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systems analysis and design

Radar Systems Analysis And Design Using MATLAB 2nd Edition Bassem R. Mahafza - Solutions

Repeat the previous problem for x(t) = exp(-1/2)cos2fol.
A pulse train y(t) is given by 2 y(t) = w(n)x(t-nt') n = 0 where x(t) = exp(-1/2) is a single pulse of duration t' and the weighting sequence is {w(n)} = {0.5, 1, 0.7}. Find and sketch the correlations R., Rw, and Ry-
Consider a Sonar system with range resolution AR = 4cm. (a) A sinu- soidal pulse at frequency fo = 100KHz is transmitted. What is the pulse width, and what is the bandwidth? (b) By using an up-chirp LFM, centered at fo, one can increase the pulse width for the same range resolution. If you want to
Hyperbolic frequency modulation (HFM) is better than LFM for high radial velocities. The HFM phase is Wh(t) 2 + th Han 000 where , is an HFM coefficient and a is a constant. (a) Give an expression for the instantaneous frequency of a HFM pulse of duration t'h. (b) Show that HFM can be approximated
(a) Give an expression for the ambiguity function for a pulse train consisting of 4 pulses, where the pulse width is t = 1s and the pulse repeti- tion interval is 7 = 10s. Assume a wavelength of 1cm. (b) Sketch the =ambiguity function contour.
(a) Write an expression for the ambiguity function of a LFM signal with bandwidth B = 10MHz, pulse width t' = 1s, and wavelength =1cm. (b) Plot the zero Doppler cut of the ambiguity function. (c) Assume a target moving towards the radar with radial velocity v, = 100m/s. What is the Doppler shift
Repeat the example on page 245 using B = 2, 5, and 10GHz.
(a) Write an expression for the ambiguity function of an LFM wave- form, where t' = 6.4s, and the compression ratio is 32. (b) Give an expres- sion for the matched filter impulse response.
A radar system uses LFM waveforms. The received signal is of the form s,(t) As(t-t)+n(t), where is a time delay that depends on range, s(t) = Rect(t/t') cos(2fot-w(t)), and w(t) = -B/'. Assume that the radar bandwidth is B = 5MHz, and the pulse width is t' = 5s. (a) Give the quadrature components
Starting with Eq. (6.89) derive Eq. (6.90).=
Prove the properties of the radar ambiguity function.
Repeat the example on pp. 226 using x(t) = u(t) exp(-at).
Compute the frequency response for the filter matched to the signal (a) x(t) = exp ; (b) x(t) = u(t)exp(-at), where a is a positive constant.
Define {x,(n) = 1,-1,1} and {xo(n) = 1, 1, -1}. (a) Compute the dis- (b) A certain radar transmits the crete correlations: Rx, Rx Rxpo and R , xxoxi- signal s(t) = x(t) cos 2fot-x(t) sin 2fot. Assume that the autocorrelation s(t) is equal to y(t) = y(t)cos2fot-yo(t) sin2fot. Compute and sketch
A certain circularly scanning radar with a fan beam has a rotation rate of 3 seconds per revolution. The azimuth beamwidth is 3 degrees and the radar uses a PRI of 600 microseconds. The radar pulse width is 2 microseconds and the radar searches a range window that extends from 15 Km to 100 Km. It
Generate curves for Swerling I, II, III, and IV type targets. (c) Repeat part (b) above when non coherent integration is used.
A certain radar has the following parameters: Peak power P = 500KW; total losses L = 12dB; operating frequency fo = 5.6GHZ; PRF f = 2KHz; pulse width = 0.5s; antenna beamwidth 0az = 2 and el = 7; noise figure F = 6dB; scan time Tse = 2s. The radar can experi- ence one false alarm per scan. (a) What
Reproduce Fig. 4.10 for Swerling II, III, and IV type targets.
Repeat Problem 4.15 for swerling IV type target.
