Channel Identification Machines

Figure 5

Channel identification in a SISO [Filter]-[Ideal IAF] circuit using a single I/O pair. (a) An input signal 𝑢 is bandlimited to 25 Hz. The order of the space is 𝐿=5. (b) The corresponding biased output of the filter 𝑣(𝑡)+𝑏. (c) The filter output in (b) is integrated by the ideal IAF neuron. Whenever the membrane potential reaches a threshold 𝛿, a spike is produced by the neuron and the potential is reset to 0. (d) The neuron generated a total of 13 spikes. (e) The identified impulse response of the filter 𝒫ℎ^∗ (red) is shown together with the original filter ℎ (dashed black) and its projection 𝒫ℎ (blue). The MSE between 𝒫ℎ^∗ and 𝒫ℎ is −77.5dB. (f)–(h) Fourier amplitude spectra of 𝐾, ℎ and 𝒫ℎ^∗. Note that supp(ℱ𝐾)=supp(ℱ𝒫ℎ^∗)=[−Ω,Ω] but supp(ℱℎ)⊃[−Ω,Ω]. In other words, 𝒫ℎ^∗∈ℋ but ℎ∉ℋ.