On the Capacity of MISO Channels with One-Bit ADCs and DACs

Transcription

1 1 On the Capacity of MISO Channels with One-Bit ADCs and DACs Yunseo Nam Heedong Do Yo-Se Jeon and Namyoon Lee Astract arxiv: v1 [cs.it] 1 Nov 18 A one-it wireless transceiver is a promising communication architecture that not only can facilitate the design of mmwave communication systems ut also can extremely diminish power consumption. The non-linear distortion effects y one-it quantization at the transceiver however change the fundamental limits of communication rates. In this paper the capacity of a multiple-input single-output (MISO) fading channel with one-it transceiver is characterized in a closed form when perfect channel state information (CSI) is availale at oth a transmitter and a receiver. One major finding is that the capacityachieving transmission strategy is to uniformly use four multi-dimensional constellation points. The four multi-dimensional constellation points are optimally chosen as a function of the channel and the signalto-noise ratio (SNR) among the channel input set constructed y a spatial lattice modulation method. As a yproduct it is shown that a few-it CSI feedack suffices to achieve the capacity. For the case when CSI is not perfectly known to the receiver practical channel training and CSI feedack methods are presented which effectively exploit the derived capacity-achieving transmission strategy. Index Terms MISO channel one-it quantization channel capacity and spatial lattice modulation. I. INTRODUCTION A. Motivation The use of large andwidths at mmwave ands is a key technology for future wireless systems supporting high data throughput [1] []. Although the large signal andwidth is ale to provide a significant improvement of data rates it complicates the design of analog and analog-digital mixed hardware components in transceiver. In particular very high-speed digitalto-analog converters (DACs) and analog-to-digital converters (ADCs) at the transceiver can lead Y. Nam H. Do Y.-S. Jeon and N. Lee are with the Department of Electrical Engineering POSTECH Pohang Gyeonguk South Korea ( {edwin6 doheedong yose.jeon nylee}@postech.ac.kr).

2 to significant power consumption [3] [5]. For example it has shown in [3] that an ADC consumes two Watts when quantizing a received signal with the sampling rate of Gsamples per second and 1-it precision per sample. In addition multiple transceiver chains each with a large amount of antenna elements i.e. hyrid-precoding [6] [7] are essentially needed to compensate a high path loss existing in mmwave channels and to offer high data rates y simultaneously sending multiple data streams. As a result the use of multiple transceiver chains in mmwave wireless systems additionally increase the circuit power consumption and the hardware cost. A simple solution for reducing the huge power consumption and the expensive hardware cost is to exploit low-precision DACs and ADCs at the transceiver. By reducing the numer of quantization its per sample the power consumption can e decreased exponentially [8] [1]. Such low-power and cost-effective solution however significantly alters the characteristics of the wireless system due to the nonlinear distortion effects of transmit and receive signals; therey it changes the fundamental limits of communication rates and the practical communication schemes achieving these limits. B. Related Works There have een an increasing research interest in oth understanding the information theoretic limits and designing the practical communication schemes using low-resolution DACs or ADCs. Considering one extreme case in which a transmitter employs infinite-precision DACs while a receiver uses one-it ADCs information theoretic limits were analyzed in [1] [17]. For example it was shown in [1] that an uniformly distriuted quadrature phase shift keying (QPSK) signaling achieves the capacity of a quantized single-input single-output (SISO) additive white Gaussian noise (AWGN) channel. For a non-coherent communication system i.e. no CSIR is availale a closed form expression of the capacity was derived as a function of the coherence time and the SNR y solving a convex optimization prolem [11]. With the derived expression it was demonstrated that the capacity-achieving input for a SISO Rayleigh-fading channel is the on-off QPSK. For the coherent communication scenario when oth CSIT and CSIR are availale the capacity expression of a MISO fading channel was derived in a closed form [1]. Specifically the capacity-achieving transmission method is to use the uniform QPSK modulation with the maximum ratio transmit (MRT) precoding. These results revealed that the use of a finite

3 3 numer of constellation points with the uniform distriution is optimal when the receiver has one-it resolution. Such capacity-achieving input distriution is different from the Gaussian input distriution which is known to e optimal when infinite-precision ADCs are employed. For the multiple-input multiple-output (MIMO) channel case an exact channel capacity expression in a closed form is still unknown [1] [17]. Instead of characterizing the exact channel capacity expression in a closed form several works focused on the asymptotic characterization of the channel capacity of a MIMO channel [1] [15]. In [1] an upper and a lower ound of the capacity were characterized y using the hyperplane-cutting argument in the extremely high SNR regime. Furthermore an asymptotical channel capacity expression was derived in the extremely low SNR regime y using a second-order expansion of the mutual information etween the input and the output of the MIMO channel [13] [15]. One important finding was that the capacity loss caused y the one-it quantization is unexpectedly small (1.96 db) compared to the case of using infinite-precision ADCs in the low SNR regime. The common limitation of the aforementioned studies however is that they ignore the effect of low-precision DACs of the transmit chains. Unlike the case of using one-it ADCs at receivers only a few works have focused on the capacity analysis for the case in which oth the receiver and the transmitter are equipped with one-it ADCs and DACs respectively. In [19] [] the channel capacity of a real-valued MIMO channel have een characterized under the assumption of perfect CSIR only. Specifically under the constraint of the inary phase shift keying (BPSK) constellation per each transmit antenna as a channel input an asymptotical expression of the constraint-capacity is derived as a function of the ratio etween the numer of transmit and receive antennas and the SNR when the numer of antennas goes to infinity. The key technique to find this capacity expression is the replica method [1] which has een widely used as a tool for statistical mechanics and information theory. Although the results in [19] [] provide a useful guidance on how the capacity ehaves in an asymptotical regime as a function of the important system parameters the enefits of exploiting CSIT are not revealed. In addition the major limitation of the works in [19] [] is the assumption of BPSK signaling per transmit antenna as channel inputs. This assumption excludes the possiility of using spatial modulation methods [] [7] where information its are jointly mapped across a set of activated indices of spatial in-phase and quadrature dimensions

