Bandwidth of a Phase-Modulated Signal

The given signal is a phase-modulated (PM) signal, represented by the equation: \[ y(t) = A_c \cos (2\pi f_c t + m(t)) \] Here, \(A_c\) is the carrier amplitude, \(f_c\) is the carrier frequency, and \(m(t)\) is the message signal, which is given as: \[ m(t) = A_m \sin (2\pi f_m t) \] Where \(A_m\) is the amplitude of the message signal and \(f_m\) is the frequency of the message signal.

Phase Modulation Fundamentals

In Phase Modulation, the phase of the carrier signal is varied directly in proportion to the instantaneous amplitude of the message signal. In this specific case, the message signal \(m(t)\) directly represents the phase deviation.

Signal Parameters: Phase and Frequency

The instantaneous phase of the modulated signal \(y(t)\) is given by: \[ \theta_i(t) = 2\pi f_c t + m(t) \] Substituting the expression for \(m(t)\): \[ \theta_i(t) = 2\pi f_c t + A_m \sin (2\pi f_m t) \] The instantaneous frequency \(f_i(t)\) is the derivative of the instantaneous phase with respect to time, divided by \(2\pi\): \[ f_i(t) = \frac{1}{2\pi} \frac{d\theta_i(t)}{dt} \] Let's calculate the derivative of \(m(t)\): \[ \frac{dm(t)}{dt} = \frac{d}{dt} (A_m \sin (2\pi f_m t)) = A_m (2\pi f_m) \cos (2\pi f_m t) \] Now, substitute this back into the instantaneous frequency equation: \[ f_i(t) = \frac{1}{2\pi} \left( \frac{d}{dt}(2\pi f_c t) + \frac{d}{dt}(A_m \sin (2\pi f_m t)) \right) \] \[ f_i(t) = \frac{1}{2\pi} (2\pi f_c + A_m (2\pi f_m) \cos (2\pi f_m t)) \] \[ f_i(t) = f_c + A_m f_m \cos (2\pi f_m t) \] This equation shows that the instantaneous frequency varies around the carrier frequency \(f_c\). The term \(A_m f_m \cos (2\pi f_m t)\) represents the frequency deviation.

Frequency Deviation and Modulation Index

The maximum frequency deviation, denoted as \(\Delta f_{max}\), is the maximum change in frequency from the carrier frequency. From the instantaneous frequency equation, the maximum value of the deviation term is when \(\cos (2\pi f_m t) = 1\): \[ \Delta f_{max} = A_m f_m \] For Phase Modulation (PM), the modulation index \(\beta_{PM}\) is the maximum phase deviation, which in this case is the maximum amplitude of \(m(t)\): \[ \beta_{PM} = A_m \] This is a key parameter for determining the bandwidth of a PM signal.

Carson's Rule for Bandwidth

Carson's Rule is widely used to estimate the bandwidth of angle-modulated signals (both FM and PM). It states that the bandwidth (BW) is approximately: \[ \text{BW} \approx 2 (\Delta f_{max} + f_m) \] Alternatively, using the modulation index \(\beta_{PM}\): \[ \text{BW} \approx 2 (\beta_{PM} + 1) f_m \] Let's substitute the expressions for \(\Delta f_{max}\) or \(\beta_{PM}\) derived above.

Carson's Rule Application

Using the first form of Carson's Rule with \(\Delta f_{max} = A_m f_m\): \[ \text{BW} \approx 2 (A_m f_m + f_m) \] \[ \text{BW} \approx 2 f_m (A_m + 1) \] Or, using the second form with \(\beta_{PM} = A_m\): \[ \text{BW} \approx 2 (A_m + 1) f_m \] Both forms yield the same result. This formula clearly shows the factors that influence the bandwidth of the phase-modulated signal.

Bandwidth Dependence Summary

From the derived bandwidth formula \(\text{BW} \approx 2 f_m (A_m + 1)\), we can conclude the following:

• The bandwidth depends on \(A_m\) (the amplitude of the message signal) because \(A_m\) is a direct term in the bandwidth formula.
• The bandwidth depends on \(f_m\) (the frequency of the message signal) because \(f_m\) is also a direct term in the bandwidth formula.

Therefore, the bandwidth of the signal \(y(t)\) depends on both the amplitude \(A_m\) and the frequency \(f_m\) of the message signal \(m(t)\).