MS_3장 CW and Pulse Radar

Prof. Y Kwag@ RSP LabKorea Aerospace Univ.

Chapter 3

Continuous Wave and Pulsed Radars


3.1. Functional Block Diagram

< CW radar block diagram >


Narrow Band Filters (NBF)

: bandwidth Doppler measurement accuracy , noise power

In practical, operating bandwidth of CW Radar is finite bandwidth

NBF bank (Doppler filter bank) is implemented using an FFT of size NFFT

- individual NBF bandwidth (FFT bin)

effective radar Doppler bandwidth

Single frequency CW radar cannot measure target range

- in order to measure target range

transmit and receive waveforms must have some sort of timing marks

- timing mark : implemented by modulating the transmit waveform

commonly technique Linear Frequency Modulation (LFM)

3.1. Functional Block Diagram

f

2/fNFFT


: loss term associated with the type of window (weighting)

3.2. CW Radar Equation

- Dwell interval : determine frequency resolution or bandwidth of the individual NBFs

DwellTf /1

fBNFFT /2

BNT FFTDwell 2/

FLkTRGTP

SNRe

iav

43

22

4

avP

winertDwellcw

FLLkTR

GGTPSNR

43

2

4

CWP iT DwellT

winL

(3.1)

(3.2)

(3.3)

(3.4)

(3.5)

- NBF bank size

- for (1), (2)

- CW radar equation : derived from high PRF Radar equation high PRF Radar equation

- (CW average transmitted power over the dwell interval),


3.3 Frequency Modulation

General formula for an FM waveform

t

mf duufktfAts0

0 2cos2cos)(

: carrier frequency, : modulating signal

A : constant, : peak freq. deviation

0f tfm2cos

peakf fk 2 peakf

(3.6)

tftfduufftft m

t

mpeak 2sin22cos22)( 00

0 (3.7)

m

peak

f

f (3.8)

- FM modulation index

- Phase



Received radar signal

)(2sin)(2cos)( 0 ttfttfAts mrr (3.9)

- Time Delay

c

Rt

2 (3.10)

Phase detector : extract target range from the instantaneous frequency



FM waveform

tftfAts m 2sin2cos)( 0 (3.11)

tfsin2jtfj2 m0 eeReAs(t) (3.12)

- Using the complex exponential Fourier series (F.S)

n

tfjn

n

tfj mm eCe22sin (3.13)

(3.14)

- F.S coefficient

dtee

Ctfjntfj

nmm

22sin

2

1



- Bessel function of the first kind of order n,

due

J nuujn

)sin(2

1)(

tfu m2

(3.15)

(3.16)),(nn JC

n

tfjn

n

tfj mm eJe22sin

)(



- Total power in the signal s(t)

(3.17)

(3.18)

2

2

2

2

1)(

2

1AJAP

n

n

- Substituting Eq.(3.16) into Eq.(3.12)

tfjn2

n

n

tfj2 m0 eJeReAs(t) )(

(3.19)

- Expanding Eq.(3.18)

n

mn tfnfJAts )22cos()()( 0



- Since for n Odd & for n Even)()( nn JJ )()( nn JJ

}..........

])cos()[cos()(

])cos()[cos()(

])cos()[cos()(

])cos()[cos()(

)({

4

3

2

1

0

tf8f2tf8f2J

tf6f2tf6f2J

tf4f2tf4f2J

tf2f2tf2f2J

eJAs(t)

m0m0

m0m0

m0m0

m0m0

tfjn2 0

(3.20)

- The spectrum of s(t)

Amplitude of the central spectral line

-

Amplitude of the nth spectral line

-

),(0 AJ

),(nAJ



For small , the Bessel functions can be approximated by

mfB )1(2 (3.21)

(3.22)

- When is small, and : significant value)(0 J )(1 J

]})cos()[cos()(

2cos)({

1

00

tf2f2tf2f2J

tfJAs(t)

m0m0

1)(0 J 21

)(1 J

- Bandwidth can be approximated using Carsons rule

(3.23), (3.24)

]})cos()[cos(2

1

2{cos 0

tf2f2tf2f2

tfAs(t)

m0m0

(3.25)


Example 3.1 & 3.2

If the modulation index is = 0.5, give an expression for the signal s(t).

- Solution: 9385.0)5.0(0 J 2423.0)5.0(1 J

]})cos()[cos()2423.0(

2cos)9385.0{( 0

tf2f2tf2f2

tfAs(t)

m0m0

Prob.3.2 ))(2000cos(100 tts(t) Output signal:Frequency deviation : 4Hz

Modulating waveform : tx(t) 6cos10

How many spectral lines will pass through a band pass filter whose

bandwidth is 58Hz centered at 1000Hz?

