Orthogonal Time Frequency Space (OTFS) 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

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 2 / 86 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

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 3 / 86 Introduction

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 4 / 86 Evolution of wireless

Mobile 4G LTE OFDMA Mobile 3G CDMA PS data, VOIP Mobile 2G TDMA Voice, SMS, PS data transfer Mobile 1G Analog FDMA Voice, SMS, CS data transfer Voice, Analog traffic

1980s, N/A 1990s, 0.5 Mbps 2000s, 63 Mbps 2010s, 300 Mbps

Waveform design is the major change between the generations

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 5 / 86 High-Doppler wireless channels

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 6 / 86 Wireless Channels - delay spread

r 2 Reflected path r3

r1 LoS path

Delay of LoS path: τ1 = r1/c

Delay of reflected path: τ2 = (r2 + r3)/c

Delay spread: τ2 τ1 −

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 7 / 86 Wireless Channels - Doppler spread

Reflected path

v cosθ θ

LoS path v

v Doppler frequency of LoS path: ν1 = fc c v cos θ Doppler frequency of reflected path: ν2 = fc c Doppler spread: ν2 ν1 −

−j2πν1t −j2πν2t TX: s(t) RX: r(t) = h1s(t τ1)e + h2s(t τ2)e − − (Monash University, Australia) OTFS modulation SPCOM 2020, IISc 8 / 86 Typical delay and Doppler spreads

Delay spread (c = 3 108m/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

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 9 / 86 Conventional modulation scheme – OFDM

OFDM - Orthogonal Frequency Division

Subcarriers

Frequency

OFDM divides the frequency selective channel into multiple parallel sub-channels

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 10 / 86 OFDM system model

Figure: OFDM Tx

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

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 11 / 86 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 ···   h0 0 ··· 0 hP−1 hP−2 ··· h1  h1 h0 ··· 0 0 hP−1 ··· h2     ......   ......   . .     ......   ...... h   P−1 r = h ∗ s =  ...... s  ......  hP−1 ......     ......   ......   . .     ......   ......  0 0 ··· hP−1 hP−2 ··· h1 h0 | {z } M×M Circulant matrix (H)

H Eigenvalue decomposition property H = F DF where D = diag[DFTM (h)]

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 12 / 86 OFDM frequency domain input-output relation

At the receiver we have

P−1 H X i r = Hs = F DFs = hi Π s i=0

 0 ··· 0 1  .  ..  where Π is the permutation matrix  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 x where x = Fs and s = FH x |{z} | {z } Diagonal matrix with subcarrier gains Tx IFFT

OFDM Pros Simple detection (one tap equalizer) Efficiently combat the multi-path effects

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 13 / 86 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

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 14 / 86 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.

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 15 / 86 Wireless channel representation

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 16 / 86 Wireless channel representation

Different representations of linear time variant (LTV) wireless channels

time-variant impulse response g(t; τ) F F

time-frequency SFFT delay-Doppler response H(t; f) h(τ; ν) response (OFDM) (OTFS)

FF B(ν; f) Doppler-variant transfer response

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 17 / 86 Wireless channel representation

The received signal in linear time variant channel (LTV) Z r(t) = g(t, τ) s(t τ)dτ generalization of LTI | {z } − → time-variant impulse response ZZ = h(τ, ν) s(t τ)ej2πνt dτdν Delay–Doppler Channel | {z } − → Delay–Doppler spreading function Z = H(t, f )S(f )ej2πft df Time–Frequency Channel | {z } → time-frequency response

Relation between h(τ, ν) and H(t, f ) ZZ  h(τ, ν) = H(t, f )e−j2π(νt−f τ)dtdf   ZZ Pair of 2D symplectic FT j2π(νt−f τ)  H(t, f ) = h(τ, ν)e dτdν 

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 18 / 86 Wireless channel representation

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

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 19 / 86 Wireless channel representation

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

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 20 / 86 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

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 21 / 86 Time-frequency and delay-Doppler responses

SFFT −−−→ ←−−−ISFFT

Channel in Time–frequency H(t, f ) and delay–Doppler h(τ, ν)

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 22 / 86 Time–Frequency and delay–Doppler grids