A certain radar has the following specifications: single pulse SNR corresponding to a reference range R = 200Km is 10dB. The probability of detection at this range is PD = 0.95.Assume a Swerling I type target. Use the radar equation to compute the required pulse widths at ranges R = 220Km, 250Km,
Derive Eq. (4.107).
Repeat example in Section 4.13 with PD = 0.8 and P = 105.
Using the equation = -SNR PD 1-e = 10(-4SNR Inu)du Pfa fa calculate PD when SNR = 10dB and P =0.01. Perform the integration numerically. fa
Consider a scanning low PRF radar. The antenna half-power beam width is 1.5, and the antenna scan rate is 35 per second. The pulse width is t = 2s, and the PRF is fr = 400 Hz. (a) Compute the radar operating band- width. (b) Calculate the number of returned pulses from each target illumina- tion.
(a) Com- pute the probability of false alarm Pfa. (b) Find the threshold voltage VT.
A certain radar utilizes 10 pulses for noncoherent integration. The sin- gle pulse SNR is 15dB and the probability of miss is P =
An X-band radar has the following specifications: received peak power 100W, probability of detection PD = 0.95, time of false alarm Tfa = 8 min, pulse width = 2s, operating bandwidth B = 2MHz, operating frequency fo = 10GHz, and detection range R = 100Km. Assume single pulse processing. (a) Compute
A radar system uses a threshold detection criterion. The probability of false alarm Pfa = 1010. (a) What must be the average SNR at the input of a linear detector so that the probability of miss is Pm = 0.15? Assume large SNR approximation (see Problem 4.6). (b) Write an expression for the pdf at
(a) Show how you can use the radar equation to determine the PRF fr, the pulse width, the peak power P,, the probability of false alarm Pfa, and the minimum detectable signal level Smin Assume the following specifica- tions: operating frequency fo = 1.5MHz, operating bandwidth B = 1MHz, noise
An L-band radar has the following specifications: operating frequency fo = 1.5GHz, operating bandwidth B = 2MHz, noise figure F = 8dB, system losses L = 4dB, time of false alarm Ta = 12 minutes, detection. 12Km, probability of detection PD = 0.5, antenna gain range R =G = 5000, and target RCS = 1m.
A pulsed radar has the following specifications: time of false alarm 10 min, probability of detection PD = 0.95, operating bandwidth B = 1MHz. (a) What is the probability of false alarm Pfa? (b) What is the single pulse SNR? (c) Assuming noncoherent integration of 100 pulses, what is the SNR
(a) Derive Eq. (4.13); (b) derive Eq. (4.15). Tf fa =
In the case of noise alone, the quadrature components of a radar return are independent Gaussian random variables with zero mean and variance w. Assume that the radar processing consists of envelope detection followed by threshold decision. (a) Write an expression for the pdf of the envelope; (b)
A certain radar uses three PRFS to resolve range ambiguities. The desired unambiguous range is R = 250Km. Select N = 43. Compute the corresponding fri fr2 fr3, Ru1, R2, and R3.
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 fr2, Ru1, and R2-
Consider an X-band radar with wavelength = 3cm and bandwidth B = 10MHz. The radar uses two PRFs, fr = 50KHz and fr2 = 55.55KHz. A target is detected at range bin 46 for fr and at bin 12 for fr2. Determine the actual target range.
A certain radar operates at two PRFS, fri and fr2, where rl Tr = (1/1) T/5 and T2 = (1/fr2) = T/6. Show that this multiple PRF scheme will give the same range ambiguity as that of a single PRF with PRI T.
Repeat Problem 3.8 when the target is 15 off the radar line of sight.
