This article has been accepted for publication in a future issue of this journal, but has not been

fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/JSAC.2017.2720398, IEEE Journal
on Selected Areas in Communications
1

Achievable Rate and Energy Efficiency of Hybrid

and Digital Beamforming Receivers with Low
Resolution ADC
Kilian Roth, Member, IEEE, Josef A. Nossek, Life Fellow, IEEE

Abstract—For 5G it will be important to leverage the available The use of high carrier frequencies above 6 GHz will go
millimeter wave spectrum. To achieve an approximately omni- hand in hand with the implementation of large antenna arrays
directional coverage with a similar effective antenna aperture [1], [2]. The support of a large number of antennas at the
compared to state of the art cellular systems, an antenna array
is required at both the mobile and basestation. Due to the mobile and base station requires a new frontend design. To
large bandwidth, the analog front-end of the receiver with a attain a similar link budget, the effective antenna aperture of
large number of antennas becomes especially power hungry. a mmWave system must be comparable to current systems
Two main solutions exist to reduce the power consumption: operating at carrier frequencies below 6 GHz. Therefore, an
Hybrid BeamForming (HBF) and Digital BeamForming (DBF) antenna array at the base and mobile station is unavoidable.
with low resolution Analog to Digital Converters (ADCs). Hybrid
beamforming can also be combined with low resolution ADCs. Since the antenna gain and thus the directivity increases with
This paper compares the spectral and energy efficiency based the aperture, an antenna array is the only solution to achieve a
on the RF-frontend configuration. A channel with multipath high effective aperture while maintaining an omnidirectional
propagation is used. In contrast to previous publication, we take coverage.
the spatial correlation of the quantization noise into account.
We show that the low resolution ADC digital beamforming is
robust to small Automatic Gain Control (AGC) imperfections. A. Related Work
We showed that in the low SNR regime the performance of DBF
even with 1-2 bit resolution outperforms HBF. If we consider the Current LTE systems have a limited amount of antennas at
relationship of spectral and energy efficiency, DBF with 3-5 bits the base and mobile stations. Since the bandwidth is relatively
resolution achieves the best ratio of spectral efficiency per power narrow, the power consumption of having a receiver RF
consumption of the RF receiver frontend over a wide SNR range.
The power consumption model is based on components reported
chain with a high resolution ADC at each antenna is still
in literature. feasible. For future mmWave mobile broadband systems, a
much larger bandwidth [3] and a large number of antennas
Index Terms—Hybrid beamforming, low resolution ADC, mil-
limeter wave, wireless communication.
are being considered [1]. The survey [4] shows that ADCs
with an extensive sampling frequency, and medium number
of effective bits consume a considerable amount of power.
I. I NTRODUCTION The ADC can be considered as the bottleneck of the receiver
The use of the available bandwidth in the frequency range [5].
of 6 to 100 GHz is considered to be an essential part of the The antenna array combined with the large bandwidth is a
next generation mobile broadband standard 5G [1]. Due to the huge challenge for the hardware implementation, essentially
propagation condition, this technology is especially attractive the power consumption will limit the design space. At the
for high data rate, low range wireless communication. This moment, analog or hybrid beamforming are considered as a
frequency range is referred to as millimeter wave (mmWave), possible solution to reduce the power consumption. Analog or
even though it contains the lower centimeter wave range. In hybrid beamforming systems highly depend on the calibration
the last years, the available spectrum and the start of the of the analog components. Another major disadvantage is the
availability of consumer grade systems led to a huge increase large overhead associated with the alignment of the Tx and
in academic and industrial research. However, to fully leverage Rx beams of the base and mobile station. Specifically, if high
the spectrum while being power-efficient, the Base Band (BB) gain is needed, the beamwidth is small and thus the acquisition
and Radio Front-End (RFE) capabilities must be drastically and constant alignment of the optimal beams in a dynamic
changed from state of the art cellular devices. environment is very challenging [6], [7], [8].
The idea of hybrid beamforming is based on the concept
Manuscript receive 12. October 2016; revised 14. March 2017 and 2. May of phase array antennas commonly used in radar applica-
2017; accepted 8. May 2017. Date of current version 16. May 2017. tion [9]. Due to the reduced power consumption, it is also
K. Roth is with Next Generation and Standards, Intel Deutschland GmbH,
Neubiberg 85579, Germany (email: {kilian.roth}@intel.com) seen as a possible solution for mmWave mobile broadband
K. Roth and J. A. Nossek are with the Department of Electrical and Com- communication[10]. If the phase array approach is combined
puter Engineering, Technical University Munich, Munich 80290, Germany with digital beamforming the phase array approach might also
(email: {kilian.roth, josef.a.nossek}@tum.de)
J. A. Nossek is with Department of Teleinformatics Engineering, Federal be feasible for non-static or quasi static scenarios. In [11],
University of Ceara, Fortaleza, Brazil it was shown that considering the inefficiency of mmWave

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on Selected Areas in Communications
2

analog
A/D
MC signal
conversion
combination
MIMO
digital
transmitter MT channel MRF E
baseband
analog
A/D
MC signal
conversion
combination

Fig. 1. System model with MT transmit antennas and MC antennas at each of the MRF E RF chains. Number of receive antennas MR is equal to
MC · MRF E .

