Outline. •Principle of Pulsed Doppler Radar. •The Doppler spectrum of the
ionospheric plasma. •Mathematics of Doppler Processing. •Pulse Compression. 3
...
Basic Radar Signal Processing Joshua Semeter, Boston University
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Why study ISR? • You get to learn about many useful things, in substantial depth.
- Plasma physics - Radar - Coding - Electronics (Power, RF, DSP) - Signal Processing
• But what if I probably won’t stay in this field? - See above!
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Outline
• Principle of Pulsed Doppler Radar • The Doppler spectrum of the ionospheric plasma • Mathematics of Doppler Processing • Pulse Compression
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The Ubiquitous deciBel The relative value of two things, measured on a logarithmic scale, is often expressed in deciBel’s (dB)
Example: SNR
Signal-to-noise ratio (dB) = 10 log 10
Factor of: 0.1 0.5 1 2 10 100 1000
Scientific Notation 10-1 100.3 100 100.3 101 102 103
dB -10 -3 0 3 10 20 30
. . 1,000,000
106
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Signal Power Noise Power
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Waves versus Pulses What do radars transmit?
Waves? Waves, modulated by “on-off” action of pulse envelope
or Pulses?
How many cycles are in a typical pulse? PFISR frequency: 449 MHz Typical long-pulse length: 480 µs 215,520 cycles! 5
Pulsed Radar Peak power
Power
Pulse length
100 µsec
1 Mega-Watt
Target Return
10-12 Watt
Inter-pulse period (IPP) 1 msec
Duty cycle =
Pulse length Pulse repetition interval
Time 10%
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)
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Distance = Time Range resolution: Set by pulse length given in units of time, τp, or length, cτp
ΔR = R2 − R1 =
cτ p 2
Maximum unambiguous range: Set by Inter-pulse Period (IPP) IPP = Interpulse period (s) PRF = pulse repetition frequence = 1/IPP (Hz) 2R/c IPP
c IPP Ru = 2
IPP
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Velocity = Frequency Transmitted signal:
After return from target:
cos(2πf o t) ⎡ ⎛ 2R ⎞⎤ cos⎢2πf 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: Substituting:
R = Ro + v o t ⎡ ⎛ 2v o ⎞ 2πf o Ro ⎤ cos⎢2π ⎜ f o + f o ⎟t + ⎥ c ⎠ c ⎦ ⎣ ⎝
− fD
constant
−2 f ov o −2v o fD = = c λo By convention, positive Doppler frequency shift
Target and radar closing 8
Two key concepts Two key concepts: Distant
Time R = cΔt 2
Velocity
e-
Frequency
v = − f D λ0 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.
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Two key concepts Two key concepts: Distant
Time R = cΔt 2
Velocity
Frequency
v = − f D λ0 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.
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J. Semeter, Boston Un
Concept of a “Doppler Spectrum”
her Doppler Spectra Two key concepts: Distant
Velocity
Frequency
v = − f D λ0 2
J. Semeter
Power (dB)
Time R =Remote cΔt 2 Sensing ENG SC700 Radar
Some Other Doppler Spectra
NEXRAD WSR-88D
gine at give Doppler spectrum
Velocity (m/s)
S. Bachma
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 ( λ of 10 to 30 cm)?