Consider a medium PRF radar on board an aircraft moving at a speed of 350 m/s with PRFS fr = 10KHz, fr2 = 15KHz, and fr3 = 20KHz; the radar operating frequency is 9.5GHz. Calculate the frequency position of a nose-on target with a speed of 300 m/s. Also calculate the closing rate of a target
In Chapter 1 we developed an expression for the Doppler shift associ- ated with a CW radar (i.e., fd = 2v/2, where the plus sign is used for clos- ing targets and the negative sign is used for receding targets). CW radars can use the system shown in Fig. P3.7 to determine whether the target is
A CW radar uses linear frequency modulation to determine both range and range rate. The radar wavelength is = 3cm, and the frequency sweep is Af=200KHz. Let to = 20ms. (a) Calculate the mean Doppler shift; (b) compute fbu and fbd corresponding to a target at range R = 350Km, which is approaching
A certain radar using linear frequency modulation has a modulation fre- quency fm 300Hz, and frequency sweep Af = 50MHz. Calculate the average beat frequency differences that correspond to range increments of 10 and 15 meters.
Consider a radar system using linear frequency modulation. Compute the range that corresponds to f = 20, 10MHz. Assume a beat frequency fb = 1200 Hz.
In Chapter 1 we developed an expression for the Doppler shift associ- ated with a CW radar (i.e., fd = 2v/2, where the plus sign is used for clos- ing targets and the negative sign is used for receding targets). CW radars can use the system shown in Fig. P3.7 to determine whether the target is
A CW radar uses linear frequency modulation to determine both range and range rate. The radar wavelength is = 3cm, and the frequency sweep is Af=200KHz. Let to = 20ms. (a) Calculate the mean Doppler shift; (b) compute fbu and fbd corresponding to a target at range R = 350Km, which is approaching
A certain radar using linear frequency modulation has a modulation fre- quency fm 300Hz, and frequency sweep Af = 50MHz. Calculate the average beat frequency differences that correspond to range increments of 10 and 15 meters.
Consider a radar system using linear frequency modulation. Compute the range that corresponds to f = 20, 10MHz. Assume a beat frequency fb = 1200 Hz.
In a multiple frequency CW radar, the transmitted waveform consists of two continuous sinewaves of frequencies f = 105KHz and f = 115KHz. Compute the maximum unambiguous detection range.
Show that J(z) = (-1)"J(z). Hint: You may utilize the relation J(=) = cos 0 cos (zsiny - ny)dy
Prove that 8 Jn (z) = 1. 11-00
Assume that a certain sequence is determined by its FFT. If the record length is 2ms and the sampling frequency is fs = 10KHz, find N.
A certain band-limited signal has bandwidth B = 20KHz. Find the FFT size required so that the frequency resolution is Af = 50Hz. Assume radix 2 FFT and a record length of 1 second.
Assume that a certain sequence is determined by its FFT. If the record length is 2ms and the sampling frequency is fs = 10KHz, find N.(b) Assume that you want to compute the modulus of the Fourier transform using a DFT of size 512 with a sampling interval of 1 second. Evaluate the modulus at
A certain band-limited signal has bandwidth B = 20KHz. Find the FFT size required so that the frequency resolution is Af = 50Hz. Assume radix 2 FFT and a record length of 1 second.
(a) Write an expression for the Fourier transform of x(t) = Rect(t/3)(b) Assume that you want to compute the modulus of the Fourier transform using a DFT of size 512 with a sampling interval of 1 second. Evaluate the modulus at frequency (80/512) Hz. Compare your answer to the theoretical value and
Compute the Z-transform for (a) x(n) = u(n); (b) x(n) = mu-n (-n).
In Fig. 2.1, let 00 P(t) = ARect(-n) Give an expression for X,(o). 11=-00
Let X(t) be a stationary random process, E[X(t)] = 1 and the auto- correlation R(t) = 3+ exp(-|t|). Define a new random variable,Compute E[Y(t)] and .
Let X be a random variable with t0 fx(x) = 0 elsewhere (a) Determine the characteristic function Cx(w). (b) Using Cx(0), validate that fx(x) is a proper pdf. (c) Use Cx(o) to determine the first two moments of X. (d) Calculate the variance of X.
