Signal Sampling Test - 3

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

None

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

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

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

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

u_c & u_s - lowpass signal component

u_c - lowpass signal component

none of the mentioned

u_s - lowpass signal component

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

None of the mentioned

\(\sum_{n=-∞}^∞ (-1)^{n+r+1} x(2nT^{'}-T^{'}) \frac{sin⁡(π/(2T^{'})) (t-2nT^{'}+T^{'})}{(π/(2T^{'}))(t-2nT^{'}+T^{'})}\)

All of the mentioned

\(\sum_{n=-∞}^∞ (-1)^n x(2nT^{'}) \frac{sin⁡(π/(2T^{'})) (t-2nT^{'})}{(π/(2T^{'}))(t-2nT^{'})}\)

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

u_s-lowpass signal component

u_c-lowpass signal component

none of the mentioned

u_c & u_s-lowpass signal component

5. What is the basic relationship between the spectrum o f the real band pass signal x(t) and the spectrum of the equivalent low pass signal x_l(t)?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

9. The frequency shift can be achieved by multiplying the band pass signal as given in equation
x(t) = \(u_c (t) cos⁡2π F_c t-u_s (t) sin⁡2π F_c t\) by the quadrature carriers cos[2πF_ct] and sin[2πF_ct] and lowpass filtering the products to eliminate the signal components of 2F_c.

False

True

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

None of the mentioned

All of the mentioned

\(\sum_{n=-∞}^∞ (-1)^{n+r+1} x(2nT^{'}-T^{'}) \frac{sin⁡(π/(2T^{'})) (t-2nT^{'}+T^{'})}{(π/(2T^{'}))(t-2nT^{'}+T^{'})}\)

\(\sum_{n=-∞}^∞ (-1)^n x(2nT^{'}) \frac{sin⁡(π/(2T^{'})) (t-2nT^{'})}{(π/(2T^{'}))(t-2nT^{'})}\)