Operator Identification and Sampling

Time–invariant communication channels are usually modelled as convolution with a fixed impulse–response function. As the name suggests, such a channel is completely determined by its action on a unit impulse. Time–varying communication channels are modelled as pseudodifferential operators or superpositions of time and frequency shifts. The function or distribution weighting those time and frequency shifts is referred to as the spreading function of the operator. We consider the question of whether such operators are identifiable, that is, whether they are completely determined by their action on a single function or distribution. It turns out that the answer is dependent on the size of the support of the spreading function, and that when the operators are identifiable, the input can be chosen as a distribution supported on an appropriately chosen grid. These results provide a sampling theory for operators that can be thought of as a generalization of the classical sampling formula for bandlimited functions. 1. Channel Models and Identification A communications channel is said to be measurable or identifiable if its characteristics can be determined by its action on a single fixed input signal. A general model for linear (time-varying) communication channels is as operators of the form Hf(x) = ∫ hH(t, x) f(x− t) dt. The function hH(t, x) is referred to as the impulse response of the channel and is interpreted as the response of the channel at time x to a unit impulse at time x − t, that is, originating t time units earlier. If hH(t, x) = hH(t) then the characteristics of the channel are time-invariant and in this case the channel is modelled as a convolution operator. Such channels are identifiable since hH(t) can be recovered as the response of the channel to the input signal δ0(t), the unit-impulse at t = 0. There are two representations of H that will be convenient for our purposes. 1. Letting ηH(t, ν) = ∫ hH(t, x) e−2πiν(x−t) dx gives Hf(x) = ∫∫ ηH(t, ν) e2πiν(x−t) f(x− t) dν dt = ∫∫ ηH(t, ν)Tt Mνf(x) dν dt. ηH(t, ν) is the spreading function of H . If supp ηH ⊆ [0, a]× [−b/2, b/2] for some a, b > 0 then a is called the maximum time-delay and b the maximum Doppler spread of the channel. 2. Letting σH(x, ξ) = ∫ hH(t, x) e dt gives Hf(x) = ∫ σH(x, ξ)f̂(ξ) e dξ. σH(x, ξ) is the Kohn-Nirenberg (KN) symbol of H and we have the relation ηH(t, ν) = ∫∫ σH(x, ξ) e−2πi(νx−ξt) dx dξ. In other words, the spreading function ηH is the symplectic Fourier transform of the KN symbol of H . In 1963, T. Kailath [3, 4, 5] asserted that for time-variant communication channels to be identifiable it is necessary and sufficient that the maximum time-delay, a, and Doppler shift, b, satisfy ab ≤ 1 and gave an argument for this assertion based on counting degrees of freedom. In the argument, Kailath looks at the response of the channel to a train of impulses separated by at least a time units, so that in this sense the channel is being “sampled” by a succession of evenly-spaced impulse responses. The condition ab ≤ 1 allows for the recovery of sufficiently many samples of hH(t, x) to determine it uniquely. Kailath’s conjecture was given a precise mathematical framework and proved in [6]. The framework is as follows. Choose normed linear spaces D(R) and Y (R) of functions or distributions on R, and a normed linear space of bounded linear operators H ⊂ L(D(R), Y (R)). Each fixed element g ∈ D(R) induces a map Φg : H −→ Y (R), H 7→ Hg. If for some g ∈ D(R), Φg is bounded above and below, that is, there are constants 0 < A ≤ B such that for all H ∈ H, A‖H‖H ≤ ‖Hg‖Y ≤ B ‖H‖H ha l-0 04 51 44 7, v er si on 1 29 J an 2 01 0 Author manuscript, published in "SAMPTA'09, Marseille : France (2009)"