Tutorial_SPCOM2020___OTFS_modulation.pdf

Orthogonal Time Frequency Space (OTFS) Modulation
and Applications
Tutorial at SPCOM 2020, IISc, Bangalore, July, 2020
Yi Hong, Emanuele Viterbo, Raviteja Patchava
Department of Electrical and Computer Systems Engineering
Monash University, Clayton, Australia
Special thanks to
Tharaj Thaj, Khoa T.Phan
(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 1 / 86

Overview I
1 Introduction
Evolution of wireless
High-Doppler wireless channels
Conventional modulation schemes (e.g., OFDM)
Effect of high Dopplers in conventional modulation
2 Wireless channel representation
Time–frequency representation
Time–delay representation
Delay–Doppler representation
3 OTFS modulation
Signaling in the delay–Doppler domain
Compatibility with OFDM architecture

Overview II
4 OTFS Input-Output Relation in Matrix Form
5 OTFS Signal Detection
Vectorized formulation of the input-output relation
Message passing based detection
Other detectors
6 OTFS channel estimation
Channel estimation in delay-Doppler domain
Multiuser OTFS
7 OTFS applications
SDR implementation of OTFS
OTFS with static multipath channels
Link to download Matlab code:
https://ecse.monash.edu/staff/eviterbo/OTFS-VTC18/OTFS_sample_code.zip

Introduction

Evolution of wireless
Voice, Analog traffic
Voice, SMS, CS data
transfer
Voice, SMS, PS data
transfer
PS data, VOIP
Mobile 1G
Analog FDMA
Mobile 2G
TDMA
Mobile 3G
CDMA
Mobile 4G LTE
OFDMA
1980s, N/A 1990s, 0.5 Mbps 2000s, 63 Mbps 2010s, 300 Mbps
Waveform design is the major change between the generations

High-Doppler wireless channels

Wireless Channels - delay spread
LoS path
Reflected path
r1
r2
r3
Delay of LoS path: τ1 = r1/c
Delay of reflected path: τ2 = (r2 + r3)/c
Delay spread: τ2 − τ1

Wireless Channels - Doppler spread
LoS path
Reflected path
v
θ
v cosθ
Doppler frequency of LoS path: ν1 = fc
v
c
Doppler frequency of reflected path: ν2 = fc
v cos θ
c
Doppler spread: ν2 − ν1
TX: s(t) RX: r(t) = h1s(t − τ1)e−j2πν1t
+ h2s(t − τ2)e−j2πν2t

Typical delay and Doppler spreads
Delay spread (c = 3 · 108
m/s)
∆rmax Indoor (3m) Outdoor (3km)
τmax 10ns 10µs
Doppler spread
νmax fc = 2GHz fc = 60GHz
v = 1.5m/s = 5.5km/h 10Hz 300Hz
v = 3m/s = 11km/h 20Hz 600Hz
v = 30m/s = 110km/h 200Hz 6KHz
v = 150m/s = 550km/h 1KHz 30KHz

Conventional modulation scheme – OFDM
OFDM - Orthogonal Frequency Division Multiplexing
Subcarriers
Frequency
OFDM divides the frequency selective channel into multiple parallel
sub-channels

OFDM system model
Figure: OFDM Tx
Figure: OFDM Rx
(*) From Wikipedia, the free encyclopedia

OFDM time domain input-output relation
Received signal – channel is constant over OFDM symbol (no Doppler)
h = (h0, h1, · · · , hP−1) – Path gains over P taps
r = h ∗ s =




















h0 0 · · · 0 hP−1 hP−2 · · · h1
h1 h0 · · · 0 0 hP−1 · · · h2
.
.
.
...
...
...
...
...
...
.
.
.
.
.
.
...
...
...
...
...
... hP−1
hP−1
...
...
...
...
...
...
.
.
.
.
.
.
...
...
...
...
...
...
.
.
.
.
.
.
...
...
...
...
...
...
.
.
.
0 0 · · · hP−1 hP−2 · · · h1 h0




















| {z }
M×M Circulant matrix (H)
s
Eigenvalue decomposition property H = FH
DF where D = diag[DFTM (h)]

OFDM frequency domain input-output relation
At the receiver we have
r = Hs = FH
DFs =
P−1
X
i=0
hi Πi
s
where Π is the permutation matrix





0 · · · 0 1
1
... 0 0
.
.
.
...
...
.
.
.
0 · · · 1 0





(notation used later as alternative representation of the channel)
At the receiver we have input-output relation in frequency domain
y = Fr = D
|{z}
Diagonal matrix with subcarrier gains
x where x = Fs and s = FH
x
| {z }
Tx IFFT
OFDM Pros
Simple detection (one tap equalizer)
Efficiently combat the multi-path effects

Effect of high multiple Dopplers in OFDM
H matrix lost the circulant structure – decomposition becomes erroneous
Introduces inter carrier interference (ICI)
ICI
0
Frequency
OFDM Cons
multiple Dopplers are difficult to equalize
Sub-channel gains are not equal and lowest gain decides the performance

Effect of high Dopplers in OFDM
Orthogonal Time Frequency Space Modulation (OTFS)(∗)
Solves the two cons of OFDM
Works in Delay–Doppler domain rather than Time–Frequency domain
——————
(*) R. Hadani, S. Rakib, M. Tsatsanis, A. Monk, A. J. Goldsmith, A. F. Molisch, and R.
Calderbank, “Orthogonal time frequency space modulation,” in Proc. IEEE WCNC, San
Francisco, CA, USA, March 2017.

