5. Communication System Parameters [12 points]

P-3.2 A signal composed of sinusoids is given by the equation x(t)=10 \cos (800 \pi t+\pi / 4)+7 \cos (1200 \pi t-\pi / 3)-3 \cos (1600 \pi t) (a) Sketch the spectrum of this signal, indicating the complex amplitude of each frequency component. Make separate plots for real/imaginary or magnitude/phase of the complex amplitudes at each frequency.

P-2.2 Figure P-2.2 (see below) is a plot of a sinusoidal wave. From the plot, determine values for the amplitude(A), phase (p), and frequency (coo) needed in the representation: x(t) = Acos(oot + p)Give the answer as numerical values, including the units where applicable.

1.1 Compute and plot the fraction of power transmitted as a function of the transmission-line loss in decibels. Do this on log paper for a loss range of 0-50 dB.

Part A: Analog Communications Review Modulation methods within Analog Communications have evolved considerably over the last century. Produce a report outlining the key analog modulation methods. Within your report discuss each of the major techniques. You are required to describe the operation of each technique, how to generate each technique and the major advantages and disadvantage of each technique. (As a guideline this should be approximately 1000 words) Part B: Single Audio Signal Transmission An audio signal described below is to be transmitted using a radio signal. Vm = 2 сos(2π * 10^3) t Propose 3 different ways to do this. Include choice of frequency, carrier frequency, bandwidth, modulation index, etc. Compare the advantages and disadvantages of each type and suggest the most appropriate method - justify all decisions. Part C: Multiple Audio Signal Wireless Transmission Three audio signals described below is to be transmitted on the same wireless channel. Signal 1 has a frequency response from 2KHz to 4KHz Signal 2 has a frequency response from 2KHz to 4KHz Signal 3 has a frequency response from 4KHz to 8KHz Design a system to achieve this. You should include the frequency response at each stage. You should also justify all key decisions.

QUESTIONS/SUGGESTIONS 1) Compute the dipole input impedance Zin for various values of the length to radius ratios: 100, 200, 500, 1000, 2000 and 25000. Choose a frequency between 950 MHz and 2.88 GHz. Present your result in a tabular form. Specify exactly the values of L and a conforming to the ratios. 2) For the above data in part 1), prepare a dipole model and obtain the full-wave results for the dipole input impedance. Compare the analytical results in part 1) against the results in part 2). Present your result in tabulated form. 3) Next, choose a fixed length of the dipole and its radius. Select a frequency fo in the range between 950 MHz and 2.88 GHz such that the dipole is exactly at the frequency of your choice. Now vary the frequency, keeping all the other data fixed, and compute the input impedance. (Though L and a are formally fixed due to the choice of fo, varying the frequency changes the kL and ka; this changes the dipole input impedance.) Plot the variations of Rin and Xin vs. kL and ka. (This is using the analytical result.) 4) Repeat the process in part 3) but using an appropriate CAD software. Compare your results against analytical results you obtained in part 3). 5) Going by the results you have obtained in parts 1) through 4), comment on the validation of the analytical formulas. How does the approximation break down? Which ratios show the best agreements between analytical and full-wave numerical results?

2.17 Consider a baseband signal: X(t) = x₁(1) + jxq(t), where x/(t) and xo(t) are baseband signals with frequency content limited to [-W, +W]. Let X₁(f) and Xo(f) be the frequency content of the signals. So X₁(f) = XQ(f) = 0 for f * [-W, W]. The energy of the lowpass complex signal is The passband signal is = f ₁8010³d₁. E₁ = x(t) = x1(1) √2 cos(2n fet) - xo(t) √2 sin(2n fet), where fe < W. The energy of the passband signal is 138 Ep = fix(0)³dt. Show that E = Ep. Hint: Derive expressions for the energy in the frequency domain for x(f). Use Parseval's theorem: fut² (1dt = fuvas,

1. (a) What is the equation for the received signal power at a monostatic radar in terms of: Pt transmit power, Gt= Gr=G antenna gain, o RCS, R range to target, f frequency, Ltot total losses? (b) What are the types of losses comprising Ltot? (c) How is receiver noise related to noise figure and other parameters? (d) If (S/N)min is the minimum detectable SNR, find the relationship between & and antenna gain when other parameters are held constant: o = g(G:other parameters)

2.12 A communication system uses BPSK modulation to transmit data bits b₁, 1 = 0, 1, 2,.... In the transmitter, a sequence of rectan- gular pulses is mixed to a carrier frequency by multiplying the rectangular pulses by √2P cos(27f₁t): s(t) = √2P bipr(t-IT) cos(2n fit). /-0,1.... At the receiver, the received signal is first mixed down to baseband by multiplying by √2/T cos(27f2t), where f₂ - f₁ = Af is the offset of the two 136 oscillators. After the signal is mixed down, it is filtered with a matched filter, that is h(t) = pr(t). The filter is sampled at time t = iT for i = 1,2,.... In addition, the signal is mixed down by multiplying by -√2/T sin(27f2t). Let ye(iT) denote the first output and y, (iT) denote the second output. Then, ye(it) = = £ $(7) √/²7/ CC cos(21f27)h(iT - t)dt ys(IT) = - = - * S(T) 7) √ sin(2727)h(iT - T)dt. 1-00 (a) In the absence of noise, evaluate the outputs y(iT) and y, (iT) in terms of bi-₁› E = PT. AfT, and į. Ignore double frequency terms in evaluating the output. That is, derive an expression for ye(iT) and y,(iT). (Useful trig identity sin(u) - sin(v) = 2 cos("") sin("").) (b) Assume you buy two crystal oscillators at a 10 MHz nominal frequency that have + 10 PPM accuracy. That is, factual = fnominal (1 ± 10/106). Assume that the data rate is 100 kbps (7 = 10-5), that the data bits are all positive ( b; = 1, i = 0, 1, 2,..., 500), and that E = 1. (i) Are the double frequency terms negligible? (ii) Plot the output of the filters y (iT) and y, (iT) as a function of į for 1 ≤i≤ 500. Assume all the data bits are +1 and f₂-fi = Af = 200 Hz and T = 10-³.

Consider an FM signal generated by modulating a 10-MHz car-rier by a 1-kHz sinusoidal waveform such that peak frequency deviation is 2.5 kHz. a. Determine the bandwidth of the modulated signal. b. If the modulating signal amplitude is doubled, determine the bandwidth of the modulated signal. c. Determine the bandwidth of the modulated signal if the modulating signal frequency is doubled. 1. Determine the bandwidth of the modulated signal if both the-amplitude and frequency of the modulating signal are doubled.

A 1-MHz carrier is frequency-modulated by a single tone of frequency 2 kHz, resulting in the peak frequency deviation of10 kHz. a.What is the bandwidth occupied by the modulated signal?Plot the spectrum of the FM signal (only side bands in the Carson's bandwidth).F b. If the amplitude of the modulating sinusoidal signal is increased by a factor of 3 and its frequency is decreased to1 kHz, how is the bandwidth of the modulated signal modified? Plot the spectrum of the FM signal. c. Repeat part (b) with the frequency of the modulating sinu-soidal signal increased to 3 kHz.