44 CHAPTER 5. PERFORMACE OF SMALL SIGAL SETS In digital communications, we usually focus entirely on the code, and do not care what encoding ma is use

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1 Performance of small signal sets Chater 5 In this chater, we show how to estimate the erformance of small-to-moderate-sized signal constellations on the discrete-time AWG channel. With equirobable signal oints in iid Gaussian noise, the otimum decision rule is a minimumdistance rule, so the otimum decision regions are minimum-distance (Voronoi) regions. We develo useful erformance estimates for the error robability based on the union bound. These are based on exact exressions for airwise error robabilities, which i n volve the Gaussian robability of error Q( ) function. An aendix develos the main roerties of this function. Finally, we use the union b o u n d estimate to find the coding gain" of small-to-moderatesized signal constellations in the ower-limited and bandwidth-limited regimes, comared to the -PAM or (M M )-QAM baselines, resectively. 5. Signal constellations for the AWG channel In general, a coding scheme for the discrete-time AWG channel model Y = X + is a method of maing an inut bit sequence into a transmitted real symbol sequence x, which i s called encoding, and a method for maing a received real symbol sequence y into an estimated transmitted signal sequence ^x, which is called decoding. Initially we will consider coding schemes of the tye considered by Shannon, namely block codes with a fixed block length. With such codes, the transmitted sequence x consists of a sequence (: : : ; x k ; x k+ ; : : : ) of -tules x k R that are chosen indeendently from some block code of length with M codewords. Block codes are not the only ossible kinds of coding schemes, as we will see when we study convolutional and trellis codes. Usually the numb e r M of codewords is chosen to be a ower of, and codewords are chosen by some encoding ma from blocks of log M bits in the inut bit sequence. If the inut bit sequence is assumed to b e an iid random sequence of equirobable bits, then the transmitted sequence will be an iid random sequence X = ( : : : ; X k ; X k+ ; : : : ) of equirobable random codewords X k. We almost always assume equirobability, because this is a worst-case (minimax) assumtion. Also, the bit sequence roduced by an efficient source coder must statistically resemble an iid equirobable bit sequence. 43

2 44 CHAPTER 5. PERFORMACE OF SMALL SIGAL SETS In digital communications, we usually focus entirely on the code, and do not care what encoding ma is used from bits to codewords. In other contexts the encoding ma is also imortant; e.g., in the Morse code" of telegrahy. If the block length and the number of codew ords M are relatively small, then a block c o d e for the AWG channel may alternatively b e called a signal set, signal constellation, or signal alhabet. A scheme in which the block length is or, corresonding to a single signaling interval of PAM or QAM, may be regarded as an uncoded" scheme. Figure illustrates some -dimensional and -dimensional signal constellations. r r r r r r r 3 r r r r r 3r r r r r r 3 r 3r r r r r r :r5 r r r r r r r 3 0:r5 r 3 3 :r5 3:r5 r 3 (a) (b) (c) (d) (e) Figure. Uncoded signal constellations: (a) -PAM; (b) ( )-QAM; (c) 4-PAM; (d) (4 4)-QAM; (e) hexagonal 6-QAM. An -dimensional signal constellation (set, alhabet) will be denoted by A = fa j ;» j» M g: Its M elements a j R will be called signal oints (vectors, -tules). The basic arameters of a signal constellation A = fa j ;» j» M g are its dimension ; its size M (number of signal oints); its average energy E(A) = M j jja j jj ; and its minimum squared distance d min(a), which i s a n elementary measure of its noise resistance. A secondary arameter is the average numb e r K min (A) of nearest neighbors (oints at distance d min (A)). From these basic arameters we can derive such arameters as: ffl The bit rate (nominal sectral efficiency) ρ = ( = ) log M b/d; P ffl The average energy er two dimensions E s = ( = )E(A), or the average energy er bit E b = E(A)=(log M ) = E s =ρ; ffl Energy-normalized figures of merit such a s d min (A)=E(A); d min (A)=E s or d min(a)=e b, which are indeendent of scale. For examle, in Figure, the bit rate (nominal sectral efficiency) of the -PAM and ( )- QAM constellations is ρ = b/d, whereas for the other three constellations it is ρ = 4 b/d. The average energy er two dimensions of the -PAM and ( )-QAM constellations is E s =, whereas for the 4-PAM and (4 4)-QAM constellations it is E s = 0, and for the hexagonal 6-QAM constellation it is E s = 8 :75. For all constellations, d min = 4. The average numbers of nearest neighb o rs a re K min = ; ; :5; 3, and 4:5, resectively.

