FFT-Spread Faster-Than-Nyquist Signaling inFrequency-Selective Fading Channel
Takumi Ishihara and Shinya Sugiura∗
Institute of Industrial Science, The University of Tokyo
Meguro-Ku, Tokyo 153-8505, Japan
Email: {t.ishihara@ieee.org, sugiura@iis.u-tokyo.ac.jp}

Abstract—In this paper, we propose novel reduced-complexity
fast Fourier transform (FFT)-spread faster-than-Nyquist (FTN)
signaling with optimal power allocation for a frequency-selective
fading channel. The information rate of the proposed FTN
signaling is approximately derived by relying on the circulant approximation of the FTN-specific intersymbol interference
matrix and noise covariance matrix. This allows us to constitute efficient calculations of precoding and weighting matrices.
The power allocation coefficients are optimized such that theapproximated information rate is maximized. Our simulation
results demonstrate that the proposed scheme achieves the bit
error ratio performance close to the conventional eigenvaluedecomposition (EVD)-precoded FTN signaling counterpart thatis optimal in terms of an achievable information rate while
significantly reducing the computational complexity as low as
the order of O(N log N ).

I. I NTRODUCTION
Conventional wireless communication systems are designed
based on the Nyquist criterion, which defines a minimum
symbol interval by T0 = 1/(2W ), where 2W represents
the bandwidth of an ideal rectangular shaping filter, andthe associated maximum symbol rate is 1/T0 . The excess
bandwidth of 2W (1+β) is induced when employing a realisticroot raised cosine (RRC) shaping filter having a roll-off factor
β > 0. Hence, the achievable spectral efficiency of Nyquist
signaling based on an RRC shaping filter becomes (1 + β)times lower than that of an idealistic rectangular shaping filter(β = 0). To overcome this limitation, the concept of fasterthan-Nyquist (FTN) signaling has been explored for more than
50 years [1–3]. In FTN signaling, a symbol interval is set
lower than that of Nyquist signaling, i.e., T = τ T0 , whereτ (0 < τ ≤ 1) represents a symbol packing ratio, hence
achieving a higher symbol rate than the Nyquist signaling
counterpart, which is, naturally, achieved at the cost of the
detrimental intersymbol interference (ISI) effects.
To efficiently demodulate ISI-contaminated FTN symbols,
several time-domain (TD) receivers have been developed [4–7]. In [4], the TD trellis-based iterative decoding algorithm was
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first developed. In [5], the TD equalizer based on the Bahl,
Cocke, Jelinek, and Raviv (BCJR) algorithm was presented.
Also, Prlja et al. in [6] conceived the reduced-state M algorithm-based BCJR (M -BCJR) algorithm to further reducethe decoding complexity. Moreover, in [7], the modified M BCJR algorithm was proposed for the sake of striking aflexible balance between the detection performance and complexity. However, the computational complexity of TD trellisbased detectors significantly increases upon increasing theeffective tap length and the constellation size, which becomesprohibitively high, especially in a realistic frequency-selective
fading channel.
By contrast, low-complexity frequency-domain (FD) FTN
receivers have been developed [8–10]. In [8], the singlecarrier FD equalization (FDE) algorithm with the aid of
cyclic prefix (CP) was first applied to FTN signaling, where
the FD weights were derived based on the minimum-meansquare-error (MMSE) criterion. Owing to the efficient fast
Fourier transform (FFT)-based implementation, practical lowcomplexity detection was achieved even in a highly dispersive
frequency-selective fading channel. Moreover, in [9], the harddecision MMSE-FDE algorithm in [8] was extended to its
soft-decision (SoD) counterpart to support iterative detection.
In [10], the low-complexity FDE algorithm, capable of joint
channel estimation and data detection, was presented for FTN
signaling in a frequency-selective fading channel.
Precoded FTN signaling relying on the matrix factorization
was proposed [11–14]. In [11], the capacity of FTN signaling
was derived based on the eigenvalue decomposition (EVD) ofan FTN-induced ISI matrix, showing that the capacity of EVDprecoded FTN signaling without power allocation is identicalto that of unprecoded FTN signaling in [15]. Furthermore,
optimal power allocation was introduced to EVD-precoded
FTN signaling [12, 16], which achieves a higher capacity thanunprecoded FTN signaling and classic Nyquist signaling whenemploying the RRC shaping filter having β > 0 for eachscheme [3, 14, 16]. Also, the capacity of EVD-precoded FTN
signaling with optimal power allocation is maximized when
τ = 1/(1 + β), which achieves the spectral efficiency equal tothat of Nyquist signaling employing an idealistic rectangular
shaping filter (β = 0) [3, 14, 16]. Note that optimal power allocation of EVD-precoded FTN signaling [12] is implementablewithout elaborate symbol truncation only for 1/(1 + β) ≤ τ ≤1 [3, 14, 16]. Furthermore, in [14], EVD-precoded FTN signaling with optimal power allocation designed for an AWGN
channel [12] was extended to that supporting a frequencyselective fading channel, and the associated information rate
was derived. However, the computational complexity of the
EVD-precoded FTN signaling scheme [14] is as prohibitively
high as O(N 3 ) per block transmission.
The novel contribution of this paper is that we propose novel
reduced-complexity FFT-spread FTN signaling with optimal
power allocation for a frequency-selective fading channel.
Owing to the low-complexity FFT-based implementation, thecomputational complexity order of the proposed FTN transmitter and receiver is as low as O(N log2 N ) per block. Morespecifically, the ISI matrix and the correlated noise covariancematrix, which are specific to received FTN signals, are approximated by circulant matrices. This allows us to decompose thereceived FTN signal into N independent parallel substreams
with the aid of FFT-based factorization, hence enabling simplified symbol-by-symbol demodulation. In our simulations, it
is demonstrated that the proposed scheme achieves the BER
performance close to that of the conventional EVD-precoded
FTN signaling counterpart [14], which is highly complex but
optimal in terms of an information rate.
II. P ROPOSED R EDUCED -C OMPLEXITY FFT- SPREAD FTN
S IGNALING WITH O PTIMAL P OWER A LLOCATIONIn this section, we propose novel reduced-complexity FFTspread FTN signaling with optimal power allocation for afrequency-selective fading channel. The block diagram of the
proposed FTN system is shown in Fig. 1. At the transmitter,
information bits are modulated onto N -length complex-valued
symbols s = [s0 , · · · , sN −1 ]T ∈ CN , where E[|sk |2 ] =σs2 (k = 0, · · · , N − 1), and E[·] is the expectation operation.
Similar to conventional FTN signaling of [8, 10], the 2ν-length
CP symbols are added to the precoded symbols x as follows:
xcp = [xcp,0 , · · · , xcp,N +2ν−1 ] ∈ CN +2ν

