250+ TOP MCQs on The Representation of Bandpass Signals & Answers

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Digital Signal Processing Multiple Choice Questions on “The Representation of Bandpass Signals”.

1. Which of the following is the right way of representation of equation that contains only the positive frequencies in a given x(t) signal?
A. X+(F)=4V(F)X(F)
B. X+(F)=V(F)X(F)
C. X+(F)=2V(F)X(F)
D. X+(F)=8V(F)X(F)
Answer: C
Clarification: In a real valued signal x(t), has a frequency content concentrated in a narrow band of frequencies in the vicinity of a frequency Fc. Such a signal which has only positive frequencies can be expressed as X+(F)=2V(F)X(F)
Where X+(F) is a Fourier transform of x(t) and V(F) is unit step function.

2. What is the equivalent time –domain expression of X+(F)=2V(F)X(F)?
A. F(+1)[2V(F)]*F(+1)[X(F)]
B. F(-1)[4V(F)]*F(-1)[X(F)]
C. F(-1)[V(F)]*F(-1)[X(F)]
D. F(-1)[2V(F)]*F(-1)[X(F)]
Answer: D
Clarification: Given Expression, X+(F)=2V(F)X(F).It can be calculated as follows
(x_+ (t)=int_{-∞}^∞ X_+ (F)e^{j2πFt} dF)
=(F^{-1} [2V(F)]*F^{-1} [X(F)])

3. In time-domain expression, (x_+ (t)=F^{-1} [2V(F)]*F^{-1} [X(F)]). The signal x+(t) is known as
A. Systematic signal
B. Analytic signal
C. Pre-envelope of x(t)
D. Both Analytic signal & Pre-envelope of x(t)
Answer: D
Clarification: From the given expression, (x_+ (t)=F^{-1} [2V(F)] * F^{-1}[X(F)]).

4. In equation (x_+ (t)=F^{-1} [2V(F)]*F^{-1} [X(F)]), if (F^{-1} [2V(F)]=δ(t)+j/πt) and (F^{-1} [X(F)]) = x(t). Then the value of ẋ(t) is?
A. (frac{1}{π} int_{-infty}^infty frac{x(t)}{t+τ} dτ)
B. (frac{1}{π} int_{-infty}^infty frac{x(t)}{t-τ} dτ)
C. (frac{1}{π} int_{-infty}^infty frac{2x(t)}{t-τ} dτ)
D. (frac{1}{π} int_{-infty}^infty frac{4x(t)}{t-τ} dτ)
Answer: B
Clarification: (x_+ (t)=[δ(t)+j/πt]*x(t))
(x_+ (t)=x(t)+[j/πt]*x(t))
(ẋ(t)=[j/πt]*x(t))
=(frac{1}{π} int_{-infty}^infty frac{x(t)}{t-τ} dτ)
Hence proved.

5. If the signal ẋ(t) can be viewed as the output of the filter with impulse response h(t) = 1/πt, -∞ < t < ∞ when excited by the input signal x(t) then such a filter is called as __________
A. Analytic transformer
B. Hilbert transformer
C. Both Analytic & Hilbert transformer
D. None of the mentioned

Answer: B
Clarification: The signal ẋ(t) can be viewed as the output of the filter with impulse response h(t) = 1/πt,
-∞ < t < ∞ when excited by the input signal x(t) then such a filter is called as Hilbert transformer.

6. What is the frequency response of a Hilbert transform H(F)=?
A. (begin{cases}&-j (F>0) \ & 0 (F=0)\ & j (F<0)end{cases})
B. (left{begin{matrix}-j & (F<0)\0 & (F=0) \j & (F>0)end{matrix}right. )
C. (left{begin{matrix}-j & (F>0)\0 &(F=0) \j & (F<0)end{matrix}right. )
D. (left{begin{matrix}j&(F>0)\0 & (F=0)\j & (F<0)end{matrix}right. )
Answer: A
Clarification: H(F) =(int_{-∞}^∞ h(t)e^{-j2πFt} dt)
=(frac{1}{π} int_{-∞}^∞ 1/t e^{-2πFt} dt)
=(left{begin{matrix}-j& (F>0)\0&(F=0) \ j& (F<0)end{matrix}right.)
We Observe that │H (F)│=1 and the phase response ⊙(F) = -1/2π for F > 0 and ⊙(F) = 1/2π for F < 0.

7. What is the equivalent lowpass representation obtained by performing a frequency translation of X+(F) to Xl(F)= ?
A. X+(F+Fc)
B. X+(F-Fc)
C. X+(F*Fc)
D. X+(Fc-F)
Answer: A
Clarification: The analytic signal x+(t) is a bandpass signal. We obtain an equivalent lowpass representation by performing a frequency translation of X+(F).

8. What is the equivalent time domain relation of xl(t) i.e., lowpass signal?
A. (x_l (t)=[x(t)+j ẋ(t)]e^{-j2πF_c t})
B. x(t)+j ẋ(t) = (x_l (t) e^{j2πF_c t})
C. (x_l (t)=[x(t)+j ẋ(t)]e^{-j2πF_c t}) & x(t)+j ẋ(t) = (x_l (t) e^{j2πF_c t})
D. None of the mentioned
Answer: C
Clarification: (x_l (t)=x_+ (t) e^{-j2πF_c t})
=([x(t)+j ẋ(t)] e^{-j2πF_c t})
Or equivalently, x(t)+j ẋ(t) =(x_l (t) e^{j2πF_c t}).

