ONE-BIT ADCS IN WIDEBAND MASSIVE MIMO SYSTEMS WITH OFDM TRANSMISSION

Christopher Mollen 1 , Junil Choi 2 , Erik G. Larsson 1 , and Robert W. Heath Jr. 2

1 Linkoping University, Dept. of Electrical Engineering, 581 83 Linkoping, Sweden2 University of Texas at Austin, Dept. of Electrical and Computer Engineering, Austin, TX 78712, USA

ABSTRACT

We investigate the performance of wideband massive MIMO basestations that use one-bit ADCs for quantizing the uplink signal. Ourmain result is to show that the many taps of the frequency-selectivechannel make linear combiners asymptotically consistent and thequantization noise additive and Gaussian, which simplifies signalprocessing and enables the straightforward use of OFDM. We alsofind that single-carrier systems and OFDM systems are affected inthe same way by one-bit quantizers in wideband systems becausethe distribution of the quantization noise becomes the same in bothsystems as the number of channel taps grows.

Index Terms— massive MIMO, OFDM, one-bit ADCs, quantiza-tion, wideband.

1. INTRODUCTION

A one-bit Analog-to-Digital Converter (ADC) is the simplest devicefor quantization of an analog signal into a digital. It is the leastpower consuming quantizer, since the power consumption of ADCsgrows exponentially with the number of bits needed to represent allquantization levels [1]. One-bit ADCs would also simplify the analogfront end, e.g., automatic gain control would become trivial. One-bit ADCs just output the sign of the input signal, all other informa-tion is discarded. The use of such radically coarse quantization hasbeen suggested for use in massive Multiple-Input Multiple-Output(MIMO) base stations, where the large number of radio chains makeshigh resolution quantization very power consuming [2]. Recent stud-ies have shown that the performance loss due to the coarse quantiza-tion of one-bit ADCs becomes less severe as the number of receivingantennas grows; and, in a massive MIMO base station that has hun-dreds of antennas, the power saving that comes with one-bit ADCsmight well outweigh this performance loss [3–5].

Pioneering work on the performance achievable with one-bitADCs was done in [6–8]. These works and most studies of one-bit ADCs since have focused on narrowband systems that havefrequency-flat channels. Whether the results also hold for wide-band systems, whose channels more realistically are modeled as fre-quency selective, is not clear from the cited literature.

Here we highlight that, when the number of channel taps is large,quantization noise due to one-bit ADCs is effectively additive andcircularly symmetric Gaussian, and affects a system that uses or-thogonal frequency division multiplexing (OFDM) the same way itaffects a single-carrier system. As a consequence OFDM can easilybe implemented in the same way as in the unquantized system. We

The research leading to these results has received funding from theEuropean Union Seventh Framework Programme under grant agreementnumber ICT-619086 (MAMMOET), the Swedish Research Council (Veten-skapsradet) and the National Science Foundation under grant number NSF-CCF-1527079.

IFFT CPFFT

CP

Combiner

IFFT CPCP

FFT

Fig. 1. The system model for both single-carrier (dashed IFFT andFFT are not used) and OFDM transmission (dashed boxes are used).

also show that the averaging effect of having multiple samples frommany antennas, which enables the use of one-bit ADCs, is even moreeffective in a wideband system, where averaging is also done overtime due to the frequency selectivity of the channel.

In contrast to the unquantized case, sampling at rates higher thanthe Nyquist-rate might lead to improved performance in a quantizedsystem [9]. The power consumption of the ADC, however, increasesproportionally with the sampling rate [1]. This study is limited tothe study of the extreme case, where sampling is done at Nyquist-rateand quantizing is done with one-bit resolution. Related work on low-order ADCs with frequency-selective channels considered nonlineardetection algorithms, either with single-carrier transmission [10] orwith OFDM [11]. In contrast, we study a general system that usessimple combiners that are linear in the quantized signals, both forsingle-carrier and OFDM transmission.

2. SYSTEM MODEL

We consider the uplink of the massive MIMO system in Figure 1,where the base station has M antennas and there are K single-antenna users. The system is modeled in complex baseband andthe signals are uniformly sampled at the Nyquist-rate with perfectsynchronization.

