Introduction - Basic Concepts

Size: px

Start display at page:

Download "Introduction - Basic Concepts"

Delphia Carpenter
5 years ago
Views:

1 IMPERIAL COLLEGE LONDON, DEPARTMENT of ELECTRICAL and ELECTRONIC ENGINEERING. COMPACT LECTURE NOTES on COMMUNICATION THEORY. Prof Athanassios Manikas, Autumn 2001 Introduction - Basic Concepts Outline: ì Digital Comm. Systems: General Block Structures ì Classification of Signals. ìauto and Cross Correlation Functions and PSD( 0). ì FT and Woodwords Notation. ì Additive White Gaussian Noise and its Modelling. ì Tail Function and its Graph. ì Some Useful Appendices.

3 1. GENERAL BLOCK STRUCTURE OF A DIGITAL COMMUNICATION SYSTEM H( f) ^ ^ ^ ^ ^ ^ E303/ISE3.2E: Introduction - Basic Concepts 2 The points A may be considered as the input of a Digital Communication System where messages consist of sequences of symbols selected from an alphabet e.g. levels of a quantizer or telegraph letters, numbers and punctuations. The objective of a Source Encoder (or data compressor) is to represent the message-symbols arriving at point A by as few digits as possible. Thus, each level (symbol) at point A is mapped, by the Source Encoder, to a unique codeword of 1s and 0s and, at point B, we get a sequence of binary digits. There are two ways to reduce the channel noise/interference effects 1. to introduce deliberately some redundancy in the sequence at point B and this is what a Discrete Channel Encoder does. This redundancy aids the receiver in decoding the desired sequence by detecting and many times correcting errors indroduced by the channel; repeat each bit of B 7 times, e.g. f or, a more sophisticated approach, use a mapper: k-bits at point B È n-bits at point B1 Ú k k n À is the rate of code or code-rate œ Vc œ n Ý : measures the amount of redundancy introduced to the data by the Vc NoteÛ channel encoder. Note also that BANDWIDTH= Å by Vc Ý If limited BANDWIDTH, then there is a need for CLEVER Ü REDUNDANCY without need to increase the BANDWIDTH. 2. to increase Transmitter's power - point T Ðoften very expensive therefore better to trade transmitter's power for channel BANDWIDTHÑ at point T : =Ð>Ñ waveform. cs The digital modulator takes cs -bits at a time at some uniform rate r cs and transmits one of Q =2 distinct waveforms = Ðt Ñ,....,sQ ÐÑ t i.e. we have an Q-ary communication system. A new waveform corresponding to a new cs-bit sequence is transmitted every T cs seconds. If cs= we have one bit at a time œ 0 È= i.e. a binary communication system 1 È= at point T ^ : noisy waveform <Ð>Ñ œ =Ð>Ñ 8Ð>Ñ. The transmitted waveform =Ð>Ñ, affected by the channel, is received at point T ^ at point B2 ^ : a binary sequence. based on the received signal <Ð>Ñ the digital demodulator has to decide which of the Q waveforms = 3Ð>Ñ has been transmitted in any given time interval X -= at point B ^ : a binary sequence. The channel decoder attempts to reconstruct the sequence at B from: ˆ the knowledge of the code used in the channel encoder, and ˆ the redundancy contained in the received data at point A ^ : message. The source decoder processes the sequence received from the output of the channel decoder and, from the knowledge of the source encoding method used, attempts to reconstruct the signal of the information source. message at point A ^ message at point A Ðdue to channel decoding errors and distortion introduced by the quantizerñ E303/ISE3.2E: Introduction - Basic Concepts 3

4 2. A SIMPLIFIED BLOCK STRUCTURE OF A DIGITAL COMMUNICATION SYSTEM A simplified and general block structure of a Digital Communication System is shown in the following page and it is common practice its quality to be expressed in terms of the accuracy with which the binary digits delivered at the output of the detector/rx (point B^ ) represent the binary digits that were fed into the digital modulator/tx (point B). It is generally taken that it is the fraction of the binary digits that are delivered back in error that is a measure of the quality of the communication system. This fraction, or rate, is referred to as the probablity of a bit error : /, or, Bit-Error-Rate BER (point B^ ). E303/ISE3.2E: Introduction - Basic Concepts 4 Discrete Information Source B Tx T H( f) Discrete Information Sink B^ Rx T^ N.B.: Quality is measured as the Bit Error Rate (BER) E303/ISE3.2E: Introduction - Basic Concepts 5

