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Introduction Radar system Resolution Pulse-Doppler radar Radar equation Target detection theory

RADAR signal processingRadar basics

Dr. Ir. Xavier Neyt

Associate Professor

Communication, Information, Systems and Sensors Departement

Royal Military Academy

April, 2012
http://find/http://goback/


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Definition

RadarTransmits an electromagnetic wave

The wave is reflected by objects

Reflection is received

Compares transmitted and reflected wavein amplitude and in phase

Requirement

The radar need to be coherent there is a coherency between the

transmitted and the received wave.

Configuration

monostatic bistatic

moving non-moving
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Measured quantities

Measurements

Time-delay Distance (Range)

Frequency shift Radial velocity

Amplitude Radar cross section (Detection)

Other quantities

Change in polarization Target identificaiton
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Elements of a radar system

SignalProcessor

GeneratorWaveform

Detection Tracking Display

Transmitter stage

Receiver stage

Local oscillator

We will concentrate on the Signal processor block

Central question:

How to process the data to maximize system performance?

I d i R d R l i P l D l d R d i T d i h
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Down-conversion: typical analog

LPF900

LPF

LPF

AD

A

D

BPFRF

Q

I

IF

LO1 LO2

Issue: preserve the phase

IF: sIF(t) = A cos(t + )I: sIF(t)cosIF =

A2 [cos(( IF)t + ) + cos(( + IF)t + )]

Q: sIF(t)sinIF =A2 [sin(( IF)t + ) + sin(( + IF)t + )]

I+jQ: A2 ej[ej(IF)t + ej(+IF)t]

I t d ti R d t R l ti P l D l d R d ti T t d t ti th


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Down-conversion: typical analog

LPF900

LPF

LPF

AD

AD

BPFRF

Q

I

IF

LO1 LO2


Advantage

Low sampling frequencyfs > B

Inconvenience

Imbalance

DC-offset of amplifier

Introduction Radar system Resolution Pulse Doppler radar Radar equation Target detection theory
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Down-conversion: Digital Down Conversion (DDC)

LPFBPF AD

LPFRF

LO1

IF

LO2

I + j Q

ej2nT


IF: sIF(t) = A cos(t + )

I+jQ: A2 ej[ej(IF)t + ej(+IF)t]

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Down-conversion: Digital Down Conversion (DDC)

LPFBPF AD

LPFRF

LO1

IF

LO2

I + j Q

ej2nT


Advantage

No I/Q imbalance

Perfect I/Q orthogonality

Inconvenience

Higher sampling frequencyfs > 2B

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Range-resolution

Definition

Range resolution: Smallest distance that must exist between twotargets to permit the discrimination of the two targets.

Pulsed radarPulse of length p

Resolution : r = p

Bandwidth: p 1B

Conclusion

The range resolution is proportional to the bandwidth of thetransmitted signal.



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Doppler frequency resolution

range of a moving target: r(t) = r(0) + vrt

vr = radial velocity

time delay: (t) = (0) + 2 vrc

t

transmitted signal: s(t) = A(t)ej(t)ejt

received signal:sr(t) = s(t (t)) = A(t (t))ej(t(t))ej(t(t))

with (0) = 0:sr(t) = s(t(12

vrc

t)) = A(t(12vrc

t))ej(t(12vrct))ejt(12

vrct)

neglect envelope A() and phase () change during

observationsr(t) = A(t)e

j(t)ejtej2vrct = A(t)ej(t)ejte2jDt

Doppler frequency

fD =1

22vrc = 2

vr



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y pp q g y

Doppler frequency resolution

Definition

Doppler freq. resolution: Smallest Doppler frequency separationthat must exist between two targets to permit the discrimination ofthe two targets.

Pulsed radar

Pulse of length p

Resolution : 1p

Conclusion

The Doppler frequency resolution is proportional to the length ofthe transmitted signal.



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y pp q g y

Pulse-Doppler radar

Situation

High range resolution: small p

High frequency resolution: large p

Contradictory requirements

Solution

Coherent pulse train

http://find/


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y g y

Pulse-Doppler radar

t

TR

p

Re(sPD)

Pulses are repeated at the Pulse Repetition Frequency (PRF).PRF = 1

TR

Mathematical expression

sPD(t) =N1

k=0

rect

t kTR

p

ejt

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Doppler-frequency resolution

1000 500 0 500 10000

0.5

1

f (Hz)

|Scw

|

1000 500 0 500 10000

0.5

1

f(Hz)

|Spd

|

Resolution

= 2 1T

= 2 1NTR

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Range ambiguities

Ambiguous range

Targets separated by Ramb will produce the same echo.

Ramb =TR c

2

Solutions

Antenna radiation pattern only applicable for down-lookingradar

Increase PRI large PRI for surveillance radars

Pulse decorrelation PRI staggering

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Spectrum of a pulse train: construction

Start from one pulse

Replicate (convolve) in time with a comb-functionIs a multiplication in the spectral domain

Consider a finite comb function

To have a pulse train of finite length.

