03. signal and spectra

7/28/2019 03. Signal and Spectra

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Signal and Spectra

Telecommunication Engineering

www.ee.ui.ac.id/wasp


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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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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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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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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

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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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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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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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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

03. signal and spectra

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