ww (Reflections and translations (or flips and slides)) Consider the signal x₁(t) = 0.5tI[0,2] (t) + (t − 3)I(2,3] (t). Sketch the following signals: (a) x₁(t), (b) x₂(t) = x₁(−—t), (c) x3 (t) = (x₁(t) + x₁(−t)). x1. Remark: This is is an even signal by construction, and is called the even part of (1)
ww (Reflections and translations (or flips and slides)) Consider the signal x₁(t) = 0.5tI[0,2] (t) + (t − 3)I(2,3] (t). Sketch the following signals: (a) x₁(t), (b) x₂(t) = x₁(−—t), (c) x3 (t) = (x₁(t) + x₁(−t)). x1. Remark: This is is an even signal by construction, and is called the even part of (1)
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![**Reflections and Translations (or Flips and Slides)**
Consider the signal \( x_1(t) = 0.5I_{[0,2]}(t) + (t - 3)I_{(2,3]}(t) \). Sketch the following signals:
(a) \( x_1(t) \),
(b) \( x_2(t) = x_1(-t) \),
(c) \( x_3(t) = \frac{1}{2} \left( x_1(t) + x_1(-t) \right) \).
*Remark:* This is an even signal by construction and is called the even part of \( x_1 \).
(d) \( x_4(t) = \frac{1}{2} \left( x_1(t) - x_1(-t) \right) \).
*Remark:* This is an odd signal by construction and is called the odd part of \( x_1 \).
(e) \( x_5(t) = x_1(t + 2) \),
(f) \( x_6(t) = x_1(2 - t) \),
(g) \( x_7(t) = x_2(t) + x_5(t) \).
**Explanation of Concepts:**
1. **Signal Notation:** The function \( I_{[a,b]}(t) \) is an indicator function which is 1 if \( t \) is in the interval \([a, b]\) and 0 otherwise. The notation \(I_{(a,b]}(t)\) indicates the interval is open at \( a \) and closed at \( b \).
2. **Even and Odd Signals:**
- An *even signal* \( x(t) \) satisfies \( x(t) = x(-t) \).
- An *odd signal* \( x(t) \) satisfies \( x(t) = -x(-t) \).
3. **Reflections and Translations:**
- \( x_2(t) = x_1(-t) \) is the reflection of \( x_1(t) \) across the \( y \)-axis.
- \( x_5(t) = x_1(t+2) \) represents a shift of \( x_1(t) \) to the left by 2 units.
-](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F099a9854-7cc3-46b9-a37c-ec0486b4a12a%2F1b291724-df29-4b82-b7cb-0bfcf23f27bd%2Fmjb8yk_processed.png&w=3840&q=75)
Transcribed Image Text:**Reflections and Translations (or Flips and Slides)**
Consider the signal \( x_1(t) = 0.5I_{[0,2]}(t) + (t - 3)I_{(2,3]}(t) \). Sketch the following signals:
(a) \( x_1(t) \),
(b) \( x_2(t) = x_1(-t) \),
(c) \( x_3(t) = \frac{1}{2} \left( x_1(t) + x_1(-t) \right) \).
*Remark:* This is an even signal by construction and is called the even part of \( x_1 \).
(d) \( x_4(t) = \frac{1}{2} \left( x_1(t) - x_1(-t) \right) \).
*Remark:* This is an odd signal by construction and is called the odd part of \( x_1 \).
(e) \( x_5(t) = x_1(t + 2) \),
(f) \( x_6(t) = x_1(2 - t) \),
(g) \( x_7(t) = x_2(t) + x_5(t) \).
**Explanation of Concepts:**
1. **Signal Notation:** The function \( I_{[a,b]}(t) \) is an indicator function which is 1 if \( t \) is in the interval \([a, b]\) and 0 otherwise. The notation \(I_{(a,b]}(t)\) indicates the interval is open at \( a \) and closed at \( b \).
2. **Even and Odd Signals:**
- An *even signal* \( x(t) \) satisfies \( x(t) = x(-t) \).
- An *odd signal* \( x(t) \) satisfies \( x(t) = -x(-t) \).
3. **Reflections and Translations:**
- \( x_2(t) = x_1(-t) \) is the reflection of \( x_1(t) \) across the \( y \)-axis.
- \( x_5(t) = x_1(t+2) \) represents a shift of \( x_1(t) \) to the left by 2 units.
-
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