dc lecture2

8/12/2019 DC Lecture2

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Book on Probability: http://web.mit.edu/afs/athena.mit.edu /user/d/i/dimitrib/www/Probability.html

Plan for Today

Classification Of Signals Spectral Density Autocorrelation Random Signals
http://web.mit.edu/afs/athena.mit.eduhttp://web.mit.edu/afs/athena.mit.edu


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Why Digital Communications? Easy to regenerate the dis tor ted s ign al

Regenerative repeaters along the transmission path can detect adigital signal and retransmit a new, clean (noise free) signalThese repeaters prevent accumulation of noise along the path

This is not possible with analog communication systemsTwo -state s ig nal representat ion

The input to a digital system is in the form of a sequence of bits(binary or M_ary)

Im m uni ty to d is tor t ion and in ter ference

Digital communication is rugged in the sense that it is more immuneto channel noise and distortion


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Why Digital Communications? Hardw are is m ore f lexib le

Digital hardware implementation is flexible and permits the use ofmicroprocessors, mini-processors, digital switching and VLSIShorter design and production cycle

L o w c o s tThe use of LSI and VLSI in the design of components and systemshave resulted in lower cost

Easier and m ore eff ic ient to m ult ip lex several dig i ta l s ig nalsDigital multiplexing techniques Time & Code Division Multiple

Access - are easier to implement than analog techniques such asFrequency Division Multiple Access

Can c om bine d i fferent s ignal types data, v oi c e, text , etc.

Data communication in computers is digital in nature whereas voicecommunication between people is analog in nature


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Why Digital Communications?

The two types of communication are difficult to combine overthe same medium in the analog domain.Using digital techniques, it is possible to combine both formatfor transmission through a common medium

Can u se packet sw i tch ing

Encry pt ion and pr ivacy techn iques are eas ier toimplementBet ter o veral l perform ance

Digital communication is inherently more efficient than analogin realizing the exchange of SNR for bandwidth

Digital signals can be coded to yield extremely low rates andhigh fidelity as well as privacy


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3. DisadvantagesRequires reliable synchronization Requ ires A/D con vers ion s a t h igh ra te

Requ i res larger bandw idth

4. Performance CriteriaProbability of error or Bit Error Rate


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Summary


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Goals in Communication System Design

To maximize transmission rate, RTo maximize system utilization, UTo minimize bit error rate, P eTo minimize required systems bandwidth, WTo minimize system complexity, C x To minimize required power, E b /N o


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Digital Signal Nomenclature

Information SourceDiscrete output values e.g. Keyboard

Analog signal source e.g. output of a microphone

CharacterMember of an alphanumeric/symbol (A to Z, 0 to 9)Characters can be mapped into a sequence of binary digitsusing one of the standardized codes such as

ASCII: American Standard Code for Information Interchange

EBCDIC: Extended Binary Coded Decimal Interchange Code


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Bits and ByteBinary Digit: Fundamental unit of information made up of 2symbols ( 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 aunit


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Digital Message or a symbolMessages constructed from a finite number of symbols; e.g., printedlanguage consists of 26 letters, 10 numbers, space and severalpunctuation marks. Hence a text is a digital message constructedfrom about 50 symbolsMorse-coded telegraph message is a digital message constructedfrom two symbols Mark and Space

M - ary A digital message constructed with M symbols

Digital WaveformCurrent or voltage waveform that represents a digital symbol

Bit Rate Actual rate at which information is transmitted per second


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Baud RateRefers to the rate at which the signaling elementsare transmitted, i.e. number of signaling elements

per second.

Bit Error RateThe probability that one of the bits is in error or

simply the probability of error


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Elements of Digital Communication

Each of these blocks represents one or more transformations Each block identifies a major signal processing function whichchanges or transforms the signal from one signal space to another

Some of the transformation block overlap in functions


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Basic Digital Communication Transformations

Formatting/Source CodingTransforms source info into digital symbols (digitization)Selects compatible waveforms (matching function)Introduces redundancy which facilitates accurate decoding despiteerrorsIt is essential for reliable communication

Modulation/DemodulationModulation is the process of modifying the info signal to facilitatetransmissionDemodulation reverses the process of modulation. It involves thedetection and retrieval of the info signalTypes

Coherent: Requires a reference info for detectionNoncoherent: Does not require reference phase information


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Coding/DecodingTranslating info bits to transmitter data symbolsTechniques used to enhance info signal so that they are lessvulnerable to channel impairment (e.g. noise, fading, jamming,interference)

Two CategoriesWaveform Coding

Produces new waveforms with better performanceStruc tured Sequences

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

Multiplexing/Multiple AccessIs synonymous with resource sharing with other users

Frequency Division Multiplexing/Multiple Access (FDM/FDMA)


