1.1 Harmonic Signals and Exponentials

Before we begin to talk about wireless, we briefly remind the reader of a previous acquaintance with three concepts that are ubiquitous in radio engineering: sinusoidal signals, complex numbers, and imaginary exponentials. The reader who is familiar with such matters can skip this section without harm.

Almost everything in radio is done by making tiny changes modulations of a signal that is periodic in time. The archetype of a smooth periodic signal is the sinusoid (Figure 1.1), typically written as the product of the angular frequency ? and time t.

Figure 1.1: Cosine and Sine Functions

Both of these functions alternate between a maximum value of 1 and minimum value of ?1; cosine starts at +1, and sine starts at 0, when the argument is zero. We can see that cosines and sines are identical except for an offset in the argument (the phase):

We say that the sine lags the cosine by 90 degrees. (Note that here, following common practice, we write angles in radians but often speak of them in degrees.) The cosine and sine are periodic with a period = (1/ f ), where f = ?/2 ? is the frequency in cycles per second or hertz.

Let us now digress briefly to discuss complex numbers, for reasons that will become clear in a page or two. Imaginary numbers, the reader will recall, are introduced to provide square roots of negative reals; the...