Problems

3.3.In some system power SIR at the matched filter output degrades 101 times as compared to power SNR, while the system can preserve its operation capability with only two-times degrading SIR. What should be changed in the signal, SNR remaining constant, if

(a)Only plain signals are allowed?

(b)Signal peak power cannot be raised (what should the signal processing gain be in this case)?

3.4.In some system a band-elimination filter neutralizes the narrowband jammer. Due to this, the matched filter SNR degrades by 3 dB.

(a)What should be done to the plain signal duration and amplitude if only 2% degradation in power SNR is tolerable (SNR in the absence of band-elimination is fixed)?

(b)Is it possible to push degradation below 2% without increasing signal power? If so, what should the signal processing gain be?

3.5.A system can operate with SNR no smaller than 10 dB. Due to a barrage jammer SNR drops to 3 dB. How could the signal parameters be changed to neutralize the jammer, if:

(a)Only plain signal of the fixed energy is allowed?

(b)Only plain signal of the same peak power can be used?

(c)Peak power and energy of the signal are fixed with no other constraints?

(d)Peak power of the signal is fixed and its bandwidth can be increased only 10 times?

Find the processing gain in cases (c) and (d).

3.6.In conflict with a barrage jammer transmitter, a system increases signal duration by 4 times with simultaneous halving of signal power and widening of the bandwidth by 50 times. The jammer transmitter is capable of increasing its power no more than 13 dB. Who will be the winner in this game?

3.7.A signal occupies two separate sub-bands of identical width W0. Total signal energy is distributed between them in the proportion 9:16. In the receiver two matched filters process both sub-band parts and their outputs are combined optimally to maximize a resultant SNR. There is a barrage jammer transmitter. What is the most harmful distribution of its total power between the signal sub-bands?

3.8.Matched filter SNR for an intended receiver equals 14 dB. The processing gain of the signal WT ¼ 400. Find:

(a)Ratio of power spectrum densities of the signal and background AWGN.

(b)SNR at the integrator output of an interceptor radiometer.

3.9.In the BPSK data transmission system error probability per bit Pe ¼ 1:5 10 3 is required. A system designer wants the SNR of the interceptor radiometer to be no greater than 10 dB per one transmitted bit duration. What processing gain per one bit would be satisfactory?

3.10. The interceptor radiometer SNR is 10 dB per signal duration, whereas the intended receiver needs SNR 12 dB. Signal duration T ¼ 100 ms. What should its minimal bandwidth be?

3.11.Two spread spectrum BPSK data transmission systems are compared. The first employs binary modulation to widen the bandwidth, providing processing gain of 100 per data bit. The second operates with a ternary modulation resulting in processing gain of 50 per data bit. Which of them has better immunity to cracking a modulation law, the data rate being the same?

3.12.Two spread spectrum BPSK data transmission systems operating at the same data rate are compared. In the first an intended receiver SNR per data bit is 12 dB, while the interceptor radiometer SNR per bit is 12 dB. For the second system these parameters are 16 and 4 dB, respectively. Which of them is more immune to breaking a modulation law?

3.13.A system designer takes care of the EMC of a developed system. The maximal SNR provided by the new system over the entire zone covered by the old ones is 20 dB. Any old system operates with a satisfactory quality if the extra power spectrum density does not exceed 10 dB compared to the background AWGN spectrum. What should the minimum processing gain of the new system be?

3.14.There are two spread spectrum systems occupying the same total bandwidth and operating inside the same geographical area. The maximal SNR over all the intersection of their operational zones are 20 and 17 dB, respectively. For their compatibility, the extra power spectrum density due to the emission of the other system should be at least 7 dB below the natural AWGN level. Find the minimal processing gain of each of the systems.

3.15.For normal functioning, the system operating at the wavelength w ¼ 30 cm needs

signal-to-AWGN ratio q ¼ 14 dB. Signal duration T ¼ 100 ms and the noise temperature of the receiver n ¼ 1000 K. The transmitter antenna has gain of 5 dB, while the receiving antenna is omnidirectional. Find the necessary transmitted power for a free-space propagation model, if the system coverage zone should have a radius no smaller than 30 km. How will this power increase for the conditions typical of mobile telephone with the distance-attenuation exponent 3.84, no other adjustment of the free-space model being necessary?

3.16.A system survives if the received voltage SNR drops no more than 4 times below the average predicted level. Find the probability of system failure due to long-term fading with the standard deviation of the decibel power content equal to 9 dB.

3.17.There are two propagation paths: LOS and one through a reflector located 3 km away from the LOS and equidistant from both receiver and transmitter. Find the standing wave period in metres and the time interval between successive power drops at a distance 12 km from the transmitter for a receiver moving at a constant speed of 60 km/h if the wavelength is 30 cm.

3.18.Is it practical to use BPSK in the channel with fast multipath Rayleigh fading? What binary transmission mode is advisable in this case?

3.19.Binary data are transmitted over a channel with lognormal long-term and Rayleigh short-term fading. Due to the long-term fading, the signal power fluctuates around

27 dB with standard deviation 12 dB. Is bit error probability of 10 3 achievable in this channel without error-control coding?

3.20.There are two Rayleigh diversity branches with identical average signal energies. Using average power SNR at the combiner output as a criterion, compare the energy gains of two combining techniques: the maximal ratio one and selection of a maximum SNR branch.

3.21.A signal of a system occupies bandwidth of 60 kHz. The delay spread of the channel is 20 ms. The total bandwidth is no greater than 300 kHz. How many frequency diversity branches may be arranged?

3.22. There are four propagation paths with lengths 5 km, 5.4 km, 5.55 km and 6 km. A system transmits data at the rate R ¼ 20 kbps. Estimate the approximate signal bandwidth and minimal processing gain necessary for arranging a 4-finger RAKE receiver.

3.23.The minimum difference of lengths of paths in a channel is 300 m. The delay spread of the channel is within 10 ms. The system transmits data using QPSK at the rate 20 kbps. What is the maximal available number of RAKE fingers? Find the necessary bandwidth and processing gain of the signal for arranging the RAKE receiver.

3.24.A RAKE receiver splits the resultant signal at the output of a Rayleigh channel into nd non-fading signals of equal power using maximal ratio combining. How does the average power SNR at the combiner output differ from the case when the RAKE algorithm is not used? In the light of the answer how can the energy gain of the RAKE technique be explained?

Matlab-based problems

3.25.Write a program illustrating the advantages of spread spectrum in countering a narrowband jammer when the receiver does not use band-elimination filtering (Figure 3.4).

(a)Form and plot two rectangular bandpass signals of the same duration T (vectors of dimension 1000): a plain pulse and an LFM pulse with deviation (40–50)/T (see Problems 2.55–2.56). Take carrier frequency to have 25–30 periods per signal duration.

(b)Form the matrix of a CW jammer with 10 rows, each row having frequency equal to the signal carrier frequency and random initial phase uniformly distributed over [ , ].

(c)Form observation matrices for both signals by adding the signals to the jammer. Set up the jammer level several (2–10) times higher than the signal level. Plot the observations.

(d)Process the observations with corresponding matched filters. Plot the filter output waveforms.

(e)Run the program, varying the jammer intensity and frequency, and explain the results.

(a)Form and plot two rectangular bandpass bit pulses of the same duration T, amplitude and carrier frequency about (30–40)/T, the first being plain and the second being LFM with deviation Wd ¼(20–30)/T.

(b)Set up a bit stream (5 to 6 bits) and arrange BPSK for both bit pulses. Plot the transmitted signals.

(c)Add two delayed copies to the transmitted signal in both cases. Take amplitudes of the copies equal to that of the transmitted signal and delays around 0.15T and 0.3T. Plot the resultant received waveforms.

(d)Calculate and plot matched filter response for each case.

(e)Comment on the results.

3.36.Based on Problem 3.35, write a program demonstrating the effect of combining the path signals in a RAKE receiver.