Wireless channel representation

Different representations of linear time variant (LTV) wireless channels
time-variant impulse response
F
F F
F
Doppler-variant transfer response
SFFT
time-frequency
(OFDM)
response
delay-Doppler
(OTFS)
response
B(ν; f)
g(t; τ)
H(t; f) h(τ; ν)

The received signal in linear time variant channel (LTV)
r(t) =
Z
g(t, τ)
| {z }
time-variant impulse response
s(t − τ)dτ → generalization of LTI
=
Z Z
h(τ, ν)
| {z }
Delay–Doppler spreading function
s(t − τ)ej2πνt
dτdν → Delay–Doppler Channel
=
Z
H(t, f )
| {z }
time-frequency response
S(f )ej2πft
df → Time–Frequency Channel
Relation between h(τ, ν) and H(t, f )
h(τ, ν) =
Z Z
H(t, f )e−j2π(νt−f τ)
dtdf
H(t, f ) =
Z Z
h(τ, ν)ej2π(νt−f τ)
dτdν







Pair of 2D symplectic FT

Delay
Doppler
0 1 2 3 4
0
-1
-2
1
2

Time-variant impulse response g(t, τ)
—————
* G. Matz and F. Hlawatsch, Chapter 1, Wireless Communications Over Rapidly Time-Varying
Channels. New York, NY, USA: Academic, 2011

Time-frequency and delay-Doppler responses
SFFT
−
−
−
→
←
−
−
−
ISFFT
Channel in Time–frequency H(t, f ) and delay–Doppler h(τ, ν)

Time–Frequency and delay–Doppler grids
Assume ∆f = 1/T
2 N
M
2
1
M∆f
1
NT
Doppler
Delay
2
M
2
∆f
T
Time
Frequency
2D SFFT
2D ISFFT
1 N
1
1
1
Channel h(τ, ν)
h(τ, ν) =
P
X
i=1
hi δ(τ − τi )δ(ν − νi )
Assume τi = lτi
1
M∆f

and νi = kνi
1
NT


OTFS Parameters
Subcarrier
spacing (∆f )
M Bandwidth
(W =M∆f )
Symbol duration
(Ts = 1/W )
delay
spread
lτmax
15 KHz 1024 15 MHz 0.067 µs 4.7 µs 71 (≈ 7%)
Carrier
frequency
(fc )
N Latency
(NMTs
= NT)
Doppler
resolution
(1/NT)
UE speed
(v)
Doppler
frequency
(fd = fc
v
c
)
kνmax
4 GHz 128 8.75 ms 114 Hz
30 Kmph 111 Hz 1 (≈ 1.5%)
120 Kmph 444 Hz 4 (≈ 6%)
500 Kmph 1850 Hz 16(≈ 25%)

OTFS modulation

OTFS modulation
ISFFT SFFT
Time-Frequency Domain
Delay-Doppler Domain
x[k; l] X[n; m] Y [n; m] y[k; l]
s(t) r(t) Wigner
Transform
Heisenberg
Transform
Channel
h(τ; ν)
Figure: OTFS mod/demod
Time–frequency domain is similar to an OFDM system with N symbols in a
frame (Pulse-Shaped OFDM)

Time–frequency domain
Modulator – Heisenberg transform
s(t) =
N−1
X
n=0
M−1
X
m=0
X[n, m]gtx(t − nT)ej2πm∆f (t−nT)
Simplifies to IFFT in the case of N = 1 and rectangular gtx
Channel
r(t) =
Z
H(t, f )S(f )ej2πft
df
Matched filter – Wigner transform
Y (t, f ) = Agrx,r (t, f ) ,
Z
g∗
rx(t0
− t)r(t0
)e−j2πf (t0
−t)
dt0
Y [n, m] = Y (t, f )|t=nT,f =m∆f
Simplifies to FFT in the case of N = 1 and rectangular grx

Time–frequency domain – ideal pulses
If gtx and grx are perfectly localized in time and frequency then they satisfy
the bi-orthogonality condition and
Y [n, m] = H[n, m]X[n, m]
where
H[n, m] =
Z Z
h(τ, ν)ej2πνnT
e−j2πm∆f τ
dτdν
Section 2.2: A discretized system model 11
t
f
T 2T
F
2F
· · ·
·
·
·
0
0
Symbol
Subcarrier
Figure 2.1: Pulse-shaped OFDM interpretation of the signaling scheme (2.13). The
shaded areas represent the approximate time-frequency support of the pulses gk,l(t).
the beginning of Section 2.2.2. Specifically, the channel coefficients h[k,l] inherit the
two-dimensional stationarity property of the underlying continuous-time system func-
tion LH(t, f) [see (2.2)]. Furthermore, the noise coefficients w[k,l] are i.i.d. CN (0,1) as
a consequence of the orthonormality of (g(t),T,F). These two properties are crucial for
the ensuing analysis.
A drawback of (2.14) is the presence of (self-)interference [the second term in (2.14)],
which makes the derivation of capacity bounds involved, as will be seen in Section 2.4.
The signaling scheme (2.13) can be interpreted as PS-OFDM [KM98], where the input
Figure: Time–frequency domain
—————
* F. Hlawatsch and G. Matz, Eds., Chapter 2, Wireless Communications Over Rapidly
Time-Varying Channels. New York, NY, USA: Academic, 2011 (PS-OFDM)

Signaling in the delay–Doppler domain
Time–frequency input-output relation
Y [n, m] = H[n, m]X[n, m]
where
H[n, m] =
X
k
X
l
h [k, l] ej2π nk
N − ml
M

ISFFT
X[n, m] =
1
√
NM
N−1
X
k=0
M−1
X
l=0
x[k, l]ej2π nk
N − ml
M

SFFT
y[k, l] =
1
√
NM
N−1
X
n=0
M−1
X
m=0
Y [n, m]e−j2π nk
N − ml
M


Delay–Doppler domain input-output relation
Received signal in delay–Doppler domain
y[k, l] =
P
X
i=1
hi x[[k − kνi
]N , [l − lτi
]]M
= h[k, l] ∗ x[k, l] (2D Circular Convolution)
0
0
0.2
0.4
0.6
5
0.8
20
1
10
15
15
10
20
5
25
(a) Input signal, x[k, l]
0
0.2
1
0.4
2
0.6
3
0.8
4 10
1
5 9
8
6
7
7
6
8
5
9 4
10 3
2
1
(b) Channel, h[k, l]
0
0
0.2
0.4
5
0.6
0.8
10
1
15
15
20
10
25
5
30
(c) Output signal, y[k, l]
Figure: OTFS signals

Fractional doppler effect
Actual Doppler may not be perfectly aligned with the grid
νi = (kνi
+ κνi
)

1
NT

, kνi
∈ Z, −1/2 κνi
1/2
Induces interference from the neighbor points of kνi
in the Doppler grid due
to non-orthogonality in channel relation – Inter Doppler Interference (IDI)
Received signal equation becomes
y(k, l) =
P
X
i=1
Ni
X
q=−Ni
hi

ej2π(−q−κνi
)
− 1
Nej 2π
N (−q−κνi
)
− N

x [[k − kνi
+ q]N , [l − lτi
]M ]