(1)= Acp x(2)

= Acp Ps,

(3)

where

[
Acp =

02ν×(N −2ν)
IN

I2ν

]
∈ R(N +2ν)×N ,(4)

and 02ν×(N −2ν) denotes the (2ν×(N −2ν))-sized zero matrix.
Furthermore, P ∈ CN ×N is a precoding matrix. The CP-added
FTN signal xcp (t) is given by∑(5)
xcp (t) =xcp,n h(t − nT ),nAt the receiver, the received signal after matched-filtering
is represented byycp (t) =

L−1
∑∑
l=0

n

hl xcp,n g(t − (l + n)T ) + η(t),(6)

By removing the 2ν samples associated with CP from thereceived sample block y = [ycp (0), ycp (T ), · · · , ycp (N + 2ν −1)]T , the N -length received block are obtained asr = [r0 , · · · , rN −1 ]T ∈ CN

(7)

= [ycp (νT ), · · · , ycp ((N + ν − 1)T )](8)= Rcp Gh,N +2ν Acp x + η,

(9)

T

where
Rcp = [0N ×ν IN 0N ×ν ] ∈ RN ×(N +2ν) ,(10)

while Gh,N +2ν is the ((N + 2ν) × (N + 2ν))-sized ISI matrix
whose kth-row and mth column entry is given by
Gh (k, m) =L−1∑

hl g(kT − (m + l)T ).

(11)

l=0

Furthermore, by assuming the condition ofg(kT ) = 0 for |k| > ν,(12)(9) is approximated by [8–10]
r ≃ Gc x + η,

(13)

where the matrix Gc ∈ CN ×N is the circulant matrix whose
EVD is given by
Gc = QH ΘQ.