3 5.. SIGAL COSTELLATIOS FOR THE AWG CHAEL Cartesian-roduct constellations Some of these relations may be exlained by the fact that an (M M)-QAM constellation is the Cartesian roduct of two M-PAM constellations. In general, a Cartesian-roduct constellation A K is the set of all sequences of K oints from an elementary constellation A; i.e., A K = f(x ; x ; : : : ; x K ) j x k Ag: If the dimension and size of A are and M, resectively, then the dimension of A 0 = A K is 0 = K and its size is M 0 = M K. Exercise (Cartesian-roduct constellations). (a) Show t h a t i f A 0 = A, then the arameters ; log 0 M ; E (A 0 ) and K min (A 0 ) of A are K times as large as the corresonding arameters of A, whereas the normalized arameters ρ; E s ; E b and d min(a) are the same as those of A. Verify that these relations hold for the (M M)-QAM constellations of Figure. otice that there is no difference between a random inut sequence X with elements from A and a sequence X with elements from a Cartesian-roduct constellation A K. For examle, there is no difference between a random M-PAM sequence and a random (M M)-QAM sequence. Thus Cartesian-roduct constellations cature in a non-statistical way the idea of indeendent transmissions. We t h us may regard a Cartesian-roduct constellation A K as equivalent to (o r a version" of) the elementary constellation A. In articular, it has the same ρ; E s ; E b and d min. We m a y further define a code over A" as a subset C ρ A K of a Cartesian-roduct constellation A K. In general, a code C over A will have a l o wer bit rate (nominal sectral efficiency) ρ than A, but a higher minimum squared distance d min. Via this tradeoff, we hoe to achieve a coding gain." Practically all of the codes that we will consider in later chaters will be of this tye. 5.. Minimum-distance decoding Again, a decoding scheme is a method for maing the received sequence into an estimate of the transmitted signal sequence. (Sometimes the decoder does more than this, but this definition will do for a start.) If the encoding scheme is a block s c heme, then it is lausible that the receiver should decode block-by-block as well. That there is no loss of otimality in block-by-block decoding can b e shown from the theorem of irrelevance, or alternatively by an extension of the exercise involving Cartesian-roduct constellations at the end of this subsection. We will now recaitulate how for block-by-block decoding, with equirobable signals and iid Gaussian noise, the otimum decision rule is a minimum-distance (MD) rule. For block-by-block decoding, the channel model is Y = X +, where all sequences are - tules. The transmitted sequence X is chosen equirobably from the M -tules a j in a signal constellation A. The noise df is where the symb o l v ariance is ff = 0 =. e jjnjj =ff (n) = ; (ßff ) = In digital communications, we are usually interested in the minimum-robability-of-error (MPE) decision rule: given a received vector y, choose the signal oint ^a A to minimize the robability of decision error Pr(E). K

4 46 CHAPTER 5. PERFORMACE OF SMALL SIGAL SETS Since the robability that a decision ^a is correct is simly the a osteriori robability (^a j y), the MPE rule is equivalent to the maximum-a-osteriori-robability (MAP) rule: choose the ^a A such that (^a j y) is m a x im um among all (a j j y); a j A. By Bayes' law, (y j a j )(a j ) (a j j y) = : (y) If the signals a j are equirobable, so (a j ) = =M for all j, then the MAP rule is equivalent t o the maximum-likelihood (ML) rule: choose the ^a A such that (y j ^a) is maximum among all (y j a j ); a j A. Using the noise df, we can write (y j a j ) = (y a j ) = e jjy a j jj =ff : (ßff ) = Therefore the ML rule is equivalent to the minimum-distance (MD) rule: choose the ^a A such that jjy ^ajj is minimum among all jjy a j jj ; a j A. In summary, under the assumtion of equirobable inuts and iid Gaussian noise, the MPE rule is the minimum-distance rule. Therefore from this o i n t forward we consider only MD detection, which is easy to understand from a geometrical oint o f v i e w. Exercise (Cartesian-roduct constellations, cont.). (b) Show that if the signal constellation is a Cartesian roduct A K, then MD detection can b e erformed by erforming indeendent MD detection on each of the K comonents of the received K -tule y = (y ; y ; : : : ; y K ). Using this result, sketch the decision regions of the (4 4)-QAM signal set of Figure (d). (c) Show that if Pr(E) is the robability of error for MD detection of A, then the robability of error for MD detection of A 0 is Show that P r(e) 0 ß K Pr(E) if Pr(E) is small. Pr(E) 0 = ( Pr(E)) K ; Examle. The K -fold Cartesian roduct A 0 = A K of a -PAM signal set A = f±ffg corresonds to indeendent transmission of K bits using -PAM. Geometrically, A 0 is the vertex set of a K -cube of side ff. For examle, for K =, A 0 is the ( )-QAM constellation of Figure (b). From Exercise (a), the K -cube constellation A 0 = A K has dimension 0 = K, size M 0 = K, bit rate (nominal sectral efficiency) ρ = b/d, average energy E(A 0 ) = Kff, a verage energy er bit E b = ff, minimum squared distance d (A 0 ) = 4ff, and average numb e r of nearest min neighb o rs K 0 (A 0 min ) = K. From Exercise (c), its robability of error is aroximately K times the single-bit error robability: Pr(E) 0 ß KQ E b = 0 : Consequently, i f w e define the robability of error er bit as P b (E) = Pr(E) 0 =K, then we obtain the curve of (4.) for all K -cube constellations: P b (E) ß Q E b = 0 ; including the ( )-QAM constellation of Figure (b). A c o d e o ver the -PAM signal set A is thus simly a subset of the vertices of a K -cube.

