1 LV Distribution, Performance and Applications Presented by: Bi Guoan School of Electrical and...

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LV Distribution, Performance and Applications

Presented by: Bi Guoan

School of Electrical and Electronic EngineeringNanyang Technological University

Singapore

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Outline

Introduction

Lv distribution (LVD)

– Definition and properties

– Performances

Application

– SAR image processing

Conclusion

Introduction Most real signals contain frequencies varying with time – known as

non-stationary or time-varying Fourier transform tells us the frequencies in the signals, but not

how they change with time. Time frequency signal analysis has been useful to truthfully reveal

the useful information matching with our observations, Example.

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Introduction A time–frequency distribution should have the following

desirable features:− High signal concentration for accurate analysis and

interpretation. − No cross-term to avoid confusing components duo to the

imperfection of the processing methods

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− Desirable mathematical properties ensure such methods beneficial to real-life applications.

− Lower computational complexity to ensure the time needed to process the signal for real-time implementations.

Introduction Well known time frequency representations (TFRs):

− STFT (Gabor transform): simple with low resolution limited to uncertainty principle (TFR).

− Local Polynomial periodogram (LPP): requires much more computation for estimation of polynomial coefficient (TFR)

− WVD or smoothed pseudo WVD: providing best signal concentration, but suffering from the cross-term for multi-component signals

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2 30 1 2( ...)j t t te

− Fractional Fourier transform: a rotation operation on TFRs to find the best signal concentration

− Wavelet transform: no direct physical meaning from the results

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Introduction

The frequencies of time varying signals may change in various ways.

Within a short time period, we assume the frequencies are changing linearly − Linear frequency modulated (LFM) signal:

− Two types of signal representations:

• Time-frequency representations (TFRs)

• Frequency chirp rate representations (FCRs)

2

1 1

exp 2K K

k k k kk k

x t x t A j f t j t

centroid frequency

chirp rate

Introduction

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− Reduced resolution and cross-terms

− Trade-off between auto-terms and cross-terms

− Better resolution than STFT

− Some small cross-terms

− No cross-term,− poor resolution

− High resolution− Heavy cross-terms

Introduction

For critical environments, such as low SNR and multiple signal components, most of these methods are not usable.

Other suitable methods are required to obtain a balanced performance among a few contradictory requirements− High signal concentration with strong noise environment− Low cross-terms among different components− High signal resolution and − Low computation complexity

LV distribution (LVD) is a new algorithm achieving the above requirements

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Outline

Introduction


– Definition and properties

– Performances

Application


Conclusion

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The LVD*

– Combines the desirable properties of linear and bilinear methods for dealing with LFM signals

– Provides peaks for each LFM components in the FCR domain

– Main merits:

• High resolution/concentration

• Asymptotic linear (very small cross-terms can be ignored)

• Reasonable computational complexity

*X. Lv, G. Bi, C. Wan, and M. Xing, Lv's distribution: Principle, implementation, properties, and performance, IEEE Transactions on Signal Processing, vol. 59, no. 8, pp. 3576--3591, Aug. 2011.

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Definition : a parametric symmetric instantaneous autocorrelation function (PSIAF) of the input signal x(t) :

A scaling operator of a phase function G with respect to (t, ) as

*

2

1

1

1 1

,2 2

exp 2 2

,i j j i

Cx

K

k k kk

K KC Cx x x x

i j i

a aR t x t x t

A j f a j a t

R t R t

: constant time-delay a

cross-terms

, ,ntG t G

h a

h : a scaling factor

nt a ht

t and are coupled together, leading to cross term effects

auto-terms

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Definition of the LVD− Perform the scaling operation on the PSIAF and obtain

− Take the 2-D FT of the previous equation with respect to and tn

2

1

1

1 1

, exp 2 [ ]

, ,i j j i

KC kx k k n

k

K KC Cx x x x

i j i

R t A j f a th

R t R t

1

1 1 1

, , , , ,n k i j

K K KC

x t x x x xk i j i

L f F F R t L f L f

2 2,k

j af kx k kL f A e f f

h

auto-term cross-term

t and are decoupled

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Definition of the LVDFor a multi-component LFM signal, the LVD is asymptotic linear as

1 1

, , .k

K K

k x xk k

x t x t L f L f

The LVD has:•three components;•very little cross-terms;•high resolution.

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Properties of the LVD– Asymptotic linear:

– Frequency and chirp rate shift: if

we have

– Resolution:

– Others: scaling, translation, convolution, etc.