(14)Furthermore, Q ∈ CN ×N in (14) represents the normalized
discrete Fourier transform (DFT)√matrix whose kth-row andmth-column entry is given by (1/ N ) exp[−2πj(k − 1)(m −1)/N ]. Additionally, Θ = diag{θ0 , · · · , θN −1 } is a diagonal
matrix having the diagonal elements calculated by the FFT ofthe first column of Gc [8, 10].
A. Approximated Mutual Information
Mutual information between the received block r and thetransmit symbols x is formulated byI(x; r) = he (r) − he (r|x)(15)= he (r) − he (η),(16)

where he (·) represents a differential entropy. The upper boundsof the differential entropies of r and η are given by [17])(he (r) ≤ log2 (πe)N E[rrH ] det
(17)
(
)
H
N
he (η) ≤ log2 (πe) E[ηη ] det ,(18)
where | · |det denotes the determinant operation. For an instantaneous fading channel, the covariance matrix of the received
block is approximately given by
E[rrH ] ≃ E[(Gc x + η)(Gc x + η)H ](19)

H= Gc E[xx ]GHc + E[ηη ]= Gc Rx GH
c + N0 G,

(20)

H

(21)

where we have Rx = E[xxH ].
As shown in [18], for the limit of N → ∞, a banded

Transmitter
SourceBit loading

PSK/QAM
modulation

OptimizationofOptimal power
allocation by

IFFT

CP
insertion

ShapingfilterFrequency-selective
channelSymbol-by-symbol
demodulationCP
removing

FFT

Sampling at

MatchedfilterReceiverFig. 1. System model of the proposed reduced-complexity FFT-spread FTN signaling scheme with optimal power allocation for a channel-uncoded scenario.

Toeplitz matrix T ∈ CN ×N , whose first column andfirst row are given by [t0 , t1 , · · · , tm , 0, · · · , 0]T and[t0 , t−1 , · · · , t−m , 0, · · · , 0], respectively, is asymptoticallyequivalent to a circulant matrix C ∈ CN ×N , having thefirst column of [t0 , t1 , · · · , tm , 0, · · · , 0, t−m , · · · , t−1 ]T .
Hence, by assuming sufficiently large N and (12), the noise
matrix G, having the Toeplitz structure, is approximatedby a circulant matrix whose first column is given by g =[g(0), g(T ), · · · , g(mT ), 0, · · · , 0, g(−mT ), · · · , g(−T )]T(0 ≤ m ≤ ⌈N/2 − 1⌉). In this paper, we assume therelationship of m = ⌈N/2 − 1⌉. This allows us to factorize
G based on the FFT as follows:G ≃ QH ΨQ,(22)

Γ = diag[γ0 , · · · , γN −1 ] ∈ RN ×N be a real-valued diagonal
matrix, (28) is maximized when the following equality holds:
QRx QH = σs2 Γ⇔ x = Q Γ s,−1
where the relationships of |A|−1|det and |IN +det = |A
AB|det = |IN + BA|det are exploited.
Then, according to Hadamard’s inequality [17, 19], theupper bound of the approximated mutual information (28)11
is maximized when Ψ− 2 ΘQRx QH ΘH Ψ− 2 is a diagonal
matrix, i.e., when QRx QH is a diagonal matrix. By letting(31)

σs2 IN .
1

P = QH Γ 2(32)

By substituting (31) into (28), the upper bound of the approximated mutual information is maximized as follows:1
σs2 − 1Ψ 2 ΘΓΘH Ψ− 2
N0
det
(
)
N
−1
∑
σs2
log2 1 +=γk θk2 ψk−1 .
N0I(x; y) ≤ log2 IN +where we have Ψ = diag[ψ0 , · · · , ψN −1 ] ∈ R, whosediagonal entries are given by the FFT of g.
Under the assumption of 1/(1 + β) < τ ≤ 1, the upperbound of mutual information I(x; r) is approximated by(πe)N E[rrH ] det(23)(πe)N |E[ηη H ]|det
Gc Rx GHc + N0 G det≃ log2(24)|N0 G|det
1−1= log2 IN +Gc Rx GH
(25)
c G
N0
det
1
H −1
≃ log2 IN +Gc Rx GH
Q
(26)
c Q Ψ
N0
det
1 H
= log2 IN +Q ΘQRx QH ΘH Ψ−1 Q (27)
N0
det
1
1 −1
H
H
−
Ψ 2 ΘQRx Q Θ Ψ 2= log2 IN +
,
N0
det
(28)