9. If we substitute the equation (x_l (t)= u_c (t)+j u_s (t)) in equation x (t) + j ẋ (t) = xl(t) ej2πFct and equate real and imaginary parts on side, then what are the relations that we obtain?
A. x(t)=(u_c (t) ,cos⁡2π ,F_c ,t+u_s (t) ,sin⁡2π ,F_c ,t); ẋ(t)=(u_s (t) ,cos⁡2π ,F_c ,t-u_c ,(t) ,sin⁡2π ,F_c ,t)
B. x(t)=(u_c (t) ,cos⁡2π ,F_c ,t-u_s (t) ,sin⁡2π ,F_c ,t); ẋ(t)=(u_s (t) ,cos⁡2π ,F_c t+u_c (t) ,sin⁡2π ,F_c ,t)
C. x(t)=(u_c (t) ,cos⁡2π ,F_c t+u_s (t) ,sin⁡2π ,F_c ,t); ẋ(t)=(u_s (t) ,cos⁡2π ,F_c t+u_c (t) ,sin⁡2π ,F_c ,t)
D. x(t)=(u_c (t) ,cos⁡2π ,F_c ,t-u_s (t) ,sin⁡2π ,F_c ,t); ẋ(t)=(u_s (t) ,cos⁡2π ,F_c ,t-u_c (t) ,sin⁡2π ,F_c ,t)

Answer: B
Clarification: If we substitute the given equation in other, then we get the required result

10. In the relation, x(t) = (u_c (t) cos⁡2π ,F_c ,t-u_s (t) sin⁡2π ,F_c ,t) the low frequency components uc and us are called _____________ of the bandpass signal x(t).
A. Quadratic components
B. Quadrature components
C. Triplet components
D. None of the mentioned
Answer: B
Clarification: The low frequency signal components uc(t) and us(t) can be viewed as amplitude modulations impressed on the carrier components cos2πFct and sin2πFct, respectively. Since these carrier components are in phase quadrature, uc(t) and us(t) are called the Quadrature components of the bandpass signal x (t).

11. What is the other way of representation of bandpass signal x(t)?
A. x(t) = Re([x_l (t) e^{j2πF_c t}])
B. x(t) = Re([x_l (t) e^{jπF_c t}])
C. x(t) = Re([x_l (t) e^{j4πF_c t}])
D. x(t) = Re([x_l (t) e^{j0πF_c t}])
Answer: A
Clarification: The above signal is formed from quadrature components, x(t) = Re([x_l (t) e^{j2πF_c t}]) where Re denotes the real part of complex valued quantity.

12. In the equation x(t) = Re([x_l (t) e^{j2πF_c t}]), What is the lowpass signal xl (t) is usually called the ___ of the real signal x(t).
A. Mediature envelope
B. Complex envelope
C. Equivalent envelope
D. All of the mentioned
Answer: B
Clarification: In the equation x(t) = Re[xl(t)e(j2πFct)], Re denotes the real part of the complex valued quantity in the brackets following. The lowpass signal xl (t) is usually called the Complex envelope of the real signal x(t), and is basically the equivalent low pass signal.

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13. If a possible representation of a band pass signal is obtained by expressing xl (t) as (x_l (t)=a(t)e^{jθ(t})) then what are the equations of a(t) and θ(t)?
A. a(t) = (sqrt{u_c^2 (t)+u_s^2 (t)}) and θ(t)=(tan^{-1}frac{u_s (t)}{u_c (t)})
B. a(t) = (sqrt{u_c^2 (t)-u_s^2 (t)}) and θ(t)=(tan^{-1}frac{u_s (t)}{u_c (t)})
C. a(t) = (sqrt{u_c^2 (t)+u_s^2 (t)}) and θ(t)=(tan^{-1}frac{u_c (t)}{u_s (t)})
D. a(t) = (sqrt{u_s^2 (t)-u_c^2 (t)}) and θ(t)=(tan^{-1}⁡frac{u_s (t)}{u_c (t)})
Answer: A
Clarification: A third possible representation of a band pass signal is obtained by expressing (x_l (t)=a(t)e^{jθ(t)}) where a(t) = (sqrt{u_c^2 (t)+u_s^2 (t)}) and θ(t)=(tan^{-1}frac{u_s (t)}{u_c (t)}).

14. What is the possible representation of x(t) if xl(t)=a(t)e(jθ(t))?
A. x(t) = a(t) cos[2πFct – θ(t)]
B. x(t) = a(t) cos[2πFct + θ(t)]
C. x(t) = a(t) sin[2πFct + θ(t)]
D. x(t) = a(t) sin[2πFct – θ(t)]
Answer: B
Clarification: x(t) = Re([x_l (t) e^{j2πF_c t}])
= Re([a(t) e^{j[2πF_c t + θ(t)]}])
= (a(t) ,cos⁡ [2πF_c t+θ(t)])
Hence proved.

15. In the equation x(t) = a(t)cos[2πFct+θ(t)], Which of the following relations between a(t) and x(t), θ(t) and x(t) are true?
A. a(t), θ(t) are called the Phases of x(t)
B. a(t) is the Phase of x(t), θ(t) is called the Envelope of x(t)
C. a(t) is the Envelope of x(t), θ(t) is called the Phase of x(t)
D. none of the mentioned
Answer: C
Clarification: In the equation x(t) = a(t) cos[2πFct+θ(t)], the signal a(t) is called the Envelope of x(t), and θ(t) is called the phase of x(t).

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