Interpolation of oversampled signals: Assume a function (t) bandlimited to ωm = π. If the sampling frequency is chosen at the Nyquist rate, ωs = 2π, the interpolation filter is the usual sinc filter with slow decay (∼ 1/t). If f(t) is oversampled, for example, with ωs = 3π, then filters with faster decay can be used for interpolating f(t) from its samples. Such filters are obtained by convolving (in frequency) elementary rectangular filters (two for H2(ω), three for H3(ω), while H1(ω) would be the usual sinc filter)
(a) Give the expression for h2(t), and verify that it decays as 1/t2.
(b) Same for h3(t), which decays as 1/t3. Show that H3(ω) has a continuous derivative. (c) By generalizing the construction above of H2(ω) and H3(ω), show that one can obtain hi(t) with decay 1/ti . Also, show that H(ω) has a continuous ( − 2)th derivative. However, the filters involved become spread out in time, and the result is only interesting asymptotically.