https://file.aiquickdraw.com/mathbot/file/1729641362371xz09f5er.jpg

Let's analyze the system given in part **(e)**:

### \( y(t) = x(t/3) \)

We will discuss its properties based on **Linearity**, **Causality**, **Time-Invariance**, **Stability**, and **Memory**:

### 1. Linearity:
- A system is linear if it satisfies both **additivity** and **homogeneity** (or scaling).
- Let’s test this by assuming two input signals \(x_1(t)\) and \(x_2(t)\) with corresponding outputs \(y_1(t)\) and \(y_2(t)\):
  
  \[
  y_1(t) = x_1(t/3) \quad \text{and} \quad y_2(t) = x_2(t/3)
  \]
  
  Now, for a linear combination of inputs \(x(t) = a x_1(t) + b x_2(t)\), the output would be:
  
  \[
  y(t) = x\left(\frac{t}{3}\right) = a x_1\left(\frac{t}{3}\right) + b x_2\left(\frac{t}{3}\right) = a y_1(t) + b y_2(t)
  \]
  
  Since the output is a linear combination of the individual outputs, **the system is linear**.

### 2. Causality:
- A system is causal if the output at time \(t\) depends only on the present and past values of the input, and not on future values.
- Here, \( y(t) = x(t/3) \). Notice that if \(t > 0\), then \( x(t/3) \) is determined by the input value at \(t/3\). 
  - If \( t/3 > t \), then \( y(t) \) depends on a future value of the input, which means the system **is not causal**. The system depends on the input at \( t/3 \), which can include future times if \( t/3 > t \).

### 3. Time-Invariance:
- A system is time-invariant if a time shift in the input results in an identical time shift in the output.
- Let’s check this property. Consider a time-shifted input \( x(t - t_0) \). The output would be:
  
  \[
  y(t) = x\left(\frac{t}{3}\right)
  \]
  
  Now, if we shift the input by \(t_0\), the new output will be:
  
  \[
  y'(t) = x\left(\frac{t - t_0}{3}\right)
  \]
  
  Compare this with the output for the shifted input \( x\left(\frac{(t - t_0)}{3}\right) \). The output has a time-dependent scaling, meaning the shift does not directly correspond in a simple manner. Therefore, **the system is not time-invariant**.

### 4. Stability:
- A system is **bounded-input, bounded-output (BIBO) stable** if a bounded input always leads to a bounded output.
- If \( x(t) \) is bounded, say \( |x(t)| \leq M \) for all \(t\), then \( y(t) = x(t/3) \) will also be bounded, as scaling the argument of \(x(t)\) does not change its boundedness.
  - Therefore, **the system is stable**.

### 5. Memory:
- A system has **memory** if the output depends on past or future inputs; otherwise, it is memoryless.
- Since \( y(t) = x(t/3) \) depends on values of \( x \) at times other than \(t\), **the system has memory**.

### Summary of Properties:
- **Linearity:** Yes
- **Causality:** No
- **Time-Invariance:** No
- **Stability:** Yes
- **Memory:** Yes

Would you like further details or clarification on any of these properties? Here are some related questions for further practice:

1. What would the system properties be if \( y(t) = x(3t) \)?
2. How does the time-scaling affect causality in general?
3. Could the system be made causal with a different transformation?
4. What are the conditions for a system to be both causal and time-invariant?
5. How would you determine the stability of a system with an impulse response?

**Tip:** Always verify the time scaling or shifts in time-invariance checks by considering how a time shift in input affects the output directly.

Discuss the properties (Linearity, Causality, Time-Invariance, Stability, and Memory) of the system y(t) = x(t/3).

Analyze the system properties of y(t) = x(t/3), including Linearity, Causality, Time-Invariance, Stability, and Memory. This problem explores core concepts in signal processing and system analysis, making it suitable for undergraduate students in engineering and related fields.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation