Radar Systems Analysis and Design Using Matlab, C03

经典的《Radar Systems Analysis and Design Using Matlab》。 第三章。 讲解详实，图文并茂，深入浅出，有理有据。 配合MATLAB源代码。The timing mark can be implemented by modulating the transmit waveformand one commonly used technique is Linear Frequency Modulation (LFM)Before we discuss lFm signals, we will first introduce the Cw radar equationand briefly address the general Frequency Modulated(FM) waveforms usingsinusoidal modulating signals3. 2. Cw Radar equationAs indicated by Fig 3.1, the CW radar receiver declares detection at the out-put of a particular Doppler bin if that output value passes the detection threshold within the detector box. Since the nbf bank is inplemented by an FFtonly finite length data sets can be processed at a time. The length of suchblocks is normally referred to as the dwell time or dwell interval. The dwellinterval determines the frequency resolution or the bandwidth of the individualNBFS. More precisely△f=1/ TOwell(3.1)Dwell ls the dwell interval. Therefore. once the maximum resolvable frequency by the nb f bank is chosen the size of the nBf bank is computed asFr=2B/△f(3.2)B is the maximum resolvable frequency by the FfT. The factor 2 is needed toaccount for both positive and negative Doppler shifts. It follows thatDwell=NFFT/2B(3.3)The Cw radar equation can now be derived from the high prf radar equation given in Eq(1.69)and repeated here as eq (3. 4)P..GSNr=(34)(4兀) RKTFLIn the case of cw radars, P is replaced by the cw average transmittedpower over the dwell interval Pcw, and Ti must be replaced by TDwell. Thus,the Cw radar equation can be written asPcutGG7SNR(3.5)(4)RkTeFLLwiwhere G, and G, are the transmit and receive antenna gains, respectively. Thefactor Lwin is a loss term associated with the type of window(weighting) usedin computing the FFT. Other terms in Eq(.5)have been defined earlier3.3. Frequency ModulationThe discussion presented in this section will be restricted to sinusoidal modulating signals. In this case, the general formula for an FM waveform can beexpressed bys(t)=Acos 2Ifo!+k cos 2rfmudu(36)fo is the radar operating frequency (carrier frequency ) cos 2Tfmt is the mod-ulating signal, A is a constant, and k= 2TAfpeak, where Afpeak is the peakfrequency deviation. The phase is given byv(t)=2fot+ 2TAfpeak cos 2Tfmnudu= 2Tfo!+ Bsin2fmt (3.7)0where B is the FM modulation index given byLet s,(t) be the received radar signal from a target at range R. It followslatr(1)=A,COs(2兀/0(t-△1)+阝sin2/n2(1-△))(39)where the delay△tis2R△c is the speed of light. CW radar receivers utilize phase detectors in order toextract target range from the instantaneous frequency, as illustrated in Fig. 3.2A good measurement of the phase detector output o(t) implies a good measurement of At, and hence ranges (t)o(t)=KicosOmAtphasedetectorigure 3.2. Extracting range from an Fm signal returnKi is a constantConsider the FM waveform s(t) given by(1)πGt+阝sin2(3.11)which can be written as25(1)=ARe(312)where Rei .] denotes the real part. Since the signal exp(iBsin 2 If, t) isperiodic with period T=l/fm, it can be expressed using the complex exponential fourier series asβsin2xfn∑jni fnt313)n=-0where the Fourier series coefficients Cn are given b1f邝jn2Ifr(3.Make the change of variable u 2Tfmt, and recognize that the Bessel function of the first kind of order n isdu(3.15)TThus, the Fourier series coefficients are Cn=Jn(B), and consequently Ec(3. 13)can now be written as=∑(B)which is known as the Bessel-Jacobi equation Fig 3.3 shows a plot of Bessefunctions of the first kind for n =0, 1, 2, 3The total power in the signal s(t) isA∑MB)=4stituting Eg.(3.16) into Eq (3.12) yieldsFigure 3.3. Plot of Bessel functions of order o, 1, 2, and 3.()=ARj2πfjni/mt∑(3.18)Expanding Eg. (3. 18)yields∑小(B)eos(2760+n2x(3.19)Finally, since j(β)=J,(B) for n odd and j(β)=-(β) for n even wecan rewrite Eq (3.19)as(t)=A,(B)cos 2 For+(320)J1(阝)Lcos(2πf0+27fn)t-COS(2Tf0-27fmn)+J2(β[cos(2xfo+4丌fn)t+cos(2f47fn)J3(β)Icos(2/0+6兀/mn)-c0S(2兀/0-67/m)1J([cos(2兀0+8mn)+co(2xf-8fm))+…}The spectrum of s(t)is composed of pairs of spectral lines centered at fo,assketched in Fig. 3. 