High SNR Capacity of Millimeter Wave MIMO systems...

High SNR Capacity of Millimeter Wave MIMO systems with

One-Bit Quantization

Jianhua Mo and Robert W. Heath Jr.

Wireless Networking and Communications Group

Department of Electrical and Computer Engineering

The University of Texas at Austin

www.profheath.org

http://www.profheath.org

(c) Robert W. Heath Jr. 2014

Why mmWave for Cellular?

Microwave

3 GHz 300 GHz

�2

28 GHz 38-49 GHz 70-90 GHz300 MHz

Huge amount of spectrum available in mmWave bands* Cellular systems live with limited microwave spectrum ~ 600MHz

29GHz possibly available in 23GHz, LMDS, 38, 40, 46, 47, 49, and E-band

Technology advances make mmWave possible Silicon-based technology enables low-cost highly-packed mmWave RFIC**

Commercial products already available (or soon) for PAN and LAN

Already deployed for backhaul in commercial products

1G-4G cellular 5G cellular

m i l l i m e t e r w a v e

* Z. Pi and F. Khan. "An introduction to millimeter-wave mobile broadband systems." IEEE Communications Magazine, vol. 49, no. 6, pp.101-107, Jun. 2011. ** T.S. Rappaport, R. W. Heath Jr., R. C. Daniels, and J. N. Murdock, Millimeter Wave Wireless Communication, Prentice Hall, 2014.


mmWave Needs Antenna Arrays

�3

antennas are small (mm)

highly directional MIMO transmission

O. El Ayach, S. Abu-Surra, S. Rajagopal, Z. Pi, and R. W. Heath, Jr., `` Spatially Sparse Precoding in Millimeter Wave MIMO Systems,’' IEEE Trans. on Wireless, 2014.

Smaller wavelength means smaller captured energy at antenna

Larger bandwidth means higher noise power and lower SNR

Use antenna arrays for TX and RX array gain

Baseband Processing

base station mobile station

Hybrid Beamforming for mmWave

Mixed signal components are power hungry Example: 2.5GS/s 14bit ADC consumes 2.39W (2013)*

Use analog or hybrid beamforming** with a few ADCs Small number of digital basebands (2 or 4)

�4

RF Chain

RF Chain

DigitalPrecoder

FBB

NBS

NMSN

RFN

S

RFChain

RF Chain

DigitalPrecoder

FBB

FRFFRF

Digital/Baseband

Analog/RF

NS

NRFBS

BS BSMS

Digital/Baseband

Analog/RF

RF Chain

RF Chain

DigitalPrecoder

FBB

NBS

NMSN

RFN

S

RFChain

RF Chain

DigitalPrecoder

FBB

FRFFRF

Digital/Baseband

Analog/RF

NS

NRFBS

BS BSMS

Digital/Baseband

Analog/RF

*B. Murmann, "ADC Performance Survey 1997-2013," [Online]. Available: http://www.stanford.edu/~murmann/adcsurvey.html ** O. El Ayach, S. Abu-Surra, S. Rajagopal, Z. Pi, and R. W. Heath, Jr., `` Spatially Sparse Precoding in Millimeter Wave MIMO Systems,'' to appear in EEE Trans. on Wireless. Available on ArXiv.

ADC

ADC

…

DAC

DAC

…

Feasible for base stations, may be too power hungry at mobile

http://www.stanford.edu/~murmann/adcsurvey.html


Alternative RX Architecture

�5



Use low power 1-bit ADCs for each I/Q channelAllows digital beamforming in baseband

Reduces power consumption

Prior work by Madhow et. al., Nossek et. al., and Fettweis et. al.Usually channel is known perfectly

2

Analog or Hybrid Beamforming

High Resolution

ADCRF

Beamforming performed in digital

1-bit ADCRF

1-bit ADCRF

1-bit ADCRF

... ...vs.

Use very coarse 1-bit ADC for each I/Q signal

1-bit flash ADC consists of only one comparator**


* B. Murmann, "ADC Performance Survey 1997-2013," [Online]. Available: http://www.stanford.edu/~murmann/adcsurvey.html ** O. El Ayach, S. Abu-Surra, S. Rajagopal, Z. Pi, and R. W. Heath, Jr., `` Spatially Sparse Precoding in Millimeter Wave MIMO Systems,’' IEEE Trans. on Wireless, 2014.

http://www.stanford.edu/~murmann/adcsurvey.html

SISO with One-Bit ADCsCapacity analysis

Real AWGN channel (Madhow TCOM09)

Oversampling (Dabeer SPAWC06, Fettweis SARNOFF12 & ITG13)

Channels with memory (Zeitler TCOM12)

Channel estimation Use dithering (Dabeer IT06 & ICC10, Zeilter TSP12)

Synchronization Blind synchronization with low-resolution ADC (Wadhwa Allerton13)

Receiver Design ML decoder and suboptimal detector for QPSK (Wang TCOM13)

�6

J. Singh, S. Punnuru and U. Madhow, “Multi-gigabit communication: the ADC bottleneck”, TCOM, 2009 O. Dabeer and U. Madhow, “Channel estimation with low precision ADC”, ICC, 2010 S. Krone and G. Fettweis, “Capacity of communications channels with 1-bit quantization at the receiver”, SARNOFF, 2012 S. Ponnuru, U. Madhow, “Joint mismatch and channel compensation for high speed OFDM receivers with Time-Interleaved ADCs”, TCOM, 2010 G. Zeitler, G. Kramer, et. al. “Bayesian parameter estimation using single-bit dithered quantization”, TSP, 2012 G. Zeitler, A. Singer and G. Kramer, “Low precision ADC for maximum information rate in channels with memory”, TCOM, 2012 L. Landau, S. Krone and C. Fettweis, “ISI design for maximum information rates with 1-bit quantization and oversampling at the receiver”, ITG 2013 Z. Wang, et al, ‘’Monobit Digital Receivers for QPSK: Design, Performance and Impact of IQ imbalances”, TCOM 2013 A. Wadhwa and U. Madhow, “Blind phase/frequency synchronization with low-resolution ADC: a Bayesian approach”, Allerton, 2013

Extensions to MIMOMIMO with one-bit ADCs

Time block coding design (Ivrlac ISIT06)

Static channel (Mezghani ISIT07)

Rayleigh fading channels (Mezghani ISIT08)

Noncoherent fading channel (Mezghani ISIT09)

Detection (Mezghani ISIT10 & MELECON12)

Channel estimation (Ivrlac ITG07，Mezghani ITG10)

Limitations with respect to mmWave Capacity analysis at low SNR using certain approximations