amplifiers and the high insertion loss of RF phase shifters, it between hybrid and digital beamforming it is important
is better to perform the phase shifting in the baseband. The to consider multipath channels. The evaluation shows
power consumption of both cases is comparable, as long as that the digital beamforming system for any resolution
the number of antennas per RF-chain remains relatively small. of the ADC always outperforms the hybrid system in
Another option to reduce the power consumption while the low SNR regime. It is important to stress that the
keeping the number of antennas constant is, to reduce the low per antenna SNR regime is very likely the practical
power consumption of the ADCs by reducing their resolution. operating point of future mmWave systems. The low
This can also be combined with hybrid beamforming. Some resolution ADC is essentially limiting the performance
of these evaluations consider only the extreme case of 1-bit in the high SNR regime. Therefore, a hybrid system with
quantization [8], [12], [5]. In [13], [14] the Analog to Digital higher resolution ADC will always at some point surpass
(A/D) conversion is modeled as a linear stochastic process. the digital system with lower resolution. We also show
Low resolution A/D conversion combined with OFDM in an that small imperfections in the AGC do not degrade the
uplink scenario are considered in [15], [16]. performance of the digital system.
In [17], [18] hybrid beamforming with low resolution A/D • By including the off-diagonal elements of the
conversion was considered. The energy efficiency / spectral quantization-error covariance matrix, the Additive
efficiency trade-off of fully-connected hybrid and digital beam- Quantization Noise Model (AQNM) is refined in this
forming with low resolution ADCs is assessed in [18]. But work. For a scenario with very low resolution ADC
in contrast as shown in the system diagram in Figure 1, (1-2 bit) and a larger number of receive antennas than
we consider a hybrid beamforming system that has exclusive transmit antennas, it is important to take this off-diagonal
antennas per RF-chain (aka. sub-array hybrid beamforming). elements into account.
In this work we concentrated on effects of the hardware con- • Energy efficiency and spectral efficiency of the given
straints at the receiver, thus we assumed the transmitter to be systems are characterized. We show that for a wide SNR
ideal. In [18], a fully-connected hybrid beamforming system range the digital beamforming system is more energy
is used, this has a large additional overhead associated with an efficient than the hybrid beamforming one. We also show
increased number of phase shifter and larger power combiners. that an A/D resolution in the range of 3-5 lead to the most
Also in this case additional amplifiers to compensate for the energy efficient receiver.
insertion-loss of the RF phase shifters and combiners are
required. In [19], analog beamforming is compared with digital C. Notation
beamforming in terms of power efficiency. Throughout the paper we use boldface lower and upper case
letters to represent column vectors and matrices. The term
B. Contribution am,l is the element on row m and column l of matrix A and
In this paper, we assess the achievable rate of hybrid and am is the mth element of vector a. The expressions A∗ , AT ,
digital beamforming with low resolution A/D conversion in a AH , and A−1 represent the complex conjugate, the transpose,
multipath environment. The paper [18] showed that a digital the Hermitian, and the inverse of the matrix A. The symbol
beamforming system is always more energy efficient than a Rab is the correlation matrix of vector a and b defined as
fully-connected hybrid beamforming system. In contrast we E[abH ]. The Discrete Fourier Transformation (DFT) F(·) and
use a hybrid beamforming system with exclusive antennas, its inverse F −1 (·) and the Fourier transformation F {·} and
which has a greatly reduced hardware complexity compared its inverse F −1 {·} are also used.
to fully-connected hybrid beamforming.
II. S YSTEM M ODEL
• The achievable rate of hybrid and digital beamforming
with low resolution ADC in a multipath environment is A. Signal Model
derived. The phase shifters of hybrid beamforming are not The signal model is shown in Figure 2. The symbols x[n],
frequency selective, therefore if considering a comparison H[n], η[n], and y[n] represent the transmit signal, channel,

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u[n] y[n] The quantization operator Qb (a) is treating the I and Q

x[n] H[n] ∗ x[n] F (·) r[n] component of each element of a vector a separately. For a real
valued, scalar input a, the output of the operation is defined
MT MR as: i i
η[n] r = Qb (a) = q j ∀ a ∈ qlj−1 qlj . (5)

Fig. 2. Signal Model. Here q j is the representative

i i of the jth quantization bin with
noise, and receive signal of a system at time n. MT transmit the input interval qlj−1 qlj . To cover a real valued input the
and MR receive antennas are used. Since we assume a channel left limit of the first interval ql0 and the right limit of the last
with multipath propagation the receive signal y[n] is defined interval qlNb are equal to −∞ and ∞ respectively. The number
as: of quantization bins Nb is equal to 2b . For real world ADC the
L−1
X difference between representatives of quantization bins qj and
y[n] = H[l]x[n − l] + η[n], (1) the size of the quantization bins are uniform. We thus limit
l=0
our evaluation to this set of quantizers. For the theoretical
where L is the maximum delay of the channel in samples. evaluation we assume Gaussian signaling. Consequently, we
The operation F (·) is defined as multiplication with the analog use the stepsize to minimize the distortion for Gaussian signals
receiver beamforming matrix W R followed by a quantization shown in [23].
operation Qb (·) with resolution of b bits: Since the actual receive power at each antenna can be
r[n] = F (y[n]) = Qb (y C [n]) = Qb (W H
R y[n]). (2) different, an AGC needs to adapt a Variable Gain Amplifier
(VGA) to generate the minimal distortion. To simplify our
We restricted the system to have MC antennas exclusively
model, we assume that the AGC is always perfectly adapting
connected to one RF front-end chain (see Figure 1). Therefore,
to the current situation. Since in practice an AGC cannot
the matrix W R has the form:
 1 accomplish this task without error, we will show the impact
...

wR 0MC 0 MC of an imperfect AGC. We model this by a relative error to the
.. perfect gain value.
0MC w2R
 
. 0 MC 
WR =   ∈ CMR ×MRF E , For the rest of the paper we define the SNR γ as:
 .. . . . . .. 
 . . . .  "  2 #
L−1
0 MC ... 0MC wM RF E P 
R E  H[l]x[n − l]
(3) l=0 2
where the vector wiR is the analog beamforming vector of the γ= 2 . (6)
E[||η[n]||2 ]
ith RF chain. We also restrict our evaluation to each RF chain
utilizing the same number of antennas MC . The vectors wiR This formula is basically just describing the average SNR at
and 0MC have dimension MC . each antenna. It is important to note that the expectation takes
The use of analog beamforming is envisioned in many the realization of the channel and realizations of x[n] into
future mobile broadband systems, especially in the mmWave account.
frequency range ([20], [21]). Since the complete channel
matrix cannot be directly observed, one practical solution is B. Channel Model
scanning different spatial direction (beams) and then select the
Dependent on the scenario, different channel models are
configuration maximizing the SNR. There are many different
used:
possibilities for selecting the optimal beam, e.g. 802.11ad is
• Finite path model with all paths arriving at the same time
using a procedure based on exhaustive search [22].
• Finite path model with exponential Power Delay Profile
For the evaluation, we assume that the antennas of each
RF chain form a Uniform Linear Array (ULA). If a planar (PDP)
wavefront is impinging on the ULA and the spacing of The channel models assume different rays impinging on the
adjacent antennas is d = λ/2, the receive signal at adjacent receiver antenna array. In the first model, they are assumed
antennas is phase shifted by φi = π sin(θi ). The angle θi is to arrive at the receiver antennas at the same time. Under the
the angle of a planar wavefront relative to the antennas of assumption of a ULA at the transmitter and receiver, a channel
the ULA. This formula assumes that a planar wavefront is consisting of K different rays can be modeled as:
impinging at the antenna array, and that the symbol duration K
1 X
is large relative to the maximum delay between two antennas. H=√ α(k)ar (φr (k))aTt (φt (k)). (7)
With the constraint of observing only a single spatial direction, KMT k=1
the receive vector wiR for an ULA antenna array takes the The vectors ar (φr (k)) and at (φt (k)) are the array steering
form: vectors at the receiver and transmitter. The phase shift between
h iH
wiR = 1, ejφi , ej2φi , · · · , ej(MC −1)φi . (4) the signal of adjacent antenna elements φr (k) and φt (k) of
path k depend on the angle of arrival θr (k) and departure
In the special case of full digital beamforming (MC = 1 and θt (k) .
therefore MRF E = MR ), W R is equal to the identity matrix h i
I of size MR × MR . aTr (φr (k)) = 1, ejφr (k) , ej2φr (k) , · · · , ej(Mr −1)φr (k) . (8)

