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Representation of Signals PPT
#1

Representation of Signals



INTRODUCTION

Signals may be classified as predictable or as unpredictable, as analog or discrete and
of finite or infinite duration. Deterministic signals are defined exactly as a function of time.
They can be periodic (such as a sinewave or squarewave) or they can be an aperiodic “one
shot” signal. Deterministic signals contain no information because their future is completely
predictable by the receiver. They are easy to model and are useful since they can provide a
reasonably accurate evaluation of communication system performance. Stochastic signals, on
the other hand, are unpredictable and thus can communicate information. Although the time
waveform of a stochastic signal is random, the signal power may be predictable. Examples of
stochastic signals are thermal noise in electronic circuits, (i.e. “background hiss”) and
information signals such as voice or music.
An analog signal, denoted x(t), is a continuous function of time and is uniquely
determined for all t. When a physical signal such as speech is converted to an electrical signal
by a microphone, we have an electrical analog of the physical waveform. An equivalent
discrete-time signal, denoted as x(kT), exists only at discrete instants. It is characterized by a
sequence of values that exist at specific times, kT, where k is an integer and T is normally a
fixed time interval. On the other hand, a continuous time signal may be restricted to a set of
discrete amplitudes. A signal that is discrete in both time and amplitude is referred to as a
digital signal. Furthermore, these discrete digital signal amplitudes can be represented by a
set of numbers (codes) and, as such, can be stored in a computer memory. Pulse code
modulation (PCM) is an example of a digital signal. These categorizations are illustrated in
Figure 2-1.

SIGNAL POWER

Instantaneous power is the voltage-current product at a specific time, p(t) = v(t) i(t),
while average power is P = 〈p(t)〉 = 〈v(t) i(t)〉 where 〈•〉 indicates time average. With
normalized power, the load resistance is assumed to be 1 Ω, current is numerically equal to
voltage and average normalized power is simply the time average of voltage squared, PN =
〈v(t)2〉. Root-mean-square (rms) voltage is the square root of normalized power, Vrms = PN
For example, the 12 volt supply in an automobile has normalized power of 144 ‘watts’.

Voltage Gain

The decibel unit of power gain has been generalized to
describe the voltage ratio in systems where the input and output
impedances are not defined. For example, an ideal transformer
with turns ratio 1:2 will have no power gain (0 dB) but will
have a voltage gain of 2 (6 dB). Voltage gain is useful to
describe electronic amplifiers which have high input impedance
and low output impedance. In this case, voltage gain is constant
but the power gain varies with the load resistance.

FREQUENCY SPECTRUM REPRESENTATION OF PERIODIC WAVEFORMS

A periodic waveform is composed of repeated copies of a waveform segment where
the segment length is equal to the repetition period. These waveforms are classed as power
signals since the duration is assumed to be infinite and there is no decay. Examples of
periodic waveforms are a sinusoid, a square wave and a triangular wave. A lengthy example
might be a continuously repeating musical recording. We are particularly interested in pulse
waveforms since they are used in sampling, an essential foundation of digital communication.
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#2
Electrical signals have time and frequency representations. In the time domain, voltage or current is expressed as a function of time, as illustrated in Figure 1. Most people are relatively comfortable with time domain signal representations. Signals measured on an oscilloscope are displayed in the time domain and digital information is often transmitted by a voltage as a function of time.

[Image: TextFig23-1.png]

Signals physically exist in the time domain and are generally expressed as a function of the time1 parameter. Due to this feature, it is not too difficult, at least in most situations of interest to us, to visualize the behavior of the signal in the time domain. In fact, it may even be possible to see the signals on an oscilloscope. But equally important is the characterization of the signals in the frequency domain or spectral domain. That is, we characterize the signal in terms of its different frequency components (or their spectrum). The Fourier analysis (Fourier series and Fourier transform) helps us reach the spectral description of the relevant signals.
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