0.5 Sampling with automatic gain control  (Page 5/19)

Implement the procedure diagrammed in [link] . Comment on the choice of sampling rates.How have you specified the LPF?

Using your code from Exercise  [link] , examine the effect of “incorrect” sampling rates bydemodulating with $f_s+\gamma$ instead of $f_s$ . This is analogous to the problem that occurs in cosinemixing demodulation when the frequency is not accurate. Is there an analogy to the phase problem that occurs,for instance, with nonzero $\Phi$ in [link] ?

Consider the part of a communication system shown in [link] .

1. Sketch the magnitude spectrum $|X_1\left(f\right)|$ of
   $$x_1\left(t\right)=w\left(t\right)\cos \left(1500\pi t\right).$$
   Be certain to give specific values of frequency and magnitude at all significant points in the sketch.
2. Draw the magnitude spectrum $|X_2\left(f\right)|$ of
   $$x_2\left(t\right)=w\left(t\right)x_1\left(t\right).$$
   Be certain to give specific values of frequency and magnitude at all significant points in the sketch.
3. Between -3750 Hz and 3750 Hz, draw the magnitude spectrum $|X_3\left(f\right)|$ of
   $$x_3\left(t\right)=x_2\left(t\right)\sum _{k=-\infty }^{\infty }\delta (t-k{T}_s).$$
   for $T_s=400\mu$ sec. Be certain to give specific values of frequency andmagnitude at all significant points in the sketch.

Consider the digital receiver shown in [link] . The baseband signal $w\left(t\right)$ has absolute bandwidth $B$ and the carrier frequency is $f_C$ . The channel adds a narrowband interferer at frequency $f_I$ . The received signal is sampled with period $T_s$ . As shown, the sampled signal is demodulated by mixing with a cosineof frequency $f_1$ and the ideal lowpass filter has a cutoff frequency of $f_2$ . For the following designs you are to decide if they are successful,i.e. whether or not the magnitude spectrum of the lowpass filter output $x_4$ is the same (up to a scale factor) as the magnitude spectrum of the sampled $w\left(t\right)$ with a sample period of $T_s$ .

1. Candidate System A: $B=7$ kHz, $f_C=34$ kHz, $f_I=49$ kHz, $T_s=\frac{1}{34}$ msec, $f_1=$ 0, and $f_2=16$ kHz.
2. Candidate System B: $B=11$ kHz, $f_C=39$ kHz, $f_I=130$ kHz, $T_s=\frac{1}{52}$ msec, $f_1=13$ kHz, and $f_2=12$ kHz.

Consider the communication system shown in [link] . In this problem you are to build a receiver from a limitednumber of components. The parts available are:

• four mixers with input $u$ and output $y$ related by
  $$y\left(t\right)=u\left(t\right)\cos \left(2\pi f_{o}t\right)$$
  and oscillator frequencies $f_o$ of 1 MHz, 1.5 MHz, 2 MHz, and 4 MHz,
• four ideal linear bandpass filters with passband ( $f_L$ , $f_U$ ) of (0.5MHz, 6MHz), (1.2 MHz, 6.2MHz),(3.4 MHz, 7.2MHz), and (4.2 MHz, 8.3MHz),
• four impulse samplers with input $u$ and output $y$ related by
  $$y\left(t\right)=\sum _{k=-\infty }^{\infty }u\left(t\right)\delta (t-k{T}_s)$$
  with sample periods of 1/7, 1/5, 1/4, and 1/3.5 microseconds.

The magnitude spectrum $\left|R\right(f\left)\right|$ of the received signal $r\left(t\right)$ is shown in the top part of [link] . The objective is to specify the bandpass filter, sampler, andmixer so that the “M”-shaped segment of the magnitude spectrum is centered at $f=0$ in the output $\left|Y\right(f\left)\right|$ with no other signals within $\pm 1.5$ MHz of the upper and lower edges.

1. Specify the three parts from the 12 provided:
   1. (i) bandpass filter passband range ( $f_L,$ $f_U$ ) in MHz
   2. (ii) sampler period $T_s$ in $\mu$ sec
   3. (iii) mixer oscillator frequency $f_o$ in MHz
2. For the three components selected in part (a), sketch the magnitude spectrum of the sampler outputbetween $-20$ and +20 MHz. Be certain to give specific values of frequency andmagnitude at all significant points in the spectra.
3. For the three components selected in part (a), sketch the magnitude spectrum of $y\left(t\right)$ between between the frequencies $-12$ and $+12$ MHz for your design. Be certain to give specific values of frequency andmagnitude at all significant points in the spectra.
4. Is the magnitude spectrum of $y\left(t\right)$ identical to the the “M-shaped” segment of $\left|R\right(f\left)\right|$ first downconverted to baseband and then sampled?