The impulse response

Feed a system a single spike. Try each System option, amplifier, echo and averager, and watch the short trace that comes back.

One measurement is enough

Tap a bridge once with a hammer and it shudders, then settles back down: one short reply to one short knock. Here’s the surprising part. Record that one reply carefully, and you’ve captured everything the bridge will ever do to any load that shakes it later.

That works because of two plain rules from the last page. Scale or add the inputs you feed a system, and the output scales or adds the same way: that’s what makes it linear. Shift an input later in time, and the output shifts by exactly the same amount, without changing shape: that’s what makes it time-invariant. A system with both properties gets a short name, LTI.

The signal-decomposition page showed that any signal can be written as a sum of scaled, shifted spikes, $x[n]=\sum_k x[k]\,\delta[n-k]$, where $\delta[n-k]$ is a spike worth $1$ at $n=k$ and $0$ everywhere else. Put that together with the two LTI rules above: once you know an LTI system’s reply to a single spike $\delta[n]$, called its impulse response and written $h[n]$, you already know its reply to every shifted, scaled spike. And since any input is nothing but a sum of those, you know its reply to any input at all.

Go back to the three systems at the top of the page. Each of those short traces is, by the last page’s two rules, the entire system: nothing else about its inner rule is needed to predict any other output.

Building the output from copies of h

Play a short three-note tune into an echoing space, and each note starts its own decaying echo the moment it sounds. By the time the third note rings out, all three echoes are overlapping in the air. That overlap is exactly what this instrument builds, one note at a time.

Each input spike, the instant it “arrives,” launches its own copy of $h[n]$, scaled by that spike’s height and shifted to start at that spike’s instant. The whole output is nothing but those launched copies, added together sample by sample.

Drag Place next spike forward one step at a time, and read Spike just placed for the copy it just launched.

Once both spikes are placed, that total matches exactly what feeding the whole input through the system at once would give, built without ever running the system again.

Measuring h in practice

Clap once in a small room and the slap of sound off close walls dies out almost at once. Clap the same way inside a church and the sound keeps rolling back at you for seconds, off walls and pillars far apart. Since $h[n]$ is nothing but a system’s reply to one spike, you can measure it on a real system the same way: feed it something spike-like, a clap, a hammer tap, and record what comes back.

Switch Room size between small and large on the same clap, and watch Rings for about.

Notice Rings for about climbs a lot for the large room: the shape and length of $h[n]$ alone already tells you something physical about the room that produced it.

The step response, and its link to h

Ease off a car’s accelerator and the speed falls away smoothly. Floor it instead and hold it there, a sudden change that stays, and the speed climbs and settles into a new cruising value: that settling trace is a step response.

Formally, a system’s step response $s[n]$ is what it gives back when fed a unit step $u[n]$ instead of a spike: the same “switch” from the signal-decomposition page, written $u[n]$, worth $0$ before $n=0$ and $1$ from $n=0$ on. Since $u[n]$ is the running sum of $\delta[n]$, and the system is LTI, $s[n]$ is exactly the running sum of $h[n]$, the same running-sum relationship from the amplitude-operations page, now applied to a whole system’s response. Running that the other way, taking $s[n]$‘s first difference, hands back $h[n]$ exactly.

Switch View between Impulse response h[n] and Step response s[n].

Notice running-summing the $h[n]$ trace reproduces the $s[n]$ trace exactly, sample for sample, and first-differencing $s[n]$ hands back $h[n]$ again.

Worked example

Take $h[n]=1,\,0.5,\,0.25$ for $n=0,1,2$ (else 0), and an input with $x[0]=2$, $x[1]=1$ (else 0), so $x[n]=2\delta[n]+1\delta[n-1]$.

Launching the copies. The spike at $n=0$ launches $2h[n]=2,1,0.5$ at $n=0,1,2$. The spike at $n=1$ launches $1\cdot h[n-1]=1,0.5,0.25$ at $n=1,2,3$.

Adding them. $y[0]=2$, $y[1]=1+1=2$, $y[2]=0.5+0.5=1$, $y[3]=0.25$, so $y=2,2,1,0.25$, exactly what the instrument above builds.

The step response. Running-summing the same $h$ gives $s[0]=1$, $s[1]=1.5$, $s[2]=1.75$. Its first difference, $s[n]-s[n-1]$, gives back $1,0.5,0.25$, exactly $h[n]$ again.

Where you’ll meet this

Audio engineers measure a concert hall’s impulse response once, with a clap or a starter pistol, then reuse it to make any recording sound as if it were played inside that hall, by the same shift-scale-and-add trick as the instrument above. Structural engineers tap a bridge or a building and read how long its impulse response keeps vibrating afterward. Both reuse one measurement for every input they’ll ever care about.

The next page gives this shifted-and-added sum its own name and its own notation: convolution.

Reference card