2.1 Elemental signals

Elemental signals are the building blocks with which we build complicated signals. By definition, elemental signals have a simple structure. Exactly what wemean by the "structure of a signal" will unfold in this section of the course. Signals are nothing more thanfunctions defined with respect to some independent variable, which we take to be time for the most part. Very interestingsignals are not functions solely of time; one great example of which is an image. For it, the independent variables are $x$ and $y$ (two-dimensional space). Video signals are functions of three variables: two spatialdimensions and time. Fortunately, most of the ideas underlying modern signal theory can be exemplified with one-dimensional signals.

Sinusoids

Perhaps the most common real-valued signal is the sinusoid.

Complex exponentials

The most important signal is complex-valued, the complex exponential.

The complex exponential defines the notion of frequency: it is the only signal that contains only one frequency component. The sinusoid consists of two frequencycomponents: one at the frequency $f_0$ and the other at $-f_0$ .

Using the complex plane, we can envision the complex exponential's temporal variations as seen in the above figure( [link] ). The magnitude of the complex exponential is $A$ , and the initial value of the complex exponential at $t=0$ has an angle of $\phi$ . As time increases, the locus of points traced by the complexexponential is a circle (it has constant magnitude of $A$ ). The number of times per second we go around the circle equals the frequency $f$ . The time taken for the complex exponential to go around the circle once is known asits period $T$ , and equals $\frac{1}{f}$ . The projections onto the real and imaginary axes of the rotating vector representing the complex exponentialsignal are the cosine and sine signal of Euler's relation ( [link] ).

Real exponentials

As opposed to complex exponentials which oscillate, real exponentials decay.

The quantity $\tau$ is known as the exponential's time constant , and corresponds to the time required for the exponential to decrease by afactor of $\frac{1}{e}$ , which approximately equals $0.368$ . A decaying complex exponential is the product of a real and a complex exponential.

Unit step

The unit step function is denoted by $u(t)$ , and is defined to be

Pulse

The unit pulse describes turning a unit-amplitude signal on for a duration of $\Delta$ seconds, then turning it off.

Square wave

The square wave $\mathrm{sq}(t)$ is a periodic signal like the sinusoid. It too has an amplitude and a period, which must be specified tocharacterize the signal. We find subsequently that the sine wave is a simpler signal than the square wave.