Basic Radar Signal Processing

Basic Radar Signal Processing
Joshua Semeter, Boston University
1

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!
2

Outline
• Principle of Pulsed Doppler Radar
•
•
The Doppler spectrum of the ionospheric plasma
Mathematics of Doppler Processing
• Pulse Compression 3

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
Signal Power
Noise Power
Scientific
Factor of: Notation dB
0.1 10 -1 -10
0.5 10 0.3 - 3
1 10 0 0
2 10 0.3 3
10 10 1 10
100 10 2 20
1000 10 3 30
.
1,000,000 10 6 60
4

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
Pulse length
100 µsec
Power
Peak power
1 Mega-Watt
Target
Return
10 -12 Watt
Inter-pulse period
(IPP) 1 msec
Time
Duty cycle =
Pulse length
Pulse repetition interval
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)
6

Distance = Time
Range resolution: Set by pulse length given in units of time, τ p , or length, cτ p
ΔR = R 2
− R 1
= cτ p
2
Maximum unambiguous range: Set by Inter-pulse Period (IPP)
2R/c
IPP
IPP
IPP = Interpulse period (s)
PRF = pulse repetition frequence
= 1/IPP (Hz)
R u
= c IPP
2
7

Velocity = Frequency
Transmitted signal: cos(2πf o
t)
After return from target:
⎡ ⎛
cos⎢
2πf o ⎜ t + 2R
⎣ ⎝ 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:
⎡ ⎛
cos⎢
2π⎜
f o
+
⎣ ⎝
R = R o
+ v o
t
f o
2v o
c
− f D
⎞
⎟ t + 2πf R ⎤
o o
⎥
⎠ c ⎦
constant
f D
= −2 f o v o
c
= −2v o
λ o
By convention, positive Doppler frequency shift
Target and radar closing
8

Two key concepts
Two key concepts:
Distant
Time
R = cΔt 2
e-
Velocity
v = − f D
λ 0
2
Frequency
A Doppler radar measures backscattered power as a function range and velocity.
Velocity is manifested as a Doppler frequency shift in the received signal.
9

Two key concepts
Two key concepts:
Distant
Time
R = cΔt 2
Velocity
v = − f D
λ 0
2
Frequency
A Doppler radar measures backscattered power as a function range and velocity.
Velocity is manifested as a Doppler frequency shift in the received signal.
10

J. Semeter, Boston Un
Concept of a “Doppler Spectrum”
er Doppler Spectra
Two key concepts:
Distant Time
R = cΔt 2
ENG SC700 Radar Remote Sensing
Velocity
v = − f D
λ 0
2
Frequency
NEXRAD WSR-88D
Power (dB)
Some Other Doppler Spectra
J. Semeter
ine
t give Doppler spectrum
Tennis ball, birds, aircraft engine
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)?
11

