Lecture 02: Digital Modulation Outline Digital-to-analog and analog-to-digital: a signal space perspective Pulse aplitude odulation (PAM), pulse shaping, and the Nyquist criterion Quadrature aplitude odulation (QAM), and the equivalent coplex aseand representation Syol apping and constellation set his lecture introduces the principle of digital odulation, which serves as the ridge etween the cyer world and the physical world. he channel noise is neglected in this lecture, and how to deodulate the signal under noise using optial detection principles is the thee of the next lecture. he asic architecture of digital odulation is depicted in Fig.. {c i } Syol Mapper {u } Pulse Shaper x (t) Up Converter x(t) coded its discrete sequence aseand wavefor passand wavefor Noisy Channel {ĉ i } Syol Deapper {û } Sapler + Filter y (t) Down Converter y(t) Figure : Digital odulation For the transitter, there are three ajor coponents: (a) Syol apping: it sequence syol sequence () Pulse shaping: syol sequence (aseand) wavefor (c) Up conversion: aseand wavefor passand wavefor o e assigned: constellation set o e assigned: pulse, andwidth to e assigned: carrier frequency In the following, we will first introduce the concept of signal space, which is a vector space with wavefor as vectors. With the concept of signal space in ind, we then develop the asic principles of converting sequences to wavefors and vice versa, using orthogonal expansions and projections. hen, we introduce a standard way of converting syol sequences to wavefors, called Pulse Aplitude Modulation (PAM), where the key coponent is pulse shaping. In pulse shaping, in order to e ale to reconstruct the syols fro wavefors at the receiver, certain condition needs to e satisfied for the wavefor. Nyquist Criterion is a sufficient condition that is siple to evaluate. Next, we switch gear to aseand-passand conversion and introduce a standard way called Quadrature Aplitude Modulation (QAM). With QAM, we then derive the equivalent coplex aseand representation. Finally, the design of syol apping is discussed, and a prevalent way called Gray Mapping is introduced. he design of constellation set is also covered, and several standard constellation sets including PSK, PAM, and QAM are introduced. Fall 207 National aiwan University Page
Prelude Fro the discussions in Lecture 0, two key eleents in digital counication systes can e identified: (a) Conversion etween its and syol sequences: Source coding: quantization; tale lookup Channel coding: syol apper; syol deapper () Conversion etween (discrete-tie) sequences and (continuous-tie) wavefors: Source coding: sapling; interpolation filter Channel coding: pulse shaping; sapling his step uilds the ridge etween the cyer and the physical world and enales us to do things digitally. We shall egin with the latter one: conversion etween sequences and wavefors, and then coe ack to the forer one later. 2 Signal space 2. Exaples of conversion etween sequences and wavefors Recall fro the course Signals and Systes, we have learned two approaches for the conversion etween sequences and wavefors: Fourier Series for tie-liited signals {x(t), t } where is an interval with length Sapling heore for and-liited signals {x(t)} with Fourier transfor x(f) = 0 for all f > W. he following tale suarizes how to do analysis (fro wavefor to sequence) and synthesis (fro sequence to wavefor). Recall that sinc(t) sin(πt) πt. Fourier Series Analysis (x(t) x[]) x[] = Sapling heore x[] = x ( ) x(t)e j2π t dt x(t) = x(t) = Synthesis (x[] x(t)) = = x[]e j 2π t x[] sinc( t ) ale : Forulas of analysis and synthesis in Fourier Series and Sapling heore he fors look siilar. It turns out that such siilarity is no coincidence, and there is an unifying principle ehind the. Such principle is called Signal Space, which can e viewed as an inner product space with wavefors identified as vectors. Below we riefly review vector space and inner product space. For ore details, please read Chapter 5 of []. 2.2 Vector space and inner product space Recall that a vector space (V, F) is a collection of vectors v V along with a set F of scalars, following soe axios. he scalar set F is usually the real field R or the coplex field C. When the scalars are Fall 207 National aiwan University Page 2
constrained to e real, the corresponding vector space is called a real vector space. If the scalars are coplex, the corresponding vector space is called a coplex vector space. In the following, we focus on coplex vector spaces, that is, the aient scalar set F is C. When the context is clear, that is, the scalar field is well-understood fro the context, we often use V to denote the vector space. A vector space follows soe axios (you can check your Linear Algera textook or Chapter 5 of []). One of the ost iportant consequences is linearity: α, β C, u, v V, αu + βv V. () Recall the following profound result fro your Linear Algera course. heore : For any finite-diensional vector space V (that is, there exists a spanning set of V with finite cardinality), the following hold: If {v,..., v } span V ut are linearly dependent, then a suset of {v,..., v } for a asis for V with n < vectors. If {v,..., v } are linearly independent ut do not span V, then there exists a asis for V with n > vectors that contains {v,..., v }. Every asis of V has the sae cardinality. 2.2. Inner product and nor One of the ost iportant kinds of vector space are Euclidean spaces R n, C n. he vectors here have the natural notion of angle and length, which can e further astracted into the concepts of inner product and nor. An inner product, on a coplex vector space V satisfies the following axios: for u, v, w V and α, β C, (a) Heritian syetry: () Heritian ilinearity: (c) Positivity: v, u = u, v. (2) αu + βv, w = α u, w + β v, w. (3) v, v 0, with equality iff v = 0. (4) A vector space with an inner product is called an inner product space. he nor v of a vector v is defined as v v, v. (5) In an inner product space, a set of vectors {ϕ, ϕ 2,...} is orthonoral if { 0 i j ϕ i, ϕ j = i = j. (6) 2.2.2 Projection For an orthonoral asis Φ = {ϕ i } of an inner product space V, the expansion of a vector v V over the asis is fairly siple: v = α i ϕ i, with α i = v, ϕ i. (7) i Furtherore, v 2 = i α i 2. More generally, we have a ore general projection theore as follows: Fall 207 National aiwan University Page 3
heore 2 (Projection): Consider a suspace S of an inner product space V and an orthonoral asis Φ of S. For any vector v V, there exists an unique v S S such that v v S, s = 0 for all s S. Furtherore, this unique v S is given as v S = i α i ϕ i, with α i = v, ϕ i. (8) Denoting v v S y v S, we can see that any vector v can e decoposed into two orthogonal parts v S and v S, with v S eing the projection of v onto S. 2.3 Signal space point of view Now, let us eploy the linear algeraic point of view to interpret the two exaples of conversion etween sequences and wavefors Fourier Series and Sapling heore. First, we view a wavefor u(t) as a vector v and define the inner product etween two wavefors as u, v u(t)v (t) dt. (9) For Fourier Series, we then identify the coplex sinusoids as the asis (so-called Fourier Basis) and rewrite the Fourier Series expansion as follows: ϕ ϕ (t) exp(j 2π t), Z, (0) x[] = x x(t) = = x(t) e j2π t dt x, ϕ () x[] e j 2π t x[]ϕ. (2) Siilarly for Sapling heore, we identify the uniforly shifted sinc functions as the asis (so-called Sinc Basis) ϕ ϕ (t) sinc( t ), Z, (3) and rewrite Sapling heore as follows: x[] = x(/ ) = {x(t) sinc( t)} t=/ = x, ϕ x x(t) = x(/ ) sinc( t ) = Exercise. = = x[]ϕ (t) (a) Show that the Fourier Basis (0) and the Sinc Basis (3) are orthonoral. () Prove the last two equalities in equation (4). x(t) sinc( t ) dt (4) (5) x[]ϕ. (6) As shown in the Exercise, oth the Fourier Basis and the Sinc Basis are orthonoral ases. Hence, we can otain a linear algeraic interpretation of analysis and sythesis of wavefor signals: Fall 207 National aiwan University Page 4
