Time-Domain and Frequency-Domain Analysis

Why two domains?

Every signal you will ever see on a scope is a waveform in time — voltage changing moment by moment. That is the time domain, and it is the natural language of oscilloscopes, TDR, and eye diagrams. But the same signal can also be described as a recipe of sine waves at different frequencies, each with its own amplitude and timing. That is the frequency domain, and it is the natural language of S-parameters, impedance plots, and channel budgets.

Think of a chord struck on a guitar. What you hear is a single sound — that is the time-domain waveform your ear receives. But a spectrum analyser would show you the individual notes and their harmonics — the frequency-domain decomposition of the same sound. Neither description is more "real" than the other; they are two views of one physical event.

The reason signal integrity uses both is that different problems are obvious in different domains. Loss rises smoothly with frequency and is easiest to see in S₂₁. A via stub resonance shows up as a sharp null in frequency but is nearly invisible on a TDR. A ground-bounce spike is obvious on a scope and hides inside a broadband impedance curve. Fluency in both domains — and knowing which bridges connect them — is what lets you pick the right instrument and the right simulation for each question.

One object, three descriptions

A channel is a single physical thing. S-parameters, an impulse response and a TDR trace are not three models of it — they are three questions put to the same object. Fix the network, the source, the receiver, the excitation, the reference planes and the bandwidth you compute over, and every one of those views has to agree, because they are all consequences of the same network.

What does not follow is that each view can be recovered from any other. Some can; several cannot, and the difference is practical rather than pedantic — it decides which measurement you have to go and take.

• Complex S₂₁ and the impulse response are the same information. One is the Fourier transform of the other. Nothing is lost either way, and that pair is the reason the time and frequency panels on this site can never disagree.
• A magnitude alone is not. |S₂₁| without its phase does not determine an impulse response. Two channels with identical loss curves and different phase behave differently, which is why this site's line model derives loss and phase from one material rather than choosing them separately.
• S₁₁ does not determine S₂₁. A matched through-line and a matched attenuator both have zero reflection; they transmit completely differently. Put a 6 dB matched attenuator in a 50 Ω line and a TDR reads a flat 50 Ω from end to end while exactly 50.0% of the launched voltage never arrives. A one-port TDR sees only what comes back, so it can find a discontinuity and still tell you nothing about how much signal gets there.
• An eye diagram is not invertible at all. It is thousands of symbol histories drawn on top of each other. You cannot recover the channel from it, or even the pattern that made it.

So the useful version of "it settles arguments" is narrower. A channel cannot have good complex S-parameters and a bad pulse response over the same band — those two really are one fact in two notations. But good insertion loss with a closed eye is entirely possible: a notch, a reflection, or a phase distortion at the wrong frequency does not have to move the loss number that a budget looks at. That is exactly the case a scalar loss budget cannot catch, and it is why this site keeps six views of one network rather than one.

Fourier: why frequencies exist at all

Back to the guitar chord: you hear one sound, but it is made of individual notes at specific frequencies. Fourier's insight is that any waveform — not just a musical chord — can be broken down this way. Any repeating signal can be built by adding sine waves at multiples of its repetition rate. And any non-repeating signal (like a single digital edge) can be represented as a continuous spread of sine waves across all frequencies.

The useful consequence is the reverse reading: a system's response to sine waves tells you its response to everything. If you know how much a channel attenuates and delays each frequency — which is exactly what S₂₁(f) tells you — you can predict what comes out for any input, because any input is just a sum of those same sine waves.

This is only true because the systems here are linear — double the input and the output doubles; add two inputs and the outputs add. Copper is linear. A driver at its limits is not, which is exactly why a statistical channel simulation stops being exact there.

Convolution: what a channel does to a signal

Imagine dropping a pebble into a still pond and watching the ripple spread and fade. That ripple pattern is the pond's "impulse response" — what it does to a single disturbance. Now imagine dropping pebbles one after another, each a bit different. The resulting pattern on the water is the sum of all those overlapping ripples, each shifted in time. That superposition is convolution.

If you know how a channel responds to a single impulse, you know how it responds to anything — because any signal is a sum of scaled, delayed impulses, and a linear system's response to a sum is the sum of the responses.

That operation is convolution, and it is the whole of what a channel does:

That last note is worth pausing on. In the frequency domain, passing a signal through a channel is just multiplication: Y(f) = X(f) · H(f). Cascading two channels is multiplying their transfer functions. In the time domain, the same operation requires the full convolution integral. This asymmetry is why channel budgets are written in dB (where multiplication becomes addition) and why frequency-domain thinking dominates link design.

Sampled once per bit, that integral becomes a sum — and that sum is ISI. There is no separate mechanism: ISI is convolution, read at the decision instants.

Impulse, step and pulse: three responses, one system

Three different inputs, and knowing which one you are looking at matters because they have different shapes and different uses.

The step response is the integral of the impulse response, and the pulse response is the difference of two steps one UI apart. So all three carry the same information — but the cursor and the ISI taps must be read off the pulse response, not the impulse, because a real transmitter sends a pulse with a finite edge and taps taken from an idealised impulse do not cancel the waveform a receiver actually sees.

