SC · Signal basics

Elementary Signals

Impulse, step, exponential are the alphabet of signals. Instead of memorizing them apart, tie them together with one relation: the step is the running sum of the impulse.

Sweep the upper limit and build the step

The narrow pulse in the middle is an impulse approximated by area 1. Slide the upper limit t to the right: area under the pulse accumulates, and that running value climbs onto the faint target step.

target step uimpulse δ

upper limit t-2.00

Accumulated area = step value

∫ δ = 0.00

With the limit left of the pulse, you have not met the impulse yet. The accumulated area is 0; the step has not started.

Area 0

The impulse is a single point of area 1

The impulse δ(t) is the limit of a pulse whose width shrinks to 0 and height grows without bound while the area is pinned at 1. So it is defined by its area, not its value. Multiplied by another signal and integrated, it sifts out that signal’s value at a single point (the sifting property).

The step is the running integral of the impulse

The unit step u(t) is 0 for t < 0 and 1 for t ≥ 0. This is exactly the impulse integrated from the left up to t. Conversely, differentiating the step gives an impulse that spikes only at the jump. Integration and differentiation bind the two into one pair.

Observeu(t) = ∫ δ(τ) dτ

The step is the impulse accumulated from the left.

Chooseδ(t) = ?

Differentiating the step gives an impulse at the jump.

Fill in∫ x(t) δ(t − t₀) dt = ?

The impulse sifts out a single value.

On your ownd/dt eᵃᵗ = ?

The exponential’s derivative is a times itself.

The exponential rate is itself

The exponential eᵃᵗ has a derivative equal to a times itself. So the sign of a decides its fate: a > 0 blows up, a < 0 decays toward 0, and a = 0 is constant. A system’s natural response always shows up in this shape.

Back to the first screen

The accumulated area the slider swept up was the step itself. While the limit crossed the impulse, the area filled from 0 to 1 and drew the step’s jump; once past, it locked at 1. The impulse and the step are one pair joined by a single integration, and the exponential is the third letter whose rate is itself. These three become the bricks for every signal ahead.

The three elementary signals are bound by calculus. The impulse δ(t) is defined by area 1 and has the sifting property ∫ x(t) δ(t − t₀) dt = x(t₀). The unit step u(t) is the accumulation of the impulse, that is u(t) = ∫ δ(τ) dτ, and conversely δ(t) = du/dt. The exponential eᵃᵗ has a derivative a times itself, so the sign of a splits growth, constant, and decay.

On to the next units

The impulse is the lead of the next chapter. The impulse response is the output when a single impulse enters a system, and from there convolution opens the way to compute the output of any input from that response. The exponential becomes the test signal that cracks systems in Laplace and Fourier. The next unit, operations on a signal, is about shifting and stretching a signal along the time axis.