Limited Feedback in Multiple-Antenna Systems with One-Bit Quantization
LLimited Feedback in Multiple-Antenna Systemswith One-Bit Quantization
Jianhua Mo and Robert W. Heath Jr.
Wireless Networking and Communications GroupThe University of Texas at Austin, Austin, TX 78712, USAEmail: { jhmo, rheath } @utexas.edu Abstract —Communication systems with low-resolution analog-to-digital-converters (ADCs) can exploit channel state informa-tion at the transmitter (CSIT) and receiver. This paper presentsinitial results on codebook design and performance analysis forlimited feedback systems with one-bit ADCs. Different fromthe high-resolution case, the absolute phase at the receiver isimportant to align the phase of the received signals when thereceived signal is sliced by one-bit ADCs. A new codebook designfor the beamforming case is proposed that separately quantizesthe channel direction and the residual phase.
I. I
NTRODUCTION
The large bandwidth of the emerging wireless network,for example, millimeter wave WLAN [1] or 5G millimeterwave cellular [2]–[4], has introduced new challenges for thehardware design. A main challenge is the power assumptionwith analog-to-digital converters (ADCs). Data summarized in[5] show that ADC power consumption increases drasticallyfor sampling frequency above megasamples per second.A direct solution to this bottleneck is to use power efficientlow-resolution – even one-bit – ADCs.The use of few- and especially one-bit ADCs radicallychanges both the theory and practice of communication, forexample, the capacity analysis [6]–[11], channel estimation[12]–[18], and symbol detection [19]–[21]. At present, theexact capacity of quantized multiple-input multiple-output(MIMO) channel is generally unknown, except for the simplemultiple-input single-output (MISO) channel and some specialcases, such as in the low or high SNR regime [6]–[8], [11].Transmitting independent QPSK signals [8] or Gaussian sig-nals [9], [10] from each antenna nearly achieves the capacity atlow SNR, but is far from the optimal at high SNR. In our work[11], where channel state information at the transmitter (CSIT)is assumed, simple channel inversion precoding (versus theusual eigenbeamforming) is nearly optimal if the channel hasfull row rank. If the channel is low rank, we proposed a newprecoding method achieving the capacity at high SNR. MIMOprecoding provides a substantial performance improvementcompared with the no-precoding case (i.e., independent QPSKor Gaussian signaling).Despite the potential gain of precoding and therefore theimportance of CSIT, there is – to our knowledge – no work thathas considered limited feedback with low-resolution ADCs.
This material is based upon work supported in part by the National ScienceFoundation under Grant No. NSF-CCF-1527079.
The results on limited feedback with infinite-resolution ADCs,for example, [22]–[24], cannot be directly extended to thelow-resolution ADCs – the fundamentals are different. Forexample, in MISO limited feedback beamforming [22], theoptimum beamformer f is phase invariant, meaning that equiv-alent performance is also achieved by f e jθ . In our capacityresults in [11], we found that the optimum beamformer was thematched filter, and in fact was not phase invariant. The reasonis that phase at the receiver is important when the receivedsignal is sliced by 1-bit ADCs; an important function of CSITis to align the phase of the received signals. One implicationis that phase-invariant Grassmannian beamforming codebooks[22] will no longer be appropriate.In this paper, we develop limited feedback schemes forsingle-input single-output (SISO) and MISO systems withone-bit ADCs. Our approach leverages recent work [12]–[18] that shows that it is possible to estimate the MIMOchannel even with one-bit ADCs at the receiver. Given anestimate of the channel, we propose a new codebook designthat explicitly incorporates the phase sensitivity required forfeedback in the one-bit ADC channel. Our proposed codebookdesign for the MISO beamforming case separately quantizesthe channel direction and the residual phase. The performanceloss incurred by the finite rate feedback compared to perfectCSIT is analyzed. Bounds of the power and capacity loss arederived. Our work provides a path to making the assumptionof CSIT in SISO and MISO systems with one-bit ADCs morerealistic. II. S YSTEM M ODEL
