2.2.5 General Expression For A Sinusoidal Waveform

We will make use of this general waveform of Equation (2.16) a number of times throughout this text. Although the equation appears simple, it provides a mathematical description of almost every sinusoidal signal encountered in communications engineering. Whether the signal represents a voice-frequency tone, an RF carrier, or a clock signal within a digital system, its instantaneous value can be described using the same general expression by selecting appropriate values for the amplitude, frequency, and phase.

For example, when we are considering modulation in Chapter 6, the general expression for a sinusoidal carrier wave is therefore:

v_{c} (t) = V_{c} \sin (ω_{c} t - ϕ_{c})

(2.17)

where v_c(t) is the instantaneous carrier voltage, V_c is the peak value of the carrier voltage, ω_cis the angular frequency of the carrier (2πf_c), and ϕ_c is the carrier phase relative to a reference frequency.

This equation shows that every sinusoidal waveform is completely described by just three independent parameters:

Amplitude, which determines the strength or magnitude of the signal.
Frequency, which determines how rapidly the waveform repeats.
Phase, which specifies the position of the waveform in time relative to a chosen reference.

Changing any one of these parameters changes the waveform while leaving the other two unchanged. This observation is particularly important because many communications techniques deliberately encode information by varying one of these three parameters. For example, amplitude modulation (AM) varies the signal amplitude, frequency modulation (FM) varies the carrier frequency, and phase modulation (PM) varies the carrier phase—see Chapter 6. Modern digital modulation schemes similarly convey information by making carefully controlled changes to one or more of these parameters.

As we progress through the book, we shall repeatedly return to this simple expression. Although practical communications signals often appear considerably more complicated than a single sine wave, many can be understood as combinations of sinusoids, each described by an equation of this form. Consequently, a thorough understanding of amplitude, frequency, and phase provides the foundation for much of the signal analysis presented in later chapters.