LOW COMPLEXITY EQUALIZATION FOR AFDM IN DOUBLY DISPERSIVE CHANNELS
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LOW COMPLEXITY EQUALIZATION FOR AFDM IN DOUBLY DISPERSIVE CHANNELS
Ali Bemani1 , Nassar Ksairi2 , Marios Kountouris1
Communication Systems Department, EURECOM, Sophia Antipolis, France
1
Mathematical and Algorithmic Sciences Lab, Huawei France R&D, Paris, France
2
Email: ali.bemani@eurecom.fr, nassar.ksairi@huawei.com, marios.kountouris@eurecom.fr
to achieve full diversity in doubly dispersive channels [1] as
arXiv:2203.01875v2 [cs.IT] 7 Mar 2022
ABSTRACT
Affine Frequency Division Multiplexing (AFDM), which opposed to existing chirp-based waveforms, for instance [3,
is based on discrete affine Fourier transform (DAFT), has 4]. Furthermore, AFDM has similar performance in terms
recently been proposed for reliable communication in high- of bit error rate (BER) with orthogonal time frequency space
mobility scenarios. Two low complexity detectors for AFDM (OTFS) [5]. However, AFDM outperforms OTFS in terms of
are introduced here. Approximating the channel matrix as pilot overhead and multiuser multiplexing overhead [2, 6].
a band matrix via placing null symbols in the AFDM frame In this paper, we propose two low complexity detection
in the DAFT domain, a low complexity MMSE detection is algorithms for AFDM taking advantage of its inherent chan-
proposed by means of the LDL factorization. Furthermore, nel sparsity. By placing some null symbols - zero padding the
exploiting the sparsity of the channel matrix, we propose a AFDM frame - in the DAFT domain, the channel matrix can
low complexity iterative decision feedback equalizer (DFE) be approximated as a band matrix. Using the approximated
based on weighted maximal ratio combining (MRC), which band matrix and inspired by the equalizer in [7] for OFDM
extracts and combines the received multipath components systems, we first design a low complexity MMSE detector
of the transmitted symbols in the DAFT domain. Simula- based on LDL factorization [8]. The overall complexity of the
tion results show that the proposed detectors have similar proposed algorithm is linear in the number of subcarriers and
performance, while weighted MRC-based DFE has lower quadratic in the bandwidth of the band matrix, which in turn
complexity than band-matrix-approximation LMMSE when depends on the the maximum delay and maximum Doppler
the channel impulse response has gaps. shift. Second, we propose a low complexity iterative decision
feedback equalizer (DFE) based on weighted maximal ratio
Index Terms— AFDM, affine Fourier transform, doubly combining (MRC) of the channel impaired input symbols re-
dispersive channels, detector, MMSE, DFE, MRC. ceived from different paths. The overall complexity of the
second algorithm is also linear in the number of subcarriers
1. INTRODUCTION and quadratic in the number of paths. We show that these two
Next-generation wireless systems (e.g., B5G/6G) are evolv- detectors have similar performance between them, whereas
ing to cater to a wide range of applications and services re- when the channel is sparse in the delay domain, weighted
quiring reliable communication in high-mobility scenarios. MRC-based DFE detector exhibits lower complexity.
This calls for new waveform design able to cope with time- 2. SYSTEM MODEL
varying channels. In this setting, existing waveforms, in par-
The AFDM block diagram is given in Fig. 1. Modulation
ticular orthogonal frequency division multiplexing (OFDM),
is produced by using DAFT at the transmitter and receiver.
lose subcarrier orthogonality, thus resulting in inter-carrier in-
DAFT is a discretized version of AFT [1, 9–11] and its kernel
terference and deteriorated system performance. 2 1 2
is equal to e−ı2π(c2 m + N mn+c1 n ) where c1 and c2 are the
Affine frequency division multiplexing (AFDM) has re-
AFDM parameters tuned to provide full delay-Doppler rep-
cently been proposed as a promising waveform for commu-
resentation of the channel in the DAFT domain. It has been
nication in time-varying channels [1, 2] showing significant
shown that tuning c1 using the the maximum Doppler shift
performance gains over OFDM. AFDM employs multiple or-
normalized with respect to the subcarrier spacing, and setting
thogonal information-bearing chirps generated using the dis-
c2 to be an arbitrary irrational number or a rational number
crete affine Fourier transform (DAFT). A key feature is that
sufficiently smaller than 1/2N , enables AFDM to achieve full
its chirp pulse parameters can be adapted to the channel char-
diversity in doubly dispersive channels [1].
