Elementary Signals and Singularity Functions

Chapter 1 set out a three-step strategy — decompose, respond, superpose — and observed that the whole art lies in choosing the building blocks. Chapter 2 sorted signals into classes. This chapter finally introduces the blocks themselves, and there are remarkably few of them.

The reason so few suffice is that engineering waveforms are overwhelmingly built from switching, decaying and oscillating. A voltage is applied at some instant: that is a step. A capacitor discharges: that is an exponential. An oscillator runs: that is a sinusoid. A signal is confined to a finite window: that is a rectangular pulse, which is just two steps. Combine, shift and scale these and you can write down almost any waveform an engineer encounters, in closed form, ready to be pushed through an integral.

One of the blocks is different in kind from the others. The unit impulse is not a signal you could ever generate or measure; it is an idealisation with infinite height, zero width and unit area. It is also the single most useful object in the subject, because a system's response to it — the impulse response — turns out to determine the system's response to everything else. Chapter 7 will prove that. This chapter builds the tool.

The unit step is the mathematical description of a switch closing at \(t = 0\). It is zero before the origin and one after it, and it is the reason the phrase "and nothing happened before we started watching" can be written as a multiplication.

Notice the small asymmetry between the two definitions. In discrete time \(u[0]\) is simply defined to be 1 and there is nothing more to say. In continuous time the value at \(t=0\) is genuinely ambiguous — the function jumps there — and different books adopt \(u(0)=1\), \(u(0)=0\) or \(u(0)=\tfrac12\). Fortunately it almost never matters: changing a signal at a single isolated instant changes no integral, and integrals are what we compute. The symmetric choice \(u(0)=\tfrac12\) is the tidiest, because it makes \(u(t) + u(-t) = 1\) hold everywhere.

The step earns its keep as a switch. Multiplying any signal by \(u(t)\) annihilates everything before the origin and leaves the rest untouched, and shifting the step moves the switching instant: \(u(t - t_0)\) turns on at \(t = t_0\). Subtracting two shifted steps isolates a window, which is how every finite-duration pulse in this book is written.

Now the strange one. The unit impulse \(\delta(t)\) — also called the Dirac delta — is meant to represent an event of vanishing duration but finite effect: a hammer blow, a lightning strike, a switching spike. We want it to be zero everywhere except at the origin, and yet to have unit area. No ordinary function can do both, and \(\delta(t)\) is therefore not a function at all. It is a generalised function, or distribution, defined entirely by what it does inside an integral.

The honest way to introduce it is as a limit. Take a rectangular pulse of width \(\Delta\) and height \(1/\Delta\), so that its area is exactly 1 no matter how small \(\Delta\) is. Now shrink \(\Delta\). The pulse becomes taller and narrower while its area stays pinned at unity, and the impulse is what that sequence tends to.

This is why the arrow notation matters. An impulse is drawn as a vertical arrow, and the number written beside it is its weight — the area — not the height of the arrow on the page. An arrow labelled \((2)\) represents \(2\delta(t)\), an impulse of twice the area, and its being drawn taller is purely a drawing convention.

The discrete-time impulse is, by contrast, completely tame. There is no limiting process and no distribution theory: \(\delta[n]\) is an ordinary sequence which is 1 at \(n=0\) and 0 everywhere else. Every property we will labour over in continuous time is either obvious or trivial for \(\delta[n]\), which is one reason discrete-time analysis is often the easier place to build intuition.

Those last two relations are worth staring at, because they are the discrete counterparts of a calculus statement. The impulse is the first difference of the step, and the step is the running sum of the impulse. In continuous time, difference becomes derivative and sum becomes integral, giving the pair that ties the two most important elementary signals together.

Four properties do essentially all the work. The first is the one you will use more than any other identity in the entire course.

The reason is easy to see from the limiting picture. The impulse is zero everywhere except in a vanishing neighbourhood of \(t_0\); over that tiny interval \(x(t)\) is essentially constant at \(x(t_0)\), so it comes out of the integral, and what remains is the impulse's area, which is 1.

The second property is the sampling or product property, and it is the pointwise version of the same idea. Multiplying a signal by a shifted impulse gives an impulse whose weight is the signal's value at that point.

The third is time scaling, and it is the one most often got wrong. Compressing the time axis by a factor \(a\) makes the impulse narrower without making it taller, so its area shrinks by \(|a|\) — and to restore the definition we must divide by \(|a|\). The modulus matters: reversing time does not change the area.

