Question:hard
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 \(-\infty<t<\infty\)?
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 \(-\infty<t<\infty\)?
Show Hint
Recall that any narrowband signal has a general envelope-phase form that covers PSK, AM, FM, and even noise as special cases.
Updated On: Jul 20, 2026
- \(x(t)\) represents a PSK modulated signal for suitable choices of \(A(t)\) and \(\theta(t)\).
- \(x(t)\) represents an amplitude modulated signal for suitable choices of \(A(t)\) and \(\theta(t)\).
- \(x(t)\) represents a band-limited Gaussian noise process.
- \(x(t)\) never represents a narrowband FM signal.
Show Solution
The Correct Option is A, B, C
Solution and Explanation
Step 1: Understanding the Concept:
A narrowband bandpass signal is one whose spectrum sits in a thin band around a carrier $f_c$. Any such signal, no matter what modulation created it, can always be split into a slowly changing size (envelope) times a cosine with a slowly changing phase shift.
Step 2: Key Formula or Approach:
Write the general narrowband form as $x(t)=A(t)\cos[2\pi f_c t+\theta(t)]$, then check whether each modulation type in the options can be produced by some valid pick of $A(t)$ and $\theta(t)$.
Step 3: Detailed Explanation:
For PSK, keep the size fixed, $A(t)=$ constant, and let $\theta(t)$ jump between a set of discrete phase values as the data bits change. This fits the general form, so PSK is possible.
For AM, set the phase term to zero, $\theta(t)=0$, and let the size carry the message, $A(t)=1+km(t)$. This also fits, so amplitude modulation is possible.
For narrowband Gaussian noise, the classical result is that such noise can be written as $n(t)=R(t)\cos[2\pi f_c t+\psi(t)]$ with $R(t)$ a random envelope and $\psi(t)$ a random phase, both slowly varying. Picking $A(t)=R(t)$ and $\theta(t)=\psi(t)$ reproduces this, so band-limited Gaussian noise is possible too.
For FM, keep the size fixed and let the phase carry the message through its integral, $\theta(t)=2\pi k_f\int m(\tau)\,d\tau$. This is again just a special case of the same form, so $x(t)$ absolutely can represent narrowband FM. The claim that it never can is therefore false.
Step 4: Final Answer:
Options (A), (B), and (C) hold, while (D) fails because narrowband FM is also one of the special cases of this general form.
\[ \boxed{\text{(A), (B), (C)}} \]
A narrowband bandpass signal is one whose spectrum sits in a thin band around a carrier $f_c$. Any such signal, no matter what modulation created it, can always be split into a slowly changing size (envelope) times a cosine with a slowly changing phase shift.
Step 2: Key Formula or Approach:
Write the general narrowband form as $x(t)=A(t)\cos[2\pi f_c t+\theta(t)]$, then check whether each modulation type in the options can be produced by some valid pick of $A(t)$ and $\theta(t)$.
Step 3: Detailed Explanation:
For PSK, keep the size fixed, $A(t)=$ constant, and let $\theta(t)$ jump between a set of discrete phase values as the data bits change. This fits the general form, so PSK is possible.
For AM, set the phase term to zero, $\theta(t)=0$, and let the size carry the message, $A(t)=1+km(t)$. This also fits, so amplitude modulation is possible.
For narrowband Gaussian noise, the classical result is that such noise can be written as $n(t)=R(t)\cos[2\pi f_c t+\psi(t)]$ with $R(t)$ a random envelope and $\psi(t)$ a random phase, both slowly varying. Picking $A(t)=R(t)$ and $\theta(t)=\psi(t)$ reproduces this, so band-limited Gaussian noise is possible too.
For FM, keep the size fixed and let the phase carry the message through its integral, $\theta(t)=2\pi k_f\int m(\tau)\,d\tau$. This is again just a special case of the same form, so $x(t)$ absolutely can represent narrowband FM. The claim that it never can is therefore false.
Step 4: Final Answer:
Options (A), (B), and (C) hold, while (D) fails because narrowband FM is also one of the special cases of this general form.
\[ \boxed{\text{(A), (B), (C)}} \]
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