Questions and Answers of Systems Analysis And Design Using MATLAB

3.4. Consider a radar system using linear frequency modulation. Compute the range that corresponds to . Assume a beat frequency.
3.5. A certain radar using linear frequency modulation has a modulation frequency, and frequency sweep . Calculate the average beat frequency differences that correspond to range increments of and
3.6. A CW radar uses linear frequency modulation to determine both range and range rate. The radar wavelength is , and the frequency sweep is. Let . (a) Calculate the mean Doppler shift; (b)compute
3.7. In Chapter 1 we developed an expression for the Doppler shift associ- ated with a CW radar (i.e.. fd = 2v/h, where the plus sign is used for clos- ing targets and the negative sign is used for
3.8. Consider a medium PRF radar on board an aircraft moving at a speed of 350 m/s with PRFs fri = 10KHz, fr2 = 15KHz, and fr3 = 20KHz; the radar operating frequency is 9.5GHz. Calculate the
3.9. Repeat Problem 3.8 when the target is 15 off the radar line of sight.
3.10. A certain radar operates at two PRFs, fri and fr2, where T = (1/f,1) = T/5 and T2 = (1/f,2) = T/6. Show that this multiple PRF scheme will give the same range ambiguity as that of a single PRF
3.11. Consider an X-band radar with wavelength = 3cm and bandwidth B = 10MHz. The radar uses two PRFs, f1 = 50KHz and fr = 55.55KHz. A target is detected at range bin 46 forf, and at bin 12 for fr2.
3.12. A certain radar uses two PRFS to resolve range ambiguities. The desired unambiguous range is R = 150Km. Select a reasonable value for N. Compute the corresponding fri fra Rai, and R.2.
3.13. A certain radar uses three PRFS to resolve range ambiguities. The desired unambiguous range is R = 250 Km. Select N = 43. Compute the corresponding fri fra fr3 R1 R2 and R3
4.1. In the case of noise alone, the quadrature components of a radar return are independent Gaussian random variables with zero mean and variance .Assume that the radar processing consists of
4.2. (a) Derive Eq. (4.13); (b) derive Eq. (4.15).
4.3. A pulsed radar has the following specifications: time of false alarm, probability of detection , operating bandwidth. (a) What is the probability of false alarm ? (b) What is the single pulse
4.4. An L-band radar has the following specifications: operating frequency, operating bandwidth , noise figure , system losses , time of false alarm , detection range , probability of detection ,
4.5. (a) Show how you can use the radar equation to determine the PRF , the pulse width , the peak power , the probability of false alarm , and the minimum detectable signal level . Assume the
4.7. A radar system uses a threshold detection criterion. The probability of false alarm . (a) What must be the average SNR at the input of a linear detector so that the probability of miss is ?
4.8. An X-band radar has the following specifications: received peak power, probability of detection , time of false alarm, pulse width , operating bandwidth , operating frequency , and detection
4.9. A certain radar utilizes 10 pulses for non-coherent integration. The single pulse SNR is and the probability of miss is . (a) Compute the probability of false alarm . (b) Find the threshold
4.10. Consider a scanning low PRF radar. The antenna half-power beam width is , and the antenna scan rate is per second. The pulse width is, and the PRF is . (a) Compute the radar operating
4.11. Using the equation calculate when and . Perform the integration numerically.
4.12. Repeat Example 4.3 with and .
4.13. Derive Eq. (4.107).
4.14. Write a MATLAB program to compute the CA-CFAR threshold value. Use similar approach to that used in the case of a fixed threshold.
4.15. A certain radar has the following specifications: single pulse SNR corresponding to a reference range is . The probability of detection at this range is . Assume a Swerling I type target. Use
4.16. Repeat Problem 4.15 for swerling IV type target.
4.18. Reproduce Fig. 4.10 for Swerling II, III, and IV type targets.
4.19. Develop a MATLAB program to calculate the cumulative probability of detection.
5.1. Derive Eq. (5.17).
5.2. Derive Eq. (5.66).
5.3. Derive Eq. (5.54).
5.4. Write a MATLAB program to perform HRR synthesis for frequency coded waveforms.
5.5. Reproduce Fig. 5.5 for . Compare your outputs. What are your conclusions?
6.1. Define and . (a) Compute the discrete correlations: , , , and . (b) A certain radar transmits the signal . Assume that the autocorrelation is equal to . Compute and sketch and .
