# 0.4 Analog (de)modulation  (Page 9/9)

 Page 9 / 9

Observe:

• Low-side injection results in symmetry in the translated message spectrum about $±{f}_{c}$ on each of the positive and negative half-axes.
• High-side injection separates the undesired images further from the lower frequency portion (which will ultimatelybe retained to reconstruct the message). This eases the requirements on the bandpass filter.
• Both high-side and low-side injection can place frequency interferers in undesirable places.This highlights the need for adequate out-of-band rejection by a bandpass filter before downconversion to IF.

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 $3{f}_{c}/4$ 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 $3{f}_{c}$ .

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 $±3000$ 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
$y\left(t\right)=u\left(t\right)\mathrm{cos}\left(2\pi {f}_{c}t\right)$
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
$y\left(t\right)=\sum _{k=-\infty }^{\infty }u\left(t\right)\delta \left(t-k{T}_{s}\right)$
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.

A friendly and readable introduction to analog transmission systems can be found in

• P. J. Nahin, On the Science of Radio , AIP Press, 1996.

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