An introduction to Orthogonal Time Frequency Space (OTFS) modulation for high mobility communications Emanuele Viterbo Department of Electrical and Computer Systems Engineering Monash University, Clayton, Australia November, 2019 Special thanks to P. Raviteja and Yi Hong (Monash University, Australia) OTFS modulation 1 / 53
Overview I Introduction 1 Evolution of wireless High-Doppler wireless channels Conventional modulation scheme – OFDM Effect of high Dopplers in conventional modulation Wireless channel representation 2 Time–frequency representation Time–delay representation Delay–Doppler representation OTFS modulation 3 Signaling in the delay–Doppler domain (Monash University, Australia) OTFS modulation 2 / 53
Overview II OTFS Input-Output Relation in Matrix Form 4 OTFS Signal Detection 5 Vectorized formulation of the input-output relation Message passing based detection (Monash University, Australia) OTFS modulation 3 / 53
Introduction (Monash University, Australia) OTFS modulation 4 / 53
Evolution of wireless Mobile 4G LTE OFDMA Mobile 3G PS data, VOIP CDMA Mobile 2G Voice, SMS, PS data TDMA 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 5 / 53
High-Doppler wireless channels (Monash University, Australia) OTFS modulation 6 / 53
Wireless Channels - delay spread r 2 Reflected path r 3 r 1 LoS path Delay of LoS path: τ 1 = r 1 / c Delay of reflected path: τ 2 = ( r 2 + r 3 ) / c Delay spread: τ 2 − τ 1 (Monash University, Australia) OTFS modulation 7 / 53
Wireless Channels - Doppler spread Reflected path v cos θ θ LoS path v Doppler frequency of LoS path: ν 1 = f c v c Doppler frequency of reflected path: ν 2 = f c v cos θ c Doppler spread: ν 2 − ν 1 r ( t ) = h 1 s ( t − τ 1 ) e − j 2 πν 1 t + h 2 s ( t − τ 2 ) e − j 2 πν 2 t TX: s ( t ) RX: (Monash University, Australia) OTFS modulation 8 / 53
Typical delay and Doppler spreads Delay spread ( c = 3 · 10 8 m/s) ∆ r max Indoor (3m) Outdoor (3km) τ max 10ns 10 µ s Doppler spread ν max f c = 2GHz f c = 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 9 / 53
Conventional modulation scheme – OFDM OFDM - Orthogonal Frequency Division Multiplexing Subcarriers Frequency OFDM divides the frequency selective channel into M parallel sub-channels (Monash University, Australia) OTFS modulation 10 / 53
OFDM system model Figure: OFDM Tx Figure: OFDM Rx (*) From Wikipedia, the free encyclopedia (Monash University, Australia) OTFS modulation 11 / 53
OFDM system model Received signal – channel is constant over OFDM symbol (no Doppler) h = ( h 0 , h 1 , · · · , h P − 1 ) – Path gains over P taps h 0 0 0 h P − 1 h P − 2 h 1 · · · · · · h 1 h 0 0 0 h P − 1 h 2 · · · · · · . . ... ... ... ... ... ... . . . . . ... ... ... ... ... ... . . h P − 1 r = h ∗ s = s . ... ... ... ... ... ... . h P − 1 . . . ... ... ... ... ... ... . . . . . . ... ... ... ... ... ... . . . . 0 0 h P − 1 h P − 2 h 1 h 0 · · · · · · � �� � M × M Circulant matrix ( H ) Eigenvalue decomposition property H = F H DF where D = diag[ DFT M ( h )] (Monash University, Australia) OTFS modulation 12 / 53
OFDM system model At the receiver we have P − 1 � r = F H DFs = h i Π i s i =0 0 · · · 0 1 ... 1 0 0 where Π is the permutation matrix . . ... ... . . . . 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 ���� Diagonal matrix with subcarrier gains OFDM Pros Simple detection (one tap equalizer) Efficiently combat the multi-path effects (Monash University, Australia) OTFS modulation 13 / 53