5 5.. SIGAL COSTELLATIOS FOR THE AWG CHAEL Decision regions Under a minimum-distance (MD) decision rule, real -sace R is artitioned into M decision regions R j ;» j» M, where R j consists of the received vectors y R that are at least as close to a j as to any other oint i n A: R j = fy R : jjy a j jj» jj y a j 0 jj for all j 0 6= jg: (5.) The minimum-distance regions R j are also called Voronoi regions. Under the MD rule, given a received sequence y, the decision is a j only if y R j. The decision regions R j cover all of -sace R, and are disjoint excet on their boundaries. Since the noise vector is a continuous random vector, the robability that y will actually fall recisely on the boundary of R j is zero, so in that case it does not matter which decision is made. by The decision region R j is the intersection of the M airwise decision regions R jj 0 defined R jj 0 = fy R : jjy a j jj» jj y a j 0 jj g: Geometrically, it is obvious that R jj 0 is the half-sace containing a j that is bounded by the erendicular bisector hyerlane H jj 0 b e t ween a j and a j 0, as shown in Figure. a j 0 ja j 0 a j y jaj 0 a j = m jaj 0 a j a jjaj 0 a j 6 r rr a j 0 '$ 6 m y Hjj 0 '$ &% a j Figure. The boundary hyerlane H jj 0 is the erendicular bisector between a j and a j 0. - Algebraically, since H jj 0 is the set of o in ts in R that are equidistant f r o m a j and a j 0, it is characterized by the following equivalent equations: jjy a j jj = jjy a j 0 jj ; hy; a j i + jja j jj = hy; a j 0 i + jja j 0 jj ; hy; a j 0 a j i = a j + a j 0 h ; a j 0 a j i = hm; a j 0 a j i: (5.) where m denotes the midvector m = ( a j + a j 0 )=. If the difference vector between a j 0 and a j is a j 0 a j and a j 0 a ffi j j!j 0 = jja j 0 a j jj is the normalized difference vector, so that jjffi j!j 0 jj =, then the rojection of any vector x onto the difference vector a j 0 a j is hx; a j 0 a j i x jaj 0 a j = hx; ffi j!j 0 iffi j!j 0 = (a j 0 a j ): jja j 0 a j jj

6 48 CHAPTER 5. PERFORMACE OF SMALL SIGAL SETS The geometric meaning of Equation (5.) is thus that y H jj 0 if and only if the rojection y jaj 0 a j of y onto the difference vector a j 0 a j is equal to the rojection m jaj 0 a j of the midvector m = ( a j + a j 0 )= o n to the difference vector a j a j 0, as illustrated in Figure. The decision region R j is the intersection of these M half-saces: R j = R jj 0 : j0 6=j (Equivalently, the comlementary region R j is the union of the comlementary half-saces R jj 0.) A decision region R j is therefore a convex olytoe bounded by ortions of a subset fh jj 0 ; a j 0 (a j )g of the boundary hyerlanes H jj 0, where the subset (a j ) A of neighb o rs of a j that contribute boundary faces to this olytoe is called the relevant subset. It is easy to see that the relevant subset must always include the nearest neighb o r s t o a j. 5. Probability of decision error The robability of decision error given that a j is transmitted is the robability that Y = a j + falls outside the decision region R j, whose center" is a j. Equivalently, it is the robability that the noise variable falls outside the translated region R j a j, whose center" is 0: Z Z Z Pr(E j a j ) = Y (y j a j ) dy = (y a j ) dy = (n) dn: R j R j R j a j Exercise (error robability i n variance). (a) Show that the robabilities of error Pr(E j a j ) are unchanged if A is translated by any vector v; i.e., the constellation A 0 = A + v has the same error robability P r (E) as A. (b) Show that Pr(E) is invariant under orthogonal transformations; i.e., the constellation A 0 = U A has the same Pr(E) as A when U is any orthogonal matrix (i.e., U = U T ). (c) Show that Pr(E) is unchanged if both the constellation A and the noise are scaled by the same scale factor ff > 0. Exercise 3 (otimality of zero-mean constellations). Consider an arbitrary signal set A = fa j ;» j» M g. Assume that all signals are equirobable. Let m(a) = M j a j b e the average signal, and let A 0 be A translated by m(a) so that the mean of A 0 is zero: A 0 = A m(a) = fa j m(a);» j» M g: Let E(A) and E(A 0 ) denote the average energies of A and A 0, resectively. W(A) (a) Show that the error robability o f a n o t i m um detector is the same for A 0 as it is for A. (b) Show th a t E(A 0 ) = E(A) jj m(a)jj. Conclude that removing the mean m(a) is always a good idea. (c) Show that a binary antiodal signal set A = f±ag is always otimal for M =. In general, there is no closed-form exression for the Gaussian integral Pr(E j a j ). However, we can obtain an uer bound in terms of airwise error robabilities, called the union bound, which is usually quite shar. The first term of the union bound, called the union bound estimate, P

