6.37 Information communication problems

Signals on transmission lines

A modulated signal needs to be sent over a transmission line having a characteristic impedance of $Z_0()=50(\Omega )$ . So that the signal does not interfere with signals others may be transmitting, it must be bandpassfiltered so that its bandwidth is 1 MHz and centered at 3.5 MHz. The filter's gain should be one inmagnitude. An op-amp filter is proposed.

1. What is the transfer function between the input voltage and the voltage across the transmission line?
2. Find values for the resistors and capacitors so that design goals are met.

Noise in am systems

The signal $\hat{s}(t)$ emerging from an AM communication system consists of two parts:the message signal, $s(t)$ , and additive noise. The plot shows the message spectrum $S(f)$ and noise power spectrum $P_N(f)$ . The noise power spectrum lies completely within the signal's band, and has a constant value thereof $\frac{N_0}{2}$ .

1. What is the message signal's power? What is the signal-to-noise ratio?
2. Because the power in the message decreases with frequency, the signal-to-noise ratio is not constantwithin subbands. What is the signal-to-noise ratio in the upper half of the frequency band?
3. A clever 241 student suggests filtering the message before the transmitter modulates it so thatthe signal spectrum is balanced (constant) across frequency. Realizing that this filtering affects the message signal, the studentrealizes that the receiver must also compensate for the message to arrive intact. Draw a block diagram ofthis communication system. How does this system's signal-to-noise ratio compare with that of the usualAM radio?

Complementary filters

Complementary filters usually have “opposite” filtering characteristics (like alowpass and a highpass) and have transfer functions that add to one. Mathematically, $H_1(f)$ and $H_2(f)$ are complementary if $$H_1(f)+H_2(f)=1$$ We can use complementary filters to separate a signal into two parts by passing it through each filter. Eachoutput can then be transmitted separately and the original signal reconstructed at the receiver. Let'sassume the message is bandlimited to $W\mathrm{Hz}$ and that $H_1(f)=\frac{a}{a+i\times 2\pi f}$ .

1. What circuits would be used to produce the complementary filters?
2. Sketch a block diagram for a communication system (transmitter and receiver) that employs complementarysignal transmission to send a message $m(t)$ .
3. What is the receiver's signal-to-noise ratio? How does it compare to the standard system that sends thesignal by simple amplitude modulation?

Phase modulation

A message signal $m(t)$ phase modulates a carrier if the transmitted signal equals $$x(t)=A\sin (2\pi f_{c}t+{\phi }_{d}m(t))$$ where ${\phi }_d$ is known as the phase deviation. In this problem, thephase deviation is small. As with all analog modulation schemes, assume that $\left|m(t)\right|< 1$ , the message is bandlimited to $W$ Hz, and the carrier frequency $f_c$ is much larger than $W$ .

1. What is the transmission bandwidth?
2. Find a receiver for this modulation scheme.
3. What is the signal-to-noise ratio of the received signal?