Assume ∆f = 1/T Delay

1 M∆f M Frequency ∆f M

2D SFFT

2 2D ISFFT 1 1 2 N Time 2 T 1 Channel h(τ, ν) 1 2 N Doppler 1 NT P X h(τ, ν) = hi δ(τ τi )δ(ν νi ) − − i=1

1  1  Assume τi = lτi M∆f and νi = kνi NT

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 23 / 86 OTFS Parameters

Subcarrier M Bandwidth Symbol duration delay lτmax spacing (∆f ) (W =M∆f ) (Ts = 1/W ) spread 15 KHz 1024 15 MHz 0.067 µs 4.7 µs 71 ( 7%) ≈

Carrier N Latency Doppler UE speed Doppler kνmax frequency (NMT resolution (v) frequency s v (fc ) = NT ) (1/NT ) (f = f ) d c c 30 Kmph 111 Hz 1 ( 1.5%) 4 GHz 128 8.75 ms 114 Hz 120 Kmph 444 Hz 4 (≈ 6%) 500 Kmph 1850 Hz 16(≈ 25%) ≈

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 24 / 86 OTFS modulation

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 25 / 86 OTFS modulation

Time-Frequency Domain x[k; l] X[n; m] Heisenberg s(t) Channel r(t) Wigner Y [n; m] y[k; l] ISFFT SFFT Transform h(τ; ν) Transform

Delay-Doppler Domain

Figure: OTFS mod/demod

Time–frequency domain is similar to an OFDM system with N symbols in a frame (Pulse-Shaped OFDM)

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 26 / 86 Time–frequency domain

Modulator – Heisenberg transform

N−1 M−1 X X j2πm∆f (t−nT ) s(t) = X [n, m]gtx(t nT )e − n=0 m=0

Simplifies to IFFT in the case of N = 1 and rectangular gtx Channel Z r(t) = H(t, f )S(f )ej2πft df

Matched filter – Wigner transform Z ∗ 0 0 −j2πf (t0−t) 0 Y (t, f ) = Ag ,r (t, f ) g (t t)r(t )e dt rx , rx −

Y [n, m] = Y (t, f ) t=nT ,f =m∆f |

Simplifies to FFT in the case of N = 1 and rectangular grx

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 27 / 86 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 ZZ H[n, m] = h(τ, ν)ej2πνnT e−j2πm∆f τ dτdν Section 2.2: A discretized system model 11

Symbol f

Subcarrier ···

2F F t 0

0 T 2T ··· Figure 2.1: Pulse-shaped OFDM interpretation of the signaling scheme (2.13). The shadedFigure areas represent: Time–frequency the approximate time-frequency support domain 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- * F. Hlawatsch and G. Matz,tion Eds.,LH(t, f ) [seeChapter (2.2)]. Furthermore, 2, theWireless noise coefficients Communicationsw[k,l] are i.i.d. CN (0,1) as Over Rapidly a consequence of the orthonormality of (g(t),T,F). These two properties are crucial for Time-Varying Channels. Newthe York, ensuing analysis. NY, USA: Academic, 2011 A drawback of (2.14) is the presence of (self-)interference [the second term(PS-OFDM) in (2.14)], which makes the derivation of capacity bounds involved, as will be seen in Section 2.4. (Monash University, Australia) The signaling scheme (2.13)OTFS can be interpreted modulation as PS-OFDM [KM98], where the input SPCOM 2020, IISc 28 / 86 symbols s[k,l] are modulated onto a set of orthogonal signals, indexed by discrete time (symbol index) k and discrete frequency (subcarrier index) l. From this perspective, the interference term in (2.14) can be interpreted as intersymbol and intercarrier interference (ISI and ICI). Fig. 2.1 provides a qualitative representation of the PS-OFDM signaling scheme.