7 5.. PROBABILITY OF DECISIO ERROR 49 is usually an excellent aroximation, and will be the basis for our analysis of coding gains of small-to-moderate-sized constellations. A l o wer bound with the same exonential behavior may be obtained by considering only the worst-case airwise error robability. 5.. Pairwise error robabilities We now show that each airwise robability has a simle closed-form exression that deends only on the squared distance d (a j ; a j 0 ) = jja j a j 0 jj and the noise variance ff = 0 =. From Figure, it is clear that whether y = a j + n is closer to a j 0 than to a j deends only on the rojection y jaj 0 a j of y onto the difference vector a j 0 a j. In fact, from (5.), an error can occur if and only if jhn; a j 0 a j ij ha j 0 a j ; a j 0 a j i jja j 0 a j jj jn jaj 0 a j j = jhn; ffi j!j 0 ij = = : jja j 0 a j jj jja j 0 a j jj In other words, an error can o c c u r if and only if the magnitude of the one-dimensional noise comonent n = n jaj 0 a j, the rojection of n onto the difference vector a j 0 a j, exceeds half the distance d(a j 0 ; a j ) = jja j 0 a j jj b e t ween a j 0 and a j. We n o w use the fact that the distribution (n) of the iid Gaussian noise vector is sherically symmetric, so the df of any one-dimensional rojection such a s n is (n ) = ßff e n =ff : In other words, is an iid zero-mean Gaussian vector with variance ff in any coordinate system, including a coordinate system in which the first coordinate axis is aligned with the vector a j 0 a j. Consequently, the airwise error robability P r fa j! a j 0 g that if a j is transmitted, the received vector y = a j + n will be at least as close to a j 0 as to a j is given simly by Prfa j! a j 0 g = ßff Z e x =ff d(a j 0 ;a j )= ff where Q( ) is again the Gaussian robability of error function. dx = Q d(a j 0 ; a j ) ; (5.3) As we have seen, the robability o f error for a -PAM signal set f±ffg is Q(ff=ff). Since the distance between the two signals is d = ff, this is just a secial case of this general formula. In summary, the sherical symmetry of iid Gaussian noise leads to the remarkable result that the airwise error robability f r o m a j to a j 0 deends only on the squared distance d (a j 0 ; a j ) = jja j 0 a j jj b e t ween a j and a j 0 and the noise variance ff. Exercise 4 (non-equirobable signals). Let a j and a j 0 b e two signals that are not equirobable. Find the otimum (MPE) airwise decision rule and airwise error robability P r fa j! a j 0 g. 5.. The union bound and the UBE The union bound on error robability is based on the elementary union bound of robability theory: if A and B are any t wo e v ents, then Pr(A [ B)» Pr(A)+ Pr(B). Thus the robability o f

8 50 CHAPTER 5. PERFORMACE OF SMALL SIGAL SETS detection error Pr(E j a j ) with minimum-distance detection if a j is sent i.e., the robability that y will b e closer to some other a j 0 A than to a j is uerbounded by the sum of the airwise error robabilities to all other signals a j 0 6= a j A : X X d(a j ; a j 0 ) Pr(E j a j )» Prfa j! a j 0 g = Q : ff a j =a 6 j 0A a j 6=a j 0 A Let D denote the set of distances b e t ween signal o i n ts in A; then we can write the union bound as X d Pr(E j a j )» K d (a j )Q ; (5.4) ff dd where K d (a j ) is the numb e r of signals a j 0 6= a j A at distance d from a j. Because Q(x) decreases exonentially as e x = (see Aendix), the factor Q(d=ff) will b e largest for the minimum Euclidean distance and will decrease raidly for larger distances. d min (A) = min jja j 0 a j jj; a j 6=a j 0 A The union bound estimate (UBE) of Pr(E j a j ) is based on the idea that the nearest neighb o r s to a j at distance d min (A) (if there are any) will dominate this sum. If there are K min (a j ) neighbors at distance d min (A) from a j, then d min (A) Pr(E j a j ) ß K min (a j ) Q : (5.5) ff Of course this estimate is valid only if the next nearest neighbors are at a significantly greater distance and there are not too many of them; if these assumtions are violated, then further terms should be used in the estimate. The union bound may b e somewhat sharened by considering only signals in the relevant subset (a j ) that determine faces of the decision region R j. However, since (a j ) includes all nearest neighbors at distance d min (A), this will not affect the UBE. Finally, if there is at least one neighb o r a j 0 at distance d min (A) from a j, then we have the airwise lower bound d min (A) Pr(E j a j ) Prfa j! a j 0 g = Q ; (5.6) ff since there must b e a detection error if y is closer to a j 0 than to a j. Thus we are usually able to obtain uer and lower bounds on Pr(E j a j ) that have the same exonent" (argument of the Q( ) function) and that differ only by a small factor of the order of K min (a j ). We can obtain similar uer and lower bounds and estimates for the total error robability Pr(E) = Pr(E j a j ); where the overbar denotes the exectation over the equirobable ensemble of signals in A. For examle, if K min (A) = K min (a j ) is the average number of nearest neighbors at distance d min (A), then the union bound estimate of Pr(E) is d min (A) Pr(E) ß K min (A)Q : (5.7) ff