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The transmit vectors aTt (φr (k)) has the same form as
aTr (φr (k)). The complex gains α(k) are circular symmetric
Gaussian distributed with zero mean and unit variance. Except
for the different normalization factor, this channel model is the 0◦
same as the one presented in [24]. The difference comes from LNA LO
the fact that the sum power of the transmit signal is constraint 90◦
to be less or equal to MT . To set the average per antenna
1 hybrid and
receive power to one we normalize the channel by √KM . mixers
The angles of arrival θr (k) and departure θt (k) are uniformly
T
LO buffer
distributed in the range of −π to π.
Since in real world scenario the different rays are reflection Fig. 3. Direct conversion receiver.
of different scatterers, the path of each of these rays from the
transmitter to the receiver has a different length. This results in
rays arriving at the receiver at different time. In a simplified
case, it can be expected that the path that arrives at a later
time have a lower power. The measurements in [25] show that
for channels at 60 GHz an exponential Power Delay Profile
(PDP) is sufficiently approximating a real world scenario. MC

1
H[l] = √ α(l)ar (φr (l))aTt (φt (l)). (9)
MT
Here we assume, that at delay l only one ray arrives at the phase analog
receiver. Here the complex gain of the ray α(l) is circular shifter combiner
symmetric Gaussian distributed with zero mean and a variance
defined according to: Fig. 4. Analog signal combination.

the power consumption for a low number of antennas per RF-

h i
2
vl = E |α(l)| = e−βl . (10)
chain is equivalent to a system utilizing RF Phase Shifters
The parameter β defines the how fast the power decays in (PS).
relation to the sample time. The additional parameters are All systems utilize the same direct conversion receiver
the maximum channel length in samples L and the number (Figure 3) to convert the signal into the analog baseband.
of present channel tabs P . This means that for all possible For each system, we assume that the Local Oscillator (LO)
present channel rays v of length L, P positions are selected is shared by the whole system. For the case of analog/hybrid
for each channel realization. At all other positions, v is equal beamforming systems, the analog baseband signals are phase
to 0. To normalize the average power, the variance vector v shifted and then combined to generate the input signal of the
is normalized by: MRF E ADCs (Figure 4).
v The A/D conversion consists of a VGA that is amplifying
vn = . (11)
||v||2 the signal to use the full dynamic range of the ADC (Figure
5). For the special case of 1-bit quantized digital beamforming
C. Power Consumption Model a VGA is not necessary. It can be replaced by a much simpler
In a future 5G millimeter Wave mobile broadband system, it Limiting Amplifier (LA).
will be necessary to utilize large antenna arrays. It is therefore The power consumption of each component, including a ref-
important to compare the power consumption of different erence, are shown in Table I. An LO with a power consumption
receiver architectures. In this section we present a power model as low as 22.5 mW is reported in [27]. The power consumption
for analog/hybrid beamforming and digital beamforming in of a LNA, a mixer including a quadrature-hybrid coupler, and
combination with low resolution ADCs. a VGA are reported in [28] as 5.4, 0.5, and 2 mW. The 90◦
Since the spectrum in the 60 GHz band can be accessed hybrid and the LO buffer reported in [29] have a combined
without a license, it got significant attention. Especially the power consumption of 3 mW. The power consumption of the
WiGig (802.11ad) standard operating in this band, increased
the transceiver RF hardware R&D activities. Many chips were
reported from industry and academia. Thus, it is safe to assume VGA ADC
that the design reached a certain maturity, and performance
figures derived from them represent the performance that is
possible for a low cost CMOS implementation today.
According to the discussion in [26], baseband or IF phase VGA ADC
shifting in contrast to RF phase shifting is assumed. This has
the advantage of increased accuracy, decreased insertion loss,
and reduced gain mismatch. In [26], the authors showed that Fig. 5. A/D conversion.

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TABLE I combination of flat fading channel, multipath channel, hybrid

C OMPONENTS WITH POWER CONSUMPTION . beamforming and digital beamforming with low resolution
label component power reference
A/D conversion. In the case of hybrid beamforming, first the
consumption analog receive vectors are calculated. Afterwards, the system
PLO LO 22.5 mW [27] including the analog combining is treated as an equivalent
PLN A LNA 5.4 mW [28] digital beamforming system.
PM mixer 0.3 mW [30]
PH 90◦ hybrid 3 mW [29]
and LO buffer A. Hybrid Beamforming Vectors
PLA LA 0.8 mW [31]
To mimic the behavior of a spatial scan, we restricted the
P1 1-bit ADC 0 mW
receive vectors wiR of the ith RF chain to Vandermonde
PP S phase shifter 2 mW [11], [26]
vectors. A practical system would have a set of predefined
PV GA VGA 2 mW [28]
beamforming configuration that are scanned for every sub-
PADC ADC 15 µW/GHz [4] [32] [33]
·fs 2ENOB array. To obtain the optimal results, all combination of beams
need to be tested by the receiver. This is a combinatorial
mixer reported in [30] is as low as 0.3 mW. The survey in problem with size growing exponentially with the number of
[4] gives a good overview of state of the art ADCs regarding receiver RF-chains. To make the problem feasible, the scan is
Effective Number Of Bits (ENOB), sampling rate, and power performed separately for each receiver RF chain. This problem
consumption. Taking the predicted curve for the Walden figure can be formulated as:
of merit in [4] for a sampling frequency of 2.5 GS/s, we get L−1
15 fJ per conversion step. A LA that consumes 0.8 mW is wR (φB )H H i [l]2 with φB ∈ B,
X
wiR (φ̂)
 i 
= arg max 2
reported in [31]. In the 1-bit quantized system, the LA (aka. wiR (φB )
l=0
Schmitt trigger) is already producing a digital signal, therefore (14)
the 1-bit ADC can be replaced by a flip flop (FF). The power with B being the set of all spatial direction φB that are
consumption of a FF is negligible compared to the rest of the scanned. The channel H i [l] contains the MC rows of H[l]
RF front-end. that belong to the antennas of the ith RF chain.
With the power consumption of the components, it is This procedure mimics the receive beam training in a prac-
possible to compute the power consumption of the overall tical system as described in [34]. For this case the transmitter
receiver front-end PR as: is sending a known reference sequence. The receiver tries
PR = PLO + MR (PLN A + PH + 2PM ) + different receiver beamforming configurations separately on
each subarray i and records the achieved channel quality
flagC (MR PP S ) +
metric. Afterwards, the configuration resulting in the best
MRF E (¬flag1bit (2PV GA + 2PADC ) + flag1bit (2PLA )) , channel is selected. In this work such a procedure is emulated
(12) by selecting the receive beamforming vector resulting in the
where flagC is indicating if analog combining is used: highest receive energy, based on the channel knowledge. This
procedure avoids lengthy numerical simulation of sequence

