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Chapter 5 Signal-Space Analysis Signal space analysis provides a mathematically elegant and highly insightful tool for the study of data transmission. 5.1 Introduction Statistical model for a genetic digital communication system Message source: A priori probabilities for information source pi P(mi ) for i 1,2,..., M Transmitter: The transmitter takes the message source output mi and (en-)codes it into a distinct signal si(t) suitable for transmission over the channel. So: pi P(mi ) P( si (t )) for i 1,2,..., M © Po-Ning [email protected] Chapter 5-2 5.1 Introduction si(t) must be a real-valued energy signal (i.e., a signal with finite energy) with duration T. Ei 0 si2 (t )dt . T Channel: The channel is assumed (in this text) linear and with a bandwidth wide enough to pass si(t) with no distortion. Zero-mean additive white Gaussian noise (AWGN) is also assumed (to facilitate the analysis). © Po-Ning [email protected] Chapter 5-3 5.1 A Mathematical Model We can then simplify the previous system block diagram to: Upon the receipt of x(t) for a duration of T, the receiver makes the best estimate of mi. (We haven’t defined what the best is.) © Po-Ning [email protected] Chapter 5-4 5.1 Criterion for the “Best” Decision Best = Minimization of the average probability of symbol error. M Pe pi P(mˆ mi | mi ) i 1 This is optimum in the minimum probability of error sense. Based on this criterion, we can begin to design the receiver that can give the best decision. © Po-Ning [email protected] Chapter 5-5 5.2 Geometric Representation of Signals Signal space concept Vectorization of the (discrete or continuous) signals removes the redundancy in the signals, and provides a compact representation for them. Determination of the vectorization basis Gram-Schmidt orthogonalization procedure © Po-Ning [email protected] Chapter 5-6 5.2 Gram-Schmidt Orthogonalization Procedure Given v1 , v2 ,, vk , how to find an orthonorma l basis for them? v (step i ) Let u1 1 . || v1 || ' u (step ii ) u2' v2 ( v2 u1 )u1. Set u2 2' . || u2 || (step iii ) For i 3,4,..., Let ui' vi ( vi ui 1 )ui 1 ( vi ui 2 )ui 2 ( vi u1 )u1. ' u Set ui i' . || ui || (step iv ) Then u1 , u2 ,, uk form an orthonorma l basis . © Po-Ning [email protected] Chapter 5-7 Properties : (i) vector : v ( v1, ,vn ) n (ii) inner product : v1 v2 v1i v2i i 1 (iii) orthogonal , if inter product 0. (iv) norm : || v || v12 vn2 ( v) orthosnorm al, if inner product 0, and indivual norm 1. (vi) linearly independen t, if none can be represente d as a linear combinatio n of others. (vii) triangle inequality : || v1 v2 |||| v1 || || v2 || . © Po-Ning [email protected] Chapter 5-8 ( viii) Cauchy Schwartz inequality : | v1 v2 ||| v1 || || v2 || with equality holds, if v1 av2 . ( xi) norm square : 2 2 2 || v1 v2 || || v1 || || v2 || 2v1 v2 . ( x) Pythagore an property : If orthogonal , 2 2 2 || v1 v2 || || v1 || || v2 || . ( xi) Matrix tr ansformati on w.r.t. matrix A : v1 Av2 . ( xii) eigenvalue s w.r.t. matrix A : solution of det A - I 0. ( xiii) eigenvecto rs w.r.t. eigenvalue : solution v of Av v . © Po-Ning [email protected] Chapter 5-9 5.2 Signal Space Concept for Continuous Functions Properties for continuous functions (i) (complex v alued) signal : z (t ) (ii) inner product : z (t ), zˆ(t ) a z (t ) zˆ* (t )dt. b (iii) orthogonal , if inter product 0. (iv) norm : z (t ) 2 | z ( t ) | dt a b ( v) orthonorma l, if inner product 0, and indivual norm 1. (vi) linearly independen t vectors, if none can be represente d as a linear combinatio n of others. (vii) triangle inequality : z (t ) zˆ(t ) || z (t ) || || zˆ(t ) || . © Po-Ning [email protected] Chapter 5-10 ( viii) Cauchy Schwartz inequality : z (t ), zˆ(t ) || z (t ) || || zˆ(t ) || with equality holds, if z (t ) a zˆ(t ). (a is a complex number.) ( xi) norm square : || z (t ) zˆ(t ) ||2 || z (t ) ||2 || zˆ(t ) ||2 z (t ), zˆ(t ) zˆ(t ), z (t ) . ( x) Pythagore an property : If orthogonal , || z (t ) zˆ(t ) ||2 || z (t ) ||2 || zˆ(t ) ||2 . ( xi) Transforma tion w.r.t. a function C (t,τ ) : n (Recall v1 j a jn v2 i .) zˆ(t ) a C (t , ) z ( )d b i 1 (xii.a) eigenvalue s and eigenfunct ions w.r.t. a continuous function C (t, ) : solutions k and { k (t )} k 1 of k k (t ) a C (t , ) k ( )d b and C (t , ) can be represente d as C (t , ) k (t ) k k ( ). k 1 © Po-Ning [email protected] Chapter 5-11 (xii.b) Give a determinis tic function {s(t ), t [0, T )} and a set of orthonorma l basis { k (t )}1k that can span s(t ). Then s(t ) ak k (t ), 0 t T , k 0 where ak 0 s(t ) k (t )dt. T (xii.c) If orthonorma l set { k (t )}1k K does not span the space, then K it is possible that sˆ(t ) ak k (t ) s(t ) for all choices of {ak }1k K . k 0 © Po-Ning [email protected] Chapter 5-12 Problem : How to minimize the “energy” of e(t ) s(t ) sˆ(t ) ? To select the coefficien ts {ak } that minimize 2 e (t )dt, K 2 2 [ s ( t ) a ( t ) ] dt k k e (t )dt k 1 a j a j K 2 s(t ) ak k (t ) j (t )dt k 1 K 2 s(t ) j (t )dt 2 ak k (t ) j (t )dt k 1 2 s(t ) j (t )dt 2a j 0. a j s(t ) j (t )dt. © Po-Ning [email protected] Chapter 5-13 s (t ) a2 s(t ), 2 (t ) 2 (t ) a1 s(t ), 1 (t ) e(t ) sˆ(t ) 1 (t ) Hence, e(t ), sˆ(t ) e(t ) sˆ(t )dt 0 Interpretation aj is the projection of s(t) onto the Yj(t)-axis. (aj)2 is the energy-projection of s(t) onto the Yj(t)-axis. © Po-Ning [email protected] Chapter 5-14 s (t ) a1 s(t ), 1 (t ) e(t ) sˆ(t ) a2 s(t ), 2 (t ) 2 (t ) 1 (t ) 2 e (t )dt e(t )[ s(t ) sˆ(t )]dt e(t ) s(t )dt e(t ) sˆ(t )dt e(t ) s(t )dt 0 e(t ) s(t )dt [ s(t ) sˆ(t )]s(t )dt K s (t )dt ak k (t ) s(t )dt k 1 2 K s (t )dt ak s(t ) k (t )dt 2 k 1 K s (t )dt a 2 k 1 © Po-Ning [email protected] 2 k Notably, K 2 ˆ s ( t ) dt a k. 2 k 1 Chapter 5-15 5.2 Signal Space Concept for Continuous Functions Completeness If every finite energy signal satisfies K 2 s ( t ) dt a k, 2 k 1 { k (t )}1k K is a complete orthonorma l set. Example. Fourier series 2 2kt 2 2kt cos sin , T T T 0 k T is a complete orthonorma l set for signals defined over [0, T ]. © Po-Ning [email protected] Chapter 5-16 5.2 Gram-Schmidt Orthogonalization Procedure Given v1 (t ), v2 (t ), , vk (t ), how to find an orthonorma l basis for them? v1 (t ) (step i ) Let u1 (t ) . || v1 (t ) || u2' (t ) (step ii ) u (t ) v2 (t ) ( v2 (t ), u1 (t )) u1 (t ). Set u2 (t ) ' . || u2 (t ) || ' 2 (step iii ) For i 3,4,..., Let ui' (t ) vi (t ) vi (t ), ui 1 (t ) ui 1 (t ) vi (t ), u1 (t ) u1 (t ). ui' (t ) Set ui (t ) ' . || ui (t ) || (step iv ) Then u1 (t ), u2 (t ), , uk (t ) form an orthonorma l basis . © Po-Ning [email protected] Chapter 5-17 5.2 Geometric Representation of Signals N si (t ) sij f j (t ), 0 t T , i 1,2,..., M j 1 { f i }iN1 orthonorma l sij 0 si (t ) f j (t )dt , T i 1,2,...M , j 1,2,..., N © Po-Ning [email protected] Chapter 5-18 5.2 Geometric Representation of Signals Through the signal space concept, si(t) (where 1 i M) can be unambiguously represented by an N-dimensional signal vector (si1, si2,…, siN) over an N-dimensional signal space. The design of transmitters becomes the selection of M points over the signal space, and the receivers make a guess about which of the M points was transmitted. In the N-dimensional signal space, length of the vector = energy of the signal angle between vectors = energy correlation