JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech II Year SIGNALS AND STOCHASTIC PROCESS Previous Question Paper (March - 2021 - R16)

The error function in signal approximation represents the difference between a signal f(t) and its approximation f'(t) constructed using a set of orthogonal basis functions. It is defined as the mean square error (MSE) over an interval (t1, t2), given by the integral of the squared difference: E = ∫[f(t) - f'(t)]^2 dt. Minimizing this error is the fundamental principle of orthogonal signal representation.

Two waveforms f1(t) and f2(t) are defined as orthogonal over an interval (t1, t2) if their inner product is zero. The condition is derived from the requirement that the energy of the sum of the signals equals the sum of their individual energies: ∫[f1(t) + f2(t)]^2 dt = ∫f1(t)^2 dt + ∫f2(t)^2 dt. Expanding the left side gives ∫f1(t)^2 dt + ∫f2(t)^2 dt + 2∫f1(t)f2(t) dt. For the equality to hold, the cross-product term must vanish, leading to the orthogonality condition:

Signal bandwidth refers to the range of frequencies over which the signal's spectral density is non-zero or above a specific threshold. System bandwidth is the range of frequencies over which a system provides a specified response, typically defined by the half-power or -3dB points.

The relationship between rise time (tr) and bandwidth (BW) is derived from the step response of a low-pass system, often modeled as an RC circuit. The step response is given by v(t) = V(1 - e^(-t/RC)). The rise time is defined as the time taken for the signal to rise from 10% to 90% of its final value. Solving 0.1 = 1 - e^(-t1/RC) and 0.9 = 1 - e^(-t2/RC) yields t1 = 0.1RC and t2 = 2.3RC. Thus, tr = t2 - t1 = 2.2RC. Since the bandwidth of an RC circuit is BW = 1/(2πRC), we have RC = 1/(2πBW). Substituting this into the rise time expression results in: