JNTU B.Tech III SIGNALS AND SYSTEMS Previous Question Paper (August - 2022 - R18)

A set of functions is orthogonal over an interval [t0, t0 + T] if the integral of the product of any two distinct functions in the set is zero. For complex exponential functions of the form φ_n(t) = e^(jnω0t), where ω0 = 2π/T, we examine the inner product:

If m ≠ n, the integral evaluates to:

This is because the exponential function is periodic with period T, resulting in the same value at the boundaries. If m = n, the integral becomes:

Since the set spans the space of periodic functions with period T, it is considered a complete orthogonal set.

An RC network is linear if it satisfies the properties of additivity and homogeneity, and time-invariant if its characteristics do not change with time. In an RC circuit, the relationship between input voltage v_i(t) and output voltage v_o(t) is governed by the linear differential equation:

1. Linearity: Since the differential operator is linear, if inputs v_i1(t) and v_i2(t) produce outputs v_o1(t) and v_o2(t), then an input a*v_i1(t) + b*v_i2(t) produces an output a*v_o1(t) + b*v_o2(t). This satisfies the superposition principle.

2. Time-Invariance: The parameters R and C are constants independent of time. If an input v_i(t) produces an output v_o(t), a time-shifted input v_i(t - τ) produces the same output shifted by the same amount, v_o(t - τ), because the system coefficients do not depend on t.

Thus, the RC network is both linear and time-invariant.