(30)

where we assume E[ss ] =Hence, the precoding
matrix to be optimized is rewritten by
H

N ×N

I(x; r) ≤ log2

(29)

Rx = σs2 QH ΓQ
H 21

(33)
(34)

k=0

B. Optimal Power Allocation
The coefficients γk are optimized to maximize the r.h.s of(34) based on the Lagrange multiplier method. The transmit
1energy of the CP-added symbols xcp = Acp QH Γ 2 s is
calculated by [14][]E = E xH
(35)
cp GN +2ν xcp= E[tr{xH
cp GN +2ν xcp }](36)= E[tr{GN +2ν xcp xH
cp }]
1
H 12
= tr{GN +2ν Acp Q Γ E[ssH ]Γ 2 QATcp }= σs2 tr{GN +2ν Acp QH ΓQATcp }= σs2 tr{ΓQATcp GN +2ν Acp QH }= σs2 tr{ΓΦ}
N
−1
∑
= σs2
γk ϕk ,

(37)
(38)
(39)
(40)
(41)
(42)

k=0

where
Φ = QATcp GN +2ν Acp QH ,(43)and GN +2ν
∈R(N +2ν)×(N +2ν) represents the
Toeplitz matrix whose first row and first column
are given by [g(0), g(−T ), · · · , g(−(N + 2ν)T )] and[g(0), g(T ), · · · , g((N + 2ν)T )]T , respectively. Moreover,
ϕk represents the kth diagonal element of Φ,1 and tr{·}represents the trace operation. If the precoding matrix be
P = IN , the average transmit energy is given by [14]E = N σs2 .

(44)

To maintain the transmit energy per block to be unchanged
regardless of P, the energy constraint of E = N σs2 is imposed.
With (44) and (42), we arrive at the energy constraint ofN−1∑

γk ϕk = N,

(45)

k=0

With (34) and (45), the Lagrange function is formulated by
(N −1
)
(
)
N
−1
∑
∑
σs2
2 −1
J=
log2 1 +γk θk ψk−αγk ϕk − N ,
N0
k=0

k=0

(46)
where α is the Lagrange multiplier. To maximize (46), thefollowing problem is solved [12]:∂J= 0, subject to γk ≥ 0.
∂γk
Finally, the optimal coefficients γk are given by
(
)
1
ψk N0
γk = max− 2 2 ,0 ,αϕk ln 2
θ k σs(47)(48)

which is calculated in the same manner as the classic waterfilling algorithm [14, 20].
C. Detection Algorithm
In this section, the detection algorithm of the proposed
FFT-spread FTN signaling scheme is presented. Let us set
the weight matrix at the receiver to W = Q, and then theapproximated received block is rewritten byrd ≃ QGc x + Qη(49)
1

= QQH ΘQQH Γ 2 s + ηq
1
2

= ΘΓ s + ηq ,

(50)
(51)

where ηq = Qη. The covariance matrix of the noise components ηq is approximately calculated byE[ηq ηqH ] = QE[ηη H ]QH
H(52)= N0 QGQ(53)≃ N0 QQH ΨQQH

(54)

= N0 Ψ.

(55)

1 Note that Φ is calculated offline since the matrices Q, A , and G
cp
N +2ν
do not include channel state information (CSI), hence known in advance oftransmission.