e, which is the main mode detected by ISRs, and the Langmuir
Longitudinal Modes in a Thermal Plasma
n, 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
ispersion relation of these modes is obtained. The real part of
me 1
3
DIAZ ET AL.: BEAM-PLASMA π
2 INSTABILITY Te
2 EFFECTS ON IS SPECTRA
relationωreads
si = −
+ exp − T e
− 3 me ω s 1
3(2)
8 m i T i 2T i π 2
2 Te
2 27
Ion-acoustic
ω si = −
+ exp − T e
− 3 ω
s
me 1 8
3
m i T i 2T i 2
s = k B (T e +3T i )/m π i is the2 ion-acoustic Te
ω si = −
+
6
ω s = C s k, where C s = 2 speed. The
exp − T e
dependence
− 3 of this mode
ω s (2)
DIAZ ET8AL.: BEAM-PLASMA m i T i INSTABILITY EFFECTS ON IS SPECTRA
k B (T e +3T2T i )/m i i
2
pheric state parameters is observed in Eq. 2. The Langmuir ismode the(1)
ion-acoustic is detected byspeed. The dependence of
= k B (T e +3T i )/m i is the
ssuming ω si ω s , k 2 on λ 2 ionospheric
ion-acoustic speed. The
De me
1 and 1
T state parameters
dependence
is observed
of this mode
er certain conditions, and the real part of its dispersion 3 relation in Eq. 2. The Langmuir mode is d
π
2
i/T Te e 2 1) can be written
ω si = −
+ exp − T e
− 3 is expressed as
heric state parameters ω s (2)
is observed in Eq. 2. The Langmuir mode is detected by
ISR8
under m i certainT conditions, i and 2T i the 2 real part of its dispersion relation is exp
ere Langmuir C s = ω L = ωpe 2 +3k 2 vthe 2
r certain conditions, k B (T e +3T and i )/m the i
real is the part ≈ ω pe + 3
ion-acoustic of its 2dispersion v theλ De k 2 , (3)
speed. relation
The dependence is expressed of this as
ω L = ωpe 2 +3k 2 vthe 2 ≈ ω pe + 3 mode
imaginary part (assuming ω
2 v theλ De k 2 ,
ionospheric state parameters Li ω
is observed L )
in Eq. 2. The Langmuir mode is detected by
ω L = ωpe 2 +3k 2 vthe 2 ≈ is
ω pe + 3 2 v theλ De k 2 , (3)
under certain conditions,
73 and the andimaginary the real part part of its (assuming dispersion ω Li
relation ω L ) is isexpressed as
aginary Maypart 29, (assuming 2010, π 6:36am ω pe
3 Li 1
D R A F T
ω Li ≈− ω
exp L ) is−
ω2 pe
− 3
ω L = ωpe 2 +3k 2 vthe 2 ≈ ω pe + 3 ω
8 k 3 vthe
3 2k 2 v 2 L . (4)
the
2
2 v theλ
De k 2 , (3)
rward model used toestimate ionospheric parameters π ωpe
3 1
ω Li ≈−
exp −
ω2 pe
π ωpe
3 1
the imaginary ω part (assuming ω 8 k 3 vthe
3 Li ≈−
exp
8 k 3 Li −
ω2 pe
ω L 2k ) is − 3 in ISR assumes that these
dominating modes in the ISR spectrum. However, 2 vthe 2 an2injected beam of particles,
v 3 the
Figure 2·4: Longitudinal modes of−a 3 plasma. Blue lines re
acoustic ω waves and red ones 2k to 2 Langmuir vthe 2 L . (4) 2waves.
74
ward model used to estimate
The forward
ionospheric
model
parameters
used to estimate ionospheric parameters in ISR assumes
in ISR assumes that these
π ωpe
3 1
ω75
≈−
exp −
ω2 pe
−
ω 3 . (4)
cular electrons, can destabilize the plasma, altering the dispersion relations and
plasma particles start to interact more strongly with the growing wav
are the dominating modes This caninsometimes the ISR be spectrum. described in However, terms of the an so-called injectedquasi-linea
beam of
des of these modes.
12
ω L .

