The communication systems described in this book differ in many ways, but they all have two things in common. In every case we have a signal, which is used to carry useful information; and in every case there is noise, which enters the system from a variety of sources and degrades the signal, reducing the quality of the communication.

Keeping the ratio between signal and noise sufficiently high is the basis for a great deal of the work that goes into the design of a communication system. This signal-to-noise ratio, abbreviated S/N and almost always expressed in decibels, is an important specification of virtually all communication systems. Let us first consider signal and noise separately, and then take a preliminary look at S/N.

**Modulated Signals**

Given the necessity for modulating a higher-frequency signal with a lower-frequency baseband signal, it is useful to look at the equation for a sine-wave carrier and consider what aspects of the signal can be varied. A general equation for a sine wave is:

e(t) = Ec sin(ωct + θ); where

e(t) = instantaneous voltage as a function of time

Ec = peak voltage of the carrier wave

ωc = carrier frequency in radians per second

t = time in seconds

θ = phase angle in radians

It is common to use radians and radians per second, rather than degrees and hertz, in the equations dealing with modulation, because it makes the mathematics simpler. Of course, practical equipment uses hertz for frequency indications. The conversion is easy. Just remember from basic ac theory

that

ω = 2πƒ (1.2); where

ω = frequency in radians per second

ƒ = frequency in hertz

A look at Equation shows us that there are only three parameters of a sine wave that can be varied: the amplitude Ec, the frequency ω, and the phase angle θ. It is also possible to change more than one of these parameters simultaneously; for example, in digital communication it is common to vary both the amplitude and the phase of the signal.

Once we decide to vary, or modulate, a sine wave, it becomes a complex waveform. This means that the signal will exist at more than one frequency; that is, it will occupy bandwidth. It is not sufficient to transmit a signal from transmitter to receiver if the noise that accompanies it is strong enough to prevent it from being understood.

All electronic systems are affected by noise, which has many sources, the most important noise component is thermal noise, which is created by the random motion of molecules that occurs in all materials at any temperature above absolute zero (0 K or −273° C). We shall have a good deal to say about noise and the ratio between signal and noise power (S/N) in later chapters.

For now let us just note that thermal noise power is proportional to the bandwidth over which a system operates. The equation is very simple:

PN = kTB; where

PN = noise power in watts

k = Boltzmann’s constant, 1.38 × 10−23 joules/kelvin (J/K)

T = temperature in kelvins

B = noise power bandwidth in hertz

Note the recurrence of the term bandwidth. Here it refers to the range of frequencies over which the noise is observed. If we had a system with infinite bandwidth, theoretically the noise power would be infinite. Of course, real systems never have infinite bandwidth.

A couple of other notes are in order. First, kelvins are equal to degrees Celsius in size; only the zero point on the scale is different. Therefore, converting between degrees Celsius and kelvins is easy:

T(K) = T(°C) + 273 (1.4); where

T(K) = absolute temperature in kelvins

T(°C) = temperature in degrees Celsius

Also, the official terminology is “degrees Celsius” or °C but just “kelvins” or K.

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