between N T signals 2 2 2 si (t ), sk (t ) || s ( t ) || s ( t ) dt s i ij 0 i cos(ik ) j 1 || si (t ) || || sk (t ) || © Po-Ning [email protected] Chapter 5-19 5.2 Geometric Representation of Signals the angle between vectors is independent of the basis used. From this view, the transmitter may be viewed as a synthesizer, which synthesizes the transmitted signal by a bank of N multipliers. the receiver may be viewed as an analyzer, which correlates (product-integrate) the common input into individual informational signal. © Po-Ning [email protected] Chapter 5-20 5.2 Geometric Representation of Energy Signals Illustration the geometric representation of signals for the case when N = 2 and M = 3 © Po-Ning [email protected] Chapter 5-21 5.2 Euclidean Distance After vectorization, we can then calculate the Euclidean distance between two signals: T 0 N ( si (t ) sk (t )) dt || si (t ) sk (t ) || ( sij skj )2 2 2 j 1 1, i j Kronecker delta function : ij 0, i j Applicatio ns : We may say that orthonorma lity means i (t ), j (t ) ij . © Po-Ning [email protected] Chapter 5-22 Example 5.1 Schwarz Inequality Cauchy-Schwarz inequality and angle between signals Cauchy-Schwarz inequality said that s1 (t ), s2 (t ) 2 || s1 (t ) ||2 || s2 (t ) ||2 with equality holds if s1 (t ) cs2 (t ). Also, angles between signals give that s1 (t ), s2 (t ) cos(12 ) || s1 (t ) || || s2 (t ) || Hence, Cauchy-Schwarz inequality is equivalently stated as: | cos(12 ) |2 1 with equality holds if 12 0 or © Po-Ning [email protected] Chapter 5-23 5.2 Basis The (complete) orthonormal basis for a signal space is not unique! So the synthesizer and analyzer for the transmission of the same informational messages are not unique! One way to determine a set of orthonormal basis is the Gram-Schmidt orthogonalization procedure. Try and practice Example 5.2 yourself! © Po-Ning [email protected] Chapter 5-24 5.3 Conversion of the Continuous AWGN Channel into a Vector Channel Influence of the AWGN noise to the signal space concept x(t ) si (t ) w(t ) where w(t) is zero-mean AWGN with PSD N0/2. After the correlator at the receiver, we obtain: x (t ), f j (t ) si (t ), f j (t ) w(t ), f j (t ) Or equivalently, x j sij w j . Notably, there is no information loss by the signal space representation. © Po-Ning [email protected] x1 , x (t ) f1 ( t ) , xN f N (t ) Chapter 5-25 5.3 Conversion of the Continuous AWGN Channel into a Vector Channel Statistics of {wj} x1 si1 w1 xN siN wN Since {sij} is deterministic, the distribution of x is a meanshift of that of w. Observe that w is Gaussian distributed because w(t) is AWGN. The distribution of w can therefore be determined by its mean vector and covariance matrix. © Po-Ning [email protected] Chapter 5-26 5.3 Conversion of the Continuous AWGN Channel into a Vector Channel Mean E[ w j ] E 0 w(t ) f j (t )dt 0 E[ w(t )] f j (t )dt 0 Covariance T E[ wi w j ] E 0 T T w(s) f (s)ds w(t) f (t)dt T 0 T 0 T i 0 j E[ w( s ) w(t )] f i ( s ) f j (t )dsdt N0 0 0 ( s t ) f i ( s ) f j (t )dsdt 2 N0 T N0 f i (t ) f j (t )dt ij 0 2 2 T © Po-Ning [email protected] T Chapter 5-27 5.3 Conversion of the Continuous AWGN Channel into a Vector Channel As a result, [w1, w2, …, wN] are zero-mean i.i.d. Gaussian distributed with variance N0/2. This shows that x is independent Gaussian distributed with common variance N0/2 and mean vector si = [si1, si2, …, siN]. Equivalently, N 1 1 2 f ( x | si ) exp ( x j sij ) N 0 j 1 N0 © Po-Ning [email protected] Chapter 5-28 5.3 Conversion of the Continuous AWGN Channel into a Vector Channel Remainder term in noise It is possible that N w' (t ) w(t ) wi f i (t ) 0 i 1 However, it can be shown