Time frequency analysis based techniques for radar signal

processing

Igor Djurović, LJubiša Stanković, Miloš Daković

Electrical Engineering Department,

University of Montenegro

Thayananthan Thayaparan,

Department of Defense, Canada

Background

• From 2003, the Research Center for Signals and Systems, from the University of Montenegro (head of the centerp Prof. Stanković) has been engagged in several projects related to the radar signal processing and application of the time-frequency analysis techniques in this field.

• Projects were funded by Department of defense Canada and we collaborated with Dr Thayanathan Thayaparan.

Topics of projects

• We analyzed multiple topics:– HF radar imaging in clutter environment based on the TF analysis

and decomposition techniques.

– Modeling radar images.

– Analysis and removal of micro-Doppler effects in ISAR images.

– Focusing and reducing motion caused effects in SAR and ISAR images.

– High-resolution radar systems.

– Noise waveform radar systems.

– SAR imaging of ships and other vessels.

– ...

My topics

• I were engaged in two main subtopics:– Removing and analysis of micro-Doppler effect from radar

images;

– Focusing and reducing motion caused effects in SAR and ISAR images.

• In this presentation three issues are highlighted:– Decomposition of radar signals by using the TF representations

with application to radar imaging in clutter environment (this technique is proposed by Dr Miloš Daković).

– Model of helicopter target developed in analysis of micro-Doppler effect.

– Focusing and reducing motion caused effects in SAR and ISAR images.

I. PARAMETER ESTIMATION BY USING DECOMPOSITION OF SIGNALS IN TIME-FREQUENCY DOMAIN WITH APPLICATION

TO RADAR SIGNALS

Wigner distribution (WD)

• Bilinear TF representation– Highly concentrated in the TF plane– Ideal representation of the linear FM signals

• Drawbacks:– cross-terms– over-sampling

• Pseudo-Wigner distribution (PWD)– window along τ coordinate– cross-terms reducing for components located in different time

intervals

*( , ) ( ) ( )2 2

jWD t x t x t e d

Wigner and pseudo-Wigner TFD• Signal with 6 components.• Auto-terms denoted with ai, cross-terms cij .• PWD reduces some cross-terms.• Cross-terms caused by signals that appear in the

same instant could not be removed.

Wigner distribution

time

freq

uenc

y a1

a2

a3

a4

c12

c13

c23

c24

c34

c14

-100 -50 0 50 100-1.5

-1

-0.5

0

0.5

1

1.5Pseudo WD, window width 96

time

freq

uenc

y a1

a2

a3

a4

c12

c23

c24

c34

-100 -50 0 50 100-1.5

-1

-0.5

0

0.5

1

1.5Pseudo WD, window width 16

timefr

eque

ncy a

1

a2

a3

a4

c34

-100 -50 0 50 100-1.5

-1

-0.5

0

0.5

1

1.5

S-method

• Based on the short-time Fourier transform (STFT)• Special cases:

– P(θ)=1the PWD with window function w(τ)w*(-τ)– P(θ)=δ(τ)the spectrogram

• Suitable selection of the window P(θ) width reduces cross-terms with the same quality of the auto-terms as in the WD

*1( , ) ( ) ( , ) ( , )

2 2 2SM t P STFT t STFT t d

( , ) ( ) ( ) jSTFT t w x t e d

S-method• Signal with 6 short components• STFT with rectangular window• S-method eliminates cross terms on different frequencies• Cross-term causes by signal on the same frequency

remains.

Spectrogram

time

freq

uenc

y

a1

a2

a3

a4

-100 -50 0 50 100

-1.5

-1

-0.5

0

0.5

1

1.5

S-method, L=16

time

freq

uenc

y

a1

a2

a3

a4

-100 -50 0 50 100

-1.5

-1

-0.5

0

0.5

1

1.5

S-method, L=32

time

freq

uenc

y

a1

a2

a3

a4

-100 -50 0 50 100

-1.5

-1

-0.5

0

0.5

1

1.5

S-method - discrete realization

• It can be realized in recursive manner. Rectangular window of width 2L+1 is used.

• Initial step in calculation is the spectrogram for L=0.

2 *

1

( , ) | ( , ) | 2Re ( , ) ( , )L

p

SM n k STFT n k STFT n k p STFT n k p

*

1( , ) ( , ) 2Re[ ( , ) ( , )]L LSM n k SM n k STFT n k L STFT n k L

Signal parameters estimation

• Estimation of signal amplitude

• Instantaneous frequency estimation

• Signal detection for high noise environment

• Estimating number of components

• Analysis of components in multicomponent signals

Decomposition of multicomponent signals

• The inverse of the WD can be calculated by eigendecomposition of appropriate matrix obtained from the WD.