Compatibility with OFDM architecture
Time-Frequency Domain (N OFDM symbols)
Delay-Doppler Domain
x[k; l] X[n; m] Y [n; m] y[k; l]
s(t) r(t)
Precoder
(ISFFT)
Decoder
(SFFT)
OFDM
Modulator
OFDM
Demodulator
Channel
H(t; f)
Figure: OTFS mod/demod
OTFS is compatible with LTE system
OTFS can be easily implemented by applying a precoding and decoding
blocks on N consecutive OFDM symbols

OTFS with rectangular pulses – time–frequency domain
Assume gtx and grx to be rectangular pulses (same as OFDM) – don’t follow
bi-orthogonality condition
Time–frequency input-output relation
Y [n, m] = H[n, m]X[n, m] + ICI + ISI
ICI – loss of orthogonality in frequency domain due to Dopplers
ISI – loss of orthogonality in time domain due to delays
————
(*) P. Raviteja, K. T. Phan, Y. Hong, and E. Viterbo, “Interference cancellation and iterative
detection for orthogonal time frequency space modulation,” IEEE Trans. Wireless Commun., vol.
17, no. 10, pp. 6501-6515, Oct. 2018. Available on: https://arxiv.org/abs/1802.05242

OTFS Input-Output Relation in Matrix Form

OTFS transmitter implementation: M = 2048, N = 128
…
IFFT
128
IFFT
128
FFT
2048
FFT
2048
IFFT
2048
IFFT
2048
…
P/S+CP
delay
(M=2048)
Doppler (N=128)
delay
frequency
(2048
subcarriers)
ISFFT
MxN
time
(128 symbols)
Heisenberg transform
time-frequency - time
(N-symbol OFDM transmitter)
.
.
.
…
…
time
(128 symbols)
XMxN
Q-QAM
MN*log2(Q) bits
IFFT
128
IFFT
128
P/S+CP
delay
(M=2048)
Doppler (N=128)
delay
time
(128 symbols)
.
.
.
…
XMxN
Q-QAM
MN*log2(Q) bits
…
Only
one CP
Time domain signal (128 symbols, 2048 samples each)
2048 samples
MN TX complexity PAPR
OTFS MN*log2(N) N
OFDM MN*log2(M) M
time
(128 symbols)
OTFS transmitter implements inverse ZAK transform (2D→1D)

OTFS: Tx matrix representation
Transmit signal at 2D time domain: ISFFT+Heisenberg+pulse shaping on
delay–Doppler
S = GtxFH
M (FM XFH
N )
| {z }
ISFFT
= GtxXFH
N
In vector form:
s = vec(S) = (FH
N ⊗ Gtx)x

OTFS receiver implementation: M = 2048, N = 128
…
FFT
128
FFT
128
remove
CP
+
S/P
delay
(M=2048)
Doppler (N=128)
delay
…
Time domain signal (128 symbols, 2048 samples each)
time
(128 symbols)
.
.
.
2048 samples
YMxN
received
Symbols
time varying
channel
Received signal at delay–Doppler domain: pulse shaping+Wigner+SFFT on
time–frequency received signal
Y = FH
M (FM GrxR)FN = GrxRFN
In vector form:
y = (FN ⊗ Grx)r
OTFS receiver implements ZAK transform (1D→2D)

OTFS: matrix representation – channel
Received signal in the time–frequency domain
r(t) =
Z Z
h(τ, ν)s(t − τ)ej2πν(t−τ)
dτdν + w(t)
Channel
h(τ, ν) =
P
X
i=1
hi δ(τ − τi )δ(ν − νi )
Received signal in discrete form
r(n) =
P
X
i=1
hi e
j2πki (n−li )
MN
| {z }
Doppler
s([n − li ]MN )
| {z }
Delay
+ w(n), 0 ≤ n ≤ MN − 1

Received signal in vector form
r = Hs + w
H is an MN × MN matrix of the following form
H =
P
X
i=1
hi Πli
∆(ki )
,
where, Π is the permutation matrix (forward cyclic shift), and ∆(ki )
is the
diagonal matrix
Π =






0 · · · 0 1
1
... 0 0
.
.
.
...
...
.
.
.
0 · · · 1 0






MN×MN
| {z }
Delay (similar to OFDM)
, ∆(ki )
=






e
j2πki (0)
MN 0 · · · 0
0 e
j2πki (1)
MN · · · 0
.
.
.
...
.
.
.
0 0 · · · e
j2πki (MN−1)
MN






| {z }
Doppler

Received signal at delay–Doppler domain
y =

(FN ⊗ Grx)H(FH
N ⊗ Gtx)

x + (FN ⊗ Grx)w
= Heffx + e
w
Effective channel for arbitrary pulses
Heff = (IN ⊗ Grx)(FN ⊗ IM )H(FH
N ⊗ IM )(IN ⊗ Gtx)
= (IN ⊗ Grx) Hrect
eff
|{z}
Channel for rectangular pulses (Gtx=Grx=IM )
(IN ⊗ Gtx)
Effective channel for rectangular pulses
Hrect
eff =
P
X
i=1
hi

(FN ⊗ IM )Πli
(FH
N ⊗ IM )

| {z }
P(i) (delay)
h
(FN ⊗ IM )∆(ki )
(FH
N ⊗ IM )
i
| {z }
Q(i) (Doppler)
=
P
X
i=1
hi P(i)
Q(i)
=
P
X
i=1
hi T(i)

OTFS: Example for computing Hrect
eff
M = 2, N = 2,MN = 4
li = 0 and ki = 0 (no delay and Doppler)
Πli =0
= I4 ⇒ P(i)
= (F2 ⊗ I2)(FH
2 ⊗ I2) = I4
∆(ki =0)
= I4 ⇒ Q(i)
= (F2 ⊗ I2)(FH
2 ⊗ I2) = I4
T(i)
= P(i)
Q(i)
= I4 ⇒ Narrowband channel
0
0
1
1

eff
li = 1 and ki = 0 (delay but no Doppler)
0
0
1
1
Πli =1
=
2
6
6
4
0 0 0 1
1 0 0 0
0 1 0 0
0 0 1 0
3
7
7
5⇒ block circulant matrix with 2 × 2 (M × M) block size
P(i)
= (F2 ⊗ I2)Π(FH
2 ⊗ I2) =
2
6
6
4
0 1 0 0
1 0 0 0
0 0 0 e−j2π 1
2
0 0 1 0
3
7
7
5
(using the block circulant matrix decomposition → generalization of circulant
matrix decomposition in OFDM)
∆(ki =0)
= I4 ⇒ Q(i)
= (F2 ⊗ I2)(FH
2 ⊗ I2) = I4
T(i)
= P(i)
⇒ T(i)
s → circularly shifts the elements in each block (size M) of s
by 1 (delay shift)

eff
li = 0 and ki = 1 (Doppler but no delay)
0
0
1
1
Πli =0
= I4 ⇒ P(i)
= (F2 ⊗ I2)(FH
2 ⊗ I2) = I4
∆(ki =1)
=
2
6
6
4
1 0 0 0
0 ej2π 1
4 0 0
0 0 ej2π 2
4 0
0 0 0 ej2π 3
4
3
7
7
5⇒ block diagonal matrix with 2 × 2 (M × M)
block size
Q(i)
= (F2 ⊗ I2)∆(1)
(FH
2 ⊗ I2) =
2
6
6
4
0 0 1 0
0 0 0 ej2π 1
4
1 0 0 0
0 ej2π 1
4 0 0
3
7
7
5
(using the block circulant matrix decomposition in reverse direction)
T(i)
= Q(i)
⇒ T(i)
s → circularly shifts the blocks (size M) of s by 1 (Doppler
shift)

eff
li = 1 and ki = 1 (both delay and Doppler)
0
0
1
1
P(i)
=
2
6
6
4
0 1 0 0
1 0 0 0
0 0 0 e−j2π 1
2
0 0 1 0
3
7
7
5
Q(i)
=
2
6
6
4
0 0 1 0
0 0 0 ej2π 1
4
1 0 0 0
0 ej2π 1
4 0 0
3
7
7
5
T(i)
= P(i)
Q(i)
⇒ T(i)
s → circularly shifts both the blocks (size M) and the
elements in each block of s by 1 (delay and Doppler shifts)

OTFS: channel for rectangular pulses
T(i)
has only one non-zero element in each row and the position and value of
the non-zero element depends on the delay and Doppler values.
T(i)
(p, q) =





e−j2π n
N ej2π
ki ([m−li ]M )
MN , if q = [m − li ]M + M[n − ki ]N and m li
ej2π
ki ([m−li ]M )
MN , if q = [m − li ]M + M[n − ki ]N and m ≥ li
0, otherwise.
Example: li = 1 and ki = 1
T(i)
=




0 0 0 ej2π 1
4
0 0 1 0
0 e−j2π 1
4 0 0
1 0 0 0





OTFS Signal Detection

Vectorized formulation of the input-output relation
The input-output relation in the delay–Doppler domain is a 2D convolution
(with i.i.d. additive noise w[k, l])
y[k, l] =
P
X
i=1
hi x[[k − kνi
]N , [l − lτi
]M ] + w[k, l] k = 1 . . . N, l = 1 . . . M (1)
Detection of information symbols x[k, l] requires a deconvolution operation
i.e., the solution of the linear system of NM equations
y = Hx + w (2)
where x, y, w are x[k, l], y[k, l], w[k, l] in vectorized form and H is the
NM × NM coefficient matrix of (1).
Given the sparse nature of H we can solve (2) by using a message passing
algorithm similar to (*)
————
(*) P. Som, T. Datta, N. Srinidhi, A. Chockalingam, and B. S. Rajan, “Low-complexity
detection in large-dimension MIMO-ISI channels using graphical models,” IEEE J. Sel. Topics in
Signal Processing, vol. 5, no. 8, pp. 1497-1511, December 2011.

Message passing based detection
Symbol-by-symbol MAP detection
b
x[c] = arg max
aj ∈A
Pr x[c] = aj y, H

= arg max
aj ∈A
1
Q
Pr y x[c] = aj , H

≈ arg max
aj ∈A
Y
d∈Jc
Pr y[d] x[c] = aj , H

Received signal y[d]
y[d] = x[c]H[d, c] +
X
e∈Id ,e6=c
x[e]H[d, e] + z[d]
| {z }
ζ
(i)
d,c → assumed to be Gaussian

Messages in factor graph
Algorithm 1 MP algorithm for OTFS symbol detection
Input: Received signal y, channel matrix H
Initialization: pmf p
(0)
c,d = 1/Q repeat
- Observation nodes send the mean and variance to variable nodes
- Variable nodes send the pmf to the observation nodes
- Update the decision
until Stopping criteria;
Output: The decision on transmitted symbols b
x[c]
(µd;e1
; σ2
d;e1
)
fe1; e2; · · · ; eSg = Id
y[d]
x[e1] x[eS]
(µd;eS
; σ2
d;eS
)
Observation node messages
y[e1]
x[c]
y[eS]
pc;e1
pc;eS
fe1; e2; · · · ; eSg = Jc
Variable node messages

Messages in factor graph – observation node messages
Received signal
y[d] = x[c]H[d, c] +
X
e∈I(d),e6=c
x[e]H[d, e] + z[d]
| {z }
ζ
(i)
d,c → assumed to be Gaussian
(µd;e1
; σ2
d;e1
)
fe1; e2; · · · ; eSg = Id
y[d]
x[e1] x[eS]
(µd;eS
; σ2
d;eS
)
Mean and Variance
µ
(i)
d,c =
X
e∈I(d),e6=c
Q
X
j=1
p
(i−1)
e,d (aj )aj H[d, e]
(σ
(i)
d,c )2
=
X
e∈I(d),e6=c



Q
X
j=1
p
(i−1)
e,d (aj )|aj |2
|H[d, e]|2
−
Q
X
j=1
p
(i−1)
e,d (aj )aj H[d, e]
2


