JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech II Year SIGNALS AND STOCHASTIC PROCESS Previous Question Paper (April/May - 2023 - R16)

Two fundamental properties of a Linear Time-Invariant (LTI) system are:

• Superposition Property: A system is linear if it satisfies both the additivity and homogeneity properties. If an input x₁(t) produces an output y₁(t) and an input x₂(t) produces an output y₂(t), then an input ax₁(t) + bx₂(t) will produce an output ay₁(t) + by₂(t), where a and b are constants.
• Time-Invariance Property: A system is time-invariant if a time shift in the input signal results in an identical time shift in the output signal. If an input x(t) produces an output y(t), then an input x(t - t₀) produces an output y(t - t₀) for any real constant t₀.

The relationship between bandwidth (BW) and rise time (t_r) is a fundamental characteristic of linear time-invariant systems, particularly in pulse-based communication and circuit design. It describes the trade-off between the speed of a system's response to a step input and its frequency range.

Consider a low-pass system with a transfer function H(f). The rise time is defined as the time taken for the step response to rise from 10% to 90% of its final value. For a system with a Gaussian frequency response, the relationship is derived as follows:

• Let the system have a bandwidth defined as the frequency where the magnitude drops to 1/√2 of its maximum value.
• The impulse response of a Gaussian filter is given by:
h(t) = αe^-πα2t2
• The step response is the integral of the impulse response, which involves the error function. By calculating the time difference between the 10% and 90% points, we obtain the rise time:
t_r = 0.35 / BW

This result, known as the 0.35 rule, indicates that the product of rise time and bandwidth is approximately constant. A system with a wider bandwidth is capable of faster transitions, resulting in a smaller rise time, while a narrow bandwidth limits the system's ability to track rapid changes, leading to a larger rise time.