1. Which of them are real signals? (Please use the property of Fourier series) 100 k x₂ (1) = [ ( 1 ) ² Σ()*** 50t k=0 2π 100 x₂ (t) = Σ cos(k)ejkt k=-100 100 jk x3 (t) = Σ / sin (1) e/k² k=-100
1. Which of them are real signals? (Please use the property of Fourier series) 100 k x₂ (1) = [ ( 1 ) ² Σ()*** 50t k=0 2π 100 x₂ (t) = Σ cos(k)ejkt k=-100 100 jk x3 (t) = Σ / sin (1) e/k² k=-100
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![### Fourier Series and Real Signals
#### 1. Real Signal Identification
**Problem:**
Identify which of the given signals \( x_1(t) \), \( x_2(t) \), and \( x_3(t) \) are real. Use the properties of Fourier series.
- \( x_1(t) = \sum_{k=0}^{100} \left(\frac{1}{2}\right)^k e^{jk\frac{2\pi}{50}t} \)
- \( x_2(t) = \sum_{k=-100}^{100} \cos(k\pi) e^{jk\frac{2\pi}{50}t} \)
- \( x_3(t) = \sum_{k=-100}^{100} j \sin\left(\frac{k\pi}{2}\right) e^{jk\frac{2\pi}{50}t} \)
**Explanation:**
To determine which signals are real, consider if the complex conjugate symmetry property of the Fourier series is satisfied. A real-valued signal should have its coefficients satisfy the condition: \( a_{-k} = a_k^* \).
#### 2. Expression of a Real Discrete Signal
**Problem:**
Given a real discrete time periodic signal \( x[n] \) with a period \( N = 5 \) and the specified nonzero Fourier series coefficients:
- \( a_0 = 1 \)
- \( a_2 = a_{-2}^* = e^{-j\frac{\pi}{4}} \)
- \( a_4 = a_{-4}^* = 2e^{-j\frac{\pi}{3}} \)
Express \( x[n] \) in the form:
\[ x[n] = A_0 + \sum_{k=1}^{\infty} A_k \sin(\omega_k n + \phi_k) \]
**Explanation:**
This involves transforming the non-zero Fourier coefficients into the sinusoidal form by separating them into amplitude and phase, utilizing Euler's formula: \( e^{j\theta} = \cos(\theta) + j\sin(\theta) \).
In this manner, the problem explores the properties associated with Fourier series that pertain to identifying real-valued signals and converting Fourier coefficients into time-domain sinusoidal expressions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F584ffa08-94ad-4eb4-a87d-fd258db9f802%2Fa9808f75-f588-4059-999e-15c0fbde492d%2F85yplwq_processed.png&w=3840&q=75)
Transcribed Image Text:### Fourier Series and Real Signals
#### 1. Real Signal Identification
**Problem:**
Identify which of the given signals \( x_1(t) \), \( x_2(t) \), and \( x_3(t) \) are real. Use the properties of Fourier series.
- \( x_1(t) = \sum_{k=0}^{100} \left(\frac{1}{2}\right)^k e^{jk\frac{2\pi}{50}t} \)
- \( x_2(t) = \sum_{k=-100}^{100} \cos(k\pi) e^{jk\frac{2\pi}{50}t} \)
- \( x_3(t) = \sum_{k=-100}^{100} j \sin\left(\frac{k\pi}{2}\right) e^{jk\frac{2\pi}{50}t} \)
**Explanation:**
To determine which signals are real, consider if the complex conjugate symmetry property of the Fourier series is satisfied. A real-valued signal should have its coefficients satisfy the condition: \( a_{-k} = a_k^* \).
#### 2. Expression of a Real Discrete Signal
**Problem:**
Given a real discrete time periodic signal \( x[n] \) with a period \( N = 5 \) and the specified nonzero Fourier series coefficients:
- \( a_0 = 1 \)
- \( a_2 = a_{-2}^* = e^{-j\frac{\pi}{4}} \)
- \( a_4 = a_{-4}^* = 2e^{-j\frac{\pi}{3}} \)
Express \( x[n] \) in the form:
\[ x[n] = A_0 + \sum_{k=1}^{\infty} A_k \sin(\omega_k n + \phi_k) \]
**Explanation:**
This involves transforming the non-zero Fourier coefficients into the sinusoidal form by separating them into amplitude and phase, utilizing Euler's formula: \( e^{j\theta} = \cos(\theta) + j\sin(\theta) \).
In this manner, the problem explores the properties associated with Fourier series that pertain to identifying real-valued signals and converting Fourier coefficients into time-domain sinusoidal expressions.
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