2.2.2.4 Outline of the information-theoretic analysis The program pursued in the next sections is to tightly bound the noncoherent capacity of the discretized channel (2.14) under an average-power and a peak-power constraint on the input symbols s[k,l]. We refer the interested reader to [DMB09,DMBon] for a discussion on the relation between the capacity of the discretized channel (2.14) and the capacity of the underlying continuous-time channel (2.1). The derivation of capacity bounds is made difficult by the presence of the interference term in (2.14). Fortunately, to establish our main results in Sections 2.4 and 2.5, it will be Signaling in the delay–Doppler domain

Time–frequency input-output relation

Y [n, m] = H[n, m]X [n, m]

where

X X j2π nk − ml  H[n, m] = h [k, l] e N M k l ISFFT

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

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

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 29 / 86 Delay–Doppler domain input-output relation

Received signal in delay–Doppler domain

P X y[k, l] = hi x[[k kν ]N , [l lτ ]]M − i − i i=1 = h[k, l] x[k, l] (2D Circular Convolution) ∗

1

1 1 0.8

0.8 0.8 0.6

0.6 0.6

0.4 0.4 0.4

0.2 0.2 0.2

0 0 0 0 0 1 5 2 5 3 20 10 4 10 10 5 9 15 6 8 15 7 7 15 15 6 10 8 20 5 10 9 4 20 10 25 5 3 5 2 25 1 30

(a) Input signal, x[k, l] (b) Channel, h[k, l] (c) Output signal, y[k, l] Figure: OTFS signals

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 30 / 86 Fractional doppler effect

Actual Doppler may not be perfectly aligned with the grid

 1  νi = (kνi + κνi ) , kνi Z, 1/2 < κνi < 1/2 NT ∈ −

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

P N i  j2π(−q−κν )  X X e i 1 y(k, l) = h x [[k k + q] , [l l ] ] i j 2π (−q−κ )− νi N τi M Ne N νi N − − i=1 q=−Ni −

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 31 / 86 Compatibility with OFDM architecture

Time-Frequency Domain (N OFDM symbols) x[k; l] X[n; m] s(t) r(t) Y [n; m] y[k; l] Precoder OFDM Channel OFDM Decoder (ISFFT) Modulator H(t; f) Demodulator (SFFT)

Delay-Doppler Domain

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

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 32 / 86 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

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 33 / 86 OTFS Input-Output Relation in Matrix Form

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 34 / 86 OTFS transmitter implementation: M = 2048, N = 128

time time (128 symbols) (128 symbols) IFFT IFFT 128 128 XMxN . XMxN . time (128 symbols) Q-QAM . … Q-QAM . … delay MN*log2(Q) bits . delay ISFFT MN*log2(Q) bits . MxN delay (M=2048) delay (M=2048) IFFT IFFT 128 128 Doppler (N=128) Doppler (N=128) FFT … FFT 2048 2048

time (128 symbols)

…

Heisenberg transform frequency M>N TX complexity PAPR

(2048 subcarriers)(2048 P/S+CP time-frequency -> time OTFS MN*log (N) N (N-symbol OFDM transmitter) 2 IFFT … IFFT OFDM MN*log2(M) M 2048 2048

P/S+CP Only 2048 samples one CP …

Time domain signal (128 symbols, 2048 samples each)

OTFS transmitter implements inverse ZAK transform (2D 1D) →

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 35 / 86 OTFS: Tx matrix representation

Transmit signal at 2D time domain: ISFFT+Heisenberg+pulse shaping on delay–Doppler H H H S = GtxFM (FM XFN ) = GtxXFN | {z } ISFFT In vector form:

H s = vec(S) = (F Gtx)x N ⊗

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 36 / 86 OTFS receiver implementation: M = 2048, N = 128

2048 samples time varying … channel Time domain signal (128 symbols, 2048 samples each)

time FFT (128 symbols) 128 remove . YMxN CP … . received + delay . Symbols

S/P (M=2048)elay

FFT d 128 Doppler (N=128)

Received signal at delay–Doppler domain: pulse shaping+Wigner+SFFT on time–frequency received signal

H Y = FM (FM GrxR)FN = GrxRFN In vector form:

y = (FN Grx)r ⊗ OTFS receiver implements ZAK transform (1D 2D) → (Monash University, Australia) OTFS modulation SPCOM 2020, IISc 37 / 86 OTFS: matrix representation – channel

Received signal in the time–frequency domain ZZ r(t) = h(τ, ν)s(t τ)ej2πν(t−τ)dτdν + w(t) − Channel P X h(τ, ν) = hi δ(τ τi )δ(ν νi ) − − i=1 Received signal in discrete form