0, MRF E = MR , MC = 1
flagC = . (13) detection with different configurations, but leads to the same
1, else
beamformer configuration.
The variable flag1bit is indicating if 1 or multibit quantization To select the values in B, we first calculated the array factor
is used. The operator ¬ represents a logic negation. In the case of the antenna array. With this array factor, we then select
of 1-bit quantization, the power consumption of the VGA is the spacing of the values φB uniform from 0 to 2π. Here we
replaced by the one of the LA and the power consumption assume isotropic minimum scattering antennas. For ULA with
of the 1-bit quantizer is neglected with the above stated spacing λ/2, the absolute value of the normalized array factor
reasoning. This formula now contains all special cases of is defined as [35, page 294]:
digital beamforming (MRF E = MR ), analog beamforming
(MR > 0 and MRF E = 1) and hybrid beamforming.


1  sin MC π2 sin (θ − φB ) 
A receiver directly designed for the 1-bit quantization digital AF =
. (15)
MC  sin π2 sin (θ − φB ) 

beamforming systems is very likely to reduce the power
consumptions even further. Due to the 1-bit quantization at the
That means that for actual arriving angle θ choosing φB = θ
end of the analog part of the receiver, the linearity required
is optimal. But this would mean that we have an infinite grid
of the circuits before is greatly reduced. This would enable
of φB . Assuming a single wavefront arriving at the receiver
specialized designs to improve the performance in terms of
and an uniformly distributed angle of the arriving signal θ, we
power consumption, which are not exploited in this work.
get the following expression for the average error ǫ:
III. ACHIEVABLE R ATE E XPRESSIONS Z2
∆  
2 1  sin M π sin (x)

C2
In this subsection achievable rate expressions for differ- 1−

 dx ≤ ǫ. (16)
∆ MC  sin π2 sin (x) 

ent scenarios are derived. The different scenarios are any 0

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TABLE II
M INIMUM NUMBER OF BEAMS NECESSARY TO ACHIEVE MEAN ERROR
Algorithm 1 Selection of the beamforming vectors.
ǫ = 0.1.
Require: H[l], MRF E and MC
MC minimum number of approximation 4MC 1: B ← {φ1 , φ2 , · · · , φ4MC }
beams 2: for i ← 1 to MRF E do
2 7 8 3: for j ← 1 to 4MC do
H
wtst · · · 1, eφj , · · · , e(MC −1)φj

4 14 16 4:
L−1
P  H i  2
8 29 32
5: pi (j) ← wtst H [l]
2
16 58 64 l=0
32 115 128 6: end for
7: ĵ ← argmax pi (j)
j
Setting a maximum allowed ǫ, we can solve the equation for h iH
the distance ∆ between two angles in the set B: 8: wR ← 1, eφĵ , · · · , e(MC −1)φĵ
i

 ∆  9: end for
Z2 
π


return wiR ∀i = {1, . . . , MRF E }
2  1  sin MC 2 sin (x)  
 10:
 1−
 dx − ǫ = 0. (17)
∆ MC  sin π2 sin (x) 

0 The input of the quantizer is assumed to be zero mean unit
variance Gaussian random variable a. The distortion factor
Equation (17) can be solved by a bisection based procedure.
ρ (Qb (·)) depends on the actual quantization operator Qb (·)
In this case we select a lower bound ∆l and an upper bound
and represents the variance of the introduced distortion. As
∆u for ∆, these values are chosen in a way to ensure that the
shown in [13], the matrix F can be calculated as:
value that solves the equation is in between them. Afterwards,
Equation (17) is solved for ∆l , ∆u and (∆l + ∆u )/2 by F = RryC R−1
yC yC . (22)
numeric integration. Based on the results of this function
evaluation we select the bounds for the next iteration of the With the definition of the distortion factor, this expression can
bisection method. A table for some configurations and the be reduced to:
minimum number of elements in B are shown Table II. It RryC = RyC r =
can be observed that for the given parameters, the minimum
− 1 (23)
(1 − ρ (Qb (·))) diag RyC yC 2 RyC yC .
number of elements can be well approximated by 4MC .
Therefore we select 4MC elements uniform in the range from Plugging (23) into (22) results in:
0 to 2π to represent the set B.
− 21
After selecting all beamforming vectors wiR , the overall F = (1 − ρ (Qb (·))) diag RyC yC . (24)
matrix W R is constructed. With W R , we can generate the The covariance matrix Ree can be calculated from:
effective channel H C [l]:
Ree = Rrr − RryC R−1
y C y C Ry C r . (25)
H C [l] = WH
R H[l]. (18)
In [13], [14], [17], [18], [19] only the diagonal elements of
and the effective noise covariance matrix RηC ηC : Rrr are used. As we showed in [36] based on the assumption
Rη C η C = W H of Gaussian signaling, it is possible to calculate the complete
R Rηη W R . (19)
matrix Rrr , which we will show changes the overall result.
The effective channel and noise covariance matrix are then We define this operation of calculating Rrr from RyC yC
input to the digital system with low resolution A/D conversion. as T (RyC yC , Qb (·)) dependent on the quantization function
Algorithm 1 shows the procedure of finding the receiver Qb (·). Using this definition and plugging (23) into (25) we
beamforming vectors wiR given the channel and the number get:
of antennas per RF chain.
2
Ree = T RyC yC , Qb (·) − (1 − ρ (Qb (·)))

− 1
− 1 (26)
B. Modeling the Quantization diag RyC yC 2 RyC yC diag RyC yC 2 .
As the model used in [13], [19], [17], we use the Bussgang Now we can calculated the effective channel H ′ [l] and noise
theorem to decompose the signal after quantization in a signal covariance matrix Rη′ η′ of the overall system including the
component and an uncorrelated quantization error e: analog combing and the quantization:
r[n] = F y C [n] + e[n], (20) H ′ [l] = F W H
R H[l], (27)
with y C [n] being the signal after the analog combining at and
the receiver equal to u[n] + η[n], where u[n] is the receive Rη ′ η ′ = F W H
R Rηη W R F
H
+ Ree . (28)
signal after the multipath channel. This is basically modeling
the deterministic process of quantization as a random process. It is important to keep in mind that since F and Ree are
The quantization distortion factor ρ (Qb (·)) is defined as: dependent on Ryy , which dependents on Rxx , thus the ef-
h i fective channel and noise covariance matrices change with the
2
ρ (Qb (·)) = E |a − Qb (a)| . (21) transmit covariance matrix. Up to now, the actual expression