4 System Model MISO systems with one-it transceiver # 1 Transmit processing One-Bit DAC One-Bit DAC One-Bit DAC One-Bit DAC Re Im Re Im RF RF # # M RF Re Im One-Bit ADC One-Bit ADC Receive processing One-Bit DAC One-Bit DAC Re Im RF Rerepresentation Fig. 1. An illustration of the MISO system with one-it transceiver. Complex-valued & vector form Real-valued & matrix form and the BPSK symols of the activated dimensions. Under the one-it DACs constraint a set of all possile channel inputs is generated using spatial lattice modulation (SLM) method [7] which is known as a generalization of spatial modulation techniques [] [6]. Since the joint information mapping y SLM maximizes the numer of possile input alphaets in general it might achieve higher transmission rates compared to the case when using the BPSK signaling per transmit antennas as in [19] []. C. Contriutions In this paper we consider a MISO channel in which a transmitter sends a message using M multiple transmit antennas (or interchangealy transmit chains) each with one-it DACs and a receiver decodes the message with a single receive antenna with one-it ADCs. Our goal in this paper is to understand the joint impact of one-it ADCs and DACs on the capacity of a MISO channel when the perfect knowledge of CSIT and CSIR is availale. In particular we characterize the capacity expression with a closed form for the MISO channel with one-it ADCs and DACs. The key idea to find the capacity expression is to construct the channel input vectors in an M-dimensional input space { 1 1} M y the joint information mapping across the transmit antennas the real and the imaginary components [7]. Then we optimize the input distriution to maximize the mutual information etween the channel inputs and outputs. We first start with the SISO channel with one-it ADCs and DACs i.e. M = 1. One major finding is that the capacity-achieving transmission method is to adaptively exploit oth QPSK and spatial modulation as a function of the channel amplitude phase and the SNR. This result is different from the capacity result of the SISO channel when using one-it ADCs ut employing infinite-precision DACs. In this case the uniform QPSK constellation with precoding to align

5 5 the channel phase is shown to e optimal. Another important oservation is that one-it CSIT is sufficient to achieve the capacity. This is quite appealing in practice ecause the receiver needs to send ack just inary information to the transmitter. Whereas for the case of the SISO channel with one-it ADCs and infinite-precision DACs infinite amount of feedack its are required to perform precoding for the phase alignment. By comparing the known capacity results we provide a complete picture on how oth one-it ADCs and DACs changes the capacity of the SISO channel. We extend the results of the SISO case into the MISO case. For the MISO channel it is possile to construct the channel input set with 9 M 1 different non-zero input vectors y the joint information mapping rule in [7]. Using this channel input set we characterize the channel capacity y deriving the capacity-achieving input distriution under the average transmit power constraint. Our major finding is that the capacity-achieving transmission strategy is to send information its y uniformly using four signal points in { 1 1} M. The optimal four signal points are selected among the SLM signals as a function of the channel and the SNR. In addition ( ) it is shown that log 9 M 1 feedack its are sufficient to achieve the capacity. This result is also different from the capacity result of the MISO channel with one-it ADCs and infinite-precision DACs in which the uniform QAM signaling with MRT eamforming is shown to e optimal. We also present a practical channel-training and CSI feedack method for the MISO channel with one-it ADCs and DACs when perfect CSI is not availale. The proposed idea is to exploit a supervised-learning approach in [8] [3]. In the phase of the channel-training the transmitter sends the channel input vectors repeatedly which are chosen from 9M 1 distinct susets of SLM signal set. Using the received signals during the channel-training the receiver empirically estimates the entropy functions for each suset. From the estimated entropy functions the receiver selects the est suset that maximizes the channel capacity. Then the index of the selected suset is sent ack to the transmitter via a finite-rate feedack link. To reduce the channel-training and CSI feedack overheads a su-optimal channel-training and CSI feedack method is also proposed in which only channel input vectors with the maximum instantaneous power level are sent during the channel-training phase. By simulations it is shown that the proposed method is ale to achieve the capacity within. its/sec/hz for all SNRs when M = and the total training-length of 3 and -it CSI feedack are employed respectively.

6 6 Our paper is organized as follows. Section II descries the system model. In Section III the capacity expression of the SISO channel with one-it ADCs and DACs is characterized. Then Section IV provides a closed form expression of the channel capacity when the transmitter has multiple antennas. When perfect CSI is not availale a practical channel-training and a CSI feedack method are presented in Section V. Some simulation results are provided to validate our results in Section VI. Section VII concludes the paper with some discussion aout possile extensions. II. SYSTEM MODEL We consider a MISO system in which a transmitter equipped with M antennas sends information its to a receiver equipped with a single antenna. As illustrated in Fig. 1 we assume one-it transceiver model where the transmitter uses one-it DACs and the receiver uses one-it ADCs to extremely reduce the power consumption. Let the channel input vector sent y the transmitter e x = [ x 1 x... x M ]. We also denote h C 1 M as a frequency-flat aseand-equivalent channel etween the transmitter and the receiver. The aseand-equivalent channel considered in this paper can include analog eamforming effects y considering the hyrid-eamforming architecture in [6] [7]. Here the frequency-flat assumption might e valid for the millimeterwave frequency ands in which delay-spreads are limited with analog eamforming in line of sight (LOS) channel environments []. When the channel input vector x is sent the complex-valued aseand received vector with one-it ADCs is ȳ = sign ( h x + z ) { 1 j 1 + j +1 j +1 + j} (1) where z C represents the additive noise distriuted as circularly-symmetric complex Gaussian random variale with zero-mean and variance of σ i.e. z CN( σ ). Here sign( ) : R {1 1} denotes the one-it quantization function with sign(c) = 1 if c and 1 otherwise. This sign function is separately applied to the real and imaginary component of each received signal.

