Applications: Signal Theory Across the Sciences

A particle's state can be described equivalently in position or momentum space:

$$\tilde{\psi}(p) = \frac{1}{\sqrt{2\pi\hbar}} \int_{-\infty}^{\infty} \psi(x)\, e^{-ipx/\hbar}\, dx$$

This is precisely the Fourier transform with the substitution $k = p/\hbar$. The inverse transform recovers $\psi(x)$ from $\tilde{\psi}(p)$.

The Heisenberg uncertainty principle is a direct consequence of the Fourier uncertainty relation we studied in Chapter 3:

$$\Delta x \cdot \Delta p \;\geq\; \frac{\hbar}{2}$$

A state that is sharply localised in position ($\Delta x$ small) must be spread out in momentum ($\Delta p$ large), and vice versa. The Gaussian wave packet saturates the bound, just as the Gaussian minimises the time-frequency uncertainty product.

The quantum propagator $K(x, t; x', 0)$ tells us how a wave function evolves:

$$\psi(x, t) = \int K(x, t;\, x', 0)\, \psi(x', 0)\, dx'$$

This is a convolution — the propagator is the Green's function (impulse response) of the Schrodinger equation. For a free particle:

$$K_{\text{free}}(x, t;\, x', 0) = \sqrt{\frac{m}{2\pi i\hbar t}}\, \exp\!\left(\frac{im(x - x')^2}{2\hbar t}\right)$$

A free scalar field is expanded in plane-wave modes, exactly as we decompose a signal into frequency components:

$$\hat{\phi}(x) = \int \frac{d^3 k}{(2\pi)^3} \frac{1}{\sqrt{2\omega_k}} \left( \hat{a}_k\, e^{ik \cdot x} + \hat{a}_k^\dagger\, e^{-ik \cdot x} \right)$$

Each mode $k$ is an independent harmonic oscillator. The creation and annihilation operators $\hat{a}_k^\dagger, \hat{a}_k$add or remove quanta of that mode — analogous to adding spectral energy at frequency $\omega_k$.

The Feynman propagator in momentum space is:

$$\widetilde{G}_F(k) = \frac{i}{k^2 - m^2 + i\epsilon}$$

This is the frequency-domain transfer function of the Klein–Gordon equation. It has a pole at $k^2 = m^2$ — the on-shell condition — playing the same role as a resonance pole in a filter's transfer function $H(s)$.

In a Fock state $|n_{k_1}, n_{k_2}, \ldots\rangle$, the occupation number $n_k$ counts the quanta (particles) in mode $k$. The expected energy distribution is:

$$\langle \hat{H} \rangle = \sum_k \omega_k \left( n_k + \tfrac{1}{2} \right)$$

This is precisely analogous to a power spectral density: the energy per mode is weighted by the number of quanta at that frequency.

The scattering matrix $S$ maps “in” states to “out” states:

$$|\text{out}\rangle = S\, |\text{in}\rangle$$

In momentum space the $T$-matrix elements $\mathcal{M}(k_{\text{in}} \to k_{\text{out}})$ describe how each spectral component scatters — a direct analogue of a linear system's transfer function $H(\omega)$.

Loop integrals in QFT diverge because they integrate over arbitrarily high momenta:

$$\int \frac{d^4 k}{(2\pi)^4} \frac{1}{(k^2 - m^2)^2} \;\to\; \infty$$

In signal-processing language this is an ultraviolet divergence — the spectral integral diverges at high frequencies. Regularisation (cutoff, dimensional) is the physicist's band-limiting.

Wilson's renormalisation group integrates out high-momentum shells$\Lambda/b < |k| < \Lambda$, then rescales:

$$Z_\Lambda[\phi_<] = \int \mathcal{D}\phi_>\; e^{-S[\phi_< + \phi_>]}$$

Each RG step is a low-pass filter that removes the highest-frequency modes from the path integral, yielding an effective action for the remaining modes. Iterating the filter produces a flow in coupling-constant space.

In thermal field theory, imaginary time $\tau \in [0, \beta]$ is periodic with period $\beta = 1/k_B T$. The Fourier decomposition is:

$$G(\tau) = \frac{1}{\beta} \sum_{n=-\infty}^{\infty} \tilde{G}(i\omega_n)\, e^{-i\omega_n \tau}, \qquad \omega_n = \frac{2\pi n}{\beta}$$

The discrete Matsubara frequencies $\omega_n$ are the exact finite-temperature analogue of the DFT frequencies $2\pi k / N$. Bosonic fields use even multiples; fermionic fields use odd multiples (anti-periodic boundary conditions).

Pulsars emit radio pulses with extraordinary regularity. Their signal is modelled as a periodic pulse train with period $P$:

$$s(t) = \sum_{n} \delta(t - nP) * h(t)$$

where $h(t)$ is the individual pulse shape. In the Fourier domain this becomes a comb spectrum modulated by $H(f)$. Deviations in pulse arrival times (timing residuals) encode information about gravitational waves, orbital companions, and the interstellar medium.

LIGO and Virgo detect gravitational waves by cross-correlating detector output with a bank of template waveforms. The matched filter maximises the SNR:

$$\text{SNR}^2 = 4 \int_0^\infty \frac{|\tilde{h}(f)|^2}{S_n(f)}\, df$$

where $\tilde{h}(f)$ is the template spectrum and $S_n(f)$ is the detector noise PSD. This is the Wiener filter from Chapter 8 applied to one of the greatest experimental achievements in physics.

The Hubble's initial spherical aberration produced blurred images. In signal terms, the observed image is a convolution:

$$I_{\text{obs}}(x, y) = I_{\text{true}}(x, y) * \text{PSF}(x, y) + n(x, y)$$

Deconvolution algorithms (Richardson–Lucy, Wiener) recover $I_{\text{true}}$ by dividing in Fourier space, with regularisation to control noise amplification at high spatial frequencies.

In magnetic resonance imaging, the raw data acquired by the scanner lives in k-space, which is the 2D (or 3D) spatial-frequency domain:

$$S(k_x, k_y) = \iint \rho(x, y)\, e^{-i2\pi(k_x x + k_y y)}\, dx\, dy$$

where $\rho(x, y)$ is the proton density (the image). The image is reconstructed by an inverse 2D FFT. Undersampling k-space introduces aliasing artifacts, governed by exactly the same Nyquist criterion from Chapter 5.

Computed tomography measures line integrals (projections) of the attenuation coefficient $\mu(x, y)$. The Fourier slice theorem states:

$$\mathcal{F}_1\{R_\theta\{\mu\}\}(\omega) = \mathcal{F}_2\{\mu\}(\omega \cos\theta,\, \omega \sin\theta)$$

Each 1D projection, when Fourier-transformed, gives a radial slice through the 2D Fourier transform of the image. Filtered back-projection reconstructs the image by applying a ramp filter $|\omega|$ in frequency space before back-projecting — a direct application of the Fourier convolution theorem.

A seismogram $u(t)$ is modelled as the convolution of the source time function $s(t)$, the Earth's impulse response (Green's function) $g(t)$, and the instrument response $i(t)$:

$$u(t) = s(t) * g(t) * i(t)$$

In the frequency domain: $U(f) = S(f)\, G(f)\, I(f)$. Removing the instrument response (deconvolution) recovers the ground motion. Water-level deconvolution adds a regularisation floor to prevent division by near-zero spectral values.

After a great earthquake, the entire Earth rings like a bell. The free oscillations (normal modes) form a discrete spectrum:

$$u(t) = \sum_k A_k \cos(\omega_k t + \phi_k)\, e^{-\alpha_k t}$$

Each mode $_nS_l$ (spheroidal) or $_nT_l$ (toroidal)is a standing wave on the sphere, characterised by overtone number $n$and angular degree $l$. Spectral analysis of months-long records resolves the splitting of these modes, revealing the Earth's 3D density structure.

The Earth's orbital parameters vary quasi-periodically, driving ice-age cycles. Power spectral analysis of deep-sea sediment cores reveals three dominant peaks:

• Eccentricity: $T \approx 100$ kyr
• Obliquity: $T \approx 41$ kyr
• Precession: $T \approx 23$ kyr

These are identified via Lomb–Scargle periodograms or multi-taper spectral analysis (to handle unevenly spaced data). The alignment of these spectral peaks with computed orbital frequencies is one of the great triumphs of paleoclimate science.

