Channel Identification Machines

Figure 7

Channel identification for ℎ(𝑡)=𝛿(𝑡). (a) Input signals 𝑢¹,𝑢² are bandlimited to 50 Hz. The order of the space 𝐿=10. (b) Biased output of the filter 𝑣¹(𝑡)+𝑏 in response to the stimulus 𝑢¹. (c) The filter output in (b) is integrated by an ideal IAF neuron. (d) The neuron generated a total of 28 spikes in response to 2 input signals. (e) The identified filter 𝒫ℎ^∗ (red) is exactly the kernel 𝐾(𝑡,0) for ℋ¹_Ω,𝐿 with Ω=2𝜋⋅10rad/s and 𝐿=10. Also shown is the original filter ℎ=𝛿 (dashed black) and its projection 𝒫ℎ=𝛿∗𝐾(⋅,0)=𝐾(⋅,0) (blue). The MSE between 𝒫ℎ^∗ and 𝒫ℎ is −87.6 dB. (f)–(h) Fourier amplitude spectra of 𝐾, ℎ and 𝒫ℎ^∗. As before, 𝒫ℎ^∗∈ℋ but ℎ∉ℋ.