Consider a MISO system with one-bit quantization, asshown in Fig. 1. There are N t antennas at the transmitter andsingle antenna at the receiver. Assuming perfect synchroniza-tion and a narrowband channel, the baseband received signalin this MISO system is y = h ∗ x + n (1)where h ∈ C N t × is the channel vector, x ∈ C N t × isthe signal sent by the transmitter, y ∈ C is the receivedsignal before quantization, and n ∼ CN (0 , is the circularlysymmetric complex Gaussian noise. The average transmitpower is P t , i.e., E [ x ∗ x ] = P t . In our system, there are twoone-bit resolution quantizers that separately quantize the real a r X i v : . [ c s . I T ] D ec /DTransmitter ... Digital ProcessingRF {1,-1} Baseband1-bitADC {1,-1}Re()Im() Nt Low-rate feedback channel h ryx
B bits
Fig. 1. A MISO system with one-bit quantization and limited feedback. At thereceiver side, there are two one-bit ADCs. There is also a low-rate feedbackpath from the receiver to the transmitter. Note that there is no limitation onthe structure of the transmitter. and imaginary part of the received signal. The output after theone-bit quantization is r = sgn ( y ) = sgn ( h ∗ x + n ) , (2)where sgn() is the signum function applied separately to thereal and imaginary parts. Therefore, the quantization output r ∈ { , − j , − , − − j } .In this paper, we assume the receiver has perfect channelstate information. This is justified by the prior work on channelestimation with one-bit quantization, for example [12]–[18].Furthermore, the feedback is assumed to delay and error free,as is typical in limited feedback problems. Adding realism tothe feedback channel is an interesting topic for future work.III. Q UANTIZED
SISO C
HANNEL WITH L IMITED F EEDBACK
We first consider the SISO channel as a special case. Since N t = 1 , the received signal is r = sgn ( y ) = sgn ( hx + n ) . (3)Denote the phase of h as ∠ h . As shown in our previous work[11, Lemma 1], the capacity-achieving scheme is to transmitrotated QPSK symbols, i.e., Pr (cid:104) x = (cid:112) P t e j ( kπ + π − ∠ h ) (cid:105) = 14 , for k = 0 , , . (4)The term − ∠ h in the transmitted signals is introduced to pre-cancel the phase rotation of the channel such that the receiverwill observe a regular QPSK signal.If h is unknown at the transmitter, then only the phaseneeds to be quantized and fed back to the transmitter. Sincethe QPSK constellation is unchanged for a 90-degree rotation,only mod (cid:0) ∠ h, π (cid:1) instead of ∠ h needs to be fed back. Nowassume B bits are used to uniformly quantize the region [0 , π ] . Uniform quantization is reasonable since for moststatistical channel models the phase of the SISO channel isuniformly distributed. The codebook is then Φ = { φ i = iπ B +1 + π B +2 , ≤ i ≤ B − } . For instance, Φ = { π , π } if B = 1 . The receiver sends the index i of ˆ φ to the transmittersuch that x H b ( Q ( √ x ))1 − H b ( Q ( √ x )) Fig. 2. The figure shows H b (cid:0) Q (cid:0) √ x (cid:1)(cid:1) and − H b (cid:0) Q (cid:0) √ x (cid:1)(cid:1) versus x .It is seen that H b (cid:0) Q (cid:0) √ x (cid:1)(cid:1) is a decreasing and convex function of x . ˆ φ = arg min φ m ∈ Φ (cid:12)(cid:12)(cid:12) mod (cid:16) ∠ h, π (cid:17) − φ m (cid:12)(cid:12)(cid:12) . (5)Based on