Capacity of the Discrete-Time AWGN Channel Under Output Quantization

Capacity of the Discrete-T ime A WGN Channel Under Output Quantization Jaspree t Singh, Onkar Dabeer an d Up amanyu Madh ow ∗ Abstract — W e in vestigate the limits of communication ov er the discrete-time Add itive White Gaussian Noise (A WGN) ch annel, when the channel output is quantized using a sm all number of bits. W e first prov ide a proof of our recent conj ecture on the optimality of a discrete in put distributi on in th is sce nario. Specifically , we sh ow that for any given outpu t quantizer choice with K quantization bins (i .e., a precision of log 2 K b its), the input distribution, under an aver age power constraint, need not hav e any more than K + 1 mass p oints to achieve the ch annel capacity . The cutting-p lane algorithm is employed to compute this capacity and to generate optimum input distributions. Numerical op timization o ver the choice of the quantizer is then perfo rmed (fo r 2 -bit and 3 -bit symmetric q uantization), and the results we obtain show that the loss due to l ow-pre cision output quantization, which is small at low signal-to-noise ratio ( SNR ) as expected, can be quite acceptable ev en fo r moderate to h igh SNR values. F or example, at S NR s up to 20 dB, 2 - 3 bit quantization achiev es 80 - 90% of the capac ity achievable using infinite-p recision quanti zation. I . I N T R O D U C T I O N Analog-to -digital con version (ADC) is an integral part of modern communica tion receiver ar chitectures based on digital signal processing (DSP). T yp ically , ADCs with 6 - 12 b its of precision ar e employed at the receiver to con vert the recei ved analog baseband signal into digital form for further processing. Howe ver, as the co mmunica tion systems scale up in spee d and bandwidth (for e.g., systems op erating in the ultrawide band or the mm-wave band ), the cost an d power consumptio n of such high precision AD C becom es pro hibitive [1]. A DSP-centric architecture nonethe less remains a ttractiv e, due to the continu - ing expon ential advances in digital electronics (M oore’ s law). It is of interest, therefor e, to understand whether DSP-centr ic design is compatible with the use of low-precision ADC. In this paper, we contin ue our investigation of the Shanno n- theoretic co mmunicatio n limits imposed b y the u se of low- precision ADC for idea l Nyquist sam pled linear modulation in A WGN. The discrete- time mem oryless A WGN- Quantized Output (A W GN-QO) chan nel model th us in duced is shown in Fig. 1. In our prior work for this chann el mod el, we have sho w n that for th e extreme scenar io of 1-b it symmetric quantization , binary antipo dal signaling ach iev es the chann el ∗ J. Singh a nd U. Madho w are with the ECE Depa rtment, UC Santa Ba rbara, CA 93106, USA. Their research was supported by the National Science Foun- dation under grant CCF-0729222 and by the Office for Nav al Research under grant N00014-06-1-0066. O. Dabeer is with the T ata Institu te of Fundamental Researc h, Mumbai 400005, India. His work was supported in part by a grant from the Dept. of Science and T echnology , Govt. of India, and in part by the Homi Bhabha Fello wship. { jsingh, madhow } @ece.ucsb.edu, onkar@tcs.tif r.res.in ADC Quantizer Q + X Y N Fig. 1. The A WGN-Quanti zed Ouput Channel : Y = Q ( X + N ) . capacity for any signal- to-noise ratio ( SNR ) [2]. For m ulti- bit quantization [3], we p rovided a du ality-based ap proach to bound the capacity from above, and e mployed the cutting - plane algo rithm to generate in put d istributions that nearly