5 INTERNET SERVICE PROVIDER B^ MODEM T B MODEM T^C L LOCAL EXCHANGE (or a street-cabinet) H A N N E POTS Network (Narrowband Network) To a POTS line Card LOCAL EXCHANGE ( or a street-cabinet) LOCAL EXCHANGE ( or a street-cabinet) To a POTS line Card E303/ISE3.2E: Introduction - Basic Concepts 6 3. Digital Transmission of Analogue Signals Continuous Information Source A ADC B Tx T H( f) Continuous Information Sink ^A DAC ^B Rx T^ N.B.: Quality is measured as the SNR (analogue signals degrade as noise level increases) E303/ISE3.2E: Introduction - Basic Concepts 7

6 INTERNET SERVICE PROVIDER MODEM A MODEM A A To a POTS line Card LOCAL EXCHANGE (or a street-cabinet) POTS Network (Narrowband Network) A^ B^ T^ B CHANNEL T LOCAL EXCHANGE ( or a street-cabinet) To a POTS line Card A^ LOCAL EXCHANGE ( or a street-cabinet) Note that like Wireline and fiber communications wireless communications are also fully digital. E303/ISE3.2E: Introduction - Basic Concepts 8 It is clear from the previous discussion that signals (representing bits) propagate through the networks. Therefore the following sections are concerned with the main properties and parameters of communication signals. E303/ISE3.2E: Introduction - Basic Concepts 9

4. Communication Signals Time Domain (TD) TRANSFORMATION Frequency Domain (FD) z-domain s-domain etc Frequency Domain (Spectrum): very important in Communications Fourier Transform TD FD signal: 1Ð>Ñ

7 4. Communication Signals Time Domain (TD) TRANSFORMATION Frequency Domain (FD) z-domain s-domain etc Frequency Domain (Spectrum): very important in Communications Fourier Transform TD FD signal: 1Ð>Ñ KÐ0Ñ system: 2Ð>Ñ=impulse response LÐ0Ñ =transfer function E303/ISE3.2E: Introduction - Basic Concepts 10 Classification of Signals (according to their description) Deterministic Signals describable by mathematical function Random Signals these are unpredictable; cannot be expressed as a function can be expressed probabilistically 3cos(2π1000 t) N.B. - Random Signals: very important in Communications E303/ISE3.2E: Introduction - Basic Concepts 11

!!! E303/ISE3.2E: Introduction - Basic Concepts 12 Classification of Signals (according to their energy) Energy= ' signal Þ.

8 Classification of Signals (according to their periodicity) Periodic Z96>= non-periodic XÐperiod) 0 J! œ X! N.B.:according to Fourier Series Theorem any periodic waveform can be represented by a sum of sinusoidals having frequencies F, 2F, 3F, etc.!!! E303/ISE3.2E: Introduction - Basic Concepts 12 Classification of Signals (according to their energy) Energy= ' signal Þ.> Energy Signals Power Signals ( Energy _ ) ( Energy œ_ ) T N.B.: Signals of finite duration are Energy Signals Periodic Signals are Power Signals E303/ISE3.2E: Introduction - Basic Concepts 3

2E: Introduction - Basic Concepts 4 Classification of Signals (according to their spectrum) LOW-PASS BAND-PASS

9 The figures below show the following parameters: peak (Volts) or peak-to-peak Energy (J) (or Power (W)) rms (Volts) Crest Factor CF E303/ISE3.2E: Introduction - Basic Concepts 4 Classification of Signals (according to their spectrum) LOW-PASS BAND-PASS Other types (or baseband) signals signals e.g. all-pass high-pass, etc -J 0 J 1 1 F F Bandwidth= J Bandwidth = F 1 where J 1 œ max. frequency of 1Ð>Ñ E303/ISE3.2E: Introduction - Basic Concepts 5