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is exact for the case of a Gaussian input signal x[n]. The Due to the linearity of the expectation operation E we can
actual distribution of e[n] is unknown. Approximating e by interchange it with the integrals:
a Gaussian with the same covariance matrix leads to simple  ∞ 
rate expressions and represents a worst case scenario. For this   Z   
statements we need to proof that E[u[n]e[n]] and E[η[n]e[n]] E ui (t)u∗j (t) = F −1 E ui (f ′ )u∗j (f ′ − f ) df ′ .
 
are equal to zero. With this choice of F we ensured that −∞
(39)
E[y C [n]eH [n]] = 0. We can expand this term to
Since the transmit signals of each frequency bin x(f )
E[y C [n]eH [n]] = E E[y C [n]eH [n]|η[n]] ,
 
(29) are independent and  have zero mean, the expectation
E ui (f ′ )u∗j (f ′ − f ) is only unequal to zero if f = 0. Since
if we plug in the definition of y C [n] we get outside of the transmission bandwidth the signal is going to
be zero, we get the following expression:
E E[u[n]eH [n]] + η[n]E[eH [n]] .
 
(30)
 
Since the first term is independent of η[n] this reduces to

 Zf2 
 ∗
 −1
 ′ ∗ ′
 ′
E ui (t)uj (t) = F δ(f ) E ui (f )uj (f ) df ,
E[u[n]eH [n]] + E[η[n]]E[eH [n]]. (31) 

f1


(40)
Since E[η[n]] is equal to zero it follows that
with the Dirac pulse δ(f ) at frequency f = 0. So if we
E[u[n]eH [n]] = 0. (32) transform this Dirac impulse back into the time domain we
get:
The proof for E[η[n]eH [n]] = 0 follows the same steps. Zf2
   
∗
E ui (t)uj (t) = E ui (f )u∗j (f ) df, (41)
C. Calculation of the Receive Covariance Matrix f1
For the calculation of the matrix F and the effective independent of time t, that also states that we still have a
noise correlation matrix Rη′ η′ , it is necessary to calculate stationary random process. Plugging the definition of ui (f )
the correlation matrix RyC yC of the signal after the analog into the equation we get the covariance matrix Ru(t)u(t) :
combining. This signal is defined as:
L−1
! Zf2
H(f )Rx(f )x(f ) H H (f )df.
X
H
y C (t) = W R H(l)x(t − τl ) + η(t) Ru(t)u(t) = (42)
l=0
(33)
f1
= WH
R (u(t) + η(t)) . Combining these results we can express the matrix RyC yC as:
Since the two random variables x and η are independent the
covariance matrix decomposes into: Ry C y C =
 
Zf2
Ry C y C = W H


R Ru(t)u(t) + Rη(t)η(t) W R . (34)
WH

H(f )Rx(f )x(f ) H H (f )df + Rη(t)η(t)  W R .

R 
The remaining matrix that needs to be calculated is Ru(t)u(t) . f1
(43)
Ru(t)u(t) = E u(t)uH (t) .
 
(35)
To simplify the notation for the derivation we evaluate the D. Problem Formulation
elements of the matrix separately:
For the given signal model, the problem of finding the
   
Ru(t)u(t) i,j = E ui (t)u∗j (t) . (36) maximum achievable rate for a multipath channel with full
Channel State Information (CSI) at the Transmitter (CSIT)
Without changing the result, we can transform this equation and the Receiver (CSIR) can be formulated as:
into the frequency domain and then transform it back. Since
the expectation E and the Fourier transformation F are linear 1
R= max I(xN , r N |H[l])
operations, we can interchange the order we perform them: N p(xN ,wiR )
      s.t. E[||x||22 ] ≤ PT x
E ui (t)u∗j (t) = F −1 F E ui (t)u∗j (t) iH
(37) h
wiR = 1, ejφi , · · · , ej(MC −1)φi ∀i ∈ {1, ..., MRF E },
   
= F −1 E F ui (t)u∗j (t) .
(44)
With the convolution property of the Fourier transformation
with xN and r N being N input/output samples of the system.
we get:
     Due to the non-linearity of the quantization and the non-trivial
E ui (t)u∗j (t) = F −1 E ui (f ) ∗ u∗j (−f ) problem of finding the optimal beamforming configurations
  ∞  wiR , we make a number of approximations that make the
(38)
 Z 
= F −1 E  ui (f ′ )u∗j (f ′ − f )df ′  . expression traceable:
• Assume x(f ) is Gaussian
 
−∞

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• For a system with CSIT SVD based precoding is used streams and is equal to rank(H(f )). If we would allow all
(SVD of the effective channel after analog combining is frequencies to separately allocate the number of streams, we
used) again have a combinatorial problem. Therefore, we check the
i
• w R are selected from the derived finite set separately for overall achievable rate for allocating j spatial streams and in
each antenna group based on an SNR criteria the end select one that has the largest achievable rate.
• Quantization is modeled as additive Gaussian noise with From Equation (28), we see that Rη′ η′ is not diagonal.
the AQNM model including the off-diagonal elements In a system, where the noise covariance matrix is known
With this simplifications the wiR are already defined and we and independent of the transmit covariance one would simply
−1
can transform the problem into a frequency domain equation. multiply the receive vector with Rη′2η′ . This does generate a
In [37], [38] the achievable rate of a digital beamforming −1
new system with a different channel Rη′2η′ H ′ (f ) and spatial
system without quantization, but considering a multi-path white noise. Afterwards, the waterfilling solution is applied to
channel is described. The solution is waterfiling across the the new channel [37]. In general, the achievable rate increases
frequency bins and the spatial streams. Since for a system with compared to a system with white noise. In a more abstract
low resolution ADCs the quantization does influence the signal way, the reason for the improvement is that channels with
relative to the total power, it is intuitive to use each frequency lower noise power can be used. Dependent on the rank of the
bin independent of each other. Since the optimization is carried channel relative to the number of the receive antennas, the
out for each frequency bin f separately, the result only is a orthogonal subchannels with highest noise power might not
lower bound to the joint optimization. be used. In a system, where the channel and the noise depend
Zf 2 on the covariance matrix of the transmit signal, it is very
R≤ max I(x(f ), r(f )|H ′ (f ))df difficult to generate precoding and reception matrices that split
Rx(f )x(f ) (45) the channel into orthogonal subchannels. Therefore, with our
f1
system, considering the correlation of the quantization noise
s.t. E[||x(f )||22 ] ≤ PT x ∀f ∈ [f1 , f2 ], leads to a decrease in achievable rate.
with x(f ), r(f ) and H ′ (f ) being the input/output signal For both calculation of the achievable rate in Equation (45)
and equivalent channel of frequency bin f . The frequency as well as the calculation of the receive signal covariance
f1 and f2 mark the borders of the band of interest in the matrix RyC yC in Equation (43), it is necessary to integrate
equivalent baseband channel. If not the whole band covered over the whole signal band from f1 to f2 . Instead of taking
by the sampling rate is available to the system, the parameters infinitely small frequency bins, we approximate the signal
f1 and f2 have to account for the oversampling. band in the interval from f1 to f2 by a finite number of
Since all signals are represented by Gaussian random frequency bins. We choose the number of frequency bins to
variables, we get the following expression for the mutual make the channel H(f ) at each frequency bin sufficiently flat.
information: This leads to a good approximation of the achievable rate.
Equation (45) is reduced to:
I(x(f ), r(f )|H ′ (f )) =
   (46) f2   
log2 det I + R−1 ′ ′H X
η ′ η ′ H (f )Rx(f )x(f ) H (f ) . log2 det I + R−1 ′ ′H
R≤ η η
′ ′ H (f )R x(f )x(f ) H (f )
f1
For the non-quantized case the optimal result is the water-
filling solution. Due to the modeling of the quantization, s.t. E[||x(f )||22 ] ≤ PT x ∀f ∈ [f1 , f2 ].
(48)
the effective noise covariance matrix Rη′ η′ and the effective
The receive signal correlation matrix can then be calculated
channel H ′ (f ) are dependent on the input covariance matrix
as:
Rx(f )x(f )
In a system without quantization, the covariance Rx(f )x(f ) Ry C y C =
 
