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DHAYABARAN et L.: TRANSMIT SIGNAL SHAPING FOR MOLECULAR COMMUNICATION
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C. Transmit Signal Shaping Using Least Squares Optimization
Analytical Impulse Signal = 13ms, A = 770
T, = 13ms, A = 770 T = 0.5s, A = 20 T = 1s, A = 10
In this section, we design an optimal transmit signal that minimizes the effect of ISI using constrained LS optimization. Let h = [h1 h2 hn]T be a vector whose elements are the samples of the diffusive channel impulse response and let
0.5s, A = 2
samples of the transmit signal. The constrained LS problem can be formulated as [14]
0.5
Time (s)
1
1.5
Fig. 2. Expected number of received molecules for Rectangular signal, Te = 1 s, r = 5 m, R = 4 m, D = 79.4 m2/s, = 0.001s, N = 10000.
min J(x = h - HxTh - Hx s.t. 1x = Ntx ∀ >> 0
(14)
Analytical Impulse Signal
T = 0.04s, A = 80 0.6s, A = 100
where 1 is a row vector with all 1's entries Hx is the response of the molecular channel to the input x, and H is the convolution matrix of size (n - m given by
= 0.6s, A = 100 = 1s, A = X
/h1 0 0 ... 0 h2 hi 0 ... 0 . . . - .. ... . - thn hn-1 hn-2 .. . hn-m/
H
(15)
0.5
Time (s)
Fig. 3. Expected number of received molecules for Exponential signal, The objective of (14) is to minimize the square Euclidean norm Te = 1 s, r = 5m, R = 4 m, D = 79.4 m/s, = 0.001s, Nx = 10000. of the error between the channel response to be designed and the ideal one. We can rewrite (14) as
where A is the molecular concentration at = 0, and is a positive scalar defining the rate of decay. To ensure that the total number of transmitted molecules remains Ntx, A must be chosen as
min Jx = h - HxTh - Hx s.t. 1x = Ntx Gxb,
(16)
aNtx
where matrix G = Im) being Im the identity matrix of size m - m, and vector b =
= V = N = P - V 1 - e - aT)
Theorem 2: For the DBMC system considered in Section II, m m the number of molecules absorbed by the receiver until time We use the technique of Lagrangian multipliers to t, corresponding to the exponential transmit signal defined solve14[14]. The Lagrangian function corresponding in 11, is obtained as to (14) is
(13) Jx, X, = h - Hx h - Hx + X1x - N + Gx - b, 17 where and = [1, 2.. 2m]T are Lagrangian where multipliers. The optimal value of x that minimizes the cost function (17) is given by [14] 1t = [1 - e - aterf(b / 1 - e) {w(+w} g2(t) = L - 1 {G2(s)} = e - (aTe)g1(t - Te)u(t - Te), = 0 - [TX - GT] (18) with b = r2 / 4D, i = 1, z = at + i / b / t, 2 = -at + where , , and A are respectively given by i / b / t, and w(z = e - 22erfc - iz. Proof: The proof is provided in Appendix B. M = [GAGT] - I [GO - b] - [1AGT][10 - Nr] The windowed exponential transmit signals for different values of A and a are shown in Fig.1. Nrecr, t and Nexpr, t, X = [1A1T] - [18 - N] + [Gf1T]1[G8 - b] which correspond to different values of A, are presented in Fig. 2 and Fig. 3 respectively. As Te decreases, or consequently A increases, both Nrecr, t) and Nexpr, t) approach the average number of received molecules for an ideal DBMC The average number of received molecules corresponding to system. It can be observed that for a similar number of LS transmit signal is shown in Fig. 4. From the figure, it can be received molecules, the exponential signal has a longer Te inferred that, as A increases, the number of molecules absorbed than the rectangular one. by the receiver gets closer to that of the ideal system.