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Chapter 2
SIGNALS AND SPECTRA
Chapter Objectives:
• Basic signal properties (DC, RMS, dBm, and power);
• Fourier transform and spectra;
• Linear systems and linear distortion;
• Band limited signals and sampling;
• Discrete Fourier Transform;
• Bandwidth of signals.
Huseyin Bilgekul
Eeng360 Communication Systems I
Department of Electrical and Electronic Engineering
Eastern Mediterranean University
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Properties of Signals & Noise
 In communication systems, the received waveform is
usually categorized into two parts:
Signal:
The desired part containing the
information.
Noise:
The undesired part
Properties of waveforms include:
• DC value,
• Root-mean-square (rms) value,
• Normalized power,
• Magnitude spectrum,
• Phase spectrum,
• Power spectral density,
• Bandwidth
• ………………..
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Physically Realizable Waveforms
 Physically realizable waveforms are practical
waveforms which can be measured in a laboratory.
 These waveforms satisfy the following conditions
• The waveform has significant nonzero values over a
composite time interval that is finite.
• The spectrum of the waveform has significant values over a
composite frequency interval that is finite
• The waveform is a continuous function of time
• The waveform has a finite peak value
• The waveform has only real values. That is, at any time, it
cannot have a complex value a+jb, where b is nonzero.
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Physically Realizable Waveforms
 Mathematical Models that violate some or all of the conditions
listed above are often used.
 One main reason is to simplify the mathematical analysis.
 If we are careful with the mathematical model, the correct
result can be obtained when the answer is properly interpreted.
Physical Waveform
Mathematical Model Waveform
The Math model in this example
violates the following rules:
1. Continuity
2. Finite duration
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Time Average Operator
Definition: The time average operator is given by,
The operator is a linear operator,
• the average of the sum of two quantities is the same as
the sum of their averages:
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Periodic Waveforms
 Definition
A waveform w(t) is periodic with period T0 if,
w(t) = w(t + T0) for all t
where T0 is the smallest positive number that satisfies this relationship
 A sinusoidal waveform of frequency f0 = 1/T0 Hertz is periodic
 Theorem: If the waveform involved is periodic, the time average
operator can be reduced to
where T0 is the period of the waveform and a is an arbitrary real constant, which
may be taken to be zero.
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DC Value
Definition: The DC (direct “current”) value of a
waveform w(t) is given by its time average, w(t). Thus,
 For a physical waveform, we are actually interested in evaluating
the DC value only over a finite interval of interest, say, from t1 to t2,
so that the dc value is
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Power
 Definition.
Let v(t) denote the voltage across a set of circuit terminals,
and let i(t) denote the current into the terminal, as shown .
The instantaneous power (incremental work divided by
incremental time) associated with the circuit is given by:
p(t) = v(t)i(t)
the instantaneous power flows into the circuit when p(t) is
positive and flows out of the circuit when p(t) is negative.
 The average power is:
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Evaluation of DC Value
 A 120V , 60 Hz fluorescent lamp wired in a high power factor configuration.
Assume the voltage and current are both sinusoids and in phase ( unity power
factor)
Voltage
DC Value of this waveform is:
VDC  v(t )  V cos 0t
1

T0

T0 / 2
T0 / 2
Current
V cos 0t  dt  0
Where,
0  2 / T0 , and
Instantenous Power
f 0  1/ T0  60 Hz
Similarly, I DC  0
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p(t) = v(t)i(t)
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Evaluation of Power
The instantaneous power is:
p(t )  V cos  0t   I cos  0t 
 1 / 2 VI 1  2 cos  0t 
The Average power is:
P  1/ 2  VI 1  2 cos 0t 
VI

2T0

T0
2
T 0
1  2 cos 0t  dt
2
VI

2
The Maximum power is:
Pmax=VI
Maximum
Power
Average
Power
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RMS Value
 Definition: The root-mean-square (rms) value of w(t) is:
 Rms value of a sinusoidal:
Wrms 
V cos(ot )
2

V
2
 Theorem:
If a load is resistive (i.e., with unity power factor), the average
power is:
where R is the value of the resistive load.
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Normalized Power
 In the concept of Normalized Power, R is assumed to be 1Ω,
although it may be another value in the actual circuit.
 Another way of expressing this concept is to say that the
power is given on a per-ohm basis.
 It can also be realized that the square root of the normalized
power is the rms value.
Definition. The average normalized power is given as follows,
Where w(t) is the voltage or current waveform
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Energy and Power Waveforms
 Definition: w(t) is a power waveform if and only if the normalized
average power P is finite and nonzero (0 < P < ∞).
 Definition: The total normalized energy is
 Definition: w(t) is an energy waveform if and only if the total
normalized energy is finite and nonzero (0 < E < ∞).
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Energy and Power Waveforms
 If a waveform is classified as either one of these types, it
cannot be of the other type.
 If w(t) has finite energy, the power averaged over infinite time
is zero.
 If the power (averaged over infinite time) is finite, the energy
if infinite.
 However, mathematical functions can be found that have both
infinite energy and infinite power and, consequently, cannot be
classified into either of these two categories. (w(t) = e-t).
 Physically realizable waveforms are of the energy type.
– We can find a finite power for these!!
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Decibel
 A base 10 logarithmic measure of power ratios.
 The ratio of the power level at the output of a circuit
compared with that at the input is often specified by
the decibel gain instead of the actual ratio.
 Decibel measure can be defined in 3 ways
•
•
•
Decibel Gain
Decibel signal-to-noise ratio
Mill watt Decibel or dBm
 Definition: Decibel Gain of a circuit is:
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Decibel Gain
 If resistive loads are involved,
Definition of dB may be reduced to,
or
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Decibel Signal-to-noise Ratio (SNR)
 Definition. The decibel signal-to-noise ratio (S/R, SNR) is:
Where, Signal Power (S) =
And, Noise Power (N) =
So, definition is equivalent to
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Decibel with Mili watt Reference (dBm)
 Definition. The decibel power level with respect to 1 mW is:
= 30 + 10 log (Actual Power Level (watts)
• Here the “m” in the dBm denotes a milliwatt reference.
• When a 1-W reference level is used, the decibel level is
denoted dBW;
• when a 1-kW reference level is used, the decibel level is
denoted dBk.
E.g.: If an antenna receives a signal power of 0.3W, what is the
received power level in dBm?
dBm = 30 + 10xlog(0.3) = 30 + 10x(-0.523)3 = 24.77 dBm
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Phasors
 Definition: A complex number c is said to be a “phasor” if it is
used to represent a sinusoidal waveform. That is,
where the phasor c = |c|ejc and Re{.} denotes the real part of the complex
quantity {.}.
 The phasor can be written as: c  x  jy  c e j  c 
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