6.2. Compute the frequency response for the filter matched to the signal(a) ; (b) , where is a positive constant.
6.3. Repeat Example 6.1 for .
6.4. Derive Eq. (6.43).
6.5. Prove the properties of the radar ambiguity function.
6.6. Starting with Eq. (6.61) derive Eq. (6.62).
6.7. A radar system uses LFM waveforms. The received signal is of the form , where is a time delay that depends on range,, and . Assume that the radar bandwidth is , and the pulse width is . (a) Give
6.8. (a) Write an expression for the ambiguity function of an LFM waveform, where , and the compression ratio is . (b) Give an expression for the matched filter impulse response.
6.9. Repeat Example 6.2 for , and .
6.10. (a) Write an expression for the ambiguity function of a LFM signal with bandwidth , pulse width , and wavelength. (b) Plot the zero Doppler cut of the ambiguity function. (c) Assume a target
6.11. (a) Give an expression for the ambiguity function for a pulse train consisting of 4 pulses, where the pulse width is and the pulse repetition interval is . Assume a wavelength of . (b) Sketch
6.12. Hyperbolic frequency modulation (HFM) is better than LFM for high radial velocities. The HFM phase is where is an HFM coefficient and is a constant. (a) Give an expression for the instantaneous
6.13. Consider a Sonar system with range resolution . (a) A sinusoidal pulse at frequency is transmitted. What is the pulse width, and what is the bandwidth? (b) By using an up-chirp LFM, centered at
6.14. A pulse train is given by where is a single pulse of duration and the weighting sequence is . Find and sketch the correlations ,, and .
6.15. Repeat the previous problem for .
6.16. Modify the function “train_ambg.m” to accommodate the case.
6.17. Using the MATLAB functions presented in this chapter, generate the exact plots that correspond to Figs. 6.13 and 6.14.
6.18. Using the function “lfm_ambg.m” reproduce Fig. 6.6b for a downchirp LFM pulse.
7.1. Starting with Eq. (7.17), prove Eq. (7.21).
7.2. The smallest positive primitive root of is ; for generate the corresponding Costas matrix.
7.3. Develop a MATLAB program to plot the ambiguity function associated with Costas codes. Use Eqs. (7.53) through (7.56). Your program should generate 3-D plots, contour plots, and zero
7.4. Consider the 7-bit Barker code, designated by the sequence . (a)Compute and plot the autocorrelation of this code. (b) A radar uses binary phase coded pulses of the form , where, , and. Assume .
7.5. (a) Perform the discrete convolution between the sequence defined in Eq. (7.59), and the transversal filter impulse response (i.e., derive Eq. (7.60). (b) Solve Eq. (7.60), and sketch the
7.6. Repeat the previous problem for and . Use Barker code of length 13.
7.7. Develop a Barker code of length 35. Consider both and .
7.8. Write a computer program to calculate the discrete correlation between any two finite length sequences. Verify your code by comparing your results to the output of the MATLAB function
7.9. Compute the discrete autocorrelation for an Frank code.
7.10. Generate a Frank code of length 8, .q = 11 γ = 2 N = 10 x(n)s(t) = r(t) cos (2πf0t)r(t) = x(0), for 0 < t < Δt r(t) = x(n), for nΔt < t < (n + 1)Δt r(t) = 0, for t > 7Δt Δt = 0.5μs
8.1. Using Eq. (8.4), determine when and .
8.2. An exponential expression for the index of refraction is given by where the altitude is in Km. Calculate the index of refraction for a well mixed atmosphere at 10% and 50% of the troposphere.
8.3. Rederive Eq. (8.34) assuming vertical polarization.
8.4. Reproduce Figs. 8.6 and 8.7 by using and (a)and (dry soil); (b) and (sea water at ); (c)and (lake water at ).
8.5. In reference to Fig. 8.9, assume a radar height of and a target height of . The range is . (a) Calculate the lengths of the direct and indirect paths. (b) Calculate how long it will take a pulse
8.6. In the previous problem, assuming that you may be able to use the small grazing angle approximation: (a) Calculate the ratio of the direct to the indirect signal strengths at the target. (b) If
