JNTU B.Tech II SIGNALS SYSTEMS AND SIGNAL PROCESSING Previous Question Paper (August/September - 2022 - R18)

The Dirac Delta function, denoted as δ(t), is a generalized function defined as having zero value everywhere except at t = 0, where it is infinitely high such that its total area equals unity. It is defined by the sifting property:

The waveform is represented by an arrow pointing upward at the origin t = 0 with a label of 1. Its key properties include:

• Sifting Property: ∫ x(t)δ(t - t₀) dt = x(t₀)
• Sampling Property: x(t)δ(t - t₀) = x(t₀)δ(t - t₀)
• Scaling Property: δ(at) = (1/|a|)δ(t)
• Even Symmetry: δ(t) = δ(-t)

A system is a physical device or a mathematical model that performs an operation on an input signal to produce an output signal. It acts as a transformation process mapping an input x(t) to an output y(t).

Signals are classified into the following categories:

• Continuous-time vs. Discrete-time: Continuous-time signals are defined for every instant of time (e.g., a sine wave x(t) = sin(ωt)), whereas discrete-time signals are defined only at specific time intervals (e.g., a sequence x[n] = [1, 2, 3]).
• Deterministic vs. Random: Deterministic signals can be described by a mathematical function (e.g., x(t) = 5cos(10πt)), while random signals are characterized by statistical properties (e.g., thermal noise in a resistor).
• Periodic vs. Aperiodic: Periodic signals repeat at regular intervals, satisfying x(t) = x(t + T), such as a square wave. Aperiodic signals do not repeat, such as a single rectangular pulse.
• Energy vs. Power: Energy signals have finite total energy and zero average power (e.g., a single pulse). Power signals have finite average power and infinite total energy (e.g., a constant DC signal or a periodic waveform).
• Even vs. Odd: Even signals satisfy x(t) = x(-t) (symmetry about the y-axis), while odd signals satisfy x(t) = -x(-t) (symmetry about the origin).