ECE 6602 - Week1 Lec2

Digital Communications

ECE 6602

Fall 2012

Instructor: Dr. Adnan A. Khan

Lec 2

6th Sep 2012

2

Sessionals : 15% (each)

Homework/Matlab (Simulink) Assignments : 15%

Simulation Project (Report and Presentations) : 15%

Abstract Submission by 4th week

Final Exam : 40%

Grading Policy

Transmitter Receiver Block Diagram

• Information source comes from higher networking

layers. It is packetized.

• Source Encoding: Finding a compact digital

representation for the data source. Includes

sampling of continuous-time signals, and

quantization of continuous-valued signals. Also

includes compression of those sources (lossy, or

lossless).


• Channel encoding refers to redundancy added to thesignal such that any bit errors can be corrected. Achannel decoder, because of the redundancy, cancorrect some bit errors.

• Modulation refers to the digital-to-analogconversion which produces a continuous-time signalthat can be sent on the physical channel. It isanalogous to impedance matching - proper matchingof a modulation to a channel allows optimalinformation transfer, like impedance matchingensured optimal power transfer.


• Channel/Media: We can chose from a few media, but

we largely can‘t change the properties of the medium

(although there are exceptions). Here are some media:

– EM Spectra: (anything above 0 Hz) Radio, Microwave,

mm-wave bands, light

– Acoustic: ultrasound

– Transmission lines, waveguides, optical fiber, coaxial cable,

wire pairs, ...

– Disk (data storage applications)


Basic Digital Communication

Transformations

• Formatting, Source Encoding

– Transforms source info into digital symbols (digitization)

– Selects compatible waveforms (matching function)

– Introduces redundancy which facilitates accurate decoding despite errors

– It is essential for reliable communication

• Modulation/Demodulation

– Modulation is the process of modifying the info signal to facilitate transmission

– Demodulation reverses the process of modulation. It involves the detection and retrieval of the info signal

– Types

• Coherent: Requires a reference info for detection

• Noncoherent: Does not require reference phase information


Transformations

• Coding/Decoding

– Translating info bits to transmitter data symbols

– Techniques used to enhance info signal so that they are less vulnerable to channel impairment (e.g. noise, fading, jamming, interference)

– Two Categories

• Waveform Coding

– Produces new waveforms with better performance

• Structured Sequences

– Involves the use of redundant bits to determine the occurrence of error (and sometimes correct it)

• Multiplexing/Multiple Access

Is synonymous with resource sharing with other users

– Frequency Division Multiplexing/Multiple Access (FDM/FDMA)


Transformations

– Time Division Multiplexing/Multiple Access (TDM/TDMA)

– Code Division Multiplexing/Multiple Access (CDM/CDMA)

– Space Division Multiple Access (SDMA)

– Polarization Division Multiple Access (PDMA)

• Spread Spreading (SS) Techniques

– It is usually used to protect privacy, protect against interference and allow flexible access to resources

– Common techniques include

• Direct Sequence (DS) Spread Spectrum - DSSS

• Frequency Hopping (FH) Spread Spectrum - FHSS

• Time Hopping (TH) Spread Spectrum - THSS

• Hybrids Techniques

– Time Division CDMA, Time Division Frequency Hopping, FDMA/CDMA, etc.,

Performance Metrics

• Analog Communication Systems

– Metric is fidelity: want m(t)m(t)

– SNR typically used as performance metric

• Digital Communication Systems

– Metrics are data rate (R bps) and probability of bit error

(Pb=p(bb))

– Symbols already known at the receiver

– Without noise/distortion/sync. problem, we will never make bit

errors

^

^

Main Points

• Transmitters modulate analog messages or bits in

case of a DCS for transmission over a channel.

• Receivers recreate signals or bits from received signal

(mitigate channel effects)

• Performance metric for analog systems is fidelity, for

digital it is the bit rate and error probability.

