IEEE Circuits and Systems Magazine - Q4 2022 - 14

RF, which are known to suffer from low scalability
in terms of required area and power efficiency when
the array size becomes large. In our recent work in
[51], [61], [76], we proposed a scalable TTD array
with delay elements implemented in baseband for a
single-symbol mmW beam training in an OFDM system.
The key idea was to exploit signal delaying in
each antenna branch to exacerbate the wideband
spatial effect (beam squint) among subcarriers in a
controllable manner, i.e., to create a frequencydependent
codebook which probes all angular directions
at once. This is achieved by properly configuring the
delay and phase taps in a TTD array.
B. TTD Codebook and DSP Algorithm Aesign
In our recent work we introduced TTD-based beamtraining
at the UE side that requires only one OFDM
symbol [51], [61], [76]. We assume that the BS with NT
antennas uses a fixed frequency-flat precoder v∈CNT
designed according to the previously estimated AoD.
The received signal Y[m] at the m-th OFDM subcarrier
can be expressed as follows
Ym mm sm mm
combiner, H m C RT
wH vw nHH ,
where w m C is a frequency-dependent UE TTD
>@ NNu
bd2 bdbd bd bd bd
>@ NR
pilot, and nImbdCN0 2
exp
(7)
is a channel matrix, s[m] is a
, NR is a noise vector at the
m-th subcarrier, respectively. The n-th element of w is
defined as
wmj ff2bdnn mn n
c
h s .
where h
nn2 1, , and φn are the gain, delay tap, and
phase tap in the n-th antenna branch, respectively.
Assuming the bandwith BW and a uniform linear
array (ULA) with a frequency-flat spatial response at the
UE side, it was shown in [61] that a codebook of rainbow
beams which cover the entire angular range
bd
/, /22
can be created by setting the delay taps n n, , as follows
 
n nn whereBW
,,
2 992 /,11 (9)
Since the taps in (9) are proportional to the Nyquist
sampling period, the codebook can be obtained without
the need to implement a fractional ADC sampling.
In such a codebook, the rainbow beam associated with
the subcarrier frequency fm is pointing in the angle θm
given as follows [61]

mmf2sinmod1 21219 ,. (10)
An example of the resulting TTD codebook for
a UE with NR = 16 antenna elements is provided in
Fig. 10. To probe D = NR = 16 directions, M = 16 subcarriers
need to be used. The codebook can be designed
to be more robust in frequency-selective channels
by mapping R different subcarriers to each probed
direction. This is achieved by increasing the delay
difference between antenna elements R times, i.e.,
9 2 R /BW [76]. With R being the codebook diversity
order and R the number of probed directions, TTD
beam training requires a waveform with M = DR loaded
subcarriers.
The pointing angles m m, , of rainbow beams can be
jointly rotated by using the frequency-flat PSs in the TTD
array. A rotation of θrot requires the phase taps sn n, ,
to be set as follows
[] (8) ss s 
2
n nn,where
 mm m
' 2rot,
2199 sin rot,. (11)
2
The rotated angles can then be expressed as
.
With this codebook and described frequencyto-angle
mapping, the information of the dominant
propagation angle can be acquired with simple frequency-domain
DSP. Given the received signal in (7), a
coarse AoA estimate θˆ
is given as ˆ 2
angle θm*
'
m*
'
corresponds to the subcarrier with the strongest
received signal power. Mathematically, the estimation
can be expressed as
ˆ ,| |22 bd
* mYm
'*
m
whereargmax.2
m
Ym
(12)
If the codebook diversity is R, the power ||
bd 2 is
averaged across all R subcarriers m mapped into the
same direction before angle estimation.
The estimation accuracy can be improved by
Figure 10. resulting ttd codebook, assuming Nr = 16,
D = 16, M = 16. subcarriersfmm, ∀ , are associated with
their corresponding anglesm m, .
14
IEEE cIrcuIts and systEms magazInE
designing a high-resolution dictionary-based algorithm
that relies on the codebook with diversity order
R [76]. The average received signal power in all probed
Fourth quartEr 2022
, where the
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