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Illustrate the instantaneous frequency for the following FM modulated waveforms: kHz S fm (t) = A cos[2πft+2nk, fm (t)dt], f = 100 MHz, k, = 10: Volt a) with m (t) = Σm rect k=1 {rect (¹-KT) m² = {-1,1} b) with m (0)= Emreal-AT) met-2,2} rect (¹~KT) m = {-4,4} c) with m (t)= Σm, rect( kul

Illustrate the instantaneous frequency for the following FM modulated waveforms: kHz S fm (t) = A cos[2πft+2nk, fm (t)dt], f = 100 MHz, k, = 10: Volt a) with m (t) = Σm rect k=1 {rect (¹-KT) m² = {-1,1} b) with m (0)= Emreal-AT) met-2,2} rect (¹~KT) m = {-4,4} c) with m (t)= Σm, rect( kul

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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The text describes the instantaneous frequency of frequency-modulated (FM) waveforms. The modulation index and signal are expressed using different values for \( m(t) \). Here's a transcription and explanation suitable for an educational website: --- **Illustration of Instantaneous Frequency for FM Modulated Waveforms** The FM 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: - \( f_c = 100 \, \text{MHz} \) is the carrier frequency. - \( k_f = 10 \, \text{kHz/Volt} \) is the frequency sensitivity. The modulation function \( m(t) \) is described for different cases: **Case a)** \[ m(t) = \sum_{k=1}^{N} m_k \, \text{rect}\left(\frac{t-kT}{T}\right), \quad m_k \in \{-1, 1\} \] **Case b)** \[ m(t) = \sum_{k=1}^{N} m_k \, \text{rect}\left(\frac{t-kT}{T}\right), \quad m_k \in \{-2, 2\} \] **Case c)** \[ m(t) = \sum_{k=1}^{N} m_k \, \text{rect}\left(\frac{t-kT}{T}\right), \quad m_k \in \{-4, 4\} \] - The term \( \text{rect}\left(\frac{t-kT}{T}\right) \) represents a rectangular pulse centered at \( kT \) with duration \( T \). - \( m_k \) are the amplitudes that change in each case, affecting the modulation depth. **Explanation:** - In Case a, \( m(t) \) switches between -1 and 1, creating a frequency deviation based on these values. - In Case b, \( m(t) \) switches between -2 and 2, leading to larger frequency deviations. - In Case c, \( m(t) \) switches between -4 and 4, resulting in even greater frequency deviations. Each case illustrates how varying \( m_k \) values influence the instantaneous
Transcribed Image Text:The text describes the instantaneous frequency of frequency-modulated (FM) waveforms. The modulation index and signal are expressed using different values for \( m(t) \). Here's a transcription and explanation suitable for an educational website: --- **Illustration of Instantaneous Frequency for FM Modulated Waveforms** The FM 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: - \( f_c = 100 \, \text{MHz} \) is the carrier frequency. - \( k_f = 10 \, \text{kHz/Volt} \) is the frequency sensitivity. The modulation function \( m(t) \) is described for different cases: **Case a)** \[ m(t) = \sum_{k=1}^{N} m_k \, \text{rect}\left(\frac{t-kT}{T}\right), \quad m_k \in \{-1, 1\} \] **Case b)** \[ m(t) = \sum_{k=1}^{N} m_k \, \text{rect}\left(\frac{t-kT}{T}\right), \quad m_k \in \{-2, 2\} \] **Case c)** \[ m(t) = \sum_{k=1}^{N} m_k \, \text{rect}\left(\frac{t-kT}{T}\right), \quad m_k \in \{-4, 4\} \] - The term \( \text{rect}\left(\frac{t-kT}{T}\right) \) represents a rectangular pulse centered at \( kT \) with duration \( T \). - \( m_k \) are the amplitudes that change in each case, affecting the modulation depth. **Explanation:** - In Case a, \( m(t) \) switches between -1 and 1, creating a frequency deviation based on these values. - In Case b, \( m(t) \) switches between -2 and 2, leading to larger frequency deviations. - In Case c, \( m(t) \) switches between -4 and 4, resulting in even greater frequency deviations. Each case illustrates how varying \( m_k \) values influence the instantaneous
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