Download Bounds for optimization of the reflection coefficient by by Arne Schneck PDF

By Arne Schneck

The aim of this ebook is twofold. Our place to begin is the layout of layered media with a prescribed mirrored image coefficient. within the first a part of this booklet we exhibit that the gap of bodily realizable mirrored image coefficients is very constrained via a couple of homes. within the moment half we contemplate a limited approximation challenge in Hardy areas. this is seen as an optimization challenge for the frequency reaction of a causal LTI method with constrained achieve.

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31). 2. CONTINUITY OF THE DIRECT PROBLEM notation. Multiplying by ϕ ∈ C ∞(0, d), d 0= u′′ (x)ϕ(x) + k 2n2 (x)u(x)ϕ(x) dx 0 ′ d ′ = u (d)ϕ(d) − u (0)ϕ(0) + 0 −u′ (x)ϕ′(x) + k 2 n2 (x)u(x)ϕ(x) dx = ik (u(d)ϕ(d) + u(0)ϕ(0)) d + 0 k 2n2 (x)u(x)ϕ(x) − u′ (x)ϕ′(x) dx − 2ikϕ(0). 33) where d Bn [u, ϕ] = ik (u(d)ϕ(d) + u(0)ϕ(0)) + 0 k 2n2 (x)u(x)ϕ(x) − u′ (x)ϕ′(x) dx. The careful reader should not confuse the Sobolev space H 1 (0, d) with the Hardy spaces H p (D) and H p (C+ ). 8. 33). 31). Proof. 33). Then u(x) = u(x) − u(0) − 1 (u(d) − u(0))x is a weak solution of d u′′(x) + k 2 n2(x)u(x) = f (x), u(0) = u(d) = 0, with f (x) = −k 2n2 (x) u(0) + d1 (u(d) − u(0))x .

E(x, t) = Ein (x)e−iωtey with Ein (x) = eikx. The layered medium gives rise to a reflected field E(x, t) = Eref (x)e−iωtey with Eref (x) = R(k)e−ikx. The number R(k) is called reflection coefficient (from the left). The reflected field is also a plane wave, but it travels in the opposite direction. Then the mapping L : Ein → Eref is an LTI system which maps eik· to R(k)eik· . Because the system should map a real incident field to a real reflected field, its impulse response should only take values in R.

Together with (c) we get for x ≥ 0 x 0 m2 (t, k)(1 − n2(t))D˙ k (x − t) dt x t ≤ 0 ≤ |k| x + x2 2 exp |k| 0 |1 − n2 (s)| ds |1 − n2(t)| (1 + |k|(x − t)) dt x 2 sup |1 − n (t)| exp |k| t∈[0,x] 0 |1 − n2 (t)| dt . 10). The estimates for m1 can be obtained in the same way. 3. 7) for m1 and m2 , one sees easily that it converges uniformly in k on compact subsets of C. Since the partial sums themselves are analytic in C, m1 (x, ·) and m2 (x, ·) are analytic in C. 4. One easily sees that since n is real-valued, the symmetry relations (or reality conditions) mj (x, −k) = mj (x, k) and uj (x, −k) = uj (x, k), j = 1, 2, hold.

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