achieved these u pper bo unds. Based o n ou r results, we con- jectured that a discrete inpu t with cardinality not exceeding the n umber of quantization b ins achieves the cap acity of the average p ower co nstrained A WGN-Q O ch annel. In this work , we prove that a discrete input is ind eed optimal, althou gh o ur result only gu arantees its card inality to be at most K + 1 , where K is the num ber of quantization bins. Our proo f is inspired by Wits enhausen ’ s re sult in [4], whe re Dubins’ theorem [5] was used to show that the cap acity o f a discrete- time memoryless channel with o utput card inality K , und er only a peak power con straint is a chiev ab le by a discr ete in put with at mo st K poin ts. The key to our proof is to show that, under output quantization, an average p ower constraint automatically induces a peak power c onstraint, af ter which we use Dubins’ theorem as d one by W itsen hausen. Alth ough not applicable to ou r setting, it is worth noting that for a Discrete Memoryle ss Chan nel, Ga llager first showed that the nu mber of inp uts with nonzero p robability mass need not exceed the number of outputs [6, p. 96 , Coro llary 3]. While the precedin g results optimize the inpu t distribution for a fixed q uantizer, c omparison with an unqu antized sy stem requires op timization over the c hoice o f the quantizer as well. W e d o this numer ically for 2 - bit and 3 -bit sym metric quantization , and use ou r numerical results to make the following encour aging o bservations: (a) Low-precision ADC incurs a relatively small loss in spectra l efficiency com pared to unqu antized observations. While this is expected for lo w SNR s, we find that even at modera tely hig h SNR s o f up to 20 dB, 2 - 3 b it ADC still achieves 80- 90 % o f the sp ectral effi- ciency attaine d using u nquan tized obser vations. These results indicate the fe asibility of system design using low-precision ADC fo r hig h bandw idth sy stems. ( b) Stand ard un iform Pulse Amplitude Modulated (P AM ) input with quantizer threshold s set to impleme nt max imum likelihood (ML ) hard decisions achieves nearly the same perf ormance as that attained by an optimal input and quantizer pair . This is usefu l from a system designer’ s point o f view , sinc e the ML q uantizer thresho lds have a simple analytical depende nce on SNR, which is an easily measurable quantity . The rest of the p aper is organized as follows. The quantized output A WGN chan nel m odel is giv en in the next section. In Section III, we show that a d iscrete input achieves the cap acity of this chan nel. Quantizer optimization resu lts are presented in Section IV, f ollowed by the conclusion s in Section V. I I . C H A N N E L M O D E L W e consider linear m odulation over a real A W GN chan nel, and assume that the Nyqu ist criterio n for no intersym bol in- terference is satisfied [7, p p. 5 0]. Symbo l rate sampling of the receiver’ s matched filter outpu t u sing a finite-precision ADC therefor e r esults in the following discrete-time mem oryless A WGN- Qu antized Output (A WGN-QO) chann el (Fig. 1) Y = Q ( X + N ) . (1) Here X ∈ R is the ch annel inpu t with distribution F ( x ) and N is N (0 , σ 2 ) . T he q uantizer Q maps the real valued input X + N to o ne of the K bins, producin g a discrete channel output Y ∈ { y 1 , · · · , y K } . W e only co nsider quantizers for which each bin is an interval of the real line. The q uantizer Q with K bin s ca n therefore be cha racterized by the set of its ( K − 1) th resholds q q q = [ q 1 , q 2 , · · · , q K − 1 ] ∈ R K − 1 , such that −∞ := q 