acteristics, making a complete delay-Doppler representation
of the channel in the DAFT domain. This enables AFDM 2.1. Modulation and Demodulation
This work has been supported by a Huawei France-funded Chair towards Consider a set of quadrature amplitude modulation (QAM)
Future Wireless Networks. symbols xk , k = 0, 1, 2, ..., N − 1. AFDM maps xk to snModulation Demodulation Heff
0 x
0 x
0 x
Q+1
0 x
d x
Add Remove
S\P ΛH IFFT ΛH P\S S\P Λc1 FFT Λc2 P\S d x
c2 c1
CPP CPP
d x
d x
d x
d x
d x
d x
=
d x
Fig. 1: AFDM modulation/demodulation block diagram. d
d x
x
x
d x
using inverse DAFT (IDAFT) as follows: d
d
x
x
d x
N −1 d x
1 X 2 1 2 d x
sn = √ xm eı2π(c2 m + N mn+c1 n ) , n = 0, · · · , N − 1. d x
N m=0 d x
d x
(1) d x
To make the channel lie in a periodic domain, a chirp- 0 x
periodic prefix (CPP) should be added to the modulated Fig. 2: Truncated parts of x and Heff
signal, defined as
sn = sN +n e−ı2πc1 (N
2
+2N n)
, n = −M, · · · , −1 (2) sparse structure i.e., there are exactly L = P non-zero en-
tries in each row and column if νi is integer and if we set
PP
where M is any integer greater than or equal to the value in c1 = 2νmax
2N
+1
. Indeed, in this case Heff = i=1 hi Hi where
samples of the maximum delay spread of the wireless chan-
2 2
2π −p2 ))
(
nel. After transmission over the channel, the received samples eı N (N c1 li −qli +N c2 (q q = (p + loci )N
Hi (p, q) =
are X∞ 0 otherwise
rn = sn−l gn (l) + wn (3) (7)
l=0 where loci = (νi + (2νmax + 1)li )N [1]. For fractional
where wn ∼ CN (0, N0 ) is an additive Gaussian noise and νi , it can be shown that Hi has in each row and column a
gn (l) is the impulse response of the time-varying channel at peak surrounded by approximately 2kν non-zero entries de-
time n and delay l, given by creasing rapidly as we move away from it. Hence, if we set
P
X c1 = 2(bνmax2N c+kν )+1
, Heff can be approximated as having
gn (l) = hi e−ı2πfi n δ(l − li ) (4) L = (2kν + 1)P non-zero entries per row and column (i.e.,
i=1
each received symbol can be approximately expressed as a
where P ≥ 1 is the number of paths, δ(·) is the Dirac delta linear combination of only a few input symbols). We pro-
function, and hi , fi and li are the complex gain, Doppler shift pose below two low complexity detection algorithms leverag-
(in digital frequencies), and the integer delay associated with ing the channel matrix sparsity.
the i-th path, respectively. We define νi , N fi , where νi ∈
[−νmax , νmax ] is the Doppler shift normalized with respect to 3. DETECTION ALGORITHMS
the subcarrier spacing. We assume that the maximum delay
of the channel satisfies lmax , max(li ) < N . The first step is to place some null symbols that allow to ap-
The DAFT domain output symbols are obtained by proximate the truncated part of Heff as a band matrix. This
N −1
also simplifies the input-output relation as the modular oper-
1 X 2 1 2 ation is no longer needed, as shown in Fig. 2. Note that these
ym = rn e−ı2π(c2 m + N mn+c1 n ) . (5)
N n=0 symbols do not entail extra overhead as they can serve not
only the proposed detection algorithms but also embedded pi-
Discarding the CPP, the input-output relation can be written lot aided channel estimation. Due to the structure of Heff and
in matrix form as Hi , the number of the null guard symbols should be greater
than Q = (lmax + 1)(2(αmax + kν ) + 1) − 1. Taking into ac-
y =Ar = Heff x + Aw (6) count the zero padding, the vector of DAFT domain received
where A = Λc2 FΛc1 is the DAFT matrix, F is the discrete samples writes as
√ y = Heff x + w e (8)
Fourier transform (DFT) matrix with entries e−ı2πmn/N / N ,
2
Λc = diag(e−ı2πcn , n = 0, 1, . . . , N −1), Heff = AHAH , where x and Heff are the truncated parts of x and Heff , re-
and H is the matrix representation of the channel. The ele- spectively (see Fig. 2). They can be expressed using the ma-
ments of y, r, x and w ∼ CN (0, N0 I) are similarly related trix T = [IN ]Q−(αmax +kν ):N −(αmax +kν )−1,: as x = Tx and
to yk , rk , xk and wk , respectively. Since A is a unitary Heff = Heff TH . Using LMMSE equalization based on (8)
matrix, we = Aw and w have the same statistics. Heff has for detection requires O(N 3 ) flops, which can be prohibitivefor large N . We thus propose two detectors with lower com- Transmitted symbols: x0 x1 x2 x3 x4 x5
h02 h13 h24 h35 h46 h57
plexity. The first is a low-complexity LMMSE based on a
h01 h12 h23 h34 h45 h56
