1. The frequency shift can be achieved by multiplying the bandpass signal as given in the equation x(t) = \(u_c (t) cos2π F_c t-u_s (t) sin2π F_c t\) by the quadrature carriers cos[2πFct] and sin[2πFct] and lowpass filtering the products to eliminate the signal components of 2Fc.

A. True

B. False

2. What is the final result obtained by substituting Fc=kB-B/2, T= 1/2B and say n = 2m i.e., for even and n=2m-1 for odd in equation x(nT)= \(u_c (nT)cos2πF_c nT-u_s (nT)sin 2πF_c nT\)?

A. \((-1)^m u_c (mT_1)-u_s\)

B. \(u_s (mT_1-\frac{T_1}{2})(-1)^{m+k+1}\)

C. \((-1)^m u_c (mT_1)- u_s (mT_1-\frac{T_1}{2})(-1)^{m+k+1}\)

D. None

3. Which low pass signal component occurs at the rate of B samples per second with even-numbered samples of x(t)?

A. uc-lowpass signal component

B. us-lowpass signal component

C. uc & us-lowpass signal component

D. none of the mentioned

4. Which low pass signal component occurs at the rate of B samples per second with odd-numbered samples of x(t)?

A. uc – lowpass signal component

B. us – lowpass signal component

C. uc & us – lowpass signal component

D. none of the mentioned

5. What is the reconstruction formula for the bandpass signal x(t) with samples taken at the rate of 2B samples per second?

A. \(\sum_{m=-\infty}^{\infty}x(mT)\frac{sin(π/2T) (t-mT)}{(π/2T)(t-mT)} cos2πF_c (t-mT)\)

B. \(\sum_{m=-\infty}^{\infty}x(mT)\frac{sin(π/2T) (t+mT)}{(π/2T)(t+mT)} cos2πF_c (t-mT)\)

C. \(\sum_{m=-\infty}^{\infty}x(mT)\frac{sin(π/2T) (t-mT)}{(π/2T)(t-mT)} cos2πF_c (t+mT)\)

D. \(\sum_{m=-\infty}^{\infty}x(mT)\frac{sin(π/2T) (t+mT)}{(π/2T)(t+mT)} cos2πF_c (t+mT)\)

6. What is the new center frequency for the increased bandwidth signal?

A. Fc‘= Fc+B/2+B’/2

B. Fc‘= Fc+B/2-B’/2

C. Fc‘= Fc-B/2-B’/2

D. None of the mentioned

7. According to the sampling theorem for low pass signals with T1=1/B, then what is the expression for uc(t) = ?

A. \(\sum_{m=-∞}^∞ u_c (mT_1)\frac{sin(\frac{π}{T_1}) (t-mT_1)}{(π/T_1)(t-mT_1)}\)

B. \(\sum_{m=-∞}^∞ u_s (mT_1-\frac{T_1}{2}) \frac{sin(\frac{π}{T_1}) (t-mT_1+T_1/2)}{(\frac{π}{T_1})(t-mT_1+\frac{T_1}{2})}\)

C. \(\sum_{m=-∞}^∞ u_c (mT_1)\frac{sin(\frac{π}{T_1}) (t+mT_1)}{(\frac{π}{T_1})(t+mT_1)}\)

D. \(\sum_{m=-∞}^∞ u_s (mT_1-\frac{T_1}{2}) \frac{sin(\frac{π}{T_1}) (t+mT_1+\frac{T_1}{2})}{(\frac{π}{T_1})(t+mT_1+\frac{T_1}{2})}\)

8. According to the sampling theorem for low pass signals with T1=1/B, then what is the expression for us(t) = ?

A. \(\sum_{m=-∞}^∞ u_c (mT_1) \frac{sin(\frac{π}{T_1}) (t-mT_1)}{(\frac{π}{T_1})(t-mT_1)}\)

B. \(\sum_{m=-∞}^∞ u_s (mT_1-\frac{T_1}{2}) \frac{sin(\frac{π}{T_1}) (t-mT_1+\frac{T_1}{2})}{(π/T_1)(t-mT_1+\frac{T_1}{2})}\)

C. \(\sum_{m=-∞}^∞ u_s (mT_1-\frac{T_1}{2}) \frac{sin(\frac{π}{T_1}) (t-mT_1-\frac{T_1}{2})}{(\frac{π}{T_1})(t-mT_1-\frac{T_1}{2})}\)

D. \(\sum_{m=-∞}^∞ u_c (mT_1) \frac{sin(\frac{π}{T_1}) (t+mT_1)}{(\frac{π}{T_1})(t+mT_1)}\)

9. What is the expression for low pass signal component uc(t) that can be expressed in terms of samples of the bandpass signal?