The users transmit the signals x[n] , (x1[n], . . . , xK [n])T atsymbol duration n over the frequency-selective channel that is de-scribed by the L-tap impulse response {H[`]}L−1

`=0 , where H[`] is anM×K-dimensional matrix. The elements {hmk[`]}L−1

`=0 at position(m, k) form the impulse response between user k and base stationantenna m. To study a wideband scenario, the number L is assumedto be in the order of tens. For example, a system that uses 15 MHzof bandwidth over a channel with 1 µs of maximum excess delay,which corresponds to a moderately frequency-selective channel, hasL = 15 channel taps, cf. [12], where the “Extended Typical UrbanModel” has a maximum excess delay of 5 µs. The received signal atantenna m at symbol duration n is

ym[n] ,K∑

k=1

L−1∑`=0

√Pkhmk[`]xk[n− `] + zm[n], (1)

where zm[n] ∼ CN (0, N0) models the thermal noise of the basestation hardware and Pk is the transmit power of user k.

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Upon reception, the in-phase and quadrature signals are sepa-rately quantized, each by a one-bit ADC:

q(ym[n]

),

1√2sign

(Re ym[n]

)+ j

1√2sign

(Im ym[n]

). (2)

Here the threshold of the one-bit ADC is assumed to be zero. Otherthresholds are also possible [13]. The arbitrary scaling of the quan-tized signal is chosen such that (2) has unit power. For convenience,we denote qm[n] , q

(ym[n]

)and q[n] , (q1[n], . . . , qM [n])T.

Both single-carrier and OFDM transmission are studied. Thetransmission is observed for a block of N symbols. During sym-bol periods n = 0, . . . , N−1, we assume that the users transmit

x[n] =

{s[n], if single-carrier1√N

∑N−1ν=0 s[ν]ej2πnν/N , if OFDM

, (3)

where s[n] = (s1[n], . . . , sK [n])T is the vector of data symbolsthat are concurrently transmitted during symbol duration n by the Kusers. We assume that E[ sk[n] ] = 0 and E

[|sk[n]|2

]= 1 for all

k, n. The users also transmit an L−1-symbol long cyclic prefix:

x[n] = x[N + n], −L < n < 0. (4)

The signal power is normalized such that E[|xk[n]|2

]= 1, ∀n, k.

3. RECEIVER COMBINING

The base station combines the received signals in a FIR filter to esti-mate the transmitted signals, just as it would have if the quantizationwere perfect. The resulting estimate is

x[n] ,(x1[n], . . . , xK [n]

)T=

1

M

`max∑`=`min

W[`]q[n− `], (5)

where `min and `max are the smallest and largest indices of the non-zero taps of the impulse response of the combiner {W[`]∈CK×M}:

W[`] ,1

N

N−1∑n=0

W[n]ej2πn`/N , ` = `min, . . . , `max, (6)

where the frequency domain combining matrices W[n] can be de-fined in a number of different ways. Three common linear com-biners are the Maximum-Ratio, Zero-Forcing, and Minimum Mean-Square-Error Combiners (MRC, ZFC, MMSEC):

W[n] =

H

H[n], if MRC

(HH[n]H[n])−1H

H[n], if ZFC

(HH[n]H[n] + ρIK)−1H

H[n], if MMSEC

, (7)

where ρ in the definition of MMSEC is a system parameter that canbe used to make the combiner more like MRC (large ρ) or more likeZFC (small ρ). The definitions are in terms of the channel spectrum:

H[n] ,L−1∑`=0

H[`]e−j2πn`/N . (8)

The element at position (k,m) of W[`] is denoted by wkm[`]. Thesymbol estimates are obtained directly as

s[n] =

{x[n], if single-carrier1√N

∑N−1ν=0 x[ν]e−j2πνn/N , if OFDM

. (9)

In the comparison of single-carrier and OFDM transmission, weassume that the channel is perfectly known to the base station. In

reality, perfect channel state knowledge is not realistic in massiveMIMO, because of the huge dimension of the channel, and more sowhen one-bit ADCs are used, due to the coarse quantization of thereceived signals. We reason that any estimation of the channel willaffect a single-carrier system the same way it will affect an OFDMsystem. Therefore, if the performance of the two systems is the samewith perfect channel state information, it should also be the samewith imperfect channel state information. Previous work [14,15] hasshown that channel state information can be acquired in a massiveMIMO system even with one-bit ADCs.

