Limited Feedback in MISO Systems with Finite-Bit ADCs
LLimited Feedback in MISO Systems with Finite-BitADCs
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 —We analyze limited feedback in systems where amultiple-antenna transmitter sends signals to single-antennareceivers with finite-bit ADCs. If channel state information (CSI)is not available with high resolution at the transmitter and theprecoding is not well designed, the inter-user interference is a bigdecoding challenge for receivers with low-resolution quantization.In this paper, we derive achievable rates with finite-bit ADCs andfinite-bit CSI feedback. The performance loss compared to thecase with perfect CSI is then analyzed. The results show that thenumber of bits per feedback should increase linearly with theADC resolution to restrict the loss.
I. I
NTRODUCTION
The wide bandwidth and large antenna arrays in future com-munication systems impose big challenges for the hardwaredesign of the receiver, which has to efficiently process multiplesignals from antennas at a much faster pace. The analog-to-digital converter (ADC) is one of the bottlenecks. At ratesabove 100 Mega samples per second, ADC power consump-tion increases quadratically with the sampling frequency [1],[2].The use of few- and especially one-bit ADCs is proposedas one approach to overcoming this challenge, for example, inthe millimeter wave multiple-input multiple-output (MIMO)channel [3]–[12] and massive MIMO channel [13]–[18]. Thiswork has shown that low resolution ADCs can be used inpractical communications. For instance, it is found that thereis nearly no performance loss (less than dB) at low SNRcompared to infinite-bit ADCs; it is possible to estimate thechannel (IID Rayleigh fading or correlated) and detect symbols(QPSK or higher-order QAM) with coarse quantization.In our previous work [19], we analyzed the single-usersingle-input single-output (SISO) and multiple-input single-output (MISO) channels with one-bit ADCs where the thetransmitter sends the capacity-achieving QPSK symbols. Ourproposed codebook design for the MISO beamforming caseseparately quantizes the channel direction and the residualphase to incorporate the phase sensitivity of QPSK symbols.This design, however, cannot be extended to the channel withmore than one-bit ADCs because the optimal signaling in thiscase is unknown.In this paper, we assume that the transmitter adopts subop-timal Gaussian signaling. Since Gaussian signaling is circular
This material is based upon work supported in part by the National ScienceFoundation under Grant No. NSF-CCF-1319556 and NSF-CCF-1527079. (a) Single-user MISO system(b) Multi-user MISO system
Fig. 1: MISO systems with finite-bit quantization and limitedfeedback. At each receiver, there are two b -bit ADCs. Thereis also a low-rate feedback path from each receiver to thetransmitter.symmetric, a single codebook quantizing the channel directionis enough. We derive the bounds of achievable rates with finite-bit ADC and finite-bit feedback. The rate and power lossesincurred by the finite rate feedback compared to perfect CSITis analyzed. Our results bridge the gap between the case ofinfinite-bit ADC [20] and one-bit ADC [19].II. S YSTEM M ODEL
In this paper, we consider single-user and multiple-userMISO systems shown in Fig. 1. The transmitter is equippedwith N t antennas, while each receiver has only one antennawith finite-bit ADCs. In our system, there are two b -bitresolution quantizers that separately quantize the real andimaginary part of the received signal. We assume that uniformquantization is applied since it is easier for implementation and a r X i v : . [ c s . I T ] D ec chieves only slightly worse performance than non-uniformcase [21].We assume that B bits are used to convey the channel direc-tion information. A codebook C = (cid:110)(cid:98) h (0) , (cid:98) h (1) , · · · , (cid:98) h (2 B − (cid:111) is shared by the transmitter and receiver. The receiver sendsback the index i of (cid:98) h ( i ) maximizing | h ∗ (cid:98) h ( i ) | where h repre-sents the channel. Then the transmitter performs beamformingbased on the feedback information. Similar to a MISO systemwith infinite-resolution ADCs, random vector quantization(RVQ), which performs close to the optimal quantization andis amenable to analysis [20], [22], is adopted to quantizethe direction of channel h . In the codebook C , each ofthe