The stationarity problem

Classical spectral analysis rests on a key assumption: the covariance between observations depends only on the lag between them, not on when they occur. Formally, \(\text{Cov}(X(t),\, X(t+h)) = \gamma(h)\).

When this fails — when \(\text{Cov}(X(t), X(t+h)) = \gamma(t,h)\) depends on \(t\) — the process is nonstationary. Its frequency content changes over time. Such data are called time-varying frequency (TVF) data. Examples include seismic signals, insect noises, and financial volatility.

Hierarchy of solutions

1. Fourier Transform (baseline)

\[G(f) = \int_{-\infty}^{\infty} e^{-2\pi ifx}\, g(x)\, dx\] Global by construction — no time localization. Useless for TVF data.

2. Short-Time Fourier Transform (STFT)

A window \(h(x-t)\) with \(|h(t)| \to 0\) as \(|t| \to \infty\) localizes the transform around time \(t\): \[G(t,f) = \int_{-\infty}^{\infty} g(x)\, h(x-t)\, e^{-2\pi ifx}\, dx\] Partial fix, but the window width is fixed. This creates an unavoidable trade-off: narrow window = good time resolution, poor frequency resolution. Wide window = the reverse. This is the signal-processing analogue of the Heisenberg uncertainty principle.

3. Wigner-Ville Spectrum

Generalizes the power spectrum to nonstationary processes: \[W(t,f) = \sum_{k=-\infty}^{\infty} \gamma\!\left(t+\tfrac{k}{2},\, t-\tfrac{k}{2}\right) e^{-2\pi ijk}\] More theoretically principled, but suffers from cross-term interference in practice (it is bilinear, so products of signal components produce spurious features).

4. Wavelets — Adaptive Resolution

Instead of a fixed window, use dilations of a single function — the mother wavelet — whose scale stretches or compresses to match the frequency of interest. High-frequency events get a narrow window; low-frequency trends get a wide one.

Fourier vs. wavelets: the key analogy

The Fourier basis is built from dilations of \(e^{ix}\): \[S_2 = \{e^{ikx},\ k \in \mathbb{Z}\}\] This works locally, but globally \(|e^{ix}| = 1\) everywhere — it never decays. Therefore \(e^{ix} \notin L^2(\mathbb{R})\), meaning \(\int_{-\infty}^{\infty} |e^{ix}|^2\, dx = \infty\).

Wavelets build on this idea but fix the decay problem. A function is called square integrable if \[\int_{-\infty}^{\infty} |g(x)|^2\, dx < \infty \quad (\text{written } g \in L^2(\mathbb{R}))\] A necessary condition for \(g \in L^2(\mathbb{R})\) is \(|g(t)| \to 0\) as \(|t| \to \infty\). This is what sine and cosine fail to satisfy — and what the mother wavelet \(\Omega\) must satisfy.

The mother wavelet

Let \(\Omega \in L^2(\mathbb{R})\) be the mother wavelet. Its entire family of basis elements is generated by two operations:

• Dilation by integer \(j\): stretches or compresses the wavelet (controls scale/frequency)
• Translation by integer \(k\): shifts it in time (controls location)

\[\Omega_{j,k}(x) = 2^{-j/2}\, \Omega(2^{-j}x - k)\] The factor \(2^{-j/2}\) normalizes so that \(\|\Omega_{j,k}\| = \|\Omega\|\) for all \(j, k\). Large \(j\) → stretched wavelet → captures low-frequency structure. Small \(j\) → compressed wavelet → captures high-frequency bursts.

Fundamental approximation result

Any \(g \in L^2(\mathbb{R})\) admits the representation: \[g(x) = \sum_{j=-\infty}^{\infty} \sum_{k=-\infty}^{\infty} d_{j,k}\, \Omega_{j,k}(x)\] with equality in the mean-square sense (\(L^2\) convergence). The wavelet transform coefficients are: \[d_{j,k} = \frac{1}{\|\Omega\|^2} \int_{-\infty}^{\infty} g(t)\, \Omega_{j,k}(t)\, dt\]

This is a projection of \(g\) onto each basis element — exactly analogous to Fourier coefficients \(\hat{g}(k) = \int g(t)\, e^{-ikt}\, dt\), but now indexed by both scale \(j\) and location \(k\).

Comparison table

Figures