2.19. Let Sx(o) be the PSD function for the stationary random process for the PSD function of X(t). Compute an Y(t) = X(t)2X(t-T). expression
(b) Determine the radius of a circle about the bulls-eye that contains 80% of the darts thrown. (c) Consider a square with side s in the first quadrant of the board. Determine s so that the probability that a dart will fall within the square is
Assume the X and Y miss distances of darts thrown at a bulls-eye dart board are Gaussian with zero mean and variance . (a) Determine the probability that a dart will fall between 0.80 and
(a) A random voltage v(t) has an exponential distribution function fy(v) = aexp(-av) where (a>0);(0v 0.5}.
Consider the network shown in Fig. P2.16, where x(t) is a random voltage with zero mean and autocorrelation function R(t) = 1 + exp(-a|t|). Find the power spectrum S,(o). What is the transfer function? Find the power spectrum S, (oo).
An LTI system has impulse response, exp(-21) t0 h(t) 0 1
Write an expression for the autocorrelation function R,(t), where and {Y} S,(@). = y(t) = 5 YRect(-5) n = 1 2 {0.8, 1, 1, 1, 0.8}. Give an expression for the density function
If x(t) = x(t)-2x(t-5)+x(t - 10), determine the autocorrela- tion functions R,,(t) and R(t) when x(t) = exp(-1/2).
Determine the quadrature components for the signal -21 h(t) = 8(t)- esinoot u(t). 00
Write an expression for the autocorrelation function R,(t), where and {Y} S,(@). = y(t) = 5 YRect(-5) n = 1 2 {0.8, 1, 1, 1, 0.8}. Give an expression for the density function2.9. Consider the signal x(t) = Rect(t/t) cos (@ot-Bt/2t) and let t = 15s and B = 10MHz. What are the quadrature components?
If x(t) = x(t)-2x(t-5)+x(t - 10), determine the autocorrela- tion functions R,,(t) and R(t) when x(t) = exp(-1/2).
Determine the quadrature components for the signal -21 h(t) = 8(t)- esinoot u(t). 00
Consider the signal x(t) = Rect(t/t) cos (@ot-Bt/2t) and let t = 15s and B = 10MHz. What are the quadrature components?
A certain radar system uses linear frequency modulated waveforms of the form x(t) = Rect() cos(cool + What are the quadrature components? Give an expression for both the modula- tion and instantaneous frequencies.
What is the power spectral density for the signal x(t) = Acos (2fot+00)
Show that (a) Rx(t) = Rx*(t). (b) If x(t) = f(t) +m and y(t) = g(t)+m2, then Ry(t) =mm2, where the average values for f(t) and g(t) are zeroes.
A periodic signal x(t) is formed by repeating the pulse x(t) = 2A((t-3)/5) every 10 seconds. (a) What is the Fourier transform of x(t). (b) Compute the complex Fourier series of x,(t)? (c) Give an expression for the autocorrelation function R (t) and the power spectrum density S().X(1) = Xe/2mnt/T
(a) Prove that p(t) and p2(t), shown in Fig. P2.3, are orthogonal over the interval (-2t2). (b) Express the signal x(t) = t as a weighted sum of p(t) and (02(t) over the same time interval.
Compute the energy associated with the signal x(t) = ARect(t/t).
Classify each of the following signals as an energy signal, as a power signal, or as neither. (a) exp(0.5t) (t0); (b) exp(-0.5t) (t0); (c) cost+cos2t (-00 0).