0 < q 1 < q 2 < · · · < q K − 1 < q K := ∞ . The resulting transition probability functions ar e gi ven by W i ( x ) = P ( Y = y i | X = x ) = Q  q i − 1 − x σ  − Q  q i − x σ  , (2) where Q ( x ) denotes the complem entary Gaussian distribution function 1 √ 2 π R ∞ x exp( − t 2 / 2) dt . The input-o utput m utual inf ormation I ( X ; Y ) , expressed explicitly as a fu nction of F is I ( F ) = Z ∞ −∞ K X i =1 W i ( x ) log W i ( x ) R ( y i ; F ) dF ( x ) , (3) where { R ( y i ; F ) , 1 ≤ i ≤ K } is the Prob ability Mass Function (PMF) o f the o utput when th e input is F . Un der an a verag e power con straint P (i.e., E [ X 2 ] ≤ P ), we wish to compute the capacity o f the channel ( 1), which is g iv en by C = sup F ∈F I ( F ) , (4) where F is th e set of all average power constrained distrib u- tions on R . I I I . D I S C R E T E I N P U T A C H I E V E S C A PAC I T Y W e first use the Karu sh-Kuhn-T ucker ( KKT) o ptimality condition to show that an av erage p ower constraint fo r the A WGN-QO chan nel au tomatically induces a constraint on the peak p ower , in the sense th at a n op timal input d istribution must h av e a bou nded support set. This fact is then exploited to show the optimality of a discrete inp ut. A. An Implicit peak power Constr aint The follo wing KKT co ndition can be deri ved fo r the A WGN-QO chan nel, using conv ex o ptimization pr inciples in a man ner similar to that in [8], [ 9]. Th e input distribution F is optimal if and o nly if the re exists a γ ≥ 0 such that K X i =1 W i ( x ) log W i ( x ) R ( y i ; F ) + γ ( P − x 2 ) ≤ I ( F ) , (5) for all x , with eq uality if x is in the support of F . The first term on the left hand side of the KKT cond ition (5) is the divergence (o r the rela ti ve entropy) between the transition an d the output PMFs. For co n venience, let us deno te it b y d ( x ; F ) . The fo llowing result co ncernin g th e behavior of d ( x ; F ) has been pr oved in [10]. Lemma 1: For the A WGN-QO channel (1) with input dis- tribution F , the d iv ergence fu nction d ( x ; F ) satisfies the following pr operties (a) lim x →∞ d ( x ; F ) = − log R ( y K ; F ) . (b) There e xists a finite c onstant A 0 such that ∀ x > A 0 , d ( x ; F ) < − log R ( y K ; F ) . Pr oof: See [10]. W e now use Lem ma 1 to prove th e main resu lt of th is subsection. Pr opo sition 1: A cap acity-achieving input distrib u tion for the average p ower co nstrained A WGN-QO chan nel ( 1) must have boun ded sup port. Pr oof: Assum e that the input distribution F ∗ achieves 1 the capacity in (4) (i.e. , I ( F ∗ ) = C ), with γ ∗ ≥ 0 being a co rrespon ding op timal Lagr ange param eter in the KKT condition . In other words, with γ = γ ∗ , and, F = F ∗ , (5) must be satisfied with an equality at every point in th e suppor t of F ∗ . W e exploit this necessary condition n ext to show that the support o f F ∗ is u pper bo unded . Specifically , we prove that there exists a finite con stant A 2 ∗ such that it is not p ossible to attain equality in (5) for any x > A 2 ∗ . Using Lemma 1 , we first let lim x →∞ d ( x ; F ∗ ) = − lo g( R ( y K ; F ∗ )) = L , an d also assume that there exists a finite constant A 0 such that ∀ x > A 0 , d ( x ; F ∗ ) < L . W e conside r two possible cases. • Case 1: γ ∗ > 0 . If C > L + γ ∗ P , then pick A 2 ∗ = A 0 . Else pick A 2 ∗ ≥ max { A 0 , p ( L + γ ∗ P − C ) / γ ∗ } . In either situation, for x > A 2 ∗ , we get d ( x ; F ∗ ) < L , and, γ ∗ x 2 > L + γ ∗ P − C . This gi ves d ( x ; F ∗ ) + γ ∗ ( P − x 2 ) < L + γ ∗ P − ( L + γ ∗ P − C ) = C . • Case 2: γ ∗ = 0 . Putting γ ∗ = 0 in the KKT co ndition (5), we get d ( x ; F ∗ ) = K X i =1 W i ( x ) log W i ( x ) R ( y i ; F ∗ ) ≤ C , ∀ x. 1 That the capacity is achie vab le can be shown using standard results from optimiza tion the ory . Fo r lack of space here, we refe r the reader to [10 ] for detai ls. Thus, L = lim x →∞ d ( x ; F ∗ ) ≤ C. Picking A 2 ∗ = A 0 , we th erefore hav e that for x > A 2 ∗ d ( x ; F ∗ ) + γ ∗ ( P − x 2 ) = d ( x ; F ∗ ) < L. = ⇒ d ( x ; F ∗ ) + γ ∗ ( P − x 2 ) < C. Combining the tw o cases, w e ha ve shown that the support of the distribution F ∗ has a finite upper bound A 2 ∗ . Using similar arguments, it can easily be shown that the suppo rt o f F ∗ has a finite lower boun d A 1 ∗ as well, wh ich implies that F ∗ has a bounde d suppo rt. B. Achievability o f Capacity b y a Discr ete Input T o sho w the o ptimality of a d iscrete input f or our p roblem, we use the following theo rem which we hav e proved in [10]. The theorem ho lds fo r ch annels w ith a finite output alp habet, under the con dition that the input is constrain ed in both peak power and av erage power . Theor em 1: Consider a stationary discrete-time memo- ryless chann el with a co ntinuou s inpu t X taking values in the b ound ed in terval [ A 1 , A 2 ] , and a d iscrete outp ut Y ∈ { y 1 , y 2 , · · · , y K } . Let th e transition pr obability function W i ( x ) = P ( Y = y i | X = x ) be continu ous in x , for each i in { 1 , .., K } . The capacity of this channel, under an average power constraint on the in put, is achievable by a discrete input with at most K + 1 poin ts. Pr oof: See [10]. Our pro of in [10] uses Dub ins’ th eorem [5], and is an extension of W itsenhausen ’ s result in [4], wherein h e sho wed that a d istribution with only K po ints would be sufficient to achieve the capacity if the average p ower of the input was not constrained . The im plicit p eak p ower constraint de riv ed in Section III-A allows us to use Theorem 1 to g et the follo wing r esult. Pr opo sition 2: The capacity of the av erage p ower con- strained A WGN-Q O channel (1) is ach iev able by a discrete input distribution with at most K + 1 poin ts o f support. Pr oof: Using notation f rom the las t sub section, let F ∗ be a n optimal distribution for (4), with the sup port of F ∗ being contained in the boun ded interval [ A 1 ∗ , A 2 ∗ ] . Define F 1 to be the set of all a verag e power constra ined d istributions whose support is co ntained in [ A 1 ∗ , A 2 ∗ ] . No te that F ∗ ∈ F 1 ⊂ F , where F is the set of all average p ower con strained distributions on R . Consider the maximization of the mutu al informa tion I ( X ; Y ) over the set F 1 C 1 = max F ∈F 1 I ( F ) . (6) Since the tra nsition prob ability fu nctions in (2) are continuo us in x , The orem 1 implies that a d iscrete distribution with at most K + 1 mass po ints achieves the m aximum C 1 in (6). Denote such a distribution by F 1 . Howe ver , since F ∗ achieves the maxim um C in (4) and F ∗ ∈ F 1 , it mu st also achieve the maximum in (6 ). This implies that C 1 = C , and that F 1 is optimal for (4), th us completing the proof. C. Capacity Computation W e hav e alr eady addressed the issue of com puting the capacity (4) in our prior work. Specifically , in [2], we ha ve shown analytically th at for th e extreme scenar io of 1-bit symmetric quantization , bin ary an tipodal signaling achieves the capacity ( at any SNR ). Multi-bit qu antization ha s bee n considered in [3], [10], where we show th at the cuttin g-plane algorithm [11] can be employed for co mputing the capacity and obtaining o ptimal input distributions. I V . O P T I M I Z AT I O N O