P X j2πki (n−li ) r(n) = hi e MN s([n li ]MN ) + w(n), 0 n MN 1 | {z }| −{z } ≤ ≤ − i=1 Doppler Delay

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 38 / 86 OTFS: matrix representation – channel

Received signal in vector form

r = Hs + w

H is an MN MN matrix of the following form × P X li (ki ) H = hi Π ∆ , i=1

where, Π is the permutation matrix (forward cyclic shift), and ∆(ki ) is the diagonal matrix

   j2πki (0)  0 0 1 e MN 0 0 ··· . j2πki (1) ···  .   MN  1 . 0 0 (k )  0 e 0  Π =   , ∆ i =  . ··· .  ......   . .. .  . . . .  . . .  j2πki (MN−1) 0 1 0 0 0 e MN ··· MN×MN | {z } | {z··· } Delay (similar to OFDM) Doppler

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 39 / 86 OTFS: matrix representation – channel

Received signal at delay–Doppler domain  H  y = (FN Grx)H(F Gtx) x + (FN Grx)w ⊗ N ⊗ ⊗ = Heffx + we Effective channel for arbitrary pulses

H Heff =( IN Grx)(FN IM )H(FN IM )(IN Gtx) ⊗ rect⊗ ⊗ ⊗ = (IN Grx) Heff (IN Gtx) ⊗ |{z} ⊗ Channel for rectangular pulses (Gtx=Grx=IM ) Effective channel for rectangular pulses

P h i rect X  li H  (ki ) H H = hi (FN IM )Π (F IM ) (FN IM )∆ (F IM ) eff ⊗ N ⊗ ⊗ N ⊗ i=1 | {z } (i) | {z } P (delay) Q(i) (Doppler) P P X (i) (i) X (i) = hi P Q = hi T i=1 i=1

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 40 / 86 rect OTFS: Example for computing Heff

M = 2, N = 2,MN = 4 1

li = 0 and ki = 0 (no delay and Doppler) 0

li =0 (i) H Π = I4 ⇒ P = (F2 ⊗ I2)(F2 ⊗ I2) = I4 0 1

(ki =0) (i) H ∆ = I4 ⇒ Q = (F2 ⊗ I2)(F2 ⊗ I2) = I4 (i) (i) (i) T = P Q = I4 ⇒ Narrowband channel

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 41 / 86 rect OTFS: Example for computing Heff

1 li = 1 and ki = 0 (delay but no Doppler) 0

0 0 0 1 0 1 21 0 0 03 Πli =1 = ⇒ block circulant matrix with 2 × 2 (M × M) block size 60 1 0 07 6 7 40 0 1 05 0 1 0 0 21 0 0 0 3 (i) H P = (F2 ⊗ I2)Π(F2 ⊗ I2)= − 2π 1 60 0 0 e j 2 7 6 7 40 0 1 0 5 (using the block circulant matrix decomposition → generalization of circulant matrix decomposition in OFDM)

(ki =0) (i) H ∆ = I4 ⇒ Q = (F2 ⊗ I2)(F2 ⊗ I2) = I4 T(i) = P(i) ⇒ T(i)s → circularly shifts the elements in each block (size M) of s by 1 (delay shift)

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 42 / 86 rect OTFS: Example for computing Heff

1 li = 0 and ki = 1 (Doppler but no delay) 0

li =0 (i) H Π = I4 ⇒ P = (F2 ⊗ I2)(F2 ⊗ I2) = I4 0 1 1 0 0 0 2π 1 20 ej 4 0 0 3 (ki =1) ∆ = 2π 2 ⇒ block diagonal matrix with 2 × 2 (M × M) 60 0 ej 4 0 7 6 2π 3 7 40 0 0 ej 4 5 block size 0 0 1 0 2 j2π 1 3 (i) (1) H 0 0 0 e 4 Q = (F2 ⊗ I2)∆ (F ⊗ I2) = 2 61 0 0 0 7 6 2π 1 7 40 ej 4 0 0 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)

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 43 / 86 rect OTFS: Example for computing Heff

1 li = 1 and ki = 1 (both delay and Doppler) 0

0 1 0 0 0 1 21 0 0 0 3 (i) P = − 2π 1 60 0 0 e j 2 7 6 7 40 0 1 0 5 0 0 1 0 2π 1 20 0 0 ej 4 3 Q(i) = 61 0 0 0 7 6 2π 1 7 40 ej 4 0 0 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)

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 44 / 86 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.