7 7 Without loss of generality we rewrite the complex-valued input-output relationship in (1) into an equivalent real-valued representation as y = sign (Hx + z) () h where H = Re h Im h h Im R M y = [ȳ Re ȳ Im ] { 1 +1} = Y x = [ x Re Im] x R M Re and z = [ z Re z Im ] R. This real-valued representation will e used in the sequel. A. Channel Input Construction When information its are encoded with one-it DACs each transmit antenna can only send a BPSK signal per in-phase (or quadrature) component. Unlike the previous modulation methods under the one-it DACs constraint [19] [] we consider spatial lattice modulation (SLM) in [7] which is known as a generalized method of the spatial modulation. The key idea of the SLM is to jointly map information its into one of 3 M 1 cuic lattice vectors in { 1 1} M y using M transmit antennas and the real and imaginary components per antenna. Each transmit signal of the SLM method represents a set of active dimensions and BPSK signals. Specifically when the m-th component of x i.e. x m is activated a BPSK signal x m { 1 +1} is sent. If the m-th component is deactivated the zero signal is sent. Since there are ( M) i possile ways to select i activated dimensions a total of M ( ) M i = 3 M (3) i i= distinct information vectors can e generated using M transmit antennas with one-it DACs. If we discard the all-zero vector 3 M 1 signal vectors can e used as channel inputs. We define the corresponding channel inputs of the SLM method as X = { 1 +1} M \ M. In general X can e regarded as a collection of spatially modulated signals with different sparsity level. With this view-point we define some properties of X. Property 1 (Instantaneous transmit power): The instantaneous transmission power (i.e. sparsity level) of any signal point in X elongs to {1... M} namely x {1... M} ( x X). ()

8 8 TABLE I LOOK-UP TABLE FOR SLM WITH CUBIC LATTICES (M = ). x T X u x T X u [±1 ] [±1 ±1 ] [ ±1 ] [±1 ±1 ] [ ±1 ] X 1 [±1 ±1] [ ±1] [ ±1 ±1 ] X [±1 ±1 ±1 ] [ ±1 ±1] [±1 ±1 ±1] [ ±1 ±1] [±1 ±1 ±1] X 3 [ ±1 ±1 ±1] [±1 ±1 ±1 ±1] X Property (Power-level signal suset): We define disjoint susets X u X each of which contains vectors in X with the same instantaneous power level u i.e. X u = { x x = u} (5) where X = M u=1 X u. The cardinality of the u-th power level suset is X u = ( M) u. Example 1 : Suppose M =. In this case it is possile to generate 8 distinct channel inputs y using the SLM method. The corresponding 8 vectors are listed in Tale I. Note that the channel input set generated y the conventional spatial modulation and spatial multiplexing are and X MG X SM = X = X respectively. Since X SM and X MG are the susets of the SLM signal set X = { 1 +1} \ with the proper real and imaginary mapping the SLM signal set in [7] generalizes the existing spatial modulation and spatial multiplexing methods in [] [6]. Property 3 (Rotationally invariant signal suset in X u ): We define X uk X u as the kth suset of X u which contains four elements that satisfy the invariant property of the 9 rotation. Since X u = ( M) u u it is possile to generate K u = ( M) u u disjoint susets X uk X u where k = {1... K u }. Here we say that a set X uk is rotationally invariant if X uk = { R x uk R 1 x uk R x uk R 3 x uk } xuk X uk (6) u

9 9 where R = M I M I M M is a 9 rotation matrix. From the definition the channel input set X can e decomposed with disjoint susets X uk as X = M u=1 K u k=1 X uk X u1 k 1 u 1 k 1 u k X u k = φ. (7) Example : When M = we otain two susets of X 1 as X 11 = X 1 = (8) In a complex-valued representation X 11 and X 1 can e equivalently written as X 11 = j j X 1 = j j. (9) In a spatial domain the rotationally invariant property of X uk implies that the elements in X uk share the same antenna activation pattern. Property (The average transmission power): For each X uk we define the corresponding proaility mass function as p uk = Pr [ x X uk ]. Then the average transmit power is E [ x ] = M u=1 u K u k=1 p uk P t (1) where P t is the average power constraint in the system. Throughout this paper we define the signal-to-noise ratio (SNR) as SNR = P t σ. (11) B. Channel Capacity In this paper we consider the channel capacity when the perfect knowledge of CSIT and CSIR are availale at the transceiver. When the channel is fixed over the duration of spanning codewords the capacity of the constant MISO channel with one-it ADCs and DACs is otained y solving the following optimization prolem: C = max Pr[x] I (x; y H = H) H = h 1 h s.t. E [ x ] Pt x X = { 1 +1} M \ M (1)

10 1 where I(x; y) = x Xy Y Pr[x]Pr[y x] log Pr[y x] Pr[y]. (13) Our goal in this paper is to find the closed form expression of the capacity y deriving the optimal input distriution over X. III. SISO CHANNEL WITH ONE-BIT TRANSCEIVER In this section we characterize the fundamental limit of the SISO channel when one-it ADCs and DACs are employed at the transceiver. Throughout this section we omit the index k for the notational simplicity i.e. X u = X u1 and x u1 = x u ecause there is an unique suset X uk in each power level set X u. A. Capacity of the SISO Channel The following theorem is the main result of this section. Theorem 1: For a given channel realization i.e. H = H let H X u functions for different power level input susets i.e. H X u = H (Q ( σ h n x u )) e the sum of inary entropy (1) where H (x) = x log x (1 x) log (1 x) for < x < 1. Then the capacity of the SISO channel with one-it ADCs and DACs is { } H X 1 if H X 1 HX {P t =1} { } C SISO = ( P t )H X 1 (P t 1)H X if H X 1 > HX {1 < P t < } { } H X if H X 1 > HX {P t =}. Proof: The proof consists of two steps. We first specify the property of the optimal input distriution which is stated in the following lemma: Lemma 1: For the SISO channel with one-it ADCs/DACs there exists a capacity-achieving input distriution which is uniform in each X u namely (15) Pr [ x = R i ] p u x u = i { 1 3} (16)