• S-method of multicomponent signal can be equal to sum of the WD of components.

• Eigenvalue decomposition of matrix obtained by using the S-method produces the eigenvectors that represent normalized signal components.

• Eigenvalues contain information on energy of components.

Discrete WD is defined as:

Inversion of the WD

4/ 2* 1

/ 2

( , ) ( ) ( )N j mk

N

m N

WD n k f n m f n m e

*( ) ( ) ( ( , ))k m

f n m f n m IFFT WD n k

1n m n 2n m n

1 2

* 1 21 2 1 2

2

( , ) ( ) ( ) ( , )2k m n n

m

n nR n n f n f n IFFT WD k

Inverse FT produces:

Now we can form matrix R with elements:

Eigenvector decompositionEigenvector decomposition of R gives:

1*

1

N

n n nn

R u uwhere λn are eigenvalues

while un are eigenvectors.

Matrix obtained based on the WD is:* R f f

From this conclusions it follows that:– Matrix R has one non-zero eigenvalue λ1 – Eigenvector u1 is proportional to analyzed signal– Reconstructed signal is:

1 1( ) ( )jf n e u n

S-method decomposition

• S-method could be equal to the sum of WDs of signal components.

1

( ) ( )M

pp

f n f n

1

( , ) ( , )M

pp

SM n k WD n k

• Matrix R is now: *

1

M

p pp

R f f

• Eigenvectors of R are proportional to signal components under condition that signal components are linearly independent and that we have no multiple eigenvalues.

Example: Decomposition of signal with multiple components

S-method

time

freq

uenc

y

0 10 20 30 400

20

40

60

80

100

120Eigenvalues for SM

Wigner distribution

time

frek

quen

cy

0 5 10 15 20 25 30 35 400

100

200

300

400Eigenvalues for WD

number

Spectrogram

time

frek

quen

cy

0 5 10 15 20 25 30 35 400

5

10

15

20

25

number

Eigenvalues for SPEC

TFR of eigenvectors for the S-method decomposition

3127109

8765

4321

Problems in decomposition

• n=(n1+n2)/2 and m=(n1-n2)/2 can introduce non-integer index (to avoid it, the TFR is oversampled).

• For very close components in the TF plane it is not possible to select L in the WD that the S-method is equal to sum of WDs. Then– Analyzed component can be contained in several

eigenvectors.

– Eigenvectors are orthogonal and it is possible to sum their TFRs.

– We need some criterion for selection of eigenvectors (or eigenvalues).

Detection of deterministic signal

frequency

Fourier transform

frequency

Fuourier transform

timefr

eque

ncy

TFR

time

freq

uenc

y

TFR

Signal+Signal+NoiseNoise

NoiseNoise

FFourier transform and TFRourier transform and TFR

TF signal detector• TFR of signal s(n) is S(n,k). • Assume that signal s(n) has deterministic component

x(n), with slowly varying amplitude and IF.• Form set of paths in the TF plane

Π={π(n):|π(n)-π(n-1)|<D}.• Calculate sum of S(n,k) values along the paths from and calculate maximum.

• Obtained maximum is compared with threshold value and based on this comparison we are making decision about existence of deterministic signal in the mixture.

Problems in TF detector realization• Decreasing of number of paths in Π

– We propose strategy for decreasing the number of paths in that decreases search complexity but it increases probability of error in detection.

– Considered paths are only those that are on the local maximum of the TFR

• Detection threshold– Proportional to noise variance (noise variance estimation is required

step).– Threshold depends on the used TFR.– Threshold depends on the “false alarm” probability.– For a given TFR and “false alarm” probability detector threshold can

be efficiently determined using statistical techniques.

Radar signals

• Radar signals are non-stationary and TFR can produce very favorable results for this signal type.

• Algorithm for decomposition has been applied for separating useful signal (radar targets return) from noise and clutter.

• The proposed algorithm has been tested on simulated and experimentally obtained signals.

• It has been shown that the TFR decomposition gives very accurate results even for high noise environment.

Model of radar signals• We derived analytical model of radar signal reflected

from the moving target.• Radar emits sequence of M linear frequency modulated

signals.• Reflected signal is delayed for 2d/c with respect to

emitted signal.• Frequency shift is proportional to target velocity.• Based on the derived model we analyzed resolution of

radar system in estimation of position and distance of targets.

• Radar clutter is also modeled.