Tennis ball, birds, aircraft engine
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de, which is the main mode detected by ISRs, and the Langmuir
Longitudinal Modes in a Thermal Plasma
on, 2003]. Using a linear approach to solve the Vlasov-Poisson DIAZ ET AL.: BEAM-PLASMA INSTABILITY EFFECTS ON IS SPECTRA
X-6
DIAZ ET AL.: BEAM-PLASMA INSTABILITY EFFECTS ON IS SPECTRA
dispersion relation of these modes is obtained. The real part of
� �� � � � �1 � �3 DIAZ ET AL.: BEAM-PLASMA EFFECTS π me 2 INSTABILITY Te 2 Te ON�3IS SPECTRA � �� 3(2) + exp − � − � ωs � 1 n relationωreads � � si = − 27 Te 8 mi Ti 2Ti π 2 me 2 Te 2 3 ωsi�= − + exp − − ωs � �� � Ion-acoustic � � � 12 � � 32 8 dependence mi Ti mode 2Ti 2 Cs = kB (Te + 3Ti )/m ion-acoustic speed. The of this π i ismthe T T 3 e e e ωsi = − + exp − − ωs (2) � -6 DIAZ ET 8AL.: BEAM-PLASMA INSTABILITY EFFECTS ON IS SPECTRA mi Cs =Ti kB (Te + 3T 2Ti )/m 2 is the ion-acoustic speed. The dependence of i where i ω = C k, (1)is detected by spheric state parameters s s is observed in Eq. 2. The Langmuir mode � = kB (Te + 3Ti )/mi is the ion-acoustic speed. The dependence of this mode � � �� on ionospheric state parameters is observed in Eq. � 1 3 � its �dispersion der certain conditions, and the�real � part of relation is expressed as 2. The Langmuir mode is d 2 2 assuming ωsi � ωs , k 2 λ2De � 1 and T /T � 1) can be written i Tee π me Te 3 ωsi = − is observed in +Eq. 2. The expLangmuir − − (2) pheric state parameters mode isωsdetected by � 8 mi certainTconditions, 2Ti the2 real part of its dispersion relation is exp i 3 ISR under and 2 2 2 2 (3) � ωL = ωpe + 3 k vthe ≈ ωpe + 2 vthe λDe k , er certain conditions, and thei real part of its dispersion relation is expressed as mode here C = k (T + 3T )/m is the ion-acoustic speed. The dependence of this s B e i � Langmuir
3 2 ω = + ≈ ω + v λ k � L pe the De , e imaginary part (assuming ωLi � ωL ) is 3 2 by ionospheric state parameters is observed in Eq. 2. The Langmuir mode is detected ω = ω2 + 3 k2 v2 ≈ ω + v λ k2, (3) 2 ωpe
L
pe
the
pe
3 k2
2 vthe
the De
2 R under certain conditions, and imaginary the real of its dispersion relation � part � 73 � and 3the part (assuming ωLi � ωL )isisexpressed as 2 ω ωpe π6:36am 1 ω ) is 3 D R A F T Maypart 29, 2010, pe � maginary (assuming ω Li L ωLi ≈ − exp − − ωL . (4) 3 2 � 3 2 8 k vthe 2k vthe 23 2 2 2 ωL = ωpe + 3 k vthe ≈ ωpe + vthe λDe k 2 , (3) � � � 3 2 � parameters2� orward model used to � estimate ionospheric in ISR assumes that these ω ω π 1 3 pe pe 3 2 Figure 2·4: Longitudinal modes of a ωpeωLi ≈3 − exp − 2 2 − plasma. ωL . Blue lines re π ωpe 1 3 3 ωLi part ≈ − (assuming exp − L ) is − acoustic ωL .8 waves (4)Langmuir kbeam vthe vthe 2waves. and red ones 2k to ddominating the imaginary 3 ωLi � ωHowever, 2 3 2 modes in the8 ISR spectrum. an injected of particles, k vthe 2k vthe 2
The forward model usedthe toinestimate ionospheric parameters in ISR assumes icular canto74destabilize the plasma, altering dispersion relations and rward electrons, model used estimate ionospheric parameters ISR assumes that these start to interact more strongly with the growing wav � plasma particles � � 3 2 ωpe π ωpe 1 3 This can sometimes terms of 75 are the dominating modes in the spectrum. However, an so-called injectedquasi-linea beam 12of exp − − ISR ω be . described in (4)the udes of these modes. ω ≈ −
otion has The to be name used. The system of equations formed fieldequation, equations. “Particle-in-Cell” comes from the way of um at high latitudes, therefore a kinetic approach, which ch includes the simulation Vlasov equation plus Maxwell’s equations, has ntities to the particles. n equation, has to be used. The system of equations formed y to obtain the wave modes propagating along the plasma. odes solve equation of motion of particles withhas the Newton-Lorentz includes thethe Vlasov equation plus Maxwell’s equations,
Simulated ISR Doppler Spectrum
ρ= = 0ρ(x1 , v1 , . . . , xN , vN ), ∇ · B ge and current densities.