The El Nino–Southern Oscillation (ENSO) is a coupled ocean–atmosphere phenomenon with a broad spectral peak at 2–7 years. Its power spectrum is characterised by:

$$S_{\text{ENSO}}(f) \propto \frac{1}{(f - f_0)^2 + \gamma^2}$$

— a Lorentzian shape centred near $f_0 \approx 1/(3.7\text{ yr})$, corresponding to a damped oscillator driven by stochastic forcing. Wavelet analysis reveals that the dominant period shifts over decadal timescales.

The sea surface elevation $\eta(x, t)$ is modelled as a random superposition of sinusoidal components. The Pierson–Moskowitz spectrum for a fully developed sea is:

$$S(\omega) = \frac{\alpha g^2}{\omega^5} \exp\!\left(-\beta \left(\frac{\omega_0}{\omega}\right)^4\right)$$

where $\omega_0 = g / U_{19.5}$ depends on wind speed. The significant wave height is related to the zeroth spectral moment: $H_{1/3} = 4\sqrt{m_0}$ where $m_0 = \int S(\omega)\, d\omega$.

Tides are decomposed into harmonic constituents — a Fourier series with astronomically determined frequencies:

$$h(t) = h_0 + \sum_{k=1}^{N} A_k \cos(\omega_k t - \phi_k)$$

The principal constituents include $M_2$ (principal lunar semidiurnal, period 12.42 h), $S_2$ (principal solar, 12.00 h), $K_1$(luni-solar diurnal, 23.93 h), and dozens more. Least-squares harmonic analysis extracts $A_k$ and $\phi_k$ from tide-gauge records, enabling tide prediction — one of the oldest applications of Fourier analysis.

X-ray diffraction from a crystal produces a pattern that is the Fourier transform of the electron density:

$$F(\mathbf{h}) = \int_{\text{cell}} \rho(\mathbf{r})\, e^{-2\pi i \mathbf{h} \cdot \mathbf{r}}\, d^3 r$$

The structure factor $F(\mathbf{h})$ is measured at discrete reciprocal-lattice points $\mathbf{h} = (h, k, l)$. Recovering$\rho(\mathbf{r})$ requires an inverse Fourier transform, but only the intensities $|F|^2$ are measured — the phases are lost. The phase problem is crystallography's version of spectral phase retrieval.

Many genes are expressed with circadian (~24 h) periodicity. Identifying cycling genes from microarray time series uses spectral methods:

$$g_i(t) = A_i \cos(2\pi t / T - \phi_i) + \bar{g}_i + \epsilon_i(t)$$

The Fisher $g$-statistic tests whether the dominant spectral peak at$f = 1/24$ h$^{-1}$ is significant against a null hypothesis of white noise. Thousands of genes in organisms from cyanobacteria to humans show significant circadian spectral power.

The MP3 codec uses a modified discrete cosine transform (MDCT) to convert time-domain audio into frequency-domain coefficients:

$$X(k) = \sum_{n=0}^{2N-1} x(n)\, w(n)\, \cos\!\left[\frac{\pi}{N}\left(n + \frac{N+1}{2}\right)\!\left(k + \frac{1}{2}\right)\right]$$

A psychoacoustic model then determines which coefficients can be coarsely quantised (or discarded) without perceptible quality loss. Frequency masking — the phenomenon that a loud tone masks nearby quieter tones — is exploited to achieve 10:1 compression ratios. This is lossy spectral coding.

Additive synthesisers build complex timbres by summing sinusoidal partials:

$$s(t) = \sum_{k=1}^{K} A_k(t)\, \sin\!\big(2\pi k f_0 t + \phi_k(t)\big)$$

This is exactly Fourier synthesis with time-varying coefficients. Each partial$k f_0$ is a harmonic of the fundamental $f_0$. By controlling the amplitude envelopes $A_k(t)$, a synthesiser can morph between timbres in real time — from a flute (few harmonics, slow attack) to a trumpet (many harmonics, fast attack).