Introduction to ISR Signal Processing Joshua Semeter, Boston University
Why study ISR?
Requires that you learn about a great many useful and fascinating subjects in substantial depth.
• Plasma physics
• Radar
• Coding (information theory)
• Electronics (Power, RF, DSP)
• Signal Processing
• Inverse theory Outline • Mathematical toolbox
• Review of basic radar concepts
• Ionospheric Doppler spectrum
• Range resolution and matched filtering
• I/Q demodulation
• Autocorrelation function (ACF) and Power Spectral Density (PSD)
Euler identity and the complex plane Imaginary
Time Real
is the “angular velocity” (radians/s) of the spinning arrow is the number of complete rotations ( radians) in one second (1/s or Hz) We need a signal that tells us how fast and in which direction the arrow is spinning. This signal is the complex exponential. Invoking the Euler identity,
I = in-phase component Q = in-quadrature component Essential mathematical operations Fourier Transform: Expresses a function as a weighted sum of harmonic functions (i.e., complex exponentials)
Convolution: Expresses the action of a linear, time-invariant system on a function.
Correlation: A measure of the degree to which two functions look alike at a given offset.
Autocorrelation, Convolution, Power Spectral Density, Wiener-Khinchin Theorem
Dirac Delta Function
&$ +", x = 0 !(t) = % '& 0, x # 0
The Fourier Transform of a train of delta functions is a train of delta functions. Harmonic Functions
Gate function
ISR spectrum Autocorrelation function (ACF)
Increasing Te
Not surprisingly, the ISR ACF looks like a sinc function… How it all hangs together. • Duality: - Gate function in the time domain represents amplitude modulation - Gate function in the frequency domain represents filtering
• Limiting cases: τ - Gate function approaches delta function as width goes to 0 with constant area - A constant function in time domain is a special case of harmonic function where frequency = 0. - A constant function in time domain is a special case of a gate function where width = infinity. frequency τ f0
f0 – ()1 ⁄ τ f0 + ()1 ⁄ τ
frequency
Figure 5.3. Amplitude spectrum for a single pulse, or a train of f0 non-coherent pulses. τ f0 – ()1 ⁄ τ f0 + ()1 ⁄ τ
Now consider the coherent gated CW waveform f3()t given by Figure 5.3. Amplitude spectrum for a single pulse, or a train of How many cycles are in a typical ISR pulse? non-coherent pulses. PFISR frequency: 449∞ MHz 215,520 cycles! Typical long-pulse length: 480 s (5.15) f3()t = f2()tnT– µ Now consider the coherent gated CW waveform f3()t given by ∑ frequency n = –∞ ∞ f Clearly f ()t is periodic, where T0 is the period (recall that f = 1 ⁄ T is the 3 r f3()t = ∑ f2()tnT– (5.15) PRF). Using the complex exponential Fourier series we can rewrite f3()t as f0 – ()1 ⁄ τ f0 + ()1 ⁄ τ n = –∞ Clearly is periodic, where is the period (recall that = 1 is the Bandwidth∞ off3()t a pulsedT signalfr ⁄ T j------2πnt PRF). Using the complex exponential Fourier series we can rewrite f ()t as T 3 Figure 5.3. Amplitudef3()t spectrum= F forne a single pulse, or a train of (5.16) ∑ ∞ non-coherent pulses. j------2πnt Spectrum of receivern = –∞ output has sinc shape, with sidelobes halfT the width of the f3()t = Fne (5.16) where the Fouriercentral series coefficients lobe and F continuously are given by diminishing in amplitude ∑above and below main lobe n n = –∞ Now consider the coherent gated CW waveform f3()t given by where the Fourier series coefficients F are given by Aτ nτπ n F = ------∞Sinc------(5.17) n T T Aτ nτπ Fn = ------Sinc ------(5.17) f3()t = f2()tnT– (5.15) T T It follows that the FT of f3()t is ∑ n = –∞ It follows that the FT of f3()t is ∞ A 1 microsecond pulse has a null- ∞ Clearly f3()t is periodic, where T is the period (recall that fr = 1Two⁄ T ispossible the bandwidth measures: F3()ω = 2π Fnδω()– 2nπfr (5.18) PRF). Using the complexto-null exponentialbandwidth∑ Fourier of the series central we can rewrite f ()tF ()asω = 2π F δω()– 2nπf (5.18) 3 3 ∑ n r lobe = 2 MHzn = –∞ 2 “null to null” bandwidthn = –∞ ∞ Bnn = j------2πnt T ! © 2000 by Chapman & Hall/CRC = © 2000 by Chapman & Hall/CRC(5.16) f3()t ∑ Fne n = –∞ “3dB” bandwidth where the Fourier series coefficients F are given by n Aτ nτπ F = ------Sinc------(5.17) n T T Unless otherwise specified, assume bandwidth refers to 3 dB bandwidth It follows that the FT of f3()t is
∞ 10 = 2 – 2 (5.18) F3()ω π ∑ Fnδω()nπfr n = –∞
© 2000 by Chapman & Hall/CRC Pulse-Bandwidth Connection
frequency fo
Shorter pulse Larger bandwidth
Faster sampling rate Larger bandwidth
Components of a Pulsed Doppler Radar
+ _ _ + _ +
Plasma density (Ne) Ion temperature (T ) Model Fit i Electron temperature (Te) Bulk velocity (Vi) 12 The deciBel (dB) The relative value of two quantities expressed on a logarithmic scale
P1 V1 SNR = 10 log10 = 20 log10 2 P2 V2 (Power Voltage )
Scientific Other forms used in radar: Factor of: Notation dB -1 0.1 10 -10 dBW dB relative to 1 Watt 0.5 100.3 -3 0 dBm dB relative to 1 mW 1 10 0 2 2 100.3 3 dBsm dB relative to 1 m of 10 101 10 radar cross section 100 102 20 dBi dB relative to isotropic 1000 103 30 . radiation 1,000,000 106 60
Pulsed Radar Pulse length 100 µsec
1 Mega-Watt Peak power Peak Power Target Return 10-14 Watt
Inter-pulse period (IPP) 1 msec Time
Pulse length Duty cycle = 10% Pulse repetition interval
Average power = Peak power * Duty cycle 100 kWatt
Pulse repetition frequency (PRF) = 1/(IPP) 1 kHz
Continuous wave (CW) radar: Duty cycle = 100% (always on) Doppler Frequency Shift
Transmitted signal:
After return from target:
To measure frequency, we need to observe signal for at least one cycle. So we will need a model of how R changes with time. Assume constant velocity:
Substituting:
constant ! f D
By convention, positive Doppler shift Target and radar are “closing”
Two key concepts
Two key concepts: e-
Distant Time e- e- R = c!t 2 R How many cycles are in a typical pulse? Velocity Frequency PFISR frequency: 449 MHz v = ! fD"0 2 Typical long-pulse length: 480 µs
A Doppler radar measures backscattered power as a function range and velocity. Velocity is manifested as a Doppler frequency shift in the received signal. Two key concepts
Two key concepts:
Distant Time R = c!t 2
Velocity Frequency
v = ! fD"0 2
A Doppler radar measures backscattered power as a function range and velocity. Velocity is manifested as a Doppler frequency shift in the received signal.
