2.1 Frequency domain problems (ticket #6364)  (Page 6/6)

Sammy falls asleep...

While sitting in ELEC 241 class, he falls asleep during a critical time when an AM receiver is being described.The received signal has the form $r(t)=A(1+m(t))\cos (2\pi f_{c}t+\phi )$ where the phase $\phi$ is unknown. The message signal is $m(t)$ ; it has a bandwidth of $W$ Hz and a magnitude less than 1( $\left|m(t)\right|< 1$ ). The phase $\phi$ is unknown. The instructor drew a diagram for a receiver on the board; Sammy slept through the description of what the unknownsystems where.

1. What are the signals $x_c(t)$ and $x_s(t)$ ?
2. What would you put in for the unknown systems that would guarantee that the final output contained themessage regardless of the phase?
   Hint Think of a trigonometric identity that would prove useful.
3. Sammy may have been asleep, but he can think of a far simpler receiver. What is it?

Jamming

Sid Richardson college decides to set up its own AM radio station KSRR. The resident electrical engineerdecides that she can choose any carrier frequency and message bandwidth for the station. A rival college decides to jam its transmissions by transmitting a high-power signal thatinterferes with radios that try to receive KSRR. The jamming signal $\mathrm{jam}(t)$ is what is known as a sawtooth wave (depicted in [link] ) having a period known to KSRR's engineer.

1. Find the spectrum of the jamming signal.
2. Can KSRR entirely circumvent the attempt to jam it by carefully choosing its carrier frequency andtransmission bandwidth? If so, find the station's carrier frequency and transmission bandwidth in termsof $T$ , the period of the jamming signal; if not, showwhy not.

Am stereo

A stereophonic signal consists of a "left" signal $l(t)$ and a "right" signal $r(t)$ that conveys sounds coming from an orchestra's left andright sides, respectively. To transmit these two signals simultaneously, the transmitter first forms thesum signal $s_{+}(t)=l(t)+r(t)$ and the difference signal $s_{-}(t)=l(t)-r(t)$ . Then, the transmitter amplitude-modulates the differencesignal with a sinusoid having frequency $2W$ , where $W$ is the bandwidth of the left and right signals. The sum signal and the modulated difference signal are added,the sum amplitude-modulated to the radio station's carrier frequency $f_c$ , and transmitted. Assume the spectra of the left and right signals are as shown .

1. What is the expression for the transmitted signal? Sketch its spectrum.
2. Show the block diagram of a stereo AM receiver that can yield the left and right signals as separateoutputs.
3. What signal would be produced by a conventionalcoherent AM receiver that expects to receive a standard AM signal conveying a message signal havingbandwidth $W$ ?

Novel am stereo method

A clever engineer has submitted a patent for a new method for transmitting two signals simultaneously in the same transmission bandwidth as commercial AM radio. As shown , her approach is to modulate the positive portion of the carrier with onesignal and the negative portion with a second.

1. Provide a more concise expression for the transmitted signal $x(t)$ than given above.
2. What is the receiver for this scheme? It would yield both $m_1(t)$ and $m_2(t)$ from $x(t)$ .
3. Find the spectrum of the positive portion of the transmitted signal.
4. Determine whether this scheme satisfies the design criteria, allowing you to grant the patent. Explainyour reasoning.

A radical radio idea

An ELEC 241 student has the bright idea of using a square wave instead of a sinusoid as an AM carrier. Thetransmitted signal would have the form $$x(t)=A(1+m(t)){\mathrm{sq}}_T(t)$$ where the message signal $m(t)$ would be amplitude-limited: $\left|m(t)\right|< 1$

1. Assuming the message signal is lowpass and has a bandwidth of $W$ Hz, what values for the square wave's period $T$ are feasible. In other words, do some combinationsof $W$ and $T$ prevent reception?
2. Assuming reception is possible, can standard radios receive this innovative AM transmission? If so, show how acoherent receiver could demodulate it; if not, show how the coherent receiver's output would becorrupted. Assume that the message bandwidth $W=5$  kHz.

Secret communication

An amplitude-modulated secret message $m(t)$ has the following form. $$r(t)=A(1+m(t))\cos (2\pi (f_c+f_0)t)$$ The message signal has a bandwidth of $W$ Hz and a magnitude less than 1 ( $\left|m(t)\right|< 1$ ). The idea is to offset the carrier frequency by $f_0$ Hz from standard radio carrier frequencies. Thus,"off-the-shelf" coherent demodulators would assume the carrier frequency has $f_c$ Hz. Here, $f_0< W$ .

1. Sketch the spectrum of the demodulated signal produced by a coherent demodulator tuned to $f_c$ Hz.
2. Will this demodulated signal be a “scrambled” version of the original? If so, how so; if not, why not?
3. Can you develop a receiver that can demodulate the message without knowing the offset frequency $f_c$ ?

Signal scrambling

An excited inventor announces the discovery of a way of using analog technology to render musicunlistenable without knowing the secret recovery method. The idea is to modulate the bandlimited message $m(t)$ by a special periodic signal $s(t)$ that is zero during half of its period, which renders the message unlistenable and superficially, atleast, unrecoverable ( [link] ).

1. What is the Fourier series for the periodic signal?
2. What are the restrictions on the period $T$ so that the messagesignal can be recovered from $m(t)s(t)$ ?
3. ELEC 241 students think they have "broken" the inventor's scheme and are going to announce it to theworld. How would they recover the original message without having detailed knowledge of the modulating signal?