JNTU B.Tech II Year SIGNALS AND STOCHASTIC PROCESS Previous Question Paper (October - 2020 - R16)

Two functions f1(t) and f2(t) are orthogonal over an interval (t1, t2) if the integral of their product, with one function complex conjugated, is zero:

Consider the complex exponential functions f_m(t) = exp(j m ω0 t) and f_n(t) = exp(j n ω0 t) over the interval T = 2π/ω0. The condition for orthogonality is:

If m ≠ n, let k = m - n (where k is a non-zero integer):

Substituting T = 2π/ω0:

Since exp(j 2π k) = cos(2π k) + j sin(2π k) = 1 + j(0) = 1 for any integer k, the expression becomes (1 - 1) / (j k ω0) = 0. Thus, the functions are orthogonal for m ≠ n.

i) Impulse Function (δ(t)): Defined as zero everywhere except at t=0, where it has an infinite peak such that its area is unity. It is the derivative of the unit step function. Step Function (u(t)): Defined as 1 for t ≥ 0 and 0 for t < 0. It represents a sudden change in signal level. Signum Function (sgn(t)): Defined as 1 for t > 0, -1 for t < 0, and 0 at t=0. It extracts the sign of the input signal.

ii) Sinusoidal Signals: These are signals defined by the form x(t) = A cos(ωt + φ) or x(t) = A sin(ωt + φ), where A is the amplitude, ω is the angular frequency (rad/s), and φ is the phase angle. They are periodic signals with period T = 2π/ω. Sinusoidal signals are fundamental in signal analysis because they are eigenfunctions of linear time-invariant (LTI) systems, meaning the output of an LTI system for a sinusoidal input is also a sinusoid of the same frequency, differing only in amplitude and phase.