9 5.3. PERFORMACE AALYSIS I THE POWER-LIMITED REGIME 5 Exercise 5 (UBE for M -PAM constellations). For an M -PAM constellation A, show that Kmin(A) = ( M )=M. Conclude that the union bound estimate of Pr(E) is M d Pr(E) ß Q : M ff Show that in this case the union bound estimate is exact. Exlain why. 5.3 Performance analysis in the o wer-limited regime Recall that the o wer-limited regime is defined as the domain in which the nominal sectral efficiency ρ is not greater than b/d. In this regime we normalize all quantities er bit," and generally use E b = 0 as our normalized measure of signal-to-noise ratio. The baseline uncoded signal set in this regime is the one-dimensional -PAM signal set A = f±ffg, or equivalently a K -cube constellation A K. Such a constellation has bit rate (nominal sectral efficiency) ρ = b/d, average energy b e r bit E b = ff, minimum squared distance d min (A) = 4 ff, and average number of nearest neighb o r s e r b i t K b (A) =. By the UBE (5.7), its error robability er bit is given by P b (E) ß Q (E b = 0 ); (5.8) where we n ow use the Q-of-the-square-root-of" function Q, defined by Q (x) = Q( x) (see Aendix). This baseline curve o f P b (E) vs. E b = 0 is lotted in Chater 4, Figure. The effective coding gain fl eff (A) of a signal set A at a given target error robability e r bit P b (E) will be defined as the difference in db between the E b = 0 required to achieve the target P b (E) w ith A and the E b = 0 required to achieve the target P b (E) with -PAM (i.e., no coding). For examle, we h a ve seen that the maximum ossible effective coding gain at P b (E) ß 0 5 is aroximately. db. For lower P b (E), the maximum ossible gain is higher, and for higher P b (E), the maximum ossible gain is lower. In this definition, the effective coding gain includes any gains that result from using a lower nominal sectral efficiency ρ < b/d, which a s w e h a ve seen can range u to 3.35 db. If ρ is held constant at ρ = b/d, then the maximum ossible effective coding gain is lower; e.g., at P b (E) ß 0 5 it is aroximately 8 db. If there is a constraint on ρ (bandwidth), then it is better to lot P b (E) vs. SR norm, esecially to measure how f a r A is from achieving caacity. The UBE allows us to estimate the effective coding gain as follows. The robability o f error er bit (not in general the same as the bit error robability!) is Pr(E) K min (A) P b (E) = ß Q log jaj log jaj 0 d min (A) since Q d min(a)= 0 = Q(d min (A)=ff). In the ower-limited regime, we define the nominal coding gain fl c (A) as dmin (A) fl c (A) = : (5.9) 4E b This definition is normalized so that for -PAM, fl c (A) =. Because nominal coding gain is a multilicative factor in the argument of the Q ( ) function, it is often measured in db. The UBE then becomes P b (E) ß K b (A)Q (fl c (A)E b = 0 ); (5.0) ;

