded time support. However, in numerical applications using finite-precision arithmetic, the distinction among the two cases can be considered to some extent as a mathematical nuance. E.g., a Gaussian function that we compute numerically over an extended range of values of the independent variable will end up, as we go farther and farther from its center, assuming such a small value to be practically indistinguishable from zero. For instance, the minimum positive number representable in double precision is 2.22507e\(-\)308 and the maximum is 1.79769e\(+\)308; positive numbers that are smaller than 2.22507e\(-\)308 are treated as zero and positive numbers greater than 1.79769e\(+\)308 are treated as \(+\infty \). For this very practical reason, talking about wavelets we will often neglect the conceptual difference between compact support and concentration, and will loosely use expressions like “waveform localized in time”, etc. In a similar way, in the frequency domain we will speak about the spectra of the wavelet functions using terms like “passband spectrum, waveform localized in the frequency domain”, and so on.

In the following discussion, for brevity we will often write simply “scale” instead of “scale factor”, provided the context does not allow any ambiguity.

The signals analyzed by CWT may be mono- or bi-dimensional, possibly with space (instead of time) as the independent variable, as in the case of image processing; in principle, the signals can even be multi-dimensional. In this book we will only deal with wavelet analysis of a time series.

Of course, in digital applications \(x(\theta )\) will become a sampled signal x[n] and the continuous variables \(\omega \) and b will be made discrete.

Here we indicate angular frequencies by \(\omega \), even if we are formally working in the continuous-time domain: this is justified because we are using an adimensional continuous time and therefore angular frequency is adimensional as well. However, as long as we do not apply some discretization scheme, \(\omega \) is not constrained by the inequality \(-\pi \le \omega < \pi \).

A complex system is defined as a system with complex-valued impulse response. In the frequency domain, real-valued signals/systems always have even-symmetric amplitude-spectrum/response and odd-symmetric phase-spectrum/response with respect to the zero frequency. Complex signals/systems do not need to have any spectral symmetry properties in general: e.g., the spectral support (region of non-zero amplitude spectrum) can basically be anything. This book focuses on real signals/systems, and for further discussion on the STFT and CWT we need no more information about complex filters. Note also that the analog filter bank related to the STFT will be a digital filter bank in sampled-signal applications.

Complex wavelets also provide phase information concerning the signal’s elementary components, even is this information is not often easy to interpret. For this reason, we will not discuss CWT phase plots.

Recall that the term spectrogram indicates instead a STFT plot versus time and frequency.

We should speak about energy per squared unit of time, but time—delay, scale factor—is adimensional here.

We may note that if the data were cyclical, as in the case of a meteo-climatic data sequence measured at fixed latitude and time as a function of longitude, there would be no need to add zeros and the COI would not exist.

The plot should also include a colorbar telling the scalogram value associated with each shade of gray. Since this is only a qualitative example, the colorbar has been omitted.

Recall that the phase angle of a complex quantity is always obtained by taking the arc tangent of the ratio of the imaginary and real parts.

Time-series modeling is common practice in economics. Logarithms possess properties that assist with model-building and with the visual display of models and data in graphs. In essence, log-linearization is a solution to the problem of reducing computational complexity in systems of numerically specified equations that need to be solved simultaneously. Log-linearization converts a nonlinear equation into an equation that is linear in terms of the log-deviations of the associated variables from their steady-state values. For small deviations from the steady state, log-deviations have a convenient economic interpretation: they are approximately equal to the percentage deviations from the steady state. Log-linearization can greatly simplify the computational burden and, therefore, help solve a model that may otherwise be intractable.