8.7. Utilizing the plots generated in solving Problem 8.4, derive an emperical expression for the Brewster’s angle.
8.8. A radar at altitude and a target at altitude , and assuming a spherical earth, calculate , , and .
8.9. Derive an asymptotic form for and when the grazing angle is very small.
8.10. In reference to Fig. 8.8, assume a radar height of and a target height of . The range is . (a) Calculate the lengths of the direct and indirect paths. (b) Calculate how long it will take a
8.11. Using the law of cosines, derive Eqs. (8.51) through (8.53).
8.12. In the previous problem, assuming that you may be able to use the small grazing angle approximation: (a) Calculate the ratio of the direct to the indirect signal strengths at the target. (b) If
8.13. In the previous problem, assuming that you may be able to use the small grazing angle approximation: (a) Calculate the ratio of the direct to the indirect signal strengths at the target. (b) If
8.14. Calculate the range to the horizon corresponding to a radar at and of altitude. Assume 4/3 earth.
8.15. Develop a mathematical expression that can be used to reproduce Figs. 8.14 and 8.15.
9.1. Compute the signal-to-clutter ratio (SCR) for the radar described in Example 9.1. In this case, assume antenna 3dB beam width , pulse width , range , grazing angle , target RCS , and clutter
9.2. Repeat Example 9.2 for target RCS , pulse width, antenna beam width ; the detection range is , and .θ3dB = 0.03radτ = 10μs R = 50Km ψg = 15°σt 0.1m2 = σ0 0.02 m2 m2 = ( ⁄ )σt 0.15m2
9.3. The quadrature components of the clutter power spectrum are, respectively, given by Compute the D.C. and A.C. power of the clutter. Let .
9.4. A certain radar has the following specifications: pulse width, antenna beam width , and wavelength . The radar antenna is high. A certain target is simulated by two point targets(scatterers).
9.5. A certain radar has range resolution of and is observing a target somewhere in a line of high towers each having RCS . If the target has RCS , (a) How much signal-to-clutter ratio should the
9.7. One implementation of a single delay line canceler with feedback is shown below:(a) What is the transfer function, ? (b) If the clutter power spectrum is, find an exact expression for the filter
9.8. Plot the frequency response for the filter described in the previous problem for .
9.9. An implementation of a double delay line canceler with feedback is shown below:(a) What is the transfer function, ? (b) Plot the frequency response for, and .
9.10. Consider a single delay line canceler. Calculate the clutter attenuation and the improvement factor. Assume that and a PRF.
9.11. Develop an expression for the improvement factor of a double delay line canceler.
9.12. Repeat Problem 9.10 for a double delay line canceler.
9.13. An experimental expression for the clutter power spectrum density is, where is a constant. Show that using this expression leads to the same result obtained for the improvement factor as
9.14. Repeat Problem 9.13 for a double delay line canceler.
9.15. A certain radar uses two PRFs with stagger ratio 63/64. If the first PRF is , compute the blind speeds for both PRFs and for the resultant composite PRF. Assume .
9.16. A certain filter used for clutter rejection has an impulse response. (a) Show an implementation of this filter using delay lines and adders. (b) What is the transfer function?(c) Plot the
9.17. The quadrature components of the clutter power spectrum are given in Problem 9.3. Let and . Compute the improvement of the signal-to-clutter ratio when a double delay line canceler is
9.18. Develop an expression for the clutter improvement factor for single and double line cancelers using the clutter autocorrelation function. Assume that the clutter power spectrum is as defined in
10.1. Consider an antenna whose diameter is . What is the far field requirement for an X-band or an L-band radar that is using this antenna?
10.2. Consider an antenna with electric field intensity in the xy-plane. This electric field is generated by a current distribution in the yzplane.The electric field intensity is computed using the
10.3. A linear phased array consists of 50 elements with element spacing. (a) Compute the 3dB beam width when the main beam steering angle is and . (b) Compute the electronic phase difference for any
10.4. A linear phased array antenna consists of eight elements spaced with element spacing. (a) Give an expression for the antenna gain pattern(assume no steering and uniform aperture weighting). (b)
10.5. In Section 10.6 we showed how a DFT can be used to compute the radiation pattern of a linear phased array. Consider a linear of 64 elements at half wavelength spacing, where an FFT of size 512
11.1. Show that in order to be able to quickly achieve changing the beam position the error signal needs to be a linear function of the deviation angle.
11.2. Prepare a short report on the vulnerability of conical scan to amplitude modulation jamming. In particular consider the self-protecting technique called “Gain Inversion.”
11.3. Consider a conical scan radar. The pulse repetition interval is .Calculate the scan rate so that at least ten pulses are emitted within one scan.

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