 −j2π n j2π ki ([m−li ]M ) e N e MN , if q = [m li ]M + M[n ki ]N and m < li  k ([m−l ] ) − − (i) j2π i i M T (p, q) = e MN , if q = [m li ]M + M[n ki ]N and m li  − − ≥ 0, otherwise.

Example: li = 1 and ki = 1

 j2π 1  0 0 0 e 4 (i) 0 010  T =  −j2π 1  0 e 4 0 0  1000

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 45 / 86 OTFS Signal Detection

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 46 / 86 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]) P X y[k, l] = hi x[[k kν ]N , [l lτ ]M ] + w[k, l] k = 1 ... N, l = 1 ... M (1) − i − i i=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.

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 47 / 86 Message passing based detection

Symbol-by-symbol MAP detection  xb[c] = arg max Pr x[c] = aj y, H aj ∈A 1  = arg max Pr y x[c] = aj , H aj ∈A Q Y  arg max Pr y[d] x[c] = aj , H ≈ aj ∈ A d∈Jc

Received signal y[d] X y[d] = x[c]H[d, c] + x[e]H[d, e] + z[d]

e∈Id ,e6=c | {z } (i) ζd,c → assumed to be Gaussian

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 48 / 86 Messages in factor graph

Algorithm 1 MP algorithm for OTFS symbol detection Input: Received signal y, channel matrix H (0) Initialization: pmf pc,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 xb[c]

y[d] y[e1] y[eS]

2 2 (µd;e1 ; σd;e1 ) (µd;eS ; σd;eS )

pc;e1 pc;eS

x[e1] x[eS] x[c]

fe1; e2; ··· ; eSg = Id fe1; e2; ··· ; eSg = Jc Observation node messages Variable node messages

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 49 / 86 Messages in factor graph – observation node messages

y[d]

2 2 (µd;e1 ; σd;e1 ) (µd;eS ; σd;eS ) Received signal X x[e1] x[eS] y[d] = x[c]H[d, c] + x[e]H[d, e] + z[d] e∈I(d),e6=c fe1; e2; ··· ; eSg = Id | {z } (i) ζd,c → assumed to be Gaussian

Mean and Variance

Q (i) X X (i−1) µd,c = pe,d (aj )aj H[d, e] e∈I(d),e6=c j=1

 2 Q Q (i) 2 X X (i−1) 2 2 X (i−1) 2 (σ ) =  p (aj ) aj H[d, e] p (aj )aj H[d, e] + σ d,c  e,d | | | | − e,d  e∈I(d),e6=c j=1 j=1

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 50 / 86 Messages in factor graph – variable node messages

y[e1] y[eS]

p p Probability update with damping c;e1 c;eS factor ∆ x[c]

(i) (i) (i−1) fe1; e2; ··· ; eSg = Jc p (aj ) = ∆ p˜ (aj ) + (1 ∆) p (aj ), aj A c,d · c,d − · c,d ∈ where

(i) Y   p˜ (aj ) Pr y[e] x[c] = aj , H c,d ∝ e∈J (c),e6=d Y ξ(i)(e, c, j) = PQ (i) e∈J (c),e6=d k=1 ξ (e, c, k)  2  (i) y[e] µe,c He,c ak ξ(i)(e, c, k) = exp − − −   (i) 2  (σe,c )

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 51 / 86 Final update and stopping criterion

Final update

(i) (i) Y ξ (e, c, j) p (aj ) = c PQ (i) e∈J (c) k=1 ξ (e, c, k) (i) xb[c] = arg max pc (aj ), c = 1, , NM. aj ∈A ··· Stopping Criterion Convergence Indicator η(i) = 1

NM   (i) 1 X (i) η = I max pc (aj ) ≥ 0.99 NM aj ∈ c=1 A Maximum number of Iterations

Complexity (linear)– (niter SQ) per symbol which is much less even compared to a linear MMSEO ((NM)2) O

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 52 / 86 Simulation results – damping factor ∆