Synthesis can e viewed as expansion over an orthonoral asis: {x[]} {ϕ (t)} x(t) = = Analysis can e viewed as projection onto an orthonoral asis x(t) ϕ (t) x[] = 3 Pulse aplitude odulation (PAM) x[]ϕ (t) (7) x(t)ϕ (t) dt. (8) With the signal space interpretation, we can see that for the conversion etween sequences and wavefors in digital odulation, there are infinitely nuer of ways to select the orthonoral asis {ϕ (t), Z}. On the other hand, a prevalent way in digital counication systes nowadays, is called Pulse Aplitude Modulation (PAM), where the asis consists of uniforly-tie-shifted pulses: ϕ (t) = p(t ), =, (9) where is called the transission interval. = which will e ade clear later. is deterined y the operational andwidth W, 3. PAM odulation pulse shaping PAM odulation follows the synthesis forula elow (using the notation in the syste diagra Fig. ): x (t) = Hence, the aove step is called pulse shaping. he pulse p(t) needs to e chosen carefully. following properties are desirale: = u p(t ). (20) Fro the point of view of the transitter, ideally the (a) ie-liited: p(t) = 0 for all t < τ with soe finite τ > 0. his is ecause each syol u arrives at the odulator at soe finite tie, say τ, so the contriution of u to the transitted wavefor x (t) cannot start until τ, which iplies p(t) = 0 for t < τ. () Band-liited: p(f) = 0 for all f > B. Otherwise, the pulse easily violates physical constraints. Reark. It is ipossile to ake a pulse wavefor p(t) tie-liited and and-liited siultaneously. In practice, frequency-doain constraints are usually ore stringent. Hence, we keep the property of eing and-liited ut replace that of eing tie-liited y eing approxiately tie-liited, that is, p(t) 0 rapidly as t. When we ipleent the pulse, usually the pulse p(t) is truncated, that is, the wave for is set to zero for all t < τ for soe τ 0. his truncation process will introduce additional error and result in noise effectively. Hence, the faster p(t) 0, the less noisy it will e. We use p(f) to denote the Fourier transfor of p(t). Fall 207 National aiwan University Page 5
3.2 PAM deodulation filtering + sapling Although for odulation using an orthonoral asis, the optial deodulation is to do projection as discussed in Section 2, in PAM we first filter the received signal y a filter q(t) and then saple the output signal at -spaced saple ties: û = y (τ)q( τ) dτ. (2) Recall that in this lecture, we assue that there is no noise (that is, y (t) = x (t)), and hence the goal of the receiver is to ensure the reconstructed syols û = u,. If this is satisfied, then we say the PAM syste with pulse p(t) and filter q(t) is free of inter-syol interference (ISI), that is, no aliasing effect happens. Since x (t) = u k p(t k ), y the linearity of convolution, we have k= û = (x q)( ) = k= u k g( k ) = k= u k g(( k) )), (22) where g(t) (p q)(t). Hence, a sufficient condition for û = u is { g(k 0 if k 0 ) = if k = 0. (23) Definition 3 (Ideal Nyquist): We say g(t) is ideal Nyquist with interval if the aove condition holds, that is, g(k ) = δ k. In other words, if the pulse p(t) and the filter q(t) are chosen such that g(t) = (p q)(t) is ideal Nyquist with interval, then there is no inter-syol interference (ISI), that is, the recovered û = u when y (t) = x (t). Coined with the desired tie-liited and and-liited properties of p(t), the desired properties for g(t) are suarized as follows: (a) ie-liited: g(t) = 0 for all t < τ with soe finite τ > 0. Recall fro the previous reark, we shall relax this criterion to approxiately tie-liited, that is, g(t) 0 rapidly as t. () Band-liited: ğ(f) = 0 for all f > B. (c) g(t) is ideal Nyquist with interval. 3.3 Nyquist criterion Let us introduce a siple equivalent condition in the frequency doain for checking the property of ideal Nyquist, as stated in the following theore. heore 4 (Nyquist criterion): g(t) is ideal Nyquist with interval if and only if its frequency doain response ğ(f) satisfies the following Nyquist Criterion: rect( f) = ( ğ f ) rect( f), (24) where the function rect(f) { 0 f 2 otherwise. (25) Fall 207 National aiwan University Page 6