ENG SC700 Radar Remote Sensing J. Semeter, Boston University Concept of a “Doppler Spectrum” Some Other Doppler Spectra NEXRAD WSR-88D Two key concepts:
Distant Time
ENG SC700 RadarR = Remotec!t 2 Sensing J. Semeter, Boston University Velocity FrequencySome Other Doppler Spectra v = ! f " 2 D 0 Power (dB) NEXRAD WSR-88D
Tennis ball, birds, aircraft engine --examples of solid objects that give Doppler spectrum Velocity (m/s)S. Bachman, MS Thesis If there is a distribution of targets moving at different velocities (e.g., electrons in the ionosphere) then there is no single Doppler shift but, rather, a Doppler spectrum. What is the Doppler spectrum of the ionosphere at UHF ( ! of 10 to 30 cm)? Tennis ball, birds, aircraft engine Incoherent Scatter Radar --examples of solid objects that give Doppler spectrum S. Bachman, MS Thesis --random thermal motion of plasma results in Doppler spectrum ..more on that in the next 2 lectures.
Incoherent Scatter Radar --random thermal motion of plasma results in Doppler spectrum ..more on that in the next 2 lectures. DIAZ ET AL.: BEAM-PLASMA INSTABILITY EFFECTS ON IS SPECTRA X-5
55 can account for the simultaneous enhancement in the two ion lines, and the simultaneous
56 ion and plasma line enhancement.
57 This purpose of this paper is to provide a unified theoretical model of modes expected in
58 the ISR spectrum in the presence of field-aligned electron beams. The work is motivated
59 by the phenomenological studies summarized above, in addition to recent theoretical
60 results–in particular, those of Yoon et al. [2003], and references therein, which suggest
61 that Langmuir harmonics should arise as a natural consequence of the same conditions
62 producing NEIALs. Although these effects have been treated in considerable detail in the
63 plasma physics literature, their implications for the field of ionospheric radio science (and
64 ISR in particular) have not yet been discussed. The conditions to detect all the modes
65 present within the IS spectrum within the same ISR is also presented in this work.
2. Plasma in Thermal Equilibrium
66 There exist two natural electrostatic longitudinal modes in a plasma in thermal equilib-
67 rium: the ion acoustic mode, which is the main mode detected by ISRs, and the Langmuir 19 68 mode [Boyd and Sanderson, 2003]. Using a linear approach to solve the Vlasov-Poisson X-6 DIAZ ET AL.: BEAM-PLASMA INSTABILITY EFFECTS ON IS SPECTRA When the plasma is warm, which meansX-6 that the thermalDIAZ velocityET AL.: ofBEAM-PLASMA the particles INSTABILITY is EFFECTS ON IS SPECTRA 69 system of equations, the dispersion relation of these modes is obtained. The real part of 19 Longitudinal1 3 Modes in a Thermal Plasma important,X-6 it can be describedDIAZET as previously AL.: BEAM-PLASMA with a2 force-balance INSTABILITY2 motion EFFECTS equation ON IS SPECTRA but this π me Te Te 3 70 the ion acoustic dispersion relation readsωsi = + exp ωs 1 3 (2) − 8 m T − 2T − 2 2 2 timeWhen with the a plasma term that is warm, accounts which for means the thermal that thei velocity thermal ofi velocity the particles, of theiπ a particles pressureme is term Te Te 3 ωsi = + exp ωs (2) Ion-acoustic 1 3 2 2 − 8 mi Ti − 2Ti − 2 important,(γpTj n itj). can Thus,where be described theCs equation= askB previously(T becomese +3Ti)π/m withi ism a theeforce-balance ion-acousticTe motion speed. equation TheTe dependence but3 this of this mode ∇ ωsi = + exp ωs (2) X-6 DIAZ− ET AL.:8 where BEAM-PLASMAmi C = Ti INSTABILITYk (T +3−T EFFECTS2)T/mi − 2is ON the IS SPECTRAion-acoustic speed. The dependence of this mode time with a termon that ionospheric accounts state for the parametersωs thermal= Csk, velocityis observed s of the in Eq. particles,B 2.e The a Langmuir pressurei i term mode(1) is detected by ∂v (γpTj nj). Thus,where theC equations = mjknB becomesj(Te +3= qTjin)j/m(Ei+isv theB ion-acoustic) γpTj nj, speed. The dependence(2.2) of this mode ∇ ISR under certain∂ conditions,t on and ionospheric the× real− part1 state∇ of its parameters dispersion3 relation is observed is expressed in Eq. as 2. The Langmuir mode is detected by 2 2 2 2 27 71 and the imaginary part (assuming ωsi ωs, k λDeπ m1e and Ti/TTee 1) can beTe written3 on ionospheric state parametersωsi = is observed in+ Eq. 2. Theexp Langmuir modeω iss detected by (2) where γp is a proportionality constant and−ISRTj 8the under temperaturemi certainT of conditions,i the species− and2jT. i − the2 real part of its dispersion relation is expressed as ∂v 2 2 2 3 2 72 as m n = q nω(E=+ v ω B+3) kγ Tv n ,ω + v λ k , (2.2) (3) j j ∂t j j L ×pe − p jthe∇ j pe 2 the De Even thoughISR under thewhere main certainC modes= conditions,k present(T +3 inT and a) warm/m theis real plasma the40 part ion-acoustic≈ can of be its obtained dispersion speed. with Therelation Equation dependence is expressed of this as mode s LangmuirB e i i 3 where γ is a proportionality constant and T the temperature of the species j. 2 2 2 2 2.2, partp of the physics of those modesj is