Chapter 5 Signal Space Analysis CHAPTER 5 SIGNAL

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Chapter 5: Signal Space Analysis CHAPTER 5 SIGNAL SPACE ANALYSIS Digital Communication Systems 2012

Chapter 5: Signal Space Analysis CHAPTER 5 SIGNAL SPACE ANALYSIS Digital Communication Systems 2012 R. Sokullu 1/45

Chapter 5: Signal Space Analysis Outline • 5. 1 Introduction • 5. 2 Geometric

Chapter 5: Signal Space Analysis Outline • 5. 1 Introduction • 5. 2 Geometric Representation of Signals – Gram-Schmidt Orthogonalization Procedure • 5. 3 Conversion of the AWGN into a Vector Channel • 5. 4 Maximum Likelihood Decoding • 5. 5 Correlation Receiver • 5. 6 Probability of Error Digital Communication Systems 2012 R. Sokullu 2/45

Chapter 5: Signal Space Analysis Introduction – the Model • We consider the following

Chapter 5: Signal Space Analysis Introduction – the Model • We consider the following model of a generic transmission system (digital source): – A message source transmits 1 symbol every T sec – Symbols belong to an alphabet M (m 1, m 2, …m. M) • Binary – symbols are 0 s and 1 s • Quaternary PCM – symbols are 00, 01, 10, 11 Digital Communication Systems 2012 R. Sokullu 3/45

Chapter 5: Signal Space Analysis Transmitter Side • Symbol generation (message) is probabilistic, with

Chapter 5: Signal Space Analysis Transmitter Side • Symbol generation (message) is probabilistic, with a priori probabilities p 1, p 2, . . p. M. or • Symbols are equally likely • So, probability that symbol mi will be emitted: Digital Communication Systems 2012 R. Sokullu 4/45

Chapter 5: Signal Space Analysis • Transmitter takes the symbol (data) mi (digital message

Chapter 5: Signal Space Analysis • Transmitter takes the symbol (data) mi (digital message source output) and encodes it into a distinct signal si(t). • The signal si(t) occupies the whole slot T allotted to symbol mi. • si(t) is a real valued energy signal (? ? ? ) Digital Communication Systems 2012 R. Sokullu 5/45

Chapter 5: Signal Space Analysis • Transmitter takes the symbol (data) mi (digital message

Chapter 5: Signal Space Analysis • Transmitter takes the symbol (data) mi (digital message source output) and encodes it into a distinct signal si(t). • The signal si(t) occupies the whole slot T allotted to symbol mi. • si(t) is a real valued energy signal (signal with finite energy) Digital Communication Systems 2012 R. Sokullu 6/45

Chapter 5: Signal Space Analysis Channel Assumptions: • Linear, wide enough to accommodate the

Chapter 5: Signal Space Analysis Channel Assumptions: • Linear, wide enough to accommodate the signal si(t) with no or negligible distortion • Channel noise is w(t) is a zero-mean white Gaussian noise process – AWGN – additive noise – received signal may be expressed as: Digital Communication Systems 2012 R. Sokullu 7/45

Chapter 5: Signal Space Analysis Receiver Side • Observes the received signal x(t) for

Chapter 5: Signal Space Analysis Receiver Side • Observes the received signal x(t) for a duration of time T sec • Makes an estimate of the transmitted signal si(t) (eq. symbol mi). • Process is statistical – presence of noise – errors • So, receiver has to be designed for minimizing the average probability of error (Pe) What is this? Pe = cond. error probability given ith symbol was sent Symbol sent Digital Communication Systems 2012 R. Sokullu 8/45

Chapter 5: Signal Space Analysis Outline • 5. 1 Introduction • 5. 2 Geometric

Chapter 5: Signal Space Analysis Outline • 5. 1 Introduction • 5. 2 Geometric Representation of Signals – Gram-Schmidt Orthogonalization Procedure • 5. 3 Conversion of the AWGN into a Vector Channel • 5. 4 Maximum Likelihood Decoding • 5. 5 Correlation Receiver • 5. 6 Probability of Error Digital Communication Systems 2012 R. Sokullu 9/45

Chapter 5: Signal Space Analysis 5. 2. Geometric Representation of Signals • Objective: To

Chapter 5: Signal Space Analysis 5. 2. Geometric Representation of Signals • Objective: To represent any set of M energy signals {si(t)} as linear combinations of N orthogonal basis functions, where N ≤ M • Real value energy signals s 1(t), s 2(t), . . s. M(t), each of duration T sec Orthogonal basis function coefficient Energy signal Digital Communication Systems 2012 R. Sokullu 10/45