11 11 for u {1 }. Proof: See Appendix A. By leveraging Lemma 1 we derive the capacity-achieving input distriution y finding the optimal proailities p 1 and p so as to maximize the mutual information under the average power constraint. We start y rewriting the input-output relationship in () for the SISO channel as y y = 1 sign ( h = 1 x + z ) 1 y sign ( h x + z ). (17) The mutual information etween the input x and the output y for a given channel H is I(x; y H = H) = H(y H = H) H(y x H = H). (18) To compute H(y H = H) in (18) we calculate the conditional proaility of y when the channel is given as H = H. We first compute the conditional proaility of y = [1 1] which is Pr [ y = [1 1] H = H ] 3 = Pr [ y = [1 1] x=r i x u H = H ] Pr [ x=r i ] x u (a) = () = u=1 i= u=1 i= u=1 ( 3 Pr [ y n = 1 x=r i x u h ] ) n Pr [ x=r i ] x u p u 3 i= Pr [ y n = 1 x=r i x u h ] n (19) where (a) follows from the conditional independence of y 1 and y for the given channel and input vectors and () is due to the result from Lemma 1. By using the following identities i.e. h 1 R = h and h R = h 1 () we further simplify 3 Pr [ y n = 1 x=r i x u h ] n =Pr[z1 > h 1 x u]pr[z > h x u]+pr[z 1 > h x u]pr[z > h 1 x u] i= +Pr[z 1 > h 1 x u]pr[z > h x u]+pr[z 1 > h x u]pr[z > h 1 x u] (a) = 1. where (a) follows from Pr[z > x] + Pr[z > x] = 1. By plugging (1) into (19) we otain Pr [ y = [1 1] H = H ] p u = = 1. () u=1 (1)

12 1 From () and the symmetry of the channel inputs we conclude that the channel outputs are uniformly distriuted regardless of the channel realizations. As a result the channel output entropy is H(y H = H) =. (3) We also compute the conditional entropy H (y x H = H) in (18). From the independence of z 1 and z we otain Then the conditional entropy is computed as H (y x H = H) (a) = H (y x H = H) = H(y 1 x h 1 ) + H(y x h ). () () = u=1 p u 3 i= { ( ) ( )} H y 1 x=r i x u h 1 + H y x=r i x u h { ( p u H y1 x=x u h ) ( 1 + H y x=x u h )} (5) u=1 where (a) follows from Lemma 1 and () comes from the identities in (). To calculate H ( y n x=x u h ) n we compute the conditional proaility of yn as ( ) Pr[y n = 1 x = x u h n ] = Pr[z n > h n x u ] =1 Q σ h n x u (6) where Q(x) = x 1 exp( t π )dt is the tail proaility of the standard Gaussian distriution. Using (6) the conditional entropy in (18) oils down to H (y x H = H) (a) = p u H (Q u=1 ( )) σ h n x u (7) where (a) is due to the symmetry property of the inary entropy function i.e. H (x) = H (1 x). As a result the mutual information of the SISO channel with one-it ADCs and DACs is I(x; y H = H) = H(y H = H) H (y n x H = H) = p u H (Q u=1 ( σ h n x u )) = p 1 H X 1 p H X. (8)

13 13 To find the capacity we need to optimize the input distriution i.e. p 1 and p. From (8) the maximization of I(x; y H = H) with respect to p 1 and p under the average power constraint p 1 + p P t is equivalent to solve the following linear programming prolem: min p 1 H X 1 p 1 p + p H X such that p 1 + p P t p 1 + p = 1 p 1 and p. (9) Using the standard simplex method we otain a closed form solution of the capacity-achieving input distriution as { } (1 ) if H X 1 HX {P t =1} (p 1 p )= { } ( P t P t 1) if H X 1 > HX {1 < P t < } (3) { } ( 1) if H X 1 > HX {P t =}. By invoking (3) into (8) the capacity of the SISO channel with one-it ADCs and DACs is given as in (15). This completes the proof. B. Implication To shed further light on the importance of Theorem 1 we explain the capacity expression for three cases. Case 1: When { } H X 1 HX the capacity-achieving transmission strategy is to use the spatial modulation method i.e. X 1 uniformly. This is ecause H X 1 H X implies that the effect of the phase alignment etween the inputs and the channel is more important than the effect of transmission power. In addition in the case of P t = 1 the optimal transmission method is also to uniformly use the input signals in X 1 regardless of channel realizations. The reason is that the use of input signals in X is infeasile to satisfy the average power constraint. In these cases the channel capacity expression in Theorem 1 oils down to ( )) C SISO = H (Q σ h n x 1 = H Q h Re σ H Q h Im σ. (31)

14 1 Fig.. The eight possile channel input vectors for a single-antenna transmitter using one-it DACs. Case : When { } H X 1 > HX and {1 < P t < } the capacity-achieving transmission method is to use all signal points generated y the spatial lattice modulation method. Here the signal points in X 1 and X are used with the proaility of P t and P t 1 respectively. To provide an intuition on the design of the practical communication scheme we can equivalently rewrite the mutual information expression in (8) as I(x; y H = H) = p 1 H X 1 p H X = p 1 ( H X 1 ) + p ( H X ). (3) From (3) one can achieve the same transmission rates y time sharing etween the spatial modulation method and the QPSK signaling with the time fractions of p 1 and p. Since the channel is etter matched with the input signals in X than in X 1 the transmitter first harnesses X uniformly during the P t 1 fractions of the entire channel uses. Since P t < it is impossile to use the inputs in X for all the channel uses to satisfy the average power constraint. Therefore for the remaining fractions of the time i.e. P t it exploits the inputs in X 1. For this case the capacity expression is ( )) ( )) C SISO = ( P t ) H (Q σ h n x 1 (P t 1) H (Q σ h n x = ( P t ) Q H h Re σ + H Q h Im σ (P t 1) Q H ( h Re + h Im ) σ +H Q ( h Re h Im ) σ. (33)