10 OPERATIONS TD Fourier Transform FD + + denotes Š convolution E303/ISE3.2E: Introduction - Basic Concepts 6 5. IMPORTANT SPECTRUM SHAPES Fourier Transform TD signal: 1Ð>Ñ 1. sinewaves/carriers FD KÐ0Ñ KÐ0Ñl ESD( 0) 9< PSD( 0Ñ N.B.: periodic waveform in TD Ê discrete spectrum in FD E303/ISE3.2E: Introduction - Basic Concepts 7

11 2. finite duration (i.e. Energy signals) E303/ISE3.2E: Introduction - Basic Concepts V DC œ t f 4. message signals 1Ð>Ñ f - J 0 J transmitted signals =Ð>Ñ ((or, received signals <Ð>ÑÑ F F f E303/ISE3.2E: Introduction - Basic Concepts 9

12 6. An important example in the TD E303/ISE3.2E: Introduction - Basic Concepts 10 FD representation of the previous example: E303/ISE3.2E: Introduction - Basic Concepts 11

13 6. Bandwidth of a signal the range of the significant frequency components in a signal waveform Examples of message signals (baseband signals) and their bandwidth: television signal bandwidth 5.5MHz speech signal bandwidth 4KHz audio signal bandwidth 8kHz to 20kHz Examples of transmitted signals (bandpass signals) and their bandwidth: to be discussed latter on Note that there are various definitions of bandwidth, e.g. 3dB bandwidth, null-to-null bandwidth, Nyquist (minimum) bandwidth E303/ISE3.2E: Introduction - Basic Concepts 12 F 0Hz--// 0 J- X J- J- X F F œ null-to-null bandwidth œ X where X œ signal's duration F œ $ db bandwidth (Energy Signal) F œ Nyquist Bandwidth œ $ X E303/ISE3.2E: Introduction - Basic Concepts 13

14 7. REDUNDANCY The degree of Redundancy autocorrelation function. in a signal is provided by its For instance the autocorrelation function of a signal 1Ð>Ñ is V Ð Ñ 11 7 Accumulator 1Ð>Ñ V Ð Ñ 11 7 if =fixed then a number NB: œ 7 V11Ð 7 Ñ œ if 7=variable (i.e. a7) then V Ð7Ñ œ a function of 7 11 E303/ISE3.2E: Introduction - Basic Concepts 14 8.SIMILARITY The degree of Similarity between two signals is given by their cross-correlation function. For instance the cross-correlation function between two signals 1Ð>Ñand 1Ð>Ñis V Ð Ñ Ð>Ñ Ac c umulator V 11 Ð Ñ 7 1Ð>Ñ 2 if =fixed then a number N.B.: œ 7 V11Ð 7 Ñ œ if 7=variable (i.e. a7) then V Ð7Ñ œ a function of 7 11 E303/ISE3.2E: Introduction - Basic Concepts 15

15 9. SUMMARY Parameters for a signal 1Ð>Ñ: I 1 œenergy (J) T1 œ Power (W) ESD 1Ð0Ñ œ Energy Spectral Density (J/Hz) PSD 1Ð0Ñ œ Power Spectral Density (W/Hz) V11Ð7Ñ œautocorrelation function N.B.: The above are normalised parameters (1 Ohm Resistor) other parameters B œ Bandwidth J1 œmax frequ. of 1Ð>Ñ CF œ Crest Factor rms (Volts) peak (Volts) E303/ISE3.2E: Introduction - Basic Concepts More On Transformations E303/ISE3.2E: Introduction - Basic Concepts 17

16 11. WOODWARD's Notation The evaluation of FT, that is FTe1Ð>Ñf œ Gf Ð Ñ = ' j 1ft gt.e Ð Ñ dt FT ekð0ñf œ gt Ð Ñ œ ' +j 1ft G Ð0Ñ.e df involves integrating the product of a function and a complex exponential - which can be difficult; so tables of useful transformations are frequently used. However, the use of tables is greatly simplified by employing Woodward's notation for certain commonly occurring situations. main advantage of using Woodward's notation: allows periodic time/frequency functions to be handled with FT rather than Fourier Series E303/ISE3.2E: Introduction - Basic Concepts rect{ t} if t ± œ! 9>2/<A3=/ ± 2. sincöt sin Ð1 t 1t Ñ 3. A { t} œ 1 t if 0 Ÿ t Ÿ 1 t if Ÿt Ÿ0 E303/ISE3.2E: Introduction - Basic Concepts 19

! _ 8= _ 4. rep T Ö gt ÐÑ gt Ð 8TÑ where 8=...,-2,-1,0,+1,+2,.