Chapter 5: Signal Space Analysis • Coefficients: • Real-valued basis functions: Digital Communication Systems

Chapter 5: Signal Space Analysis • Coefficients: • Real-valued basis functions: Digital Communication Systems 2012 R. Sokullu 11/45

Chapter 5: Signal Space Analysis • The set of coefficients can be viewed as

Chapter 5: Signal Space Analysis • The set of coefficients can be viewed as a Ndimensional vector, denoted by si • Bears a one-to-one relationship with the transmitted signal si(t) Digital Communication Systems 2012 R. Sokullu 12/45

Chapter 5: Signal Space Analysis Figure 5. 3 (a) Synthesizer for generating the signal

Chapter 5: Signal Space Analysis Figure 5. 3 (a) Synthesizer for generating the signal si(t). (b) Analyzer for generating the set of signal vectors si. Digital Communication Systems 2012 R. Sokullu 13/45

Chapter 5: Signal Space Analysis So, • Each signal in the set si(t) is

Chapter 5: Signal Space Analysis So, • Each signal in the set si(t) is completely determined by the vector of its coefficients Digital Communication Systems 2012 R. Sokullu 14/45

Chapter 5: Signal Space Analysis Finally, • The signal vector si concept can be

Chapter 5: Signal Space Analysis Finally, • The signal vector si concept can be extended to 2 D, 3 D etc. Ndimensional Euclidian space • Provides mathematical basis for the geometric representation of energy signals that is used in noise analysis • Allows definition of – Length of vectors (absolute value) – Angles between vectors – Squared value (inner product of si with itself) Matrix Transposition Digital Communication Systems 2012 R. Sokullu 15/45

Chapter 5: Signal Space Analysis Figure 5. 4 Illustrating the geometric representation of signals

Chapter 5: Signal Space Analysis Figure 5. 4 Illustrating the geometric representation of signals for the case when N 2 and M 3. (two dimensional space, three signals) Digital Communication Systems 2012 R. Sokullu 16/45

Chapter 5: Signal Space Analysis Also, What is the relation between the vector representation

Chapter 5: Signal Space Analysis Also, What is the relation between the vector representation of a signal and its energy value? • …start with the definition of average energy in a signal…(5. 10) • Where si(t) is as in (5. 5): Digital Communication Systems 2012 R. Sokullu 17/45

Chapter 5: Signal Space Analysis • After substitution: • After regrouping: • Φj(t) is

Chapter 5: Signal Space Analysis • After substitution: • After regrouping: • Φj(t) is orthogonal, so finally we have: The energy of a signal is equal to the squared length of its vector Digital Communication Systems 2012 R. Sokullu 18/45

Chapter 5: Signal Space Analysis Formulas for two signals • Assume we have a

Chapter 5: Signal Space Analysis Formulas for two signals • Assume we have a pair of signals: si(t) and sj(t), each represented by its vector, • Then: Inner product of the signals is equal to the inner product of their vector representations [0, T] Inner product is invariant to the selection of basis functions Digital Communication Systems 2012 R. Sokullu 19/45

Chapter 5: Signal Space Analysis Euclidian Distance • The Euclidean distance between two points

Chapter 5: Signal Space Analysis Euclidian Distance • The Euclidean distance between two points represented by vectors (signal vectors) is equal to ||si-sk|| and the squared value is given by: Digital Communication Systems 2012 R. Sokullu 20/45

Chapter 5: Signal Space Analysis Angle between two signals • The cosine of the

Chapter 5: Signal Space Analysis Angle between two signals • The cosine of the angle Θik between two signal vectors si and sk is equal to the inner product of these two vectors, divided by the product of their norms: • So the two signal vectors are orthogonal if their inner product si. Tsk is zero (cos Θik = 0) Digital Communication Systems 2012 R. Sokullu 21/45

Chapter 5: Signal Space Analysis Schwartz Inequality • Defined as: • accept without proof…

Chapter 5: Signal Space Analysis Schwartz Inequality • Defined as: • accept without proof… Digital Communication Systems 2012 R. Sokullu 22/45

Chapter 5: Signal Space Analysis Outline • 5. 1 Introduction • 5. 2 Geometric

Chapter 5: Signal Space Analysis Outline • 5. 1 Introduction • 5. 2 Geometric Representation of Signals – Gram-Schmidt Orthogonalization Procedure • 5. 3 Conversion of the AWGN into a Vector Channel • 5. 4 Maximum Likelihood Decoding • 5. 5 Correlation Receiver • 5. 6 Probability of Error Digital Communication Systems 2012 R. Sokullu 23/45

