signals for conveying information

Titne Domain Concepts

Viewed as a function of time, an electromagnetic signal can be either analog or
digital. An analog signal is one in which the signal intensity varies in a smooth fashion
over time. In other words, there are no breaks or discontinuities in the signal.
A digital signal is one in which the signal intensity maintains a constant level for
some period of time and then changes to another constant level.



Examples of both kinds of signals. The analog signal might represent speech, and the
digital signal might represent binary Is and Os.
The simplest sort of signal is a periodic signal, in which the same signal pattern
repeats over time. Figure 2.2 shows an example of a periodic analog signal (sine
wave) and a periodic digital signal (square wave). Mathematically, a signal set) is
defined to be periodic if and only if
set + T) = set) -00 < t < +00
where the constant T is the period of the signal (T is the smallest value that satisfies
the equation). Otherwise, a signal is aperiodic.
The sine wave is the fundamental analog signal. A general sine wave can be
represented by three parameters: peak amplitude (A), frequency (j), and phase (¢).
The peak amplitude is the maximum value or strength of the signal over time;
typically, this value is measured in volts. The frequency is the rate [in cycles per
second, or Hertz (Hz)] at which the signal repeats. An equivalent parameter is the

period (1) of a signal, which is the amount of time it takes for one repetition; therefore,
T = 1/f. Phase is a measure of the relative position in time within a single
period of a signal, as illustrated later.
The general sine wave can be written
s( t) = A sin(21Tft + cP) (2.1)
A function with the form of Equation (2.1) is known as a sinusoid. Figure 2.3 shows
the effect of varying each of the three parameters. In part (a) of the figure, the frequency
is 1 Hz; thus the period is T = 1 second. Part (b) has the same frequency and
phase but a peak amplitude of 0.5. In part (c) we have f = 2, which is equivalent to
T = 1/2. Finally, part (d) shows the effect of a phase shift of 1T14 radians, which is 45
degrees (21T radians = 3600 = 1 period).
In Figure 2.3 the horizontal axis is time; the graphs display the value of a signal
at a given point in space as a function of time. These same graphs, with a change of
scale, can apply with horizontal axes in space. In that case, the graphs display the
value of a signal at a given point in time as a function of distance. For example, for a
sinusoidal transmission (say, an electromagnetic radio wave some distance from a
radio antenna or sound some distance from loudspeaker) at a particular instant of
time, the intensity of the signal varies in a sinusoidal way as a function of distance
from the source.
There is a simple relationship between the two sine waves, one in time and one
in space. The wavelength (A) of a signal is the distance occupied by a single cycle, or,
put another way, the distance between two points of corresponding phase of two
consecutive cycles. Assume that the signal is traveling with a velocity v. Then the
wavelength is related to the period as follows: A = vT. Equivalently, Af = v.Ofparticular
relevance to this discussion is the case where v = c, the speed of light in free
space, which is approximately 3 X 108 m/s.

Frequency DOluain Concepts

In practice, an electromagnetic signal will be made up of many frequencies. For
example, the signal
s(t) = (4/1T) x (sin(21Tft) + (1/3)sin(21T(3f)t))
is shown in Figure 2Ac. The components of this signal are just sine waves of frequencies
f and 3f; parts (a) and (b) of the figure show these individual components.
There are two interesting points that can be made about this figure:
'" The second frequency is an integer multiple of the first frequency. When all of
the frequency components of a signal are integer multiples of one frequency,
the latter frequency is referred to as the fundamental frequency.
• The period of the total signal is equal to the period of the fundamental frequency.
The period of the component sin(21Tft) is T = 1/f, and the period of
s(t) is also T, as can be seen from Figure 2Ac.
It can be shown, using a discipline known as Fourier analysis, that any signal
is made up of components at various frequencies, in which each component is a
sinusoid. By adding together enough sinusoidal signals, each with the appropriate

amplitude, frequency, and phase, any electromagnetic signal can be constructed. Put
another way, any electromagnetic signal can be shown to consist of a collection of
periodic analog signals (sine waves) at different amplitudes, frequencies, and phases.
The importance of being able to look at a signal from the frequency perspective
(frequency domain) rather than a time perspective (time domain) should become
clear as the discussion proceeds.