“…k−kν i +q] i N x [[k − k ν i + q] N , [l − l τ i ] M ] l < l τ i , 2: OTFS INPUT-OUTPUT RELATION IN DELAY-DOPPLER DOMAIN FOR IDEAL PULSESThe received signal y[k, l] for the ideal pulses, from(11) and(13), can be written asy[k, l] n,m [n, m]X[n, m]e −j2π nk N − ml M .By substituting the ISFFT equation from(2), y[k, l] can be expanded as in from(51) to (53).Here, h w [k − k ′ , l − l ′ ] can be seen as the value of h w (ν, τ ) sampled at ν = k−k ′ N T , τ = l−l ′ M ∆f .The value of h w (ν, τ ) can be obtained as from(54)to (56), by substituting H n,m [n, m] from (10), which completes the proof. APPENDIX C PROOF OF THEOREM 2: OTFS INPUT-OUTPUT RELATION IN DELAY-DOPPLER DOMAIN FOR RECTANGULAR PULSES We start with expanding y[k, l] in (11) using the Y [n, m] for rectangular pulses in (24) as in (57).…”