Experimental data

• High frequency surface wave radar (HFSWR) is used in experiments.

• Target was King Air 200 above the sea on small altitude (emphatic radar clutter)– Operating frequency: 5.672 MHz– Bandwidth: 125 kHz– Pulse repetition frequency: 9.17762 Hz– Number of pulses: 256– Coherent integration time: 27.89 s – Number of positions: 69– Total experiment duration: 33 min.

• The main source of clutter were signals reflected from sea surface.

Decomposition of experimental data: Algorithm

1. Calculate the STFT of oversampled radar signal.2. Calculate S-method for a given L.3. Form matrix R.4. Perform eigendecomposition of R5. Calculate TFR of eigenvectors and decide if it is signal

caused by target or by clutter (or noise).6. If radar target is not detected repeat step 2 with smaller L.7. TFR of target is obtained as a sum of TFRs of

eigenvectors that correspond to the detected target components.

Selection of Eigenvectors

• Criterion based on the concentration measure:– S-method of the eigenvector– S-method of target is cross-term free– Clutter signal has emphatic cross-terms– Cross-terms have oscillatory nature

• Criterion based on amplitude of target signal:– Signal reflected from target has slowly varying amplitude– Based on the experimental data we concluded that clutter

components have fast varying amplitude

Example 1: Constant target velocity

TFR of received signal (log)

time

freq

uenc

y

TFR of target signal

time

freq

uenc

y

-0.5 0 0.5

Fourier transform

normalized frequency

ampl

itude

0 MTr

-100

-50

0

50

100

Estimated target velocity

time

0 5 10 15 200

0.2

0.4

0.6

0.8

1Eigenvalue

number

0 5 10 15 200

2

4

6

8measure

number

Example 1: TFR of eigenvectors

1211109

8765

4321

Example 2: Target with nonstationary motion

TFR of received signal (log)

time

freq

uenc

y

TFR of target signal

time

freq

uenc

y

-0.5 0 0.5

Fourier transform

normalized frequency

ampl

itude

0 MTr

-100

-50

0

50

100

Target velocity

time

0 5 10 15 200

0.2

0.4

0.6

0.8

1Eigenvalues

number

0 5 10 15 200

1

2

3

4

5

6measure

number

Example 2: TFR of eigenvectors

1211109

8765

4321

Example 3: Nonstationary motion with small velocity

TFR of received signal (log)

time

freq

uenc

y

TFR of target signal

time

freq

uenc

y

-0.5 0 0.5

Fourier transform

normalized frequency

ampl

itude

0 MTr

-100

-50

0

50

100

target velocity

time

0 5 10 15 200

0.2

0.4

0.6

0.8

1Eigenvalues

number

0 5 10 15 200

2

4

6

8

10

12measure

number

Example 3: TFR of eigenvectors

1211109

8765

4321

Example of Signal DetectionExperimental data, SNR=-8 dB

TFR of filtered signal

0 2 4 6 8 10 12 140

0.01

0.02

0.03

0.04

0.05

0.06

0.07Criterion – TF detection

number of paths0 100 200 300 400 500

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35Fourier transform and referent level

optimal path

Conclusion

• It is developed theoretical model of decomposition of multicomponent signal to separate signal components.

• The algorithm for decomposition has been applied to simulated and real signals.

• Noise influence to decomposition has been analyzed.

• Method for detection of deterministic signals in heavy noise based on the TFR has been developed.

Published Papers on this Topic

• LJ. Stanković, T. Thayaparan, M. Daković, "Signal Decomposition by Using the S-method With Application to the Analysis of HF Radar Signals in Sea-Clutter," IEEE Trans. on Signal Processing, Vol.54, No.11, Nov. 2006. 

• LJubisa Stankovic, Thayananthan Thayaparan, Milos Dakovic, “Algorithm for signal decomposition by using the S-method”, 13th EUSIPCO Conference, Antalya, Turkey.

II. SEPARATION OF MICRO-DOPPLER EFFECT AND

STATIONARY BODY FOR HELICOPTER SIGNALS BY

USING THE SPECTROGRAM AND L-STATISTICS

Model of helicopter signal

• Modeled effects (model of UH-1D Iroquois):– Main body – fuselage (stationary reflector points)

– Main rotor

– Main rotor flashes

– Tail rotor flashes

Signal is sampled with t=1/48000s and considered within interval of 400ms.

Rigid body is modeled as sinusoidal components at: -10.3kHz, -2.5kHz, 2.3kHz and 2.7kHz. In addition components at 0.4kHz are modulated time tones added to the data tape.