quantization vectors is independently chosen from theisotropic distribution on the Grassmannian manifold G ( N t , [23].Different from our previous work [19] where capacity-achieving QPSK signaling was adopted, we assume that Gaus-sian signaling is used at the transmitter. Although Gaussiansignalling is suboptimal, it is amenable for analyses and closeto optimal at low and medium SNR [7], [9].In this paper, we assume the channel follows IID Rayleighfading. The extension to correlated channel model is aninteresting topic for future work. We also assume the receiverhas perfect channel state information. This is justified by theprior work on channel estimation with low resolution ADCs,for example [5], [6], [11], [12]. Furthermore, the feedback isassumed to be delay and error free, as is typical in limitedfeedback problems.III. S INGLE - USER
MISO C
HANNEL WITH F INITE - BIT
ADC
S AND L IMITED F EEDBACK
We first consider a single-user MISO system with finite-bit quantization, as shown in Fig. 1a. Assuming perfect syn-chronization and a narrowband channel, the baseband receivedsignal in this MISO system is y = h ∗ v s + n, (1)where h ∈ C N t × is the channel vector, v ∈ C N t × ( (cid:107) v (cid:107) = 1) is the beamforming vector, s is the Gaussian distributed sym-bol sent by the transmitter, y ∈ C is the received signal beforequantization, and n ∼ CN (0 , σ ) is the circularly symmetriccomplex Gaussian noise. The average transmit power is P t ,i.e., E [ | s | ] = P t .The output after the finite-bit quantization is r = Q ( y ) = Q ( h ∗ v s + n ) , (2)where Q ( · ) is the finite-bit quantization function appliedseparately to the real and imaginary parts.By Bussgang’s theorem [7], [24], [25], the quantizationoutput can be decoupled into two uncorrelated parts, i.e., r = (1 − η b ) y + n Q (3) = (1 − η b ) h ∗ v s + (1 − η b ) n + n Q , (4)where η b = E [ | r − y | ] E [ | y | ] is the normalized mean squared er-ror and n Q is the quantization noise with variance σ = η b (1 − η b ) E [ | y | ] = η b (1 − η b ) (cid:0) | h ∗ v | P t + σ (cid:1) . Therefore,the effective noise n ef (cid:44) (1 − η b ) n + n Q has variance η b (1 − η b ) (cid:0) | h ∗ v | P t (cid:1) + (1 − η b ) σ . The values of η b (1 ≤ b ≤ arelisted in Table I. The resulting signal-to-quantization and noiseratio (SQNR) at the receiver is SQNR = (1 − η b ) P t | h ∗ v | (1 − η b ) σ + σ = (1 − η b ) P t | h ∗ v | η b P t | h ∗ v | + σ . (5)Denote the achievable rate with b -bit ADC and B -bitfeedback as R ( b, B ) . In this paper, ‘ b = ∞ ’ represents thecase of full-precision ADCs, while ‘ B = ∞ ’ represents thecase of perfect CSIT. Assuming that the noise n Q follows theworst-case Gaussian distribution, the average achievable ratewith perfect CSIT and conjugate beamforming is R ( b, ∞ ) = E h (cid:34) log (cid:32) − η b ) P t (cid:107) h (cid:107) η b P t (cid:107) h (cid:107) + σ (cid:33)(cid:35) (6) ( a ) ≤ log − η b ) P t E (cid:104) (cid:107) h (cid:107) (cid:105) η b P t E (cid:104) (cid:107) h (cid:107) (cid:105) + σ (7) ( b ) = log (cid:18) − η b ) P t N t η b P t N t + σ (cid:19) (8)where ( a ) follows from the concavity of the function f ( x ) =log (cid:16) axbx + c (cid:17) ( a > , b > , c > when x > , ( b ) follows from the assumption of IID Rayleigh fading channel.In the low and high SNR (cid:0) P t σ (cid:1) regimes, the averageachievable rate with perfect CSIT is approximated as, R ( b, ∞ ) ≈ log (cid:16) (1 − η b ) P t N t σ (cid:17) , when P t σ is small, log (cid:16) η b (cid:17) , when P t σ is large.(9)It is seen that the high SNR rate is limited by the signal-to-quantization ratio (SQR) defined as SQR (cid:44) η b . Since η b ≈ π √ − b when b ≥ [26], the achievable rate at high SNR is R ( b, ∞ ) ≈ b − log π √ (10) ≈ b − . bps/Hz . (11)The values of log (cid:16) η b (cid:17) are given in Table I.Averaging over the RVQ codebooks, the achievable rateunder limited feedback is R ( b, B ) (12) = E h , C (cid:34) log (cid:32) − η b ) P t | h ∗ v | η b P t | h ∗ v | + σ (cid:33)(cid:35) (13) ( a ) ≈ log − η b ) P t N t (cid:16) − B β (cid:16) B , N t N t − (cid:17)(cid:17) η b P t N t (cid:16) − B β (cid:16) B , N t N t − (cid:17)(cid:17) + σ (14) ( b ) ≥ log − η b ) P t N t (cid:16) − − BN t − (cid:17) η b P t N t (cid:16) − − BN t − (cid:17) + σ (15)ABLE I: The optimum uniform quantizer for a Gaussian unit-variance input signal [21] Resolution b η b π − π ( ≈ .
