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

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 F_c. 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 X_l(F)= ?
A. X₊(F+F_c)
B. X₊(F-F_c)
C. X₊(F*F_c)
D. X₊(F_c-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 x_l(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) = x_l(t) e^j2πF_ct 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 u_c and u_s 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 u_c(t) and u_s(t) can be viewed as amplitude modulations impressed on the carrier components cos2πF_ct and sin2πF_ct, respectively. Since these carrier components are in phase quadrature, u_c(t) and u_s(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) = R_e([x_l (t) e^{j2πF_c t}])
B. x(t) = R_e([x_l (t) e^{jπF_c t}])
C. x(t) = R_e([x_l (t) e^{j4πF_c t}])
D. x(t) = R_e([x_l (t) e^{j0πF_c t}])
Answer: A
Clarification: The above signal is formed from quadrature components, x(t) = R_e([x_l (t) e^{j2πF_c t}]) where R_e denotes the real part of complex valued quantity.

12. In the equation x(t) = R_e([x_l (t) e^{j2πF_c t}]), What is the lowpass signal x_l (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) = R_e[x_l(t)e^(j2πF_ct)], R_e denotes the real part of the complex valued quantity in the brackets following. The lowpass signal x_l (t) is usually called the Complex envelope of the real signal x(t), and is basically the equivalent low pass signal.

13. If a possible representation of a band pass signal is obtained by expressing x_l (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 x_l(t)=a(t)e^(jθ(t))?
A. x(t) = a(t) cos[2πF_ct – θ(t)]
B. x(t) = a(t) cos[2πF_ct + θ(t)]
C. x(t) = a(t) sin[2πF_ct + θ(t)]
D. x(t) = a(t) sin[2πF_ct – θ(t)]
Answer: B
Clarification: x(t) = R_e([x_l (t) e^{j2πF_c t}])
= R_e([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[2πF_ct+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[2πF_ct+θ(t)], the signal a(t) is called the Envelope of x(t), and θ(t) is called the phase of x(t).