Give step-by-step solution with explanation and final answer:1 Question1
A continuous time signal x(¢) is shown in Figure 1. Sketch and label carefully
each of the following signals
Figure 1: x(¢)

Question

Give step-by-step solution with explanation and final answer:1 Question1
A continuous time signal x(¢) is shown in Figure 1. Sketch and label carefully
each of the following signals
Figure 1: x(¢)

1. x(t-2) and x(t +2)

2. x(-8),x(1 -t) and x(~1-£)

3. x(31), 3x(t) and x(3t + 1)

4. x(t/2),x(t/2 + Vand x(-t/2 +2)

5. (x(0) + x(-0)u(t-1)

6. X(0)[8(-3) + 5(¢ + 2)]

7. Write a Matlab code to draw the signal x(t)

1

VIPSolutions · Accepted Answer

Let's solve **Question 1** step by step:

---

## **Given Signal $ x(t) $**
From the graph:
- Defined from $ t = 1 $ to $ t = 4 $.
- At $ t = 1 $, $ x(1) = 1 $
- Increases linearly to $ x(3) = 3 $
- Decreases linearly to $ x(4) =  $

---

### **1. Time Shifting: $ x(t-2) $ and $ x(t+2) $**

#### **a. $ x(t-2) $**
- This **shifts the signal right** by 2 units.
- Original $ x(t) $ is nonzero for $ t \in [1, 4] $, so $ x(t-2) $ is nonzero for $ t-2 \in [1, 4] \implies t \in [3, 6] $.

#### **b. $ x(t+2) $**
- This **shifts the signal left** by 2 units.
- Nonzero for $ t+2 \in [1, 4] \implies t \in [-1, 2] $.

---

#### **Summary Table:**
| Signal      | Nonzero for $ t $ | Description       |
|-------------|---------------------|------------------|
| $ x(t-2) $ | $ 3 \leq t \leq 6 $ | Shift right by 2 |
| $ x(t+2) $ | $ -1 \leq t \leq 2 $ | Shift left by 2  |

---

### **2. Time Reversal and Shifting**
#### **a. $ x(-t) $**
- **Time reversal**: Flip signal about $ t =  $.
- Nonzero for $ -t \in [1, 4] \implies t \in [-4, -1] $

#### **b. $ x(1-t) $**
- **Reverse and shift**: Substitute $ t \leftarrow 1-t $
- Nonzero for $ 1-t \in [1, 4] \implies -t \in [, 3] \implies t \in [-3, ] $

#### **c. $ x(-1-t) $**
- **Reverse and shift**: Substitute $ t \leftarrow -1-t $
- Nonzero for $ -1-t \in [1, 4] \implies -t \in [2, 5] \implies t \in [-5, -2] $

---

#### **Summary Table:**
| Signal         | Nonzero for $ t $ |
|----------------|---------------------|
| $ x(-t) $    | $ -4 \leq t \leq -1 $ |
| $ x(1-t) $   | $ -3 \leq t \leq  $  |
| $ x(-1-t) $  | $ -5 \leq t \leq -2 $ |

---

### **3. Time Scaling and Shifting**
#### **a. $ x(3t) $**
- **Compresses** the signal by a factor of 3.
- Nonzero for $ 3t \in [1, 4] \implies t \in [1/3, 4/3] $

#### **b. $ 3x(t) $**
- **Amplitude scaling**: Multiply all values by 3, support is unchanged ($ t \in [1, 4] $)

#### **c. $ x(3t+1) $**
- **Scale and shift**: Let $ \tau = 3t+1 $
- Nonzero for $ 3t+1 \in [1, 4] \implies 3t \in [, 3] \implies t \in [, 1] $

---

#### **Summary Table:**
| Signal           | Nonzero for $ t $ | Description         |
|------------------|---------------------|---------------------|
| $ x(3t) $      | $ 1/3 \leq t \leq 4/3 $ | Compressed          |
| $ 3x(t) $      | $ 1 \leq t \leq 4 $     | Amplitude scaled    |
| $ x(3t+1) $    | $  \leq t \leq 1 $     | Compressed/shifted  |

---

### **4. Time Scaling and Shifting**
#### **a. $ x(t/2) $**
- **Expands** the signal by a factor of 2.
- Nonzero for $ t/2 \in [1, 4] \implies t \in [2, 8] $

#### **b. $ x(t/2+1) $**
- **Expand and shift**: $ t/2 + 1 \in [1, 4] \implies t/2 \in [, 3] \implies t \in [, 6] $

#### **c. $ x(-t/2+2) $**
- **Expand, reverse, and shift**: $ -t/2 + 2 \in [1, 4] \implies -t/2 \in [-1, 2] \implies t/2 \in [-2, 1] \implies t \in [-4, 2] $

---

#### **Summary Table:**
| Signal               | Nonzero for $ t $ |
|----------------------|---------------------|
| $ x(t/2) $         | $ 2 \leq t \leq 8 $ |
| $ x(t/2+1) $       | $  \leq t \leq 6 $ |
| $ x(-t/2+2) $      | $ -4 \leq t \leq 2 $ |

---

### **5. Addition with Step Function**
#### $ [x(t) + x(-t)]u(t-1) $
- $ x(t) + x(-t) $: Add the original and time-reversed signals.
- $ u(t-1) $: **Zero for $ t < 1 $**, one for $ t \geq 1 $.
- So, output is  for $ t < 1 $, and for $ t \geq 1 $, it's the sum.

---

### **6. Delta Function Expression**
#### $ x(t)[\delta(t-3) + \delta(t+2)] $
- This is nonzero only at $ t = 3 $ and $ t = -2 $ (because delta function is zero everywhere except its argument zero).
- At $ t = 3 $: Output is $ x(3) $.
- At $ t = -2 $: Output is $ x(-2) $, but from the graph, $ x(t) $ is only defined for $ t \in [1,4] $; so $ x(-2) =  $.

---

### **7. MATLAB Code to Draw $ x(t) $**
```matlab
t = [1 3 4];
x = [1 3 ];
figure;
plot([1 3 4],[1 3 ],'b','LineWidth',2);
xlabel('t');
ylabel('x(t)');
title('Signal x(t)');
grid on;
axis([ 5  3]);
```

---

## **Summary Table**

| Part         | Key Step/Result                                      | Support (t values)     |
|--------------|------------------------------------------------------|------------------------|
| 1            | Time shift ±2                                        | [3,6] and [-1,2]       |
| 2            | Time reversal and shifts                             | [-4,-1], [-3,], [-5,-2]|
| 3            | Scaling/compression/shifting                         | [1/3,4/3], [1,4], [,1]|
| 4            | Expanding and shifting                               | [2,8], [,6], [-4,2]   |
| 5            | Add x(t) and x(-t), zero for t

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