Week 12 — Signals and Systems: LTI, Fourier, and Sampling

Overview

Signals and systems is the continuous/discrete-time counterpart to linear algebra: an LTI system is a linear operator, convolution is how it acts, and the Fourier transform diagonalizes it — complex exponentials are the eigenfunctions. This week develops signals and system properties, convolution and impulse response, Fourier series and transforms in both continuous and discrete time, frequency response, and the sampling theorem that connects the two worlds. It is the foundation for DSP (Week 13) and the wave/optics material (Week 18), and it rests on the complex-analysis vocabulary of Week 11. The delta function, the impulse response, and the Fourier transforms of objects like the constant and the impulse train are made rigorous by distribution theory (Strichartz), which sits underneath the whole construction.

Readings

OWN Ch 1 — Signals and systems. Continuous vs discrete time, energy/power, transformations of the independent variable, system properties (linearity, time-invariance, causality, stability).
OWN Ch 2 — LTI systems. Convolution, impulse and step response.
OWN Ch 3 — Fourier series of periodic signals.
OWN Ch 4–5 — Continuous- and discrete-time Fourier transforms. Properties, filtering, modulation.
OWN Ch 6 — Time/frequency characterization. Magnitude/phase, group delay.
OWN Ch 7 — Sampling. The sampling theorem, aliasing, reconstruction.
Strichartz Ch 1–5 — Distributions and the Fourier transform. Test functions and distributions (generalized functions), the Dirac delta and Heaviside step, distributional derivatives (why a discontinuous signal can be differentiated), convolution of distributions, and the Fourier transform of tempered distributions. The rigorous foundation under the impulse response, the convolution theorem, and the Fourier transforms used throughout this week.

Key Concepts

Signals, systems, and LTI

Signals are functions of time, continuous \(x(t)\) or discrete \(x[n]\). A system is an operator mapping input to output; the central class is linear time-invariant (LTI): linear (superposition) and time-invariant (a time shift of the input shifts the output). An LTI system is completely characterized by its impulse response \(h\).

Convolution

The output of an LTI system is the convolution of the input with the impulse response:

\[y(t) = (x*h)(t) = \int_{-\infty}^{\infty} x(\tau)\,h(t-\tau)\,d\tau, \qquad y[n] = \sum_{k} x[k]\,h[n-k].\]

Causality means \(h\) is zero for negative argument; BIBO stability means \(h\) is absolutely summable/integrable. Convolution is the time-domain action; the transforms below turn it into multiplication.

Distributions: making the delta rigorous

The impulse response presumes an “impulse” \(\delta\) with \(\int\delta = 1\) concentrated at a single point — no ordinary function does this. Distribution theory (Strichartz) supplies it: a distribution is a continuous linear functional on smooth, rapidly decaying test functions, and \(\delta\) is the one that evaluates a test function at the origin, \(\langle\delta,\varphi\rangle = \varphi(0)\). In this framework the impulse response \(h\) is genuinely the system’s response to \(\delta\), and every distribution can be differentiated — so the derivative of the discontinuous Heaviside step is exactly \(\delta\), and discontinuous signals acquire well-defined spectra. The Fourier transform extends to tempered distributions, which is what legitimizes \(\mathcal{F}\{1\} = 2\pi\delta(\omega)\) and the transforms of \(e^{j\omega_0 t}\), \(\cos\omega_0 t\), and the impulse train rather than leaving them formal. Convolution with \(h\) and multiplication by \(H(j\omega)\) — the engine of this whole week — are theorems here, not heuristics.

Fourier series and transforms

A periodic signal expands in harmonically related complex exponentials (Fourier series). An aperiodic signal uses the Fourier transform:

\[X(j\omega) = \int_{-\infty}^{\infty} x(t)\,e^{-j\omega t}\,dt, \qquad X(e^{j\omega}) = \sum_n x[n]\,e^{-j\omega n}.\]

The decisive property is the convolution theorem: convolution in time is multiplication in frequency, \(y = x*h \Leftrightarrow Y(j\omega) = X(j\omega)H(j\omega)\). So \(H(j\omega)\) — the frequency response — tells you exactly how each frequency is scaled and phase-shifted. Complex exponentials are the eigenfunctions of LTI systems, with eigenvalue \(H(j\omega)\); this is the Fourier analogue of diagonalization (Week 1). Other key properties: linearity, time/frequency shifting, the modulation property, and Parseval’s relation (energy is preserved across domains).

Magnitude, phase, group delay

A system’s effect splits into a magnitude response \(|H(j\omega)|\) (which frequencies are passed or attenuated — the notion of a filter) and a phase response \(\angle H(j\omega)\). Group delay \(-\frac{d}{d\omega}\angle H(j\omega)\) measures the delay of the signal envelope; linear phase means no waveform distortion.

Sampling

The sampling theorem (Nyquist–Shannon): a signal bandlimited to \(W\) Hz is exactly recoverable from samples taken at rate \(f_s > 2W\). Below that rate, high frequencies fold down and masquerade as low ones — aliasing. Reconstruction is ideally sinc interpolation; in practice an anti-alias filter precedes sampling. This theorem is the bridge between the continuous world and the discrete DSP of Week 13.

Connections

Backward: complex exponentials, the eigenfunctions of LTI systems, are the analytic functions of Week 11; the delta function and the Fourier transform of distributions rest on the measure-theoretic and \(L^2\) foundation of Week 5 (MIR, Strichartz).
Forward: Week 13 adds the Laplace/z-transforms (poles, zeros, stability) and the discrete computational tools (DFT/FFT, filter design).
Across courses: Sampling, filtering, and frequency response under sensor processing and control loops (Course 3), DSP internals (Course 3); the same Fourier machinery reappears as Fourier optics (Week 18).