Properties Of Systems | Example 1

Summary

In this video, the instructor tests a system, defined by the equation Y(T) = X(T)^2, for five essential properties: linearity, time-invariance, memorylessness, causality, and stability. Through a series of steps, it is shown that the system is nonlinear, time-invariant, memoryless, causal, and stable. The explanation includes detailed tests for each property, such as using superposition for linearity and bounded-input bounded-output analysis for stability. The video provides a comprehensive and clear breakdown of these fundamental system properties, helping viewers understand how to analyze and categorize different types of systems.

Takeaways

• 😀 The system under consideration is defined as Y(T) = X^2(T), and the task is to evaluate it for various properties.
• 😀 Linearity of the system is tested by applying the principles of superposition and homogeneity, ultimately concluding that the system is nonlinear.
• 😀 Time invariance is tested by shifting the input and observing if the output shifts identically. The system is found to be time-invariant.
• 😀 The system is memoryless, as the output at any given time only depends on the present value of the input and does not rely on past or future values.
• 😀 The system is causal, meaning the output depends only on the present or past inputs, and not future values.
• 😀 Stability of the system is tested by examining if a bounded input results in a bounded output (BIBO stability). The system is found to be stable.
• 😀 For linearity, after testing with scaled and added inputs, it is concluded that the system fails to be linear.
• 😀 The system is not linear because the output for a combination of inputs does not equal the combination of outputs for each input.
• 😀 Time invariance ensures the system behaves consistently over time, and the system does not change with a time shift in the input.
• 😀 Bounded-input, bounded-output (BIBO) stability is confirmed, as the output remains bounded when the input is finite and bounded.

Q & A

What does it mean for a system to be linear?

A system is linear if it satisfies two properties: superposition (the system's response to a sum of inputs is the sum of the responses to each input) and homogeneity (the system's response to a scaled input is the scaled response).

How can we test if the system is linear using the given example?

To test if the system is linear, the script tests two different inputs, scales them, and then adds them. If the output matches the same operation applied to the sum of inputs, the system would be linear. In this case, the system is nonlinear, as the sum of the squared inputs does not equal the square of the sum of the inputs.

What does 'time invariance' mean in the context of a system?

A system is time-invariant if a shift in the input signal results in the same shift in the output signal. Essentially, the system's behavior does not change over time.

How was time invariance tested in the script?

Time invariance was tested by shifting the input by a constant time and checking if the output shifted by the same amount. Since both the input and output shifted identically, the system was determined to be time-invariant.

What does it mean for a system to be memoryless?

A memoryless system's output at any time depends only on the input at the same time, not on past or future values of the input.

How was it determined that the system is memoryless?

The system was considered memoryless because the output at any given time only depended on the present input (i.e., Y(t) = X(t)²), without reference to past or future inputs.

What is the definition of a causal system?

A system is causal if its output at any given time depends only on the present and past values of the input, but not on future values.

Why is the system considered causal in this case?

The system is causal because its output at any time only depends on the current input and not on any future inputs.

What does 'stability' mean for a system?

A system is stable if bounded inputs (finite values) always produce bounded outputs. This is known as Bounded-Input Bounded-Output (BIBO) stability.

How was stability tested in the script?

Stability was tested by applying a bounded input (a finite value for X(t)) and checking if the output remained finite (bounded). Since the squared output was also finite, the system was deemed stable.

What properties of the system were tested in the video?

The video tested the system for five properties: linearity, time invariance, memorylessness, causality, and stability.