IEEE Circuits and Systems Magazine - Q4 2022 - 9

at the edges of the frequency band, as the frequency
deviates the most from the center frequency. As the error
is inversely proportional to the frequency, at the lower
side of the frequency band error will be larger than the
higher side of the band and this maximum error can be
found as:
'Tmax sin (4)
¨ fcBW / SS
11
2
§
©
fc
')
·
¹
¸ sin
§
©
¨
')
·
¹
¸
where BW is the signal bandwidth and BW/fc is called
the fractional bandwidth. In Fig. 5, the maximum error
in the angular domain for a PS-based SSP unit is plotted
versus the ideal intended AoA, for three cases of
fractional bandwidths. In this plot the AoA range is limited
to ±60°, since for larger AoAs the actual AoA can be
as high as ±90° and the error in those cases does not
reflect the severity of the PS-based implementation. As
it can be seen, this error can get as high as 22° which
results in non-alignment with the intended transmitter
and consequently loss in the desired signal (beamforming
case) or imperfect cancellation (beam-nulling case).
The frequency-dependent approximation of a TTD element
with a PS results in frequency-dependent beamforming
gain that acts as a bandpass filter [20], [22],
and imperfect frequency-dependent beam-nulling that
results in wideband interference leakage [23], [52].
The beamforming gain in a PS-based N-element
receiver can be modeled as the absolute value of the
inner product Gf
2 waH
, where w is a receive beamformer
and a is a spatial response vector. Assuming a
phase difference of 9s between neighboring PSs, the
n-th element of w is defined as []w n
On the other hand, using (1), the n-th element of a is
defined as []a n
2jnexp1 9A . Expressed as a sum
of complex exponentials, the gain G(f) becomes
Gf
N
¦e
jn 1 '')I
n 1
In Fig. 6, the normalized beamforming gain G(f)/N for
intended AoA of 45° versus normalized frequency f/fc of
the input signal for three cases of N = 4,16,64, are plotted.
The beamforming gain in a PS-based implementation
acts as bandpass filter for the desired signal. Similar to
a filter, the 3-dB fractional bandwidth (FBW3 dB) can be
defined as the fractional bandwidth where the beamforming
gain drops 3 dB compared to the maximum
value (N). As proven in [53], [54], for large values of N, the
FBW3 dB is given as:
FBW3 dB 5
Fourth quartEr 2022
1.772
N sin
(6)
Figure 6. Ps-based normalized beamforming gain versus
normalized frequency.
IEEE cIrcuIts and systEms magazInE
9
(5)
Figure 5. generalized Ps-based implementation in an
N-element Paa.
2jnexp1 9s .
For larger arrays FBW3 dB becomes smaller and the
approximation of a TTD element with a PS becomes less
valid. Similar to the outcome of the beam squinting error,
as the intended AoA increases, the FBW3 dB gets smaller
and the effective error increases.
B. Beam-Squint for Beam-Training Mode
Beam-training is a part of the initial access (IA) protocol
in mmW networks that has a goal to align the
beamforming directions and realize the maximum
gain between the base station (BS) and user equipment
(UE) [55], [56]. The majority of existing mmW
beam-training algorithms was designed for PAAs
which frequency-flat PS in all antenna branches to
steer/combine the signal in a desired direction. With
a single mmW chain, power-efficient analog PAAs can
synthesize only one spatial beam with all frequency
components being aligned in the same direction, as
illustrated in Figure 7(a). Thus, the existing beamtraining
schemes with analog PAAs include various
types of exhaustive beam sweeping where different
beam candidates are sequentially probed to find
the dominant AoA or angle-of-departure (AoD) [57][60].
The required number of probing beams linearly
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