4. QUANTIZATION NOISE

By showing that the estimates of the linear combiners in (7) are con-sistent also with one-bit ADCs, we will show that the orthogonalityof the transmit symbols are preserved and that OFDM easily can beimplemented in massive MIMO when the number of taps is large.

4.1. Consistency of Linear Combiners

The following theorem states that, in the limit of infinite number ofantennas, the estimates of linear combiners converge to a value fromwhich the transmit signal can be recovered, i.e., that linear combinersare consistent.

Theorem 1. In a Rayleigh fading channel with uniform delay pro-file, hmk[`] ∼ CN (0, 1

L) IID for all m, k and `, the linear combiners

in (7) are consistent, i.e., given the transmit signals {xk[n]}, thereexists a deterministic invertible function g : C → C such that

xk[n]a.s.−→ g(xk[n]), M → ∞. (10)

Proof. We note that all combiners in (7) converge to the MRC as thenumber of antennas tends to infinity for an IID Rayleigh fading chan-nel because the propagation is favorable [16]. We therefore considerthe MRC estimate of the k-th user:

xk[n] =1

M

M∑m=1

L−1∑`=0

h∗mk[`]qm[n+ `]. (11)

Since the terms in the sum are IID, by the law of large numbers,

limM→∞

xk[n] =

L−1∑`=0

E[h∗mk[`]qm[n+ `]

](12)

=

L−1∑`=0

E[h∗mk[`]E

[q(√

Pkhmk[`]xk[n]+uk[n, `]) ∣∣ hmk[`]

]],

(13)

where

uk[n, `] ,K∑

k′=1

L−1∑`′=0

(k′,`′)6=(k,`)

√Pk′hmk′ [`′]xk′ [n+`−`′] + zm[n+`′].

(14)

We note that uk[n, `] ∼ CN (0, Ik[n, `]), where

Ik[n, `] ,1

L

K∑k′=1

L−1∑`′=0

(k′,`′)6=(k,`)

Pk′∣∣xk′ [n+ `− `′]

∣∣2 +N0. (15)

It is noted that, for any given complex numbers h and x andstochastic variable u ∼ CN (0, σ2), it is true that

√2Re

(E[q(hx+ u

) ∣∣ h])= Pr(Rehx > −Reu | h)− Pr(Rehx < −Reu | h) (16)= 1− 2Pr(Rehx < −Reu | h) (17)

= 1− 2Q

(√2Rehx

σ

). (18)

The imaginary part can be rewritten in the same way.We now let hRe , Rehmk[n] and hIm , Imhmk[n] be the IID

real and imaginary parts of the channel coefficient, hRe ∼ hIm ∼N (0, 1

2L). To simplify the mathematical notation, we initially as-

sume that xk[n] is real in the following steps—the complex inter-fering transmit signals xk′ [n′], (k′, n′) 6= (k, n) are still arbitrary.

Then the variable a , xk[n]√

2PkIk[n,`]

is real and the expected valuein (13) is given by

1√2E[(hRe − jhIm

)(1− 2Q(ahRe) + j − j2Q(ahIm)

)](19)

= −√2E[hRe Q(ahRe) + hIm Q(ahIm)

+ j(hRe Q(ahIm)− hIm Q(ahRe)

)](20)

= −2√2E[hRe Q(ahRe)

]. (21)

In the first step, we used (18) and its imaginary counterpart. In thelast step, the two terms of the imaginary part of the expectation arezero, because the real and imaginary parts hRe and hIm are indepen-dent and zero mean, and the two terms of the real part are the same,because they are identically distributed.

If we let fh(h) be the probability density function of a N (0, 12L

)distributed random variable, then (13) becomes

limM→∞

xk[n] = −2√2

L−1∑`=0

∞∫−∞

fh(h)hQ

(xk[n]

√2Pk

Ik[n, `]h

)dh

(22)

, g′(xk[n]). (23)

The steps in (19)–(21) can be repeated for xk[n] with arbitrary mod-ulus. It can then be seen that, for a general transmit signal xk[n],

limM→∞

xk[n] =xk[n]

|xk[n]|g′(|xk[n]|

), g(xk[n]

). (24)

Because of this relation between the functions g and g′, it is enoughto prove that g′ : R → R is monotonic to prove that g also is invert-ible. We consider the derivative of g′:

d

dxg′(x) =

L−1∑`=0

2√2Pk√

πIk[n, `]

∞∫−∞

fh(h)h2e

−Pkx2h2

Ik[n,`] dh (25)

> 0, ∀x. (26)

Here we used the fact that ddx

Q(x) = − 1√2π

e−x2

2 . Since g′ ismonotonically increasing, it is invertible. Because of (24), g is in-vertible too.