band approximation of Heff . The second is a weighted MRC- h00 h11 h22 h33 h44 h55
based DFE exploiting the sparse representation of the com- Received symbols: y0 y1 y2 y3 y4 y5 y6 y7
munication channel provided by AFDM. b20 b21 b22 b23 b24 b25
b10 b11 b12 b13 b14 b15
3.1. Low complexity MMSE detection b00 b01 b02 b03 b04 b05
Detection: W-MRC W-MRC W-MRC W-MRC W-MRC W-MRC
To recover the data symbols x, considering (8), the following
MMSE equalization is used
x̂ = HH H
eff (Heff Heff + N0 IN )
−1
y. (9) Estimated symbols: x̂0 x̂1 x̂2 x̂3 x̂4 x̂5
Although (9) involves matrix inversion, the matrix M = Fig. 3: Weighted MRC operation for N = 8 with a 3-path
Heff HH eff + N0 IN is a Hermitian band matrix with lower channel with Q = 2 where W-MRC stands for weighted MRC
and upper bandwidth Q. Thus, M−1 can be computed us- and hji = Heff (i, j).
ing LDL factorization. Algorithm 1 can be performed to
efficiently equalize the received signal. The computational for the combining. Considering the structure of Heff , it can
Algorithm 1: Low complexity MMSE detection be seen that each received symbol yk is given by
1 Construct the matrix Heff = Heff TH L−1
Construct the band matrix M = Heff HH
X
2 eff + N0 IN yk = Heff (k, pik )xpik (10)
3 Compute the LDL factorization of M = LDLH i=0
where L is a lower triangular matrix with Q sub
where pik is the column index of the i-th path coefficient in
diagonals and D is a diagonal matrix
row k of matrix Heff . Let bik be the channel impaired in-
4 Solve the triangular system Lf = y
put symbol xk in the received samples yqki after canceling
5 Solve the diagonal system Dg = f
6 Solve the triangular system LH d = g the interference from other input symbols, where qki is the
row index of the i-th path coefficient in column k of ma-
7 Calculate x̂ = HHeff d
trix Heff . In each iteration, assuming estimates of the in-
cost of the proposed detection algorithm is evaluated in terms put symbols xk are available either from the current iteration
of complex additions (CAs), complex multiplications (CMs) (for pjqi < k, j = 0, ..., L − 1) or previous iteration (for
k
and complex divisions (CDs). The first step does not need pjqi > k, j = 0, ..., L − 1), bik can be written as
any complex operation since Heff is truncated from Heff . k
In step 2, every element of Heff HHeff requires at most Q + 1
j (n)
X
bik = yqki − HH i
eff (qk , pq i )x̂pj
CMs and Q CAs. Considering that Heff HH eff is Hermitian and pj i k
k
qi
k
2 (Q + 3Q)N CMs, 2 (Q + Q)N CAs, and QN CDs. Steps
1 2 1 2 q
k
4 and 6 can be solved by band forward and backward sub- where superscript (n) denotes the n-th iteration. Considering
stitutions [8] and each of them has QN CMs and QN CAs. bik for all paths i = 0, 1, ..., L − 1, weighted MRC (as op-
Step 5 can be solved using N CDs since D is a diagonal posed to pure MRC [12]) can be performed. The output of
matrix and the last step requires (Q + 1)N CMs and QN the weighted MRC for estimating xk is given by
CAs. Thus, the algorithm requires (Q2 + 6Q + 2)N CMs,
gk
(Q2 + 4Q + 1)N CAs and (Q + 1)N CDs, which amounts ck = (12)
to (2Q2 + 11Q + 4)N complex operations in total. dk + γ −1
3.2. Weighted MRC-based DFE detection where L−1
X
gk , HH i i
eff (qk , k)bk , (13)
As mentioned in Section 2, Heff has L non-zero entries per
i=0
column. This feature enables us to propose a weighted MRC-
based detector where each data symbol is detected from the L−1
X
weighted MRC of its L channel-impaired received copies. d, |HH i
eff (qk , k)|
2
(14)
Fig. 3 shows an example of this detector for AFDM with i=0
N = 8 and a 3-path channel with Q = 2. The proposed and γ is the signal-to-noise ratio (SNR). Let D(.) denote the
detector is iterative, where in each iteration, the estimated in- decision on the symbol estimate ck , i.e, x̂nk = D(ck ). In this
ter symbol interference is canceled in the branches selected paper, we consider x̂nk = ck . The estimated symbols are then35
used for the next iteration. The algorithm continues until the 10 -2
maximum number of iterations is reached or the updated input
30
Average number of iteration
symbol vector is close enough to the previous one as summa-
25
rized in Algorithm 2. 20
BER
15
Algorithm 2: Weighted MRC-based DFE detection 10 -3
10
Data: Heff , d, y, x̂0 = 0 5
1 for n = 1 : niter do 0
10 -3 10 -2 10 -1 10 0 10 -3 10 -2 10 -1 10 0
2 for k = 0 : N-Q-1 do
3 for i = 0 : L-1 do
X (a) BER variation vs. . (b) Iterations vs. .
