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7/28/2019 03. Signal and Spectra
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Signal and Spectra
Telecommunication Engineering
www.ee.ui.ac.id/wasp
7/28/2019 03. Signal and Spectra
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The basic knowledge of signal and spectra has been given in the
signal and system course such as classification of signal, Fourierrepresentation, autocorrelation
However, we will review those topics, but this lecture emphasize
on: spectral density
random signal
bandwidth problems
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Classification of Signals
Deterministic and random signals Deterministic: there is no uncertainty with respect to its
value at any time
Can you give examples of deterministic signal?
Random: there is some degree of uncertainty before thesignal actually occurs
Can you give examples of random signal?
Deterministic signal is represented by using mathematical
expression
Random signal is represented by using random process
theorem
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Classification of Signals
Periodic and nonperiodic signals A signal is called periodic in time if
is fundamental period
Otherwise, it is called non periodic
0T
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Classification of Signals
The average power dissipated by the signal during theinterval is
A signal said energy signal iff it has nonzero but finiteenergy for all time, where
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Classification of Signals
In real world, we always transmit signals having finiteenergy
If we refer to periodic signals, they have infinite energy
(why?) we have to define power signal
A signal is said power signal iff it has finite but nonzeropower for all time, where
An energy signal has finite energy but zero average power
A power signal has finite average power but infinite
energy
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Spectral Density
Spectral density characterizes the distribution of thesignals energy or power in the frequency domain
This concept is important when considering filtering in
communication systems
We need to be able to evaluate the signal and noise atthe filter output
The energy spectral density (ESD) or power spectral
density (PSD) is used in the evaluation
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Energy Spectral Density
The energy in the time domain can be related tofrequency domain as follows
The ESD is denoted as
Then, the total energy can be expressed as
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Energy Spectral Density
ESD describes the signal energy per unit bandwidth,therefore, it is measured in joules/hertz
There are equal energy from both positive and negative
frequency (why?)
The energy spectral density is symmetrical in frequencyabout the origin
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Power Spectral Density
A periodic signal with period has average power
PSD of the periodic signal is a real, even, and nonnegative
function of frequency, defined as
Note that PSD of a periodic signal is a discrete functionof frequency
0
T
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Power Spectral Density
The average normalized power of real-valued signal is
If the signal is nonperiodic signal, it cannot be expressed
by a Fourier series, and if it is a nonperiodic power signal,it may not have a Fourier transform we need to
truncate the signal
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Autocorrelation of An Energy Signal
Autocorrelation: matching of a signal with a delayedversion of itself
The autocorrelation function of a real-valued energy
signal:
The autocorrelation function gives a measure how closely
the signal matches a copy of itself as the copy is shifted
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Autocorrelation of A Periodic (Power) Signal
The autocorrelation function of a real-valued powersignal is defined as
When the signal is periodic, the time average is takenover a single period
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Noise in Telecommunication System
Noise: unwanted electrical signals that are always presentin electrical systems
Noise source: man-made and natural
Man-made noise:
Spark-plug ignition noise Switching transients
Other radiating electromagnetic signals
Natural noise:
Atmosphere
The sun
Other galactic sources
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Noise in Telecommunication System
One common natural noise: thermal noise It is caused by the thermal motion of electrons in all
dissipative componentsresistors, wires
Thermal noise is described as Gaussian random process
so it is characterized by the Gaussian probability densityfunction
where is the variance ofn
The normalized Gaussian pdf of zero mean has
2
1
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Noise in Telecommunication System
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Noise in Telecommunication System
We will represent a random signal as the sum of aGaussian noise random variable and a dc signal
The pdf of z is expressed as
Random signal dc component Gaussian noise random variable
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Noise in Telecommunication System
The Gaussian distribution is used as the system noisemodel because of the central limit theorem
The central limit theorem states that under very general
conditions the probability distribution of the sum of j
statistically independent random variable approaches theGaussian distributions as j , no matter what the
individual distribution functions may be
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White Noise
The primary spectral characteristic of thermal noise isthat its power spectral density is the same for all
frequencies
A simple model for thermal noise assumes that its power
spectral density is flat for all frequencies
The factor of 2 is included to indicate that it is two-sided
power spectral density
The noise power has a uniform spectral density is called
white noise
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White Noise
The average power of white noise is
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White Noise
Thermal noise is a Gaussian process and the samples areuncorrelated, the noise samples are also independent
Therefore, the effect on the detection process of a
channel with additive white Gaussian noise (AWGN) is
that the noise affects each transmitted symbolindependently memoryless channel
+x
n
y = x + n
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Bandwidth
We assume that communication system has bandlimitedchannels, means that no signal power whatever is allowed
outside the defined band
The problem is that strictly bandlimited signals are not
realizable
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Bandwidth
There are several definitions of bandwidth: Half-power bandwidth
Equivalent rectangular or noise equivalent bandwidth
Null-to-null bandwidth
Fractional power containment bandwidth Bounded power spectral density
Absolute bandwidth
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Bandwidth