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Spectrum of a pulse train: 1 pulse

0 0.005 0.01 0.015 0.02 0.025 0.03

0

0.2

0.4

0.6

0.8

1

t (s)

1 0.5 0 0.5 1

x 104

0

0.2

0.4

0.6

0.8

1

f (Hz)

The spectrum of a (single) pulse is sin x/x-shaped.

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Spectrum of a pulse train: comb function

0 0.005 0.01 0.015 0.02 0.025 0.030

0.2

0.4

0.6

0.8

1

t (s)

1 0.5 0 0.5 1

x 104

0

0.2

0.4

0.6

0.8

1

f (Hz)

The spectrum of a comb-function is also a comb function.Temporal spacing: TRSpectral spacing: 1

TR



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Spectrum of a pulse train: Infinite pulse train

0 0.005 0.01 0.015 0.02 0.025 0.03

0

0.2

0.4

0.6

0.8

1

f (Hz)

1 0.5 0 0.5 1

x 104

0

0.2

0.4

0.6

0.8

1

f (Hz)

The spectrum of a pulse train is composed of peaks.

http://find/


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Spectrum of a pulse train: Infinite pulse train

0 0.005 0.01 0.015 0.02 0.025 0.030

0.2

0.4

0.6

0.8

1

f (Hz)1000 500 0 500 1000

0

0.2

0.4

0.6

0.8

1

f (Hz)

The spectrum of a pulse train is composed of peaks. causes ambiguities



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Doppler ambiguities

Target with velocity v and v + 2TR will have same Dopplerfrequency.

Velocity of target with velocity > vmax will be erroneouslyestimated.

Ambiguous velocity

Maximum unambiguous velocity:vmax =

14

TR

Solutions

Decrease the PRI Large PRF for Doppler radars

Use multiple PRI sequentially

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Spectrum of a pulse train: N-pulses

Multiply infinite pulse train with a square window of duration N TR

0 0.005 0.01 0.015 0.02 0.025 0.030

0.2

0.4

0.6

0.8

1

t (s)

1 0.5 0 0.5 1

x 104

0

0.2

0.4

0.6

0.8

1

f (Hz)

The spectrum of the window is sin x/x-shaped (but very narrow).

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Multiply infinite pulse train with a square window

0 0.005 0.01 0.015 0.02 0.025 0.030

0.2

0.4

0.6

0.8

1

t (s)1000 500 0 500 10000

0.2

0.4

0.6

0.8

1

f (Hz)

The spectrum of the window is sin x/x-shaped (but very narrow).

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The spectrum of the infinite pulse train is convolved with thespectrum of the square window.

0 0.005 0.01 0.015 0.02 0.025 0.030

0.2

0.4

0.6

0.8

1

t (s)

1 0.5 0 0.5 1

x 104

0

0.2

0.4

0.6

0.8

1

f (Hz)

The spectrum is non-zero between the peaks.



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The spectrum of the infinite pulse train is convolved with thespectrum of the square window.

0 0.005 0.01 0.015 0.02 0.025 0.030

0.2

0.4

0.6

0.8

1

t (s)1000 500 0 500 1000

0

0.2

0.4

0.6

0.8

1

f (Hz)

The spectrum is non-zero between the peaks. will lead to leakage of a target onto another

will limit detection of weak targets

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Spectrum of a pulse train: Comparison

1000 500 0 500 10000

0.2

0.4

0.6

0.8

1

f (Hz)1000 500 0 500 1000

0

0.2

0.4

0.6

0.8

1

f (Hz)1000 500 0 500 1000

0

0.2

0.4

0.6

0.8

1

f (Hz)

Summary: Pulse train

High range resolutionThere are range ambiguities

High Doppler resolution

There are Doppler (velocity) ambiguities

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Radar equation

Transmitted power

Pt

Power density at range R

Pd = PtG

4R2

Radar cross section

= PrtPd

Received power

Pr = PrtAe

4R2

Antenna effective area

Ae =G2

4

Radar equation

Pr = PtG

4R2 G

2

(4)2R2= Pt

G22

(4)3R4

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Radar equation

Radar equation

Pr = PtG22

(4)3R4

Links radar parameters to predict performanceTargets with large RCS are easier to detect (obvious)A doubling of the range imply multiplying the power by 16!

