Certainly. Below is a professional, detailed, and complete solution according to your specified guidelines. --- **1. Introduction and Conceptual Background** In digital signal processing, analog signals are converted into digital form through sampling and quantization. Figure 2 illustrates a basic digital signal processing system, where the continuous-time signal \( x_c(t) \) is sampled by an ideal Analog-to-Digital (A/D) converter at intervals of \( T \) seconds, producing the discrete sequence \( x[n] \). This sequence can then be reconstructed into a continuous-time signal \( x_r(t) \) using an ideal Digital-to-Analog (D/A) converter. The process of sampling is mathematically described by replacing the time variable \( t \) with \( nT \), where \( n \) is an integer index. Aliasing is a phenomenon that occurs when the sampling frequency is insufficient to capture the changes in the original signal, leading to distortion in the reconstructed signal. According to the Nyquist-Shannon sampling theorem, to avoid aliasing, the sampling frequency \( f_s \) must be at least twice the maximum frequency present in the signal. **Explanation:** Understanding the process of sampling and reconstruction is crucial for analyzing digital signal processing systems. The conversion between continuous and discrete representations forms the foundation for all digital processing operations. Awareness of aliasing and the Nyquist criterion is essential to preserve the integrity of the original signal during sampling and reconstruction. This conceptual grounding ensures accurate interpretation of the steps involved in solving the problems related to sampling, frequency components, and reconstruction. --- **2. Presentation of Relevant Formulas** **a. Sampling Formula:** For a continuous-time signal \( x_c(t) \), the sampled discrete-time signal is given by: \[ x[n] = x_c(nT) \] where \( T \) is the sampling period and \( n \) is an integer index. **b. Reconstruction Formula:** The reconstructed continuous-time signal from the discrete sequence is: \[ x_r(t) = x[n] \big|_{n = \frac{t}{T}} \] **c. Cosine Signal Sampling:** Given \( x_c(t) = \cos(2\pi f_0 t) \), after sampling: \[ x[n] = \cos(2\pi f_0 nT) \] **d. Nyquist-Shannon Sampling Theorem:** To avoid aliasing, \[ f_s > 2 f_0 \] where \( f_s = \frac{1}{T} \) is the sampling frequency and \( f_0 \) is the highest frequency in the signal. **Explanation:** Each formula above is directly applicable to the posed questions. The sampling formula provides the mathematical mechanism for converting continuous to discrete signals. The reconstruction formula allows mapping back to continuous time. The Nyquist criterion dictates the minimum sampling frequency required to preserve all information from the original signal, thereby preventing aliasing. These formulas are critical for ensuring the accuracy of sampling and reconstruction in digital systems. --- **3. Detailed Step-by-Step Solution** **(a) Calculation for \( f_0 = 1\,\text{kHz} \), \( T = 200\,\mu s \):** Given: \[ x_c(t) = \cos(2\pi f_0 t),\quad f_0 = 1\,\text{kHz},\quad T = 200\,\mu s = 200 \times 10^{-6}\,\text{s} \] **Step 1: Find \( x[n] \)** \[ x[n] = x_c(nT) = \cos(2\pi \cdot 1000 \cdot n \cdot 200 \times 10^{-6}) \] \[ = \cos(2\pi \cdot 1000 \cdot 0.0002 \cdot n) \] \[ = \cos(2\pi \cdot 0.2 \cdot n) \] \[ = \cos(0.4\pi n) \] **Step 2: Express \( x_r(t) \)** \[ x_r(t) = x[n] \big|_{n = \frac{t}{T}} = \cos\left(0.4\pi \cdot \frac{t}{T}\right) \] \[ = \cos\left(0.4\pi \cdot \frac{t}{200 \times 10^{-6}}\right) \] **Explanation:** The calculation follows the direct application of the sampling formula by substituting the provided values for \( f_0 \) and \( T \). The result is simplified to its most concise form, expressing the sampled cosine in discrete and reconstructed forms. --- **(b) Calculation for \( f_0 = 5\,\text{kHz} \), \( T = 125\,\mu s \):** Given: \[ x_c(t) = \cos(2\pi f_0 t),\quad f_0 = 5\,\text{kHz},\quad T = 125\,\mu s = 125 \times 10^{-6}\,\text{s} \] **Step 1: Find \( x[n] \)** \[ x[n] = x_c(nT) = \cos(2\pi \cdot 5000 \cdot n \cdot 125 \times 10^{-6}) \] \[ = \cos(2\pi \cdot 5000 \cdot 0.000125 \cdot n) \] \[ = \cos(2\pi \cdot 0.625 \cdot n) \] \[ = \cos(1.25\pi n) \] **Step 2: Express \( x_r(t) \)** \[ x_r(t) = x[n] \big|_{n = \frac{t}{T}} = \cos\left(1.25\pi \cdot \frac{t}{T}\right) \] \[ = \cos\left(1.25\pi \cdot \frac{t}{125 \times 10^{-6}}\right) \] **Explanation:** This step involves substituting the new values for frequency and sampling period into the same process, reinforcing the universality of the sampling principle and how it adapts to different signal parameters. --- **(c) Aliasing and Minimum Sampling Period for \( f_0 = 20\,\text{kHz} \):** Given: \[ x_c(t) = \cos(2\pi f_0 t),\quad f_0 = 20\,\text{kHz} \] **Step 1: Nyquist Condition** To avoid aliasing: \[ f_s > 2 f_0 = 2 \times 20,000 = 40,000\,\text{Hz} \] **Step 2: Find Maximum Admissible Sampling Period \( T \)** \[ f_s = \frac{1}{T} \implies T < \frac{1}{40,000} \] \[ T < 2.5 \times 10^{-5}\,\text{s} \] \[ T < 25.00\,\mu s \] **Step 3: Express Numerical Answer in Engineering Notation** The sampling period \( T \) needs to be less than \( 25.00\,\mu s \). **Explanation:** The Nyquist criterion is applied to determine the minimum sampling frequency required. The inequality is rearranged to solve for the maximum permissible sampling period, ensuring no aliasing occurs for the given signal frequency. The answer is presented in standard engineering notation with appropriate precision and units, as required. --- **Conclusion** - **(a)** For \( f_0 = 1\,\text{kHz} \), \( T = 200\,\mu s \): \[ x_r(t) = \cos\left(0.4\pi \cdot \frac{t}{200 \times 10^{-6}}\right) \] - **(b)** For \( f_0 = 5\,\text{kHz} \), \( T = 125\,\mu s \): \[ x_r(t) = \cos\left(1.25\pi \cdot \frac{t}{125 \times 10^{-6}}\right) \] - **(c)** To avoid aliasing for \( f_0 = 20\,\text{kHz} \), the sampling period \( T \) must be less than: \[ T < 25.00\,\mu s \] These results ensure correct digital processing and reconstruction for the specified signal scenarios, maintaining signal integrity and preventing aliasing.