Frequency (kHz)
obtain the wave modes propagating along the plasma. ∂fj (t, x, v) qj ∂fj (t, x, v) · + (E + v × B) · =0 (2.3) ∂x mj ∂v (PIC): fj (t, x,Particle-in-cell v) qj ∂fj (t, x, v) + (E + v × B) · =0 (2.3) ∂x d v mqj ∂v i i −∂B) + v × B(x )) 20 d = (E(x for i = (2.4) 1, . . . , N (2.49) i i i ∇ × E = dt mi ∂t −∂B 20 ∇×E= (2.4) 96 of ∂t quations (Equations 2.4 and 2.7) together with the prescribed rule ρ 1 ∂E ∇ · E = (2.6) ∇ × B = µ0 J + 2 (2.5) �0 c ∂t J ρ 1 ∂E ∇ · E = (2.6) ∇ × B = µ0 �J + 2 (2.5) 3000 0 c ∂t Langmuir (“Plasma Line”) ∇·B=0 (2.7)
(2.50) Ion-acoustic (“Ion Line”) (2.51) 0
(2.7)
J = densities. J(x1 , v1 , . . . , xN , vN ). ge andSimple current rules yield � � � qj ncomplex qj behavior fj d3 v (2.8) j = � jcharge and j current � � density of the medium at certain iteration. A 3 q j nj = qj fj d v (2.8) � j j PIC simulation is given in Figure 2·8. �the � -3000 qj nj vj = qj fj v d3 v, (2.9) 20 40 60 80 100 � j ly are classified depending on dimensionality of the code andkon � � = 2the π λ (1/m) 3 qj nj vj = qj fj v d v, (2.9) (a) ce-velocity distribution function of the species j, �0 and j
ations used. The codes solving a whole set of Maxwell’s equations are
120 13 13
ISR Measures a Cut Through This Surface 77
31
96
Frequency (kHz)
3000
150
0
-3000
0
20
40
60
k = 2π λ (a)
120
-150
20
40
60
k = 2π
80 100 λ (1/m)
120
Sondrestrom
EISCAT UHF
AMISR, MHO
Ion-acoustic “lines” are broadened by Landau damping
80 100 (1/m)
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The ISR model
where: Te/Ti
area ~ Ne
Ti/mi Vi
From Evans, IEEE Transactions, 1969
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Incoherent Averaging Normalized ISR spectrum for different integration times at 1290 MHz
1 sample
30 samples
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 process.
1 Uncertainties ∝ Number of Samples 600 samples
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Components of a Pulsed Doppler Radar
_
+ +
_ _
+
Model Fit
Plasma density (Ne) Ion temperature (Ti) Electron temperature (Te) Bulk velocity (Vi)
17 17
�
�
its spectrum. "! For a real waveform we have "!
Essential Mathematical Operations !
!
�
�
ply states that the total energy in a waveform is equal to the total 2 2 Euler: u (x ) dx = 2 | U ( y ) | dy (2.26) n its spectrum. For a real waveform we have "! !
Fourier:
�
0
!
�
2
u (x )spectrum dx = 2 of |aUreal ( y ) |waveform. dy ) = U ("y )* for the "!
2
(2.26)
0
e Wiener-Khinchine Relation y ) =Convolution: U ("y )* for the spectrum of a real waveform. that the autocorrelation function of a waveform is given by the Fourier transform of its power spectrum. For a waveform u with he Wiener-Khinchine Relation e) spectrum U, the power spectrum is | U | 2, and from R2 and R3 Correlation: t U *( f ) is the transform of u *("t ), so we have es that the autocorrelation function of a waveform is given by the Fourier transform of its power spectrum. For a waveform u with 2 2 f ) | from R2 (2.27) u (t ) # u *("t )$ U ( f spectrum ) � U *( f is) |=U| |U,(and and R3 de) spectrum U, the power hat U *( f ) is the transform of u *("t ), so we have Wiener-Khinchine Theorem: ing out the convolution, we have 2 (2.27) u (t ) # u *("t ) $ U ( f ) � U *( f ) = | U ( f ) | !
