GATE Academy Plus: Signals and Systems Lecture Notes

Module 1: Properties of Systems

Completed Properties

• Causal Systems
• Non-Causal Systems
• Time Variant Systems
• Time Invariant Systems
• Static Systems
• Dynamic Systems
• Linear Systems
• Non-Linear Systems

Today's Topic: Invertible and Non-Invertible Systems

Invertible Systems

• Definition: A system is invertible if it produces distinct outputs for distinct inputs. This means:
  • For every unique input, there is a unique output.
  • The output can be used to determine the input (one-to-one mapping).
  • Example: If the relationship is given as ( y(t) = 4x(t) ), then ( x(t) = \frac{1}{4}y(t) ) shows that it is invertible.
  • If an invertible system has an inverse system, cascading the two yields the original input as output.
  • Mathematically, the overall transfer function should equal 1, represented as ( x(t) \cdot H(t) \cdot H^{-1}(t) = x(t) ).

Non-Invertible Systems

• Example: ( y(t) = x^2(t) ), where multiple inputs yield the same output (e.g., ( x(t) = 1 ) and ( x(t) = -1 ) both give ( y(t) = 1 )).

Test Signals for Invertibility

• Use functions like ( u(t) ) or impulse signals ( \delta(t) ) to assess the system's invertibility through their effects on the output.

Stability of Systems

• Bounded Input Bounded Output (BIBO) Stability: A system is stable if bounded inputs lead to bounded outputs.
  • Impulse Response: Should approach zero as time tends to infinity.
  • Criteria: Continuous-time systems' impulse responses must be absolutely integrable; discrete-time systems must be absolutely summable.

Key Concepts in Stability

• Transient Response: Should tend to zero for the system to be stable.
• Pole Location: For Laplace transforms, poles must lie in the left-half of the s-plane for stability.
  • Example: If the transient contains positive exponential terms, the system is unstable.

Homework Assignments

• Practice problems regarding stability, invertibility, and other system properties.

Next Topic: Convolution

• Focus on Linear Time Invariant (LTI) systems.
• Types of Convolution:
  • Convolution Integral for continuous time signals.
  • Convolution Summation for discrete time signals.

Convolution Definition

• Continuous Time Convolution: ( y(t) = \int_{-\infty}^{\infty} x(\tau) h(t - \tau) d\tau )
• Commutative Property: Order of convolution does not change the result.
• Associative Property: Order of operations can be changed when convolving multiple signals.
• Distributive Property: Allows for expansion over sums of signals.

Linear Operations on Convolution

• Derivative operations affect the output accordingly.
• Example: If ( y(t) = x(t) \ast h(t) ), then ( \frac{dy(t)}{dt} = \frac{dx(t)}{dt} \ast h(t) = x(t) \ast \frac{dh(t)}{dt} ).

Time Invariance Property

• The output shifts in time when the inputs are shifted in time.
• Example: ( y(t-t_0) = x(t-t_0) \ast h(t-t_0) ).

Conclusion

• Convolution is crucial for determining outputs in LTI systems and will be explored further in upcoming lectures.