Received power must be larger than (thermal) noise power

High transmitted power PtAntenna with high gain (high directivity) GReceiver with low noisePreferably long wavelengths

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Signal model

Sampled baseband transmitted signal (delayed):

s(k) = A(k)ej(k)

A(k) = amplitude of the transmitted signal(k) = phase of the transmitted signal

Received (baseband) echo (in the presence of a target):

y(k) = s(k) + n(k)

= target (complex) reflectivityn(k) = noise

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Signal model: vector notation

Consider a sample sequence of length N

s = [s(0), s(1), . . . , s(N 1)]T

y = [y(0), y(1), . . . , y(N 1)]T

n = [n(0), n(1), . . . , n(N 1)]T

Received signal

y = s + n



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Signal model: noise

Gaussian noiseNoise is characterized by covariance matrix

R = E{nn}

Noise probability density function (PDF)

p(n) =1

N|R|en

R1n

Can take any noise into account, including non-white noise

White noise:

R = 2I

p(n) = 1N2Nn

e

n2

2n

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Signal model: target reflectivity

Model of the complex amplitude of the reflected signal

Deterministic constant

Steady target (Marcum model)

Not very realistic

Leads to easy analytical results

Stochastic quantity

Fluctuating target (Swerling models)

PDF of:

p() =1

2e ||2

2


S l d l b b l d f
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Signal model: probability density function

Two hypotheses: H0 and H1

H0: No target is present

sr = n

p(sr|H0) =1

N|R|es

rR

1sr

H1: A target is present

sr = s + n

p(sr|H1) =1

N|R|e(srs)

R1(srs)


D i i lik lih d
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Decision: likelihood

LikelihoodL(H0) = p(y|H0)

L(H1) = p(y|H1)

Likelihood ratio

(y) =L(H1)

L(H0)

Likelihood ratio test

(y)H1

H0

Maximize the probability of detection for a given probability offalse alarm.


T i i PDF
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Test statistics: PDF

Likelihood ratio

(y) = esR1yeyR1se||2sR1s

With sR1y = (yR1s)

(y) = e2Re(sR1y)e||

2sR1s

and taking the logarithm

(y) = 2Re(sR1y)||2sR1s.

Test statistic

T(y) = |wy|2

with

w = kR1s and k = 1sR1s


T t t ti ti Si l t i ti
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Test statistics: Signal to noise ratio

Consider the result of the filter w: = wy

Noise only noise powery = n

Power = E{||2} = 1(due to the normalization with k)

Signal only signal powery = s

Power = E{||2} = ||2sR1s

y = s

Signal to noise ratio

SNR = ||2sR1s


T t t ti ti PDF
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Consider z = wyz|H0 CN(0, 1)

z|H1 CN(

k, 1)

thus

Re(z) N(, 1) & Im(z) N(, 1)

T(y) = |z|2 = Re(z)2 + Im(z)2

T(y) is chi-squared distributed

Test statistic PDF

T(y)

12

22 H0

12

22 (2||

2sR1s) H1




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Marcum target (constant target amplitude)

0 10 20 30 40 500

0.01

0.02

0.03

0.04

0.05

0.06

0.07

T(y)

PDF(T(y))

H0

H1

The test statistics distribution is different for H0 and H1makes the discrimination between H0 and H1 possible

The separation depends on the signal to noise ratio SNR.


Test statistics: PDF (sidenote)
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Test statistics: PDF (sidenote)

H0

T(y) 1

222

A chi-square with 2 degrees of freedom = exponential distribution

H1

T(y) 1

222 (2||

2sR1s)

A non-central chi-square with 2 degrees of freedom = Rayleighdistribution


Test statistics: Probability of false alarm
http://find/


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Test statistics: Probability of false alarm

Probability of false alarm

PFA = Q() =

+

p(T|H0)dT

= Q1(PFA)

0 10 20 30 40 500

0.01

0.02

0.03

0.04

0.05

0.06

0.07

T(y)

PD

F(T(y))

PFA


Test statistics: Probability of detection
http://find/


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Test statistics: Probability of detection

Probability of detection

PD =

+

p(T|H1)dT

0 10 20 30 40 500

0.01

0.02

0.03

0.04

0.05

0.06

0.07

T(y)

PDF(T(y))

PD


Performance
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Performance

0 5 10 15 20

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SINR(dB)

PD

PFA = 10

4

PFA

= 105

PFA

= 106

PFA=10

7

Performance

Depends critically on the SNR

even a small decrease in SNR will reduce PD


Performance
http://find/


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Performance

Comparison of PD for steady and fluctuating targets

0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SINR(dB)

PD

Marcum model

SwerlingI model

large SNR: steady targets are easier to detect

low SNR: fluctuating targets are easier to detectdue to fluctuation, target signal may exceed the treshold


Performances: multiple fluctuating pulses
http://find/


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Performances: multiple fluctuating pulses

Probability of detection as afunction of the number of pulses

0 5 10 15 20

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SINR(dB)

PD

1 radar

2 radars

3 radars4 radars

Increase of performancedue to

Incoherent integrationgain

Diversity gain


Performance
http://find/


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Performance

What are the performance of a particular filter w?

Performance is measured in terms ofSNRloss

SNRloss =SNRoutput

SNRinput

direct link between SNRloss and detection performance
http://find/

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