Question

Consider a real, narrowband signal \(x(t) = A(t)\cos[2\pi f_c t + \theta(t)]\) where the maximum frequency components of \(A(t)\) and \(\theta(t)\) are \(f_M\) and \(f_c\) (\(=1000f_M\)), respectively. Which of the following statements is/are correct for \(-\infty<t<\infty\)?

• A. \(x(t)\) represents a PSK modulated signal for suitable choices of \(A(t)\) and \(\theta(t)\).
• B. \(x(t)\) represents an amplitude modulated signal for suitable choices of \(A(t)\) and \(\theta(t)\).
• C. \(x(t)\) represents a band-limited Gaussian noise process.
• D. \(x(t)\) never represents a narrowband FM signal.

Hint:

Recall that any narrowband signal has a general envelope-phase form that covers PSK, AM, FM, and even noise as special cases.

Answer 1

The Correct Option is A, B, C

Step 1: Recall the general form of a narrowband signal.
Any real narrowband (bandpass) signal can be written in the canonical envelope and phase form
\[ x(t) = A(t)\cos[2\pi f_c t + \theta(t)] \] where \(A(t)\) is the slowly varying envelope and \(\theta(t)\) is the slowly varying phase, both bandlimited compared with the carrier frequency \(f_c\).

Step 2: Check option (A), PSK.
In phase shift keying the envelope is held constant, so we pick \(A(t)=\text{constant}\), and the data is carried entirely by discrete jumps in \(\theta(t)\) (for example \(0\) and \(\pi\) for BPSK). This is exactly a special case of the given form.
So option (A) is correct.

Step 3: Check option (B), amplitude modulation.
In amplitude modulation we set \(\theta(t)=0\) and let the envelope carry the message,
\[ A(t) = 1+k\,m(t) \] where \(m(t)\) is the message signal. This is again a special case of the same canonical form.
So option (B) is correct.

Step 4: Check option (C), band-limited Gaussian noise.
A narrowband Gaussian noise process also has a well known canonical envelope-phase decomposition,
\[ n(t) = R(t)\cos[2\pi f_c t + \psi(t)] \] where \(R(t)\) is a Rayleigh distributed envelope and \(\psi(t)\) is a uniformly distributed phase, both of which are bandlimited random processes. Picking \(A(t)=R(t)\) and \(\theta(t)=\psi(t)\) reproduces this.
So option (C) is correct.

Step 5: Check option (D), narrowband FM.
In narrowband FM the envelope is held constant, \(A(t)=\text{constant}\), and the phase carries the message through
\[ \theta(t) = 2\pi k_f\int m(\tau)\,d\tau \] This is again a special case of the same canonical form, so \(x(t)\) can represent a narrowband FM signal. The claim in option (D) that it never does so is false.
So option (D) is incorrect.

Step 6: Final conclusion.
The signal \(x(t)=A(t)\cos[2\pi f_c t+\theta(t)]\) is the general canonical form for any narrowband signal, so it can represent PSK, AM, and band-limited Gaussian noise, and it can also represent narrowband FM, which rules out option (D).
\[ \boxed{\text{(A), (B), (C)}} \]