V E R Q U A N T I Z E R Until now , we h av e addressed th e problem of capacity com- putation giv en a fixed quantizer . In this section, we con sider the issue of quantizer optimization, while restricting atten tion to symmetric quan tizers only . Giv en the symmetric n ature of the A WGN noise and the power constrain t, it seems intu iti vely plausible that restriction to symm etric quantizers shou ld not be sub -optimal from the point of vie w of optimizing over the quantizer choice in ( 1), although a proof of this c onjecture has eluded us. A Simple Ben chmark: While an optimal quantizer (with a correspo nding optimal inpu t) provides the a bsolute comm u- nication limits for our m odel, fro m a system designer’ s pe r- spectiv e, it would also be usefu l to ev aluate the per forman ce degradation if we use some standard input constellations and quantizer choices. W e take the following input and quan tizer pair as ou r benchmark strate gy : fo r K-bin qu antization, consider eq uispaced u niform K-P AM (Pulse Amplitude Mod- ulated) input d istribution, with qua ntizer thresho lds as the mid-po ints of the input ma ss point locations (i. e., ML hard decisions). W ith the K -po int un iform inp ut, we have the entropy H ( X ) = log 2 K bits for any SNR . Also, it is easy to see that as SN R → ∞ , H ( X | Y ) → 0 f or the be nchmark input-q uantizer pair . Therefore, our benchmark scheme is near- optimal if we operate in the h igh SNR regime. The ma in issue to in vestigate ahe ad, th erefore is: at lo w to mo derate SNR s, how mu ch g ain does an optimal qu antizer choice provide over the benchmark. In all the results that follow , we take the no ise variance σ 2 = 1 . Howe ver , the results are scale inv ar iant in the sense that if bo th P and σ 2 are scaled by the same factor R (thus keeping the SNR uncha nged), then there is an equiv a lent quantizer (o btained by scaling the threshold s by √ R ) that giv es an iden tical performance. N U M E R I C A L R E S U LT S A. 2 -bit Symmetric Quantization A 2 -b it symmetric q uantizer is ch aracterized by a sing le parameter q , with {− q , 0 , q } being the quantizer thresholds. Hence we use a b rute fo rce search over q to optimize the quantizer . In Fig. 2, we plot the variation of th e channel capacity (computed using the cuttin g-plane alg orithm) as a function of the parameter q at various SNR s. W e ob serve that for any SNR , ther e is an optimal cho ice of q th at maximizes the capacity . At h igh SNR s, the o ptimal q is seen to increase monoto nically with SNR , which is n ot surp rising sin ce the 0 1 2 3 4 5 6 7 0 0.5 1 1.5 2 Quantizer threshold ’q’ Capacity (bits / channel use) −5 dB 0 dB 3 dB 7 dB 10 dB 15 dB Fig. 2. 2-bit symmetric quantizati on : channel capacit y versus the quantiz er threshold q (noi se var iance σ 2 = 1 ). SNR (dB) − 10 − 5 0 7 15 1 -bit optimal 0 . 0449 0 . 1353 0 . 3689 0 . 9020 0 . 9974 2 -bit optimal 0 . 0613 0 . 1792 0 . 4552 1 . 0981 1 . 9304 2 -bit benchmark 0 . 0527 0 . 1658 0 . 4401 1 . 0639 1 . 9211 T ABLE I M U T UA L I N F O R M A T I O N ( I N B I T S / C H A N N E L U S E ) AT D I F F E R E N T SNR S . benchm ark quantizer ’ s q scales as √ SNR an d is known to be near-optimal at hig h SNR s. Comparison wit h the benchmark: In T ab le I, we compare the perform ance o f the optimal s olution obtained as abov e with the be nchmark sch eme. The cap acity with 1-bit q uantization is also shown for r eference. While being n ear-optimal at moderate to high SNR s, the bench mark schem e is seen to perfor m fairly well at low SNR s also. For instance, at − 10 dB SNR , it a chieves 86% of the capacity achieved with an op timal 2 -bit quantizer and inp ut pair . Fro m a practical standpoin t, th ese results imply that the benc hmark sche me, which require s negligible computationa l effort (due to its well- defined d ependen ce on SNR ), ca n be employed e ven at small SNR s while incurring an accep table loss of pe rforman ce. 0 0.2 0.4 PMF 0 0.2 0.4 0 0.2 0.4 0 0.2 0.4 0 0.2 0.4 0 0.2 0.4 0 dB 4 dB 7 dB 10 dB 15 dB 20 dB X Fig. 3. 