Static and Dynamic Systems

Summary

This lecture delves into the concepts of static and dynamic systems, emphasizing their interrelation and the importance of understanding past, present, and future inputs. The instructor uses the notation X(T) for input, Y(T) for output, and a block for the system, focusing on the output-input relationship dictated by the system's nature. Examples are provided to illustrate how outputs depend on past inputs in certain systems, while others are based on present or future inputs. The lecture clarifies that static systems only consider present input values, exemplified by Y(T) = 2X(T), contrasting with dynamic systems that may also depend on past or future inputs, such as Y(T) = X(T) + X(T-1). The lecture concludes with a problem-solving approach to reinforce understanding, encouraging interactive learning through homework assignments.

Takeaways

• 📚 The lecture introduces the concepts of static and dynamic systems, emphasizing their close relationship and the importance of understanding past, present, and future inputs.
• 🔍 The basic notation used in the lecture includes representing input as X_T, output as Y_T, and the system as a block, focusing on the relationship between output and input based on the system's type.
• 🕒 The terms 'past input,' 'present input,' and 'future input' are crucial for defining static and dynamic systems, with examples provided to illustrate their dependencies.
• 📉 In the first case, Y_T equals X_{T-1}, indicating the output depends on the past input, showcasing how the system can introduce a delay in the output relative to the input.
• 📈 The second case, Y_T equals X_T, demonstrates the output's dependency on the present input, highlighting systems where output is directly influenced by the current input.
• 🔮 The third case, Y_T equals X_{T+1}, represents the output depending on the future input, which is not possible in practical scenarios but serves to differentiate system types theoretically.
• 🏛 A static system is defined as one where the output depends only on the present values of input, with an example given where Y_T equals twice X_T.
• 🌐 A dynamic system is characterized by an output that depends on past or future values of input at any instant, with an example provided where Y_T equals X_T plus X_{T-1}, indicating both present and past dependencies.
• 📝 The lecture clarifies that coefficients, such as in the example Y_T equals e^(-T+1) × X_T, do not affect the classification of a system as static or dynamic; only the dependency on input values matters.
• 🔑 The lecture concludes with an assignment for the audience to solve an example and encourages participation by posting answers in the comment section for further discussion in upcoming lectures.

Q & A

What are the two types of systems discussed in the lecture?

The two types of systems discussed in the lecture are static and dynamic systems.

Why are static and dynamic systems discussed together in the lecture?

Static and dynamic systems are discussed together because they are closely related.

What are the three terms that are important to define static and dynamic systems?

The three terms that are important to define static and dynamic systems are past input, present input, and future input.

How are the input, output, and system represented in the lecture?

In the lecture, the input is represented by X_T, the output by Y_T, and the system is represented by a block.

What does the system's output depend on in the case of Y_T = X_{T-1}?

In the case of Y_T = X_{T-1}, the system's output depends on the past input.

What is the significance of the input being X_T even when the output is X_{T-1} or X_{T+1}?

The input remains X_T because it is the current input at time T, and the system's properties determine how it is transformed into the output X_{T-1} or X_{T+1}.

Can you provide an example from the lecture that illustrates the concept of past input?

An example from the lecture is when Y_0 = X_{-1}, which shows the output at time 0 depends on the input at time -1, hence it is a past input.

What is the definition of a static system according to the lecture?

A static system is defined as a system where the output depends only on the present values of input.

How can you determine if a system is static by using the time variable?

You can determine if a system is static by checking if the output Y_T is dependent only on X_T for any value of T, including 0, 1, or any other value.

What is the definition of a dynamic system according to the lecture?

A dynamic system is defined as a system where the output depends on past or future values of input at any instant of time.

Why is the system in the example Y_T = e^{-T+1} * X_T considered static despite having a time-dependent coefficient?

The system Y_T = e^{-T+1} * X_T is considered static because the output Y_T is dependent on the present input X_T, and the coefficient e^{-T+1} does not change the dependency on the present input.