lost in the over simplificationωL = ofω thepe +3 motionk v ωpe + vtheλDek , (3) 73 andon the ionospheric imaginary part state (assuming parametersωLi is observedωL) is in Eq.3 2. The Langmuir modethe ≈ is detected2 by ω = ω2 +3k2 v2 ω + v λ k2, (3) equation.Even2.4 though When The the main the Particle-in-Cell temperature modes present ofL in a a plasmaMethod warmpe plasma is finite canthe and≈ be theobtainedpe thermal2 the withDe velocity Equation of the ISR under certain conditions,73 and the and imaginary the real part part of its (assuming dispersionω relationω ) is is expressed as 2.2,particles part of is the comparable physics of to those the modesphase velocity is lost in of the3 the over propagating simplification wave,2 of the the interaction motionLi of L π ωpe 1 ωpe 3 DRAFT73 and the imaginary May29,2010,6:36am part (assuming ω ω ) is DRAFT The simulator uses a particle-in-cellωLi (PIC)Li methodexpL for both theω ionsL. and electrons.(4) This equation. When the temperature of a plasma≈− is8 finitek3 v3 and the− thermal2k2v2 − velocity23 of the the particles and the wave becomes important. Some2the of the2 typical2 interactionsthe are Landau2 ωL = ωpe +3k vthe ωpe + vtheλDek , (3) particlesaccurately is comparable models to the all phase dynamics, velocity including of the propagating thermal wave, eff≈ects, the at interaction2 the cost3 of substantial com-2 damping and74 microinstabilities.The forward model Those used phenomena to estimate can ionospheric be explained parameters only through in ISRπ aω assumes motionpe 1 that theseωpe 3 ω3 ω2ωLi exp ωL. (4) the particlesputer and time. the wave The becomes idea of important. the PIC Some method,π pe of the1 described typical interactions inpe detail≈−3 are in Landau8 booksk3 v3 by Birdsall− 2k and2v2 − 2 equation that73 takesand into the account imaginary theωLi space-velocity part (assuming distributionωexp ω ) of is the particlesω formingL. the (4) the 75 are the dominating modes in the ISR3 spectrum.3 Li L However,2 2 an injected beam of particles, ≈− 8 k vthe − 2k vthe − 2 dampingLangdon and microinstabilities. (1985), Hockney Those phenomena and Eastwood can be (1988) explained or only Tajima through (1988), a motion is simple: The code plasma. This equation is the Boltzman equation,74 The which forward becomes model Vlasov used equation to estimate (Equation ionospheric parameters in ISR assumes that these 74 76 in particular electrons, can destabilize the plasma, altering the dispersion relations and equation that takesThe into forward account model the space-velocity used to estimate distribution ionospheric of the parameters particles forming in ISR the assumes that these simulates the motion of plasmas particles in3 continuous phase2 space, whereas moments of 2.3) in absence of collisions. π ω 1 ω 3 75 are the dominatingpe modespe in the ISR spectrum. However, an injected beam of particles, 77 amplitudes of these modes.ωLi exp ωL. (4) plasma. This75 equationare the dominatingis the Boltzman modes equation, in the≈− which ISR becomes8spectrum.k3 v3 Vlasov However,− equation2k2v an2 (Equation injected− 2 beam of particles, Landauthe distribution damping and such microinstabilities as densities andare important currentsthe arein determining computed the on shapediscrete of the points (or cells) 2.3) in absence of collisions. 76 in particular electrons, can destabilize the plasma, altering the dispersion relations and incoherentfrom76 scatterin the74 particular position radarThe forward spectrumelectrons, and velocity model at can high used of destabilize the latitudes, to particles. estimate the therefore plasma, ionospheric The macro-forcea altering kinetic parameters approach, the acting dispersion in which onISR theassumes relations particles that and is these Landau damping3. Current and microinstabilities Model of the are Langmuir important Decayin determining Process the for shape NEIAL of the Formation uses a Vlasov equation as motion equation,77 hasamplitudes to be used. of The these system modes. of equations formed calculated77 amplitudes75 are from the of the dominating these field modes. equations. modes in The the ISRname spectrum. “Particle-in-Cell” However,Figure an comes 2 injected4: fromLongitudinal beam the ofway particles, modes of of a plasma. Blue lines relate to ion incoherent scatter radar spectrum at high latitudes, therefore a kinetic approach, which· The model presented by Forme et al. [1993] to explainacoustic NEIALs is waves a two and step red process. ones to Langmuir waves. by Equationsassigning 2.3 macro-quantities to 2.9, which includes to the the simulation Vlasov equation particles. plus Maxwell’s equations, has uses a Vlasov equation76 in asparticular motion equation, electrons, has can to be destabilize used. The the system plasma, of equations altering formed the dispersion relations and First, a beam-plasma instabilitySimulated enhances Langmuir Waves ISR (LW). Doppler Second, if the enhance- Spectrum to be solved3. self-consistently Current Model to obtain of the the Langmuir3. wave Current modes Decay propagating Model Process of along the for the Langmuir NEIAL plasma. Formation Decay Process for NEIAL Formation by EquationsIn 2.3 general to 2.9, PICwhich codes includes solve the the Vlasov equation equation of motionplus Maxwell’s of particles equations, with has the Newton-Lorentz 77 amplitudes of these modes. plasma particles start to interact more strongly with the growing wave, e.g., by heating. ment of LW is high enough then the enhanced LW can decay, enhancing Ion Acoustic to be solvedforce self-consistentlyThe model presented to obtain the by waveForme modesThe et al. model propagating[1993] presented to explain along by the NEIALsForme plasma. is et a al. two[1993] step process. to explain NEIALs is a two step process. ∂fj(t, x, v) ∂fj(t, x, v) qj ∂Thisfj(t, x can, v) sometimes be described in terms of the so-called quasi-linear saturation within + v + (E + v B) = 0 (2.3) Waves∂t (IAWs) and· counter-propagating∂x m LWs.× · The plasma-beam∂v process involves three First,3. a beam-plasma Current Model instability of theFirst, enhancesj Langmuir a beam-plasma Langmuir Decay Waves instability Process (LW). for enhances Second, NEIAL if Langmuir the Formation enhance- Waves (LW). Second, if the enhance- ∂f (t, x, v) ∂f (t, x,Particle-in-cellv) q (PIC):∂f (t,thex, v Vlasov) theory. jspecies:+ thermalv j electrons,+ thermalj (E + v ions,B) and anj electron= 0 beam with (2.3) a bulk velocity of ∂t · ∂x m × · ∂v dmentxi ofThe LW model is high presented enoughd vi then byqjmenti Forme the∂ ofB enhanced LWet al. is[1993] high LW toA enough can wayexplain decay, of then categorizing NEIALs enhancing the enhanced is a Ion plasma two Acoustic step LW instabilities process. can decay, is to enhancing divide them Ion between Acoustic macroscopic (con- = vi and 20= E =(E−(xi)+vi B(xi)) for i =1(2.4),...,N (2.49) dtvb. By assuming smalldt perturbationsmi ∂t and small× damping/growth (vthe, vthi, vb ω/k), ∇× figurational) and microscopic (kinetic) instabilities. The division is the same as within WavesFirst, (IAWs) a beam-plasma and counter-propagating20 instabilityWaves∂B enhances (IAWs) LWs. and Langmuir The counter-propagating plasma-beam Waves (LW). process Second, LWs.involves if theThe three enhance- plasma-beam process involves three linearization of Vlasov-poisonE = − system can be used to find the dispersion(2.4) relation of the and the Maxwell’s equations∇× (Equations∂t 2.4 and 2.7)plasma together theory with in the general. prescribed A macroinstabilityrule96 of is something that can be described by species:ment thermal of LW electrons, is high enough thermalρ then ions,1 ∂ theE and enhanced an electron LW canbeam decay, with enhancing a bulk velocity Ion Acoustic of EB==species:µ0J + thermal electrons, thermal(2.6) ions,(2.5) and an electron beam with a bulk velocity of calculation of ρ and J ∇∇×· 0 c2 ∂t macroscopic equations in the configuration space. Examples of a macroinstability are the Waves (IAWs) and counter-propagatingρ 1 ∂E LWs. The plasma-beam process involves three vb. By assuming small perturbationsE = and small damping/growth(2.6) (vthe, vthi, vb ω/k), DRAFTB = µ0Jv+b. May29,2010,6:36am By2 assuming small perturbations3000 (2.5) andDRAFT small damping/growth (vthe, vthi, vb ω/k), ∇×∇ · 0 c ∂t Rayleigh-Taylor, Farley-Buneman and Kelvin-Helmholtz instabilities. On the other hand, linearizationspecies: of thermal Vlasov-poison electrons,B = 0 system thermal can ions, be used and to an find electron the dispersion (2.7) beam with relation a bulkLangmuir of velocity the (“Plasma of Line”) ∇ · linearization of Vlasov-poisona microinstability system takes can place be used in the to ( findx, v)-space the dispersion and depends relation on the of the actual shape of the ρ = ρ(x , v ,...,x , v ), (2.50) Coupling is completevb. via By charge assuming and current smallB = perturbations densities. 01 1 andN smallN damping/growth (2.7) (vthe, vthi, vb ω/k), ∇ · distribution function.Ion-acoustic A consequence (“Ion Line”) of a microinstability is a greatly enhanced level of J = J(x , v ,...,x , v ). 0 (2.51) Coupling is completeDRAFTlinearization via charge ofand Vlasov-poisonSimple current densities.rules May29,2010,6:36am1 system yield1 canN befluctuationsN used to find in the plasmadispersion associated DRAFT relationwith of the the unstable mode. These fluctuations are called DRAFT3 May29,2010,6:36am DRAFT ρ = qj ncomplexj = qj behaviorfj d v (2.8) j j microturbulence. Microturbulence can lead to enhanced radiation from the plasma and to ρ and J are the charge and current density3 of the medium at certain iteration. A ρ = qj nj = qj fj d v (kHz) Frequency (2.8) DRAFT May29,2010,6:36amenhanced scattering of particles, DRAFTresulting in anomalous transport coefficients, e.g., anoma- simplified scheme of thej PIC simulationj is given in Figure 2 8. J = q n v = q f v d3v, · -3000(2.9) j j j j j lous electric and thermal20 conductivities. 40 60 Examples80 100 of microinstability120 are the beam-driven, PIC codes usuallyj are classifiedj depending on dimensionality of the code and on the 3 k = 2! " (1/m) 20 J = qj nj vj = qj fj v d v, (2.9) (a) where fj(x, v) represents the space-velocity distribution function of theion species acousticj, and0 and electrostatic ion cyclotron instabilities. set of Maxwell’s equationsj used.j The codes solving a whole set of Maxwell’s equations are µ0 are the permitivity and permeability of the air respectively, and c is the speed of light. where fj(xcalled, v) represents electromagnetic the space-velocity codes; electrostatic distribution function ones solve of the just species the Poissonj, 0 and equation. µ Theare the complexity permitivity of this and system permeability of equations of the isair evident respectively, and the and quasi-linearc is the speed approach of light. is 0 Specifically the code used in this work can perform two and three dimensional simu- usedThe to obtain complexity an approximated of this system solution. of equations The traditional is evident anddevelopment the quasi-linear of the quasi-linear approach is theoryused to of obtain waves an in approximated plasmas follows solution. a well Theestablished traditional procedure development (Krall, of 1974; the quasi-linear Nicholson, 1983):theory First, of waves electromagnetic in plasmas follows fields, a and well in established the case of procedure warm plasmas (Krall, the 1974; space-velocity Nicholson, distribution1983): First, of electromagnetic the particles, are fields, linearized; and in then the case the linearof warm Vlasov plasmas equation the space-velocity is subjected todistribution a Fourier/Laplace of the particles, analysis are in linearized;space/time, then yielding the linear fluctuating Vlasov particles equation distributions is subjected whichto a Fourier/Laplace are used to settle analysis the current in space/time, density (J yielding) and electric fluctuating field ( particlesE) relation. distributions Usually (b) thewhich conductivity are used to tensor settle (σ the) is current obtained density from this (J) relation; and electric Fourier field analyzed (E) relation. in both Usually space andthe conductivitytime, Faraday’s tensor and ( Ampere’sσ) is obtained equations from are this combined relation; Fourierto yield analyzed a dispersion in both equation. space Theand solution time, Faraday’s of this dispersion and Ampere’s equation equations relates are frequency combinedω and to yield wavevector a dispersionk and equation. thereby determinesThe solution the of normal this dispersion modes of equation the plasma; relates thus frequency the final stepω and is towavevector insert thek conductivityand thereby tensordetermines (which the brings normal the modes plasma of properties) the plasma; into thus the the dispersion final step relation is to insert (which the states conductivity waves maintensor features) (which brings to obtain the the plasma plasma properties) waves. This into isthe the dispersion path that relation is followed (which in this states section. waves mainFollowing features) this to obtainpath, the the linearization plasma waves. of the This fields is the and path space-velocity that is followed distribution in this section. func- tionFollowing comes first this and path, is used the togetherlinearization with of a the Fourier/Laplace fields and space-velocity space/time distribution transform of func- the (c) (d) tion comes first and is used together with a Fourier/Laplace space/time transform of the Figure 4 5: Simulated incoherent scatter spectrum (for periodic boundary · conditions), obtained integrating 120 angular independent spectra,(a) as a function of the frequency and the wavenumber. (b) As a function of fre- quency for the wavenumber k 54 m 1(or radar frequency of 1300 MHz), ∼ − ∼ which is similar to the wavenumber of Sondrestrom. (c) and (d) are close ups of the negative and positive Langmuir modes, respectively. 77 ISR Measures96 a Cut Through This Surface
3000 150 39
0 0 Frequency (kHz) Frequency
-3000 20 40 60 80 100 120 -150 20 40 60 80 100 120 k = 2! " (1/m) k = 231! " (1/m) (a) EISCAT UHF EISCAT Sondrestrom AMISR, MHO AMISR,
Ion-acoustic “lines” are broadened by Landau damping
Figure 2 7: The top figure shows the ion acoustic line of the inco- · (b) herent scattering spectrum (in log scale) for various wavenumbers (ks = 4πfradar/c). In the bottom figure the normalized ion acousticFigure line of 3 6: theSimulated in- ISR spectra for many scatter wave numbers with 5 · coherent scattering spectrum (normalized to the maximal spectral10 macroparticles power of (top plot). Simulated and theoretical ISR spectrum for The ISRthree dimodelfferent scatter wave numbers with 105 macroparticles (bottom plot). a radar operated at 527 MHz) is shown for radars with operation frequencies of 527 MHz (close to radar frequency of AMISR ), 879 MHz (close to radar frequency of EISCAT, Tromso) and 1289 MHz (close to radar frequency of Sondrestrom).
Table 2.1: Summary of Plasma parameters used to calculate theoretical IS spectra showed in Figure 2 7 · Plasma Parameter(c) Symbol Value (d) 27 Ion Mass where:mi 1.6726 10− kg × 30 e−FigureMass 4 5: Simulatedme incoherent3.3452 scatter10− kg spectrum (for periodic boundary · × Ionconditions), Temperature obtainedT integratingi 1000 120K angular independent spectra,(a) as a Te/Ti e−functionTemperature of the frequencyTe and the2000 wavenumber.K (b) As a function of fre- area ~ Ne 11 3 Ionquency Density for the wavenumberni k 5410 m1/m1(or radar frequency of 1300 MHz), ∼ 11 − 3 ∼ e−whichDensity is similar to thene wavenumber10 1/m of Sondrestrom. (c) and (d) are close ups of the negative and positive Langmuir modes, respectively. Ti/mi
Vi
From Evans, IEEE Transactions, 1969
Figure 2 5: The top figure shows an Incoherent Scattering Spectrum, · including the three lines. The middle figure shows a zoom to the ion acoustic line, which is the focus of this research. The bottom figure shows the autocorrelation function ρ(τ) of the ion acoustic line. f+ is the Doppler frequency associated with the ion acoustic phase velocity. Pr " Pt % " (()) % " GA % SNR = = $ 2 ' $ 2 ' $ ' Pn # 4! R & # 4! R & # KTBNsys & P Received power ISRr = in a nutshell
Pn = Received noise power Here’s what we measure: Pt = Transmitted powTheer “radar equation” P P (()" ) " G%A % Pr "r P"t % "t (%()" ) % %(GA= Radar cross section SNRSNR= == = $ 2 ' $ 2 ' $ ' P#$ # 42 !&' #$R & # 42 !&' $R & KTB'N Pn n 4! R 4! R # KTG#BN= s yAs &ntseysnna& gain P = Received power Pr = Rer ceived power A = Antenna area
P P Ren =c eReivecde inoiveds enoi powse epowr er n = kB = Boltzman's constant P = Transmitted power Pt = Tt ransmitted power T = Temperature ( = ( Ra=da Rar cdarosr sc rsosecst isonection B = Bandwidth G G A=nte Annante gannain gain = Nsys = System noise temperature A = AA=nte Annante annarea area Here’sk the Bol theory:tzman's constant kB = BBol= tzman's constant T = T Te=m Tpeermatpeurreature B = BBa=ndw Baindwdth idth N System noise tem69perature Nsys = s ySs ys= tem noise temperature
Incoherent Averaging
Normalized ISR spectrum for different integration times at 1290 MHz
1 sample We are seeking to estimate the power spectrum of a Gaussian random process. This requires that we sample and average many independent “realizations” of the 30 samples process.
1 Uncertainties ! Number of Samples 600 samples
Figure 3 2: Simulated IS spectra for different number of independent spec- · tra integrated at an operation frequency of 1289 MHz. Top plot shows an 24 average of one spectrum. The middle plot shows an average of 30 inde- pendent spectra. The bottom plot shows an average of 600 independent spectra. RCS: hard target versus electrons
!"#$%&'&"()!"(&*'+,$") Incoherent integration
1 Uncertainties ! Number of Samples Range-time analysis Range
A single pulse is transmitted at t = 0. • The front end propagates along the line r = ct. • The back end follows along a parallel line.
Instantaneous position of the transmitted pulse
Time
Modulation envelope
Range-time analysis
• The scattered wave propagates back to the radar (r=0) at speed c. Range • Scattered signals from different parts of the pulse arrive at the A point target is represented by a antenna at different times. horizontal line in this diagram
t1 t2 t3 t4 t5
t2 t3 t4 Time
Modulation envelope Sampling a signal require time-integration
We send a pulse of duration !. How should we listen for the echo?
Convolution of two rectangle functions
Input Output
Value at a single point
• To determine range to our target, we only need to find the rising edge of the pulse we sent. So make T1< • Could make T1>>T2, then we’re integrating noise in time domain. • So how long should we close the switch? Sampling the received signal Range • The output from the radar receiver represents a convolution A point target is represented by a of the backscattered signal and the horizontal line in this diagram receiver impulse response. Time Modulation envelope Computing the ACF • We can get a larger peak signal out of the receiver by integrating Range longer for each sample A point target is represented by a • What does this mean in terms of horizontal line in this diagram receiver bandwidth? Time Modulation envelope Computing the ACF • We don’t get any more gain in signal amplitude once our integration time matches our pulse length Range • Or, stated alternately, when our A point target is represented by a receiver bandwidth matches the horizontal line in this diagram bandwidth of our pulse. Time Modulation envelope The bandwidth-noise connection The matched filter is a filter whose impulse response, or transfer function, is determined by a given signal, in a way that will result in the maximum attainable signal-to-noise ratio at the filter output when both the signal and white noise are passed through it. The optimum bandwidth of the filter, B, turns out to be very nearly equal to the inverse of the transmitted pulse width. To improve range resolution, we can reduce ! (pulse width), but that means increasing the bandwidth of transmitted signal = More noise! Matched Filter + How should we choose h(t) H(f) such that the output SNR is maximal? Assuming white Gaussian noise, it can be shown that max SNR is when Pulse compression and matched filtering “If you know what you’re looking for, it’s easier to find.” c. Kernel Problem: Find the precise location of the target in the image. Solution: Correlation Barker codes + off _ Volume target (e.g., the ionosphere) Working principle of Barker codes • In Barker-coded experiments • The resulting range- the sampling interval is equal to amplitude ambiguity the bit length. function contains a range • The signal is filtered by a centre peak and Range (digital) matched filter. side bands. • The impulse response of the • The height of the matched filter is a mirror centre peak is equal image of the modulation–++ to the number of bits envelope. + –_+ in the code. • The matched filter + + – • The range-amplitude uses n samples to+ + + – + ambiguity function is + – – function Range-ambiguity make a single ++ – + equal to the auto- filtered sample – + correlation function (n = the num- + of the modulation ber of bits envelope. in the range-amplitude code). ambiguity function + + + – + + + + – + Time sample no 1 2 3 4 5 time + + + + + + + + Modulation – – envelope0 T 2T 5T decoded sample no 1 Doppler Revisited Transmitted signal: ( " 2R%+ After return from target: cos*2 !f o$ t + '- ) # c &, To measure frequency, we need to observe signal for at least one cycle. So we will need a model of how R changes with time. Assume constant velocity: R = Ro + vot Substituting: ( " 2vo % 2!foRo + cos*2 !$ fo + f o ' t + - ) # c & c , constant ! f D !2 f ovo !2vo fD = = c "o By convention, positive Doppler frequency shift Target and radar closing Doppler Detection: Intuitive Approach Phasor diagram is a graphical representation of a sine wave I & Q components* I => in-phase component Acos(") Consider strobe light as Q => in-quadrature cosine reference wave component at same frequency but Asin(") with initial phase = 0 *relative to reference signal Doppler Detection: Intuitive Approach Closing on target – positive Doppler shift e- Transmitted Received !1 !2 Target’s Doppler frequency shows up as a pulse-to-pulse shift in phase. I and Q Demodulation The fundamental output of a pulsed Doppler radar is a time series of complex numbers. I and Q Demodulation in Frequency Domain Transmitted signal Frequency domain cos(2!f ot) -fo 0 fo Doppler shifted cos(2 ( f f )t) ! o + D -fo-fD fo+fD Mixed (multiplied) with carrier cos(2!f ot) -f -f -f f f +f o D D D o D Cosine is even function, so sign of fD (and, hence, velocity) is lost. What we need instead is: This analysis holds if we have a Doppler spectrum as well. exp( j2!f D t) = cos(2!fD t) + j sin(2!f D t) fD The analytic signal exp( j 2 ! f D t )cannot be measured directly, but the cos and sin components via mixing with two oscillators with same frequency but orthogonal phases. The components are called “in phase” (or I) and “in quadrature” (or Q): Aexp( j2!f D t) = I + jQ FFT 42 Example: Doppler Shift of a Meteor Trail • Collect N samples of I(tk) and Q(tk) from a target • Compute the complex FFT of I(tk)+jQ(tk), and find the maximum in the frequency domain • Or compute “phase slope” in time domain. Meteor Echo I & Q 58 Fourier Transforms in RadarVoltage and Signal Processing As a check, we note that if we used a single rounding function r, with transform R, the expression in (3.30) reduces to Meteor Echo Power sinc f T sinc f T R( f ) r e !2ifTr ! s e !2ifTs (3.31) 2if 2if which (with Tr = !T/2, Ts = T/2, Tr = 1, and Ts = 2) is seen, from (3.11), to be exactly the result of smoothing the asymmetrical trapezoidal pulse with the function r. (dB) SNR Time (s) 3.8 Regular Train of Identical RF Pulses This waveform could represent, for example, an approximation to the output of a radar transmitter using a magnetron triggered at regular intervals. The waveform isDoes defined by this strategy work for ISR? Typicalu(t) =ion-acousticrepT [rect (t/) cos velocity: 2f 0 t ] 3 km/s (3.32) Doppler shift at 450 MHz: 10kHz where the pulses of length of a carrier at frequency f 0 are repeated at the pulse repetitionCorrelation interval T and shown time: in Figure1/10kHz 3.17. = 0.1 ms We note thatRequired the rep operator PRF to applies probe to a productionosphere of two functions, (500km range): 300 Hz so the transform will be (by R8b) a comb version of a convolution of the transforms of these functions. We could express the cosine as a sum of exponentials, but more conveniently we use P7a in which this has already been done.Plasma Thus has (from completely P3a, P8a, R8b, decorrelated and R5) we obtain by the time we send the next pulse. Alternately,U( f ) = (/2T the) comb Doppler1/T [sinc ( f !shiftf 0 ) +issinc well ( f +beyondf 0 ) ] (3.33) the max unambiguous Doppler defined by the Inter-Pulse Period T. This spectrum is illustrated (in the positive frequency region) in Figure 3.18. Thus we see that the spectrum consists of lines (which follows fromPulse Spectra 59 the repetitive nature of the waveform) at intervals 1/T, with strengths given Figure 3.17 Regular train of identical RF pulses. Figure 3.18 Spectrum of regular RF pulse train. by two sinc function envelopes centered at frequencies f 0 and !f 0 . As discussed in Chapter 2, the negative frequency part of the spectrum is just the complex conjugate of the real part (for a real waveform) and provides no extra information. (In this case the spectrum is real, so the negative frequency part is just a mirror image of the real part.) However, as explained in Section 2.4.1, the contribution of the part of the spectrum centered at !f 0 in the positive frequency region can only be ignored if the waveform is sufficiently narrowband (i.e., if f 0 >> 1/, the approximate bandwidth of the two spectral branches). An important point about this spectrum, which is very easily made evident by this analysis, is that, although the envelope of the spectrum is centered at f 0 , there is, in general, no spectral line at f 0 . This is because the lines are at multiples of the pulse repetition frequency (PRF) (1/T ), and only if f 0 is an exact multiple of the PRF will there be a line at f 0 . Returning to the time domain, we would not really expect power at f 0 unless the carrier of one pulse were exactly in phase with the carrier of the next pulse. For there to be power at f 0 , there should be a precisely integral number of wavelengths of the carrier in the repetition interval T ; that is, the carrier frequency should be an exact multiple of the PRF. This is the case in the next example. 