A source with equivalent temperature To =290K is followed by three amplifiers with specifications shown in the table below. Amplifier 1 2 3 F, dB G, dB Te You must compute 12 350 10 22 15 35 Assume a bandwidth of 150KHz. (a) Compute the noise figure for the three cascaded amplifiers. (b) Compute
A certain radar has losses of 5 dB and a receiver noise figure of 10 dB. This radar has a detection coverage requirement that extends over 3/4 of a hemisphere and must complete it in 3 second. The base line target RCS is 6 dBsm and the minimum SNR is 15 dB. The radar detection range is less than 80
A certain radar has losses of 5 dB and a receiver noise figure of 10 dB. This radar has a detection coverage requirement that extends over 3/4 of a hemisphere and must complete it in 3 second. The base line target RCS is 6 dBsm and the minimum SNR is 15 dB. The radar detection range is less than 80
A radar has the following parameters: Peak power P = 50KW ; total losses L = 5dB; operating frequency fo = 5.6GHz; noise figure F = 10dB pulse width=10s; PRF f = 2KHz; antenna beamwidth 0 = 1 and Oel 5. (a) What is the antenna gain? (b) What is the effective aperture if the aperture efficiency is
A radar has the following parameters: Peak power P, losses L = 5dB; operating frequency fo = = 65KW ; total 8GHz; PRF fr = 4KHz; duty cycle d = 0.3; circular antenna with diameter D = 1m; effective aperture is 0.7 of physical aperture; noise figure F = 8dB. (a) Derive the various parameters needed
Compute the amount of Doppler filter straddle loss for the filter defined by H(f) 1 2 Assume half-power frequency fdB = 500Hz 1+af = 350 Hz. and crossover frequency fe =
Compute the amount of antenna pattern loss for a phased array antenna whose two-way pattern is approximated by f(y) = [exp(-2ln2(y/03dB))]where 03 dB is the 3dB beam width. Assume circular symmetry.
Consider an antenna with a sinx/x pattern. Let x = (rsin0)/2, where r is the antenna radius, & is the wavelength, and 0 is the off-boresight angle. Derive Eq. (1.116). Hint: Assume small x, and expand sinx/x as an infinite series.
Repeat the previous problem when there is 0.1dB/Km atmospheric attenuation.
An X-band airborne radar transmitter and an air-to-air missile receiver act as a bistatic radar system. The transmitter guides the missile toward its tar- get by continuously illuminating the target with a CW signal. The transmitter has the following specifications: peak power P = 4KW; antenna gain
Using Fig. 1.28 derive an expression for R.. Assume 100% synchro- nization between the transmitter and receiver.
A radar with antenna gain G is subject to a repeater jammer whose antenna gain is GJ. The repeater illuminates the radar with three fourths of the incident power on the jammer. (a) Find an expression for the ratio between the power received by the jammer and the power received by the radar; (b)
A certain radar is subject to interference from an SSJ jammer. Assume the following parameters: radar peak power P, = 55KW, radar antenna gain G = 30dB, radar pulse width = 2s, radar losses L = 10dB, jammer power P, 150W, jammer antenna gain G, = 12dB, jammer bandwidth B = 50MHz, and jammer losses
Assume PJ = 200 W, G = 15dB, and L = 2dB. Assume G' = 12dB and R, = 25Km.
derive an expression for R.. Assume 100% synchro- nization between the transmitter and receiver.1.25. Compute (as a function of BJ/B) the crossover range for the radar in Problem
Using Fig.
A radar with antenna gain G is subject to a repeater jammer whose antenna gain is GJ. The repeater illuminates the radar with three fourths of the incident power on the jammer. (a) Find an expression for the ratio between the power received by the jammer and the power received by the radar; (b)
A certain radar is subject to interference from an SSJ jammer. Assume the following parameters: radar peak power P, = 55KW, radar antenna gain G = 30dB, radar pulse width = 2s, radar losses L = 10dB, jammer power P, 150W, jammer antenna gain G, = 12dB, jammer bandwidth B = 50MHz, and jammer losses
Assume PJ = 200 W, G = 15dB, and L = 2dB. Assume G' = 12dB and R, = 25Km.
Compute (as a function of BJ/B) the crossover range for the radar in Problem
Assume P = 100 W, G, = 10dB, and L = 2dB.

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