Chapter 5: Signal Space Analysis Gram-Schmidt Orthogonalization Procedure Assume a set of M energy

Chapter 5: Signal Space Analysis Gram-Schmidt Orthogonalization Procedure Assume a set of M energy signals denoted by s 1(t), s 2(t), . . , s. M(t). 1. Define the first basis function starting with s 1 as: (where E is the energy of the signal) (based on 5. 12) 2. Then express s 1(t) using the basis function and an energy related coefficient s 11 as: 3. Later using s 2 define the coefficient s 21 as: Digital Communication Systems 2012 R. Sokullu 24/45

Chapter 5: Signal Space Analysis 4. If we introduce the intermediate function g 2

Chapter 5: Signal Space Analysis 4. If we introduce the intermediate function g 2 as: Orthogonal to φ1(t) 5. We can define the second basis function φ2(t) as: 6. Which after substitution of g 2(t) using s 1(t) and s 2(t) it becomes: • Note that φ1(t) and φ2(t) are orthogonal that means: Digital Communication Systems 2012 R. Sokullu (Look at 5. 23) 25/45

Chapter 5: Signal Space Analysis And so on for N dimensional space…, • In

Chapter 5: Signal Space Analysis And so on for N dimensional space…, • In general a basis function can be defined using the following formula: • where the coefficients can be defined using: Digital Communication Systems 2012 R. Sokullu 26/45

Chapter 5: Signal Space Analysis Special case: • For the special case of i

Chapter 5: Signal Space Analysis Special case: • For the special case of i = 1 gi(t) reduces to si(t). General case: • Given a function gi(t) we can define a set of basis functions, which form an orthogonal set, as: Digital Communication Systems 2012 R. Sokullu 27/45

Chapter 5: Signal Space Analysis Outline • 5. 1 Introduction • 5. 2 Geometric

Chapter 5: Signal Space Analysis Outline • 5. 1 Introduction • 5. 2 Geometric Representation of Signals – Gram-Schmidt Orthogonalization Procedure • 5. 3 Conversion of the AWGN into a Vector Channel • 5. 4 Maximum Likelihood Decoding • 5. 5 Correlation Receiver • 5. 6 Probability of Error Digital Communication Systems 2012 R. Sokullu 28/45

Chapter 5: Signal Space Analysis Conversion of the Continuous AWGN Channel into a Vector

Chapter 5: Signal Space Analysis Conversion of the Continuous AWGN Channel into a Vector Channel • Suppose that the si(t) is not any signal, but specifically the signal at the receiver side, defined in accordance with an AWGN channel: • So the output of the correlator (Fig. 5. 3 b) can be defined as: Digital Communication Systems 2012 R. Sokullu 29/45

Chapter 5: Signal Space Analysis deterministic quantity contributed by the transmitted signal si(t) random

Chapter 5: Signal Space Analysis deterministic quantity contributed by the transmitted signal si(t) random quantity sample value of the variable Wi due to noise Digital Communication Systems 2012 R. Sokullu 30/45

Chapter 5: Signal Space Analysis Now, • Consider a random process X 1(t), with

Chapter 5: Signal Space Analysis Now, • Consider a random process X 1(t), with x 1(t), a sample function which is related to the received signal x(t) as follows: • Using 5. 28, 5. 29 and 5. 30 and the expansion 5. 5 we get: which means that the sample function x 1(t) depends only on the channel noise! Digital Communication Systems 2012 R. Sokullu 31/45

Chapter 5: Signal Space Analysis • The received signal can be expressed as: NOTE:

Chapter 5: Signal Space Analysis • The received signal can be expressed as: NOTE: This is an expansion similar to the one in 5. 5 but it is random, due to the additive noise. Digital Communication Systems 2012 R. Sokullu 32/45

Chapter 5: Signal Space Analysis Statistical Characterization • The received signal (output of the

Chapter 5: Signal Space Analysis Statistical Characterization • The received signal (output of the correlator of Fig. 5. 3 b) is a random signal. To describe it we need to use statistical methods – mean and variance. • The assumptions are: – X(t) denotes a random process, a sample function of which is represented by the received signal x(t). – Xj(t) denotes a random variable whose sample value is represented by the correlator output xj(t), j = 1, 2, …N. – We have assumed AWGN, so the noise is Gaussian, so X(t) is a Gaussian process and being a Gaussian RV, X j is described fully by its mean value and variance. Digital Communication Systems 2012 R. Sokullu 33/45