General channel model instead of the mmWave channel

�7

M. Ivrlac and J. Nossek, “Challenges in Coding for Quantized MIMO systems”, ISIT, 2006 M. Ivrlac and J. Nossek, “On MIMO channel estimation with single-bit signal quantization”, ITG 2007 A. Mezghani, and J. Nossek, “On ultra-wideband MIMO systems with 1-bit quantized outputs: performance analysis and input optimization”, ISIT, 2007 A. Mezghani, and J. Nossek, “Analysis of Rayleigh-fading Channels with 1-bit Quantized Output”, ISIT, 2008 A. Mezghani, and J. Nossek, “Analysis of 1-bit output noncoherent fading channels in the low SNR regime”, ISIT 2009 A. Mezghani, R. Ghiat and J. Nossek, “Transmit Processing with low resolution DAC” ICECS, 2009

Contributions

�8



Use low power 1-bit ADCs for each I/Q channelAllows digital beamforming in baseband

Reduces power consumption

Prior work by Madhow et. al., Nossek et. al., and Fettweis et. al.Usually channel is known perfectly

2

Analog or Hybrid Beamforming

High Resolution

ADCRF


1-bit ADCRF

1-bit ADCRF

1-bit ADCRF

... ...vs.

1) Insights into the SIMO and MIMO high SNR capacity

2) A novel co-phasing strategy for mmWave MIMO systems

TX

ADC

ADC

ADC

MIMO Point-to-Point Channel

Assumptions Nr antennas at the receiver, 2Nr one-bit ADCs

Channel is known to both the receiver and the transmitter

A simple upper bound of the capacity is 2Nr

�9

What is the performance loss compared to high-resolution ADC?

Meeting Report

Jianhua MoWireless Networking and Communications Group

December 9, 2013

1 Research Progressr = Q(Hx+ n) (1)

In the meeting, we first talked about the research progress.

1. I obtained some numerical results about the upper bound of the MIMO channel capacity athigh SNR.

Table 1: The number of the possible distinguishable input symbols.HHHHHHNr

Nt 1 2 3 4 5 6 7 8

1 4 4 4 4 4 4 4 42 8 16 16 16 16 16 16 163 12 52 64 64 64 64 64 644 16 128 240 256 256 256 256 2565 20 260 764 1004 1024 1024 1024 10246 24 464 2048 3632 4072 4096 4096 40967 28 756 4760 11624 15628 16356 16384 163848 32 1152 9888 32768 55648 64384 65504 65536

Denote the number of distinguishable input symbols as K(Nr, Nt). From Table 1, we findthat

• If Nt � Nr, then K(Nr, Nt) = 2

2Nr . (We can prove this.)

• K(Nr, 1) = 4Nr. (We can prove this.)

• If Nr is even, K(Nr,Nr2 ) = 2

2Nr�1.

• K(Nr, Nt) +K(Nr, Nr �Nt) = 2

2Nr .

2. I have proposed a method to find all the distinguishable input symbols. There are at most2

2Nr possible distinguishable input symbols. Denote v = [vT1 vT

2 ]T as a 2Nr ⇥ 1 vector with

elements 2 {�1, 1} where v1 and v2 are Nr ⇥ 1 vectors.

1

Transmitter

ADC

ADC

RF

RF

RF

{1,-1}

{1,-1}

{1,-1}

ADC

Digital Baseband Processing

H

AWGN

�10

SIMO Channel (1)

A/D

Transmitter ... ...... Digital

Processing

RF

RF

...

{1,-1}

Baseband

ADC {1,-1}

Re()

Im()

A/D

{1,-1}ADC {1,-1}

Re()

Im()

Fig. 1. System model.

We define a one-bit quantization function Q as follows.Assume that r = {ri, 1 i Nr} and y = {yi, 1 i Nr}.If

<(ri) = sgn(<(yi)), (2)=(ri) = sgn(=(yi)), (3)

for 1 i Nr, then we denote that r = Q(y). Here, sgn(x)is the sign function and ri 2 {1 + j, 1� j,�1 + j,�1� j}.

After the one-bit quantization at the receiver, we obtain theoutput r = Q(y). Consequently, the channel capacity withone-bit quantization is

C = max

p(x):tr(E(xx⇤))PI(x; r) (4)

where P is the average power constraint at the transmitter.

III. HIGH SNR CAPACITY OF MIMO CHANNEL WITHONE-BIT QUANTIZATION

The mutual information in (4) is in the form of multipleintegrals; obtaining a closed form expression for the optimiza-tion problem seems to be a challenge. Thus, we derive anapproximation of the channel capacity in the high SNR regime.

A. SISO Channel with One-Bit Quantization

First, we deal with the very special case when Nr = Nt = 1.The channel coefficient now is a scalar denoted by h.

Lemma 1. The capacity of the SISO channel with one-bitquantization is achieved by rotated QPSK signaling, i.e.,

Pr{x =

pPej(k⇡+

⇡4 �\h)} =

1

4

, for k = 0, 1, 2, 3, (5)

and is given by

CSISO(P ) = 2

⇣1�H

⇣Q(|h|

pP )

⌘⌘, (6)

where H(p) = �p log2 p� (1�p) log2(1�p) and Q(·) is thetail probability of the standard normal distribution.

Proof: Without loss of optimality, we can assume that thetransmitted signal is x = e�j\hx̂. The outputs of the one-bitquantizer will be <(r) = sgn(|h|<(x̂) + <(n)) and =(r) =

sgn(|h|=(x̂)+=(n)). Therefore, the channel is decoupled intotwo real channels with the same channel gain. For each realchannel, it is proven in [8, Theorem 2] that binary antipodalsignaling is optimal. Therefore, the optimal input for the SISOchannel is rotated QPSK signaling.

As P ! 1, it follows that Q(|h|pP ) ! 0 and

limP!1 CSISO(P ) = 2 bps/Hz.

B. SIMO Channel with One-Bit Quantization

In the SIMO channel with Nr antennas at the receiver, thereare at most 22Nr possible quantization outputs. Therefore, 2Nr

is a simple upper bound for the channel capacity. This upperbound, unfortunately, cannot be approached when Nr is largerthan one, as shown in the following proposition.

Proposition 1. The capacity of the SIMO channel with one-bitquantization at high SNR, denoted as CSIMO(Nr), satisfies

log2(4Nr) CSIMO(Nr) log2 (4Nr + 1) .