A. \(\sum_{n=-∞}^∞ (-1)^{n+r+1} x(2nT^{‘}-T^{‘}) \frac{sin(π/(2T^{‘})) (t-2nT^{‘}+T^{‘})}{(π/(2T^{‘}))(t-2nT^{‘}+T^{‘})}\)

B. \(\sum_{n=-∞}^∞ (-1)^n x(2nT^{‘}) \frac{sin(π/(2T^{‘})) (t-2nT^{‘})}{(π/(2T^{‘}))(t-2nT^{‘})}\)

C. All of the mentioned

D. None of the mentioned

10. What is the expression for low pass signal component us(t) that can be expressed in terms of samples of the bandpass signal?

A. \(\sum_{n=-∞}^∞ (-1)^{n+r+1} x(2nT^{‘}-T^{‘}) \frac{sin(π/(2T^{‘})) (t-2nT^{‘}+T^{‘})}{(π/(2T^{‘}))(t-2nT^{‘}+T^{‘})}\)

B. \(\sum_{n=-∞}^∞ (-1)^n x(2nT^{‘}) \frac{sin(π/(2T^{‘})) (t-2nT^{‘})}{(π/(2T^{‘}))(t-2nT^{‘})}\)

C. All of the mentioned

D. None of the mentioned

11. What is the Fourier transform of x(t)?

A. X (F) = \(\frac{1}{2} [X_l (F-F_C.+X_l^* (F-F_C.]\)

B. X (F) = \(\frac{1}{2} [X_l (F-F_C.+X_l^* (F+F_C.]\)

C. X (F) = \(\frac{1}{2} [X_l (F+F_C.+X_l^* (F-F_C.]\)

D. X (F) = \(\frac{1}{2} [X_l (F-F_C.+X_l^* (-F-F_C.]\)

12. What is the basic relationship between the spectrum of the real bandpass signal x(t) and the spectrum of the equivalent low pass signal xl(t)?

A. X (F) = \(\frac{1}{2} [X_l (F-F_C.+X_l^* (F-F_C.]\)

B. X (F) = \(\frac{1}{2} [X_l (F-F_C.+X_l^* (F+F_C.]\)

C. X (F) = \(\frac{1}{2} [X_l (F+F_C.+X_l^* (F-F_C.]\)

D. X (F) = \(\frac{1}{2} [X_l (F-F_C.+X_l^* (-F-F_C.]\)

13. Which of the following is the right way of representing of the 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)

14. 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)]

15. 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)

16. 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. \(\frac{1}{π} \int_{-\infty}^\infty \frac{4x(t)}{t-τ} dτ\)

17. 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

18. 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. \(\left\{\begin{matrix}j&(F>0)\\0 & (F=0)\\j & (F<0)\end{matrix}\right. \)

19. 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)

20. 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

21. 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)=\(u_c (t) \,cos2π \,F_c \,t+u_s (t) \,sin2π \,F_c \,t\); ẋ(t)=\(u_s (t) \,cos2π \,F_c \,t-u_c \,(t) \,sin2π \,F_c \,t\)

B. x(t)=\(u_c (t) \,cos2π \,F_c \,t-u_s (t) \,sin2π \,F_c \,t\); ẋ(t)=\(u_s (t) \,cos2π \,F_c t+u_c (t) \,sin2π \,F_c \,t\)

C. x(t)=\(u_c (t) \,cos2π \,F_c t+u_s (t) \,sin2π \,F_c \,t\); ẋ(t)=\(u_s (t) \,cos2π \,F_c t+u_c (t) \,sin2π \,F_c \,t\)

D. x(t)=\(u_c (t) \,cos2π \,F_c \,t-u_s (t) \,sin2π \,F_c \,t\); ẋ(t)=\(u_s (t) \,cos2π \,F_c \,t-u_c (t) \,sin2π \,F_c \,t\)

22. In the relation, x(t) = \(u_c (t) cos2π \,F_c \,t-u_s (t) sin2π \,F_c \,t\) 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