+ σ2

Messages in factor graph – variable node messages
Probability update with damping
factor ∆
p
(i)
c,d (aj ) = ∆ · p̃
(i)
c,d (aj ) + (1 − ∆) · p
(i−1)
c,d (aj ), aj ∈ A
y[e1]
x[c]
y[eS]
pc;e1
pc;eS
fe1; e2; · · · ; eSg = Jc
where
p̃
(i)
c,d (aj ) ∝
Y
e∈J (c),e6=d
Pr

y[e] x[c] = aj , H

=
Y
e∈J (c),e6=d
ξ(i)
(e, c, j)
PQ
k=1 ξ(i)(e, c, k)
ξ(i)
(e, c, k) = exp



− y[e] − µ
(i)
e,c − He,c ak
2
(σ
(i)
e,c )2




Final update and stopping criterion
Final update
p(i)
c (aj ) =
Y
e∈J (c)
ξ(i)
(e, c, j)
PQ
k=1 ξ(i)(e, c, k)
b
x[c] = arg max
aj ∈A
p(i)
c (aj ), c = 1, · · · , NM.
Stopping Criterion
Convergence Indicator η(i)
= 1
η(i)
=
1
NM
NM
X
c=1
I

max
aj ∈A
p(i)
c (aj ) ≥ 0.99

Maximum number of Iterations
Complexity (linear) – O(niter SQ) per symbol which is much less even
compared to a linear MMSE detector O((NM)2
)

Simulation results – damping factor ∆
∆
0.2 0.4 0.6 0.8 1
BER
10-5
10-4
10-3
10-2
10-1
100
OTFS, 120 Kmph
4-QAM, SNR = 18 dB
∆
0.2 0.4 0.6 0.8 1
Average
no.
of
iterations
10
15
20
25
30
35
40
45
OTFS, 120 Kmph
4-QAM, SNR = 18 dB
Figure: Variation of BER and average iterations no. with ∆. Optimal for ∆ = 0.7

Simulation results – OTFS vs OFDM with ideal pulses
SNR in dB
5 10 15 20 25 30
BER
10-5
10-4
10-3
10-2
10-1
100
OTFS, Ideal, 30 Kmph
OFDM, 30 kmph
OFDM, 120 kmph
OFDM, 500 kmph
4-QAM
Figure: The BER performance comparison between OTFS with ideal pulses and OFDM
systems at different Doppler frequencies.

Simulation results – IDI effect
Ni
0 5 10 15 20
BER
10-5
10-4
10-3
10-2
10-1
100
OTFS, 18 dB, 120 Kmph
4-QAM
Figure: The BER performance of OTFS for different number of interference terms Ni
with 4-QAM.

Simulation results – Ideal and Rectangular pulses
SNR in dB
5 10 15 20 25 30
BER
10-5
10-4
10-3
10-2
10-1
100
OTFS, Rect., WC, 30 Kmph
OTFS, Rect., WO, 30 Kmph
OTFS, Ideal
OFDM, 500 kmph
14.2 14.3 14.4
×10-4
3.795
3.8
Figure: The BER performance of OTFS with rectangular and ideal pulses at different
Doppler frequencies for 4-QAM.

Simulation results – Ideal and Rect. pulses - 16-QAM
SNR in dB
10 15 20 25 30 35
BER
10-4
10-3
10-2
10-1
100
OTFS, Ideal
OFDM
16-QAM
Figure: The BER performance of OTFS with rectangular and ideal pulses at different
Doppler frequencies for 16-QAM.

Simulation results – Low latency
SNR in dB
20 25 30 35 40
BER
10-4
10-3
10-2
10-1
OTFS, Rect., WC, 30 Kmph, N = 16, M = 128
OTFS, Rect., WC, 120 Kmph, N = 16, M = 128
OTFS, Ideal, N = 16, M = 128
OTFS, Ideal, N = 128, M = 512
OFDM, N = 16, M = 128
16-QAM
Figure: The BER performance of OTFS with rectangular pulses and low latency
(N = 16, Tf ≈ 1.1 ms).

Matlab code
OTFS sample code.m
→ OTFS modulation – 1. ISFFT, 2. Heisenberg transform
X = fft(ifft(x).’).’/sqrt(M/N); % ISFFT
s mat = ifft(X.’)*sqrt(M); % Heisenberg transform
s = s mat(:);
→ OTFS channel gen – generates wireless channel
output: (delay taps,Doppler taps,chan coef)
→ OTFS channel output – wireless channel and noise
L = max(delay taps);
s = [s(N*M-L+1:N*M);s];% add one cp
s chan = 0;
for itao = 1:taps
s chan = s chan+chan coef(itao)*circshift([s.*exp(1j*2*pi/M...
*(-L:-L+length(s)-1)*Doppler taps(itao)/N).’;zeros(L,1)],delay taps(itao));
end
noise = sqrt(sigma 2/2)*(randn(size(s chan)) + 1i*randn(size(s chan)));
r = s chan + noise;
r = r(L+1:L+(N*M));% discard cp

Matlab code
→ OTFS demodulation – 1. Wiegner transform, 2. SFFT
r mat = reshape(r,M,N);
Y = fft(r mat)/sqrt(M); % Wigner transform
Y = Y.’;
y = ifft(fft(Y).’).’/sqrt(N/M); % SFFT
→ OTFS mp detector – message passing detector

Other detection methods
We will present a new low-complexity detection method at WCNC2020
on Tuesday in the Session T1-S7: Waveform and modulation
Output OTFS signal: y = Hx + w
1 MMSE detection:
x̂ = (HH
H + λI)
−1
HH
y
Provides diversity but high complex O((NM)3
)
2 OTFS FDE (frequency domain equalization) in [1]
Equalization in time–frequency domain (one-tap) and apply the SFFT
Low complexity equalizer
Phase shifts can’t be applied and bad performance at high Dopplers
Small improvement on OFDM
——————–
[1]. Li Li, H. Wei, Y. Huang, Y. Yao, W. Ling, G. Chen, P. Li, and Y. Cai, “A simple two-stage
equalizer With simplified orthogonal time frequency space modulation over rapidly time-varying
channels,” available online: https://arxiv.org/abs/1709.02505.

