0.4 Analog (de)modulation  (Page 9/9)

Consider the system described in [link] . The message $w\left(t\right)$ has a bandwidth of 22kHz and a magnitude spectrum as shown.The message is upconverted by a mixer with carrier frequency $f_c$ . The channel adds an interferer $n$ . The received signal $r$ is downconverted to the IF signal $x\left(t\right)$ by a mixer with frequency $f_r$ .

1. With $n\left(t\right)=0$ , $f_r=36$ kHz, and $f_c=83$ kHz, indicate all frequency ranges (i)-(x) that include any partof the IF passband signal $x\left(t\right)$ . (i) 0-20 kHz,(ii) 20-40 kHz, (iii) 40-60 kHz,(iv) 60-80 kHz, (v) 80-100 kHz,(vi) 100-120 kHz, (vii) 120-140 kHz,(viii) 140-160 kHz, (ix) 160-180 kHz,(x) 180-200 kHz
2. With $f_r=36$ kHz and $f_c=83$ kHz, indicate all frequency ranges (i)-(x) that include anyfrequency that causes a narrowband interferer $n$ to appear in the nonzero portions of the magnitude spectrum ofthe IF passband signal $x\left(t\right)$ .
3. With $f_r=84$ kHz and $f_c=62$ kHZ, indicate every range (i)-(x) that includes anyfrequency that causes a narrowband interferer $n$ to appear in the nonzero portions of the magnitude spectrum ofthe IF passband signal $x\left(t\right)$ .

A transmitter operates as a standard AM with suppressed carrier transmitter (as in AM.m ). Create a demodulation routine that operates in two steps:by mixing with a cosine of frequency $3f_c/4$ and subsequently mixing with a cosine of frequency $f_c/4$ . Where must pass/reject filters be placed in orderto ensure reconstruction of the message? Let $f_c=2000$ .

Consider the schematic shown in [link] with the absolute bandwidth of the baseband signal $x_1$ of 4 kHz, $f_1=28$ kHz, $f_2=20$ kHz, and $f_3=26$ kHz.

1. What is the absolute bandwidth of $x_2\left(t\right)$ ?
2. What is the absolute bandwidth of $x_3\left(t\right)$ ?
3. What is the absolute bandwidth of $x_4\left(t\right)$ ?
4. What is the maximum frequency in $x_2\left(t\right)$ ?
5. What is the maximum frequency in $x_3\left(t\right)$ ?

Using your M atlab code from Exercise  [link] , investigate the effect of a sinusoidal interference:

1. at frequency $\frac{f_c}{6}$ ,
2. at frequency $\frac{f_c}{3}$ ,
3. at frequency $3f_c$ .

Consider the PAM communication system in [link] . The input $x_1\left(t\right)$ has a triangular baseband magnitude spectrum. The frequency specifications are $f_1=100$ kHz, $f_2=1720$ kHz, $f_3=1940$ kHz, $f_4=1580$ kHz, $f_5=1720$ kHz, $f_6=1880$ kHz, and $f_7=1300$ kHz.

1. Draw the magnitude spectrum $|X_5\left(f\right)|$ between $\pm 3000$ kHz. Be certain to give specific values of frequency andmagnitude at all breakpoints and local maxima.
2. Specify values of $f_8$ and $f_9$ for which the system can recover the original message without corruption with $M=2$ .

This problem asks you to build a receiver from a limited number of components.The parts available are:

1. two product modulators with input $u$ and output $y$ related by
   $$y\left(t\right)=u\left(t\right)\cos \left(2\pi f_{c}t\right)$$
   and carrier frequencies $f_c$ of 12 MHz and 50 MHz
2. two linear bandpass filters with ideal rectangular magnitude spectrum of gain one between $-f_U$ and $-f_L$ and between $f_L$ and $f_U$ and zero elsewhere with ( $f_L$ , $f_U$ ) of (12MHz, 32MHz) and (35MHz, 50MHz).
3. two 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 1/15 and 1/12 microseconds
4. one square law device with input $u$ and output $y$ related by
   $$y\left(t\right)=u^2\left(t\right)$$
5. three summers with inputs $u_1$ and $u_2$ and output $y$ related by
   $$y\left(t\right)=u_1\left(t\right)+u_2\left(t\right).$$

The spectrum of the received signal is illustrated in [link] . The desired baseband output of the receivershould be a scaled version of the triangular portion centered at zero frequency with no other signalsin the range between $-8$ and 8 MHz. Using no more than four parts from the 10 available,build a receiver that produces the desired baseband signal. Draw its block diagram.Sketch the magnitude spectrum of the output of each part in the receiver.