u *("t ) =
!
u (t " t % ) u *("t % ) dt % =
u (s ) u (s " t ) ds = r (t )
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Dirac Delta Function ⎧⎪ +∞, x = 0 δ (t) = ⎨ ⎪⎩ 0, x ≠ 0
The Fourier Transform of a train of delta functions is a train of delta functions.
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Harmonic Functions
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Gate function
ISR spectrum
Autocorrelation function (ACF)
Increasing Te
Not surprisingly, the ISR ACF looks like a sinc function… 21
Fourier Transform of an RF Pulse τ
frequency
f0 f0 – ( 1 ⁄ τ )
f0 + ( 1 ⁄ τ )
The F.T. of a simple pulse is fora async function Figure 5.3.RF Amplitude spectrum single pulse, or a train of shifted to non-coherent pulses. the carrier frequency (by convolution property and Now consider the coherent gated CW waveform f ( t ) given by definition of delta function). 3
∞
f3 ( t ) =
∑ f ( t – nT ) 2
(5.15)
n = –∞
Clearly f 3 ( t ) is periodic, where T is the period (recall that f r = 1 ⁄ T is the PRF). Using the complex exponential Fourier series we can rewrite f ( t ) as
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Fourier Transform of a Finite Pulse Train T NT
frequency
2 ------NT Figure 5.5. Amplitude spectrum for a coherent pulse train of finite length.
The F.T. of a finite train of RF pulses is a decaying train of sync functions with width inversely proportional to the total number of pulses, separation 5.3. Linear Frequency and Modulation Waveforms inversely Frequency or phase modulated waveforms can be used to achieve much proportional to the Interpulse Period (IPP). wider operating bandwidths. Linear Frequency Modulation (LFM) is commonly used. In this case, the frequency is swept linearly across the pulse width, either upward (up-chirp) or downward (down-chirp). The matched filter bandwidth is proportional to the sweep bandwidth, and is independent of the pulse width. Fig. 5.6 shows a typical example of an LFM waveform. The pulse width
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Thus we see that the spectrum consists of lines (which follows from the repetitive nature of the waveform) at intervals 1/T, with strengths given
Fourier Transform of an Long Pulse Train
Pulse Spectra
59
Figure 3.17 Regular train of identical RF pulses.
TheFigure F.T. of infinite trainRF of RF pulses is a line spectrum 3.18 an Spectrum of regular pulse train. with lines separated by the pulse repetition frequency. by two sinc function envelopes centered at frequencies f and !f . As
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Bandwidth of a pulsed signal Spectrum of receiver output has sinc shape, with sidelobes half the width of the central lobe and continuously diminishing in amplitude above and below main lobe
A 1 microsecond pulse has a nullto-null bandwidth of the central lobe = 2 MHz
Two possible bandwidth measures: “null to null” bandwidth
2 Bnn = τ
“3dB” bandwidth
Unless otherwise specified, assume bandwidth refers to 3 dB bandwidth
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Pulse-Bandwidth Connection
Shorter pulse
Larger bandwidth
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Strategies for radar reception We send a pulse of duration τ. How should we listen for the echo? Convolution of two rectangle functions
Output Receiver
• • • •
Input Value at a single point
To determine range, we only need to find the rising edge of the pulse we sent. So make T1T2, then we’re integrating noise in time domain. So how long should we close the switch? 27
Detection of a signal embedded in noise
Exponential pulse buried in random noise. Sine the signal and noise overlap in both time and frequency domains, the best way to separate them is not obvious.
