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. --------------j2πnt 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 = --------------j2πnt T ! © 2000 by Chapman & Hall/CRC = © 2000 by Chapman & Hall/CRC(5.16) f3()t ∑ Fne n = –∞ “3dB” bandwidth where the Fourier series coefficients Fn are given by 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 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).