Model of moving parts

• Main rotor:

• Flashes:

AROT=19kHz

TROT=175ms

TTAIL=35.8ms

Filters for flashes

• hFL_M(t) and hFL_T(t) are impulse responses of filters with frequency responses:

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-100

-50

0

50

100

-20 -15 -10 -5 0 5 10 15 200

1

2

3

4x 10

4

real [x(t)]

t[s]

|X(f)|

f[kHz]

Simulated signal

Fourier transform

Spectrogram and L-statistics

• Separation of the m-D effect and stationary body influence will be performed by using the spectrogram:

• For a given frequency spectrogram samples are sorted from the smallest toward the bigger: S(n)() S(n+1)() where S(n)({STFT(t, ), for a given }.

2

( , ) ( ) ( )exp( )SPEC t x t w j d

Spectrogram

smallest samples (average of smallest samples for a given frequency) used for detection of stationary patterns.

region used for detection of tail blades (stationary patterns are removed)

detection of effects associated with main blades

detection of stationary patterns

detection of tail blades

detection of main blades

stationary signal pattern main rotor flashes

tail rotor flashes rotating blades

Separation requires additional processing in time domain and pattern recognition tools currently under investigation.

Characteristics of the algorithm• Current setup with proposed algorithm parameters and for given

example works accurate for SNR10dB.

• Two ingredients of the algorithm:– spectrogram (common and its implementation could be assumed to be fast)

– sorting of samples (fast sorting procedures such as quicksort or insertion sort should be employed).

• There is a room for improvement of the algorithm in terms of accuracy and adaptivity but all kind of optimization requires training on real data.

• Proposed example is simulated according to:S. L. Marple: "Special time-frequency analysis of helicopter Doppler radar data", in Time-Frequency Signal Analysis and Processing, ed. B. Boashash, Elsevier 2004.

Published Paper on this Topic

• LJ. Stanković, T. Thayaparan, I. Djurović: "Time-frequency representation based approach for separation of target rigid body and micro-Doppler effects in ISAR imaging", IEEE Transactions on Aerospace and Electronics, accepted for publication.

III. IMPROVING RADAR IMAGES FOR SAR AND

ISAR SYSTEMS

Introduction

• ISAR (Inverse Synthetic Aperture Radar) images are commonly obtained by a 2D Fourier transform of the dechirped reflected signal.

• Longer time interval gives better image resolution.• Target points with high velocity changes within

the considered time interval are blurred.• By using time-frequency analysis methods

sharpness of ISAR images can be improved without reducing resolution.

ISAR model

Analytic CW Radar Signal Model

ConsiderConsider radar signal radar signal model in the form ofmodel in the form of series of series of M chirps: chirps:

Each chirp is Each chirp is a a linear frequency modulated signal:linear frequency modulated signal:

ISAR imaging

The ISAR image The ISAR image P(m’,n’) is obtainedis obtained by by 2D 2D DDFTFT

DDemodulated emodulated filtered filtered received signalreceived signal component is of the form component is of the form

Fourier transform of the Doppler part

Consider Doppler part of the received signalConsider Doppler part of the received signal::

aand its Fourier transformnd its Fourier transform::

where where w(t) is window defining the considered is window defining the considered Coherent Intergration Time (Coherent Intergration Time (CIT).).

Denote Fourier transform of the window Denote Fourier transform of the window w(t) by by W(ω)

Time varying distance Taylor expansion of the time varying distanceTaylor expansion of the time varying distance

reduces reduces Fourier transformFourier transform to to

with spreading factorwith spreading factor

SAR Model

SAR Model

• SAR model is similar to the ISAR with difference that it is assumed that radar is moving and that target is non-moving.

• Motion of target causes spreading of components but also dislocation from the proper position.

• We will demonstrate technique for SAR imaging based on the polynomial FT with couple comments and simulations for ISAR images.

PFT – some basic informations

• The polynomial FT (PFT) is introduced several times in science.

• Detailed statistical study has been provided by Katkovnik.

• It is defined as:

• For polynomial phase signal:

the PFT is ideally concentrated on =a1, i=ai, i=2,...,k.

22 2( ; ,..., ) ( )exp( ... )k

k kt

X x t j t j t j t

1( ) exp

k lll

x t A j a t

21 2

( , ,..., )ˆ ˆ( ,..., ) arg max | ( ; ,..., ) |

kk ka a X

PFT - Introduction

• Since the PFT can be calculationally demanding we will consider the PFT of the second order:

• We assume that the second-order nonlinearity is enough for compensating motion caused effects but also we propose the order adaptive PFT form in the case that we need to increase the PFT order.