Theorem 1 tells us that if Ik[n, `] is known, then xk[n] can be de-tected error-free as M → ∞. Since knowing Ik[n, `] requires someknowledge of the interfering symbols {xk′ [n′]}, (k′, n′) 6= (k, n),determining it perfectly is only possible in a single-user frequency-flat channel, where Ik[n, `] = N0. In a wideband system, however,the function g does not depend on Ik[n, `], only its mean, and gooddetection can be achieved without knowledge of the interfering sym-bols. This is formalized in the following theorem.

Theorem 2. In a wideband system, the function g approaches thedeterministic linear function

g(xk[n])a.s.−→√

2

πxk[n]

√Pk

N0 +∑K

k′=1 Pk′, L → ∞. (27)

−20 0 20

−20

0

20

quad

ratu

resi

gnal

M=1000, K=1, L=1, no noise

−20 0 20

−20

0

20

M=1000, K=1, L=1, N0= −10 dB

−40 −20 0 20 40

−40

−20

0

20

40

in-phase signal

quad

ratu

resi

gnal

M=10000, K=2, L=1, no noise

−20 0 20

−20

0

20

in-phase signal

M=1000, K=1, L=30, no noise

Fig. 2. The MRC estimates (blue) and the 16-QAM transmit signals(red) with M antennas, K users, L channel taps. The power Pk = 1,∀k, and the channel is IID Rayleigh fading hmk[`] ∼ CN (0, 1

L).

The channel is known at the base station.

Proof. First, we note that the interference becomes deterministic:

Ik[n, `]a.s.−→ N0 +

K∑k′=1

Pk′ , L → ∞. (28)

By evaluating the integral in (25), it is seen that the derivative con-verges pointwise:

d

dxg′(x) =

√2

π

L−1∑`=0

√Pk

Ik[n, `]

√L(

L+ Pkx2

Ik[n,`]

)3/2 (29)

a.s.−→√

2Pk

π(N0 +

∑Kk′=1 Pk′

) , L → ∞. (30)

The sum in (29) is dominated by the constant function√

PkImin

, where

Imin , min{Ik[n, `], L = 1, . . .}, for any realization {xk[n]}, i.e.,

L−1∑`=0

√Pk

Ik[n, `]

√L(

L+ Pkx2

Ik[n,`]

)3/2 ≤√

Pk

Imin, ∀L. (31)

Because the dominating function is integrable over any finite inter-val [0, x], the limit of g′(x) can be obtained by integration of thelimit of its derivative (30) according to the theorem of dominatedconvergence. Because g′(0) = 0, this completes the proof.

The consistency and the behavior of the function g can be seenin the examples given in Figure 2. In the upper left system, no am-plitude information can be recovered, even if the number of antennasis large, because the variance Ik[n, `] = 0. In this case, dithering thereceived signal prior to quantization helps in recovering the ampli-tude, see the upper right system, where noise has been added. Thefunction g is nonlinear but amplitude information is still recoverablewith enough antennas at the base station. In the lower left system,the variance Ik[n, `] assumes three distinct values depending on thepower of the transmit signal of the second user, which results in threepossible points for each symbol estimate. In the lower right wide-band system however, the function g is deterministic and linear.

−2 −1 0 1 2−2

−1

0

1

2

wid

eban

dsy

stem

K=

5,L

=15

quad

ratu

resi

gnal

single carrier

−2 −1 0 1 2−2

−1

0

1

2

OFDM

−5 0 5

−5

0

5

in-phase signal

narr

owba

ndsy

stem

K=

2,L

=2

quad

ratu

resi

gnal

−5 0 5

−5

0

5

in-phase signalFig. 3. Symbol estimates sk[n] after one-bit quantization and ZFC ina massive MIMO base station with 128 antennas that serves K usersover an L-tap Rayleigh fading channel, Pk = 1,∀k, and N0 = 0.