Proof: Denote the SIMO channel as h =

[h1, h2, · · · , hNr ]T . When the phase of the transmitted

symbol x is around \x = k⇡/2 � \hi(k = 0, 1, 2, 3; i =

1, 2, · · · , Nr), one element of the one-bit quantization outputwill change its sign. There are at most 4Nr such phases,denoted as � = {�i, 1 i 4Nr}. Following the derivationsin [7] and [9], it can be shown the high SNR capacity isachieved with transmit symbols from three categories:

1) The symbol zero;2) The symbols with phases in �;3) The symbols with phases not in �.For the zero symbol, Pr(r|x = 0) = 2

�2Nr for eachpossible r. For the symbols with phases not in �, Pr(r|x) =1(r = Q(hx)) where 1(·) is the indicator function. At lastconsider the symbols with phases in �. If \x = �\h1,then Pr(r = [1 + j,Q(hx)T2:Nr

]

T |x) = Pr(r = [1 �j,Q(hx)T2:Nr

]

T |x) = 1/2. The other conditional probabilitiescan be derived similarly. Therefore, the transition probabilitymatrix from these three kinds of input symbols to the outputis

Pr(r|x) =

2

42

�2Nr · · · · · · 2�2Nr

T4Nr⇥4Nr 04Nr⇥(22Nr�4Nr)

I4Nr⇥4Nr 04Nr⇥(22Nr�4Nr)

3

5 (7)

where T is a 4Nr⇥4Nr circulant matrix with the first row as[1/2, 1/2, 0, · · · , 0].

Assume that these three kinds of symbols are transmittedwith probabilities p0, p1 and 1 � p0 � p1, respectively. Theresulting mutual information, denoted as f(p0, p1), is as shownat the top of the next page. The channel capacity can be com-puted by searching the optimal p0 and p1, denoted as p⇤0 andp⇤1, which maximizes the mutual information f(p0, p1). It turnsout that @f(p0, p1)/@p1 < 0 and thus p⇤1 = 0. Therefore, thereare at most 4Nr + 1 possible input symbols in the capacity-achieving distribution and an upper bound of the capacity islog2(4Nr + 1). The lower bound log2(4Nr) is achieved bysetting p0 = 0 and p1 = 0, i.e., f(0, 0) = log2(4Nr).

Corollary 1. When Nr is large, the capacity of SIMO channelwith one-bit quantization at high SNR is

CSIMO(Nr) ⇡ log2(4Nr + 1).

Proof: When Nr is large,

f(p0, 0) ⇡ �(1� p0) log21� p04Nr

� p0 log2 p0.

The SIMO receiver with one-bit ADCs is like a phase detector The whole range is divided into 4Nr regions

The optimal alphabet contains 4Nr-PSK (irregular) symbols and zero

−1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

In Phase

Qua

drat

ure

8−PSK constellationLines with phases in Φ

Fig. 3. In our proposed strategy, the transmitter sends 8-PSK symbols and� = {⇡

8 + k ⇡4 , k = 0, 1, · · · , 7} when Nr = 2.

To obtain beamforming gain, matched filter beamforming isused at the transmitter and the resulting channel is equivalentto a SIMO channel. The transmitter sends 4Nr-PSK symbolssince the receiver can at most distinguish 4Nr received signalsin the high SNR regime. Now we propose a receiver structurefor this system. In each antenna, a phase shifter is installedbefore the quantizers to rotate the received signal. The outputof the quantizer is

r = Q (br � (↵ar ('r)x+ n)) , (12)

where br = [ej 0 , ej 1 , ..., ej Nr�1]

T represents the rotationoperation implemented by phase shifters.

In the proof of Proposition 1, we see that the SIMO receiveracts as a phase detector defined by �. The whole range[0, 2⇡] is divided into 4Nr regions {[�m,�m+1](0 m 4Nr � 2) and [�4Nr�1,�0]} assuming 0 �0 < �1 <· · · < �4Nr�1 < 2⇡. A simple and heuristic design is tolet the phases of transmitted 4Nr-PSK symbols, which are{ k⇡2Nr

, 0 k 4Nr � 1}, fall into each of the 4Nr regions.Therefore, we want � = { ⇡

4Nr+ k ⇡

2Nr, 0 k 4Nr � 1}.

This can be achieved by designing br such that

b� ar('r) =1pNr

ej⇡

4Nr[1, ej

⇡2Nr , ej2

⇡2Nr , · · · , ej(Nr�1) ⇡

2Nr]

T .

We plot the case when Nr = 2 in Fig. 3. The transmitter sends8-PSK symbols. By appropriately designing br, the resulting� is {⇡8 + k ⇡4 , k = 0, 1, · · · , 7}. In Fig. 3, each of the 8-PSKsymbols is on the angular bisector of the lines with phases in�.

V. SIMULATION RESULTS

A. SIMO Channel with One-Bit Quantization

In the SIMO channel, we obtain the capacity-achievinginput distribution using the cutting plane method [21, Sec. IV-A]. For this method, we take a fine quantized discrete grid onthe region {x : �3

pP <{x} 3

pP ,�3

pP ={x}

3

pP} as the possible inputs and optimize their probabilities.

In Fig. 4, we shown a simple case when h = [ej⇡/8, e�j⇡/8]

T .It is interesting to find that the optimal input constellation

In Phase

Qua

drat

ure

−6 −4 −2 0 2 4 6−6

−4

−2

0

2

4

6

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

0.12

Fig. 4. The optimal input distribution of the SIMO channel when h =[ej⇡/8, e�j⇡/8]T . The transmission power P = 10 and the achieved rate isabout 2.52 bps/Hz.

contains the rotated 8-PSK symbols and the symbol zero. Forother general channels, the optimal constellation may not beregular.

B. MIMO Channel with One-Bit Quantization

In Fig. 5, we plot the achievable rate when Nt = Nr = 2.The channel coefficients are generated from CN (0, 1) distribu-tion independently and the results are obtained by averagingover 100 different channel realizations. The input alphabet,which contains 2

2Nr= 16 input symbols, are constructed

by the method used in the proof of Proposition 2. The inputsymbols are obtained by solving

x =

pP

H�1y

kH�1yk , (13)

where y = [±1± j, ±1± j]

T . These symbols are transmittedwith equal probabilities 1/16 or with probabilities optimizedby the Blahut-Arimoto algorithm [22]. We can see that thesetwo curves are very close in Fig. 5. The channel capacitywithout quantization is computed using the usual waterfillingapproach. When the SNR is less than 5dB, the gap between thecurves with and without quantization is small. When the SNRis larger than 5dB, the achievable rate with one-bit quantizationapproaches the upper bound 4 bps/Hz. In Fig. 5, we also plotthe low SNR capacity approximation given by [15, Eq. (18)].We see that when SNR is less than �5 dB, the curve of lowSNR approximation is close to the other two curves of one-bit quantization. In the high SNR regime, however, it will benegative and far away from the other curves.