the feedback index i , the transmitter sends rotatedQPSK signals with uniform probabilities, i.e., Pr (cid:104) x = (cid:112) P t e j ( kπ + π − ˆ φ ) (cid:105) = 14 , for k = 0 , , . (6)The received signal after quantization is r = sgn (cid:16) | h | (cid:112) P t e j ( kπ + π − ˆ φ + ∠ h ) + n (cid:17) . (7)The channel has four possible inputs and four possible out-puts. This is a discrete-input discrete-output channel. There-fore the channel capacity is C fbSISO = 2 − H b (cid:18) Q (cid:18)(cid:114) P t | h | sin (cid:16) π − θ (cid:17)(cid:19)(cid:19) −H b (cid:18) Q (cid:18)(cid:114) P t | h | cos (cid:16) π − θ (cid:17)(cid:19)(cid:19) (8) = 2 − H b (cid:18) Q (cid:18)(cid:113) P t | h | (1 − sin 2 θ ) (cid:19)(cid:19) −H b (cid:18) Q (cid:18)(cid:113) P t | h | (1 + sin 2 θ ) (cid:19)(cid:19) , (9)where θ := ˆ φ − mod (cid:0) ∠ h, π (cid:1) is the quantization error, H b ( p ) := − p log p − (1 − p ) log (1 − p ) is the binary entropyfunction, and Q ( · ) is the tail probability of the standard normaldistribution.In Fig. 2, we plot the function H b ( Q ( √ x )) . Since H b ( Q ( √ x )) is decreasing with x , it follows that C fbSISO ≥ (cid:18) − H b (cid:18) Q (cid:18)(cid:113) P t | h | (1 − sin 2 | θ | ) (cid:19)(cid:19)(cid:19) . (10)he channel capacity with perfect CSIT is [11, Lemma 1] C SISO = 2 (cid:18) − H b (cid:18) Q (cid:18)(cid:113) P t | h | (cid:19)(cid:19)(cid:19) . (11)Comparing (10) and (11), the power loss factor is − sin 2 | θ | .We want to minimize the power loss, or equivalently max-imize the term − sin 2 | θ | . Since the quantization error θ ∈ (cid:2) − π B +2 , π B +2 (cid:3) by the uniform quantization scheme, itfollows that − sin 2 | θ | > − sin π B +1 . When B = 1 , − sin 2 | θ | ≥ − √ , which means that there is at most .
33 dB power loss. In addition, as θ is uniformly distributed,the average power loss is E θ [1 − sin 2 | θ | ] (12) = 1 π B +2 (cid:90) π B +2 − π B +2 (1 − sin 2 | θ | ) d θ (13) = 1 − sin (cid:0) π B +2 (cid:1) π B +2 (14) ( a ) > − π B +2 (15) > − − B , (16)where ( a ) follows from sin x < x for < x < π . Therefore,the average power loss is at most with only one bitfeedback . In the simulation, we will show that with only onebit feedback, the performance is close to that with perfectCSIT.IV. Q UANTIZED
MISO C
HANNEL WITH L IMITED F EEDBACK
Similar to the MISO system with infinite-resolution ADCs,random vector quantization (RVQ), which performs close tothe optimal quantization and is amenable to analysis [23],[25], is adopted to quantize the direction of channel h .We assume that B out of the total B bits are used toconvey the channel direction information. The codebook is W = (cid:8) v , v , · · · , v B − (cid:9) where each of the quantizationvectors is independently chosen from the isotropic distributionon the Grassmannian manifold G ( N t , [22]. The receiversends back the index of v maximizing | h ∗ v | .Besides the channel direction information, the remaining B = B − B bits are used to quantized the phase of theequivalent channel, i.e., ∠ ( h ∗ v ) (denoted as residual phase afterwards). The second codebook quantizing the residualphase is Φ = { φ i = iπ B +1 + π B +2 , ≤ i ≤ B − } . Thereceiver feeds back the index i of ˆ φ such that ˆ φ = arg min φ m (cid:12)(cid:12)(cid:12) mod (cid:16) ∠ ( h ∗ v ) , π (cid:17) − φ m (cid:12)(cid:12)(cid:12) . (17)The transmitter adopts matched filter beamforming andQPSK signaling based on the feedback bits, i.e., Pr (cid:104) x = (cid:112) P t v e j ( kπ + π − ˆ φ ) (cid:105) = 14 , for k = 0 , , . (18) A tighter bound is by evaluating (14) with B = 1 . The received signal after quantization is r = sgn (cid:16)(cid:112) P t h ∗ v e j ( kπ + π − ˆ φ ) + n (cid:17) . (19)Similar to the SISO case, this channel is also a discrete-input discrete-output channel. The capacity