408616 DIGITAL COMMUNICATION
Lecture-2&3
Chapter 1
1.2 CLASSIFICATION OF SIGNALS
▪ Deterministic & Random Signals
▪ Periodic and Non-Periodic Signals
▪ Analog and Discrete Signals (Discrete vs Digital)
▪ Energy and Power Signals
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1.2.1 DETERMINISTIC & RANDOM SIGNALS
▪ Deterministic signal: No uncertainty with respect to the signal
value at any time.
▪ Random signal: Some degree of uncertainty in signal values
before it actually occurs.
▪ Thermal noise in electronic circuits due to the random
movement of electrons
▪ Reflection of radio waves from different layers of ionosphere
▪ Interference
Random waveforms/ Random processes when examined over a
long period may exhibit certain regularities that can be
described in terms of probabilities and statistical averages.
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1.2.2 PERIODIC & NON-PERIODIC SIGNALS
▪ A signal 𝒙(𝒕) is called periodic in time if there exists a constant
𝑇0 > 0 such that
𝑥 𝑡 = 𝑥 𝑡 + 𝑇0 . for −∞ < 𝒕 < ∞ (1.2)
𝑡 denotes time , 𝑇0 is the period of 𝑥(𝑡).
A periodic signal A non-periodic signal
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1.2.3 ANALOG AND DISCRETE SIGNALS
▪ An analog signal 𝑥(𝑡) is a continuous function of time; that is, 𝑥(𝑡) is
uniquely defined for all 𝑡
▪ A discrete signal 𝑥(𝑘𝑇) is one that exists only at discrete times; it is
characterized by a sequence of numbers defined for each time, 𝑘𝑇,
where 𝑘 is an integer or 𝑇 is a fixed time interval.
A discrete signal
Analog signals
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1.2.4 ENERGY AND POWER SIGNALS
▪ The performance of a communication system depends on the received
signal energy; higher energy signals are detected more reliably (with
fewer errors) than are lower energy signals
▪ x(t) is classified as an energy signal if, and only if, it has nonzero but
finite energy (0 < Ex < ∞) for all time, where:
Ex =
2
| x (t ) | dt (1.7)
−
▪ An energy signal has finite energy but zero average power.
▪ Signals that are both deterministic and non-periodic are classified as
energy signals
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1.2.4 ENERGY AND POWER SIGNALS
▪ Power is the rate at which energy is delivered.
▪ A signal is defined as a power signal if, and only if, it has finite
but nonzero power (0 < 𝑃𝑥 < ∞) for all time, where
1 ∞
𝑃𝒙 = lim 𝑇→∞ 𝑇 −∞ 𝑥 2 𝑡 𝑑𝑡 (1.8)
▪ Power signal has finite average power but infinite energy.
▪ As a general rule, periodic signals and random signals are
classified as power signals
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1.2.5 THE UNIT IMPULSE FUNCTION
▪ Dirac delta function δ(t) or impulse function is an abstraction—
an infinitely large amplitude pulse, with zero pulse width, and
unity weight (area under the pulse), concentrated at the point
where its argument
is zero.
(t) dt = 1
−
(1.9)
(1.10)
(t) = 0 for t 0
(t) is bounded at t = 0 (1.11)
▪ Sifting or Sampling Property
(1.12)
x(t ) (t-t
−
0 )dt = x(t 0 )
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1. 3.1 ENERGY SPECTRAL DENSITY (ESD)
▪ Energy spectral density describes the signal energy per unit
bandwidth measured in joules/hertz.
▪ Represented as 𝜓𝑥 (𝑓), it equals the squared magnitude spectrum
𝜓𝑥 𝑓 = 𝑋(𝑓) 2 (1.14)
▪ According to Parseval’s theorem, the energy of 𝑥(𝑡) 𝐸𝑥 is given as:
∞
𝐸𝑥 = −∞ 𝜓𝑥 𝑓 𝑑𝑓 (1.15)
▪ Therefore:
∞ ∞
𝐸𝑥 = −∞ 𝑥 2 𝑡 𝑑𝑡 = −∞ 𝑋(𝑓) 2 𝑑𝑓 (1.13)
▪ The Energy spectral density is symmetrical in frequency about
origin and total energy of the signal x(t) can be expressed as:
∞ 9
𝐸𝑥 = 2 0 𝜓𝑥 𝑓 𝑑𝑓 (1.16)
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1.3.2. POWER SPECTRAL DENSITY (PSD)
▪ The power spectral density (PSD) function Gx(f ) of the periodic
signal x(t) is a real, even, and nonnegative function of frequency
that gives the distribution of the power of x(t) in the frequency
domain.