As shown in (51) and (55), the received block and thenoise correlation matrix are approximately diagonalized. This
allows us to carry out efficient sub-optimal symbol-by-symbol
demodulation while reducing the effects of the correlated
noises.
In the proposed FFT-spread FTN signaling, the matrix G
is approximated by QH ΨQ, where the approximation may
cause the ill-conditioned problem for around the boundary ofτ = 1/(1 + β). Our extensive simulations revealed that forτ = 1/(1+β), the approximated eigenvalues may be negative,
depending on m. To avoid this, in this paper, we impose further
limitation on the packing ratio τ , i.e., 1/(1+β)+δ ≤ τ ≤ 1, toensure all the positive eigenvalues, where δ ∈ R is a predefined
small value. In our simulations, δ was set as 0.02.
In our proposed FFT-spread FTN signaling with optimal
power allocation, the eigenvalues θk and ϕk are the FFT
coefficients, while the multiplications of the precoding matrix
1
P = QH Γ 2 and the weighting matrix W = Q are efficiently
computed by the IFFT and FFT, respectively. Furthermore, thepower allocation coefficients (48) are computed by the orderof O(N ). Thus, the computational complexity of the proposed
FFT-spread FTN transceiver is given by O(N log N ).
III. S IMULATION R ESULTS
In this section, we provide the simulation results to investigate the BER performance of the proposed FFT-spread
FTN signaling scheme. We considered the frequency-selective
Rayleigh block fading channel with the L-tap, and the channel
coefficients hl were randomly generated according to thecomplex-valued Gaussian distribution of CN (0, 1/L). The
basic system parameters of N = 1024 and β = 0.25 were
set throughout the simulations.
In our BER calculations, we considered the three-stage
serially-concatenated turbo coding architecture of [14], where
the half-rate recursive systematic coding (RSC) encoder having the constraint length of 2 and octal generator polynomial(3, 2) was employed as the outer encoder, while the unity
rate coding encoder was used as the inner encoder [21]. The
number of outer decoding iteration and that of inner decoding
iteration were set to 40 and 2, respectively [14]. Moreover,
the target interleaver length was set to around 200000 bits.2
Similar to conventional EVD-precoded FTN signaling [14],the log-likelihood ratio (LLR) of the received symbols is
efficiently calculated since the equivalent channel matrix andthe noise covariance matrix are approximately diagonalized.
The transmission rate of the proposed half-rate turboencoded FFT-spread FTN signaling system is given by
N −1

Rt =

∑
1
1
1
·
·
·
bk [bps/Hz],(56)2 2W (1 + β) (N + 2ν)τ T0
k=0

where the coefficient 1/2 represents the coding rate of the
RSC code, and bk denotes the number of bits assigned onto
2 Similar to [14], multiple transmit blocks having the N symbols wereconcatenated by ignoring the detrimental IBI effects for simplicity.

Nyquist (τ = 1), β = 0.25, with power allocation
FTN (τ = 0.9), β = 0.25, with power allocation
FTN (τ = 0.82), β = 0.25, with power allocation

FTN (τ = 0.9), β = 0.25, with power allocation
FTN (τ = 0.9), β = 0.25, w/o power allocation

100

100

10-1

10-1

1 [bps/Hz]

BER

BER

1 [bps/Hz]
2 [bps/Hz]

10-2

2 [bps/Hz]

10-2

10-3

10-3

10-4

10-4
0

1

2

3

4

5

6

7

8

9

10 11

12

SNRSNR( [dB]) [dB]
Fig. 2. BER performance of the proposed FTN signaling scheme with optimal
power allocation. The roll-off factor was fixed to β = 0.25, while the packing
ratio was given by τ = 0.9 and 0.82. The other system parameters were set
to N = 1024, L = 20, and ν = 30, and the target transmission rate was set
to Rt = 1 and 2.

the kth symbol. The part of the symbols is deactivated when
the associated power allocation coefficients were γk = 0 as
an optimization result of (48). Hence, the number of bits bkchange depending on CSI to maintain the target transmissionrate, which is referred to as bit-loading in this paper.3
Fig. 2 shows the BERs of the three-stage turbo-coded FFTspread FTN signaling scheme with optimal power allocation,
where the packing ratio was given by τ = 0.9 and 0.82, andother system parameters were given by L = 20, and ν = 30.
The target transmission rate was set to Rt = 1 and 2. For
comparison, the BER curves of Nyquist signaling counterpart(τ = 1) employing the same RRC shaping filter were plotted,
where we set 2ν = L = 20 [22, 23]. Observe in Fig. 2that the proposed scheme achieved better BER performance
than its Nyquist signaling counterpart, and the proposed FTN
scheme with τ = 0.82 achieved the best BER performance.
More specifically, upon increasing the target transmission ratefrom Rt = 1 to 2, the performance advantage of the proposed
scheme with τ = 0.82 over the Nyquist signaling counterpart
increased from 1.2 dB to 1.8 dB.
Fig. 3 shows the BERs of the proposed schemes with andwithout optimal power allocation. For the proposed schemewithout power allocation, we set γk = 1 (k = 0, · · · , N − 1)which dispenses with CSI for precoding, unlike the proposed
3 In our simulations, binary phase-shift keying (BPSK) was used for all
the activated symbols. Then, quadrature PSK (QPSK) was used to replace
the BPSK-modulated symbols from the first activated symbol to the last
one until the target transmission rate was achieved. The same operation was
repeated using 16-point quadrature amplitude modulation (QAM), 64-QAM,
256-QAM, and 1024-QAM until the target transmission rate was achieved.