The fourth property is really a warning: whenever you meet \(\delta(at - b)\), factor before doing anything else. Write \(\delta(at-b) = \delta\!\big(a(t - b/a)\big) = \tfrac{1}{|a|}\delta(t - b/a)\), so the impulse sits at \(t = b/a\) and carries weight \(1/|a|\). Skipping this step is the single most common error in impulse problems, and Worked Example 1 shows it in action.

The step and the impulse are two members of a chain. Integrate the impulse and you get the step; integrate the step and you get the unit ramp, a signal that is zero until the origin and then rises with unit slope; integrate again and you get the parabolic signal. Going the other way, differentiating the impulse gives the doublet \(\delta'(t)\), a pair of opposite impulses infinitesimally close together.

The practical value of this chain is that piecewise-linear signals become sums of shifted ramps. Any waveform made of straight segments can be written as a combination of \(r(t - t_i)\) terms, one for each point where the slope changes, with the coefficient equal to the change in slope there. Differentiating that expression once gives steps, and once more gives impulses — which is exactly the sequence you need when computing the response of a circuit to a trapezoidal drive. Worked Example 3 does this end to end.

The doublet is rarer but appears whenever an impulse is differentiated, and it has its own sifting rule with a sign flip: \(\int x(t)\,\delta'(t-t_0)\,dt = -x'(t_0)\). Its discrete counterpart, \(\delta[n] - \delta[n-1]\), is entirely ordinary.

Three more named signals appear often enough to be worth naming, and all three are built from steps. The rectangular pulse or gate function confines a signal to a finite window; the triangular pulse tapers instead of cutting sharply; and the signum function extracts the sign of a signal and nothing else.

If the step and the impulse describe switching, the exponential describes everything that happens afterwards. A capacitor discharging, a current decaying in an inductor, an unstable system running away, an oscillator ringing down — all of them are exponentials, and Chapter 1 already gave the deep reason they matter: complex exponentials pass through LTI systems unchanged in shape.

The real exponential \(x(t) = Ce^{at}\) has three behaviours according to the sign of \(a\). For \(a \lt 0\) it decays; for \(a \gt 0\) it grows without bound; for \(a = 0\) it is a constant. When it decays, the quantity \(\tau = 1/|a|\) is the time constant: the signal falls to about 37% of its initial value in one time constant, and to under 2% in four, which is the usual engineering definition of "settled".

That single expression contains the whole family. The real part \(\sigma\) sets the envelope — decaying, constant or growing — and the imaginary part \(\omega_0\) sets the oscillation. With \(\sigma = 0\) you get a pure sinusoid; with \(\omega_0 = 0\) a pure exponential; with both non-zero, a damped sinusoid \(Ce^{\sigma t}\cos(\omega_0 t + \phi)\), which is the natural response of essentially every second-order circuit you will ever analyse.

The discrete-time story runs in parallel with one genuine surprise. The natural form is \(x[n] = C\alpha^n\), which decays when \(|\alpha| \lt 1\), grows when \(|\alpha| \gt 1\), and — a behaviour with no continuous-time analogue — alternates in sign when \(\alpha\) is negative. Writing \(\alpha = e^{j\Omega_0}\) gives the discrete-time complex exponential \(e^{j\Omega_0 n}\), and here the surprise appears.

We now have the payoff, and it is the reason this chapter sits where it does. Consider an arbitrary discrete-time signal \(x[n]\). By the sampling property, \(x[k]\delta[n-k]\) is a sequence that equals \(x[k]\) at \(n=k\) and is zero everywhere else — a single sample, isolated. Add up one such term for every \(k\) and you have reconstructed the entire signal, exactly.

Read that alongside Chapter 1's strategy and the plan for the next few chapters writes itself. The decomposition step is done: the building blocks are shifted impulses, and the coefficients are the signal's own samples. The respond step asks what a system does to a single impulse — its impulse response \(h[n]\). The superpose step then adds those responses together, and because the system is linear and time-invariant, the sum takes one specific form. That form is convolution, and it is the whole of Chapter 7.

Problems 1 to 4 drill the definitions; 5 to 8 build and manipulate signals; 9 and 10 ask you to prove the properties you have been using. Sketch before you compute wherever a waveform is involved.