that w’(t) is orthogonal to si(t) for 1 i M. Hence, w’(t) will not affect the decision error rate on message i. w' (t ), si (t ) 0 with probabilit y 1. © Po-Ning [email protected] Chapter 5-29 5.4 Likelihood Functions An equivalent signal-space channel model m mi , 1 i M s c(m) x s w mˆ d ( x) {m1 ,..., mM } The best decision function d( ) that minimizes the decision error is: d ( x ) mi , if P{mi | x} P{mk | x} for all 1 k M arg max P{m | x} m{ m1 ,...,mM } This is the maximum a posteriori probability (MAP) decision rule. © Po-Ning [email protected] Chapter 5-30 5.4 Likelihood Functions With equal prior probabilities, d ( x ) arg max P{m | x} m{ m1 ,...,mM } arg max P{m1 | x}, P{m2 | x},..., P{mM | x} P{mM | x} f ( x ) P{m1 | x} f ( x ) P{m2 | x} f ( x ) arg max , ,..., 1/ M 1/ M 1/ M P{m1 | x} f ( x ) P{m2 | x} f ( x ) P{mM | x} f ( x ) arg max , ,..., P ( m ) P ( m ) P ( m ) 1 2 M arg max f ( x | m1 ), f ( x | m2 ),..., f ( x | mM ) f(x|mi) is named the likelihood function given that mi is transmitted Hence, the above rule is named the maximum-likelihood decision rule. © Po-Ning [email protected] Chapter 5-31 5.4 Likelihood Functions MAP rule = ML rule, if equal prior probability is assumed. In practice, it is more convenient to work on the loglikelihood functions, defined by d ( x ) arg max f ( x | m1 ), f ( x | m2 ),..., f ( x | mM ) arg max log f ( x | m1 ), log f ( x | m2 ),..., log f ( x | mM ) Why log-likelihood functions are more convenient? The decision function becomes “sum of Euclidean distances” in AWGN channel. © Po-Ning [email protected] Chapter 5-32 d ( x ) arg max log f ( x | mi ) arg max log f ( x | si ) 1i M 1 i M N arg max log 1i M j 1 1 1 2 exp ( x j sij ) N 0 N0 1 1 2 arg max log N 0 ( x j sij ) 1i M 2 N0 j 1 N arg min 1 i M N 2 ( x s ) j ij j 1 arg min || x si ||2 1 i M Upon receipt of received signal point x, find the signal point si that is closest in Euclidean distance to x. © Po-Ning [email protected] Chapter 5-33 5.5 Coherent Detection of Signals in Noise: Maximum Likelihood Decoding Signal constellation Set of M signal points in the signal space Example. Signal constellation for 2B1Q code decision region for s1 © Po-Ning [email protected] decision region for s2 decision region for s3 decision region for s4 Chapter 5-34 5.5 Coherent Detection of Signals in Noise: Maximum Likelihood Decoding Decision regions for N = 2 and M = 4 © Po-Ning [email protected] Chapter 5-35 5.5 Coherent Detection of Signals in Noise: Maximum Likelihood Decoding Usually, s1, s2, …, sM are named the message points. The received signal point x then wanders about the transmitted message point in a Gaussian-distributed random fashion. © Po-Ning [email protected] Chapter 5-36 5.5 Coherent Detection of Signals in Noise: Maximum Likelihood Decoding Constant-energy signal constellation The ML decision rule can be reduced to an innerproduct. 2 d ( x ) arg 1min || x s || i i M 2 2 arg 1min || x || 2 x , s || s || i i i M 2 x, si Ei arg 1min i M arg max x, si , if Ei is constant. 1 i M © Po-Ning [email protected] Chapter 5-37 5.6 Correlation Receiver If signals do not have equal energy, we can use 1 d ( x ) arg max x, si Ei . 