IEEE Circuits and Systems Magazine - Q2 2019 - 42

channel G SA and G FH . When system is aware of potential
interference, digital precoding stage can be used to effectively suppress them. A practical way to acquire the information of effective channel is via a training procedure
where BS and UE use quantized analog beamformer to
exchange pilot symbols and estimate effective channel
G SA and G FH . This training procedure is similar to the
multi-beam scheme proposed for the next generation of
mmW indoor system [49]. Meanwhile, the gain reduction
due to finite phase shifter resolution is not severe either.
In fact, the gain degradation is lower bounded by
0.68 dB, 0.16 dB and 0.04 dB with Q = 3, 4, 5 bits quantization of phase shifters and does not scale with the array
size or multiplexing level. An analysis that supports these
numbers is provided in the Appendix B. Equivalently, the
gain degradation is bounded by 0.16 dB so long as angle
error of phase shifters are no larger than 11.25 degree.
Such specifications are not difficult to meet in state-ofthe-art devices as it will be discussed in Section V-C.
F. Simulation Results
In this subsection, simulation results are presented to
show the required design parameters to reach SE target
in three array architectures.

In the simulation, 3D mmW MIMO channel between BS
and U UEs are generated according to mmW sparse scattering model [54]. The channel between BS and each UE
consists of 20 multi-path rays in 3 multipath cluster and
LOS cluster, if exists, is 10 dB stronger than the rest. Angle of arrival (AOA) and angle of departure (AOD) of clusters are uniform random variables within azimuth range
[-60c, 60c] and elevation range [-30c, 30c]. Azimuth and
elevation AOA and AOD of rays within a cluster have random deviations from the cluster specific AOA and AOD,
and they follow zero mean Laplacian distribution with 10c
standard deviation. In dense urban MBB, a scheduler is
assumed such that the LOS paths of all target receivers
are unique [55]. The mean SE is evaluated by taking average of SINR in (4) over U UEs and use Shannon capacity
formula, i.e., SE = R Uu = 1 log 2 (1 + SINR u). The data streams
used in the simulation are Gaussian distributed and their
magnitudes are truncated such that PAPR is 10 dB.
With ideal hardware, the required transmit power
P (out) to reach SE target with various antenna size N
and number of data streams U in three architectures
are shown in Figure 3.
We first focus on how transmit power changes with
parameter N and U. Increasing array size N is effective

Appendix B: Impact of Phase Shifter Quantization Error and Random Error on Beamforming Gain
Consider a linear phased array system with N antenna elements that steers a beam towards direction c in a
2D plane. Beamforming vector is given by [e jz 1, f, e jz N ],
where z n = (n - 1) r sin (c). In the next, we derive beamforming gain at the main lobe for system with ideal and
non-ideal phase shifters.
Let us denote the signal at the n th elements as
w n with w n = 1/ N , 6n when all phase shifters are
ideal. Clearly, the phase shifter needs to be set such
that signals are constructively added in the intended
direction, i.e., w n e z n = 1/ N , and the beamforming
gain is

Gl =

N

/ (w nl e jz )
n

2

N

/ (w n e j } ) e j z

=

n =1

= 1
N

N

/ e j}

2
n

n =1

n

2
n

n =1

= 1
N

N

2

N

2

N

N

/ cos (} n) + j / sin (} n)

n =1

N

2

n =1
2

= 1 = / cos (} n)G + 1 = / sin (} n)G
N n =1
N n =1
$ 1 = / cos ^} n hG
N n =1
$ N cos 2 ^e h

where the second inequality is valid so long as Q $ 1, i.e.,
} n # r/2, 6n.
Therefore the gain reduction is bounded by

G=

N

/ w n e jz

2
n

n =1

=N

10 log 10 ; G E # -20 log 10 8cos ` rQ jB [dB]
Gl
2
The above derivation shows that the gain drop in the

42

When all phase shifters are non-ideal, the signal at the

main lobe is less than 0.68 dB, 0.16 dB and 0.04 dB

n th element is denoted as w nl =w n exp (j } n), where } n is

with Q = 3 to 5 bits quantization. Besides, these values

the phase error due to quantization and random imple-

are independent from the antenna size N. Equivalently,

mentation impairment. With Q bits quantization, the phase

when phase shifter implementation error is less than

error } n is bounded as } n # e where e = r/2 Q . The

e = 22.5c, 11.25c, and 5.625c, gain drop is also bounded

corresponding beamforming gain is

by 0.68 dB, 0.16 dB and 0.04 dB, respectively.

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