Complex impedance: phase is not decoration

Impedance has a magnitude and a phase, and the phase says what kind of thing you are looking at. +90° is an inductor, −90° a capacitor, 0° a resistor.

On a PDN that is the whole diagnosis. An impedance peak happens where one branch has become inductive while its neighbour is still capacitive — and the phase plot is where you see that handover happening. A magnitude plot alone tells you there is a peak; the phase tells you what made it.

Propagation delay and group delay

Two different delays, and the difference is the whole of dispersion.

• Propagation delay is how long the signal takes to get there: length divided by velocity. One number.
• Group delay is the delay of the envelope at each frequency, τ_g = −dφ/dω. It is a curve.

For a uniform lossless line they are equal and flat — every frequency arrives together. Structure in the group delay means different frequencies arrive at different times, which smears an edge. That is dispersion, and dispersion is ISI by another name.

Sampling and aliasing

Everything discussed so far assumes continuous signals — smooth curves defined at every instant. In practice, both simulations and instruments work with samples: discrete snapshots taken at regular intervals. Sampling introduces a fundamental limit.

Sampling a continuous signal at rate f_s can only represent frequencies below f_s/2 (the Nyquist frequency). Anything above that limit folds back and appears as a lower frequency that was never there — an alias. You cannot tell the alias from the real signal after the fact, which is why the damage is permanent.

This matters in four places on this site, and they are easy to miss:

• A simulation's grid. Too few samples per UI and the model aliases its own channel.
• An S-parameter file's frequency step. The step sets the time window before the response wraps around onto itself. A 10 MHz step gives 100 ns — fine for a board, marginal for a long cable.
• A scope. An equivalent-time sampling scope reconstructs a repetitive waveform from many triggers, which is exact for repetitive signals and silently wrong for anything that is not.
• A converter in a working system. An analog-to-digital converter samples whatever reaches its input or its reference at the instants its clock defines, so a clock switching at 4.1 MHz beside a 1 MS/s converter arrives as something near 127 kHz — inside the band, and indistinguishable from signal. Lab D lets you move the aggressor and watch where it lands.

Statistical confidence

The last bridge is between what you observed and what you can claim. If you see no errors in N symbols, the honest statement is a bound, not a value:

Which is why reaching 10⁻¹² directly needs about 3×10¹² symbols, and why every published bathtub curve is part measurement and part extrapolation. The arithmetic is easy; the discipline is saying which part is which.

Go deeper — what the transforms assume, and where each one breaks

Every bridge on this page rests on assumptions, and knowing which one fails tells you which result to distrust.

Linearity and time-invariance. Fourier and convolution both need them. A DFE is non-linear by construction — it makes a decision — and an adapting receiver is time-varying. So a statistical simulation built on superposition is an approximation whose error is in the tail, which is an uncomfortable place for it to be. See statistical versus time-domain.

Causality. A magnitude response alone does not define a physical channel. Attenuation and phase belong to each other, and a filter given magnitude without the corresponding phase produces output before its input. This is not a technicality: a magnitude-only channel model makes equalisation look better than it is, which is precisely the direction that hurts. This site's channel model builds a minimum-phase response for that reason.

Finite records. A discrete Fourier transform treats its record as periodic. A signal that ends somewhere other than where it began has a discontinuity at the wrap point, and the energy that injects lands everywhere — including before the event, where it reads as an acausal response that is really circular convolution. The PDN lab uses a current pulse rather than a step for exactly this reason.

What a sampled grid cannot hold. A PDN's impedance rises like jωL at high frequency, so Z(f) on its own transforms to L·δ′(t) — a doublet. The mathematics is fine: the derivative of a delta is a perfectly good distribution, and a transient solver handles the L·di/dt term directly without ever forming it. What fails is the numerical step of putting it on a sampled grid, where a doublet has no finite representation. The fix is to transform the product Z(f)·I(f), because a real current has a finite rise time and supplies the roll-off. The lesson generalises, and it is about the grid rather than about the transform: if a transform is misbehaving, ask whether the thing you are sampling is bounded.

And one that is purely bookkeeping. Cascading networks by adding decibels assumes each stage is matched to the next. Real stages are not, energy bounces between their mismatches, and the error appears as ripple rather than as an offset. Cascade the full complex networks and the ripple is handled correctly.

In the real world

The practical value of knowing these bridges is that it tells you which instrument to reach for. A smooth loss slope is obvious in S₂₁ and invisible on a TDR. A 30 Ω via field is obvious on a TDR and shows up in S₂₁ only as gentle ripple. A stub resonance is a spike in group delay, a null in S₂₁, and almost nothing on a TDR at all.

So the habit is to look in more than one domain before concluding — and, when a simulation and a measurement disagree, to find the domain where the disagreement is largest, because that is where the wrong assumption lives.

Related

• SI/PI Units, Conventions, and Reference Planes — the vocabulary these transforms use
• Intersymbol Interference (ISI) and Channel Memory — convolution read at the sampling instants
• S-Parameters: Insertion Loss, Return Loss, and Mixed Mode — the frequency-domain description
• Bathtub curves and BER extrapolation — where the confidence arithmetic lands
• Lab B: Channel Response, ISI, and Eye Diagrams — all of these bridges on one network, moving together