2 -bit symmetric quanti zation : optimal input distrib ution and quantize r at variou s SNR s (the dashed vertica l lines depict the locations of the quantizer threshold s). SNR (dB) − 10 0 5 10 20 3 -bit optimal 0 . 0667 0 . 4817 0 . 9753 1 . 5844 2 . 8367 3 -bit benchmark 0 . 0557 0 . 4707 0 . 9547 1 . 5332 2 . 8084 T ABLE II M U T UA L I N F O R M AT I O N ( I N B I T S / C H A N N E L U S E ) A T D I F F E R E N T SNR S . Optimal Inp ut Distributions : The optimal in put distributions (given by th e cutting -plane alg orithm) co rrespon ding to th e optimal quan tizers obtained above are dep icted in Fig . 3, fo r different SNR values. The locations of the optim al qu antizer thresholds are also shown (by the dashed vertical lines). Binary signaling is found to be optimal at low SNR s, an d the numb er of mass po ints increases (first to 3 and then to 4 ) with increasing SNR . Fu rther in crease in SNR eventually leads to the uniform 4 -P AM input, thus approach ing th e capacity bound of 2 bits. I t is worth no ting th at a ll the o ptimal in puts we obtained have 4 or less mass points, where as Prop osition 2 is looser a s it guarantees the achiev a bility of capacity using at most 5 points. B. 3-bit Symmetric Quantization For 3 -bit symmetric quantization , we n eed to op timize over a space of 3 parameters : { 0 < q 1 < q 2 < q 3 } , with the quantizer threshold s being {± q 1 , ± q 2 , ± q 3 } . Instead of brute force search , we use an altern ate optimizatio n proced ure fo r joint optimization of the in put and the quan tizer in this case. Due to lack of space, we ref er the reader to [10] for d etails, and proceed directly to the numer ical results. (T able II) Comparison with th e benchmark: As for 2 -bit quantization considered earlier, we find that the b enchmar k scheme per- forms quite well at lo w SNR s with 3 -bit quantization also. At − 10 dB SNR , for instance, the benchmark scheme achieves 83% of th e capa city achiev able w ith an o ptimal qu antizer choice. T able II gives the compariso n for different SNR s. Optimal Input Dis tributions : Although not depicted here, we again observe (as for the 2 -bit case) that the optimal inputs obtained all h ave at most K points ( K = 8 in this case), while Proposition 2 guaran tees the achiev ability of capacity by at most K + 1 points. Of course, Proposition 2 is applica ble to an y quantizer choice (and n ot just optimal sym metric quantizers that we consider in this section), it still leaves us with the question wh ether it can be tightened to guaran tee achiev ab ility of capacity with at most K points. C. Comparison with Unquantized Observations W e now comp are the capa city results obtained above with the case whe n the receiver ADC has infinite precision . T ab le III provid es these resu lts, and the corresp onding plo ts ar e shown in Fig . 4. W e ob serve that at low SNR s, lo w -precision quantization is a very feasible optio n. For instance , at - 5 dB SNR , even 1 -bit receiver q uantization ach iev es 6 8% of the capacity achiev able with infin ite-precision. 