1 2

2 2

1 1 2 2 1 2, ,x xLVD c x t c x t c LVD f c LVD f

22 1 0 0 0exp 2 expx t x t t j f t j t

0 0

1

2 22 0 0 0,j t j f

xLVD x t e e LVD f t f

1 2,f

T T

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Performance of the LVD– Representation comparison

LVD

RWTRAT

FrFT

LVD: no obvious cross-terms high concentration

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Performance of the LVD– Resolution

Three LFM components with differences in centroid frequencies by 2 Hz and in chirp rates by 2 Hz/s ---- only the LVD can distinguish them

LVD: most narrow main-lobe width; highest resolution.

?

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Performance of the LVD– Representation errors: mean square error (MSE)

3 2 2

, , 22 4 4

147 36 294,

98 72s

LVD f LVD

s

N f NMSE MSE

N SNRN f SNR

MSEs of the LVD are lowest, and approach to the Cramer-Rao lower bounds (CRLBs) at lowest SNR

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Performance of the LVD– LFM signal Detection:

The threshold in LVD is

and theoretical detection probability is

faP 1 1l l l fath m P

: false alarm probability

: cumulative distribution functionof a standard normal distribution

x

1 l lsd

l

th mP

ls l sm m L

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Performance of the LVD– LFM signal Detection:

-14 -9 -6

Input sample length N = 512

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Performance of the LVD– Computational complexity

They are all in the same order of , indicating that LVD provides better performance without introducing more computational complexity.

2 22

2 22

2 22

: 2 log

: 4 log

: 2 log

s s s

s s s

s s s s

PFT N N N

FrFT N N N

LVD N N N N

Ns : input length

22logs sO N N

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Outline

Introduction


– Definition, properties

– Performances

Application


Conclusion

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Synthetic Aperture Radar (SAR) Fundermental

swathswath

heightheight

synthetic aperturesynthetic aperture

Time-Frequency Based Moving Target Imaging

After clutter suppression, the phase history of each range cell containing one or multiple moving targets can be modeled as:

1

2

0

exp 2I

i i ii

s t A j f t j t

iA is the amplitude if is the Doppler centroid frequency

i is the Doppler chirp rateI is the target number

• Due to the target velocity in along- and cross-track resulting in Doppler frequency shift and Doppler frequency modulation, respectively, the moving target image will be shifted and unfocused.

• The moving target phase history has the linear frequency modulation (LFM) form. Time-frequency analysis is more suitable.

• Under the relationship between the centroid frequency and chirp rate and the velocity in along- and cross-track, the motion parameters can be expressed by time-frequency distribution (TFD).

– L. Yang, G. Bi, M. Xing and L. Zhang, “Airborne SAR moving target signatures and imagery based on LVD,” To appear in IEEE Transactions on Geoscience and Remote Sensing, 2015.

Azimuth (m)

Rang

e (m

)

-150 -100 -50 0 50 100 150

-70

-60

-50

-40

-30

-20

-10

0

10

20

30

MT1

MT2 MT3

MT4MT5

ST1

ST2

Azimuth samples

Rang

e sa

mple

s

10 20 30 40 50 60

5

10

15

20

25

30

Azimuth samples

Rang

e sa

mple

s

10 20 30 40 50 60

5

10

15

20

25

30

Fig.1 (a)

Point Moving Targets Simulation(MT1-MT5 are moving targets; ST1-ST2 are stationary targets; MT1-MT3 are located in the same range cell, and MT4-MT5 are located in the other same range cell) (Green rectangles are the shifted position of the moving targets; red crosses are the true position of the moving targets) (Fig. 1(a) is the simulation result; Fig. 1(b) is the defocused moving targets image of MT4; Fig. 1(c) is the refocused moving targets image of MT4)

Since the velocity of the moving target in cross-track will induce the position shift in azimuth direction, and the velocity in along-track will cause defocusing in azimuth direction, therefore, by applying the LVD for the SAR moving target imaging, the targets are relocated to the true (original) position and refocused to obtain the true SAR moving targets.

Although there are more than one target are located in the two separated range cell, the SAR moving targets are focused into high quality.

Fig.1 (c)

Fig.1 (b)

Fig.1 (a)Fig.1 (a)

Fig. 1(a) is the SAR image with the moving targets by the raw SAR data; Fig. 1(b) is the defocused moving targets image of T6; Fig. 1(c) is the refocused moving targets image of T6;

Fig.1 (b) Fig.1 (c)

Fig.1 (b) Fig.1 (c)

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Thank you!

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