408616 DIGITAL COMMUNICATION

Lecture-5
Chapter 1
1.5.4. POWER SPECTRAL DENSITY & AUTOCORRELATION
 A random process X(t) can generally be classified as a power signal
having a power spectral density (PSD) 𝐺𝑋(𝑓 ).
 The PSD enables evaluation of signal power that will pass through a
system having known frequency characteristics.
 Principal features of PSD functions

𝐺𝑋 𝑓 = 𝐺𝑋 −𝑓 symmetrical in τ about zero

𝐺𝑋 𝑓 ≥ 0 PSD is always real valued

𝑅𝑋 𝜏 ↔ 𝐺𝑋 (𝑓) autocorrelation and PSD form a Fourier

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Plot of autocorrelation function reveals bandwidth occupancy

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 1.5.5. NOISE IN COMMUNICATION SYSTEMS
 The term noise refers to unwanted electrical signals that are always
present in electrical systems; e.g spark-plug ignition noise,
switching transients, and other radiating electromagnetic signals.
 Can describe thermal noise as a zero-mean Gaussian random
process.
 A Gaussian process n(t) is a random function whose amplitude at
any arbitrary time t is statistically characterized by the Gaussian
probability density function
1 1 𝑛 2
𝑝 𝑛 = exp − (1.40)
𝜎 2𝜋 2 𝜎

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1.5.5.1 WHITE NOISE
 The primary spectral characteristic of thermal noise is that its
power spectral density is the same for all frequencies of interest in
most communication systems’ Power spectral density Gn(f )
N (1.42)
G ( f )  0 watts / hertz
n
2
Autocorrelation function of white noise is
N (1.43)
R ( )  1{G ( f )}  0  ( )
n n
2 
N0
 The average power Pn of white noise is infinite p(n)   2 df  
(1.44)

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 The effect on the detection process of a channel with additive
white Gaussian noise (AWGN) is that the noise affects each
transmitted symbol independently.
 Such a channel is called a memoryless channel.
 The term “additive” means that the noise is simply
superimposed or added to the signal

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 1.6 SIGNAL TRANSMISSION THROUGH LINEAR
SYSTEMS
Input Output

Linear system
 Deterministic signals:

 Random signals:

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 DISTORTION-LESS TRANSMISSION
 All the frequency components of the signal not only arrive with an
identical time delay, but also are amplified or attenuated equally.

Y ( f )  KX ( f )e  j 2 ft0
 The time delay 𝑡0 is related to the phase shift and radian frequency as:
 (radians)
t0 
2 f (radians/sec)
Therefore, for distortion less transmission, phase shift must be
proportional to frequency in order for the time delay of all
components to be identical.
 A characteristic often used to measure delay distortion of a signal is
called envelope delay or group delay 𝜏 (𝑓). Mathematically
1 d ( f )
( f )  
2 df
Therefore, for distortion less transmission, the group delay should be
a constant
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 FILTERING
 Ideal filters: ℎ(𝑡 − 𝑡0 )

Non-causal
Low-pass Infinite bandwidth

Band-pass

 Realizable filters:
RC filters Butterworth filter

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NON IDEAL FILTERING

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1.7 BANDWIDTH OF DIGITAL DATA

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1.7.2 THE BANDWIDTH DILEMMA
 Allbandwidth criteria have in
common the attempt to specify a
measure of the width, W, of a
nonnegative real-valued spectral
density defined for all frequencies
𝑓 < ∞
 The single-sided power spectral
density for a single heterodyned
pulse xc(t) takes the analytical form:

(1.73)

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DIFFERENT BANDWIDTH (BW) CRITERIA

(a) Half-power bandwidth

(c) Null-to-null BW
(d) Fractional power
containment BW
(e) Bounded PSD BW
(f) Absolute BW.

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NUMERICAL PROBLEMS

 All Examples
 End Chapter Problems 1.2, 1.4, 1.6, 1.7, 1.8, 1.9, 1.10, 1.11, 1.15, 1.20
 End Chapter Theoretical Questions

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