### Radar Imaging with Compressed Sensing

```Radar Imaging with Compressed
Sensing
Yang Lu
April 2014
Imperial College London
Outline
• Introduction to Synthetic Aperture Radar
(SAR)
• Background of Compressed Sensing
• Reconstruct Radar Image by CS methods
Introduction to SAR
Important elements of SAR
1. Range Resolution and Azimuth Resolution
2. Chirp signal and Matched Filter
Range Resolution and Azimuth
Resolution of SAR
Range Resolution 1
• Pulse signal (constant frequency signal)
Range Resolution 2
• The resolution related with pulse width
=
×
2
: pulse width
c : speed of pulse
2 : that is a round trip
( slant range resolution)
Range Resolution 3
• If the incident angle is
• Then the ground range resolution will be
=

=
×
2
Azimuth Resolution 1
• Assume two points with same range
Can’t distinguish A from B
if they are in the radar

beam at the same time
Azimuth Resolution 2
• The azimuth resolution defined by  :
×
=  ×  =

=

R: the slant range
: the wavelength
L: the length of antenna
LFM Signal
• Linear Frequency Modulated Signal (Chirp Signal)

=
2

Where   =
1   ≤
1
2
0 ℎ
t: a time variable (fast time)
T: duration of the signal
K:is the chirp rate
So the bandwidth of the signal is:
=
Matched Filter
• The output of the a Matched Filter is:
=   ⨂ℎ
+∞
=
−∞
ℎ  −  du
: the received signal and ⨂ means convolution
ℎ  = ∗ (−)
() : the duplicated signal of the original signal and
∗ means complex conjugate
Matched Filter (example)
• If   =

2
After 0 delay, we receive the signal
− 0
=
( − 0 )2

The reference signal will be
−
∗
ℎ  =  − =
−(−)2

=
− 2

Matched Filter (example)
• Output signal of the matched filter
≈ (( − 0 ))
1
−
0
1
+
0
0
So 3dB width of the main lobe=
0.886

≈
1

Range resolution improved
• The range resolution improved
Now we can distinguish B from C
Range resolution improved
Original ground range resolution:
=

=
×
2
Now replace  with 3dB main lobe width=
1

Finally, the improved ground range resolution
will be :

=
=
=
2 2
Phase difference
2 × 2
=

: phase difference between
the transmitted and the
2: the distance (round trip)
: the wavelength of the
transmitted signal
SAR Azimuth Resolution
• The phase change of the radar signal will be
4()
=

By Pythagorean theorem
=
02 + ()2
2 2
≈ 0 +
20
: a time variable (slow time)
:the speed of plane
Polarimetry (J.V. Zyl and Y. Kim)
SAR Azimuth Resolution
Substitute   ≈ 0 +
2 2
20
4() 40 2 2 2
=
≈
+

0
The instantaneous frequency change of this
1
2 2
signal is   =
=

2

0
Which also can be considered as LFM signal
And the total time  ≈
0

=
0

SAR Azimuth Resolution
The  =   =
2

The time resolution will be
1

=

2
So the azimuth resolution (in distance) will be
1

=  ×
=
2
2D signal of the target
• One target have two equations-one is in the
range direction (variable: fast time t) and
another is in the azimuth direction
(variable :slow time )
• If consider the signal on the two direction
simultaneously, that will be a 2-dimensional
signal with variable t and .
2D signal of the target
•  , η = 0  (
−40
2()
−
)

()

×
0 : a complex constant

t = rect( )

()
2
() ≈

×
(−
2() 2
)

()
0 : the centre frequency (carrier frequency)
Signal Energy
2D signal space
• The received signals are stored in the signal
space
Digital Processing of Synthetic Aperture Radar Data:
Algorithms and Implementation ( G.Cumming and H.Wong)
SAR impulse response
• If we ignore the constant 0 of  , η , we get the
impulse response of SAR sensor:
• ℎ , η =  ( −
−40
()

2()
)

×
(−
2() 2
)

×
The received signal of the ground model can be built as
the convolution of the ground reflectivity with the SAR
impulse response (with additive white noise):
,  =  ,  ⊗ ℎ ,  + (, )
•  ,  =  ,  ⊗ ℎ ,  + (, )
Radar algorithms are trying to obtain the ground
reflectivity function  ,  based on the
Range-Doppler Algorithm
Chirp scaling algorithm
Omega-K algorithms
Background of Compressed Sensing
Assume an N-dimensional signal  has a Ksparse representation () in the basis Ψ
= Ψ
If we have a measurement matrix Φ ( × ) to
measure and encode the linear projection of the
signal we get measurements
= Φ = ΦΨ
If  < , there will be enormous possible
solutions. And we want the sparsest one.
