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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