Channels

• A channel can typically be modeled as a linearfilter with the addition of noise. The noisecomes from a variety of sources, butpredominantly:– Thermal background noise: Modeled as Gaussian,

and white; thus it is referred to as additive whiteGaussian noise (AWGN).

– Interference from other transmitted signals. Theseother transmitters whose signals we cannotcompletely cancel, we lump into the ‗interference‘category. These may result in non-Gaussian noisedistribution, or non-white noise spectral density.

• The linear filtering of the channel result from the

physics and EM of the medium. For example,

attenuation in telephone wires varies by frequency.

• Narrowband wireless channels experience fading

that varies quickly as a function of frequency.

• Wideband wireless channels display multipath,

due to multiple time-delayed reflections,

diffractions, and scattering of the signal off of the

objects in the environment.

• All of these can be modeled as linear filters.

• The filter may be constant, or time-invariant, ifthe medium, the TX and RX do not move orchange.

• However, for mobile radio, the channel maychange very quickly over time. Even forstationary TX and RX, in real wireless channels,movement of cars, people, trees, etc. in theenvironment may change the channel slowly overtime.

System Model

Digital Signal Nomenclature

• Information Source

– Discrete output values e.g. Keyboard

– Analog signal source e.g. output of a microphone

• Character

– Member of an alphanumeric/symbol (A to Z, 0 to 9)

– Characters can be mapped into a sequence of binary digits using

one of the standardized codes such as

• ASCII: American Standard Code for Information Interchange

• EBCDIC: Extended Binary Coded Decimal Interchange Code


• Bits and Byte

– Binary Digit: Fundamental unit of information made up of 2

symbols (0 and 1)

– A group of 8 bits

• Binary Stream

– A sequence of binary digits, e.g., 10011100101010

• Symbol

– A digital message made up of groups of k-bits considered as a

unit


• Digital Message

– Messages constructed from a finite number of symbols; e.g., printed

language consists of 26 letters, 10 numbers, ―space‖ and several

punctuation marks. Hence a text is a digital message constructed from

about 50 symbols

– Morse-coded telegraph message is a digital message constructed from

two symbols ―Mark‖ and ―Space‖

• M - ary

– A digital message constructed with M symbols

• Digital Waveform

– Current or voltage waveform that represents a digital symbol

• Bit Rate

– Actual rate at which information is transmitted per second


• Baud Rate

– Refers to the rate at which the signaling elements are

transmitted, i.e. number of signaling elements per

second.

• Bit Error Rate

– The probability that one of the bits is in error or

simply the probability of error

Classification Of Signals

Deterministic and Random Signals

• A signal is deterministic means that there is no uncertainty with respect to its value at any time.

• Deterministic waveforms are modeled by explicit mathematical expressions, example:

x(t) = 5 Cos 10t

• A signal is random means that there is some degree of uncertainty before the signal actually occurs.

• 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.

Periodic and 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 < ∞

t denotes time

T0 is the period of x(t).

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

T is a fixed time interval.

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 = lim ∫ x2(t) dt = ∫ x2(t) dt

• An energy signal has finite energy but zero average power.

• Signals that are both deterministic and non-periodic are classified as energy signals

T→∞ -T/2

T/2

- ∞

∞

Example

26

Classify the following as power as energy signal:

x (t) = A cos 2πf0t for –T0/2 ≤ t ≤ T0/2 & 0 elsewhere

Solution:

Energy signals have finite energy

Energy signals are both deterministic and non-periodic

If we integrate x (t) over this range, then it will have a

finite result. So it has finite energy.

Look at the range of x (t). It is defined over a specific range

(–T0/2 ≤ t ≤ T0/2), after which the signal vanishes. So it is

Non-Periodic.

Hence it is an Energy Signal

• 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

Px = lim 1/T ∫ x2(t) dt

• Power signal has finite average power but infinite energy.