tion equation, has to be used. The system of equations formed
m at high latitudes, therefore a kinetic approach, which
htities includes
equation,
to the
has
simulation Vlasov equation
to be used. The
particles. plus Maxwell’s equations, has
system of equations formed
to obtain the wave modes propagating along the plasma.
ncludes the Vlasov equation plus Maxwell’s equations, has
field equations. The name “Particle-in-Cell” comes from the way of
Simulated ISR Doppler Spectrum
des solve the equation of motion of particles with the Newton-Lorentz
obtain the wave modes propagating along the plasma.
· ∂f j(t, x, v)
+ q j
(E + v × B) · ∂f j(t, x, v)
= 0 (2.3)
∂x m j ∂v
f j (t, x, Particle-in-cell v)
+ q j (PIC):
(E + v × B) · ∂f j(t, x, v)
= 0 (2.3)
∂x m j ∂v
d v i
dt = q i
20
∇×E (E(x i )+v i × B(x i )) for i =1,...,N (2.49)
m
= −∂B
(2.4)
i
uations (Equations ∂t 2.4 and 2.7) together with the prescribed rule 96of
J
∂t
20
∇×E = −∂B
(2.4)
∇×B = µ 0 J + 1 ∂E
∇ · E = ρ c 2 (2.6) (2.5)
0 ∂t
∇×B ∇ · E = = µ 0 J ρ + 1 ∂E
(2.6)
c 2 3000 (2.5)
0 ∂t
∇ · B = 0 (2.7)
Frequency (kHz)
ρ = ρ(x 1 , v 1 ,...,x N , v N ), (2.50)
Ion-acoustic (“Ion Line”)
J = J(x 1 , v 1 ,...,x N , v N ). (2.51)
e and current ∇ · B = densities. 0 (2.7)
e and current densities.
0
Simple
q j n j = rules yield
complex q j behavior f j d 3 v (2.8)
q j n j = j
q j f j d 3 v (2.8)
the j PIC simulation j is given in Figure 2·8.
q j n j v j =
q j f j v d 3 v, -3000 (2.9)
j
q j
jcharge and current density of the medium at certain iteration.
y are classified depending on dimensionality of the code and on the
q j n j v j = f j v d 3 v, (2.9)
(a)
e-velocity distribution function of the species j,
tions used. j The codes solving a whole set of 0 and
Maxwell’s equations are
Langmuir (“Plasma Line”)
A
20 40 60 80 100 120
k = 2π λ (1/m)
13
13

ISR Measures a Cut Through This Surface
77
96
31
3000
150
Frequency (kHz)
0
0
-3000
20 40 60 80 100 120
k = 2π λ
(a)
(1/m)
-150
20 40 60 80 100 120
k = 2π λ (1/m)
Ion-acoustic “lines”
are broadened by
Landau damping
AMISR, MHO
EISCAT UHF
Sondrestrom
14

The ISR model
where:
Te/Ti
area ~ Ne
Ti/mi
Vi
From Evans, IEEE Transactions, 1969
15

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.
Uncertainties ∝
1
Number of Samples
600 samples
16
16

Components of a Pulsed Doppler Radar
_
+
+
+
_
_
Model Fit
Plasma density (N e )
Ion temperature (T i )
Electron temperature (T e )
Bulk velocity (V i )
17
17

ts spectrum. For a real waveform we have
"!
"!
Essential Mathematical Operations
!
"!
ly states that the total energy in a waveform is equal to the total
Euler:
its spectrum. For a real waveform
|U( y)| 2 we have
dy (2.26)
u(x) 2 dx = 2 ! 0
Fourier:
!
"!
) = U("y)* for the spectrum of |U( a real y)| waveform.
2 dy (2.26)
u(x) 2 dx = 2 ! 0
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
ourier transform of its power spectrum. For a waveform u with
) espectrum Wiener-Khinchine U, the power Relation spectrum is |U | 2 , and from R2 and R3
U *( Correlation: f ) is the transform of u*("t), so we have
s that the autocorrelation function of a waveform is given by the
Fourier transform of its power spectrum. For a waveform u with
e) spectrum u(t) # u*("t) U, the$ power U( f spectrum ) U *( f is ) |U = |U( | 2 , and f )| 2 from R2 (2.27) and R3
at U *( f ) is the transform of u*("t), so we have
Wiener-Khinchine Theorem:
ng out the convolution, we have
u(t) # u*("t) $ U( f ) U *( f ) = |U( f )| 2 (2.27)
!
!
u*("t) = u(t " t%)u*("t%) dt% = u(s)u(s " t) ds = r(t) 18

Dirac Delta Function
δ(t) =
⎧⎪
⎨
⎩⎪
+∞, x = 0
0, x ≠ 0
The Fourier Transform of a train of delta functions is a train of delta functions.
19