4. The spacing between adjacent spectral lines is fn. Thecentral spectral line has an amplitude equal to Ao(B), while the amplitude ofthe nth spectral line is Aj, (B)Figure 3. 4. Amplitude line spectra sketch for FM signal.As indicated by eg (3.20)the bandwidth of fm signals is infinite. Howeverthe magnitudes of spectral lines of the higher orders are small, and thus thebandwidth can be approximated using Carson's ruleB≈2(β+1)mn(321)whenβ is smal, only Jo(β)and/1(β) have significant values.Thus,wemay approximate Eg. (3. 20)b)S()≈A{J0(B)cos201+J1(β)(322)I coS(2fo+ rfm)t-Cos (2tfo-2Ifm)t]Finally, for small B, the Bessel functions can be approximated by(B)≈11(β)≈(324)Thus, Eq (3.22) may be approximated bys(t)=A cos 2 Ifot+B[cos(2fo+ 2I/m)t-cos(2Ifo-2Im)tIs (3.25)Example 3. 1 If the modulation index is B=0.5, give an expression for thesignSolution: From Bessel function tables we get Jo(0.5)=0.9385 andJ(0.5)=0.2423; then using Eq (3. 17)we get≈A{(0.9385)c0s2f0t+(0.2423)L cos (2Ifo+ 2rfm)t-Cos(2Ifo-2Ifm)t]]Example 3.2: Consider an FM transmitter with output signals(t)=100cos(2000Tt+(P(t)). The frequency deviation is 4Hz, and themodulating waveform is x(t)=10 cos 16Tt. Determine the Fm signal bandwidth. How many spectral lines will pass through a band pass filter whosebandwidth is 58Hz centered at 1000Hz?Solution: The peak frequency deviation is Afpeak =4X10 = 40Hz. It follows thateak40Using Eg. (3.16)we getB=2(β+1)m=2×(5+1)×8=961乙However, only seven spectral lines pass through the band pass filter as illustrated in the figure shown below▲ amplitude/100§38§frequency3.4. Linear FM(LFM) Cw radarCw radars may use lFm waveforms so that both range and doppler information can be measured. In practical cw radars the lfm waveform cannot becontinually changed in one direction, and thus periodicity in the modulation isnormally utilized. Fig 3.5 shows a sketch of a triangular LFM waveform. Themodulation does not need to be triangular; it may be sinusoidal, saw-tooth, orsome other form. The dashed line in Fig 3.5 represents the return waveformTron a stationary target at range R. The beat frequency fh is also sketched inFig. 3.5. It is defined as the difference(due to heterodyning) between the transmitted and received signals. The time delay At is a measure of target range, asdefined in Eg.( 3.10)rf0+△ffmf△ttim(frequencyitted signaldashed receivedtimFigure 3.5. Transmitted and received triangular lfm signals and beatfrequency for stationary target.In practice, the modulating frequency fm is selected such that(326)The rate of frequency change, f, is△f2fm, Af(327)where Af is the peak frequency deviation. The beat frequency fh is given by2RAb= AtfEg. (3.28) can be rewritten asC2R329)Equating Eqs. (3. 27)and(3.29)and solving for fbyfr4Rfm△f(3.30)Now consider the case when Doppler is present (i.e, non-stationary target)The corresponding triangular LFM transmitted and received waveforms aresketched in Fig. 3.6, along with the corresponding beat frequency. As beforethe beat freis definedfh=freceived-fiadmitted(331)When the target is not stationary the received signal will contain a Dopplershift term in addition to the frequency shift due to the time delay At. In thiscase, the Doppler shift term subtracts from the heat frequency during the positive portion of the slope. Alternatively, the two terms add up during the negative portion of the slope. Denote the beat frequency during the positive(up)and negative( down) portions of the slope, respectively, as fu and fbdIt follows that2R,2R(332入where R is the range rate or the target radial velocity as seen by the radar. Thefirst term of the right-hand side of Eq (3.32) is due to the range delay definedby Eg (3.28), while the second term is due to the target Doppler. SimilarlyR. 2R(333)frequency△LIlletrequencysolid: transmitted signaldashed; received signaltimeFigure 3.6. Transmited and received lFM signals and beat frequency, for amoving target.