C. MmWave Channel with One-Bit Quantization

In this part, we evaluate the performance of our proposedtransmission strategy in a 2 ⇥ 2 channel with single path.We assume that the angles of departure 't and angles ofarrival 'r are uniformly distributed. The complex path gains↵ is Gaussian distributed. The inter-element spacing of thereceiver antenna array is set to one quarter of the wavelength.The transmitter employs 8-PSK signaling. If there is no phaseshifter, the received complex signals are directly quantized by

Antenna1

Antenna 2

Antenna 3

Input symbols

CSI is known to TX and RX

f(p0, p1) :=

✓�1 + p0 � p1 � p0

4Nr

4

Nr

◆log2

✓1� p0 � p1

4Nr+

p04

Nr+ p1

◆� 2p1 �

8Nr2

4

Nrp0 �

4

Nr � 4Nr

4

Nrp0 log2 p0 (8)

2 3 4 5 6 73

3.2

3.4

3.6

3.8

4

4.2

4.4

4.6

4.8

5

Nr

Hig

h SN

R C

apac

ity (b

ps/H

z)

Numerical Resultslog2(4Nr)log2(4Nr + 1)

Fig. 2. The high SNR capacity and its lower bound and upper bound.

It turns out that p⇤0 =

14Nr+1 and f(p⇤0, 0) ⇡ log2(4Nr + 1).

In Fig. 2, we plot the high SNR capacity obtained by numer-ically maximizing the mutual information function f(p0, p1).The lower bound log2(4Nr) and upper bound log2(4Nr + 1)

are also plotted. It is shown that the high SNR capacityconverges to log2(4Nr + 1) when Nr � 6.

For the special case when \hm = \hn + k⇡/2, k 2{0, 1, 2, 3} for some m 6= n, the total number of distinguish-able input symbols will be less than 4Nr. We usually assumethat the channel coefficients are generated from continuous dis-tribution. Thus with probability one, \hm 6= \hn+k⇡/2, k 2{0, 1, 2, 3} for m 6= n.

In many systems, the zero symbol is not included aspart of the constellation due to peak-to-average power ratio(PAPR) issues. Therefore, in the high SNR regime, only 4Nr

distinguishable symbols are employed as the channel inputsand there will be 4Nr different quantization outputs corre-sponding to each input symbol. The resulting achievable ratewill approach to log2(4Nr) as transmission power increases.

C. MIMO Channel with One-Bit Quantization

Proposition 2. For the MIMO channel with one-bit quantiza-tion, the high SNR capacity, denoted as CMIMO(H), satisfies,

2rank(H) CMIMO(H) 2Nr.

Proof: First, since there are at most 22Nr possible quan-tization outputs at the receiver, CMIMO(H) 2Nr.

Now we denote the rank of the channel matrix as n =

rank(H). Without loss of generality, assume that the first nrows are linearly independent and define A as a n⇥Nt matrixconsisting of these n rows. Consider the quantization outputsof the first 2n quantizer, i.e., r1:n := {ri, 1 i n}. For eachof the 2

2n possible values of r1:n, there is a corresponding

input symbol x = A⇤(AA⇤

)

�1y such that Q(y) = r1:nassuming there is no noise. By transmitting these 2

2n signalswith equal probability, a lower bound of the channel capacityin the high SNR regime, which is 2n, is achieved. Thus theproposition is proved.

Corollary 2. If the channel coefficients are independentlygenerated from a continuous distribution, then

2min{Nr, Nt} CMIMO(H) 2Nr.

Proof: When the channel coefficients are independentlygenerated from a continuous distribution, then with probabilityone, rank(H) = min{Nr, Nt}. Inserting this into (9), weobtain Corollary 2.

IV. MMWAVE CHANNEL WITH ONE-BIT QUANTIZATION

Scattering tends to be lower in the mmWave band comparedwith lower frequencies and thus rank(H) may be less thanmin{Nr, Nt}. In this section we assume a ray-based channelmodel with L paths [18], [19]. Denote ↵`, �r`, �t` as thestrength, the angle of arrival and the angle of departure ofthe `th path, respectively. We also assume that uniform lineararrays are deployed at the transmitter and receiver. The arrayresponse vector at the receiver can be written as

ar(�r`) =1pNr

[1, ej✓, ej2✓, ..., ej(Nr�1)✓]

T (9)

where ✓ =

2⇡� d sin('r`). Herein, � is the wavelength and d

is the inter-element spacing. The array response vector at thetransmitter at('t`) is defined similarly. Hence, the channelmatrix is,

H =

LX

`=1

↵`ar('r`)a⇤t ('t`). (10)

Corollary 3. For a MIMO system with channel model in (10),2min{L,Nr, Nt} CMIMO(H) 2Nr.

Proof: For the channel model shown in (10), rank(H) =

min{L,Nr, Nt} [20]. Combining this with Proposition 2, weobtain Corollary 3.

In mmWave systems, the number of multipaths is limit-ed due to the sparse scattering structure. Meanwhile, largeantenna arrays are usually deployed to obtain beamforminggain for combatting the path loss. Hence, in the case whenL < min{Nt, Nr}, we have 2L CMIMO(H) 2Nr.Therefore, multipath is helpful in improving the lower boundof high SNR capacity in a scattering-limited environment.

Next, we consider such a mmWave MIMO channel withonly one path and propose a transmission strategy. If L = 1,the channel (10) degenerates to

H = ↵ar('r)a⇤t ('t). (11)

�11

SIMO Channel (2)

A/D

Transmitter ... ...... Digital

Processing

RF

RF

...

{1,-1}

Baseband

ADC {1,-1}

Re()

Im()

A/D

{1,-1}ADC {1,-1}

Re()

Im()

Fig. 1. System model.

We define a one-bit quantization function Q as follows.Assume that r = {ri, 1 i Nr} and y = {yi, 1 i Nr}.If

<(ri) = sgn(<(yi)), (2)=(ri) = sgn(=(yi)), (3)

for 1 i Nr, then we denote that r = Q(y). Here, sgn(x)is the sign function and ri 2 {1 + j, 1� j,�1 + j,�1� j}.

After the one-bit quantization at the receiver, we obtain theoutput r = Q(y). Consequently, the channel capacity withone-bit quantization is

C = max

p(x):tr(E(xx⇤))PI(x; r) (4)

where P is the average power constraint at the transmitter.

III. HIGH SNR CAPACITY OF MIMO CHANNEL WITHONE-BIT QUANTIZATION

The mutual information in (4) is in the form of multipleintegrals; obtaining a closed form expression for the optimiza-tion problem seems to be a challenge. Thus, we derive anapproximation of the channel capacity in the high SNR regime.

A. SISO Channel with One-Bit Quantization

First, we deal with the very special case when Nr = Nt = 1.The channel coefficient now is a scalar denoted by h.

Lemma 1. The capacity of the SISO channel with one-bitquantization is achieved by rotated QPSK signaling, i.e.,

Pr{x =

pPej(k⇡+

⇡4 �\h)} =

1

4

, for k = 0, 1, 2, 3, (5)

and is given by

CSISO(P ) = 2

⇣1�H

⇣Q(|h|

pP )

⌘⌘, (6)

where H(p) = �p log2 p� (1�p) log2(1�p) and Q(·) is thetail probability of the standard normal distribution.