Harmonic Functions
20

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
f 0
frequency
f 0
– ( 1 ⁄ τ)
f 0 + ( 1 ⁄ τ)
The F.T. of a simple RF pulse is a sync function shifted to
Figure 5.3. Amplitude spectrum for a single pulse, or a train of
non-coherent pulses.
the carrier frequency (by convolution property and
Now consider the coherent gated CW waveform
definition of delta function).
f 3 () t = f 2 ( t – nT)
n
=
∞
∑
– ∞
f 3 () t
given by
(5.15)
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
22

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, and separation inversely
5.3. Linear Frequency Modulation Waveforms
proportional to the Interpulse Period (IPP).
Frequency or phase modulated waveforms can be used to achieve much
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
23

Thus we see that the spectrum consists of lines (which follows from
the Fourier repetitive nature Transform of the waveform) of an at intervals Long 1/T, Pulse with strengths Train given
Pulse Spectra
59
Figure 3.17 Regular train of identical RF pulses.
The F.T. of an infinite train of RF pulses is a line spectrum
Figure 3.18 Spectrum of regular RF pulse train.
with lines separated by the pulse repetition frequency.
by two sinc function envelopes centered at frequencies f and !f . As
24

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
B nn
= 2 τ
“3dB” bandwidth
Unless otherwise specified, assume
bandwidth refers to 3 dB bandwidth
25
25

Pulse-Bandwidth Connection
Shorter pulse
Larger bandwidth
26

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 T 1 T 2 , 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…
30

Transmitted signal: cos(2πf o
t)
Doppler Revisited
After return from target:
⎡ ⎛
cos⎢
2πf o ⎜ t + 2R
⎣ ⎝ 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:
⎡ ⎛
cos⎢
2π⎜
f o
+
⎣ ⎝
R = R o
+ v o
t
f o
2v o
c
− f D
⎞
⎟ t + 2πf R ⎤
o o
⎥
⎠ c ⎦
constant
f D
= −2 f o v o
c
= −2v o
λ o
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
32

Doppler Detection: Intuitive Approach
Closing on target – positive Doppler shift
e-
Transmitted
Received
φ 1
φ 2
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)
0
f o
-f o
f o +f D
Doppler shifted
cos(2π( f o
+
f D
)t)
-f o -f D
f D
cos(2πf o
t)
cos(2πf D
t)
-f o -f D
Cosine is even function, so sign of f D (and, hence, velocity) is lost.
What we need instead is:
exp( j2πf D
t) = cos(2πf D
t) + j sin(2πf D
t)
-f D f D
f o +f D
exp( j2πf D
t)
The analytic signal
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
• Collect N samples of I(t k ) and Q(t k ) from a target
• Compute the complex FFT of I(t k )+jQ(t k ), and find the maximum in the
frequency domain
• Or compute “phase slope” in time domain.
Meteor Echo I & Q
Meteor Echo Power
SNR (dB)
Voltage
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.
φ 1
φ 2
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
Alternately, the Doppler shift is well Pulse beyond Spectra the max unambiguous
Doppler defined by the Inter-Pulse Period T.
59
Figure 3.18 Spectrum of regular RF pulse train.
38

The ISR target is “overspread”
f d >>1/τ (Doppler changes significantly during one pulse)
– Must sample multiple times per pulse
– Result: Doppler can be determined from single pulse.
Range
Time
τ S
39

ISR Receiver: Doppler filter bank
Doppler Filter Bank
Doppler Range
BPF1
BPF2
BPF
f L ,f H
G
BPF3
BPF4
F1 F2 …
Choose f L and f H such that
f L f 0 +f dmax
BPF5
BPF6
Practical Problem: It is hard to make narrow band (High Q) RF filters:
Q =
f 0
f H
− f L
40

ISR Receiver: I and Q plus decorrelation
LPF
I(t i ) “in phase”
BPF
f L ,f H
G
Power
splitter
π/2 phase
shift.
LPF
Q(t i ) “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”)
ρ(τ ) = V(t)V * (t + τ )
S
FFT
Power spectrum is Fourier
Transform of the ACF
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
ΔR = cτ 2 =
c
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