• As a general rule, periodic signals and random signals are

classified as power signals

Energy and Power Signals

T→∞

-T/2

T/2

Example

28

Classify the following as power as energy signal:

x (t) = A cos 2πf0t for -∞ < t < ∞

Solution:

Energy signals have infinite energy

Energy signals are both Random and periodic.

This signal is extended over the entire space, so it has

infinite energy.

We know that cosine function is periodic with period 2π.

Hence it is a Power Signal

• 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.

∫ δ (t) dt = 1

δ (t) = 0 for t ≠ 0

δ (t) is bounded at t =0

• Sifting or Sampling Property

∫ x(t) δ (t – t0) dt = x(t0)

The Unit Impulse Function

- ∞

∞

- ∞

∞

Spectral Density

• The spectral density of a signal characterizes the distribution of the

signal‘s energy or power in the frequency domain.

• This concept is particularly important when considering filtering in

communication systems while evaluating the signal and noise at

the filter output.

• The energy spectral density (ESD) or the power spectral density

(PSD) is used in the evaluation.

Energy Spectral Density (ESD)

• Energy spectral density describes the signal energy per unit bandwidth

measured in joules/hertz.

• Represented as ψx(f), the squared magnitude spectrum

ψx(f) = |X(f)|2

• According to Parseval‘s theorem, the energy of x(t):

Ex = ∫ x2(t) dt = ∫ |X(f)|2 df

• Therefore:

Ex = ∫ ψx (f) df

• The Energy spectral density is symmetrical in frequency about origin

and total energy of the signal x(t) can be expressed as:

Ex = 2 ∫ ψx (f) df

- ∞

∞

- ∞

∞

- ∞

∞

0

∞

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 |Cn|2 δ (f – nf0)

• Whereas the average power of a periodic signal x(t) is represented as:

Px = 1/T ∫ x2(t) dt = Σ |Cn|2

• Using PSD, the average normalized power of a real-valued signal is represented as:

Px = ∫ Gx(f ) df = 2 ∫ Gx(f ) df

-T0/2

T0/2

n=-∞

∞

- ∞

∞

0

∞

n=-∞

∞

Example

33

Determine the energy spectral density of a square pulse x (t) =

rect (t/T), where rect (t/T) equals 1 for –T/2 < t < T/2, and equals

0 elsewhere.

Solution:

ESD = |X (f)|2

Where X (f) is the Fourier Transform of x (t)

X (f) = T sinc (fT)

|X (f)|2 = T2 sinc2 (fT)

So ESD = T2 sinc2 (fT)

Example

34

Find the average normalized power of:

x (t) = A cos 2πf0t for -∞ < t < ∞

Solution:

Px = 1/T0 ∫ x2(t) dt

Px = 1/T0 ∫ A2cos2 (2πf0t) dt

Px = A2/2

-T0/2

T0/2

T0/2

-T0/2

Example

35

Find the average normalized energy of:

x (t) = A cos 2πf0t for –T0/2 ≤ t ≤ T0/2 & 0 elsewhere

Solution:

Ex = ∫ x2(t) dt

Ex = ∫ A2cos2 (2πf0t) dt

Ex = A2T0/2

-T0/2

T0/2

T0/2

-T0/2

Autocorrelation

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:

Rx(τ) = ∫ x(t) x (t + τ) dt for - ∞ < τ < ∞

• 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.