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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) TechniquesIt is usually used to protect privacy, protect against interference andallow flexible access to resourcesCommon techniques include

Direct Sequence (DS) Spread Spectrum - DSSSFrequency Hopping (FH) Spread Spectrum - FHSS

Time Hopping (TH) Spread Spectrum - THSSHybrids Techniques

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


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Digital vs Analog performance criteria

Analog performanceFidelitySNR

% distortionExpected MSE b/w tx and rx waves

Digital

P E


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2. Periodic and Non-periodic Signals

A signal x( t ) is called periodic in time if there exists a constantT 0 > 0 such that

x(t) = x(t + T 0 ) for - < t < (1.2)t denotes time

T 0 is the period of x (t ).


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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 eachtime, kT, where

k is an integerT is a fixed time interval.


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4. Energy and Power Signals

The performance of a communication system depends on thereceived signal energy; higher energy signals are detected morereliably (with fewer errors) than are lower energy signals

x (t ) is classified as an energy signal if, and only if, it has nonzerobut finite energy (0 < E

x < ) for all time, where:

E x = lim x2(t) dt = x2(t) dt (1.7)

An energy signal has finite energy but zero average power.

Signals that are both deterministic and non-periodic are classifiedas energy signals

T -T/2

T/2

-


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Power is the rate at which energy is delivered

A signal is defined as a power signal if, and only if, it has finitebut nonzero power (0 < P x < ) for all time, where

P x = lim 1/T x2(t) dt (1.8)

Power signal has finite average power but infinite energy

As a general rule, periodic signals and random signals areclassified as power signals

4. Energy and Power Signals

T -T/2

T/2


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Spectral Density

The spectral density of a signal characterizes the distribution ofthe signals energy or power in the frequency domain.

This concept is particularly important when considering filtering in

communication systems while evaluating the signal and noise atthe filter output.

The energy spectral density (ESD) or the power spectral density(PSD) is used in the evaluation.


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1. Energy Spectral Density (ESD)

Energy spectral density describes the signal energy per unitbandwidth measured in joules/hertz.Represented as x(f), the squared magnitude spectrum

x(f) = |X(f)| 2 (1.14) According to Parsevals theorem, the energy of x(t):

E x = x2(t) dt = |X(f)| 2 df (1.13)Therefore:

E x = x (f) df (1.15)

The Energy spectral density is symmetrical in frequency aboutorigin and total energy of the signal x(t) can be expressed as:

E x = 2 x (f) df (1.16)

-

-

-

0


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2. Power Spectral Density (PSD)

The power spectral density (PSD) function G x(f ) of the periodicsignal x (t ) is a real, even, and nonnegative function of frequencythat gives the distribution of the power of x (t ) in the frequencydomain.PSD is represented as:

G x(f ) = |C n|2 (f nf 0) (1.18)Whereas the average power of a periodic signal x(t) isrepresented as:

P x = 1/T x2(t) dt = |C n |2 (1.17)

Using PSD, the average normalized power of a real-valuedsignal is represented as:

P x = G x(f ) df = 2 G x(f ) df (1.19)

-T0/2

T0/2

n=-

-

0

n=-


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Autocorrelation1. Autocorrelation of an Energy Signal

Autocorrelation is the counterpart of spectral densities in timedomainCorrelation is a matching process; autocorrelation refers to thematching of a signal with a delayed version of itself.

Autocorrelation function of a real-valued energy signal x (t ) isdefined as:

Rx( ) = x(t) x (t + ) dt for - < < (1.21)The autocorrelation function R x( ) provides a measure of howclosely the signal matches a copy of itself as the copy is shifted

units in time.Rx( ) is not a function of time; it is only a function of the timedifference between the waveform and its shifted copy.

-


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1. Autocorrelation of an Energy Signal

The autocorrelation function of a real-valued energy signal hasthe following properties:

Rx( ) = R x(- ) symmetrical in about zero

Rx( ) Rx(0) for all maximum value occurs at the originRx( ) x (f) autocorrelation and ESD form a

Fourier transform pair, as designatedby the double-headed arrows

Rx(0 ) = x2(t) dt value at the origin is equal to theenergy of the signal


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2. Autocorrelation of a Power Signal

Autocorrelation function of a real-valued power signal x (t ) isdefined as:

Rx( ) = lim 1/T x(t) x (t + ) dt for - < < (1.22)

When the power signal x (t ) is periodic with period T 0, theautocorrelation function can be expressed as

Rx(

) = lim 1/T 0 x(t) x (t +

) dt for -

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