Proof he proof of this theore is a siple application of the aliasing theore. First, define s(t) ( ) t g( ) sinc. (26) Hence, g(t) is ideal Nyquist s(t) = sinc ( ) t rect( f) = g( )e j2π f rect( f). o coplete the proof, it reains to show the following equality: ( ğ f ) = g( )e j2π f. (27) Oserving that the left-hand side (LHS) of (27) is the convolution of ğ(f) and the frequency doain ipulse train δ(f / ), we see the Inverse Fourier ransfor (IF) of the LHS of (27) is the ultiplication of the IF of ğ(f) and the IF of the ipulse train: ( ) g(t) δ(t ) = g( )δ(t ). (28) On the other hand, the IF of the right-hand side (RHS) of (27) is also equal to (28). Proof is coplete. In Fig. 2, an illustrative exaple of the Nyquist criterion is given. ğ(f + / ) ğ(f) ğ(f / ) f 2 W 2 W Figure 2: Illustration of the Nyquist Criterion (frequency doain) 3.4 Band-edge syetry and excessive andwidth It can een seen fro the illustration in Fig. 2 that the actual andwidth B and the operational andwidth W need to satisfy B W. he reason why the operational andwidth W is usually strictly saller than the actual andwidth B is to allow soother transition in the frequency doain so that the pulse vanishes to zero faster. Hence, the excessive andwidth is defined as B W. On the other hand, we do not want to ake B W too large. A typical choice for the effective andwidth B is to satisfy W B. In this case, one can easily verify the Nyquist Criterion (24) ecoes the following and-edge syetry condition: ğ (W ) + ğ(w + ) =, [0, W ] (29) Fall 207 National aiwan University Page 7
which is equivalent to { Re {ğ(w )} + Re {ğ(w + )} = I {ğ(w )} = I {ğ(w + )}, [0, W ] (30) his can e proved y revoking the fact that oth p(t) and q(t) are real-valued, and so does g(t). he proof is left as ani{ exercise. See} the I{ illustration in} Fig. 3. Re{ğ(W ĝ(w ) )} Re{ğ(f)} ĝ(f) f ĝ(w Re{ğ(W + ) + )} 0 W B Figure 3: Illustration of and-edge syetry (odifieded fro Figure 6.3 of []). Exaple 5 (Sinc pulse): he tie-doain and frequency-doain representation of a sinc pulse is given elow: ( ) t g(t) = sinc, (3) he operational andwidth W is equal to the actual andwidth B. ğ(f) = rect(f ). (32) Exaple 6 (Raised cosine pulse): he tie-doain and frequency-doain representation of a raised cosine pulse is given elow ( ) π 4 sinc 2β, if t = 2β g β (t) = sinc ( ) t cos( πβt ) (33), otherwise 4 β2 t 2 2 if f β 2 ğ β (f) = 0 if f > +β (34) 2 cos 2 ( π β 2β ( f 2 )) β +β if 2 < f 2 he excessive andwidth B W = ( + β)w W = βw is paraeterized y β [0, ], called the rolloff factor. he larger it is, the soother it transits fro to 0 in the frequency doain, and hence converges to zero faster in the tie doain. In practice, rolloffs as sharp as 5% to 0% is used. Reark. Engineers often define the rolloff factor as B W. You can easily check that oth the sinc pulse and the raised cosine pulse satisfy the and-edge syetry condition (29). his is left as an exercise. Exercise 2. (a) Prove the and-edge syetry condition (29) for real g(t) when W B. () Check the Fourier transfor of the raised cosine pulse g β (t) in (33) is indeed the ğ β (f) in (34). Fall 207 National aiwan University Page 8
(c) Check that the raised cosine pulse satisfy the and-edge syetry condition and hence ideal Nyquist with interval. (d) Show that the sinc pulse converges to 0 as t in the order of /t, while the raised cosine pulse converges to zero in the order of /t 3. 