Question

A narrowband FM does not have the following feature

• A. It has two sidebands
• B. Both sidebands are equal in amplitude
• C. It does not show amplitude variations
• D. Both sidebands have same phase difference with respect to carrier

Hint:

NBFM (Narrowband FM) has \(\beta \ll 1\).
NBFM signal \(s(t) \approx A_c \cos(\omega_c t) - A_c \beta \sin(\omega_m t) \sin(\omega_c t)\).
This can be expanded to show a carrier and two sidebands. The LSB and USB have opposite phases relative to each other (or, one is \(180^\circ\) phase shifted w.r.t the carrier's phase contribution compared to the other).
NBFM has a constant envelope (no amplitude variations).

Answer 1

The Correct Option is D

A narrowband FM (NBFM) signal, for a single-tone modulation \(m(t) = A_m \cos(\omega_m t)\), can be approximated as:

\( s_{NBFM}(t) \approx A_c \cos(\omega_c t) - A_c \beta \sin(\omega_m t) \sin(\omega_c t) \) 

where \(\beta = \frac{k_f A_m}{\omega_m}\) is the modulation index (and \(\beta \ll 1\)).

Using the product-to-sum identity: \(\sin A \sin B = \frac{1}{2}[\cos(A - B) - \cos(A + B)]\).

Let \(A = \omega_c t\) and \(B = \omega_m t\). Using the identity:

\(\sin(\omega_m t) \sin(\omega_c t) = \frac{1}{2}[\cos((\omega_c - \omega_m)t) - \cos((\omega_c + \omega_m)t)]\).

Thus, we get:

\( s_{NBFM}(t) \approx A_c \cos(\omega_c t) - \frac{A_c \beta}{2} [\cos((\omega_c - \omega_m)t) - \cos((\omega_c + \omega_m)t)] \)

Expanding this:

\( s_{NBFM}(t) \approx A_c \cos(\omega_c t) - \frac{A_c \beta}{2} \cos((\omega_c - \omega_m)t) + \frac{A_c \beta}{2} \cos((\omega_c + \omega_m)t) \)

The components of the NBFM signal are:

• Carrier: \(A_c \cos(\omega_c t)\)
• Lower Sideband (LSB) at \(\omega_c - \omega_m\): \(-\frac{A_c \beta}{2} \cos((\omega_c - \omega_m)t) = \frac{A_c \beta}{2} \cos((\omega_c - \omega_m)t + \pi)\) or \(\frac{A_c \beta}{2} \cos((\omega_m - \omega_c)t)\). Phase relative to carrier: \(\pi\) or \(180^\circ\) out of phase.
• Upper Sideband (USB) at \(\omega_c + \omega_m\): \(+\frac{A_c \beta}{2} \cos((\omega_c + \omega_m)t)\). Phase relative to carrier: \(0^\circ\) or in phase.

Let's analyze the features:

• (a) "It has two sidebands": True. The signal has a Lower Sideband (LSB) and Upper Sideband (USB).
• (b) "Both sidebands are equal in amplitude": True. The amplitude of both the LSB and USB is \(A_c \beta / 2\).
• (c) "It does not show amplitude variations": True. NBFM (like all FM) is a constant envelope modulation, meaning its amplitude \(A_c\) remains constant, and the approximation shows this.
• (d) "Both sidebands have the same phase difference with respect to the carrier": False. The LSB is \(180^\circ\) out of phase with the carrier, while the USB is in phase with the carrier. Therefore, the phase difference with respect to the carrier is not the same for both sidebands.

The LSB is \(+\frac{A_c \beta}{2} \cos((\omega_c - \omega_m)t + \pi)\), which is \(180^\circ\) out of phase with the carrier. The USB is \(+\frac{A_c \beta}{2} \cos((\omega_c + \omega_m)t)\), which is in phase with the carrier (\(0^\circ\)).

The phase differences are \(180^\circ\) for the LSB and \(0^\circ\) for the USB. These phases are not the same, so option (d) is false.

The question asks for the feature that NBFM does *not* have. The correct answer is:

Both sidebands have the same phase difference with respect to the carrier

Final Answer:

Both sidebands have the same phase difference with respect to the carrier