$ Ðt 8TÑ Å also known as sampling function E303/ISE3.

shifting Ðsee for instance the following tableñ shifting:

rect t T t š 7 T t t-7 Ÿ t 7 Ÿ Ÿ T Ÿ Ÿ T Ÿ i.e. 7 Ÿ t Ÿ 7+ i.e. Ÿ t Ÿ i.

17 ! _ 8= _ 4. rep T Ö gt ÐÑ gt Ð 8TÑ where 8=...,-2,-1,0,+1,+2,...! _ 8= _ 5. comb T Ö gt ÐÑ gð8t Ñ. $ Ðt 8TÑ Å also known as sampling function E303/ISE3.2E: Introduction - Basic Concepts 20 N.B.: we can generate any desired rect function by scaling and shifting Ðsee for instance the following tableñ shifting: scaling: shifting+scaling: gt= ÐÑ rectðt 7Ñ gt= ÐÑ rectš gt= ÐÑ rect t T t š 7 T t t-7 Ÿ t 7 Ÿ Ÿ T Ÿ Ÿ T Ÿ i.e. 7 Ÿ t Ÿ 7+ i.e. Ÿ t Ÿ i.e. 7 Ÿ t Ÿ 7+ T T T T effects of temporal scaling: as FT becomes narrower and amplitude rises -function at T Ä_ Ê Ê $ 0 frequency when T Ä_ E303/ISE3.2E: Introduction - Basic Concepts 21

12. Additive White Gaussian Noise (AWGN) Í Types

18 12. Additive White Gaussian Noise (AWGN) Í Types of Channel Signals =Ð>Ñ œ bandpass <Ð>Ñ œ bandpass 83Ð>Ñ œ allpass 8Ð>Ñ œ bandpass SNR in at point T^ œ T desired T at point noise T^ E303/ISE3.2E: Introduction - Basic Concepts 22 Comments on 8 Ð>Ñ 3 E303/ISE3.2E: Introduction - Basic Concepts 23

19 Comments on 8Ð>Ñ E303/ISE3.2E: Introduction - Basic Concepts 24 E303/ISE3.2E: Introduction - Basic Concepts 25

20 13. Q-function or Tail(T)-function Consider a random signal BÐ>Ñ with an amplitude probability density function pdf B(x). Then the probability that the amplitude of BÐ>Ñ is greater than 3 Volts (say) is given as follows: _ PrÐBÐ>Ñ $Z Ñ œ pdf (x) Þ. x $Z B If pdf B(x) = Gaussian of mean. B and standard deviation 5B (notation used: pdf B(x) œ N(. B, 5B)), then the above area is defined as the Q-function (or Tail-function) _ i.e.: pdf B(x) Þ. x? B œ Tš $Z l$. l 5 Note: if. =0 and 5 =1 then pdf (x) Þ. x? œ Te3f B B B $Z B _ E303/ISE3.2E: Introduction - Basic Concepts 26

21 14. Tail Function Graph Te f The graph below shows the Tail function B which represents the area from B to _ of the Gaussian probability density function N(0,1), i.e. Te f ' B œ È. expš.c 1 B _ C p B TeBf TeBf p B B Note that if B 'Þ& then B may be approximated by B Þexpš Tef Tef È 1 ÞB E303/ISE3.2E: Introduction - Basic Concepts 27