3.5 Choosing {p(t )} as an orthonoral set Now we have seen the desired properties of g(t) (p q)(t): approxiately tie-liited, and-liited, and ideal Nyquist with interval. Earlier in Section 2, we also deonstrated the effectiveness of using the wavefor expansion over and projection onto an orthonoral asis of a signal space to interpret the conversion etween sequences and wavefors. What reains in this section, is to show that for the -space shifted pulses {p(t )}, it fors an orthonoral asis if and only if p(f) 2 satisfy the Nyquist criterion. Let us egin with soe siple oservations. Since ğ(f) = p(f) q(f), we can siply choose p(f) = q(f) = ğ(f). With this choice, we have the following nice theore relating ISI-free condition and orthogonality: heore 7: Suppose ğ(f) = p(f) 2 and satisfies the Nyquist Criterion with interval. hen, {p(t ) : Z} for an orthonoral set. Conversely, if {p(t ) : Z} for an orthonoral set, then p(f) 2 satisfies the Nyquist Criterion. Proof o ake ğ(f) = p(f) 2, we need to choose q(f) = p (f) q(t) = p ( t). Plugging in g(t) = (p q)(t), we get g(k ) = p(t)p ( k + t) dt = If g(t) is ideal Nyquist with interval, then we can conclude that p(t ), p(t n ) = p(t )p (t n ) dt = p(t)p (t k ) dt. (35) p(τ)p (τ (n ) ) dτ = { n = 0 n. (36) Conversely, if {p(t )} for an orthonoral set, we can show that g(k ) = if and only if k = 0 using a siilar calculation. Proof coplete. Suary o su up, for PAM odulation and deodulation, the following principles are usually used for the design of the pulse p(t) at the transitter and the corresponding filter q(t) at the receiver: (a) p(f) = q (f) (phase offset) () p(f) 2 satisfies the Nyquist Criterion (c) If p(t) R (which is norally the case), then q(f) = p (f) = p( f) and hence q(t) = p( t). (d) For faster decay in the tie-doain (less approxiation error) in t = need larger roo for soother transition fro to 0 in the frequency doain. Exercise 3. Ipleent and siulate a PAM odulator/deodulator for the ideal aseand channel y (t) = x (t) using raised cosine pulses in your coputer. Fall 207 National aiwan University Page 9
4 Quadrature aplitude odulation (QAM) 4. QAM odulation In any counication systes, the transitted signals have to oey certain constraints in the frequency doain and can only occupy a certain passand frequency and. Hence, at the transitter it is necessary to convert the aseand signal to a passand signal. In this course, we introduce a widely used approach called quadrature aplitude odulation (QAM), which is essentially a coination of two individual ranches of PAM wavefors, say, x (I) (t) and x (Q) (t). One ranch is ixed with a cosine wavefor with center frequency f c, while the other is ixed with a sine wavefor with the sae center frequency: By defining a coplex aseand wavefor x(t) = x (I) (t) 2 cos(2πf c t) x (Q) (t) 2 sin(2πf c t) (37) we have an equivalent representation for the passand signal: x (t) x (I) (t) + jx (Q) (t), (38) x(t) = 2Re {x (t) exp(j2πf c t)}. (39) Conventionally, x (I) (t) and x (Q) (t) are called the in-phase part and the quadrature part of the coplex asedand wavefor x (t). Recall that oth x (I) (t) and x (Q) (t) are PAM wavefors: and hence the coplex asedand wavefor x (I) (t) = x (Q) (t) = u (I) p(t ) (40) u (Q) p(t ) (4) x (t) = u p(t ), (42) where u u (I) + ju (Q) C is a coplex syol. In words, the QAM odulation can e viewed purely in the coplex doain and the equivalent realdoain ipleentation, as illustrated in Fig. 4. he ixing with the coplex sinusoid 2 exp(j2πf c t) (or the ixing with the cosine/sine wavefors in the real-doain ipleentation) is called up-conversion. 4.2 QAM deodulation As for the QAM deodulation, one should split the received signal y(t) into two ranches. One ranch ais to reconstruct the in-phase syol sequence, while the other ais to reconstruct the quadrature part. For the in-phase part, the received signal is first ultiplied y the cosine wavefor 2 cos(2πf c t), low-pass filtered with one-sided andwidth B, filtered y q(t), and then uniforly sapled with interval. For the quadrature part, the received signal is first ultiplied y the sine wavefor 2 sin(2πf c t), low-pass filtered with one-sided andwidth B, filtered y q(t), and then uniforly sapled with interval. he ixing of the passand signal with cosine and the sine wavefors followed y low-pass filtering with one-sided andwidth B is called down-conversion. Note that the concatenation of the low-pass filter rect( f 2B ) and the PAM deodulation filter q(f) is always equal to q(f). Hence, these two locks can e erged into one. See Fig. 5 for an illustration. Fall 207 National aiwan University Page 0
I-Hsiang Wang Principle of Counications Lecture 02 2 cos(2 fc t) {u(i) } PAM p(t) (I) x (t) 2 sin(2 fc t) {u(q) } PAM p(t) x(t) (Q) x (t) (a) Real-doain ipleentation 2 exp(j2 fc t) {u } PAM p(t) x (t) Re{ } x(t) () Equivalent coplex aseand odel Figure 4: QAM odulation: PAM odulations + up-conversion Fall 207 National aiwan University Page
I-Hsiang Wang Principle of Counications Lecture 02 2 cos(2 fc t) = (I) y (t) LPF { f B } Filter q(t) {u (I) } 2 sin(2 fc t) y(t) = (Q) y (t) LPF { f B } Filter q(t) {u (Q) } (a) Real-doain ipleentation 2 exp( j2 fc t) = Step Filter {f 0} y(t) Filter q(t) y (t) {u } () Equivalent coplex-doain ipleentation 2 cos(2 fc t) = Filter q(t) y(t) {u (I) } 2 sin(2 fc t) = Filter q(t) {u (Q) } (c) Siplified ipleentation Figure 5: QAM deodulation: down-conversion + PAM deodulations Fall 207 National aiwan University Page 2
Exercise 4. Show that in the equivalent coplex-doain ipleentation of QAM deodulation in Fig. 5(), the coplex aseand signal y (t) is indeed equal to 4.3 Frequency-doain illustration y (I) (t) + jy (Q) (t). (43) In the slides, we use a series of pictures to illustration the idea of up conversion and down conversion via a frequency-doain approach. It is useful to recall the following properties of a signal s(t) and its Fourier F transfor s(f): for the pair s(t) s(f), Shift: s(t t 0 ) Reversal: s( t) Conjugate: s (t) F F exp( j2πt 0 f) s(f); s(f f 0 ) exp(j2πf 0 t)s(t). F s( f). F s ( f); s F (f) s ( t). For s(t) R, s(f) = s ( f), that is, Re{ s(f)} = Re{ s( f)} I{ s(f)} = I{ s( f)} s(f) = s( f) s(f) = s( f) od 2π (44) aking the real part: Re{s(t)} F 2 ( s(f) + s ( f)). 4.4 Orthonoral asis expansion of passand wavefors Effectively, the transitted passand signal in (37) can e further expanded as follows: x(t) = u (I) p(t ) 2 cos(2πf c t) k u (Q) k p(t k ) 2 sin(2πf c t). (45) When {p(t ) : Z} is chosen to e an orthonoral set, let us identify p(t ) ϕ (t) p(t ) 2 cos(2πf c t) ψ (I) (t) p(t ) 2 sin(2πf c t) ψ (Q) (t), (46) and rewrite x(t) = u (I) ψ (I) (t) + u (Q) ψ (Q) (t). (47) he following theore tells us that the aove (47) is also an expansion over an orthonoral set. heore 8: Consider an orthonoral set of wavefors {ϕ (t) : Z}. Assue the Fourier transfor exists for each ϕ (t) and is and-liited, that is, ϕ (f) = 0, f > B. (48) hen for a center frequency f c > B, {ψ (I) (t), ψ (Q) (t) Z} also for an orthonoral set, where ψ (I) (t) ϕ (t) 2 cos(2πf c t), ψ (Q) (t) ϕ (t) 2 sin(2πf c t). (49) Fall 207 National aiwan University Page 3
Proof Proof is left as an exercise (see Hoework ). One property that you ight find useful for the proof is the Plancherel theore stated elow: for two (finite-energy) wavefors u(t) and v(t) with Fourier transfors ŭ(f) and v(f) respectively, u(t)v (t) dt = 5 Design of constellation set and syol apping ŭ(f) v (f) df. (50) So far, we have een focusing on the last two layers of digital odulation, naely, conversion of sequences to aseand wavefors, and conversion fro aseand to passand. With QAM, it is now clear that the syols to e transitted can e conveniently represented y coplex nuers, that is, u C. In this last section, we address the question regarding how to convert a sequence of coded its {c i } to a sequence of coplex syols {u }. A standard way to do this is to group a fixed nuer (say l) of its and ap the to a corresponding syol elonging to a finite alphaet A with cardinality A = M 2 l : (c, c 2,..., c l ) u A {a, a 2,..., a M } C {0, } l (5) he jargon in digital counications for the syol alphaet A is constellation set, and its eleent is usually called a constellation point. he constellation set A = {a, a 2,..., a M } is a suset of M = 2 l constellation points in the coplex plane. he ain design question is, how to put these M points on the coplex plane. It turns out the design largely deterines the perforance of deodulation when noise is present, that is, the channel is not perfect anyore. How to do deodulation in the presence of noise, called detection, will e discussed in the next lecture. For this lecture, we introduce several coon constellation sets, along with a popular apping rule called Gray apping. 