15 15 Fig. 3. The channel capacity comparison when M = 1 as a function of the channel phase magnitude and the SNR when using two different channel input sets i.e. X 1 and X. Case 3: For the case of { } H X 1 > HX and {P t =} the uniform QPSK signaling achieves the capacity ecause the input vectors in X are etter aligned with the channel vector than those in X 1 and the transmission power is sufficient to meet the average power constraint. In this case the capacity ecomes C SISO = H (Q ( )) σ h n x = H Q ( h Re + h Im ) σ Q H ( h Re h Im ) σ. (3) Remark (Transmission strategy according to the SNR): For the SISO fading channel the region of the channel realizations satisfying the condition of H X 1 > HX is depicted in Fig. 3. This figure elucidates that the optimal transmission strategy depends on the SNR. For example when we set h = + j and the noise variance σ = 1 the spatial modulation method i.e. X 1 achieves a higher spectral efficiency than the QPSK signaling i.e. X. Whereas at the lower SNR i.e. σ = 9 the use of QPSK signaling achieves a higher spectral efficiency than the spatial modulation method. As the SNR increases the region of the channel values where the use of the spatial modulation method is the optimal is also expanded. This implies that the phase alignment etween the channel and the inputs ecomes more dominant in the capacity than the instantaneous transmission power at the high SNR regime. Corollary 1: One-it CSIT achieves the capacity of the SISO channel with one-it ADCs and DACs. Proof: The proof is evident from the interpretation of Theorem 1. To achieve the capacity the receiver needs to send ack one-it feedack information that indicates whether to use the

16 16 spatial modulation method or not. Since the transmitter knows the average power constraint P t it is possile to use the capacity-achieving transmission strategy with the one-it feedack information. Corollary : Without CSIT the capacity using one-it ADCs and DACs is H X 1 if {P t = 1} C SISO CSIR = ( P t )H X 1 (P t 1)H X if {1 < P t < } (35) H X if {P t = }. Proof: Since the transmitter has no information whether the channel is etter aligned to X 1 or X the optimal strategy is to transmit X as much as possile which uses more instananous power than X 1. The capacity loss due to the lack of CSIT is therefore ( ) { } C SISO CSIT = (P t 1) H X HX 1 if H X 1 < HX { } (36) if H X 1 > HX. In Fig. 3 it has een shown that the portion of H X 1 < HX increases with the SNR. From (36) the capacity gap due to the lack of CSIT i.e. C SISO CSIT also increases. In the low SNR regime however the loss disappears; this implies that the impact of CSIT is negligile. C. Capacity Loss Analysis y One-Bit DACs In this susection we characterize the capacity loss y the use of one-it DACs at the transmitter. To accomplish this we compare our capacity result in Theorem 1 with the result in [1] where infinite-precision DACs are used at the transmitter. When infinite-precision DACs are employed the SISO capacity expression with the perfect CSIT and CSIR is derived in [1]. The capacity-achieving input is the rotated QPSK namely { [ ] 1 h Re h Im X ADCs = x x X }. (37) h Re + h Im h Im h Re By perfectly align the channel inputs to the channel direction the channel can e decoupled into two real-valued su-channels each with the same channel gain. For the decoupled channels the BPSK signaling is shown to e optimal which leads to the the simple capacity expression as ( ( ))) C SISO ADCs (1 = H Q σ ( h Re + h Im ). (38)

17 17 Fig.. Phase misalignment etween the channel and the constellation points. By using (15) and (38) the capacity loss y the use of one-it DACs for the SISO channel when P t = is characterized as in the following corollary. Corollary 3: The capacity loss C SISO DACs = CSISO ADCs CSISO is ( ( )) H X 1 H Q ( h σ Re + h Im ) if C SISO DACs = H X H ( Q ( )) ( h σ Re + h Im ) if { H X 1 < HX { H X 1 > HX To provide an intuition we derive an upper ound of the loss in (39). As illustrated in Fig. the capacity of the SISO channel with one-it ADCs and DACs can e equivalently rewritten as ( ( )) ( ( )) C SISO = max { H Q σ ( h Re + h Im ) cos θ H Q σ ( h Re + h Im ) sin θ H (Q ( σ ( h Re + h Im ) cos ( π θ ) )) H ( Q } }. (39) ( )) } ( π ) σ ( h Re + h Im ) sin θ () where θ denotes the phase of h [31]. To derive an upper ound of (39) we derive a lower ound of C SISO y considering a suoptimal strategy which exploits X 1 or X y simply oserving θ i.e. X 1 X if if sin ( π θ) < sin θ sin ( π θ) > sin θ. (1)

18 18 The threshold in (1) can e easily otained as ˆθ 6.56 which satisfies tan ˆθ = 1. Note that the threshold is iased to use X due to the different instantaneous transmission power levels. From (1) the lower ound of () is otained as ( ( ))) C SISO (a) (1 H SISO Q ( h σ R Re + h Im ) sin θ if θ > 6.56 ( ( ))) (1 H Q ( h σ Re + h Im )(1 sin θ) if θ < 6.56 where (a) is ecause H (Q( x)) is a decreasing function with respective to x. Compared to the capacity with infinite-precision DACs in (38) the power loss factor is at most () P loss = max { sin θ 1 sin θ } 7dB. (3) In addition if we assume that the channel phase is uniformly distriuted i.e. Rayleigh fading channel environments the power loss due to the use of one-it DACs in an ergodic sense is { E h [P loss ] = ˆθ } π (1 sin θ)dθ + sin θdθ 3.dB. () π This result reveals that the loss due to the use of one-it DACs with one-it feedack is not significant compared to the case of using infinite-it DACs with the infinite amount of feedack its. ˆθ IV. CAPACITY OF MISO CHANNELS WITH ONE-BIT TRANSCEIVER In this section we extend the results in Section III for the MISO channel with one-it ADCs and DACs. A. Capacity of the MISO Channel We first present our main result of this section. Theorem : For a given channel realization i.e. H = H the capacity of the MISO channel with one-it ADCs and DACs is C MISO = M K u u=1 k=1 p uk HX uk (5) where p uk is the optimal solution of the following linear programming prolem:

19 19 min p 11...p MKM such that M K u u=1 k=1 M u=1 M p uk H X uk K u u p uk P t k=1 K u u=1 k=1 p uk = 1 p uk. (6) Proof: The proof resemles with that of the SISO case. Using the rotationally invariant property of the optimal input distriution shown in Lemma 1 we consider the input distriution which is uniformly distriuted over four symmetric input vectors in X uk i.e. Pr [ x = R i ] p uk x uk = i { 1 3} (7) for u {1... M}. Recall that the input-output relationship in () for the case of the MISO channel is y 1 sign ( h = 1 x + z ) 1 y sign ( h x + z ). (8) Then the mutual information etween the input x and the output y when the channel is given y H = H is I(x; y H = H) = H(y H = H) H(y x H = H). (9) To compute H(y H = H) in (9) we first compute Pr [ y = [1 1] H = H ] M K u 3 = Pr [ y = [1 1] x = R i x uk H = H ] Pr [ x = R i ] x uk u=1 k=1 i= M K u (a) = () = u=1 k=1 i= M K u u=1 k=1 ( 3 Pr [ y n = 1 x = R i x uk h ] ) n Pr [ x = R i ] x uk p uk 3 i= Pr [ y n = 1 x = R i x uk h ] n (5) where (a) is ecause the independence of z 1 and z and () follows from Lemma 1. By expansion we rewrite 3 i= Pr [ y n = 1 x = R i x uk h ] n = 1 (51)

20 using the similar approach in (1). Therefore we otain Pr [ y = [1 1] H = H ] = M K u u=1 k=1 p uk = 1. (5) By symmetry the channel outputs are uniformly distriuted regardless of the channel values. Consequently the channel output entropy ecomes H(y H = H) =. (53) Now we calculate the conditional entropy given x as a form of the weighted sum of inary entropy functions as H (y n x H = H) = = = M K u u=1 k=1 M K u p uk 3 i= u=1 k=1 M K u p uk u=1 k=1 { ( ) ( )} H y 1 x = R i x uk h 1 + H y x = R i x uk h { ( p uk H y1 x = x uk h ) ( 1 + H y x = x uk h )} H (Q ( )) σ h n x uk. (5) As a result the mutual information of the MISO channel with one-it transceivers is I(x; y H = H) = H(y H = H) H (y n x H = H) = M K u u=1 k=1 p uk H (Q ( )) σ h n x uk = M K u u=1 k=1 p uk H X uk. (55) By invoking the optimal distriution from (6) into (55) we otain the capacity expression in Theorem. This completes the proof. B. Implication The capacity expression in (5) is unwieldy to attain a clear intuition ecause the optimal solution of p uk depends on the average power constraint P t. To avoid this we focus on the case when the average transmission power is sufficient to use the signals sets with the instantaneous power of M i.e. P t = M. In this case the capacity expression is simplified as the following corollary. Corollary : When P t = M the capacity of the MISO channel with one-it transceivers is { } C MISO = min H X uk. (56) uk

21 1 Proof: When P t = M any signal SLM suset X uk can e harnessed i.e. the timesharing technique is not necessarily needed to satisfy the average transmit power constraint. Consequently the optimal transmission strategy is to use four rotationally invariant SLM signal points i.e. X uk that minimizes H X uk Corollary 5: When P t = M log ( 9 M 1 the capacity of the MISO channel with one-it transceivers. depending on the channel and the SNR. ) feedack its for CSIT are sufficient to achieve Proof: The capacity is achievale y sending ack the index of the suset X uk that yields the minimum value of the average inary entropy functions H X uk from a receiver. Since there exists M ( M ) ( ) u=1 u u = 9M 1 numer of susets log 9 M 1 feedack its are enough to achieve the capacity in (56). Remark 3 (Capacity loss y the use of one-it DACs): When infinite-resolution DACs are employed the capacity is achieved y the QPSK modulation with MRT precoding as shown in [1]. In this case the MISO channel is equivalent to a SISO channel with the channel gain of h 1. Therefore the capacity is otained as ( (1 H C MISO ADCs = With (57) the capacity loss due to the use of DACs is { C MISO DACs = min uk H X uk Q } H (Q ( ))) σ h 1. (57) ( )) σ h 1. (58) It is notale that the capacity in (57) can only e achievale when infinite amount of CSIT feedack its are availale. Whereas when using one-it transceivers the capacity is achievale ( ) with log 9 M 1 feedack its. Remark (Generalization to the multi-it DACs): For the case in which multi-it DACs are employed at the transmitter it is possile to generate the channel input set using the SLM method. For example when two-its DACs are used possile channel input set is X = { 1 1 } M /{} where X = M ( M ) i= i 5 i 1 = 5 M 1. In this case there exists 5 M 1 numer of susets i.e. X uk each with four SLM signal points that have the same antenna activation pattern. By deriving the optimal time-sharing transmission scheme among the 5M 1 numer of possile susets under the average power constraint one can find the capacity expression for the MISO channel with multi-it DACs and one-it ADCs.