4.2. Distribution of Error Due to Quantization

The error after quantization is given by:

ek[n] , xk[n]− E[g(xk[n])

∣∣ xk[n]]= dk[n] + rk[n], (32)

where

dk[n] ,1

M

M∑m=1

L′−1∑`=0

wkm[`]qk[n− `]− g(xk[n]) (33)

rk[n] , g(xk[n])− E[g(xk[n])

∣∣ xk[n]]. (34)

The first term becomes circularly symmetric Gaussian when thenumber of antennas grows large according to the central limit the-orem. The second term is a term that is proportional to the transmitsignal xk[n]. The error, thus, consists of two parts: one circularlysymmetric Gaussian dk[n] and one radial distortion rk[n].

In a narrowband system, where the term Ik[n, `] can vary sig-nificantly, the radial distortion is prominent. In a wideband systemhowever, the radial distortion is negligible because of Theorem 2.When only the circularly symmetric error is present, the error due toquantization has the same distribution in the time and frequency do-main and there is no difference in the performance between single-carrier and OFDM transmission. The radial distortion, however, iseasier to equalize if the symbols are in the time domain than if theyare in the frequency domain. This can be seen in Figure 3. In thewideband system, we see that single-carrier and OFDM perform thesame: the variance and distribution of the quantization noise is thesame. In the narrowband system, the radial error due to quantizationis not negligible, because the interference power Ik[n, `] varies be-tween estimates. It can be seen that the symbols of the single-carriersystem is easier to distinguish than those of the OFDM system. Wealso see that the error variance due to quantization can be smallerin a wideband system than in a narrowband system with the sameamount of antennas at the base station, at least if OFDM is used.

5. NUMERICAL EXAMPLES

To verify the feasibility of OFDM, we have done a numerical studyof a massive MIMO system that uses linear receive combining. The

5 10 50 100 50010000

2

4

6

K=5, L=15

K = 30,L = 15

number of antennas M

rate

peru

ser[

bpcu

] zero-forcing combiningmaximum-ratio comb.

0 10 20 300

2

4

6

M = 128, K = 5

M = 128, K = 30

number of taps L

Fig. 4. The rate in a system with M antennas that serves K usersover an L-tap Rayleigh fading channel. The power is Pk/N0 =10 dB,∀k. The curves for single-carrier and OFDM transmission co-incide. The dashed curve shows the performance of the same systemwith perfect quantization (the rate of unquantized ZFC is mostly out-side the drawn range).

channel has been modeled as IID Rayleigh with a constant powerdelay profile, i.e., hmk[`] ∼ CN (0, 1/L). The symbol estimate canbe divided into

sk[n] = ask[n] + z′k[n], (35)

where the choice a = E[ s∗k[n]sk[n] ] minimizes the variance of thenoise term z′k[n] [17]. Then we can define a Signal-to-Interference-Quantization-and-Noise Ratio as

SIQNR ,|a|2

E[ |z′k[n]|2 ]. (36)

Because E[ s∗k[n]z′k[n] ] = 0, we can achieve the following rate by

treating all uncorrelated noise z′k[n] as Gaussian [18]

R , log2(1 + SIQNR) [bpcu]. (37)

This rate was computed for different numbers of antennas anddifferent numbers of channel taps in Figure 4. It can be seen that therate increases as the number of antennas increases and that one-bitADCs with linear combining becomes feasible in the massive MIMOregime. Furthermore, we see that the rate also increases with thenumber of channel taps. The improvement saturates when Ik[n, `]becomes deterministic. With a large K, this happens sooner, com-pare the improvement when K = 5 and K = 30 as L grows inFigure 4. This suggests that one-bit ADCs perform the same or bet-ter in frequency-selective channels compared to frequency-flat chan-nels, and that wideband systems even can improve the performanceof one-bit ADCs.

6. CONCLUSION

In wideband massive MIMO systems, one-bit ADCs affect the per-formance of single-carrier and OFDM transmission in the same way,which means that many results for single-carrier systems carry overalso to OFDM systems. The frequency selectivity of the widebandchannel helps to spread the effect of the quantization, so that allsymbols are affected in the same way. We proved that this makesthe estimates of linear combiners consistent and the noise circularlysymmetric Gaussian, which makes OFDM easy to implement. Fur-ther, we note that the quantization noise has two parts: one radial andone additive circularly symmetric Gaussian. Only the latter is signif-icant in a wideband system, where we have shown that the frequencyselectivity of the channel makes radial distortion negligible.

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