6.37 Information communication problems  (Page 2/8)

Digital amplitude modulation

Two ELEC 241 students disagree about a homework problem. The issue concerns the discrete-time signal $s(n)\cos (2\pi f_{0}n)$ , where the signal $s(n)$ has no special characteristics and the modulation frequency $f_0$ is known. Sammy says that he can recover $s(n)$ from its amplitude-modulated version by the same approach used in analog communications. Samanthasays that approach won't work.

1. What is the spectrum of the modulated signal?
2. Who is correct? Why?
3. The teaching assistant does not want to take sides. He tells them that if $s(n)\cos (2\pi f_{0}n)$ and $s(n)\sin (2\pi f_{0}n)$ were both available, $s(n)$ can be recovered. What does he have in mind?

Anti-jamming

One way for someone to keep people from receiving an AM transmission is to transmit noise at the same carrierfrequency. Thus, if the carrier frequency is $f_c$ so that the transmitted signal is $A_T(1+m(t))\sin (2\pi f_{c}t)$ the jammer would transmit $A_{J}n(t)\sin (2\pi f_{c}t+\phi )$ . The noise $n(t)$ has a constant power density spectrum over the bandwidth of the message $m(t)$ . The channel adds white noise of spectral height $\frac{N_0}{2}$ .

1. What would be the output of a traditional AM receiver tuned to the carrier frequency $f_c$ ?
2. RU Electronics proposes to counteract jamming by using a different modulation scheme. The scheme'stransmitted signal has the form $A_T(1+m(t))c(t)$ where $c(t)$ is a periodic carrier signal (period $\frac{1}{f_c}$ ) having the indicated waveform . What is the spectrum of the transmitted signal with the proposed scheme?Assume the message bandwidth $W$ is much less than the fundamental carrier frequency $f_c$ .
3. The jammer, unaware of the change, is transmitting with a carrier frequency of $f_c$ , while the receiver tunes a standard AM receiver to aharmonic of the carrier frequency. What is the signal-to-noise ratio of the receiver tuned to theharmonic having the largest power that does not contain the jammer?

Secret comunications

A system for hiding AM transmissions has the transmitter randomly switching between two carrier frequencies $f_1$ and $f_2$ . "Random switching" means that one carrier frequency is used for some period of time, switches tothe other for some other period of time, back to the first, etc. The receiver knows what the carrierfrequencies are but not when carrier frequency switches occur. Consequently, the receiver must be designed toreceive the transmissions regardless of which carrier frequency is used. Assume the message signal hasbandwidth $W$ . The channel adds white noise of spectral height $\frac{N_0}{2}$ .

1. How different should the carrier frequencies be sothat the message could be received?
2. What receiver would you design?
3. What signal-to-noise ratio for the demodulated signal does your receiver yield?

Am stereo

Stereophonic radio transmits two signals simultaneously that correspond to what comes out of the left and rightspeakers of the receiving radio. While FM stereo is commonplace, AM stereo is not, but is much simpler tounderstand and analyze. An amazing aspect of AM stereo is that both signals are transmitted within the samebandwidth as used to transmit just one. Assume the left and right signals are bandlimited to $W$ Hz. $$x(t)=A(1+m_l(t))\cos (2\pi f_{c}t)+A{m}_r(t)\sin (2\pi f_{c}t)$$

1. Find the Fourier transform of $x(t)$ . What is the transmission bandwidth and how does it compare with that of standard AM?
2. Let us use a coherent demodulator as the receiver, shown in [link] . Show that this receiver indeed works: It produces the left and right signalsseparately.
3. Assume the channel adds white noise to the transmitted signal. Find the signal-to-noise ratioof each signal.