Given the signal m(t) = 10 cos 2000rt cos 8000nt (a) What is the minimum sampling rate based on the low-pass uniform sampling theorem?
Given the signal m(t) = 10 cos 2000rt cos 8000nt (a) What is the minimum sampling rate based on the low-pass uniform sampling theorem?
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Quantization and Resolution
Quantization is a methodology of carrying out signal modulation by the process of mapping input values from an infinitely long set of continuous values to a smaller set of finite values. Quantization forms the basic algorithm for lossy compression algorithms and represents a given analog signal into digital signals. In other words, these algorithms form the base of an analog-to-digital converter. Devices that process the algorithm of quantization are known as a quantizer. These devices aid in rounding off (approximation) the errors of an input function called the quantized value.
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![**Signal Analysis and Sampling Rate Determination**
**Given Signal:**
\[ m(t) = 10 \cos(2000\pi t) \cos(8000\pi t) \]
**Problem:**
(a) What is the minimum sampling rate based on the low-pass uniform sampling theorem?
---
**Explanation:**
The problem presents a mathematical representation of a signal \( m(t) \), which is the product of two cosine functions with frequencies determined by \( 2000\pi \) and \( 8000\pi \). To analyze the given signal, we need to determine the minimum sampling rate according to the low-pass uniform sampling theorem, also known as the Nyquist theorem. The theorem states that the sampling rate must be at least twice the highest frequency present in the signal to accurately reconstruct it without aliasing.
The task involves identifying the highest frequency component in the modulated signal, evaluating the resultant frequency spectrum, and calculating the minimum required sampling rate to ensure the data is captured accurately.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fba17cfa5-7b98-4692-8eea-0a66a76af76a%2Fb0dc5086-7333-4fa7-a470-ec2cad9af39b%2Fubxzbp8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Signal Analysis and Sampling Rate Determination**
**Given Signal:**
\[ m(t) = 10 \cos(2000\pi t) \cos(8000\pi t) \]
**Problem:**
(a) What is the minimum sampling rate based on the low-pass uniform sampling theorem?
---
**Explanation:**
The problem presents a mathematical representation of a signal \( m(t) \), which is the product of two cosine functions with frequencies determined by \( 2000\pi \) and \( 8000\pi \). To analyze the given signal, we need to determine the minimum sampling rate according to the low-pass uniform sampling theorem, also known as the Nyquist theorem. The theorem states that the sampling rate must be at least twice the highest frequency present in the signal to accurately reconstruct it without aliasing.
The task involves identifying the highest frequency component in the modulated signal, evaluating the resultant frequency spectrum, and calculating the minimum required sampling rate to ensure the data is captured accurately.
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