1 i M 2 to implement the ML rule. The receiver is coherent because the receiver requires to be in perfect synchronization with the transmitter (more specifically, the integration must begin at the right time instance). © Po-Ning [email protected] Chapter 5-38 N productintegrators or correlators x1 x2 Correlation receiver xN demodulator or detector © Po-Ning [email protected] decision maker Chapter 5-39 N productintegrators or correlators x1 x2 xN matched filter © Po-Ning [email protected] decision maker Chapter 5-40 5.6 Equivalence of Correlation and Matched Filter Receivers The correlator and matched filter can be made equivalent. Specifically, xi 0 x(t )i (t )dt x( )hi (T )d T if hi (t ) i (T t ) and implicitly i (t ) is zero outside 0 t T . © Po-Ning [email protected] Chapter 5-41 5.7 Probability of Symbol Error Average probability of symbol error M Pe 1 Pc 1 P( mi ) P( d ( x ) mi | mi transmitt ed) i 1 1 1 M 1 1 M 1 1 M M P(d ( x ) m | m i 1 M i i transmitt ed) 2 2 Pr || x s || min || x s || mi transmitt ed i j i 1 1 j M , j i M f ( x | s )dx i 1 Z i i where Z i x N :|| x si ||2 min || x s j ||2 . © Po-Ning [email protected] 1 j M , j i Chapter 5-42 5.7 Invariance of Probability of Symbol Error Probability of symbol error is invariant with respect to basis change (i.e., rotation and translation of the signal space). Specifically, SER (symbol error rate) only depends on the relative Euclidean distances between the message points. 1 Pe 1 M M 2 2 Pr || x s || min || x s || mi transmitt ed i j i 1 © Po-Ning [email protected] 1 j M , j i Chapter 5-43 5.7 Invariance of Probability of Symbol Error Specifically, if Q is a reversible transform (matrix), such as rotation, then x :|| x s || min x :|| Qx Qs || N 2 i 1 j M , j i N 2 i || x s j ||2 min || Qx Qs j ||2 1 j M , j i If a signal constellation is rotated by an orthonormal transformation, where Q is an orthonormal matrix, then the probability of symbol error Pe incurred in maximum likelihood signal detection over an AWGN channel is completely unchanged. © Po-Ning [email protected] Chapter 5-44 5.7 Invariance of Probability of Symbol Error A pair of signal constellation for illustrating the principle of rotational invariance. © Po-Ning [email protected] Chapter 5-45 5.7 Invariance of Probability of Symbol Error The invariance in SER for translation can be likewise proved. Is the transmission power the same for both constellation? © Po-Ning [email protected] Chapter 5-46 5.7 Minimum Energy Signals Since SER is invariant to rotation and translation, we may rotate and translate the signal constellation to minimize the transmission power without affecting SER. M Eg pi || si ||2 i 1 M Find a and Q such that Eg (a, Q) pi || Q( si a) ||2 is minimized. i 1 But Q does not change the norm (i.e., transmission power). Thus, we only need to determine the right a. © Po-Ning [email protected] Chapter 5-47 5.7 Minimum Energy Signals Determine the optimal a. M Eg (a) pi || si a ||2 i 1 pi || si ||2 2aT si || a ||2 M i 1 M pi || si || 2a pi si || a ||2 i 1 i 1 M 2 M T aoptimal pi si and Eg (aoptimal ) pi || si ||2 i 1 © Po-Ning [email protected] M i 1 M ps i 1 2 i i Chapter 5-48 5.7 Minimum Energy Signals So subfigure (a) below has minimum average energy. © Po-Ning [email protected] Chapter 5-49 5.7 Union Bound on the Probability of Error Union bound P( A B ) P( A) P( B ) 1 Pe 1 M Pr|| x s || M 2 i i 1 2 min || x s || mi transmitt ed j 1 j M , j i || x si ||2 || x s1 ||2 1 1 1 Pr || x s ||2 || x s ||2 mi transmitt ed i M M i 1 M i 1 j i M M || x si ||2 || x s1 ||2 1 Pr m transmitt ed 2 2 || x si || || x sM || i M i 1 j i 1 M M Pr || x si ||2 || x s j ||2 mi transmitt ed M i 1 j 1, j i M © Po-Ning [email protected] Chapter 5-50 5.7 Union Bound on the Probability of Error || x s1 ||2 || x s2 ||2 2 2 || x s1 || || x s3 || || x s ||2 || x s ||2 1 4 || x s || || x s || || x s || || x s || || x s || || x s || 2 1 2 2 2 © Po-Ning [email protected] 1 2 3 2 1 2 4 Chapter 5-51 5.7 Union Bound on the Probability of Error 1 Pe M M M P (s , s ) i 1 j 1, j i 2 i j where P2 ( si , s j ) Pr || x si ||2 || x s j ||2 mi transmitt ed . Notably, given mi transmitt ed, x is Gaussian distribute d with mean si . si sj w 1 si s j 2 © Po-Ning [email protected] Chapter 5-52 5.7 Union Bound on the Probability of Erroror Hence, For N = 1, P2 ( si , s j ) Pr || x si ||2 || x s j ||2 mi transmitt ed 1 Pr w | si s j | 2 d ij /2 v2 1 dv, where d ij | si s j | exp N 0 N0 d ij 1 erfc 2 N 2 0 © Po-Ning [email protected] , where erfc(u) 2 u exp( z 2 )dz. Chapter 5-53 5.7 Union Bound on the Probability of Error For N = 2, P2 ( si , s j ) Pr || x si ||2 || x s j ||2 mi transmitt ed 1 Pr w1 d ij and w2 don' t care , where d ij || si s j ||2 2 d ij /2 v2 1 dv exp N 0 N0 d ij 1 erfc 2 N 2 0 , where erfc(u) 2 u exp( z 2 )dz. The same formula is valid for any N. © Po-Ning [email protected] Chapter 5-54 5.7 Union Bound on the Probability of Error Consequently, the union bound for symbol error rate is: 1 Pe M 1 P ( s , s ) 2 i j M i 1 j 1, j i M M d ij 1 erfc 2 N i 1 j 1, j i 2 0 M M The above bound can be further simplified when additional condition is given. For example, if the signal constellation is circularly symmetric in the sense that “{di1, di2, …, diM} is a permutation of {dk1, dk2, …, dkM} for i k,” then d ij 1 Pe erfc 2 N j 1, j i 2 0 M © Po-Ning [email protected] Chapter 5-55 5.7 Union Bound on the Probability of Error Another simplification of union bound Define the minimum distance of a signal constellation as: d min 1i Mmin d ij ,1 j M ,i j Then by the strict decreasing property of erfc function, d ij erfc 2 N 0 1 Pe M d ij 1 erfc 2 N i 1 j 1, j i 2 0 M M © Po-Ning [email protected] erfc d min 2 N 0 1 M d min 1 erfc 2 N i 1 j 1, j i 2 0 M M M 1 d min erfc 2 N 2 0 Chapter 5-56 5.7 Union Bound on the Probability of Error We may use the bound for erfc function to realize the relation between SER and dmin. erfc (u ) exp( u 2 ) d min M 1 Pe erfc 2 N 2 0 for u 0.608131 2 M 1 d min 2 , if d min exp 1.47929 N 0 . 2 4N0 Conclusion: SER decreases exponentially as the squared minimum distance. © Po-Ning [email protected] Chapter 5-57 5.7 Relation between BER and SER The information bits are transmitted in group of log2M bits to form an M-ary symbol. This gives the result that a large symbol error rate (SER) may not cause a large bit error rate (BER). For example, a symbol error (for large M) may be due to only 1 bit error. Optimistically, if every symbol error is due to a single bit error, then (assuming n symbols are transmitted) n SER SER BER . n log 2 ( M ) log 2 ( M ) © Po-Ning [email protected] SER In general, BER . log 2 ( M ) Chapter 5-58 5.7 Relation between BER and SER Pessimistically, if every symbol error causes log2M bit errors, then (assuming n symbols are transmitted) n log 2 M SER BER SER . n log 2 M In general, BER SER. Summary: SER BER SER log 2 M © Po-Ning [email protected] Chapter 5-59 5.7 Relation between BER and SER If the statistics for “number of bit error patterns causes one symbol error” is known, we can then determine the exact relation between BER and SER. M 1 BER n SER # (b j ) P( b j ) j 1 n log 2 M where # (b j ) number of 1' s in b j , and b j represents one binary permutatio n of log 2 M bit pattern. Here, a 1’s in bj means a bit error is occurred in the corresponding position; hence, all-zero pattern is excluded because it represents no symbol error. © Po-Ning [email protected] Chapter 5-60 5.7 Relation between BER and SER Example. If all bit error patters are equally likely, then M 1 M 1 BER n SER # (b j ) P( b j ) j 1 n log 2 M SER ( M 1) log 2 M log 2 M u 1 SER # ( b j ) (1 /( M 1)) j 1 log 2 M k k log 2 M u (Note u k 2k 1.) u 1 u u log 2 M 2log M 1 SER ( M 1) log 2 M 2 M /2 SER M 1 © Po-Ning [email protected] Chapter 5-61