The Standard Signals
Impulse, step and ramp are one family related by calculus — and the exponential earns its place for a reason nothing else can claim.
Skip to the animationThe impulse, step and ramp are one family related by integration, and the exponential joins them for a different reason entirely — it is the only input shape an LTI system cannot reshape, which is what makes every transform in the subject possible.
The calculus family
- 1Unit impulse δ(t) — zero everywhere but the origin, unbounded there, with area exactly 1. Not a function but a *distribution*: the limit of a narrowing rectangle whose area is held at one.
- 2Unit step u(t) — the integral of the impulse. Zero before the origin, one after: a switch closing.
- 3Unit ramp r(t) — the integral of the step. Zero before, then rising with slope 1.
When an impulse is drawn, the arrow's height represents its area, not its value — which is why it is labelled 1 rather than infinity. And since the responses inherit the relationship of the inputs, a system's step response is the integral of its impulse response.
The sifting property
The impulse's defining behaviour is what it does inside an integral: ∫ x(t)·δ(t − t₀) dt = x(t₀). It sifts out one value.
Read the other way round, x(t) = ∫ x(τ)·δ(t − τ) dτ says any signal is a continuum of scaled, shifted impulses — which is the decomposition the convolution integral is built on.
The exponential, and why it is different
e^(−at) falls by the same fraction in every equal interval. After one time constant τ = 1/a it is at 37%, and within 2% of its final value after about 4τ. The sign of a decides decay or growth, and that sign becomes the stability criterion.
Euler's identity e^(jωt) = cos ωt + j sin ωt makes the sinusoid a complex exponential too. Writing e^(st) with s = σ + jω covers decay, growth and oscillation in one symbol — which is why the Laplace transform is written in s.
The real reason it is special: a complex exponential into an LTI system comes out as the same exponential multiplied by a number, H(s). It is an eigenfunction, and nothing else is. Decomposing a signal into exponentials therefore turns convolution into multiplication, which is the whole of Fourier and Laplace.
Discrete versions
δ[n] is simply 1 at n = 0 and 0 elsewhere — an ordinary finite sequence needing no distribution theory, so the discrete case is cleaner than the continuous one for once. Step, ramp and exponential all carry over, with aⁿ as the discrete exponential.
The numbers you will be asked for
- Sifting property
∫ x(t)·δ(t − t₀) dt = x(t₀)
The impulse's actual definition.
- Family relations
δ(t) = du/dt · u(t) = dr/dt
One chain under differentiation and integration.
- Euler
e^(jωt) = cos ωt + j·sin ωt
Why one function covers oscillation and decay.
- Time constant
τ = 1/a · 37% after τ, 2% after 4τ
The practical settling rule.
- Eigenfunction property
e^(st) → H(s)·e^(st)
True for LTI systems and for nothing else.
Advantages and disadvantages
Advantages
- Three of the four are one family, so there is less to remember than it appears.
- The impulse decomposes any signal, which gives convolution.
- The complex exponential unifies growth, decay and oscillation in one symbol.
- Being an eigenfunction, it makes every transform method possible.
Disadvantages
- The continuous impulse is not a function and needs distribution theory to be rigorous.
- An ideal impulse cannot be generated, so h(t) is measured with an approximation to one.
- The eigenfunction property fails the moment a system is non-linear or time-varying.
- Time constants describe first-order behaviour only.
Watch it work
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