Question

The approximation of the integral in y(t) = \(\int_{t_0}^t y'(τ)dt+y(t_0)\) by the Trapezoidal formula at t = nT and t0=nT-T yields equation?

(a) y(nT) = \(\frac{T}{2} [y^{‘} (nT)+y^{‘} (T-nT)]+y(nT-T)\)

(b) y(nT) = \(\frac{T}{2} [y^{‘} (nT)+y^{‘} (nT-T)]+y(nT-T)\)

(c) y(nT) = \(\frac{T}{2} [y^{‘} (nT)+y^{‘} (T-nT)]+y(T-nT)\)

(d) y(nT) = \(\frac{T}{2} [y^{‘} (nT)+y^{‘} (nT-T)]+y(T-nT)\)

I had been asked this question in an interview.

This interesting question is from IIR Filter Design by the Bilinear Transformation topic in chapter Discrete Time Systems Implementation of Digital Signal Processing

Answer 1

The correct answer is (b) y(nT) = \(\frac{T}{2} [y^{‘} (nT)+y^{‘} (nT-T)]+y(nT-T)\)

The explanation: By integrating the equation,

y(t) = \(\int_{t_0}^t y^{‘} (τ)dt+y(t_0)\) at t=nT and t0=nT-T we get equation,

y(nT) = \(\frac{T}{2} [y^{‘} (nT)+y^{‘} (nT-T)]+y(nT-T)\).