28
Most important consideration: Match the bandwidth of the signal you are looking for
29
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 τ (pulse width), but that means increasing the bandwidth of transmitted signal = More noise…
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Doppler Revisited Transmitted signal:
After return from target:
cos(2πf o t) ⎡ ⎛ 2R ⎞⎤ cos⎢2πf 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: Substituting:
R = Ro + v o t ⎡ ⎛ 2v o ⎞ 2πf o Ro ⎤ cos⎢2π ⎜ f o + f o ⎟t + ⎥ c ⎠ c ⎦ ⎣ ⎝
− fD
constant
−2 f ov o −2v o fD = = c λo By convention, positive Doppler frequency shift
Target and radar closing 31
Doppler Detection: Intuitive Approach Phasor diagram is a graphical representation of a sine wave
I & Q components* I => in-phase component Acos(φ) Q => in-quadrature component Asin(φ)
Consider strobe light as cosine reference wave at same frequency but with initial phase = 0
*relative to reference signal
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Doppler Detection: Intuitive Approach Closing on target – positive Doppler shift
eTransmitted Received
φ1
φ2
Target’s Doppler frequency shows up as a pulse-to-pulse shift in phase.
33
I and Q Demodulation
The fundamental output of a pulsed Doppler radar is a time series of complex numbers.
34
I and Q Demodulation in Frequency Domain Transmitted signal
Frequency domain
cos(2πf o t) -fo
0
fo
Doppler shifted
cos(2π ( f o + f D )t)
-fo-fD
cos(2πf o t)
fo+fD
cos(2πf D t) -fo-fD
-fD
fD
fo+fD
Cosine is even function, so sign of fD (and, hence, velocity) is lost. What we need instead is:
exp( j2πf D t) = cos(2πf D t) + j sin(2πf D t)
fD The analytic signal exp( j2πf 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( j2πf D t) = I + jQ
35 35
Example: Doppler Shift of a Meteor Trail
Meteor Echo I & Q Voltage
•
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 Power SNR (dB)
• •
Time (s)
36
Does this strategy work for ISR? Typical ion-acoustic velocity: 3 km/s Doppler shift at 450 MHz: 10kHz Correlation time: 1/10kHz = 0.1 ms Required PRF to probe ionosphere (500km range):
300 Hz
Plasma has completely decorrelated by the time we send the next pulse.
φ1
φ2
37
Does this strategy work for ISR? Typical ion-acoustic velocity: 3 km/s Doppler shift at 450 MHz: 10kHz Correlation time: 1/10kHz = 0.1 ms Required PRF to probe ionosphere (500km range):
300 Hz
Pulse beyond Spectra the max unambiguous Alternately, the Doppler shift is well Doppler defined by the Inter-Pulse Period T.
59
Figure 3.18 Spectrum of regular RF pulse train. 38
The ISR target is “overspread” fd>>1/τ (Doppler changes significantly during one pulse) – –
Must sample multiple times per pulse Result: Doppler can be determined from single pulse.
Range
τS Time
39
ISR Receiver: Doppler filter bank Doppler Filter Bank
Doppler Range
BPF1 BPF2
BPF fL,fH
BPF3
G
Choose fL and fH such that fLf0+fdmax
BPF4
F1 F2 …
BPF5 BPF6
Practical Problem: It is hard to make narrow band (High Q) RF filters:
f0 Q= f H − fL
40
ISR Receiver: I and Q plus decorrelation LPF
BPF fL,fH
G
I(ti) “in phase”
π/2 phase shift.
Power splitter
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 + τ ) *
ρ(τ ) =
FFT
Power spectrum is Fourier Transform of the ACF
S 41
Pulse compression and matched “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
42
Range detection: revisited cτ c ΔR = = 2 2B
τ = Pulse length B = Bandwidth
• For high range resolution we want short pulse large bandwidth
• For high SNR we want long pulse small bandwidth • Long pulse also uses a lot of the duty cycle, can’t listen as long, affects maximum range
• The Goal of pulse compression is to increase the
bandwidth (equivalent to increasing the range resolution) while retaining large pulse energy.
43
Linear Frequency Modulation (LFM or
44
Matched filter detection of a chirp
45
Barker codes + off
_
46
Matched filtering of Barker Code 3 bit code
47