100 45 OTFS, 120 Kmph 40 OTFS, 120 Kmph 10-1 35 -2 4-QAM, SNR = 18 dB 4-QAM, SNR = 18 dB 10 30

BER 25 10-3 20 10-4 15 Average no. of iterations

10-5 10 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 ∆ ∆ Figure: Variation of BER and average iterations no. with ∆. Optimal for ∆ = 0.7

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 53 / 86 Simulation results – OTFS vs OFDM with ideal pulses

100 OTFS, Ideal, 30 Kmph OTFS, Ideal, 120 Kmph OTFS, Ideal, 500 Kmph -1 4-QAM 10 OFDM, 30 kmph OFDM, 120 kmph OFDM, 500 kmph 10-2 BER 10-3

10-4

10-5 5 10 15 20 25 30 SNR in dB Figure: The BER performance comparison between OTFS with ideal pulses and OFDM systems at different Doppler frequencies.

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 54 / 86 Simulation results – IDI effect

100 OTFS, 18 dB, 120 Kmph OTFS, 18 dB, 500 Kmph OTFS, 15 dB, 120 Kmph 10-1

4-QAM 10-2 BER 10-3

10-4

10-5 0 5 10 15 20 N i

Figure: The BER performance of OTFS for different number of interference terms Ni with 4-QAM.

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 55 / 86 Simulation results – Ideal and Rectangular pulses

100

10-1

10-2 BER -3 10 ×10-4 3.8 OTFS, Rect., WC, 30 Kmph OTFS, Rect., WC, 120 Kmph OTFS, Rect., WC, 500 Kmph 10-4 OTFS, Rect., WO, 30 Kmph OTFS, Rect., WO, 120 Kmph 3.795 OTFS, Rect., WO, 500 Kmph OTFS, Ideal 14.2 14.3 14.4 OFDM, 500 kmph 10-5 5 10 15 20 25 30 SNR in dB

Figure: The BER performance of OTFS with rectangular and ideal pulses at different Doppler frequencies for 4-QAM.

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 56 / 86 Simulation results – Ideal and Rect. pulses - 16-QAM

100 OTFS, Rect., WC, 30 Kmph OTFS, Rect., WC, 120 Kmph 16-QAM OTFS, Rect., WC, 500 Kmph OTFS, Ideal -1 10 OTFS, Rect., WO, 120 Kmph OFDM

10-2 BER

10-3

10-4 10 15 20 25 30 35 SNR in dB Figure: The BER performance of OTFS with rectangular and ideal pulses at different Doppler frequencies for 16-QAM.

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 57 / 86 Simulation results – Low latency

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 10-2

BER 16-QAM

10-3

10-4 20 25 30 35 40 SNR in dB Figure: The BER performance of OTFS with rectangular pulses and low latency (N = 16, Tf ≈ 1.1 ms).

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 58 / 86 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

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 59 / 86 Matlab code

OTFS – 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 →

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 60 / 86 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: −1 xˆ = (HH H + λI) 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.

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 61 / 86 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

yˆ(i)j+1 = y − Hˆxj + H(:, i)ˆx(i)j   ˆx(i)j+1 = arg min yˆ(i)j+1 − H(:, i)a a∈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.

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 62 / 86 OTFS channel estimation

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 63 / 86 Channel estimation in time–frequency domain

(lτi , kνi ) ((delay,Doppler)) values are obtained from the baseband time domain signal equation

P X j2πνi (t−τi ) y(t) = hi x(t τi )e − i=1

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.

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 64 / 86 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 M−1 N−1 X X 1  k − n l − m  xˆ [k, l] = x [n, m] h , + v [k, l]. q p MN wqp NT M∆f q m=0 n=0 ————— 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.

