Week 8 — LTI Systems, Convolution, and Fourier Analysis

Overview

For seven weeks we built and measured circuits. Now we step up a level of abstraction and ask: what is a circuit, as a system? The answer — for the linear circuits of Weeks 1–6 — is that it is a linear time-invariant (LTI) system, completely characterized by a single function, its impulse response \(h(t)\). Everything else follows: the output to any input is the convolution \(y=x*h\), and in the frequency domain that convolution becomes simple multiplication by the frequency response \(H(j\omega)\) — which is exactly the transfer function you measured in Week 6. This week is the theoretical keystone that retroactively explains every measurement you have made and sets up the entire discrete-time second half.

The Fourier transform is the protagonist. The deep reason it matters was seeded in Week 5: complex exponentials are the eigenfunctions of LTI systems, so decomposing a signal into exponentials (Fourier) and letting the system scale each one independently (multiply by \(H(j\omega)\)) is the natural way to analyze any LTI system. A filter “shapes a spectrum” — and now you can say precisely what that means and compute it. On the bench, you will use the scope’s FFT to see the spectrum of real signals, watch a filter reshape it, and verify the convolution/multiplication duality with your own captured data.

This week unifies Weeks 4–6 (poles, transients, frequency response are all facets of \(h(t)\) and \(H(j\omega)\)) and is the direct prerequisite for Week 9 (sampling is a Fourier-domain phenomenon) and Week 10 (the DFT is the computable version of all this).

Readings

O-S&S Ch. 2: LTI systems, the impulse response, and convolution (continuous and discrete). Extract: why \(h\) fully characterizes an LTI system and how the convolution integral arises from superposition + time-invariance.
O-S&S Ch. 3: Fourier series for periodic signals — the representation of a periodic signal as a sum of harmonics, and how LTI systems act on each harmonic. Extract: the eigenfunction property in action.
O-S&S Ch. 4: The continuous-time Fourier transform, its properties (linearity, time/frequency shift, scaling, convolution theorem, duality, Parseval), and the frequency response of LTI systems. Extract: the transform pairs and the convolution theorem.
CAD Ch. 11: Fourier analysis applied to circuits — using the spectrum to find a circuit’s steady-state response to non-sinusoidal periodic inputs. Extract: the hardware tie-back.

Key Concepts

LTI systems and the impulse response

A system is linear (superposition holds) and time-invariant (a time-shifted input produces an identically time-shifted output). Such a system is completely described by its impulse response \(h(t)\) — the output when the input is a Dirac impulse \(\delta(t)\). Why completely? Because any input can be written as a continuum of shifted, scaled impulses, and by linearity + time-invariance the output is the same combination of shifted, scaled copies of \(h\). That combination is the convolution integral.

Convolution

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

Convolution is the time-domain operation of an LTI system. The mechanics — flip, shift, multiply, integrate — are worth doing by hand a few times to build intuition, but the real payoff is the next idea: convolution is awkward in time and trivial in frequency. The RC low-pass of Week 6 has \(h(t)=\frac1\tau e^{-t/\tau}u(t)\); convolving an input with this decaying exponential is exactly the “smoothing/averaging” a low-pass does.

Fourier series and the eigenfunction property

A periodic signal of period \(T\) is a sum of harmonics:

\[ x(t)=\sum_{k=-\infty}^{\infty} a_k e^{jk\omega_0 t},\qquad \omega_0=\frac{2\pi}{T},\qquad a_k=\frac1T\int_T x(t)e^{-jk\omega_0 t}\,dt. \]

Feed this into an LTI system: because each \(e^{jk\omega_0 t}\) is an eigenfunction, the output is \(\sum_k a_k H(jk\omega_0)e^{jk\omega_0 t}\) — the system simply scales each harmonic by \(H\) at that harmonic’s frequency. This is exactly why a square wave through a low-pass comes out rounded: the high harmonics are attenuated. You can now predict the output waveform harmonic by harmonic.

The Fourier transform and the convolution theorem

For aperiodic signals, the sum becomes an integral:

\[ X(j\omega)=\int_{-\infty}^{\infty} x(t)e^{-j\omega t}\,dt,\qquad x(t)=\frac1{2\pi}\int_{-\infty}^{\infty}X(j\omega)e^{j\omega t}\,d\omega. \]

The single most important property for us is the convolution theorem:

\[ y=x*h \quad\Longleftrightarrow\quad Y(j\omega)=X(j\omega)\,H(j\omega). \]

Convolution in time = multiplication in frequency. This is why frequency-domain thinking dominates signal processing: filtering is just multiplying spectra. And \(H(j\omega)\) — the Fourier transform of the impulse response — is precisely the frequency response you Bode-plotted in Week 6. Three things you measured separately (transient \(h\), ringing poles, Bode \(H\)) are now one object viewed three ways.