Proof: Without loss of optimality, we can assume that thetransmitted signal is x = e�j\hx̂. The outputs of the one-bitquantizer will be <(r) = sgn(|h|<(x̂) + <(n)) and =(r) =

sgn(|h|=(x̂)+=(n)). Therefore, the channel is decoupled intotwo real channels with the same channel gain. For each realchannel, it is proven in [8, Theorem 2] that binary antipodalsignaling is optimal. Therefore, the optimal input for the SISOchannel is rotated QPSK signaling.

As P ! 1, it follows that Q(|h|pP ) ! 0 and

limP!1 CSISO(P ) = 2 bps/Hz.

B. SIMO Channel with One-Bit Quantization

In the SIMO channel with Nr antennas at the receiver, thereare at most 22Nr possible quantization outputs. Therefore, 2Nr

is a simple upper bound for the channel capacity. This upperbound, unfortunately, cannot be approached when Nr is largerthan one, as shown in the following proposition.

Proposition 1. The capacity of the SIMO channel with one-bitquantization at high SNR, denoted as CSIMO(Nr), satisfies

log2(4Nr) CSIMO(Nr) log2 (4Nr + 1) .

Proof: Denote the SIMO channel as h =

[h1, h2, · · · , hNr ]T . When the phase of the transmitted

symbol x is around \x = k⇡/2 � \hi(k = 0, 1, 2, 3; i =

1, 2, · · · , Nr), one element of the one-bit quantization outputwill change its sign. There are at most 4Nr such phases,denoted as � = {�i, 1 i 4Nr}. Following the derivationsin [7] and [9], it can be shown the high SNR capacity isachieved with transmit symbols from three categories:

1) The symbol zero;2) The symbols with phases in �;3) The symbols with phases not in �.For the zero symbol, Pr(r|x = 0) = 2

�2Nr for eachpossible r. For the symbols with phases not in �, Pr(r|x) =1(r = Q(hx)) where 1(·) is the indicator function. At lastconsider the symbols with phases in �. If \x = �\h1,then Pr(r = [1 + j,Q(hx)T2:Nr

]

T |x) = Pr(r = [1 �j,Q(hx)T2:Nr

]

T |x) = 1/2. The other conditional probabilitiescan be derived similarly. Therefore, the transition probabilitymatrix from these three kinds of input symbols to the outputis

Pr(r|x) =

2

42

�2Nr · · · · · · 2�2Nr

T4Nr⇥4Nr 04Nr⇥(22Nr�4Nr)

I4Nr⇥4Nr 04Nr⇥(22Nr�4Nr)

3

5 (7)

where T is a 4Nr⇥4Nr circulant matrix with the first row as[1/2, 1/2, 0, · · · , 0].

Assume that these three kinds of symbols are transmittedwith probabilities p0, p1 and 1 � p0 � p1, respectively. Theresulting mutual information, denoted as f(p0, p1), is as shownat the top of the next page. The channel capacity can be com-puted by searching the optimal p0 and p1, denoted as p⇤0 andp⇤1, which maximizes the mutual information f(p0, p1). It turnsout that @f(p0, p1)/@p1 < 0 and thus p⇤1 = 0. Therefore, thereare at most 4Nr + 1 possible input symbols in the capacity-achieving distribution and an upper bound of the capacity islog2(4Nr + 1). The lower bound log2(4Nr) is achieved bysetting p0 = 0 and p1 = 0, i.e., f(0, 0) = log2(4Nr).

Corollary 1. When Nr is large, the capacity of SIMO channelwith one-bit quantization at high SNR is

CSIMO(Nr) ⇡ log2(4Nr + 1).

Proof: When Nr is large,

f(p0, 0) ⇡ �(1� p0) log21� p04Nr

� p0 log2 p0.

When Nr>6, the capacity converges to log (4Nr+1)

The zero symbol is usually not transmitted due to PAPR issues log (4Nr) is achieved by transmitting 4Nr distinguishable symbols

The gap between log(4Nr) and log(4Nr+1) is small when Nr is large

−1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

In Phase

Qua

drat

ure



8 + k ⇡4 , k = 0, 1, · · · , 7} when Nr = 2.


r = Q (br � (↵ar ('r)x+ n)) , (12)





4Nr+ k ⇡

2Nr, 0 k 4Nr � 1}.


b� ar('r) =1pNr

ej⇡

4Nr[1, ej

⇡2Nr , ej2

⇡2Nr , · · · , ej(Nr�1) ⇡

2Nr]

T .





pP <{x} 3

pP ,�3

pP ={x}

3




In Phase

Qua

drat

ure

−6 −4 −2 0 2 4 6−6

−4

−2

0

2

4

6

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

0.12







x =

pP

H�1y

kH�1yk , (13)

where y = [±1± j, ±1± j]




�12

Find the optimal inputs using the cutting plane algorithm* Discrete grid on the region

!Optimize the input distribution over this grid

The constellation is not regular for general channel conditions

SIMO Channel (3)

*J. Huang and S. Meyn, “Characterization and computation of optimal distributions for channel coding,” IEEE Trans. Inf. Theory, vol. 51, no. 7, pp. 2336–2351, 2005.

−1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

In Phase

Qua

drat

ure



8 + k ⇡4 , k = 0, 1, · · · , 7} when Nr = 2.


r = Q (br � (↵ar ('r)x+ n)) , (12)





4Nr+ k ⇡

2Nr, 0 k 4Nr � 1}.


b� ar('r) =1pNr

ej⇡

4Nr[1, ej

⇡2Nr , ej2

⇡2Nr , · · · , ej(Nr�1) ⇡

2Nr]

T .





pP <{x} 3

pP ,�3

pP ={x}

3




In Phase

Qua

drat

ure

−6 −4 −2 0 2 4 6−6

−4

−2

0

2

4

6

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

0.12


In Phase

Qua

drat

ure

−20 −15 −10 −5 0 5 10 15 20−20

−15

−10

−5

0

5

10

15

20

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

0.12

Fig. 5. The optimal input distribution of the SIMO channel when h =[1 2e�j⇡/3]T . The transmission power P = 10 dB and the achievable rateis 3.0050 bps/Hz.






x =

pP

H�1y

kH�1yk , (13)

where y = [±1± j, ±1± j]

T . These symbols are transmittedwith equal probabilities 1/16 or with probabilities optimizedby the Blahut-Arimoto algorithm [22]. We can see that thesetwo curves are very close in Fig. 6. The channel capacitywithout quantization is computed using the usual waterfillingapproach. When the SNR is less than 5dB, the gap between thecurves with and without quantization is small. When the SNR