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 65 / 86 If we transmit

xp[n, m] = 1 if (n, m) = (np, mp)

= 0 (n, m) = (np, mp), ∀ 6 as pilot from the pth antenna, the received signal at the qth antenna will be   1 k np l mp xˆ [k, l] = h − , − + v [k, l]. q MN wqp NT M∆f q

1 k l  ˆ MN hwqp NT , M∆f and thus Hqp 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

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 66 / 86 Figure: Illustration of pilots and channel response in delay-Doppler domain in a 2×1 MIMO-OTFS system

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 67 / 86 SISO OTFS system with integer Doppler

N − 1 N − 1 55555555555 55555555555

kp + 2kν 5 5 55 5 5 5 5 5 5 5

kp + kν 5 5 55 5 ⊞ ⊞ ⊞ 5 5 5 k kp p 5 5 5 5 5 ⊞ ⊞ ⊞ 5 5 5

kp − kν 5 5 5 5 5 ⊞ ⊞ ⊞ 5 5 5

kp − 2kν 5 5 55 5 5 5 5 5 5 5 1 1 55555555555 0 0 55555555555

0 1 lp − lτ lp lp + lτ M − 1 0 1 lp − lτ lp lp + lτ M − 1

(a) Tx symbol arrangement (: pilot; ◦: (b) Rx symbol pattern (O: data detection, guard symbols; ×: data symbols) : channel estimation)

Figure: Tx pilot, guard, and data symbols and Rx received symbols

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 68 / 86 SISO OTFS system with fractional Doppler

N − 1 N − 1 55555555555 ^ 55555555555 kp + 2kν + 2k k k k^ 5 5 55 5 ⊞ ⊞ ⊞ 5 5 5 kp + 2kν p + ν +

kp + kν 5 5 55 5 ⊞ ⊞ ⊞ 5 5 5 k ⊞ ⊞ ⊞ kp p 5 5 5 5 5 5 5 5

kp − kν 5 5 5 5 5 ⊞ ⊞ ⊞ 5 5 5 ^ kp − 2kν kp − kν − k 5 5 55 5 ⊞ ⊞ ⊞ 5 5 5 ^ kp − 2kν − 2k 55555555555 0 0 55555555555

0 1 lp − lτ lp lp + lτ M − 1 0 1 lp − lτ lp lp + lτ M − 1

(a) Tx symbol arrangement (: pilot; ◦: (b) Rx symbol pattern (O: data detection, guard symbols; ×: data symbols) : channel estimation) Figure: Tx pilot, guard, and data symbols and Rx received symbols

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 69 / 86 MIMO OTFS system with fractional Doppler

N − 1 ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ^ kp + 2kν + 2k ⋄ ⋄ ⋄ ⋄ ⊕ ⊕ ⊕ ⊕ ⋄ ⋄ ⋄ ⋄ ⊕ ⊕ ⊕ ⊕

kp ⋄ ⋄ ⋄ ⋄ ⊕ ⊕ ⊕ ⊕ ⋄ ⋄ ⋄ ⋄ ⊕ ⊕ ⊕ ⊕ ^ kp − 2kν − 2k ⋄ ⋄ ⋄ ⋄ ⊕ ⊕ ⊕ ⊕ ⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄ ⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕ 0 ⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄ ⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕

0 lp−lτ lp M − 1 0 lp−lτ lp +lτ +1 M − 1 0 lp−lτ lp +2lτ +2 M − 1

(a) Antenna 1 (×: antenna 1 (b) Antenna 2 (♦: antenna 2 (c) Antenna 3 (⊕: antenna data symbol) data symbol) 3 data symbol)

Figure: Tx pilot, guard, and data symbols for MIMO OTFS system (: pilot; ◦: guard)

N − 1 5 5 5 5 5 5 5 5 5 5 5 55555555555 5 5 5 5 5 5 5 5 5 5 5 ^ kp + kν + k 5 5 5 ⊞ ⊞ ⊠ ⊠ ⊗ ⊗ 5 5

kp 5 5 5 ⊞ ⊞ ⊠ ⊠ ⊗ ⊗ 5 5 ^ kp − kν − k 5 5 5 ⊞ ⊞ ⊠ ⊠ ⊗ ⊗ 5 5 5 55 5 5 5 5 5 5 5 5 55555555555

0 55555555555

0 lp lp +lτ +1 M − 1

Figure: Rx symbol pattern at antenna 1 of MIMO OTFS system (O: data detection, , , ⊗: channel estimation for Tx antenna 1, 2, and 3, respectively)

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 70 / 86 Multiuser OTFS system – uplink

N − 1 ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ^ kp + 2kν + 2k ⋄ ⋄ ⋄ ⋄

kp ⋄ ⋄ ⊕ ⊕ ⊕ ⊕ ^ kp − 2kν − 2k ⊕ ⊕ ⊕ ⊕ ⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕ 0 ⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕

0 lp−lτ lp M − 1 0 lp−lτ lp +lτ +1 M − 1 0 lp−lτ lp +2lτ +2 M − 1

(a) User 1 (×: user 1 data (b) User 2 (♦: user 2 data (c) User 3 (⊕: user 3 data symbol) symbol) symbol)

Figure: Tx pilot, guard, and data symbols for multiuser uplink OTFS system (: pilot; ◦: guard symbols)

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 71 / 86 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 2 Channel estimation threshold is 3σp, where σp = 1/SNRp is effective noise power of the pilot signal

10-1 kˆ = 2 -1 10 ˆ 30 Kmph k = 5 120 Kmph -2 Full Guard 10 500 Kmph Ideal 10-2 Ideal

10-3 BER 10-3 N = 128, M = 512, l = 20,

BER τ N = 128, M = 512, l = 20, SNR = 50 dB, 4-QAM τ 10-4 p SNR = 40 dB, 4-QAM p 10-4

10-5 10 12 14 16 18 -5 10 SNR in dB 10 12 14 16 18 d SNR in dB d (a) BER for estimated channels of different (b) BER for estimated channels of Integer Dopplers Fractional Doppler

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 72 / 86 OTFS applications

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 73 / 86 OTFS SDR implementation block diagram

Information bits OTFS Modulation

x[k,l] X[n,m] s[n]

QAM Heisenberg Add cyclic Pulse Add ISFFT modulation Transform prefix Shaping Preamble Channel OTFS Demodulation

y[k,l] Y[n,m] r[n] Preamble Reciever Channel Wigner Remove Detection and SFFT Matched Estimation Transform cyclic prefix Frame Filter Synchronization

Message QAM Information passing demodulation bits detector

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 74 / 86 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.

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 75 / 86 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.

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 76 / 86 CFO at the receiver will introduce a Doppler shift to all the received symbols. 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.

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 77 / 86 Error performance

OTFS Modem Performance OTFS vs OFDM (4-QAM) 100 100

10-1 10-1

10-2 10-2 BER

BER / FER 10-3 10-3

10-4 10-4 frame error rate(16-QAM) OTFS-static channel bit error rate(16-QAM) OFDM-static channel frame error rate(4-QAM) OTFS-mobile channel bit error rate(4-QAM) OFDM-mobile channel 10-5 10-5 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 Tx gain in dB Tx gain in dB

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 78 / 86 OTFS with static multipath channels (zero Doppler)

Received signal

H (size MN 1) y = (FN IM )H(F IM )x + w × ⊗ N ⊗ e zero Doppler ↓ (size M 1) yn = H˘ nxn + wn, for n = 0, , N 1 × e ··· − Equivalent to A-OFDM (asymmetric OFDM) in (*)

˘ −j2π n Hn structure for M L h0 0 ··· h1e N  ≥ −j2π n h1 h0 ··· h2e N  ˘   Hn =  . . . .   ......  0 ··· h h 1 0 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.

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 79 / 86 OTFS with static multipath channels (zero Doppler)

100 N L M = 1, OFDM c = 1024, = 72, 16-QAM M = 2 M = 4 M = 128, 256, 1024; MP 10-1 CPSC, MMSE M = 128; A-OFDM, ZF M = 128; A-OFDM, MMSE AWGN 10-2 BER

10-3

10-4 15 20 25 30 35 40 SNR in dB

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.

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 80 / 86 ReferencesI

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.

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 81 / 86 ReferencesII

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.

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 82 / 86 ReferencesIII

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: https://arxiv.org/abs/1709.02505. 15 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.

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 83 / 86 ReferencesIV

16 T. Zemen, M. Hofer, D. Loeschenbrand, and C. Pacher, “Iterative detection for orthogonal precoding in doubly selective channels,” available online: https://arxiv.org/pdf/1710.09912.pdf. 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: https://arxiv.org/abs/1805.02209. 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.

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 84 / 86 ReferencesV

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.

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 85 / 86 Thank you!!!

Questions?

(Monash University, Australia) OTFS modulation SPCOM 2020, IISc 86 / 86