Spectra of common signals and Parseval

Key pairs to know cold: impulse ↔︎ constant, constant ↔︎ impulse, rectangular pulse ↔︎ sinc, sinc ↔︎ rectangle (ideal filter), Gaussian ↔︎ Gaussian, cosine ↔︎ pair of impulses. The rectangle↔︎sinc pair is the seed of windowing effects in Week 10. Parseval’s theorem (\(\int|x(t)|^2dt=\frac1{2\pi}\int|X(j\omega)|^2 d\omega\)) says energy is conserved between domains — the basis of “energy spectral density” and of reading signal power off an FFT.

Bandwidth and the time–frequency tradeoff (revisited)

The scaling property \(x(at)\leftrightarrow \frac1{|a|}X(j\omega/a)\) formalizes Week 6’s tradeoff: compressing in time stretches in frequency. A short pulse has a wide spectrum; a narrowband signal is long in time. This is not an engineering limitation but a mathematical fact about the Fourier transform — and it returns as the uncertainty between time resolution and frequency resolution in the DFT/STFT (Week 10).

Theory Exercises

Derive the convolution integral from linearity + time-invariance, starting from the sifting property \(x(t)=\int x(\tau)\delta(t-\tau)d\tau\).
Compute by hand the convolution of a rectangular pulse with itself (→ triangle) and of a step with the RC impulse response \(h(t)=\frac1\tau e^{-t/\tau}u(t)\) (→ the Week-4 step response). Confirm the latter matches your Week-4 measurement.
Compute the Fourier series of a square wave; show the harmonic amplitudes fall as \(1/k\) and only odd harmonics survive.
Pass that square wave through an RC low-pass with \(\omega_c\) between the 1st and 3rd harmonic; predict the output waveform from the scaled harmonics and compare to a scope capture.
Prove the convolution theorem \(x*h \leftrightarrow X\cdot H\) from the Fourier transform definition.
Derive the rectangle↔︎sinc transform pair and state its consequence for ideal (brick-wall) filters and for windowing.
State Parseval’s theorem and use it to relate the energy of a signal to the area under \(|X(j\omega)|^2\).

Lab / Bench Work

Impulse/step response as \(h\): Re-capture the Week-4 RC step response and numerically differentiate it (in Python) to estimate \(h(t)\); compare to the analytic \(\frac1\tau e^{-t/\tau}u(t)\).

Square wave through a filter — harmonics in action: Drive an RC (or active) low-pass with a square wave and capture the output. Use the scope’s FFT to display the input spectrum (line at the fundamental and odd harmonics) and the output spectrum (high harmonics attenuated). Confirm the time-domain rounding corresponds to the missing harmonics.

Verify the convolution theorem numerically: Capture an input \(x(t)\) and the filter output \(y(t)\) on two channels. In Python, compute \(X(f)\), \(Y(f)\) via FFT and confirm \(Y(f)/X(f)\approx H(f)\) matches your Week-6 measured Bode plot. This is the convolution theorem confirmed with your own bench data.

Qucs-S (ngspice): Use a .tran of a square-wave-driven filter plus FFT, or the ngspice .four (Fourier) analysis, to cross-check the harmonic amplitudes.

Measurement Methodology

Scope FFT setup: choose an appropriate window (Hanning for general spectra, flat-top for amplitude accuracy), set the span/center, and note the frequency resolution = sample rate / record length. Understand that the displayed FFT is a DFT of a finite record — a preview of Week 10’s windowing/leakage issues.
Amplitude calibration: the FFT magnitude depends on the window; use the flat-top window when reading harmonic amplitudes quantitatively, and verify the fundamental’s level against the time-domain peak-to-peak.
Aliasing in the scope itself: the scope samples internally — ensure its sample rate is high enough that displayed FFT bins aren’t aliased (a direct foretaste of Week 9).
Reconcile: measured harmonic amplitudes vs Fourier-series prediction; \(Y/X\) from captured data vs the Week-6 Bode \(H\); differentiated step response vs analytic \(h(t)\).

Expected baselines: Square-wave harmonics at the fundamental and odd multiples with \(1/k\) amplitude falloff; output harmonics scaled by the measured \(|H|\) at each harmonic frequency. \(Y/X\) from captures matching the Week-6 Bode curve within a few percent over the passband.

Connections

This week is the theoretical hub. It explains Week 4 (the impulse/step response is \(h\)), Week 5 (eigenfunctions), and Week 6 (\(H(j\omega)\) is the Fourier transform of \(h\)) as one framework. The convolution theorem is the workhorse of all filtering, analog and digital. The rectangle↔︎sinc pair and the time–frequency tradeoff become windowing and spectral leakage in Week 10. Most importantly, the Fourier transform of a sampled signal — and what sampling does to a spectrum — is the entire content of Week 9. This is also the applied counterpart to Course 5’s signals/transforms phase.