Phase Modulation Quiz

The maximum phase deviation in PM is given by Δθ = k_p × m_p, where k_p is the phase deviation constant and m_p is the peak amplitude of the modulating signal.

Here, k_p = 2 rad/V and m_p = 5 V, so Δθ = 2 × 5 = 10 rad.

For phase modulation, the instantaneous phase is directly proportional to the modulating signal: θ_i(t) = ω_ct + k_pm(t).

This differs from frequency modulation where the instantaneous phase involves integration of the modulating signal.

The modulation index β for PM is the maximum phase deviation. In the expression s(t) = A_ccos(ω_ct + βsin(ω_mt)), β is the coefficient of the sine term.

Here, β = 10, which is the maximum phase deviation in radians.

Carson's rule for bandwidth estimation is: BW ≈ 2(Δf + f_m).

For PM, Δf = β × f_m = 5 × 5 kHz = 25 kHz.

So BW ≈ 2(25 kHz + 5 kHz) = 2 × 30 kHz = 60 kHz.

The peak frequency deviation in PM is given by Δf = k_p × (dm(t)/dt)_max / (2π).

For m(t) = 2cos(2π×2×10³t), dm(t)/dt = -2×2π×2×10³sin(2π×2×10³t).

The maximum value of |dm(t)/dt| is 2×2π×2×10³ = 8π×10³ V/s.

So Δf = (3 × 8π×10³) / (2π) = 12×10³ Hz = 12 kHz.

According to Bessel function analysis, the number of significant sideband pairs is approximately β + 1.

For β = 4, significant sideband pairs ≈ 4 + 1 = 5.

This means we have the carrier and 5 pairs of sidebands with significant amplitude.

The instantaneous frequency is f_i(t) = (1/2π) × dθ/dt, where θ is the total phase.

For s(t) = 10cos(2π×10⁷t + 8sin(2π×10⁴t)), the phase is θ(t) = 2π×10⁷t + 8sin(2π×10⁴t).

dθ/dt = 2π×10⁷ + 8×2π×10⁴cos(2π×10⁴t).

At t = 0, dθ/dt = 2π×10⁷ + 16π×10⁴ = 2π(10⁷ + 0.00016×10⁷) ≈ 2π×10⁷.

So f_i(0) ≈ 10 MHz.

In PM, if the modulating frequency is doubled while keeping amplitude constant, the modulation index β remains the same (since β = k_pm_p).

However, the frequency deviation Δf = β × f_m doubles.

Using Carson's rule: BW ≈ 2(Δf + f_m) ≈ 2(2Δf + 2f_m) = 2×2(Δf + f_m) = 2×original bandwidth.

So the bandwidth doubles.

For phase modulation, the relationship between phase deviation and frequency deviation is: Δf = Δθ × f_m.

This comes from the fact that the maximum frequency deviation occurs when the phase is changing at its maximum rate, which is Δθ × ω_m = Δθ × 2πf_m, and dividing by 2π gives Δf = Δθ × f_m.

The modulation index for PM is β = k_p × m_p, where m_p is the peak amplitude of the modulating signal.

Here, k_p = 1 rad/V and m_p = 5 V, so β = 1 × 5 = 5.