Compressed Sensing
• CS theory tells us that when the matrix A=ΦΨ
has the Restricted Isometry Property (RIP), then it
is indeed possible to recover the K-sparse signal
from a set of measurement
= ((/))
But RIP condition is hard to check. An alternative
way is to measure the mutual coherence
,
=   = max
≠  2
2
denotes the  ℎ column of matrix A
Compressive Radar Imaging (R. Baraniuk and P.Steeghs)
Compressed Sensing
• We want  to be small (incoherence)
CS theory has shown that many random
measurement matrices are universal in the
sense that they are incoherent with any fixed
basis Ψ with high probability
Compressive Radar Imaging (R. Baraniuk and P.Steeghs)
methods
• When RIP/incoherency holds, the signal  can
be recovered exactly from  by solving an 1
minimization problem as:
=   1 . .  = ΦΨ
methods
• If the measurement matrix Φ is a causal,
quasi-Toeplitz matrix , the results also show
good performance.
(Right shift distance=

)
Causal, quasi-Toeplitz Matrix
(Example)

If M=4, N=8 then right shift distance D=
=2
1 0
0 0
0 0 0 0
3 2 1 0
0 0 0 0
5 4 3 2 1 0
0 0
7 6 5 4 3 2 1 0
is the  ℎ element of a pseudo-random
sequence
Causal, quasi-Toeplitz Matrix
The measurements of the signal will be
= Φ

=
−  ()
=1
,  =  ,  ⊗ ℎ ,  + (, )
For simplicity, just consider 1D range imaging model and
ignore the noise
=   ⊗ ℎ
Under this condition, ℎ  can be considered as the
=
=
2

() ( − )
is the time delay. A is the attenuation.
Assume, the target reflectivity function () is ksparse in some basis.
The PN or Chirp signals transmitted as radar
waveforms  (t) form a dictionary that is
incoherent with the time, frequency and timefrequency bases.
• Let the transmitted radar signal be the PN
signal
• The target reflectivity generated from N
Nyquist-rate samples () n=1,…,N via   =

∆
where 0 ≤  ≤ ∆
• The PN signal generated from a N-length
Bernoulli ∓1 vector   via   =

∆
=  () ( − )
And we sample it every ∆ second
=   |=×∆
∆
=
=
(∆ − )
0
∆

0

=

∆
∆
( − )
( − )
=1

=
(−1)∆
−  ()
=1
()
The target reflectivity function can be recovered
by using an OMP greedy algorithm

y()
Compressive Radar Imaging (R. Baraniuk and P.Steeghs)
Another Example (2-dimensional)
The 2D received signal of a point target
, η = 0  (
2()
−
)

−40
()

×
(−
2() 2
)

×
If ignore the antenna pattern   =1, 0  ( −
2()
)

be a constant ( ) which is the radar cross
section of point target
Another Example (2-dimensional)
The approximate received signal will be
−40
(, )

2(, ) 2
(−
)

, η,  = ( )
×
For a measurement scene Ω ( =  ×  ) , the recorded echo
signal will be

, η =
, η,
=1
: samples on the azimuth direction (slow time samples)
: samples on the range direction (fast time samples)
i: the ℎ point target in the scene
Discrete format of the scene

, η =

=1

=
=1
−40
(,)

×
2(,)
(−  )2

2(,)
(,)
(−  )2 −40

−∅ (,)
=
=1
where
−∅(,) =
(−
2(,) 2
(,)
) −40

Discrete format of the scene

−∅ (,) = (,)
, η =
=1
(,)
−∅1 (,)
−∅2 (,)

=
⋮
−∅ (,)
(1,1)
(1,1)
(2,1)
⋮
= (,1)
(2,1)
⋮
= (,1)
(1,2)
⋮
(,)
(1,2)
⋮
(,)
Discrete format of the scene
=  +
: complete measurement matrix of SAR echo signal
According to CS theory, we only need a small set of  to
successfully recover the sparse signal  with high probability.
Randomly select  = ((/)) rows of matrix A by using
random selection matrix Φ
Discrete format of the scene
We assume that  have a sparse representation in a certain
basisΨ (for example, a set of K point targets corresponds to a
sparse sum of delta functions as in  =
=1  ( − )),
then we have
=  +  =  +  =  +
Where Θ=
, Θ and  are complex
=  +
+   =   +     +
= Re  Re  − Im  Im()
+     +
So we have
=     −   ()
=     +   ()
We define signal ,  and  as

=
, =

( )
−

, =

Final Format
=  +
Sparest solution can be solved by 1 norm
minimization
min   1 . .  −  2 <
Simulation Results
D
ERS Ship Image
Results
SNR=20dB
Noise free
RD algorithm
d
CS algorithm
CS algorithm
SNR=10dB
Reference
•
•
•
•
•
•