JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech II SIGNALS AND SYSTEMS Previous Question Paper (March - 2021 - R18)

The Unit Impulse Function, denoted by δ(t), is defined as:

Key properties include:

• Sampling Property: The product of a function x(t) and an impulse at t₀ extracts the value of the function at that point.
  x(t)δ(t - t₀) = x(t₀)δ(t - t₀)
• Sifting Property: Integration of the product of a function and a shifted impulse yields the function value at the shift.
  ∫x(t)δ(t - t₀)dt = x(t₀)
• Scaling Property:
  δ(at) = (1/|a|)δ(t)

Proof of Sifting Property: Since δ(t - t₀) is zero everywhere except at t = t₀, the integral ∫x(t)δ(t - t₀)dt reduces to ∫x(t₀)δ(t - t₀)dt. Pulling the constant x(t₀) out, we get x(t₀)∫δ(t - t₀)dt, which equals x(t₀) * 1 = x(t₀).

A signal f(t) can be approximated by a linear combination of a set of orthogonal functions {φ_n(t)} over an interval (t₁, t₂):

The coefficients c_n are determined by minimizing the Mean Square Error (MSE), resulting in:

Example: Approximating a constant signal f(t) = 1 over (0, 1) using a single orthogonal function φ₁(t) = 1.

• Calculate c₁: ∫₀¹(1)(1)dt / ∫₀¹(1)²dt = 1 / 1 = 1.
• The approximation is f(t) ≈ 1(1) = 1.
• If we added φ₂(t) = sin(2πt), since it is orthogonal to φ₁(t) over the interval, the coefficient c₂ would be ∫₀¹(1)sin(2πt)dt = 0, confirming the best approximation is the average value.