Other detection methods
3 OTFS MMSE-PIC (parallel ISI cancellation) in [2]
First applies the equalization in time–frequency domain (one-tap) and then
applies successive cancellation with coding
Successive cancellation
ŷ(i)j+1 = y − Hx̂j + H(:, i)x̂(i)j
x̂(i)j+1 = arg min
a∈A

ŷ(i)j+1 − H(:, i)a

Moderate complexity
Better performance than [1] but still struggles with the high Doppler
4 MCMC sampling [3]
Approximate ML solution using Gibbs sampling based MCMC technique
High complexity O(niter NM) compared to message passing (O(niter SQ))
(Does not take advantage of sparsity of the channel matrix)
——————–
[2]. T. Zemen, M. Hofer, and D. Loeschenbrand, “Low-complexity equalization for orthogonal
time and frequency signaling (OTFS),” available online: https://arxiv.org/pdf/1710.09916.pdf.
[3]. K. R. Murali and A. Chockalingam, “On OTFS modulation for high-Doppler fading
channels,” in Proc. ITA’2018, San Diego, Feb. 2018.

OTFS channel estimation

Channel estimation in time–frequency domain
(lτi
, kνi
) ((delay,Doppler)) values are obtained from the baseband time
domain signal equation
y(t) =
P
X
i=1
hi x(t − τi )ej2πνi (t−τi )
PN based pilots and 2D matched filter matrix is used to determine (lτi
, kνi
)
Highly complex
—————
1 A. Fish, S. Gurevich, R. Hadani, A. M. Sayeed, and O. Schwartz, “Delay-Doppler channel
estimation in almost linear complexity,” IEEE Trans. Inf. Theory, vol. 59, no. 11, pp.
7632-7644, Nov. 2013.
2 K. R. Murali, and A. Chockalingam, “On OTFS modulation for high-Doppler fading
channels,” in Proc. ITA’2018, San Diego, Feb. 2018.

Channel estimation using impulses in the delay-Doppler
domain
Each transmit and receive antenna pair sees a different channel having a
finite support in the delay-Doppler domain
The support is determined by the delay and Doppler spread of the channel
The OTFS input-output relation for pth transmit antenna and qth receive
antenna pair can be written as
x̂q[k, l] =
M−1
X
m=0
N−1
X
n=0
xp[n, m]
1
MN
hwqp

k − n
NT
,
l − m
M∆f

+ vq[k, l].
—————
1 P. Raviteja, K.T. Phan, and Y. Hong, “Embedded Pilot-Aided Channel
Estimation for OTFS in Delay-Doppler Channels”, IEEE Trans. on Veh. Tech.,
March 2019 (Early Access).
2 M. K. Ramachandran and A. Chockalingam, “MIMO-OTFS in high-Doppler
fading channels: Signal detection and channel estimation,” available online:
https://arxiv.org/abs/1805.02209.
3 R. Hadani and S. Rakib, “OTFS methods of data channel characterization and
uses thereof.” U.S. Patent 9 444 514 B2, Sept. 13, 2016.

If we transmit
xp[n, m] = 1 if (n, m) = (np, mp)
= 0 ∀ (n, m) 6= (np, mp),
as pilot from the pth antenna, the received signal at the qth antenna will be
x̂q[k, l] =
1
MN
hwqp

k − np
NT
,
l − mp
M∆f

+ vq[k, l].
1
MN hwqp
k
NT , l
M∆f

and thus Ĥqp can be estimated , since np and mp are
known at the receiver a priori
Impulse at (n, m) = (np, mp) spreads only to the extent of the support of the
channel in the delay-Doppler domain (2D convolution)
If the pilot impulses have sufficient spacing in the delay-Doppler domain, they
will be received without overlap

Figure: Illustration of pilots and channel response in delay-Doppler domain in a 2×1
MIMO-OTFS system

SISO OTFS system with integer Doppler
0 1 M − 1
lp
0
1
kp
N − 1
lp − lτ lp + lτ
kp − 2kν
kp + 2kν
(a) Tx symbol arrangement (: pilot; ◦:
guard symbols; ×: data symbols)
0 1 M − 1
lp
0
1
kp
N − 1
lp − lτ lp + lτ
kp − kν
kp + kν
5 5 5 5 5 5 5 5 5 5 5
5 5 5 5 5 5 5 5 5 5 5
5 5 5 5 5 5
5 5 5 5 5 5
5 5 5 5 5 5
5 5 5 5 5 5 5 5 5 5 5
5 5 5 5 5 5 5 5 5 5 5
5 5 5
5 5 5
5 5 5
5 5 5
⊞ ⊞ ⊞
⊞ ⊞ ⊞
⊞ ⊞ ⊞
5 5 5 5 5
5 5
5 5
5 5
5 5 5 5 5
(b) Rx symbol pattern (O: data detection,
: channel estimation)
Figure: Tx pilot, guard, and data symbols and Rx received symbols

SISO OTFS system with fractional Doppler
0 1 M − 1
lp
0
kp
N − 1
lp − lτ lp + lτ
kp − 2kν
kp + 2kν
kp + 2kν + 2^
k
kp − 2kν − 2^
k
(a) Tx symbol arrangement (: pilot; ◦:
guard symbols; ×: data symbols)
0 1 M − 1
lp
0
kp
N − 1
lp − lτ lp + lτ
kp − kν
kp + kν
5 5 5 5 5 5 5 5 5 5 5
5 5 5 5 5 5 5 5 5 5 5
5 5 5 5 5 5
5 5 5 5 5 5
5 5 5 5 5 5
5 5 5 5 5 5 5 5 5 5 5
5 5 5 5 5 5 5 5 5 5 5
5 5 5
5 5 5
5 5 5
5 5 5
⊞ ⊞ ⊞
⊞ ⊞ ⊞
⊞ ⊞ ⊞
5 5
5 5
5 5
5 5
5 5
kp + kν + ^
k
kp − kν − ^
k ⊞ ⊞ ⊞
⊞ ⊞ ⊞
(b) Rx symbol pattern (O: data detection,
: channel estimation)
Figure: Tx pilot, guard, and data symbols and Rx received symbols