22 15. Appendices Appendix-1: Expectation, Moments and Stationarity EXPECTATION Consider a r.s. ZÐt Ñ characterized by its amplitude pdf i.e. pdf. The expected or mean value of VÐtÑ is defined as follows: _ X š Vt Ð Ñ ' v. pdf Ðv.dv Ñ. _ V Consider a function of the r.s. VÐt Ñ i.e. f{ VÐtÑ} and the pdf of VÐt Ñ i.e. pdfv. The expected or mean value of f{ VÐtÑ} is defined as follows: _ Xš f{ Vt Ð Ñ} ' f{ v}. pdf Ðv.dv Ñ _ V V V PROPERTIES OF EXPECTATION if V 0 then Xš V 0 Xš c.v =c. Xš V Xš V +V = Xš V + Xš V Xš 1 =1; Xš c =c If Ðt,Vt Ñ Ð Ñ=statistical independent then X š f{ Vt Ð Ñ}.f{ ÐtÑ} = Xš f{ Vt Ð Ñ}. X š f{ ÐtÑ} MOMENTS of a RANDOM SIGNAL Consider a r.s. VÐÑ t: definitions: st st X š Vt Ð Ñ.V=. V is called 1 moment X š Vt Ð Ñ. V 1 central moment of Vt Ð Ñ of Vt Ð Ñ st nd X š Vt Ð Ñ. V is called 2 moment X š ŠVt Ð Ñ. V 2 central moment of Vt Ð Ñ of Vt Ð Ñ k th th X š Vt Ð Ñ. kv is called k moment X š ŠVt Ð Ñ. V k central moment of Vt Ð Ñ of Vt Ð Ñ k nd the 2 central moment is called variance of the r.s. Vt ÐÑ An important property of the variance of the r.s. i.e. 5 = X š ÐVÐt Ñ. Ñ = ' Ðv. Ñ. pdf V Ðv Ñ.dv V V V V Vt: ÐÑ 2 V 5 =. V. Ðproof for youñ STATIONARITY: a r.s. is said to be stationary in the STRICT-SENSE iff all its averages are time invariant. i.e. ZÐtÑ and ZÐt+ 7Ñ have the same statistics a7. a r.s. is said to be stationary in the WIDE-SENSE iff the mean and the autocorrelation function are time invariant i.e. X š ZÐt Ñ =. V Xš ZÐtÑ. ZÐt+ 7Ñ =R Ð7Ñ VV i.e a function only of the time difference 7 This is a less demanding form of stationarity Clearly, a STRICT-SENSE stationary process is also WIDE-SENSE stationary but the converse is not guaranteed. E303/ISE3.2E: Introduction - Basic Concepts 28

23 Appendix-2: pdf's with Extensive Applications in Communications: UNIFORM DISTRIBUTION If the r.v. is equal likely to take on any value within a given range (e.g. a to +a) BINOMIAL DISTRIBUTION Consider an experiment having only 2 possible outcomes A and B(e.g. 0 and 1), mutually exclusive. That means that a r.v. takes only 2 possible values, the first (A, say) with probability p and the second (B) with probability :. Repeat the experiment n-times and take a r.v.. The Probability that the event A, say, will happen exactly B-times in 8-trials will be given by: 8 B pdf ÐBÑ= Š p. Ð p) B 8 8x B BxÐ8 BÑx 8 B where Š = =binomial coef B-times N.B.: B-times mean œ8: variance œ 8:Ð :Ñ 8-trials POISSON DISTRIBUTION In the Binomial Distribution if 8 Ä_, while : Ä0, then the Binomial is approximated by the Poisson Distribution given by - - pdf ÐBÑ= e. B Bx where - œ8: N.B.: In practice the approximation is used if n8 50 while 8: 5 GEOMETRIC DISTRIBUTION Consider an experiment having only 2 possible outcomes (A, B), say, mutually exclusive. That means that the random variable takes only 2 values and Pr( =A)= p Pr( =B)=1 p Repeat the experiment up to the time you obtain an A (say), and take a r.v.. The probability that the event A will obtained after B B's is given by: B pdf ÐBÑ œ Ð p Ñ.p B-times E303/ISE3.2E: Introduction - Basic Concepts 29

