# 0.4 Analog (de)modulation  (Page 6/9)

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TRUE or FALSE: A small, fixed phase offset in the receiver demodulatingAM with suppressed carrier produces an undesirable low frequency modulated version of the analog message.

Try different frequency offsets gam $=$ $\left[.014$ , $0.1$ , $1.0$ , $10\right]$ . How well does the recovered message $m\left(t\right)$ match the actual message $w\left(t\right)$ ? For each case, what is the spectrum of $m\left(t\right)$ ? Hint: look over more than just the first $1/10$ second to see the effect.

Consider the system shown in [link] . Show that the output of the system is $2{A}_{0}w\left(t\right)cos\left(2\pi {f}_{c}t\right)$ , as indicated.

Create a M atlab routine to implement the square-law mixing modulator of [link] .

1. Create a signal $w\left(t\right)$ that has bandwidth 100 Hz
2. Modulate the signal to 1000 Hz.
3. Demodulate using the AM demodulator from AM.m (to recover the original $w\left(t\right)$ ).

Exercise [link] essentially creates a transmitter and receiver based on the square-law modulator(rather than the more standard mixing modulator). Using this system:

1. Show how the received signal degrades if the phase of the cosine wave is not known exactly.
2. Show how the received signal degrades if the frequency of the cosine wave is not exact.
3. Show how the received signal degrades if the bandpass filter is not centered at the specified frequency.

Consider the transmission system of [link] . The message signal $w\left(t\right)$ has magnitude spectrum shown in part (a).The transmitter in part (b) produces the transmitted signal $x\left(t\right)$ which passes through the channel in part (c). The channel scales thesignal and adds narrowband interferers to create the received signal $r\left(t\right)$ . The transmitter and channel parameters are ${\Phi }_{1}=0.3$ radians, ${f}_{1}=24.1$ kHz, ${f}_{2}=23.9$ kHz, ${f}_{3}=27.5$ kHz, ${f}_{4}=29.3$ kHz, and ${f}_{5}=22.6$ kHz. Thereceiver processing $r\left(t\right)$ is shown in [link] (d). All bandpass and lowpass filters are considered ideal with a gainof unity in the passband and zero in the stopband.

1. Sketch $|R\left(f\right)|$ for $-30$ kHz $\le f\le$ 30kHz. Clearly indicate the amplitudes and frequencies of key points in the sketch.
2. Assume that ${\Phi }_{2}$ is chosen to maximize the magnitude of $y\left(t\right)$ and reflects the value of ${\Phi }_{1}$ and the delays imposed by the two ideal bandpass filters that form thereceived signal $r\left(t\right)$ . Select the receiver parameters ${f}_{6}$ , ${f}_{7}$ , ${f}_{8}$ , and ${f}_{9}$ , so the receiver output $y\left(t\right)$ is a scaled version of $w\left(t\right)$ .

An analog baseband message signal $w\left(t\right)$ has all energy between $-B$ and $B$ Hz. It is upconverted to the transmitted passband signal $x\left(t\right)$ via AM with suppressed carrier

$x\left(t\right)=w\left(t\right)cos\left(2\pi {f}_{c}t+{\Phi }_{c}\right)$

where a carrier frequency ${f}_{c}>10B$ . The channel is a pure delay and the received signal $r$ is $r\left(t\right)=x\left(t-d\right)$ where the delay $d=n{T}_{c}+{T}_{c}/\alpha$ is an integer multiple $n\ge 0$ of the carrier period ${T}_{c}$ ( $=1/{f}_{c}$ ) plus a fraction of ${T}_{c}$ given by $\alpha >1$ . The mixer at the receiver is perfectly synchronizedto the transmitter so that the mixer output $y\left(t\right)$ is

$y\left(t\right)=r\left(t\right)cos\left(2\pi {f}_{c}t+{\Phi }_{r}\right).$

The receiver mixer phase may not match the transmitter mixer phase ${\Phi }_{c}$ . The receiver lowpass filters $y$ to produce

$v\left(t\right)=\mathrm{LPF}\left\{y\left(t\right)\right\}$

where the lowpass filter is ideal with unity passband gain, linear passband phase with zero phase at zero frequency, and cutoff frequency $1.2B$ .

1. Write a formula for the receiver mixer output $y\left(t\right)$ as a function of ${f}_{c}$ , ${\Phi }_{c}$ , $d$ , $\alpha$ , ${\Phi }_{r}$ , and $w\left(t\right)$ (without use of $x$ , $r$ , $n$ , or ${T}_{c}$ ).
2. Determine the amplitude of the minimum and maximum values of $y\left(t\right)$ for $\alpha =4$ .
3. For $\alpha =6$ , $n=42$ , ${\Phi }_{c}=0.2$ radians, and ${T}_{c}=20\phantom{\rule{4pt}{0ex}}\mu$ sec, determine ${\Phi }_{r}$ that maximizes the magnitude of the maximum and minimum values of $v\left(t\right)$ .

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