−1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

In Phase

Qua

drat

ure



8 + k ⇡4 , k = 0, 1, · · · , 7} when Nr = 2.


r = Q (br � (↵ar ('r)x+ n)) , (12)





4Nr+ k ⇡

2Nr, 0 k 4Nr � 1}.


b� ar('r) =1pNr

ej⇡

4Nr[1, ej

⇡2Nr , ej2

⇡2Nr , · · · , ej(Nr�1) ⇡

2Nr]

T .





pP <{x} 3

pP ,�3

pP ={x}

3




In Phase

Qua

drat

ure

−6 −4 −2 0 2 4 6−6

−4

−2

0

2

4

6

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

0.12


In Phase

Qua

drat

ure

−20 −15 −10 −5 0 5 10 15 20−20

−15

−10

−5

0

5

10

15

20

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

0.12



h = [ej⇡/8, e�j⇡/8]

T h = [1 2e�j⇡/3]

T

h =

✓ej⇡/8

e�j⇡/8

◆

h =

✓1

2e�j⇡/3

◆





x =

pP

H�1y

kH�1yk , (13)

where y = [±1± j, ±1± j]

T . These symbols are transmittedwith equal probabilities 1/16 or with probabilities optimized

−1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

In Phase

Qua

drat

ure



8 + k ⇡4 , k = 0, 1, · · · , 7} when Nr = 2.


r = Q (br � (↵ar ('r)x+ n)) , (12)





4Nr+ k ⇡

2Nr, 0 k 4Nr � 1}.


b� ar('r) =1pNr

ej⇡

4Nr[1, ej

⇡2Nr , ej2

⇡2Nr , · · · , ej(Nr�1) ⇡

2Nr]

T .





pP <{x} 3

pP ,�3

pP ={x}

3




In Phase

Qua

drat

ure

−6 −4 −2 0 2 4 6−6

−4

−2

0

2

4

6

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

0.12


In Phase

Qua

drat

ure

−20 −15 −10 −5 0 5 10 15 20−20

−15

−10

−5

0

5

10

15

20

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

0.12



h = [ej⇡/8, e�j⇡/8]

T h = [1 2e�j⇡/3]

T

h =

✓ej⇡/8

e�j⇡/8

◆

h =

✓1

2e�j⇡/3

◆





x =

pP

H�1y

kH�1yk , (13)

where y = [±1± j, ±1± j]

T . These symbols are transmittedwith equal probabilities 1/16 or with probabilities optimized

MIMO with One-bit ADCs (1)

The upper bound is 2Nr since there are 2Nr one-bit ADCs

The lower bound is achieved by construction Can find at least 2 rank(H) distinguishable input symbols.

�13

f(p0, p1) :=

✓�1 + p0 � p1 � p0

4Nr

4

Nr

◆log2

✓1� p0 � p1

4Nr+

p04

Nr+ p1

◆� 2p1 �

8Nr2

4

Nrp0 �

4

Nr � 4Nr

4

Nrp0 log2 p0 (8)

2 3 4 5 6 73

3.2

3.4

3.6

3.8

4

4.2

4.4

4.6

4.8

5

Nr

Hig

h SN

R C

apac

ity (b

ps/H

z)




14Nr+1 and f(p⇤0, 0) ⇡ log2(4Nr + 1).














)








ar(�r`) =1pNr

[1, ej✓, ej2✓, ..., ej(Nr�1)✓]

T (9)

where ✓ =



H =

LX

`=1

↵`ar('r`)a⇤t ('t`). (10)






H = ↵ar('r)a⇤t ('t). (11)

f(p0, p1) :=

✓�1 + p0 � p1 � p0

4Nr

4

Nr

◆log2

✓1� p0 � p1

4Nr+

p04

Nr+ p1

◆� 2p1 �

8Nr2

4

Nrp0 �

4

Nr � 4Nr

4

Nrp0 log2 p0 (8)

2 3 4 5 6 73

3.2

3.4

3.6

3.8

4

4.2

4.4

4.6

4.8

5

Nr

Hig

h SN

R C

apac

ity (b

ps/H

z)




14Nr+1 and f(p⇤0, 0) ⇡ log2(4Nr + 1).














)








ar(�r`) =1pNr

[1, ej✓, ej2✓, ..., ej(Nr�1)✓]

T (9)

where ✓ =



H =

LX

`=1

↵`ar('r`)a⇤t ('t`). (10)






H = ↵ar('r)a⇤t ('t). (11)

The high SNR capacity is closely related to the channel realization.

MIMO with One-bit ADCs (2)

�14

Achievable rate in a 2 X 2 MIMO channel Two curves of one-bit quantization approach the upper bound 4bps/Hz

Low SNR approximation is close to one-bit curves when SNR<-5dB

−1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

In Phase

Qua

drat

ure



8 + k ⇡4 , k = 0, 1, · · · , 7} when Nr = 2.


r = Q (br � (↵ar ('r)x+ n)) , (12)





4Nr+ k ⇡

2Nr, 0 k 4Nr � 1}.


b� ar('r) =1pNr

ej⇡

4Nr[1, ej

⇡2Nr , ej2

⇡2Nr , · · · , ej(Nr�1) ⇡

2Nr]

T .





pP <{x} 3

pP ,�3

pP ={x}

3



T .It is interesting to find that the optimal input constellationcontains the rotated 8-PSK symbols and the symbol zero. For

In Phase

Qua

drat

ure

−6 −4 −2 0 2 4 6−6

−4

−2

0

2

4

6

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

0.12


other general channels, the optimal constellation may not beregular.





x =

pP

H�1y

kH�1yk , (13)

where y = [±1± j, ±1± j]



In this part, we evaluate the performance of our proposedtransmission strategy in a 2 ⇥ 2 channel with single path.We assume that the angles of departure 't and angles ofarrival 'r are uniformly distributed. The complex path gains↵ is Gaussian distributed. The inter-element spacing of thereceiver antenna array is set to one quarter of the wavelength.The transmitter employs 8-PSK signaling. If there is no phaseshifter, the received complex signals are directly quantized bythe one-bit ADCs. If the receiver has one phase shifter on

−10 −5 0 5 100

1

2

3

4

5

6

7

8

9

SNR (dB)

Rate

(bps

/Hz)

Without quantization, waterfillingOne−bit quantization, equally probableOne−bit quantization, BA AlgorithmOne−bit quantization, low SNR approx.

Fig. 5. Achievable rate of the 2⇥ 2 MIMO channel.

−10 −5 0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

SNR (dB)

Achi

evab

le R

ate

(bps

/Hz)

w/o phase shifterswith phase shifters

Fig. 6. The achievable rate with the proposed transmission strategy in ammWave 2⇥ 2 channel with only one path.

the one-bit ADCs. If the receiver has one phase shifter oneach receiver antenna, the phase shifters will rotate the phaseof the input signals such that � = {⇡

8 + k ⇡4 , k = 0, 1, · · · , 7}.

