SC · Signal basics

Elementary Signals

The impulse, the step and the exponential are the alphabet of signals. Rather than memorizing the three separately, tie them together with one relationship, that a step is impulses piled up, and they stay with you far more firmly.

Sweep the upper limit and build the step

The narrow rectangular pulse in the middle, 0.5 wide and 2 tall, is an approximation of an impulse with area 1. Move the integration limit t to the right with the slider, and the area under the pulse fills in gold while its running total climbs along the faint dashed target step.

target step uimpulse δ

upper limit t-2.00

Accumulated area = step value

∫ δ = 0.00

The limit is still to the left of the pulse and has not met the impulse yet. The accumulated area is 0, and the step has not even started.

Area 0

The impulse is a single point of area 1

The impulse δ(t) is the limit in which the width is squeezed to 0 and the height grows without bound while the area is held at 1. That is why it is defined by its area rather than its values. Multiplied by another signal and integrated, it picks out just one point value of that signal, which is called 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 integral of the impulse from the far left up to t. Conversely, differentiating the step gives an impulse that rises only at the instant of the jump, so integration and differentiation tie the two into a 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 derivative of the exponential is a times itself.

The exponential rate is itself

The derivative of an exponential signal eᵃᵗ is a times the signal itself, so the sign of a decides its fate. If a > 0 it grows without limit, if a < 0 it dies away toward 0, and if a = 0 it stays constant. In the small figure alongside, red is a = 0.95 and blue is a = −0.95, and the natural response of a linear time-invariant system always appears in this exponential form.

Back to the first screen

The accumulated area that the slider swept up was the step itself. While the limit crossed the impulse, the area filled from 0 to 1 and drew the jump of the step, and once past it stayed at 1. The impulse and the step are a pair linked by a single integration, and the exponential is the third letter, the one whose rate of change is proportional to itself. These three become the bricks from which every signal ahead is built.

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

On to the next units

The impulse is the lead character of the next chapter. The impulse response is a system's output when a single impulse goes in, and from there convolution opens the way to computing the output for any input from the impulse response. The exponential becomes the test signal for solving systems in Laplace and Fourier analysis. In the next unit, operations on a signal, we learn to shift and stretch signals along the time axis.