24 GAUSSIAN DISTRIBUTION pdf ÐBÑ œ É 15 Ð. Ñ x 5.expŠ pdf of thermal noise= Gaussian Central Limit Theorem: The pdf of a linear combination of n statistically independent and identically distributed rvs (with finite mean and variance) tends to a Gaussian pdf as 8Ä_. n i.e. if!! 3. 3 = where =rvs &! =coefficients 3=1 3 3 then as n Ä_ ( or a large number) Ê pdf =Gaussian N.B.: if pdf =Gaussian for 0 Ÿ i Ÿ n then pdf =Gaussian for any n 3 CHI-SQUARE DISTRIBUTION Consider that you have a r.v. with pdf =gaussian N(0, 5 ) if is another r.v. such that = B then pdf ÐBÑ œ. expš B 0 É 1B 5 5 N.B.: if = and pdf =gaussian pdf =chi-square (see a mathematical handbook) n 3 Ê RAYLEIGH DISTRIBUTION Consider that you have two r.v's, with pdf =gaussian N(0, 5 ) (i.e these 2 r.v's are 3 Gaussian r.v.'s) If is another r.v. such that = + then pdf =Rayleigh È i.e. pdf ÐBÑ œ B. expš B B! e.g. pdf 5 5 envelope of Gaussian noise = Rayleigh Appendix-3:Four Useful Combinatorial-Analysis Properties 1. Factorial: 8x $ ÞÞÞÞ Ð8 Ñ Ð8 Ñ 8 with 0x N.B.: if 8=large then 8x œ È / (Stirling's approximation) 2. FUNDAMENTAL CONCEPT: If an even can happen in any one of 8 ways and if, when this has occurred, another event can happen in any one of 8 ways, then the No. of ways in which both events can happen in the specified order is 8 n. 3. PERMUTATIONS: A permutation of 8 different objects taken 3 objects at a time is an arrangment of 3 out of 8 objects with attention given to the order of the arrangment. 8x No. of permutations of 8objects taken 3at a time = Ð8 3Ñx 4. COMBINATIONS: A combination of 8 different objects taken 3 objects at a time is an selection of 3 out of 8 objects with no attention given to the order of the arrangment. No. of different permutations of objects taken at a time = 8 3 8x 3x Ð8 3Ñx E303/ISE3.2E: Introduction - Basic Concepts 30

25 Appendix-4:General Block Structure of an Analogue Communication System

26 FOURIER TRANSFORMS - TABLES DESCRIPTION FUNCTION TRANSFORM 1 Definition gt ÐÑ _ Gf ÐÑ= ' j 1ft gt.e ÐÑ dt _ DESCRIPTION FUNCTION TRANSFORM if 14 Rectangular function rect{ t} t ± sin 1f œ sincðf Ñ=! 9>2/<A3=/ 1f ± 2 Scaling 3 Time shift 4 Frequency shift 5 Complex conjugate 6 Temporal derivative 7 Spectral derivative 8 Reciprocity 9 Linearity 10 Multiplication 11 Convolution 12 Delta function 13 Constant t T g Ð Ñ T. GÐfTÑ gt Ð TÑ Gf Ð Ñ. e j 1Ft gðñ t. e GÐf FÑ * * g ÐÑ t G Ð fñ 8 d dt j 1fT. gðñ t Ðj f Ñ. GÐÑ f 8 1 Ð j2 t Ñ.gÐÑ t. GÐÑ f 1 8 d df Gt ÐÑ gð fñ A. gðñ t B. hðñ t A. GÐÑ f B. HÐÑ f gðñ t. hðñ t GÐÑ f * HÐÑ f gt ÐÑ* ht ÐÑ Gf ÐÑ. Hf ÐÑ $ ÐÑ t $ ÐfÑ Sinc function 16 Unit step function 17 Signum function 18 Decaying exponential Ðtwo-sidedÑ 19 Decaying exponential Ðone-sidedÑ sincðñ t rectðñ f ß >! ut ÐÑ= œ ÐÑ f!ß >! $ 1 jf ß >! j sgnðñ t = œ ß >! 1f e e t t.uðñ t + Ð 1fÑ j 1f + Ð 1fÑ 1t Gaussian function e e 21 Lambda function 1 t if 0 t Ÿ 1 A{ t} œ Ÿ 1 t if Ÿt Ÿ0 sinc ÐfÑ 22 Repeated function T T T rep egðñ t f = gðñ t * rep e$ ÐÑ t f. comb egðñ f f 23 Sampled function comb egt ÐÑf = gt. ÐÑrep e ÐÑ t f. rep egf ÐÑf T T $ T T T

Digital Modulators & Line Codes

Digital Modulators & Line Codes Professor A. Manikas Imperial College London EE303 - Communication Systems An Overview of Fundamental Prof. A. Manikas (Imperial College) EE303: Dig. Mod. and Line Codes