22 V. CHANNEL TRAINING AND FEEDBACK In this section we propose a practical capacity-achieving downlink transmission technique including channel training and feedack methods for MISO channels with one-it transceiver. For simplicity we focus on the case of P t = M which enales us to use the simple capacity expression in Corollary. The key idea of our proposed strategy is to empirically learn the entropy functions for each SLM susets i.e. H X uk for u k y repeatedly sending the input vectors in X uk as a training sequence. This idea extends an implicit channel-learning method developed in our prior work [8] [3]. This strategy allows the receiver to estimate H X uk downlink channel itself h directly instead of estimating the C 1 M. In [8] [9] this implicit channel-learning method is shown to e effective compared to the direct channel estimation ecause the accuracy of the direct channel estimation method is poor when using one-it ADCs at the receiver. Then using the estimate of H X uk the receiver determines the est SLM suset which achieves the capacity. Once the optimal index of the SLM suset is found the receiver sends it ack to the transmitter via a finite-rate feedack link. To accomplish the proposed strategy we present two channeltraining-and-feedack methods referred to as full training and dominant-set training. A. Full Training In the full training method the transmitter first sends L repetitions of all constellation vectors {x uk } uk for k {1... K u } and u {1... M} as a training sequence. In this case the training length ecomes 9M 1 L. This training sequence allows the receiver to otain L received vectors namely {y uk [l]} L l=1 associating with each channel input vector x uk. Using these L oservations the receiver empirically computes the likelihood function of y n for given x uk as ( ) Pr[y n = 1 x = x uk h n ] = Pr[z n > h n x k ] =1 Q σ h n x uk 1 L 1 {yukn [l]=1} (59) L where y ukn [l] is the n-th element of y uk [l] and 1{ } is an indicator function that yeilds 1 if an event is true and zero otherwise. From (59) the receiver also computes the empirical entropy function for the input set X uk as H X uk = H (Q ( σ h n x uk )) (a) H ( 1 L l=1 L 1 { y ukn [l] = 1 }) (6) l=1

23 3 where (a) follows from H (x) = H (1 x). Since we assume that the transmit power is sufficient (i.e. P t = M) as in Corollary the index of the capacity-achieving input set is determined as ( (u k 1 L ) = argmin H 1 { y ukn [l] = 1 }). (61) uk L Finally the receiver sends ack the est index (u k ) to the transmitter via a finite-rate feedack link which requires the rate of log M u=1 K u 3.17M feedack its/sec. One feature of the full training method is that when the numer of training repetitions L is sufficiently large the est input set determined from (61) is indeed the capacity-achieving input set. B. Dominant-Set Training One drawack of the full training method is that its training overhead exponentially increases with the numer of transmit antennas i.e. 9 M 1 L. This overwhelming overhead drastically reduces the throughput of wireless systems particularly when the numer of transmit chains is large. To resolve this prolem we also present a dominant-set training method which requires l=1 a less training overhead than the full training method. The idea of the proposed dominant-set-training method is to empirically estimate H X Mk the channel input sets which have the maximum instantaneous transmission power {X Mk } k { 1 +1} M. Notice that the channel input sets X Mk for have a more chance to e selected as the capacity-achieving input set in the low SNR regime ecause they use a higher instantaneous power than the other input sets. Using this fact in the dominant-set training method the transmitter sends L repetitions of the channel input vectors {x Mk } k only. Due to the reduced numer of the possile channel inputs the training overhead is consideraly reduced to K M L = M 1 L which is significantly less than that of the full-training method. From the reduced training sequence the receiver computes the empirical entropy functions for {X Mk } k and then determines the index of the est input set as ( k 1 L = argmin H 1 { y Mkn [l] = 1 }). (6) k L Finally the receiver sends ack the est index k to the transmitter via a feedack link with a finite rate of log K M = M its/sec/hz. One feature of the dominant-training method is that it is ale to diminish oth the training overhead and the numer of feedack its compared to the full training method at the cost of the performance degradation. l=1

24 .5 Spectral efficiency [ps/hz] Infinite-precision ADCs/DACs One-it ADCs + infinite-precision DACs One-it ADCs/DACs One-it ADCs/DACs (CSIR only) SNR [db] Fig. 5. precision. Ergodic capacity comparison of the MISO fading channel when M = with the different types of ADCs and DACs VI. SIMULATION RESULTS In this section using simulations we evaluate the ergodic capacity of the MISO channel with one-it transceivers. In our simulation we define the SNR = M σ y setting P t = M. To evaluate the ergodic capacity each element of the channel is assumed to e distriuted as a circularly-symmetric complex Gaussian random variale with zero-mean and unit variance i.e. CN( 1). Effects of one-it ADCs and DACs: We first evaluate the ergodic capacity of the Rayleigh MISO channels. To provide a complete picture on how the ADCs and DACs affect the capacity of the MISO channel we consider the four possile cominations: 1) infinite-precision ADCs and DACs ) one-it ADCs and infinite-precision DACs 3) one-it ADCs and DACs ) oneit ADCs and DACs with CSIR only. As can e seen in Fig. 5 the spectral efficiency when using oth infinite-precision ADCs and DACs increases with the SNR. Whereas the spectral efficiencies for the other cases are limited y its/sec/hz in the high SNR ecause of a finite numer of the channel input or output values imposed y one-it ADCs or DACs. One interesting oservation is that when the perfect knowledge of CSIT is given the capacity loss y the use of one-it DACs is aout db in the mid SNR region compared to the case when the infiniteprecision DACs are employed at the transmitter. In addition it is oserved that for the wireless system using one-it transceivers exploiting CSIT is ale to improve the spectral efficiency

25 5 Spectral efficiency [ps/hz] M={18}. One-it ADCs+infinite resolution DACs One-it ADCs/DACs SNR [db] Fig. 6. Ergodic capacity comparison of the MISO fading channel when increasing M. 1.8 Spectral efficiency [ps/hz] Capacity with perfect CSIR/CSIT Dominant-set training (L=) Dominant-set training (L=1) Full training (L=) Full training (L=1) Capacity with CSIR only SNR [db] Fig. 7. Ergodic capacity comparison of the MISO fading channel when M = for different channel training methods. significantly compared to the case where CSIR is only availale. Effects of the numer of transmit antenna chains: Fig. 6 illustrates how the numer of the transmit antennas M each with one-it DACs changes the ergodic capacity. As M increases it is shown that the capacity increases consideraly similar to the case when the transmitter uses infinite-precision DACs. The capacity improvement is possile ecause of the transmit diversity gain. Since the numer of possile channel input sets exponentially increases with M it is highly likely to find a channel input set that is well aligned with the channel direction. Effects of imperfect CSIR and CSIT: We also characterize how the imperfect CSIR and CSIT have an effect on the capacity of a MISO Rayleigh fading channel when the proposed