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 14 / 53
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 15 / 53
Wireless channel representation (Monash University, Australia) OTFS modulation 16 / 53
Wireless channel representation Different representations of linear time variant (LTV) wireless channels time-variant impulse response g ( t; τ ) F F time-frequency delay-Doppler SFFT H ( t; f ) h ( τ ; ν ) response response (OFDM) (OTFS) F F B ( ν ; f ) Doppler-variant transfer response (Monash University, Australia) OTFS modulation 17 / 53
Wireless channel representation The received signal in linear time variant channel (LTV) � r ( t ) = g ( t , τ ) s ( t − τ ) d τ → generalization of LTI � �� � time-variant impulse response � � s ( t − τ ) e j 2 πν t d τ d ν → Delay–Doppler Channel = h ( τ, ν ) � �� � Delay–Doppler spreading function � S ( f ) e j 2 π ft df → Time–Frequency Channel = H ( t , f ) � �� � time-frequency response Relation between h ( τ, ν ) and H ( t , f ) � � H ( t , f ) e − j 2 π ( ν t − f τ ) dtdf h ( τ, ν ) = � � Pair of 2D symplectic FT h ( τ, ν ) e j 2 π ( ν t − f τ ) d τ d ν H ( t , f ) = (Monash University, Australia) OTFS modulation 18 / 53
Wireless channel representation 2 1 Doppler 0 -1 -2 0 1 2 3 4 Delay (Monash University, Australia) OTFS modulation 19 / 53
Wireless channel representation 2 1 Doppler 0 -1 -2 0 1 2 3 4 Delay (Monash University, Australia) OTFS modulation 20 / 53
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 21 / 53
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 22 / 53
Time–Frequency and delay–Doppler grids Assume ∆ f = 1 / T Delay 1 M M ∆ f Frequency ∆ f M 2D SFFT 2D ISFFT 2 1 1 2 N Time 2 T 1 Channel h ( τ, ν ) 1 2 N Doppler 1 NT � P h ( τ, ν ) = h i δ ( τ − τ i ) δ ( ν − ν i ) i =1 � � � 1 � 1 Assume τ i = l τ i and ν i = k ν i M ∆ f NT (Monash University, Australia) OTFS modulation 23 / 53
OTFS Parameters Subcarrier Bandwidth Symbol duration delay M l τ max spacing (∆ f ) ( W = M ∆ f ) ( T s = 1 / W ) spread 15 KHz 1024 15 MHz 0.067 µ s 4.7 µ s 71 ( ≈ 7%) Carrier Latency Doppler UE speed Doppler N k ν max frequency ( NMT s resolution ( v ) frequency v ( f c ) = NT ) (1 / NT ) ( f d = f 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 24 / 53
OTFS modulation (Monash University, Australia) OTFS modulation 25 / 53
OTFS modulation Time-Frequency Domain x [ k; l ] X [ n; m ] s ( t ) r ( t ) Y [ n; m ] y [ k; l ] Channel Wigner Heisenberg ISFFT SFFT h ( τ ; ν ) Transform 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 26 / 53
Time–frequency domain Modulator – Heisenberg transform N − 1 � M − 1 � X [ n , m ] g tx ( t − nT ) e j 2 π m ∆ f ( t − nT ) s ( t ) = n =0 m =0 Simplifies to IFFT in the case of N = 1 and rectangular g tx Channel � H ( t , f ) S ( f ) e j 2 π ft df r ( t ) = Matched filter – Wigner transform � rx ( t ′ − t ) r ( t ′ ) e − j 2 π f ( t ′ − t ) dt ′ g ∗ Y ( t , f ) = A g rx , r ( t , f ) � Y [ n , m ] = Y ( t , f ) | t = nT , f = m ∆ f Simplifies to FFT in the case of N = 1 and rectangular g rx (Monash University, Australia) OTFS modulation 27 / 53
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