In Fig. 6, we compare the performances of the above tworeceiver structures. We see that the system with our proposedreceiver structure can achieve higher rate in medium and highSNR regimes. Note that the phase shifters are potentially aspower efficient as one-bit ADCs [23]. Therefore, our proposedreceiver structure achieves much better performance with littleincrease of power consumption. The gains may be smaller forlarger numbers of receive antennas.

VI. CONCLUSION

In this paper, we studied the capacity of point-to-pointMIMO channel with one-bit quantization at the receiver. Wefound that the high SNR capacity of SIMO channel increaselinearly with log2(Nr). For the MIMO channel, the high SNRcapacity is lower bounded by the rank of the channel. We alsoconsidered the mmWave MIMO channel with limited scatter-ing. We showed that multipath is helpful in improving thelower bound of the high SNR capacity. Finally, we proposeda power efficient receiver structure where the phase shiftersand one-bit ADCs are used to achieve higher data rate.

REFERENCES

[1] R. Walden, “Analog-to-digital converter survey and analysis,” IEEE J.Sel. Areas Commun., vol. 17, no. 4, pp. 539–550, 1999.

[2] B. Murmann, “ADC performance survey 1997-2013.” [Online].Available: http://www.stanford.edu/⇠murmann/adcsurvey.html

[3] S. Vitali, G. Cimatti, R. Rovatti, and G. Setti, “Adaptive time-interleavedADC offset compensation by nonwhite data chopping,” IEEE Trans.Circuits Syst. II, vol. 56, no. 11, pp. 820–824, 2009.

[4] S. Ponnuru, M. Seo, U. Madhow, and M. Rodwell, “Joint mismatchand channel compensation for high-speed OFDM receivers with time-interleaved ADCs,” IEEE Trans. Commun., vol. 58, no. 8, pp. 2391–2401, 2010.

[5] T. Sundstrom, B. Murmann, and C. Svensson, “Power dissipationbounds for high-speed nyquist analog-to-digital converters,” IEEE Trans.Circuits Syst. I, vol. 56, no. 3, pp. 509–518, 2009.

[6] J. Singh, S. Ponnuru, and U. Madhow, “Multi-gigabit communication:the ADC bottleneck,” in IEEE International Conference on Ultra-Wideband,, 2009, pp. 22–27.

[7] O. Dabeer, J. Singh, and U. Madhow, “On the limits of communicationperformance with one-bit analog-to-digital conversion,” in IEEE 7thWorkshop on Signal Processing Advances in Wireless Communications,2006, pp. 1–5.

[8] J. Singh, O. Dabeer, and U. Madhow, “On the limits of communicationwith low-precision analog-to-digital conversion at the receiver,” IEEETrans. Commun., vol. 57, no. 12, pp. 3629–3639, 2009.

[9] S. Krone and G. Fettweis, “Capacity of communications channels with1-bit quantization and oversampling at the receiver,” in 2012 35th IEEESarnoff Symposium (SARNOFF), 2012, pp. 1–7.

[10] G. Zeitler, A. Singer, and G. Kramer, “Low-precision A/D conversionfor maximum information rate in channels with memory,” IEEE Trans.Commun., vol. 60, no. 9, pp. 2511–2521, 2012.

[11] O. Dabeer and U. Madhow, “Channel estimation with low-precisionanalog-to-digital conversion,” in 2010 IEEE International Conferenceon Communications (ICC), 2010, pp. 1–6.

[12] G. Zeitler, G. Kramer, and A. Singer, “Bayesian parameter estimationusing single-bit dithered quantization,” IEEE Trans. Signal Process.,vol. 60, no. 6, pp. 2713–2726, 2012.

[13] A. Wadhwa and U. Madhow, “Blind phase/frequency synchronizationwith low-precision ADC: a Bayesian approach,” in Proc. of 51st AllertonConference on Communication Control and Computing, 2013.

[14] M. Ivrlac and J. Nossek, “Challenges in coding for quantized mimo sys-tems,” in 2006 IEEE International Symposium on Information Theory,July 2006, pp. 2114–2118.

[15] A. Mezghani and J. Nossek, “On ultra-wideband MIMO systems with 1-bit quantized outputs: Performance analysis and input optimization,” inIEEE International Symposium on Information Theory, 2007, pp. 1286–1289.

[16] ——, “Analysis of Rayleigh-fading channels with 1-bit quantized out-put,” in IEEE International Symposium on Information Theory, 2008,pp. 260–264.

[17] ——, “Analysis of 1-bit output noncoherent fading channels in the lowSNR regime,” in IEEE International Symposium on Information Theory,2009, pp. 1080–1084.

[18] O. El Ayach, S. Rajagopal, S. Abu-Surra, Z. Pi, and R. Heath, “Spatiallysparse precoding in millimeter wave mimo systems,” IEEE Trans.Wireless Commun., vol. PP, no. 99, pp. 1–15, 2014.

[19] A. Alkhateeb, O. El Ayach, G. Leus, and R. W. Heath Jr, “Hybridprecoding for millimeter wave cellular systems with partial channelknowledge,” in Proc. of Information Theory and Applications (ITA)Workshop, 2013.

[20] G. Raleigh and J. Cioffi, “Spatio-temporal coding for wireless com-munication,” IEEE Trans. Commun., vol. 46, no. 3, pp. 357–366, Mar1998.

[21] J. Huang and S. Meyn, “Characterization and computation of optimaldistributions for channel coding,” IEEE Trans. Inf. Theory, vol. 51, no. 7,pp. 2336–2351, 2005.

[22] R. Blahut, “Computation of channel capacity and rate-distortion func-tions,” IEEE Trans. Inf. Theory, vol. 18, no. 4, pp. 460–473, 1972.

[23] C. Doan, S. Emami, D. Sobel, A. Niknejad, and R. Brodersen, “Designconsiderations for 60 GHz CMOS radios,” IEEE Commun. Mag., vol. 42,no. 12, pp. 132–140, Dec 2004.