26 6 methods explained in Section V are employed. The numers of repetitions per a channel input vector are set to e L = 1 and L = respectively. Since there is a total of 16 distinct SLM susets i.e. X uk when M = the full training method with L = 1 and L = requires the training overheads of 16 and 38 respectively. The corresponding feedack amount of the full-training method ecomes 1.68 its which is sent to the transmitter via an error-free feedack link. Notice that the CSIT otained y the full-training method is imperfect even if we use the error-free feedack link. This is ecause the optimal index of the channel input suset can e chosen with errors y the channel training procedure. Whereas the dominantset training method is ale to drastically diminish the overheads for the channel training and feedack. Specifically this method with L = 1 and L = needs the training overheads of 6 and 18 respectively and 6-it feedack is needed. As can e seen in Fig. 7 the proposed dominant-set training method achieves a close performance to the capacity even with a reasonale numer of training overheads. One interesting oservation is that the dominant-set training method even outperforms the full training method. This count-intuitive result is ecause the full training method considers all possile input sets and the most of them are unlikely to e chosen as the capacity-achieving input set if the average transmit power is set to e P t = M. This effect is shown to e more critical in a low SNR regime in which the proaility of the incorrect input selection is high. From the numerical results it is oserved that the dominant-set training method not only reduces the training and feedack overheads ut also can improve the achievale spectral efficiency compared to the full training method in a practical scenario. VII. CONCLUSION The key features of future wireless systems are the use of a large signal andwidth and massive antennas to support very high data transmission. Exploiting such wide-and signal and massive antenna arrays in the design of wireless systems may cause extremely high power consumption and device costs. This paper focused on the simplest yet effective solution y using one-it DACs and ADCs at the transmit and receive chains. Most notaly information-theoretical limits of the MISO channel using one-it transceiver were characterized in closed forms when oth perfect CSIT and CSIR are availale. The key finding was that the capacity-achieving scheme is a time sharing technique among multiple SLM susets. With the found capacity expressions it

27 7 has een also revealed that a finite-rate of CSIT feedack is sufficient. Aside from the capacity characterization a practical transmission strategy was also presented using a supervised-learning approach when oth imperfect CSIR and CSIT are given. Using simulations it was demonstrated that the proposed strategy achieves a close performance to the capacity with a reasonale numer of training and feedack overheads. An important direction for future work would e to characterize the fundamental limit of the MIMO channel using one-it transceivers. For the MIMO channel from the result of Lemma 1 we conjecture that the capacity-achieving transmission method is to use multiple SLM susets in which the elements of each suset are uniformly distriuted. Another important extension is to design multi-user precoding methods for downlink multi-user MIMO systems with one-it DACs [3] [3] y leveraging the SLM method [7]. APPENDIX A PROOF OF LEMMA 1 Lemma 1: For a fixed MIMO channel H = H R N M with one-it ADCs and DACs there exists a capacity-achieving input distriution which is uniform in each X uk i.e. for u {1... M}. Pr[x = R i x uk ] = p uk i { 1 3} (63) Proof: For any rotation matrix i.e. R i and any initial input distriution i.e. x Pr [x] we define a distriution for the rotated input y Pr i [x] = Pr [R i x]. We first show that the mutual information etween x and y is preserved under the input rotation. As an example with the input distriution of Pr 1 [x] we otain y = sign H Re H Im M I M H Im H Re I M M = sign N I N H Re H Im I N N H Im H Re = N I N sign H Re H Im I N N H Im H Re x Re x Im x Re x Im x Re x Im +z = sign H Im H Re x Re +z H Re H Im x Im +z = sign N I N H Re H Im x Re + N I N z I N N H Im H Re x Im I N N + N I N z. (6) I N N

28 8 Since the noise is circularly-symmetric we otain I 1 (x; y H = H) = I (x; y H = H) where I i (x; y H = H) denotes the mutual information etween the input and the output under Pr i [x]. In a similar manner it is possile to show I i (x; y H = H) = I (x; y H = H) (65) for i { 1 3}. Next we define a convex comination of Pr i [x] as Pr[x] = 1 3 Pr i [x]. (66) Since the mutual information is a concave function y the Jensen s inequality we otain an upper ound of I (x; y H = H) as I (x; y H = H) = 1 i= 3 I i (x; y H = H) I (x; y H = H). (67) i= As a result if Pr [x] is a capacity-achieving distriution Pr[x] also achieves the capacity which is uniformly distriuted in X uk y the definition in (66). This completes the proof. Example 3 : To give a clear understanding on Lemma 1 when M = 1 we suppose the following input distriutions Pr i [x]: [ 1 ] [ ] [ 1 ] [ ] Pr x = Pr x = Pr x = Pr x = 1 1 ( 1 = ) ( ) ( ) ( 3 and ). (68) } {{ }} {{ }} {{ }} {{ } Pr [x] Pr 1 [x] From the property in (65) these four input distriutions achieve the same mutual information etween the input and the output. As in (66) y averaging the four input distriutions we otain the input distriution Pr[x] as Pr[x] = 1 Pr [x] 3 Pr i [x] = i= Pr 3 [x] ( ). (69) Here the inequality in (67) ensures that the input distriution in (69) achieves the higher mutual information than the input distriutions in (68). Furthermore as Pr[x] is an input distriution that is averaged over a rotation matrix R it automatically satisfies the uniform property stated in Lemma 1.

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