is derived as C fbMISO = 2 − H b (cid:18) Q (cid:18)(cid:114) P t | h ∗ v | sin (cid:16) π − θ (cid:17)(cid:19)(cid:19) −H b (cid:18) Q (cid:18)(cid:114) P t | h ∗ v | cos (cid:16) π − θ (cid:17)(cid:19)(cid:19) (20) = 2 − H b (cid:16) Q (cid:16)(cid:112) P t (cid:107) h (cid:107) cos β (1 − sin 2 θ ) (cid:17)(cid:17) −H b (cid:16) Q (cid:16)(cid:112) P t (cid:107) h (cid:107) cos β (1 + sin 2 θ ) (cid:17)(cid:17) , (21)where θ := ˆ φ − mod (cid:0) ∠ ( h ∗ v ) , π (cid:1) and cos β := | h ∗ v |(cid:107) h (cid:107) . Alower bound of the capacity is C fbMISO ≥ (cid:16) − H b (cid:16) Q (cid:16)(cid:112) P t (cid:107) h (cid:107) cos β (1 − sin 2 | θ | ) (cid:17)(cid:17)(cid:17) . (22)The channel capacity with perfect CSIT, derived in [11], is C MISO = 2 (cid:16) − H b (cid:16) Q (cid:16)(cid:112) P t (cid:107) h (cid:107) (cid:17)(cid:17)(cid:17) . (23)Comparing (22) and (23), we want to maximize the term cos β (1 − sin 2 | θ | ) to minimize the power loss. Averagingover the codebook W and the residual phase θ , we have E W ,θ (cid:2) cos β (1 − sin 2 | θ | ) (cid:3) (24) ( a ) = E W (cid:2) cos β (cid:3) E θ [1 − sin 2 | θ | ] (25) ( b ) ≥ (cid:16) − − B N t − (cid:17) (cid:0) − − B (cid:1) (26)where ( a ) is by noting that | h ∗ v | and ∠ ( h ∗ v ) are independentfor RVQ, ( b ) follows from the facts E W (cid:2) cos β (cid:3) > − − B N t − [23, Lemma 1] and E θ [1 − sin 2 | θ | ] > − − B proved in(16). From (26), it is seen that when B = N t − and B = 1 ,the average power loss is at most .Another performance metric is capacity loss, which may bemore important than power loss. We now analyze the capacityloss caused by limited feedback. Note that the capacity of thequantized system saturates to / Hz at high SNR, which isa difference from the unquantized systems. The capacity lossincurred by finite-rate feedback is C loss = C MISO − C fbMISO (27) ≤ − C fbMISO (28) ≤ H b (cid:16) Q (cid:16)(cid:112) P t (cid:107) h (cid:107) cos β (1 − sin 2 | θ | ) (cid:17)(cid:17) . (29)To ensure C loss ≤ (cid:15) , we required P t (cid:107) h (cid:107) cos β (1 − sin 2 | θ | ) ≥ δ, (30)here H b (cid:16) Q (cid:16) √ δ (cid:17)(cid:17) = (cid:15) . Plugging in (26) and assuming E (cid:2) (cid:107) h (cid:107) (cid:3) = N t , we obtain (cid:16) − − B N t − (cid:17) (cid:0) − − B (cid:1) ≥ δP t N t . (31)In (31), we see that given fixed capacity loss, the requirednumber of feedback bits actually decreases with the transmitsignal power. This is in striking contrast with the unquantizedMISO systems.In Fig. 2, it is shown that H b (cid:0) Q (cid:0) √ (cid:1)(cid:1) ≈ . . Therefore,if the numbers of feedback bits satisfy (cid:16) − − B N t − (cid:17) (cid:0) − − B (cid:1) ≥ P t N t , (32)then the capacity loss is less than . / Hz , or equivalently of the upper bound (which is / Hz ) is achieved.V. S IMULATION R ESULTS
In this section, we evaluate the performance of the proposedlimited feedback scheme. We compute the capacity for eachchannel realization then averaged over 1000 channel realiza-tions with Rayleigh fading, i.e., h ∼ CN ( , I N t ) . In thefigures, SNR(dB) (cid:44)
10 log P t since the noise is assumedto have unit variance.In Fig. 3, we compare the capacities of SISO channel withperfect CSIT, limited feedback and no CSIT. As shown inthe figure, the capacity with two bits feedback is almost sameto that with perfect CSIT. In addition, even with single bitfeedback, the power loss is very small, i.e., less than 1 dB .Without CSIT, the capacity loss is much larger, especially athigh SNR. Taking into account both the feedback overhead andthe capacity loss, it is reasonable to set B = 1 in practice.Figs. 4 - 6 show the performance of the proposed limitedfeedback scheme in MISO channel. The entries of channel h follow the IID circularly symmetric complex Gaussiandistribution with unit variance.In Fig. 4, we plot the capacities of MISO with fourantennas and four bits feedback. Three different allocationsof the feedback bits are compared. It is found that the case‘ B = 3 , B = 1 ’ has the best performance with power lossaround , which is consistent with our analysis in (26)which states the power loss is upper bounded by . InFig. 