10 5 CHAPTER 5. PERFORMACE OF SMALL SIGAL SETS where K b (A) = K min (A)= log jaj is the average numb e r of nearest neighb o r s e r transmitted bit. ote that for -PAM, this exression is exact. Given fl c (A) and K b (A), we m a y obtain a lot of the UBE (5.0) simly by m o ving the baseline curve (Figure of Chater 4) to the left by fl c (A) (in db), and then u by a factor of K b (A), since P b (E) is lotted on a log scale. (This is an excellent reason why error robability curves are always lotted on a log-log scale, with SR measured in db.) Thus if K b (A) =, then the effective coding gain fl eff (A) is equal to the nominal coding gain flc (A) for all P b (E), to the accuracy of the UBE. However, if K b (A) >, then the effective coding gain is less than the nominal coding gain by an amount which deends on the steeness of the P b (E) vs. E b = 0 curve at the target P b (E). At P b (E) ß 0 5, a rule of thumb which i s fairly accurate if K b (A) is not too large is that an increase of a factor of two in K b (A) costs about 0. db in effective coding gain; i.e., fl eff (A) ß fl c (A) (0:)(log K b (A)) (in db): (5.) A more accurate estimate may be obtained by a lot of the union bound estimate (5.0). Exercise 6 (invariance of coding gain). Show that the nominal coding gain fl c (A) of (5.9), the UBE (5.0) of P b (E), and the effective coding gain fl eff (A) are invariant t o scaling, orthogonal transformations and Cartesian roducts. 5.4 Orthogonal and related signal sets Orthogonal, simlex and biorthogonal signal sets are concrete examles of large signal sets that are suitable for the o wer-limited regime when bandwidth is truly unconstrained. Orthogonal signal sets are the easiest to describe and analyze. Simlex signal sets are believed to be otimal for a given constellation size M when there is no constraint on dimension. Biorthogonal signal sets are slightly more bandwidth-efficient. For large M, all become essentially equivalent. The following exercises develo the arameters of these signal sets, and show that they can achieve reliable transmission for E b = 0 within 3 db from the ultimate Shannon limit. The drawback of these signal sets is that the number of dimensions (bandwidth) becomes very large and the sectral efficiency ρ very small as M!. Also, even with the fast" Walsh-Hadamard transform (see Chater, Problem ), decoding comlexity is of the order of M log M, which increases exonentially with the number of bits transmitted, log M, and thus is actually slow." Exercise 7 (Orthogonal signal sets). An orthogonal signal set is a set A = fa j ;» j» M g of M orthogonal vectors in M R with equal energy E(A); i.e., ha j ; a j 0 i = E(A)ffi jj 0 (Kronecker delta). (a) Comute the nominal sectral efficiency ρ of A in bits er two dimensions. Comute the average energy E b er information bit. (b) Comute the minimum squared distance d (A). Show that every signal has K min (A) = min M nearest neighb o r s. (c) Let the noise variance be ff = 0 = er dimension. Show that the robability of error of Actually, it can b e shown that with otimum detection orthogonal signal sets can aroach the ultimate Shannon limit on E b = 0 as M! ; h o wever, the union bound is too weak to rove this.

11 5.4. ORTHOGOAL AD RELATED SIGAL SETS 53 an otimum detector is bounded by the UBE Pr(E)» (M )Q (E(A)= 0 ): (d) Let M! with E b held constant. Using an asymtotically accurate uer bound for the Q ( ) function (see Aendix), show that Pr(E)! 0 rovided that E b = 0 > ln (.4 db). How close is this to the ultimate Shannon limit on E b = 0? What is the nominal sectral efficiency ρ in the limit? Exercise 8 (Simlex signal sets). Let A be an orthogonal signal set as above. (a) Denote the mean of A by m(a). Show that m(a) 6= 0, and comute jjm(a)jj. The zero-mean set A 0 = A m(a) (as in Exercise ) is called a simlex signal set. It is universally believed to be the otimum set of M signals in AWG in the absence of bandwidth constraints, excet at ridiculously low S R s. (b) For M = ; 3; 4, sketch A and A 0. (c) Show that all signals in A 0 have the same energy E(A 0 ). Comute E(A 0 ). Comute the inner roducts ha j ; a j 0 i for all a j ; a j 0 A 0. (d) [Otional]. Show that for ridiculously low SRs, a signal set consisting of M zero signals and two a n tiodal signals f±ag has a lower Pr(E) than a simlex signal set. [Hint: see M. Steiner, The strong simlex conjecture is false," IEEE Transactions on Information Theory,. 7-73, May 994.] Exercise 9 (Biorthogonal signal sets). The set A 00 = ±A of size M consisting of the M signals in an orthogonal signal set A with symb o l energy E(A) and their negatives is called a biorthogonal signal set. (a) Show that the mean of A 00 is m(a 00 ) = 0, and that the average energy er symb o l i s E(A). (b) How much greater is the nominal sectral efficiency ρ of A 00 than that of A, in bits er two dimensions? (c) Show that the robability of error of A 00 is aroximately the same as that of an orthogonal signal set with the same size and average energy, for M large. (d) Let the numb er of signals b e a ower of : M = k. Show that the nominal sectral efficiency is ρ(a 00 ) = 4 k k b/d, and that the nominal coding gain is fl (A 00 c ) = k=. Show that the number of nearest neighb o r s i s K min (A 00 ) = k. Examle (Biorthogonal signal sets). Using Exercise 9, we can estimate the effective coding gain of a biorthogonal signal set using our rule of thumb (5.), and check its accuracy against a lot of the UBE (5.0). k The k = 6 biorthogonal signal set A has dimension = = 8, rate k = 4 b/sym, and nominal sectral efficiency ρ(a) = b/d. With energy E(A) er symbol, it has E b = E(A)=4 and d min(a) = E(A), so its nominal coding gain is fl c (A) = d min(a)=4e b = (3.0 db); The numb e r of nearest neighbors is K min (A) = k = 4, so K b (A) = 4 =4 = 3 :5, and the estimate of its effective coding gain at P b (E) ß 0 5 by our rule of thumb (5.) is thus fl eff (A) ß 3 (0:) = :6 db:

12 54 CHAPTER 5. PERFORMACE OF SMALL SIGAL SETS A more accurate lot of the UBE (5.0) may be obtained by shifting the baseline curve (Figure of Chater 4) left by 3 d B a n d u b y h a lf a v ertical unit (since 3:5 ß 0), as shown in Figure 3. This lot shows that the rough estimate fl eff (A) ß :6 db is quite accurate at P b (E) ß 0 5. Similarly, the 64-biorthogonal signal set A 0 has nominal coding gain fl c (A 0 ) = 3 (4.77 db), K b (A 0 ) = 6=6 ß 0, and effective coding gain fl eff (A 0 ) ß (0.) = 4. db by our rule of thumb. The 56-biorthogonal signal set A 00 has nominal coding gain fl (A 00 c ) = 4 (6.0 db), K b (A 00 ) = 54=8 ß 3, and effective coding gain fl eff (A 00 ) ß 6-5(0.) = 5.0 db by our rule of thumb. Figure 3 also shows lots of the UBE (5.0) for these two signal constellations, which show that our rule of thumb c o n tinues to be fairly accurate.

13 5.5. PERFORMACE I THE BADWIDTH-LIMITED REGIME Biorthogonal signal sets Uncoded PAM biorthogonal M=6 biorthogonal M=64 biorthogonal M=56 0 P b (E) Eb/0 [db] Figure 3. P b (E) vs. E b = 0 for biorthogonal signal sets with k = 6 ; 64 and Performance in the bandwidth-limited regime Recall that the bandwidth-limited regime is defined as the domain in which the nominal sectral efficiency ρ is greater than b/d; i.e., the domain of nonbinary signaling. In this regime we normalize all quantities er two dimensions," and use SR norm as our normalized measure of signal-to-noise ratio. The baseline uncoded signal set in this regime is the M -PAM signal set A = fff±; ±3; : : : ; ±(M )g, or equivalently the (M M )-QAM constellation A. Tyically M is a o wer of. Such a constellation has bit rate (nominal sectral efficiency) ρ = log M b/d and minimum squared distance d min(a ) = 4 ff. As shown in Chater 4, its average energy er two dimensions is ff (M ) d (A)( ρ min ) E s = = : (5.) 3 6 The average number of nearest neighbors er two dimensions is twice that of M -PAM, namely K s (A) = 4( M )=M, which raidly aroaches K s (A) ß 4 as M becomes large. By the UBE (5.7), the error robability er two dimensions is given by P s (E) ß 4Q (3SR norm ): (5.3) This baseline curve o f P s (E) vs. SR norm was lotted in Figure of Chater 4.

14 56 CHAPTER 5. PERFORMACE OF SMALL SIGAL SETS In the bandwidth-limited regime, the effective coding gain fl eff (A) of a signal set A at a given target error rate P s (E) will b e defined as the difference in db b e t ween the SR norm required to achieve the target P s (E) with A and the SR norm required to achieve the target P s (E) with M -PAM or (M M )-QAM (no coding). For examle, we saw from Figure of Chater 4 that the maximum ossible effective coding gain at P s (E) ß 0 5 is aroximately 8.4 db, which is about 3 db less than in the ower-limited regime (due solely to the fact that the bandwidth is fixed). The effective coding gain is again estimated by the UBE, as follows. The robability of error e r t wo dimensions is P r( E) K d min (A) min P Q (A) s (E) = ß : 0 In the bandwidth-limited regime, we define the nominal coding gain fl c (A) as ( ρ )d min (A) fl c (A) = : (5.4) 6E s This definition is normalized so that for M -PAM or (M M )-QAM, fl c (A) =. Again, fl c (A) is often measured in db. The UBE (5.0) then becomes P s (E) ß K s (A)Q (3fl c (A)SR norm ); (5.5) where K s (A) = K min (A)= is the average numb e r of nearest neighb o r s e r two dimensions. ote that for M -PAM or (M M )-QAM, this exression reduces to (5.3). Given fl c (A) and K s (A), we m a y obtain a lot of (5.5) by m o ving the baseline curve (Figure of Chater 4) to the left by fl c (A) (in db), and u by a factor of K s (A)=4. The rule of thumb that an increase of a factor of two i n K s (A) o ver the baseline K s (A) = 4 costs about 0. db in effective coding gain at P s (E) ß 0 5 may still be used if K s (A) is not too large. Exercise 6 (invariance of coding gain, cont.) Show that in the bandwidth-limited regime the nominal coding gain fl c (A) of (5.4), the UBE (5.5) of P s (E), and the effective coding gain fleff (A) a r e i n variant to scaling, orthogonal transformations and Cartesian roducts. 5.6 Design of small signal constellations The reader may n o w l i k e to try to find the best constellations of small size M in dimensions, using coding gain fl c (A) as the rimary figure of merit, and K min (A) as a secondary criterion. Exercise 0 (small nonbinary constellations). (a) For M = 4, the ( )-QAM signal set is known to be otimal in = dimensions. Show however that there exists at least one other inequivalent t wo-dimensional signal set A 0 with the same coding gain. Which signal set has the lower error coefficient" K min (A)? (b) Show that the coding gain of (a) can b e imroved in = 3 dimensions. [Hint: consider the signal set A 00 = f(; ; ); (; ; ); ( ; ; ); ( ; ; ) g.] Sketch A 00. What is the geometric name of the olytoe whose vertex set is A 00? (c) For M = 8 and =, roose at least two good signal sets, and determine which one is better. [Oen research roblem: Find the otimal such signal set, and rove that it is otimal.] (d) [Oen research roblem.] For M = 6 and =, the hexagonal signal set of Figure (e), Chater 4, is thought to be near-otimal. Prove that it is otimal, or find a better one.