−1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

In Phase

Qua

drat

ure



8 + k ⇡4 , k = 0, 1, · · · , 7} when Nr = 2.


r = Q (br � (↵ar ('r)x+ n)) , (12)





4Nr+ k ⇡

2Nr, 0 k 4Nr � 1}.


b� ar('r) =1pNr

ej⇡

4Nr[1, ej

⇡2Nr , ej2

⇡2Nr , · · · , ej(Nr�1) ⇡

2Nr]

T .





pP <{x} 3

pP ,�3

pP ={x}

3



T .It is interesting to find that the optimal input constellationcontains the rotated 8-PSK symbols and the symbol zero. For

In Phase

Qua

drat

ure

−6 −4 −2 0 2 4 6−6

−4

−2

0

2

4

6

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

0.12


other general channels, the optimal constellation may not beregular.





x =

pP

H�1y

kH�1yk , (13)

where y = [±1± j, ±1± j]



In this part, we evaluate the performance of our proposedtransmission strategy in a 2 ⇥ 2 channel with single path.We assume that the angles of departure 't and angles ofarrival 'r are uniformly distributed. The complex path gains↵ is Gaussian distributed. The inter-element spacing of thereceiver antenna array is set to one quarter of the wavelength.The transmitter employs 8-PSK signaling. If there is no phaseshifter, the received complex signals are directly quantized bythe one-bit ADCs. If the receiver has one phase shifter on

The input alphabet is constructed by solving

where

BA: Blahut-Arimoto Algorithm

mmWave ChannelRay-tracing model with L paths

�15

Multipath is helpful in improving the lower bound of high SNR capacity

f(p0, p1) :=

✓�1 + p0 � p1 � p0

4Nr

4

Nr

◆log2

✓1� p0 � p1

4Nr+

p04

Nr+ p1

◆� 2p1 �

8Nr2

4

Nrp0 �

4

Nr � 4Nr

4

Nrp0 log2 p0 (8)

2 3 4 5 6 73

3.2

3.4

3.6

3.8

4

4.2

4.4

4.6

4.8

5

Nr

Hig

h SN

R C

apac

ity (b

ps/H

z)




14Nr+1 and f(p⇤0, 0) ⇡ log2(4Nr + 1).














)








ar(�r`) =1pNr

[1, ej✓, ej2✓, ..., ej(Nr�1)✓]

T (9)

where ✓ =



H =

LX

`=1

↵`ar('r`)a⇤t ('t`). (10)






H = ↵ar('r)a⇤t ('t). (11)

f(p0, p1) :=

✓�1 + p0 � p1 � p0

4Nr

4

Nr

◆log2

✓1� p0 � p1

4Nr+

p04

Nr+ p1

◆� 2p1 �

8Nr2

4

Nrp0 �

4

Nr � 4Nr

4

Nrp0 log2 p0 (8)

2 3 4 5 6 73

3.2

3.4

3.6

3.8

4

4.2

4.4

4.6

4.8

5

Nr

Hig

h SN

R C

apac

ity (b

ps/H

z)




14Nr+1 and f(p⇤0, 0) ⇡ log2(4Nr + 1).














)








ar(�r`) =1pNr

[1, ej✓, ej2✓, ..., ej(Nr�1)✓]

T (9)

where ✓ =



H =

LX

`=1

↵`ar('r`)a⇤t ('t`). (10)

Corollary 3. For a mmWave MIMO system with L paths,

2min{L,Nr, Nt} CMIMO(H) 2Nr.





H = ↵ar('r)a⇤t ('t). (11)

rank(H) = min{L, Nr, Nt} for fading channels

Proposed Transmission Strategy (1)

Single path mmWave MIMO channel Transmitter used matched filter beamforming and 4Nr-PSK signaling

Phase shifters are placed on each antenna before the one-bit ADCs

The shifters will rotate the signals to make them distinguishable

�16

Nr=2

−1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

In Phase

Qua

drat

ure



8 + k ⇡4 , k = 0, 1, · · · , 7} when Nr = 2.


r = Q (br � (↵ar ('r)x+ n)) , (12)





4Nr+ k ⇡

2Nr, 0 k 4Nr � 1}.


b� ar('r) =1pNr

ej⇡

4Nr[1, ej

⇡2Nr , ej2

⇡2Nr , · · · , ej(Nr�1) ⇡

2Nr]

T .





pP <{x} 3

pP ,�3

pP ={x}

3




In Phase

Qua

drat

ure

−6 −4 −2 0 2 4 6−6

−4

−2

0

2

4

6

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

0.12







x =

pP

H�1y

kH�1yk , (13)

where y = [±1± j, ±1± j]




ADC

ADCADC

RF

RF

{1,-1}

{1,-1}

ADCDigital

Baseband Processing

ej�1

ej�2

digital controlled phase shifters

Proposed Transmission Strategy (2)

�17

Proposed strategy achieves higher rate in medium and high SNR regimes.

−10 −5 0 5 100

1

2

3

4

5

6

7

8

9

SNR (dB)

Rate

(bps

/Hz)

Without quantization, waterfillingOne−bit quantization, equally probableOne−bit quantization, BA AlgorithmOne−bit quantization, low SNR approx.

Fig. 5. Achievable rate of the 2⇥ 2 MIMO channel.

−10 −5 0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

SNR (dB)

Achi

evab

le R

ate

(bps

/Hz)

w/o phase shifterswith phase shifters

Fig. 6. The achievable rate with the proposed transmission strategy in ammWave 2⇥ 2 channel with only one path.

the one-bit ADCs. If the receiver has one phase shifter oneach receiver antenna, the phase shifters will rotate the phaseof the input signals such that � = {⇡

8 + k ⇡4 , k = 0, 1, · · · , 7}.

In Fig. 6, we compare the performances of the above tworeceiver structures. We see that the system with our proposedreceiver structure can achieve higher rate in medium and highSNR regimes. Note that the phase shifters are potentially aspower efficient as one-bit ADCs [23]. Therefore, our proposedreceiver structure achieves much better performance with littleincrease of power consumption. The gains may be smaller forlarger numbers of receive antennas.

VI. CONCLUSION

In this paper, we studied the capacity of point-to-pointMIMO channel with one-bit quantization at the receiver. Wefound that the high SNR capacity of SIMO channel increaselinearly with log2(Nr). For the MIMO channel, the high SNRcapacity is lower bounded by the rank of the channel. We alsoconsidered the mmWave MIMO channel with limited scatter-ing. We showed that multipath is helpful in improving thelower bound of the high SNR capacity. Finally, we proposeda power efficient receiver structure where the phase shiftersand one-bit ADCs are used to achieve higher data rate.

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Single path mmWave 2 X 2 channel !

Transmitter employs 8-PSK signaling

Conclusions

Analyzed the high SNR channel capacity with one-bit ADCs Found that the SIMO channel capacity increase linearly with log(Nr)

Derived the closed-form SIMO channel capacity when Nr is large

Provided the bounds for MIMO channel capacity

Studied the mmWave MIMO channel capacity at high SNR Found that multipath is helpful in improving the lower bound of capacity

Proposed a power efficient transmission strategy that achieves higher rate

Future work Combined hybrid beamforming and 1-bit ADCs

Channel estimation in the mmWave system with one-bit ADCs

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High SNR Capacity of Millimeter Wave MIMO systems...

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