2( ; ) ( )exp( )t

X x t j t j t

Notation

• Set of received chirps will be denoted as: s0(t,m).

• Standard radar image obtained by the 2D FT is:

0

0

( , ) ( , )exp( )

( , )exp( )

t m t mt m

t mm

S s t m j t j m

S m j m

where

0 0( , ) ( , )exp( )t tt

S m s t m j t

SAR imaging algorithm

• For each m– Let (t,m)=s0(t,m) and

I=1 and .– While radar return (t,m) contains significant energy

• Calculate SI(t,m) =R(t,m) for (t,m) representing well-focused component (target) and SI(t,m)=0 otherwise.

• Non-focused components are updated as: R(t,m)SI(t,m)- R(t,m). Then we calculate:

• Set II+1.• For (for various chirp rates from set )

– Calculate:

• Endfor

( , ) ( , )exp( )t ttR m t m j t

ˆ 0I 1

2

2ˆ( , ) { ( , )exp( )}t It m IFT R m j t

3

2( , ) ( , )exp( )t tt

R m t m j t j t

SAR Imaging algorithm

• Estimate the chirp-rate of the radar return:

– Endwhile

• Endfor

• Radar image is calculated as:

( , )ˆ ˆ( , ) arg max | ( , ) |

It

I t tR m

ˆ( , ) ( , )It tR m R m

1

( , ) ( , )exp( )I

t m J t mJ m

S S m j m

Comments on the algorithm1. A technique for determination of chirp returns with

significant energy has been developed. This technique works accurately for images with small noise and for some noise environments. Chirps with small energy are not processed since it is assumed that they have not moving components.

2. Technique for determination of well-focused components has been developed. When we cannot detect highly concentrated component we can use the third order PFT to get better concentration:

2 3ˆ , ˆ( , ) ( , )exp( )

I t t It

R m t m j t j t j t ˆ ,

( , )

ˆ ˆ( , ) arg max | ( , ) |I I

tI t tR m

2 3ˆˆ( , ) { ( , )exp( )}t I It m IFT R m j t j t

Comments on the algorithm

3. Set of chirp rates can be selected based on information of maximal velocity and acceleration of targets. Chirp-rates in the set could be non-equidistantly spaced.

4. This technique does not solve problem of displacement radar targets from proper position due to motion caused effects. For handling this problem some classical techniques for motion estimation from video-signals processing are commonly used. The PFT imaging does not require the estimation of chirp rates for each frame since it can be assumed that the chirp rates varies very slowly.

Examples

• We considered model of Environment Canada’s airborne CV 580 SAR system.– Operating frequency 5.3GHz (C-Band of the CV 580 SAR).

– Bandwidth 25MHz.

– Pulse repetition time Tr=1/300s.

– M=256 pulses within one revisit.

– Platform (aircraft) velocity 130m/s.

– Altitude 6km.

• 8 targets: 4 stationary and 4 nonstationary

• Two trials: non-noisy trial and noisy trial.

Standard imaginingAll target are non-moving 4 moving targets

PFT imaging Advanced TFR imaging but with spurious cross-terms

Standard imagining of noise imageAll target are non-moving 4 moving targets

PFT imaging

Noise only chirps are removed from the image

Advanced TFR imaging

Application to ISAR

• This technique can be applied to ISAR systems but with couple differences.

• Radar target in the case of the ISAR radars could have several close reflectors on quite small distance.

• It can happen that all reflectors of the target have the same chirp rates but for some complicate maneuvers chirp rates could be quite different.

• Some combining of results achieved for various chirps is here desirable.

Application to ISAR

• Other differences in the ISAR radars are velocity of target and different radar operating frequency and bandwidths in this case.

• All these differences cause that some more robust concentration measure is required in the PFT technique applied on the ISAR and that some combination of information related to the chirp rates between adjacent radar chirps is also discussed.

• Details of this research are published in: I. Djurović, T. Thayaparan, LJ. Stanković: "Adaptive Local Polynomial Fourier Transform in ISAR", Journal of Applied Signal Processing, vol. 2006, Article ID 36093, 2006.

• Here we will demonstrate some of results.

Simulated example

Standard FT based radar image

Adaptive chirp rate and filtered adaptive chirp-rate

PFT based image

B727 Image

Simulated image with complicated motion

Algorithm based on radar image segmentation applied.

DFT based image

Segmentation based on two values of the algorithm parameter

Simulated image with complicated motion

Algorithm based on radar image segmentation with adaptive selection of segmentation algorithm parameter applied.