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What is the instantaneous frequency of the following FM modulated waveforms: S fm (t) = A cos[ 2лft + 2nk, m (r)dr] a) with m (t) = 0 b) with m (t) = u(t) c) with m (t)= a.u(t)

What is the instantaneous frequency of the following FM modulated waveforms: S fm (t) = A cos[ 2лft + 2nk, m (r)dr] a) with m (t) = 0 b) with m (t) = u(t) c) with m (t)= a.u(t)

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
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**Title: Instantaneous Frequency of FM Modulated Waveforms** **Introduction:** In frequency modulation (FM), the frequency of the carrier wave is varied according to the information signal. Understanding the instantaneous frequency is crucial for analyzing FM signals. Below, we explore the instantaneous frequency for a given FM modulated waveform, under different conditions. **FM Modulated Waveform:** The expression for a frequency-modulated signal is given by: \[ s_{fm}(t) = A_c \cos\left[ 2\pi f_c t + 2\pi k_f \int_0^t m(\tau) \, d\tau \right] \] where: - \( A_c \) is the amplitude of the carrier signal - \( f_c \) is the carrier frequency - \( k_f \) is the frequency sensitivity - \( m(t) \) is the message signal - The integral denotes the accumulated phase shift due to the modulation **Scenarios:** a) **\( m(t) = 0 \):** In this case, the message signal is zero. Therefore, the integral term becomes zero, resulting in no modulation effect. The instantaneous frequency is the carrier frequency \( f_c \). b) **\( m(t) = u(t) \):** Here, the message signal is a unit step function, \( u(t) \). The integral of a step function from 0 to \( t \) results in a linearly increasing function. Therefore, the instantaneous frequency will increase linearly over time at a rate determined by \( k_f \). c) **\( m(t) = a \cdot u(t) \):** When the message signal is a scaled unit step function \( a \cdot u(t) \), the integral will also result in a linear function but scaled by the factor \( a \). Consequently, the rate of increase of the instantaneous frequency will be proportional to \( a \cdot k_f \). **Conclusion:** Understanding these scenarios with different message signals helps in comprehending how the instantaneous frequency of an FM signal is influenced by the nature and characteristics of the message signal. This analysis is fundamental for efficient FM communication system design.
Transcribed Image Text:**Title: Instantaneous Frequency of FM Modulated Waveforms** **Introduction:** In frequency modulation (FM), the frequency of the carrier wave is varied according to the information signal. Understanding the instantaneous frequency is crucial for analyzing FM signals. Below, we explore the instantaneous frequency for a given FM modulated waveform, under different conditions. **FM Modulated Waveform:** The expression for a frequency-modulated signal is given by: \[ s_{fm}(t) = A_c \cos\left[ 2\pi f_c t + 2\pi k_f \int_0^t m(\tau) \, d\tau \right] \] where: - \( A_c \) is the amplitude of the carrier signal - \( f_c \) is the carrier frequency - \( k_f \) is the frequency sensitivity - \( m(t) \) is the message signal - The integral denotes the accumulated phase shift due to the modulation **Scenarios:** a) **\( m(t) = 0 \):** In this case, the message signal is zero. Therefore, the integral term becomes zero, resulting in no modulation effect. The instantaneous frequency is the carrier frequency \( f_c \). b) **\( m(t) = u(t) \):** Here, the message signal is a unit step function, \( u(t) \). The integral of a step function from 0 to \( t \) results in a linearly increasing function. Therefore, the instantaneous frequency will increase linearly over time at a rate determined by \( k_f \). c) **\( m(t) = a \cdot u(t) \):** When the message signal is a scaled unit step function \( a \cdot u(t) \), the integral will also result in a linear function but scaled by the factor \( a \). Consequently, the rate of increase of the instantaneous frequency will be proportional to \( a \cdot k_f \). **Conclusion:** Understanding these scenarios with different message signals helps in comprehending how the instantaneous frequency of an FM signal is influenced by the nature and characteristics of the message signal. This analysis is fundamental for efficient FM communication system design.
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