Question

8-4) Lathi & Ding, Prob. P.5.1-10 (a) A first-order-hold circuit can also be used to reconstruct a signal g) from its samples. The impulse response of this circuit is h(f) = Δ(2,-2) where Ts is the sampling interval. Consider a typical sampled signal R() and show that this circuit performs the linear interpolation. In other words, the filter output consists of sample tops connected by straight-line segments. Follow the procedure discussed in Sec. 5.1.2 (Fig. 5.6) for a typical signal g(1) (b) Determine the transfer function of this filter and its amplitude response, and compare it with the ideal filter required for signal reconstruction.

Answer 1

When the signal X (t) is reconstructed using FOH each sample generates a triangle of width 2 T and centered at the sampling instant. The height of the triangle will be equal to the sample value. Sketch the output of FOH circuit is shown below, The resulting signal consists of straight line segments joining the sample tops, Thus, it is shown that FOH Circuit performs linear interpolation. (b) The frequency response of the circuit is given by the transfer function H (w) is the Fourier transform of h (t). H (w) = F [delta (t/ 2T)] H (w) = T sin c^2 (wT /2) Hence, the frequency response is H (w) = T sinc^2 (wT /2) The amplitude response is given by, Vertical-bar H (w) Vertical-bar . Since, H (w) is positive for all w the amplitude response is also given by H (w) Thus the amplitude response is Vertical-bar H (w) Vertical-bar = T sinc^2 (wT /2)