JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech III Year SIGNALS AND SYSTEMS Previous Question Paper (February/March - 2022 - R18)

Two complex functions f₁(t) and f₂(t) are said to be orthogonal over an interval (t₁, t₂) if the integral of the product of one function and the complex conjugate of the other over that interval is zero.

Mathematically, the condition is defined as:

where f₂^*(t) denotes the complex conjugate of f₂(t). If the functions are real-valued, the conjugate operation has no effect, and the condition simplifies to the integral of the product of the two functions being zero. This property is fundamental in signal analysis, particularly in Fourier series and transform theory, where it allows for the decomposition of signals into a set of orthogonal basis functions, such as sinusoids or complex exponentials, ensuring that each component is independent of the others.

Signals and systems are classified based on their mathematical properties and behavior:

Classification of Signals:

1. Continuous-time vs. Discrete-time: Continuous signals are defined for all values of t, while discrete signals are defined only at specific time instants n.
2. Deterministic vs. Random: Deterministic signals can be described by a mathematical formula, whereas random signals are characterized by statistical properties.
3. Periodic vs. Aperiodic: Periodic signals repeat their pattern at regular intervals, while aperiodic signals do not.
4. Energy vs. Power: Energy signals have finite total energy and zero average power, while power signals have finite average power and infinite energy.
5. Even vs. Odd: Even signals satisfy f(t) = f(-t), and odd signals satisfy f(t) = -f(-t).

Classification of Systems:

1. Linear vs. Non-linear: Linear systems satisfy the principle of superposition (additivity and homogeneity).
2. Time-invariant vs. Time-variant: A system is time-invariant if a time shift in the input results in an identical time shift in the output.
3. Causal vs. Non-causal: Causal systems depend only on present and past inputs, whereas non-causal systems depend on future inputs.
4. Stable vs. Unstable: A system is stable (BIBO) if every bounded input produces a bounded output.
5. Static vs. Dynamic: Static systems have no memory (output depends only on current input), while dynamic systems possess memory.