2 - bit q uantization at the same SNR provides as much as 90% o f the in finite- precision ca pacity . Such high figu res are und erstandable, since if noise do minates the message signal, incre asing the quan tizer SNR (dB) − 5 0 5 10 15 1-bit ADC 0 . 1353 0 . 3689 0 . 7684 0 . 9908 0 . 9999 2-bit ADC 0 . 1792 0 . 4552 0 . 8889 1 . 4731 1 . 9304 3-bit ADC 0 . 1926 0 . 4817 0 . 9753 1 . 5844 2 . 2538 Unquanti zed 0 . 1982 0 . 5000 1 . 0286 1 . 7297 2 . 5138 T ABLE III C A PAC I T Y ( I N B I T S / C H A N N E L U S E ) AT VAR I O U S SNR S . −5 0 5 10 15 20 0 0.5 1 1.5 2 2.5 3 3.5 4 SNR (dB) Capacity (Bits/Channel Use) Infinite precision ADC 1−bit ADC 2−bit ADC 3−bit ADC Fig. 4. Capaci ty with 1-bit, 2-bit, 3-bit, and infinite-pre cision ADC. precision beyond a poin t does not help much in distinguishing between different sign al levels. Howe ver, we surpr isingly fin d that e ven if we consider moderate t o high SNR s, the loss due to low-precision sampling is still very acceptab le. At 10 dB SNR , for example, the cor respondin g ratio f or 2 -bit quantizatio n is still a very high 85% , while at 20 dB, 3 -bit quantization is enou gh to achieve 8 5% of the infinite-pr ecision capacity . Similar encour aging results hav e been repo rted earlier in [12], [13] also. Howe ver , the inp ut alphabe t in these works was taken as b inary to b egin with, in which case th e go od perfor mance with low-precision outp ut quantization is perh aps less surprising. On th e o ther han d, if we fix the spectral e fficiency to that attained by an unquan tized system at 10 dB (which is 1 . 73 bits/channel use), we find tha t 2 -b it quantizatio n incurs a loss of 2 . 30 dB (see T able IV). From a practical viewpoint, this penalty in p ower is more significant compar ed to the 15% loss in spectral efficiency o n using 2 -bit quantization at 10 dB SNR . This sugg ests, for e x ample, tha t the impact of lo w-p recision ADC s hould be weathered by a moderate reductio n in the spec- tral efficienc y , rather than by increasing the transmit power . Spectra l Effic ienc y (bits per channel use) 0 . 25 0 . 5 1 . 0 1 . 73 2 . 5 1-bit ADC − 2 . 04 1 . 79 − − − 2-bit ADC − 3 . 32 0 . 59 6 . 13 12 . 30 − 3-bit ADC − 3 . 67 0 . 23 5 . 19 11 . 04 16 . 90 Unquanti zed − 3 . 83 0 . 00 4 . 77 10 . 00 14 . 91 T ABLE IV SNR ( I N D B ) R E Q U I R E D F O R A G I V E N S P E C T R A L E FFI C I E N C Y . V . C O N C L U S I O N S Our Shann on-theo retic in vestigation in dicates the feasibility of low-precision ADC for design ing f uture high- bandwidth commun ication sy stems su ch as those operating in UWB and mm-wav e ba nd. The small r eduction in sp ectral efficiency due to low-precision ADC is acceptable in such systems, giv en that the av ailable ba ndwidth is plen tiful. Curren t researc h is therefor e focussed on developing ADC-constrained algor ithms to perform receiver tasks such as carrier and timing sy nchro- nization, channel estimation an d equalization. An unresolved techn ical issue concer ns the numb er o f mass points req uired to achieve capacity . While we have shown that the capacity for the A WGN ch annel w ith K -bin output quantization is achiev able by a discrete input distribution with at mo st K + 1 po ints, numer ical compu tation of op timal inpu ts reveals that K mass p oints are sufficient. Can this be proven analytically , at least for symmetric qu antizers? Ar e symm etric quantizers optimal? Another problem for futur e in vestigation is whether our r esult regardin g the o ptimality of a discrete input can be g eneralized to other cha nnel mod els. 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