Chapter 5: Signal Space Analysis Mean Value • Let Wj, denote a random variable,

Chapter 5: Signal Space Analysis Mean Value • Let Wj, denote a random variable, represented by its sample value wj, produced by the jth correlator in response to the Gaussian noise component w(t). • So it has zero mean (by definition of the AWGN model) • …then the mean of Xj depends only on sij: Digital Communication Systems 2012 R. Sokullu 34/45

Chapter 5: Signal Space Analysis Variance • Starting from the definition, we substitute using

Chapter 5: Signal Space Analysis Variance • Starting from the definition, we substitute using 5. 29 and 5. 31 Autocorrelation function of the noise process Digital Communication Systems 2012 R. Sokullu 35/45

 • It can be expressed as: (because the noise is stationary and with

• It can be expressed as: (because the noise is stationary and with a constant power spectral density) Chapter 5: Signal Space Analysis • After substitution for the variance we get: • And since φj(t) has unit energy for the variance we finally have: • Correlator outputs, denoted by Xj have variance equal to the power spectral density N 0/2 of the noise process W(t). Digital Communication Systems 2012 R. Sokullu 36/45

Chapter 5: Signal Space Analysis Properties (without proof) • Xj are mutually uncorrelated •

Chapter 5: Signal Space Analysis Properties (without proof) • Xj are mutually uncorrelated • Xj are statistically independent (follows from above because Xj are Gaussian) • and for a memoryless channel the following equation is true: Digital Communication Systems 2012 R. Sokullu 37/45

Chapter 5: Signal Space Analysis • Define (construct) a vector X of N random

Chapter 5: Signal Space Analysis • Define (construct) a vector X of N random variables, X 1, X 2, …XN, whose elements are independent Gaussian RV with mean values sij, (output of the correlator, deterministic part of the signal defined by the signal transmitted) and variance equal to N 0/2 (output of the correlator, random part, calculated noise added by the channel). • then the X 1, X 2, …XN , elements of X are statistically independent. • So, we can express the conditional probability of X, given si(t) (correspondingly symbol mi) as a product of the conditional density functions (fx) of its individual elements fxj. NOTE: This is equal to finding an expression of the probability of a received symbol given a specific symbol was sent, assuming a memoryless channel Digital Communication Systems 2012 R. Sokullu 38/45

Chapter 5: Signal Space Analysis • …that is: • where, the vector x and

Chapter 5: Signal Space Analysis • …that is: • where, the vector x and the scalar xj, are sample values of the random vector X and the random variable Xj. Digital Communication Systems 2012 R. Sokullu 39/45

Chapter 5: Signal Space Analysis Vector x and scalar xj are sample values of

Chapter 5: Signal Space Analysis Vector x and scalar xj are sample values of the random vector X and the random variable Xj Vector x is called observation vector Scalar xj is called observable element Digital Communication Systems 2012 R. Sokullu 40/45

Chapter 5: Signal Space Analysis • Since, each Xj is Gaussian with mean sj

Chapter 5: Signal Space Analysis • Since, each Xj is Gaussian with mean sj and variance N 0/2 • we can substitute in 5. 44 to get 5. 46: Digital Communication Systems 2012 R. Sokullu 41/45

Chapter 5: Signal Space Analysis • If we go back to the formulation of

Chapter 5: Signal Space Analysis • If we go back to the formulation of the received signal through a AWGN channel 5. 34 The vector that we have constructed fully defines this part Only projections of the noise onto the basis functions of the signal set {si(t)Mi=1 affect the significant statistics of the detection problem Digital Communication Systems 2012 R. Sokullu 42/45

Chapter 5: Signal Space Analysis Finally, • The AWGN channel, is equivalent to an

Chapter 5: Signal Space Analysis Finally, • The AWGN channel, is equivalent to an Ndimensional vector channel, described by the observation vector Digital Communication Systems 2012 R. Sokullu 43/45

Chapter 5: Signal Space Analysis Outline • 5. 1 Introduction • 5. 2 Geometric

Chapter 5: Signal Space Analysis Outline • 5. 1 Introduction • 5. 2 Geometric Representation of Signals – Gram-Schmidt Orthogonalization Procedure • 5. 3 Conversion of the AWGN into a Vector Channel • 5. 4 Maximum Likelihood Decoding • 5. 5 Correlation Receiver • 5. 6 Probability of Error Digital Communication Systems 2012 R. Sokullu 44/45

Chapter 5: Signal Space Analysis Maximum Likelihood Decoding • to be continued…. Digital Communication

Chapter 5: Signal Space Analysis Maximum Likelihood Decoding • to be continued…. Digital Communication Systems 2012 R. Sokullu 45/45