would be chosen according to the right singular vectors of Xf2
H(f ) to split the channel in orthogonal subchannels [37]. In WH
R
 H(f )Rx(f )x(f ) H H (f ) + Rη(t)η(t)  W R .
this scenario Rx(f )x(f ) would be equal to f1
(49)
Rx(f )x(f ) = V (f )S(f )V H (f ), (47) The channel H(f ) or the effective channel H ′ (f ) at the
frequency bins f can be calculated from the channel tabs H[l]
where V (f ) are the eigenvectors of H H (f )H(f ). The di-
via the DFT F(·):
agonal matrix S(f ) represents the power allocation to the
subchannels. The optimal allocation in a system without H(f ) = F(H[l]). (50)
quantization follows the waterfilling solution. Since Rη′ η′
and H ′ (f ) actually depend on Rx(f )x(f ) , it is difficult to We now have all the necessary mathematical tools to ap-
separate the channel into orthogonal subchannels. To make the proximate the achievable rate of a multipath channel including
evaluation traceable, we use the suboptimal precoding vector quantization effects at the receiver. Algorithm 2 describes our
V (f ). For the matrix S(f ), we test all different possibilities approximation of the achievable rate for these type of systems.
of allocating equal to power to 1 to Smax spatial streams. This approximation is modeling a point to point closed
The number Smax is the maximum possible number of spatial loop spatial multiplexing system. There are many different

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Algorithm 2 Approximation of the achievable rate of a culation of Rrr and therefore Rη′ η′ . The model considering
quantized system with noise covariance matrix Rηη , multipath only the diagonal elements of Rrr was used in [19] and [17].
channel H[l] and sum power constraint PT x and quantization For the evaluation a channel of the first channel model is
function Qb (·) with resolution of b bits in the frequency band used. Here K = 7 separate paths are received at the same
from f1 to f2 . time. Different number of transmit and receive antennas are
used. From Figure 6, we see that the model considering the off
h ηη , H[l], PTix , f1 , f2 and Qb (·)
Require: R diagonal elements (ND) has a significant lower performance
2
1: ρ ← E |a − Qb (a)| compared to the model only considering the diagonal elements
L−1
P
 (D). In fact, for the case of only one transmit antenna (Mt = 1)
2: Smax ← rank H[l] and 1-3 bit A/D conversion, the achievable rate is not maxi-
l=0
3: H(f ) ← F(H[l]) mized at the highest SNR possible, but rather at a finite SNR
4: [V (f ) D(f )] ← eig(H H (f )H(f )) ∀f ∈ [f1 , f2 ] between 0 and 10 dB.
5: for j ← 1 to Smax do As discussed in Section III, if we compare the results in
6: S←0 Figure 6 to the ones in Figure 7. Considering the off diagonal
7: [S]i,i ← PTj x ∀i = {1, . . . , j} elements has only a large influence if the number of receive
antennas is larger than the number of transmit antennas. This
8: Rx(f )x(f ) ← V (f )SV H (f ) ∀f ∈ [f1 , f2 ]
f2 effect can be explained in the following ways: After spatial
H(f )Rx(f )x(f ) H H (f ) + Rη(t)η(t)
P
9: Ryy ← whitening, the power distribution of the effective noise is more
f1 non-uniform relative to the system that considers only the diag-
10: Rrr ← T (Ryy , Qb (·)) onal component. Since the actual channel and noise covariance
11: Ree ← Rrr − matrix depends on the precoding matrix, it is not possible to
−1 −1
(1 − ρ)2 diag (Ryy ) 2 Ryy diag (Ryy ) 2 decompose the channel into orthogonal subchannels with equal
1
−
12: F ← (1 − ρ)diag (Ryy ) 2 SNR. Thus, we cannot avoid using the channel with high noise
13: Rη′ η′ ← Ree + F Rηη F H variance and therefore the overall performance does degrade in
14: H ′ [l] ← F H[l] ∀l ∈ {0, . . . , L − 1} the quantization noise limited, high SNR regime. This effect is
15: H ′ (f ) ← F(H ′ [l]) only dominating the performance in the case of high SNR and
16: A(f ) ← I + R−1 ′
η ′ η ′ H (f )Rx(f )x(f ) H
′H
(f ) very low resolution quantization. The peak in the achievable
∀f ∈ [f1 , f2 ] rate comes from the fact that at a certain SNR the noise
f2
17: R(j) =
P
log2 (det (A(f ))) provides dithering to randomize this structural performance
f1 degradation. At the minimum variance noise, where sufficient
18: end for dithering is provided, is the peak in the performance. This
19: Rmax ← max R(j) effect is called statistic resonance and can be found in many
j
20: return Rmax non-linear systems [39].
Another important thing to mention is that in a system with
simple modification possible to change the modeled system. multipath propagation and white noise, the covariance matrix
The following are a non-exhaustive list of examples: Ryy of the receive signal is approximated diagonal. This leads
• Systems without CSIT to a diagonal matrix Rrr and therefore spatial white noise of
• Systems with imperfect channel estimation the quantized system.
• Systems with multiple terminals communication with
base station B. Influence of AGC Imperfection
• Systems with constraint feedback
In this evaluation, we show the influence of AGC im-
• Systems with multiple terminals and a basestation
perfections on the performance. To simplify the evaluation,
Most of these systems can be modeled by changing the we choose a SISO system with the simple multipath model
constraints on the precoding matrix Rx(f )x(f ) and the channel described in the signal model with the parameters L = 32,
model. P = 16 and β = 0.35. For an imperfect AGC, the assumption
that the receive signal r[n] and the quantization error e[n]
IV. E VALUATION R ESULTS are independent is no longer satisfied. Since all our formulas
In this section we evaluate the derived expression for dif- for modeling the quantization are based on this assumption
ferent scenarios. We always include a rate evaluation without E ri [n]eHi [n] = 0 ∀i = {1, . . . , MR }, they are no longer
quantization. For the system without quantization we apply valid in the case of an imperfect adapted AGC. We can enforce
the waterfilling solution separate for each frequency bin. For this orthogonality by scaling the signal r[n]. The scaling factor
all scenarios the results show the average achievable rate in ζ is equal to:
bps/Hz averaged over 1000 channel realizations. E [aQb (a)]
ζ= , (51)
E [Qb (a)2 ]
A. Comparison to Diagonal Approximation with a being a real Gaussian random variable with unit
This part of the evaluation compares difference in perfor- variance and zero mean. After this scaling we can use the
mance when considering the non-diagonal elements in the cal- derived formulas as before.