3.9 Carrier Gated by a Regular Pulse Train This waveform would be used, for example, by a pulse Doppler radar. A continuous stable frequency source is gated to produce the required pulse train (Figure 3.19). Again we take T for the pulse repetition interval, for ISR Receiver: I and Q plus correlation 31 LPF I(ti) “in phase” 31 BPF Power #/2 phase G splitter shift. fL,fH LPF Q(ti) “quadrature” We have time series of V(t) =I(t) + jQ(t), how do I compute the Doppler spectrum? Estimate the autocorrelation function (ACF) by computing products of complex voltages (“lag products”) V(t)V *(t + ! ) Power spectrum is Fourier R (! ) = FFT Transform of the ACF vv S Autocorrelation function and power spectrum Te/Ti Ion temperature (Ti) to ion area ~ Ne mass (mi) ratio from the width of the spectra • The purpose of a monostatic radar Ti/mi is to measure the range profile of the Electron to ion signal autocorrelation function (acf) temperature ratio (Te/Ti) • The acf is sampled at certain Vi from “peak-to-valley” ratio intervals of delay. • Convention: lags are numbered as 0, 1, 2, .... • Zero lag Electron (= ion) density is equal to signal power. • The range from total area (corrected and lag resolutions of the for temperatures) measurement are determined by the radar modulation zero lag (=signal average power) and sampling rate. Line-of-sight ion 1st lag velocity (Vi) from bulk Doppler shift 2nd lag Our goal is to compute lags 3rd lag Figure 2 5: The top figure shows an Incoherent Scattering Spectrum, · including the three lines. The middle figure shows a zoom to the ion acoustic Figureline, 2 which5: The is the top focus figure of this shows research. an Incoherent The bottom Scattering figure shows Spectrum, the · includingautocorrelation the three functionlines. Theρ(τ middle) of the figure ion acoustic shows line. a zoomf+ tois the Dopplerion acoustic line,frequency which is associated the focus with of the this ion research. acoustic phase The velocity. bottom figure shows the autocorrelation function ρ(τ) of the ion acoustic line. f+ is the Doppler frequency associated with the ion acoustic phase velocity. Computing the ACF (and, hence, spectrum) RRaannggee-t-itmimee d diaiaggraramm Range Range-time diagram zero lag RangeRange Range Time P S P P Time S S P Length of RF pulse TimeTime s Sample Period (typically ~ 1/10 pulse length) P P LengthLength of of RFRF pulse pulse s s SampleSample Period Period (typically (typically ~ 1/10~ 1/10pulsepulse length) length) Computing the ACF (and, hence, spectrum) RRaannggee-t-itmimee d diaiaggraramm Range Range-time diagram Range-time diagram 1st lag 0-lag Range RRaannggee-t-itmimee d diaiaggraramm RangeRange Range P 0-lag0-lag S Time Time P RangeRange S P P Time S S P Length of RF pulse TimeTime s Sample Period (typically ~ 1/10 pulse length) P P LengthLength of of RFRF pulse pulse s s SampleSample Period Period (typically (typically ~ 1/10~ 1/10pulsepulse length) length) P P S S TimeTime Range-time diagram 0-lag Range RRaannggee-t-itmimee d diaiaggraramm P 0-lag0-lag S Time RangeRange P P S S TimeTime Computing the ACF (and, hence, spectrum) RRaannggee-t-itmimee d diaiaggraramm Range Range-time diagram 2nd lag RangeRange Range Time P S P P Time S S P Length of RF pulse TimeTime s Sample Period (typically ~ 1/10 pulse length) P P LengthLength of of RFRF pulse pulse s s SampleSample Period Period (typically (typically ~ 1/10~ 1/10pulsepulse length) length) Computing the ACF (and, hence, spectrum) RRaannggee-t-itmimee d diaiaggraramm Range Range-time diagram Range-time diagram 2nd lag Uncorrelated (contributes to noise) Correlated volumes (signal) 0-lag Uncorrelated (contributes to noise) Range RRaannggee-t-itmimee d diaiaggraramm RangeRange Range P 0-lag0-lag S Time Time P RangeRange S P P Time S S P Length of RF pulse TimeTime s Sample Period (typically ~ 1/10 pulse length) P P LengthLength of of RFRF pulse pulse s s SampleSample Period Period (typically (typically ~ 1/10~ 1/10pulsepulse length) length) P P S S TimeTime Range-time diagram 0-lag Range RRaannggee-t-itmimee d diaiaggraramm P 0-lag0-lag S Time RangeRange P P S S TimeTime Lag-product!"#$%&'()*$+",&-.$ matrix Bibliography ISR tutorial material: • http://www.eiscat.se/groups/Documentation/CourseMaterials/ Radar signal processing • Mahafza, Radar Systems Analysis and Design Using MATLAB • Skolnik, Introduction to Radar Systems • Peebles, Radar Principles • Levanon, Radar Principles • Blahut, Theory of Remote Image Formation • Curlander, Synthetic Aperture Radar: Systems and Signal Analysis Background (Electromagnetics, Signal Processing: • Ulaby, Fundamentals of Engineering Electromagnetics • Cheng, Field and Wave Electromagnetics • Oppenheim, Willsky, and Nawab, Signals and Systems • Mitra, Digital Signal Processing: A Computer-based Approach For fun: •http://mathforum.org/mbower/johnandbetty/frame.htm http://mathforum.org/johnandbetty/