0

1

2

3

4

5

6

7

8

9

10 11

12

SNR
( [dB] ) [dB]
SNR
Fig. 3. BER performance of the proposed FTN signaling schemes with andwithout optimal power allocation. The system parameters were set to β =0.25, τ = 0.9, N = 1024, L = 20, and ν = 30. The target transmission
rate was set to Rt = 1 and 2.

scheme with power allocation. This implies that the proposed
scheme without power allocation is the open-loop system,
while the proposed scheme with power allocation is the closedloop one. The packing ratio was set to τ = 0.9, and othersystem parameters were the same as those used in Fig. 2. As
shown in Fig. 3, our optimal power allocation allows us tohave the better BER performance, where the gap decreased
upon increasing the transmission rate. More specifically, thegaps were 0.9 dB and 0.4 dB for Rt = 1 and 2, respectively.
Furthermore, Fig. 4 shows the BER comparisons between
the proposed scheme with and without optimal power allocation, the EVD-precoded FTN signaling scheme with optimal
power allocation [14], and the open-loop SoD-FDE-aided FTN
signaling scheme with soft cancellation [10]. Note that thecomputational complexity in the proposed scheme is as low asO(N log N ), while those of the EVD-precoded FTN signaling
scheme and the open-loop SoD-FDE-aided FTN signaling
scheme with soft cancellation are O(N 3 ) and O(N 2 ), respectively. The system parameters were set to τ = 0.82, L = 20,
and ν = 30. The target transmission rate was set to Rt = 0.461
for each scheme. In Fig. 4, it was observed that the proposed
scheme with optimal power allocation outperformed two otheropen-loop schemes, i.e., SoD-FDE-aided unprecoded FTN
signaling scheme [10] and the proposed FTN scheme without
power allocation, where the associated performance gaps were
2.1 dB and 3.2 dB, respectively. Naturally, the EVD-precoded
FTN signaling with optimal power allocation [14] exhibitedthe best performance owing to the optimality in terms of anachievable information rate, which is achieved at the costof prohibitively high computational complexity. The penaltyof the proposed scheme with optimal power allocation over

R EFERENCES

Proposed FFT-spread FTN signaling with power allocation
Proposed FFT-spread FTN signaling w/o power allocation
EVD-precoded FTN signaling with power allocation
SoD-FDE-aided unprecoded FTN signaling

100

BER

10-1

10-2

10-3

10-4
-4

-3

-2

-1

0

1

2

SNRSNR( [dB]) [dB]Fig. 4. BER comparisons between the proposed FTN signaling schemeswith and without optimal power allocation, the EVD-precoded FTN signaling
scheme with optimal power allocation [14], and the open-loop SoD-FDE-aided
FTN signaling scheme with soft cancellation [10]. The system parameters
were set to β = 0.25, τ = 0.82, N = 1024, L = 20, ν = 30, andRt = 0.461.

the EVD-precoded FTN signaling scheme with optimal power
allocation [14] is imposed due to the circulant matrix approximation and the power penalty caused by the CP insertion. Notethat a sufficiently long guard interval has to be added for the
EVD-precoded FTN signaling benchmark [14] to eliminate thedetrimental IBI effects, while it was ignored in our simulations.
IV. C ONCLUSIONSIn this paper, we proposed the novel reduced-complexity
FFT-spread FTN signaling with optimal power allocation for afrequency-selective fading channel. Under the circulant matrix
approximation of the equivalent channel matrix and the noise
covariance matrix, the approximated information rate of the
proposed scheme was formulated. Then, the optimal power
allocation coefficients were derived by maximizing the approximated information rate. Our simulation results demonstrated
that the proposed FTN signaling scheme achieved the BER
performance close to the EVD-precoded FTN signaling bound,which is optimal in terms of an information rate while reducing
the transceiver’s computational complexity imposed by thecalculations of precoding and weighting matrices.
ACKNOWLEDGEMENT
The work of T. Ishihara was supported in part by theJapan Society for the Promotion of Science (JSPS) KAKENHI (Grant Number 20K22410). The work of S. Sugiura
was supported in part by the Japan Science and Technology
Agency (JST) Precursory Research for Embryonic Science andTechnology (PRESTO) (Grant Number JPMJPR1933).

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