(n)
bik = yqi − HH i j
eff (qk , pq i )x̂ j
k p i
Fig. 4: BER variation and the average number of iterations
k
j q
p i k k
q
k 100
5 end
PL−1
6 gk = i=0 HH i i
eff (qk , k)bk
gk 10-1
7 ck = dk +γ −1
(n) (n)
8 x̂k = ck or x̂k = D(ck )
BER
9 end 10-2
10 if ||x̂(n) − x̂(n−1) || < then EXIT;
11 end
AFDM, MMSE
10-3
Computing the complexity of Algorithm 2 is straightfor- AFDM, MRC-based DFE
AFDM, low-complexity MMSE
ward as it has only scalar operation. From step 3 to step 11, it OFDM, MMSE
requires L2 CMs, L2 CAs and 1 CD. Therefore, its total com- 10-4
OFDM, low-complexity MMSE [7]
plexity is niter (2L2 +1)(N −Q). In simulations, we observed 0 5 10 15 20
that the algorithm typically converges within 15 iterations. In SNR in dB
the longer version of the article, convergence of x̂(n) to the Fig. 5: BER performance comparison between AFDM and
LMMSE estimate x̂ defined in (9) is proved. OFDM systems using different detectors
The complexity of the two proposed algorithms is re-
markably smaller than maximum likelihood (ML) and linear
constant, and the algorithm converges within 14 iterations, as
MMSE detectors, which have exponential O(|A|N ) and cu-
shown in Fig. 4b. In Fig. 5, we plot the BER performance of
bic O(N 3 ) complexity, respectively, with A representing the
AFDM and OFDM for LMMSE, low-complexity MMSE [7]
QAM alphabet. Moreover, Algorithm 2 has lower complexity
and weighted MRC-based DFE detectors. First, we observe
than Algorithm 1 when the channel impulse response has
that AFDM outperforms OFDM, thanks to achieving full di-
gaps. This is due to the fact that its complexity only depends
versity and as every information symbol is received through
on the number of non-zero elements in each column of Heff ,
multiple independent non-overlapping paths. Second, we ob-
i.e L, instead of Q ≥ L.
serve that both proposed detection algorithms have similar
4. SIMULATION RESULTS performance between them, while conventional MMSE de-
tection has slightly better performance at the cost of higher
In this section, we simulate the uncoded BER performance of
complexity.
AFDM over doubly dispersive channels. The following pa-
rameters are used: carrier frequency fc = 4 GHz, number
of subcarriers N = 128, and AFDM frame length 330 µs. 5. CONCLUSION
Path delays are fixed, and considering Jakes Doppler spec-
trum for each channel realization, the Doppler shift of the i-th We proposed two low complexity detection algorithms for
path is generated using νi = νmax cos(θi ), where θi is uni- zero-padded AFDM. First, a low complexity MMSE detector
formly distributed over [−π, π] with 4-QAM signaling. The which makes use of band LDL factorization was derived.
maximum Doppler shift is νmax = 1, which corresponds to a Second, an iterative weighted MRC-based DFE detector,
maximum speed of 810 km/h. Fig. 4a shows the BER perfor- which exploits the channel sparsity, was proposed. Our
mance of AFDM using the proposed weighted MRC-based results showed that both detectors have comparable perfor-
DFE detector for different values of at SNR = 20 dB. We mance as exact LMMSE while their complexity order is
can see that below = 0.01, the performance remains almost linear, instead of cubic, in the number of subcarriers.6. REFERENCES [12] T. Thaj and E. Viterbo, “Low complexity iterative rake
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