10 log (1 − η b ) -1.9613 -0.5429 -0.1527 -0.0414 -0.0109 -0.0029 -0.0007 -0.0002 log (cid:16) η b (cid:17) where β ( · , · ) is a beta function. ( a ) follows from E (cid:2) (cid:107) h (cid:107) (cid:3) = N t and cos ( ∠ ( h , v )) = 1 − B β (cid:16) B , N t N t − (cid:17) [22], ( b ) follows from the inequality B β (cid:16) B , N t N t − (cid:17) ≤ − BN t − [20].In the low and high SNR regimes, the average achievablerate with limited feedback is R ( b, B ) ≈ log (1 − η b ) P t N t (cid:18) − − BN t − (cid:19) σ , if P t σ is small, log (cid:16) η b (cid:17) , if P t σ is large.(16)Comparing R ( b, ∞ ) in (9) and R ( b, B ) in (16), we find thatat low SNR, the power loss between R ( b, ∞ ) and R ( b, B ) is ≈
10 log (cid:16) − − BN t − (cid:17) dB. The result is similar to the casewith infinite-bit ADCs [22], [27]. In contrary, at high SNR,both R ( b, ∞ ) and R ( b, B ) approach the same upper boundand the rate loss due to limited feedback is zero.At last, the achievable rate with infinite-bit ADC and perfectCSIT is known as R ( ∞ , ∞ ) = log (cid:0) P t N t σ (cid:1) . We findthat at low SNR, the power loss incurred by the finite-bitADC is
10 log (1 − η b ) dB while that by limited feedbackis
10 log (cid:16) − − BN t − (cid:17) dB.IV. M ULTI -U SER
MISO C
HANNEL WITH F INITE - BIT
ADC
S AND L IMITED F EEDBACK
We now consider a multi-user MISO channel shown inFig. 1b where a N t -antenna transmitter sends signals to K (1 < K ≤ N t ) single-antenna receivers. The quantizationoutput at the k -th receiver is r k = Q (cid:32) √ ρ K (cid:88) i =1 h ∗ k v i s i + n k (cid:33) (17) = (1 − η b ) √ ρ h ∗ k v i s i + (1 − η b ) √ ρ K (cid:88) i =1 ,i (cid:54) = k h ∗ k v i s i + (1 − η b ) n k + n Q (18)where ρ (cid:44) P t K is the power allocated to each user, v i is thebeamforming vector for user i , n k ∼ CN (0 , σ ) is the circu-larly symmetric complex Gaussian noise, and the quantizationnoise n Q has variance η b (1 − η b ) (cid:16) ρ (cid:80) Ki =1 ,i (cid:54) = k | h ∗ k v i | + σ (cid:17) .Therefore, the signal-to-interference, quantization and noiseratio (SIQNR) at the k -th receiver is SIQNR k = (1 − η b ) ρ | h ∗ k v i | η b ρ | h ∗ k v k | + ρ (cid:80) Ki =1 ,i (cid:54) = k | h ∗ k v i | + σ . (19) If there is perfect CSIT and the transmitter designs zero-forcing beamforming v ZF i (1 ≤ i ≤ K ) such that h ∗ k v ZF i = 0 for k (cid:54) = i , the average rate per user is R ZF ( b, ∞ ) = E H (cid:34) log (cid:32) − η b ) ρ (cid:12)(cid:12) h ∗ k v ZF k (cid:12)(cid:12) η b ρ (cid:12)(cid:12) h ∗ k v ZF k (cid:12)(cid:12) + σ (cid:33)(cid:35) (20) ( a ) ≤ log (cid:18) − η b ) ρ ( N t − K + 1) η b ρ ( N t − K + 1) + σ (cid:19) (21)where E H (cid:104)(cid:12)(cid:12) h ∗ k v ZF k (cid:12)(cid:12) (cid:105) = E H (cid:2) (cid:107) h k (cid:107) (cid:3) E H (cid:104) | (cid:101) h ∗ k v ZF k | (cid:105) = E H (cid:2) (cid:107) h k (cid:107) (cid:3) E H (cid:104) cos ∠ (cid:16)(cid:101) h k , v ZF k (cid:17)(cid:105) = N t N t − K +1 N t = N t − K + 1 where (cid:101) h k = h (cid:107) h (cid:107) .In the case without perfect CSIT, each receiver feedsback B bits as the index of the quantized channel (cid:98) h k ,then the transmitter designs zero-forcing precoding based on (cid:98) h k (1 ≤ k ≤ K ) . The average achievable rate is shown in (22)-(25).In (22)-(25), we use the equality E H , C (cid:104) | h ∗ k v i | (cid:105) = E H (cid:104) (cid:107) h k (cid:107) (cid:105) E H , C (cid:20)(cid:12)(cid:12)(cid:12)(cid:101) h ∗ k v i (cid:12)(cid:12)(cid:12) (cid:21) = N t N t − B β (cid:16) B , N t N t − (cid:17) [20]and the lower bound of E H , C (cid:104) | h ∗ k v k | (cid:105) as follows. E (cid:104) | h ∗ k v k | (cid:105) ≥ E (cid:20)(cid:12)(cid:12)(cid:12) h ∗ k (cid:98) h k (cid:12)(cid:12)(cid:12) (cid:21) E (cid:20)(cid:12)(cid:12)(cid:12)(cid:98) h ∗ k v k (cid:12)(cid:12)(cid:12) (cid:21) = E (cid:104) | h k | (cid:105) E (cid:20)(cid:12)(cid:12)(cid:12)(cid:101) h ∗ k (cid:98) h k (cid:12)(cid:12)(cid:12) (cid:21) E (cid:20)(cid:12)(cid:12)(cid:12)(cid:98) h ∗ k v k (cid:12)(cid:12)(cid:12) (cid:21) (26) ( a ) = N t (cid:18) − B β (cid:18) B , N t N t − (cid:19)(cid:19) N t − K + 1 N t , where ( a ) follows from the equalities E (cid:20)(cid:12)(cid:12)(cid:12)(cid:101) h ∗ k (cid:98) h k (cid:12)(cid:12)(cid:12) (cid:21) = 1 − B β (cid:16) B , N t N t − (cid:17) [22] and E (cid:20)(cid:12)(cid:12)(cid:12)(cid:98) h ∗ k v k (cid:12)(cid:12)(cid:12) (cid:21) = N t − K +1 N t .Therefore, the rate loss incurred by limited feedback is ∆ R ZF ( b ) = R ZF ( b, ∞ ) − R ZF ( b, B ) (27)and has an upper bounded shown in (28).When the SNR (cid:16) P t σ = Kρσ (cid:17) is low, the performance loss is ∆ R ZF ( b ) ≈ log (cid:18) − η b ) ρ ( N t − K + 1) σ (cid:19) (29) − log − η b ) ρ ( N t − K + 1) (cid:16) − − BN t − (cid:17) σ . It is found there is a power loss ≈
10 log (cid:16) − − BN t − (cid:17) dBwhich is similar to the single-user case shown in Section III. ZF ( b, B ) = E H , C (cid:34) log (cid:32) − η b ) ρ | h ∗ k v k | η b ρ | h ∗ k v k | + ρ (cid:80) Ki =1 ,i (cid:54) = k | h ∗ k v i | + σ (cid:33)(cid:35) (22) ≈ log − η b ) ρ E (cid:104) | h ∗ k v k | (cid:105) η b ρ E (cid:104) | h ∗ k v k | (cid:105) + ρ (cid:80) Ki =1 ,i (cid:54) = k E (cid:104) | h ∗ k v i | (cid:105) + σ (23) = log − η b ) ρ ( N t − K + 1) (cid:16) − B β (cid:16) B , N t N t − (cid:17)(cid:17) η b ρ ( N t − K + 1) (cid:16) − B β (cid:16) B , N t N t − (cid:17)(cid:17) + ρ ( K − N t N t − B β (cid:16) B , N t N t − (cid:17) + σ (24) ≥ log − η b ) ρ ( N t − K + 1) (cid:16) − − BN t − (cid:17) η b ρ ( N t − K + 1) (cid:16) − − BN t − (cid:17) + ρ ( K − N t N t − − BN t − + σ (25) ∆ R ZF ( b ) = log (cid:18) − η b ) ρ ( N t − K + 1) η b ρ ( N t − K + 1) + σ (cid:19) − log − η b ) ρ ( N t − K + 1) (cid:16) − − BN t − (cid:17) η b ρ ( N t − K + 1) (cid:16) − − BN t − (cid:17) + ρ ( K − N t N t − − BN t − + σ (28)At high SNR, the rate loss is ∆ R ZF ( b ) ≈ log (cid:32) − η b η b C C + 1 (cid:33) (30)where C (cid:44) ( N t − K + 1) (cid:16) − − BN t − (cid:17) and C (cid:44) ( K − N t N t − − BN t − . To guarantee that the rate loss is less than D , the number of feedback bits B should be large enoughsuch that − η b η b C /C +1 < D − .When b ≥ , − η b ≈ as shown in Table I. If B (cid:29) N t − , C C + 1 ≈ ( N t − K +1)( N t − N t ( K − BN t − . In this case, to keep the rateloss constant, we want the following term η b BN t − ≈ π √ b − BN t − = 2 π √ (cid:16) b − B N t − (cid:17) (31)to be less than a constant. Therefore, if the ADC resolution b increase bit, the number of feedback bits B should increase N t − . V. S IMULATION R ESULTS