- ∞

∞

Autocorrelation of an Energy Signal

• The autocorrelation function of a real-valued energy signal has the

following properties:

Rx(τ) = Rx(-τ) symmetrical in 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 designated

by the double-headed arrows

Rx(0) = ∫ x2(t) dt value at the origin is equal to the

energy of the signal

Autocorrelation of a Power Signal

• Autocorrelation function of a real-valued power signal x(t) is

defined as:

Rx(τ) = lim 1/T ∫ x(t) x (t + τ) dt for - ∞ < τ < ∞

• When the power signal x(t) is periodic with period T0, the

autocorrelation function can be expressed as

Rx(τ) = lim 1/T0 ∫ x(t) x (t + τ) dt for - ∞ < τ < ∞

-T/2

T/2

T→∞

-T0/2

T0/2

Autocorrelation of a Power Signal

• The autocorrelative function of a real-valued periodic signal has

the following properties similar to those of an energy signal:

Rx(τ) = Rx(-τ) symmetrical in 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

Rx(0) = 1/T0 ∫ x2(t) dt value at the origin is equal to the

average power of the signal-T0/2

T0/2

Example

40

Find the autocorrelation function of x (t) = A cos (2πf0t + φ)

in terms of its period T0 = 1/f0.

Solution:

Now use some trigonometric identities

Rx(τ) = 1/T0 ∫ x(t) x (t + τ) dt for - ∞ < τ < ∞

T0/2

-T0/2

Rx(τ) = 1/T0 ∫ A2 cos (2πf0t + φ) cos (2πf0t + φ + τ) dt

T0/2

-T0/2

Example

41

Expand the last term using

cos (a + b) = cos (a) cos (b) – sin (a) sin (b)

Take a = 2πf0t + φ & b = τ

Then use:

o cos (x) cos (y) = ½ [cos (x + y) + cos (x - y)]

o cos (x) sin (y) = ½ [sin (x + y) + sin (x - y)]

After applying these identities we will get:

Rx (τ) = A2 cos (2f0τ) / 2

Note that autocorrelation is independent of time.

Example

42

Now find the average normalized power of given signal x (t)

using autocorrelation function:

Rx (τ) = A2 cos (2f0τ) / 2

Using Px = Rx (0)

Px = Rx (0) = A2 cos (2f0(0)) / 2

Px = Rx (0) = A2 cos (0) / 2

Px = Rx (0) = A2/2

Random Signals

Random Variables

• All useful message signals appear random; that is, the receiver does not know, a priori, which of the possible waveform have been sent.

• Let a random variable X(A) represent the functional relationship between a random event A and a real number.

• The (cumulative) distribution function FX(x) of the random variable X is given by

Fx(x) = P(X ≤ x)

• Another useful function relating to the random variable X is the probability density function (pdf)

Px(x) = d Fx(x) / dx

Ensemble Averages

• The first moment of a probability distribution of a random variable X is called mean value mX, or expected value of a random variable X

• The second moment of a probability distribution is the mean-square value of X

• Central moments are the moments of the difference between X and mXand the second central moment is the variance of X

• Variance is equal to the difference between the mean-square value and the square of the mean

mX = E{X} = ∫ x Px(x) dx- ∞

∞

E{X2} = ∫ x2 Px(x) dx- ∞

∞

Var(X) = E{(X – mX)2 }

= ∫ (x – mX)2 Px(x) dx- ∞

∞

Var(X) =E{X2} –E{X}2

Random Processes

• A random process X(A, t) can be viewed as a function of two

variables: an event A and time.

Statistical Averages of a Random

Process• A random process whose distribution functions are continuous can be

described statistically with a probability density function (pdf).

• A partial description consisting of the mean and autocorrelation function are

often adequate for the needs of communication systems.

• Mean of the random process X(t) :

E{X(tk)} = ∫ x Px(x) dx = mx(tk)

• Autocorrelation function of the random process X(t)

Rx(t1,t2) = E{X(t1) X(t2)} = ∫ ∫ xt1xt2Px(xt1,xt2) dxt1dxt2

- ∞

∞

Stationary

• A random process X(t) is said to be stationary in the strict sense if

none of its statistics are affected by a shift in the time origin.

• A random process is said to be wide-sense stationary (WSS) if two

of its statistics, its mean and autocorrelation function, do not vary

with a shift in the time origin.