5. Constellation sets We introduce two failies of constellation sets: standard PAM/QAM constellation sets, and phase-shiftkeying (PSK) constellation sets. (a) he M = 2 l -ary standard PAM constellation set consists of uniforly spaced points on the real line and takes the following for: { } A PAM,2 l ±d, ±3d,..., ±(2 l )d, (52) where 2d is the iniu distance and deterines the average transit energy of the constellation set, which we discuss in ore details later in this section. () he M = 2 2l -ary standard QAM constellation set is siply the direct product of two 2 l -ary standard PAM constellation sets, which takes the following for: { } A QAM,2 2l a (I) + ja (Q) a (I), a (Q) A PAM,2 l. (53) (c) he M-ary PSK constellation set consists of angularly uniforly spaced points on a sphere in the coplex plane and takes the following for: { ( A PSK,M d exp j 2π ) M k k = 0,,..., M }. (54) See Fig. 6 and 7 for the illustration of these constellation sets. Fall 207 National aiwan University Page 4
gray 0 nongray gray 00 0 0 00 0 0 d d 3d d d 3d (a) 2-PAM () 4-PAM Figure 6: Standard PAM constellation sets 000 00 0 00 3d( + j) 0 00 d 2 ( + j) 00 0 0 0 000 000 00 0 00 d( j) 0000 000 00 000 00 0 00 (a) 6-QAM () 8-PSK Figure 7: Standard QAM and PSK constellation sets Fall 207 National aiwan University Page 5
5.2 Syol apping Definition 9 (Gray Mapping): Gray apping assigns the 2 l possile coinations of ordered l its to constellation points in a way such that there is only one-it difference etween nearest neighoring points. In Fig. 6(), one can see the coparison etween a Gray apping and a non-gray (natural) apping. Gray apping has the enefit of saller axial it error proaility when the channel is suject to noise. his will e ade clear in the next lecture. 5.3 Design principles It is quite ovious that the larger M is, the faster the it streas can e delivered (rate is higher). However, in the following, we will see that for a given constellation set, the transit power of the passand wavefor is deterined y the distance paraeter d. On the other hand, as we intuitively argue elow (and ade ore rigorous in the next lecture), the capaility to coat noise also depends on d. Hence, under a given power and reliaility constraint, the rate cannot e too large. Finding the optial tradeoff will e the suject of Lecture 04: reliale counications. In the following we view wavefor as vectors in the signal space. As entioned in Section 4.4, the passand wavefors can also e expanded over an orthonoral set. Hence, for a passand wavefor x(t) x = x[]ϕ, (55) its total energy is x(t) 2 dt = x 2 = where the last inequality is due to the projection theore. x[] 2. (56) Now, suppose in the total nuer of syols sent out is K. Let all the constellation points are sent with equal chance. As K tends to infinity, the average energy per syol ecoes E s li x[] 2 [ = E K K U Unif{A} U 2 ] alost surely. (57) In Hoework, you are asked to derive E s for standard PAM and QAM constellations. What you will see is that E s depends on the nuer of constellation points M and the iniu distance d in etween constellation points. Under a fixed power constraint, on one hand increasing d in can give etter protection to the syste under noise. On the other hand, to transit at a higher rate, decreasing d in can increase M and hence pack ore its into one syol. Such a rate-reliaility tradeoff will e ade clear and foral in Lecture 03. References [] R. G. Gallager, Principles of Digital Counication. Caridge University Press, 2008. Fall 207 National aiwan University Page 6