In this section, we compute the achievable rate for eachchannel realization then average over 1000 channel realiza-tions with Rayleigh fading, i.e., h k ∼ CN ( , I N t ) . In thefigures, SNR(dB) (cid:44)
10 log P t σ .In Fig. 2, we show the average achievable rates with perfectCSIT and limited feedback in a single-user MISO channel.First, the rate with perfect CSIT converges to log (cid:16) η b (cid:17) , whichis . , . , . , . bps/Hz when b = 1 , , , . Notethat these values are less than the theoretical upper bound b bps/Hz because Gaussian signaling is suboptimal. Second, athigh SNR (for instance, 10 dB when b = 1 , 20 dB when b = 4 ), there is almost no rate loss between the perfectCSIT and limited feedback cases since the quantization noise -20 -10 0 10 20 SNR (dB) R a t e ( bp s / H z ) Perfect CSITLimited Feedback
Fig. 2: The achievable rate of a single-user MISO system withCSIT and limited feedback when N t = 16 and B = 8 .dominates the AWGN noise in this regime. Third, in the lowSNR regime ( < dB), we see there is a constant horizontaldistance between each pair of solid curve and dashed curvewhich implies that there is a constant power loss incurred bylimited feedback. This is because the AWGN noise dominatesthe performance and the results from previous work assuminginfinite-bit ADCs [22], [27] can apply.In Fig. 3, we show the achievable rates in a multi-userMISO channel. The number feedback bits is chosen as B =2( N t − b −
12 = 6 b − . First, apart from the single-usercase, there is a gap at high SNR between the case of perfectCSIT and limited feedback due to the inter-user interference.Second, the gaps between each pair of curves are all around
20 -10 0 10 20 30 40
SNR (dB) R a t e ( bp s / H z ) Perfect CSITLimited Feedback b = 3, 4, 5B = 6,12,18
Fig. 3: The achievable rate of a multi-user MISO system withperfect CSIT and limited feedback when N t = 4 and K = 2 .When there is perfect CSIT, the figure shows three cases where b = 3 , , . When there is limited feedback, the figure showsthree cases where ‘ b = 3 , B = 6 ’, ‘ b = 4 , B = 12 ’ and‘ b = 5 , B = 18 ’. . bps/Hz, which verifies our analytical result in (31) statingthat B should increase N t − bits if b increases one bit.Third, at low SNR ( < dB), the power loss is very smallfor three cases, which validates our results in (29) saying thatthe power loss is
10 log (cid:16) − − BN t − (cid:17) dB, which is around − . dB when B = 6 , − . dB when B = 12 , and − . dB when B = 18 . VI. C ONCLUSIONS
In this paper, we analyzed the achievable rate in MISOsystems with finite-bit ADC and limited feedback. For boththe single-user and multi-user channels, the results are similarto those with infinite-bit ADC at low SNR except for anadditional power loss
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