15 5.7. SUMMARY: PERFORMACE AALYSIS AD CODIG GAI Summary: Performance analysis and coding gain The results of this chater may be summarized very simly. In the ower-limited regime, the nominal coding gain is fl c (A) = d (A)=4E b. To the accuracy min of the UBE, P b (E) ß K b (A)Q (fl c (A)E b = 0 ). This curve may b e lotted by moving the o wer-limited baseline curve P b (E) ß Q (E b = 0 ) to the left by fl c (A) in db and u by a factor of K b (A). An estimate of the effective coding gain at P b (E) ß 0 5 is fl eff (A) ß fl c (A) (0:)(log K b (A)) db. In the bandwidth-limited regime, the nominal coding gain is fl c (A) = ( ρ )d (A)=6E s. min To the accuracy of the UBE, P s (E) ß K s (A)Q (3fl c (A)SR norm ). This curve may be lotted by m o ving the bandwidth-limited baseline curve P s (E) ß 4Q (3SR norm ) to the left by fl c (A) in db and u by a factor of K s (A)=4. An estimate of the effective coding gain at P s (E) ß 0 5 is fl eff (A) ß fl c (A) (0:)(log K s (A)=4) db. Aendix: The Q function The Gaussian robability of error (or Q) function, defined by Z Q(x) = e y = dy; x ß arises frequently in error robability calculations on Gaussian channels. In this aendix we discuss some of its roerties. As we have seen, there is very often a square root in the argument of the Q function. This suggests that it might have b e e n more useful to define a Q-of-the-square-root-of" function Q (x) su ch that Q (x ) = Q(x); i.e., Z Q (x) = Q( x) = e y = dy: ß From now o n w e will use this Q function instead of the Q function. For examle, our baseline curves for -PAM and (M M )-QAM will be P b (E) = Q (E b = 0 ); P s (E) ß 4Q (3SR norm ): The Q or Q functions do not have a closed-form exression, but must be looked u in tables. on-communications texts usually tabulate the comlementary error function, namely Z erfc(x) = e y dy: x ß Evidently Q(x) = erfc(x= ), and Q (x) = erfc( x=). The main roerty o f t h e Q or Q function is that it decays exonentially with x according to Q (x ) = Q(x) ß e x = : x

16 58 CHAPTER 5. PERFORMACE OF SMALL SIGAL SETS The following exercise gives several ways to rove this, including uer bounds, a lower bound, and an estim ate. Exercise A (Bounds on the Q function). (a) As discussed in Chater 3, the Chernoff b o u n d on the robability that a real random variable Z exceeds b is given by PrfZ bg» e s(z b) ; s 0 (since e s(z b) when z b, and e s(z b) 0 otherwise). When otimized over s 0, the Chernoff exonent is asymtotically correct. Use the Chernoff bound to show that (b) Integrate by arts to derive the uer and lower bounds Q (x )» e x = : (5.6) e x = Q (x ) < ; (5.7) ßx e x = Q (x ) > : (5.8) x ßx (c) Here is another way to establish these tight uer and lower bounds. By using a simle change of variables, show t h a t Then show that Z x Q (x y ) = e ex xy dy: ß 0 y y» ex» : Putting these together, derive the bounds of art (b). '$ x- &% For (d)-(f), consider a circle of radius x inscribed in a square of side x as shown below. (d) Show that the robability that a two-dimensional iid real Gaussian random variable X with variance ff = er dimension falls inside the square is equal to ( Q (x )). (e) Show that the robability that X falls inside the circle is e x =. [Hint: write r = X (x) in o l a r coordinates: i.e., R (r; ) =. ß re You can then comute the integral R ß R x d dr R (r; ) in closed form.] 0 0 (f) Show that (d) and (e) imly that when x is large, Q (x )» e x = : 4