Give step-by-step solution with explanation and final answer:Problem 3 : Present the rationale for whether each of the following filters are: a) Linear, b) Time Invariant, c)
Stable, and d) Dynamic.

Question

Give step-by-step solution with explanation and final answer:Problem 3 : Present the rationale for whether each of the following filters are: a) Linear, b) Time Invariant, c)
Stable, and d) Dynamic.
Lym) =xmxm-1)
IL. ym) =u(m-1a™Vxm) forla|

VIPSolutions · Accepted Answer

# Problem 3: Filter Properties Analysis

Let's analyze each filter for the following properties:
- **Linearity**
- **Time Invariance**
- **Stability**
- **Dynamism (Static or Dynamic)**

---

## I. $ y(n) = x(n)x(n-1) $

### a) **Linearity**
- **Test:** For linearity, the system must satisfy additivity and homogeneity.
- **Check:**  
  $ y_1(n) = x_1(n)x_1(n-1) $  
  $ y_2(n) = x_2(n)x_2(n-1) $  
  For input $ a x_1(n) + b x_2(n) $:  
  $$
  y(n) = [a x_1(n) + b x_2(n)][a x_1(n-1) + b x_2(n-1)]
  $$
  This expands to cross-terms, so **not linear**.

### b) **Time Invariant**
- **Test:** Substitute $ x(n-k) $ as input.
- **Check:**  
  Input $ x(n-k) $:  
  $$
  y(n) = x(n-k)x(n-k-1)
  $$
  Output is delayed by $ k $. So, **time-invariant**.

### c) **Stable**
- **Test:** BIBO (Bounded Input, Bounded Output).
- **Check:**  
  Product of two bounded signals is bounded. **Stable**.

### d) **Dynamic**
- **Test:** Output depends on past/future inputs.
- **Check:**  
  Depends on $ x(n) $ and $ x(n-1) $. **Dynamic**.

---

## II. $ y(n) = u(n-1)\alpha^{(n-1)}x(n) $, $ |\alpha| < 1 $

### a) **Linearity**
- **Check:** Directly proportional to $ x(n) $, so **linear**.

### b) **Time Invariant**
- **Check:**  
  For $ x(n-k) $:  
  $$
  y(n) = u(n-1)\alpha^{(n-1)}x(n-k)
  $$
  For time-invariant system, delay must only shift output. Here, the step and exponent are functions of $ n $, not $ n-k $.
  Therefore, **not time-invariant**.

### c) **Stable**
- **Check:**  
  $ u(n-1) $ is 1 for $ n \geq 1 $, and $ |\alpha| < 1 $ makes the coefficient bounded.  
  So, **stable**.

### d) **Dynamic**
- **Check:**  
  Output depends only on present input $ x(n) $. **Static**.

---

## III. $ y(n) = u(n)\cos(x(n)) $

### a) **Linearity**
- **Check:**  
  Cosine is non-linear. **Not linear**.

### b) **Time Invariant**
- **Check:**  
  $ x(n-k) $ yields $ y(n) = u(n)\cos(x(n-k)) $, while delaying output gives $ u(n-k)\cos(x(n-k)) $.
  Thus, **not time-invariant**.

### c) **Stable**
- **Check:**  
  Cosine of any real number is bounded by [-1, 1], $ u(n) $ is 0 or 1. So, **stable**.

### d) **Dynamic**
- **Check:**  
  Only depends on $ x(n) $. **Static**.

---

## **Summary Table**

| System                              | Linear | Time Invariant | Stable | Dynamic |
|--------------------------------------|:------:|:--------------:|:------:|:-------:|
| $ y(n) = x(n)x(n-1) $              |   No   |      Yes       |  Yes   |  Yes    |
| $ y(n) = u(n-1)\alpha^{n-1}x(n) $  |  Yes   |      No        |  Yes   |  No     |
| $ y(n) = u(n)\cos(x(n)) $          |   No   |      No        |  Yes   |  No     |

---

### **Explanation Provided Above** for each step and property. Let me know if you need a breakdown of each test!

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Give step-by-step solution with explanation and final answer:Problem 3 : Present the rationale for whether each of the following filters are: a) Linear, b) Time Invariant, c) Stable, and d) Dynamic. Lym) =xmxm-1) IL. ym) =u(m-1a™Vxm) forla|

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