5, we show another example with 16 antennas and 16bits feedback. It is shown that the case ‘ B = 15 , B = 1 ’ isthe best one. We therefore conclude that more bits should beassigned to feed back the channel direction information.In Fig. 6, we show the capacity loss for different values of B and B . As expected, the capacity loss decreases as B and B increase. We also find that at high SNR, the capacity lossesincurred by limited feedback converge to zero. For instance,when the transmitter power is larger than
11 dB , even with B = B = 1 , the capacity loss is less than . / Hz ,which implies that the capacity with only two bit feedbackachieves of the capacity with perfect CSIT. The resultverifies our analysis in (32). Note that the capacity loss at lowSNR is also small since C MISO is small. -10 -5 0 5 10 15 20 25 30
SNR (dB) C apa c i t y ( bp s / H z ) Perfect CSITB=2B=1No CSIT
Fig. 3. The capacity of a SISO system with CSIT, no CSIT and feedback. -20 -10 0 10 20 30
SNR (dB) C apa c i t y ( bp s / H z ) Perfect CSITB =3, B =1B =2, B =2B =1, B =3No CSIT Fig. 4. The capacity of a MISO system with CSIT, no CSIT and limitedfeedback when N t = 4 . VI. C
ONCLUSIONS
In this paper, we developed an approach for limited feed-back in SISO and MISO channels with one-bit ADCs. Forthe SISO channel, only the phase of the channel is quantizedwhile in the MISO channel, the channel direction and residualphase are both quantized and fed back to the transmitter. Weevaluated the power and capacity losses incurred by the useof limited feedback. Based on our analyses and simulationresults, we made two important observations. First, feedingback only one bit for the phase (or residual phase in MISOchannel) is enough to guarantee good performance in the ex-amples considered. Second, when the capacity of the quantizedchannel is saturated, the required number of feedback bitsguaranteeing a small capacity loss decreases with SNR.There are several potential directions for future work. Ournumerical results were based on the IID Gaussian channel withsmall numbers of antennas. In mmWave systems - a promisingapplication of one-bit ADCs - the channels will likely be
20 -10 0 10 20 30
SNR (dB) C apa c i t y ( bp s / H z ) Perfect CSITB =15, B =1B =8, B =8B =1, B =15No CSIT Fig. 5. The capacity of a MISO system with CSIT, no CSIT and limitedfeedback when N t = 16 . -20 -10 0 10 20 30 SNR (dB) C apa c i t y l o ss ( bp s / H z ) B =4, B =2B =4, B =1B =3, B =1B =2, B =1B =1, B =1 No CSIT
Fig. 6. The capacity loss C loss = C MISO − C fbMISO incurred by finite-ratefeedback in a MISO system with N t = 4 . correlated depending on the number of scattering clusters andthe angle spread. It would be interesting to develop tech-niques that also work for large correlated channels. A naturalextension of our work would be to MIMO communicationchannels. This is complicated by the complicated structure ofthe capacity-optimum signaling distribution and the potentialfor different choices of precoders (see [11, Section IV]).Another possible direction is to combine the two separatestages, channel estimation and limited feedback, together. Inthis case, the feedback bits are decided directly by the ADCoutputs instead of the estimated CSI at the receiver.R EFERENCES[1] T. Baykas, C.-S. Sum, Z. Lan, J. Wang, M. Rahman, H. Harada, andS. Kato, “IEEE 802.15.3c: the first IEEE wireless standard for data ratesover 1 Gb/s,”
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