▪ PSD is represented as:
𝐺𝑥 𝑓 = σ∞ 2
𝑛=−∞ |𝐶𝑛 | 𝛿(𝑓 − 𝑛𝑓0 ) (1.18)
▪ Whereas the average power of a periodic signal x(t) is represented
as:
1 𝑇 /2
𝑃𝑥 = 𝑇 −𝑇0 /2 𝑥 2 𝑡 𝑑𝑡 = σ∞ 2
𝑛=−∞ |𝐶𝑛 | 𝛿(𝑓 − 𝑛𝑓0 ) (1.17)
0 0
▪ Using PSD, the average normalized power of a real-valued signal is
represented as:
1 ∞ ∞
𝑃𝑥 = 𝑇 −∞ 𝐺𝑥 𝑓 𝑑𝑓 = 2 0 𝐺𝑥 𝑓 𝑑𝑓 (1.19)
0
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1.4 AUTOCORRELATION
1.4.1 AUTOCORRELATION OF AN ENERGY SIGNAL
▪ Correlation is a matching process; autocorrelation refers to the
matching of a signal with a delayed version of itself.
▪ Autocorrelation function of a real-valued energy signal 𝑥(𝑡) is
defined as:
R x ( ) = x(t) x (t + ) dt
−
for - < <
(1.21)
▪ The autocorrelation function Rx(τ) provides a measure of how closely
the signal matches a copy of itself as the copy is shifted τ units in
time.
▪ Rx(τ) is not a function of time; it is only a function of the time
difference τ between the waveform and its shifted copy.
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EXAMPLE: CORRELATION
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The practical implementation of a correlation function
corresponds to a multiplication of the two signals followed by an
integrate function. Dwell time denotes the time needed to
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compute one integration result.
AUTOCORRELATION OF AN ENERGY SIGNAL
▪ The autocorrelation function of a real-valued energy signal has the
following properties:
𝑅𝑥 𝜏 = 𝑅𝑥 −𝜏 symmetrical in τ about zero
𝑅𝑥 𝜏 ≤ 𝑅𝑥 0 for all 𝜏 maximum value occurs at the
origin
𝑅𝑥 𝜏 𝜓𝑥 (𝑓) autocorrelation and ESD form a
Fourier transform pair, as shown
by the double-headed arrows
∞
value at the origin is equal to
𝑅𝑥 0 = න 𝑥 2 𝑡 𝑑𝑡
−∞
energy of the signal
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1.4.2 AUTOCORRELATION OF A POWER SIGNAL
▪ Autocorrelation function of a real-valued periodic power signal x(t)
is defined as:
1 𝑇/2
𝑅𝑥 𝜏 = lim 𝑇 −𝑇/2 𝑥 𝑡 𝑥 𝑡 + 𝜏 𝑑𝑡 for −∞ < 𝜏 < ∞. (1.22)
𝑇→∞
▪ When the power signal 𝑥(𝑡) is periodic with period 𝑇0 , the
autocorrelation function can be expressed as
1 𝑇 /2
𝑅𝑥 𝜏 = 𝑇 −𝑇0 /2 𝑥 𝑡 𝑥 𝑡 + 𝜏 𝑑𝑡 for −∞ < 𝜏 < ∞. (1.23)
0 0
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AUTOCORRELATION OF A POWER SIGNAL
▪ The autocorrelation function of a real-valued periodic signal has
the following properties similar to those of an energy signal:
𝑅𝑥 𝜏 = 𝑅𝑥 −𝜏 symmetrical in τ about zero
𝑅𝑥 𝜏 ≤ 𝑅𝑥 0 for all 𝜏 maximum value occurs at the origin
𝑅𝑥 𝜏 𝐺𝑥 (𝑓) autocorrelation and PSD form a Fourier
transform pair, as shown by the double-
headed arrows
1 𝑇0/2 2 value at the origin is equal to average
𝑅𝑥 0 = න 𝑥 𝑡 𝑑𝑡 power of the signal
𝑇0 −𝑇0/2
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The relationship between signal
and its autocorrelation function.
The autocorrelation function can
be computed from signal directly,
as indicated in the red box, or
indirectly by first calculating
Fourier transform (FT), squaring
its absolute value to obtain power
spectrum, and computing reverse
Fourier transform (RFT). Power
spectrum is a histogram of signal
power at each sampled
frequency. The indirect route is
much faster.
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Problem 1.7 ) Plot of part d
f=-15:0.05:15;
len=length(f);
for z=1:1:len
X(z)=exp(-2*pi*(f(z)^2)-10);
end
plot(f,X)
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PROBLEMS
▪ P 1.1-1.10
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