E{X(t)} = mx= a constant

Rx(t1,t2) = Rx (t1 – t2)

Autocorrelation of a Wide-Sense

Stationary Random Process

• For a wide-sense stationary process, the autocorrelation function

is only a function of the time difference τ = t1 – t2;

Rx(τ) = E{X(t) X(t + τ)} for - ∞ < τ < ∞

• Properties of the autocorrelation function of a real-valued wide-

sense stationary process are

Time Averaging and Ergodicity

• When a random process belongs to a special class, known as an ergodic process, its time averages equal its ensemble averages.

• The statistical properties of such processes can be determined by time averaging over a single sample function of the process.

• A random process is ergodic in the mean if

mx = lim 1/T ∫ x(t) dt

• It is ergodic in the autocorrelation function if

Rx(τ) = lim 1/T ∫ x(t) x (t + τ) dt

-T/2

T/2

T→∞ -T/2

T/2

T→∞

Power Spectral Density and

Autocorrelation

• A random process X(t) can generally be classified as a power signal

having a power spectral density (PSD) GX(f )

• Principal features of PSD functions

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 or atmosphere,

the sun, and other galactic sources.

• 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

(1.40)

Noise in Communication Systems

• The normalized or standardized Gaussian density function of a zero-

mean process is obtained by assuming unit variance.

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 )

• Autocorrelation function of white noise is

• The average power Pn of white noise is infinite

• 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

Ideal Filters

• For the ideal low-pass filter transfer function with bandwidth Wf = fu

hertz can be written as:

Ideal low-pass filter

H(f) = | H(f) | e-j (f)

Where

| H(f) | = { 1 for |f| < fu

0 for |f| fu }

e-j (f) = e-j2 f t0

Ideal Filters

• The impulse response of the ideal low-pass filter:

h(t) = F-1 {H(f)}

= H(f) e-j2 f t df (1.61)

= e-j2 f t0 e-j2 f t df

= e-j2 f (t - t0) df

= 2fu * sin 2fu(t – t0)/ 2fu(t – t0)

= 2fu * sinc 2fu(t – t0)

(1.62)

- ∞

∞

- fu

fu

- fu

fu

Bandwidth Of Digital DataBaseband versus Bandpass

• An easy way to translate the

spectrum of a low-pass or baseband

signal x(t) to a higher frequency is to

multiply or heterodyne the baseband

signal with a carrier wave cos 2fct

• xc(t) is called a double-sideband

(DSB) modulated signal

xc(t) = x(t) cos 2fct (1.70)

• From the frequency shifting theorem

Xc(f) = 1/2 [X(f-fc) + X(f+fc) ] (1.71)

• Generally the carrier wave frequency

is much higher than the bandwidth of

the baseband signal

fc >> fm and therefore WDSB = 2fm

Bandwidth

• Theorems of

communication and

information theory are

based on the assumption

of strictly bandlimited

channels

• The mathematical

description of a real signal

does not permit the signal

to be strictly duration

limited and strictly

bandlimited.

Bandwidth

• All bandwidth 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 pulse xc(t) takes the

analytical form:

Different Bandwidth Criteria

(a) Half-power bandwidth.

(b) Equivalent rectangular or

noise equivalent

bandwidth.

(c) Null-to-null bandwidth.

(d) Fractional power

containment bandwidth.

(e) Bounded power spectral

density.

(f) Absolute bandwidth.

Feedback Quiz

1) Differentiate between a discrete signal and digital signal.

2) What do you understand by Sampling and Quantization

in digital communications?

3) What do you understand by Modulation and

Demodulation in digital communications?

4) What is the difference between Baseband and Passband

Modulation?

5) What is the Nyquist Criteria for perfect reconstruction of

sampled signal?

6) Why we cannot send ideal square pulses on channel

7) What is a random variable?

8) What is a basis function ?

ECE 6602 - Week1 Lec2

Documents

Transcript of ECE 6602 - Week1 Lec2