MIMO OTFS system with fractional Doppler
0 M − 1
0
kp
N − 1
kp + 2kν + 2^
k
lp−lτ lp
kp − 2kν − 2^
k
(a) Antenna 1 (×: antenna 1
data symbol)
0 M − 1
lp−lτ lp +lτ +1
⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄
⋄ ⋄
⋄ ⋄
⋄ ⋄
⋄ ⋄
⋄ ⋄ ⋄ ⋄
⋄ ⋄
⋄ ⋄
⋄ ⋄
⋄ ⋄
⋄
⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄
⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄
⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄
(b) Antenna 2 (♦: antenna 2
data symbol)
0 M − 1
lp−lτ lp +2lτ +2
⊕ ⊕
⊕ ⊕
⊕ ⊕
⊕ ⊕
⊕ ⊕
⊕ ⊕
⊕ ⊕ ⊕ ⊕
⊕ ⊕
⊕ ⊕
⊕ ⊕
⊕ ⊕
⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕
⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕
⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕
⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕
(c) Antenna 3 (⊕: antenna
3 data symbol)
Figure: Tx pilot, guard, and data symbols for MIMO OTFS system (: pilot; ◦: guard)
0 M − 1
0
kp
N − 1
lp lp +lτ +1
5
⊞ ⊞
⊞ ⊞
⊞ ⊞
⊠ ⊗
⊠
⊠ ⊠
⊠ ⊠
⊗
⊗ ⊗
⊗ ⊗
5
5 5
5 5
5 5
5 5
5 5 5 5
5 5
5 5
5 5
5 5
5 5 5 5 5 5 5 5 5
5 5 5 5 5 5 5 5 5 5 5
5 5 5 5 5 5 5 5 5 5 5
5 5 5 5 5 5 5 5 5 5 5
5 5 5 5 5 5 5
5 5 5 5 5 5 5
5
5
5
kp + kν + ^
k
kp − kν − ^
k
Figure: Rx symbol pattern at antenna 1 of MIMO OTFS system (O: data detection,
, , ⊗: channel estimation for Tx antenna 1, 2, and 3, respectively)

Multiuser OTFS system – uplink
0 M − 1
0
kp
N − 1
kp + 2kν + 2^
k
lp−lτ lp
kp − 2kν − 2^
k
(a) User 1 (×: user 1 data
symbol)
0 M − 1
lp−lτ lp +lτ +1
⋄ ⋄ ⋄ ⋄
⋄ ⋄
⋄ ⋄
⋄ ⋄
⋄
⋄ ⋄ ⋄ ⋄ ⋄
(b) User 2 (♦: user 2 data
symbol)
0 M − 1
lp−lτ lp +2lτ +2
⊕ ⊕
⊕ ⊕ ⊕ ⊕
⊕ ⊕
⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕
⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕
(c) User 3 (⊕: user 3 data
symbol)
Figure: Tx pilot, guard, and data symbols for multiuser uplink OTFS system (: pilot; ◦:
guard symbols)

SISO-OTFS performance with the estimated channel
Simulation parameters: Carrier frequency of 4GHz, sub-carrier spacing of
15KHz, M = 512, N = 128, 4-QAM signaling, LTE EVA channel model
Let SNRp and SNRd denote the average pilot and data SNRs
Channel estimation threshold is 3σp, where σ2
p = 1/SNRp is effective noise
power of the pilot signal
SNRd
in dB
10 12 14 16 18
BER
10-5
10-4
10-3
10-2
10-1
30 Kmph
120 Kmph
500 Kmph
Ideal
N = 128, M = 512, lτ
= 20,
SNRp
= 40 dB, 4-QAM
(a) BER for estimated channels of different
Integer Dopplers
SNRd
in dB
10 12 14 16 18
BER
10-5
10-4
10-3
10-2
10-1
k̂ = 2
k̂ = 5
Full Guard
Ideal
N = 128, M = 512, lτ
= 20,
SNRp
= 50 dB, 4-QAM
(b) BER for estimated channels of
Fractional Doppler

OTFS applications

OTFS modem SDR implementation block diagram
x[k,l]
OTFS Modulation
QAM
modulation
ISFFT
Heisenberg
Transform
Add cyclic
prefix
Pulse
Shaping
Add
Preamble
Preamble
Detection and
Frame
Synchronization
Reciever
Matched
Filter
Remove
cyclic prefix
Wigner
Transform
SFFT
Channel
Estimation
Information
bits
Message
passing
detector
QAM
demodulation
OTFS Demodulation
Channel
X[n,m]
Y[n,m]
y[k,l]
s[n]
r[n]
Information
bits

Experiment setup and parameters
The wireless propagation channel can be observed in real time using LabView
GUI at the RX while receiving the OTFS frames.

OTFS received pilot in a real indoor wireless channel
DC Offset manifests itself as a constant signal in the delay-Doppler plane
shifted by Doppler equal to CFO.

OTFS received pilot in a partially emulated indoor mobile
channel
Doppler paths were added to the TX OTFS waveform and transmitted it into
a real indoor wireless channel for a time selective channel.

Error performance
5 10 15 20 25 30 35
Tx gain in dB
10-5
10-4
10-3
10-2
10-1
100
BER
/
FER
OTFS Modem Performance
frame error rate(16-QAM)
bit error rate(16-QAM)
frame error rate(4-QAM)
bit error rate(4-QAM)
0 5 10 15 20 25 30 35
Tx gain in dB
10-5
10-4
10-3
10-2
10-1
100
BER
OTFS vs OFDM (4-QAM)
OTFS-static channel
OFDM-static channel
OTFS-mobile channel
OFDM-mobile channel

OTFS with static multipath channels (zero Doppler)
Received signal
(size MN × 1) y = (FN ⊗ IM )H(FH
N ⊗ IM )x + e
w
↓ zero Doppler
(size M × 1) yn = H̆nxn + e
wn, for n = 0, · · · , N − 1
Equivalent to A-OFDM (asymmetric OFDM) in (*)
H̆n structure for M ≥ L
H̆n =





h0 0 · · · h1e−j2π n
N
h1 h0 · · · h2e−j2π n
N
.
.
.
...
...
.
.
.
0 · · · h1 h0





M×M
Achieves maximum diversity when M ≥ L (max. delay)
⇐⇒ N parallel CPSC transmissions each of length M
————
(*) J. Zhang, A. D. S. Jayalath, and Y. Chen, “Asymmetric OFDM systems based on layered
FFT structure,” IEEE Signal Proces. Lett., vol. 14, no. 11, pp. 812-815, Nov. 2007.

OTFS with static multipath channels (zero Doppler)
SNR in dB
15 20 25 30 35 40
BER
10-4
10-3
10-2
10-1
100
M = 1, OFDM
M = 2
M = 4
M = 128, 256, 1024; MP
CPSC, MMSE
M = 128; A-OFDM, ZF
M = 128; A-OFDM, MMSE
AWGN
Nc
= 1024, L = 72, 16-QAM
Figure: BER of OTFS for different M with MN = Nc = 1024, L = 72, and 16-QAM
————
(*) P. Raviteja, Y. Hong, and E. Viterbo, “OTFS performance on static multipath channels,”
IEEE Wireless Commun. Lett., Jan. 2019, doi: 10.1109/LWC.2018.2890643.

References I
1 R. Hadani, S. Rakib, M. Tsatsanis, A. Monk, A. J. Goldsmith, A. F. Molisch,
and R. Calderbank, “Orthogonal time frequency space modulation,” in Proc.
IEEE WCNC, San Francisco, CA, USA, March 2017.
2 R. Hadani, S. Rakib, S. Kons, M. Tsatsanis, A. Monk, C. Ibars, J. Delfeld, Y.
Hebron, A. J. Goldsmith, A.F. Molisch, and R. Calderbank, “Orthogonal time
frequency space modulation,” Available online:
https://arxiv.org/pdf/1808.00519.pdf.
3 R. Hadani, and A. Monk, “OTFS: A new generation of modulation addressing
the challenges of 5G,” OTFS Physics White Paper, Cohere Technologies, 7
Feb. 2018. Available online: https://arxiv.org/pdf/1802.02623.pdf.
4 R. Hadani et al., “Orthogonal Time Frequency Space (OTFS) modulation for
millimeter-wave communications systems,” 2017 IEEE MTT-S International
Microwave Symposium (IMS), Honololu, HI, 2017, pp. 681-683.
5 A. Fish, S. Gurevich, R. Hadani, A. M. Sayeed, and O. Schwartz,
“Delay-Doppler channel estimation in almost linear complexity,” IEEE Trans.
Inf. Theory, vol. 59, no. 11, pp. 7632–7644, Nov 2013.

References II
6 A. Monk, R. Hadani, M. Tsatsanis, and S. Rakib, “OTFS - Orthogonal time
frequency space: A novel modulation technique meeting 5G high mobility and
massive MIMO challenges.” Technical report. Available online:
https://arxiv.org/ftp/arxiv/papers/1608/1608.02993.pdf.
7 R. Hadani and S. Rakib. “OTFS methods of data channel characterization
and uses thereof.” U.S. Patent 9 444 514 B2, Sept. 13, 2016.
8 P. Raviteja, K. T. Phan, Q. Jin, Y. Hong, and E. Viterbo, “Low-complexity
iterative detection for orthogonal time frequency space modulation,” in Proc.
IEEE WCNC, Barcelona, April 2018.
9 P. Raviteja, K. T. Phan, Y. Hong, and E. Viterbo, “Interference cancellation
and iterative detection for orthogonal time frequency space modulation,”
IEEE Trans. Wireless Commun., vol. 17, no. 10, pp. 6501-6515, Oct. 2018.
10 P. Raviteja, K. T. Phan, Y. Hong, and E. Viterbo, “Embedded delay-Doppler
channel estimation for orthogonal time frequency space modulation,” in Proc.
IEEE VTC2018-fall, Chicago, USA, August 2018.

References III
11 P. Raviteja, K. T. Phan, and Y. Hong, “Embedded pilot-aided channel
estimation for OTFS in delay-Doppler channels,” IEEE Transactions on
Vehicular Technology, May 2019.
12 P. Raviteja, Y. Hong, E. Viterbo, and E. Biglieri, “Practical pulse-shaping
waveforms for reduced-cyclic-prefix OTFS,” IEEE Trans. Veh. Technol., vol.
68, no. 1, pp. 957-961, Jan. 2019.
13 P. Raviteja, Y. Hong, and E. Viterbo, “OTFS performance on static
multipath channels,” IEEE Wireless Commun. Lett., Jan. 2019, doi:
10.1109/LWC.2018.2890643.
14 Li Li, H. Wei, Y. Huang, Y. Yao, W. Ling, G. Chen, P. Li, and Y. Cai, “A
simple two-stage equalizer with simplified orthogonal time frequency space
modulation over rapidly time-varying channels,” available online:
15 T. Zemen, M. Hofer, and D. Loeschenbrand, “Low-complexity equalization
for orthogonal time and frequency signaling (OTFS),” available online:

References IV
16 T. Zemen, M. Hofer, D. Loeschenbrand, and C. Pacher, “Iterative detection
for orthogonal precoding in doubly selective channels,” available online:
17 K. R. Murali and A. Chockalingam, “On OTFS modulation for high-Doppler
fading channels,” in Proc. ITA’2018, San Diego, Feb. 2018.
18 M. K. Ramachandran and A. Chockalingam, “MIMO-OTFS in high-Doppler
fading channels: Signal detection and channel estimation,” available online:
19 A. Farhang, A. RezazadehReyhani, L. E. Doyle, and B. Farhang-Boroujeny,
“Low complexity modem structure for OFDM-based orthogonal time
frequency space modulation,” in IEEE Wireless Communications Letters, vol.
7, no. 3, pp. 344-347, June 2018.
20 A. RezazadehReyhani, A. Farhang, M. Ji, R. R. Chen, and B.
Farhang-Boroujeny, “Analysis of discrete-time MIMO OFDM-based
orthogonal time frequency space modulation,” in Proc. 2018 IEEE
International Conference on Communications (ICC), Kansas City, MO, USA,
pp. 1-6, 2018.

References V
21 P. Raviteja, Y. Hong, E. Viterbo, E. Biglieri, “Effective diversity of OTFS
modulation,” IEEE Wireless Communications Letters, Nov. 2019.
22 Tharaj Thaj, Emanuele Viterbo, “OTFS Modem SDR Implementation and
Experimental Study of Receiver Impairment Effects,” 2019 IEEE International
Conference on Communications Workshops (ICC 2019), Shanghai.
23 Tharaj Thaj and Emanuele Viterbo, “Low Complexity Iterative Rake Detector
for Orthogonal Time Frequency Space Modulation” in Proceedings of WCNC
2020, Seoul.

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