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16 3
NQ D 8b NQ
14 ND 8b D 7b 2b
2.5
ND 7b D 6b 2b 10%
avg. achievable rate [bps/Hz]

avg. achievable rate [bps/Hz]

12 ND 6b D 5b 2b -10%
ND 5b D 4b 2 2b 20%
10 ND 4b D 3b 2b -20%
ND 3b D 2b 2b 40%
8 ND 2b D 1b 1.5 2b -40%
ND 1b 2b 80%
6 2b -80%
1
1b
4
0.5
2
increasing resolution
0 0
−30 −20 −10 0 10 20 30 −30 −20 −10 0 10 20 30
SNR [dB] SNR [dB]
Fig. 6. MR = 32 and MT = 1 digital beamforming with different resolution Fig. 8. SISO system with imperfect AGC and 2 bit ADC resolution.
of the ADC.
80
50
DBF HBF
NQ D 8b DBF 8b HBF 8b
ND 8b D 7b DBF 7b HBF 7b
avg. achievable rate [bps/Hz]

40 ND 7b D 6b
avg. achievable rate [bps/Hz]

60 DBF 6b HBF 6b
ND 6b D 5b DBF 5b HBF 5b
ND 5b D 4b DBF 4b HBF 4b
30 ND 4b D 3b DBF 3b HBF 3b
ND 3b D 2b 40 DBF 2b HBF 2b
ND 2b D 1b DBF 1b HBF 1b
20 ND 1b

20
10

increasing resolution 0
0 −30 −20 −10 0 10 20 30
−30 −20 −10 0 10 20 30
SNR [dB]
SNR [dB]
Fig. 9. MR = 8 and MT = 64 MC = 4 different resolution of the ADC b.
Fig. 7. MR = 8 and MT = 8 digital beamforming with different resolution
of the ADC. C. Downlink (DL) Point to Point Scenario
The graphs in Figure 8 show the average achievable rate
In this subsection, a downlink like scenario is evaluated. A
with 2 bit resolution and different offset relative to the optimal
basestation with 64 antennas (MT = 64) is transmitting to
power at the VGA output. The power after the VGA is defined
a mobilestation with 8 antennas (MR = 8). For the channel
as
model the following parameters are used: L = 32, P = 16,
Ω = Ωmq (1 − ǫAGC ), (52)
β = 0.35. For the hybrid beamforming system MC ∈ {2, 4, 8}
where Ωmq and ǫAGC are the signal variance resulting in the and therefore MRF E ∈ {4, 2, 1} is used.
minimal distortion and the AGC error. Figure 9 shows the average achievable rate for the case of
The graphs shows that an error in the range of -20% to 20% MC = 4 and ADC resolution b ∈ {1, · · · , 8}. The rate curves
has only a minor impact on the performance. But as soon of the systems including an ADC clearly converge to the
as the error is larger than 20%, the performance decreases ones assuming no quantization, for higher resolution in both
dramatically. Ultimately, the quantization converges to 1-bit cases of hybrid and digital beamforming. Especially in the
quantization and therefore also our achievable rate converges low SNR regime (below 0 dB), the performance of the digital
to the one of 1-bit quantization. We can also observe the beamforming systems with low resolution ADC (1-3 bit) are
performance penalty for a larger negative or positive error is very close to the performance without quantization. These
different. systems clearly outperform a hybrid beamforming system in

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80
DBF DBF HBF
HBF MRF E 4 DBF 8b HBF 8b
25 HBF MRF E 2 DBF 7b HBF 7b

avg. achievable rate [bps/Hz]

energy efficiency [bps/J]

HBF MRF E 1 60 DBF 6b HBF 6b

DBF 5b HBF 5b
DBF 4b HBF 4b
20 DBF 3b HBF 3b
40 DBF 2b HBF 2b
DBF 1b HBF 1b

15
20

increasing resolution
10 0
1.5 2 2.5 3 3.5 4 −30 −20 −10 0 10 20 30
avg. achievable rate [bps/Hz] SNR [dB]
Fig. 10. MR = 8 and MT = 64 SNR -15 dB different resolution of the
Fig. 12. MR = 64 and MT = 8 MC = 4 different resolution of the ADC
ADC b.
b.

4.5
DBF
100 HBF MRF E 4
HBF MRF E 2 4
energy efficiency [bps/J]

HBF MRF E 1
energy efficiency [bps/J]

80
3.5

60
3
increasing resolution
40
2.5 DBF
HBF MRF E 32
20 HBF MRF E 16
increasing resolution 2
HBF MRF E 8
2 4 6 8 10 12 14 16
2 2.5 3 3.5 4
avg. achievable rate [bps/Hz]
avg. achievable rate [bps/Hz]
Fig. 11. MR = 8 and MT = 64 SNR 0 dB different resolution of the ADC
b. Fig. 13. MR = 64 and MT = 8 SNR -15 dB different resolution of the
ADC b.
this SNR regime. In this evaluation a 4-bit ADC is enough to
outperform the hybrid system over the whole SNR range. In the lower SNR case, the energy efficiency peaks at 3-bit
Since these system have a different power consumption, we ADC resolution. The higher the SNR gets, the larger the ADC
also have to compare the results in terms of energy efficiency. resolution that maximizes the energy efficiency. These results
Here we define the energy efficiency (EE) as the average show that even when perfect hybrid beamforming without the
achievable rate R divided by the power consumption of the beam-alignment overhead is considered a digital beamforming
RF-front-end PR : system is more energy efficient.
R
EE = . (53)
PR D. Uplink (UL) Point to Point Scenario
Figure 10 and 11 show the energy efficiency over the achiev- For the configuration of the system, the same parameters as
able rate for MC ∈ {2, 4, 8} and the resolution of the ADC in the DL like setup in the previous subsection are used. The
b ∈ {1, · · · , 8} with SNR ∈ {-15 dB, 0 dB}. For both cases, only difference is that in this case the antenna configuration
the digital beamforming achieves a higher data rate and also a is MR = 64 and MT = 8.
higher energy efficiency. In the -15 dB SNR case, the differ- Figure 12 shows the achievable rate for this case. We
ence in energy efficiency is not substantial but in the 0 dB SNR observe that the penalty of hybrid beamforming is less severe
there is a large gap between hybrid and digital beamforming. than in the DL case. The reason is that in this case, the side

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18 Future extensions should consider the following points. For

DBF the hybrid beamforming case, the evaluation only shows the
16 HBF MRF E 32 result if the beams are already aligned. As shown in [6], this
HBF MRF E 16 can be considered to be a large overhead. A possible future
mobile broadband system operating at mmWave frequencies
energy efficiency [bps/J]

14 HBF MRF E 8
will definitely suffer from additional other RF-frontend related
constraints. Especially considering the inefficiency of the PA
12 that have to operate close to the saturation and therefore
introduce distortion to the signal. Also phase noise scales
10 approximately with carrier frequency squared and thus, has to
be considered for mmWave systems. This will lead to a limited
8 constellation size, which will then bound the overall spectral
efficiency. In this evaluation, we also ignored the necessary
reference overhead for channel estimation. Especially for a
6 high order spatial multiplexing, this is not negligible and will
increasing resolution essentially limit the number of spatial data streams. The chan-
4 nel model is assuming an omnidirectional minimum scattering
4 6 8 10 12 14 16
antenna. Including this consideration into the evaluation would
avg. achievable rate [bps/Hz] lead to a result that is more close to a practical evaluation. A
Fig. 14. MR = 64 and MT = 8 SNR 0 dB different resolution of the ADC dynamic multi-user environment would also provide for an
b. interesting comparison between hybrid beamforming and low
of the system with less antennas (the mobilestation) has no resolution ADC digital beamforming.
constraints on the front-end which is the exact opposite in
the DL. This means that the number of spatial streams is ACKNOWLEDGMENT
in all cases just limited by the 8 possible streams of the This work was supported by the European Commission in
mobilestation. Therefore, the penalty of hybrid beamforming is the framework of the H2020-ICT-2014-2 project Flex5Gware
less and the achievable rate of hybrid and digital beamforming (Grant agreement no. 671563).
rise with the same slope for the case without quantization.
In the low to medium SNR, the low resolution ADC R EFERENCES
digital beamforming systems perform better than the hybrid [1] F. Boccardi et al., “Five disruptive technology directions for 5G,” IEEE
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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/JSAC.2017.2720398, IEEE Journal
on Selected Areas in Communications
13

[15] C. Studer and G. Durisi, “Quantized massive MU-MIMO-OFDM up- Kilian Roth is since 2014 pursuing a Ph.D. degree
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one-bit ADCs,” IEEE Trans. Wireless Commun., vol. 16, no. 1, pp. 87– is with the Next Generation and Standards Depart-
100, Jan. 2017. ment of Intel Deutschland GmbH since 2014. From
[17] J. Mo et al., “Hybrid architectures with few-bit ADC receivers: 2013 to 2014 he was a Intern at Intel Deutschland
Achievable rates and energy-rate tradeoffs,” CoRR, vol. abs/1605.00668, GmbH. He graduated from Munich University of
2016. [Online]. Available: http://arxiv.org/abs/1605.00668 Applied Sciences (MUAS) in 2011 and obtained a
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efficiency: A comprehensive comparison of beamforming schemes with participant of the projects Flex5GWare and mm-
low resolution ADCs,” CoRR, vol. abs/1607.03725, 2016. [Online]. Magic funded by the EC under the H2020 Frame-
Available: http://arxiv.org/abs/1607.03725 work Program. His main research interests are signal processing algorithms
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Josef A. Nossek (S’72-M’74-SM’81-F’93-LF’13)
wireless transmitters,” Ph.D. dissertation, EECS Department, University
received the Dipl.-Ing. and Dr. techn. degrees in
of California, Berkeley, May 2014. [Online]. Available: http://www.
electrical engineering from Vienna University of
eecs.berkeley.edu/Pubs/TechRpts/2014/EECS-2014-42.html
Technology, Austria, in 1974 and 1980, respectively.
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He joined Siemens AG, Germany, in 1974, where
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he was engaged in filter design for communication
2659, Dec. 2008.
systems. From 1987 to 1989, he was the Head of the
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Radio Systems Design Department, where he was
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Int. Symp. on Radio-Frequency Integration Technology (RFIT) 2012,
processing into digital microwave radio. From 1989
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to 2016, he has been the Head of the Institute of
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Circuit Theory and Signal Processing (NWS) at the Technical University
transceivers,” Ph.D. dissertation, EECS Department, University of
Munich, Germany. In 2016, he joined the Universidade Federal do Ceara,
California, Berkeley, Dec 2011. [Online]. Available: http://www.eecs.
Fortaleza, Brasil as a Full Professor. He was the President Elect, President,
berkeley.edu/Pubs/TechRpts/2011/EECS-2011-132.html
and Past President of the IEEE Circuits and Systems Society in 2001, 2002,
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and 2003, respectively. He was the President of Verband der Elektrotechnik,
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Mannesmann Mobilfunk (currently Vodafone) Innovations Award in 1998,
Silicon Monolithic Integrated Circuits in RF Systems (SiRF) 2009, San
and the Award for Excellence in Teaching from the Bavarian Ministry for
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Science, Research and Art in 1998. From the IEEE Circuits and Systems
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Society, he received the Golden Jubilee Medal for Outstanding Contributions
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to the Society in 1999 and the Education Award in 2008. He received the
San Francisco, CA, USA, Feb. 2015, pp. 1–3.
Order of Merit of the Federal Republic of Germany (Bundesverdienstkreuz
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am Bande) in 2008. In 2009 he became member of the National Academy of
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Engineering in Germany (acatech), in 2011 he received the IEEE Guillemin-
(VLSI-Circuits) 2016, Honolulu, HI, USA, June 2016, pp. 1–2.
Cauer Best Paper Award and in 2013 the honorary doctorate (Dr. h. c.) from
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the Peter Pazmany Catholic University, Hungary. He received the VDE Ring
multi-resolution beamforming training in 802.11ay,” Nov. 2016.
of Honor in 2014 and the TUM Emeritus of Excellence in 2016.
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