- Solution: Hzf peak 40104

58

40

m

peak

f

f

HzfB m 968)15(2)1(2

Prob.3.1


3.4 Linear FM (LFM) CW Radar

LFM CW radar range & Doppler information

Fig 3.5. Transmitted and received triangular LFM signals and beat frequency for stationary

target



(3.26)- Modulating frequency

021tm

f

- Rate of frequency change(3.27)

- Beat frequency

fff mff

t

f

m 2

)0 2/1(

fftfcR

b 2

(3.28)

Eq.(3.28) rewritebR

c ff2

(3.29)

(3.30)c

fRff mb

4

When Doppler is present.

dtransmittereceivedb fff (3.31)



Fig 3.6. Transmitted and received LFM signals and beat frequency, for a moving

target.

Positive slope Doppler shift term subtracts from the beat frequency

Negative slope the two terms add up



-Beat frequency during positive slope fbu

-Beat frequency during negative slope fbd

Rf

c

Rfbu

22

Rf

c

Rfbd

22

(3.32)

(3.33)

Range )(4

bdbu fff

cR

(3.34)

(3.35)

Range rate )(4

bubd ffR

-Maximum time delay 0max 1.0 tt (3.36)

(3.37)-Maximum range

mfctcR

41.0

2

1.0

max0


3.5 Multiple Frequency CW Radar

Multiple frequency scheme (CW radar)

- very adequate range measurement, without using frequency modulation

-Waveform : tfAs(t) 02sin

)2sin( 0 tfA(t)s rr-Received signal :

(3.38)

(3.39)

-phase : cRf 202 (3.40)

-Solving for R

44 0

f

cR (3.41)

Maximum unambiguous range occurs when is maximum.

R is limited to impractical small values.

2


3.5 Multiple Frequency CW Radar

Two CW signals

tfA(t)s 111 2sin

tfA(t)s 222 2sin

(3.42)

(3.43)

Received signals from moving target

)2sin( 1111 tfA(t)s rr

)2sin( 2222 tfA(t)s rr

(3.44)

(3.45)

Phase difference between the two received signals

fc

Rff

c

R

4)(

41212

(3.46)

f

cR

2(3.47)

Maximum unambiguous range 2


3.6 Pulsed Radar

Pulsed Radar

- Transmit & receive a train of modulated pulsed.

- Two way time delay between a Transmitted and Received pulse

extract range information.

- If accurate range measurements are available between consecutive pulses

Doppler frequency extracted from the range rate

Defined the pulsed radar waveform

carrier frequency : depend on the design requirements and radar mission.

pulse width : related to the BW and defines the range resolution.

modulation : difference modulation techniques are usually utilized to

enhance the radar performance.

PRF : must be chosen to avoid Doppler and range ambiguities.

tRR


PRF Classification & Agility

Radar system employ low, medium, and high PRF schemes.

Low PRF : accurate, long, unambiguous range measurements,

but, severe Doppler ambiguities.

Medium PRF : must resolve both range and Doppler ambiguities. but, provide

adequate average transmitted power as compare to low PRFs.

High PRF : superior average transmitted power and excellent clutter rejection

capability. but, extremely ambiguous range

- Radar system utilizing high PRFs are often called Pulsed Doppler Radar (PDR)

- Moving Target Indicator (MTI) radar use the PRF agility known as PRF staggering

PRF agility use to avoid blind speed

use to avoid range and Doppler ambiguities

use to prevent jammers from locking onto the radars PRF PRF jitter


Pulsed Radar Block Diagram

range gate : implemented as filters that open and

close at time intervals

that correspond to the

detection range.

NBF bank : implemented using an FFT, individual filter

BW = FFT freq.

resolution


3.7 Range and Doppler Ambiguities

Range and Doppler Ambiguities

Fig 3.8. Spectra of transmitted and received wavwforms and Doppler bank.


3.7 Range and Doppler Ambiguities

Range and Doppler Ambiguities

- Range ambiguous : Second pulse is transmitted prior to the return of the first pulse.

- Radars operational requirements radar PRF chose.

ex. long-range search(surveillance) low-PRF

- Line spectrum of a train of pulses has sinx/x envelope Line spectra are separated by the PRF(fr).

- The Doppler filter bank is capable or resolving target Doppler as long as the anticipated Doppler

shift is less than one half the bandwidth of the individual filters

)48.3(2

2 maxmaxr

dr

vff


Multiple PRF

Doppler ambiguous;

If the target Doppler freq. is high enough to make an adjacent spectral line

move inside the Doppler band of interest.

Detecting high speed target Require high PRF

Detecting the high speed target by using long range radar

range and Doppler ambiguous.

resolving by using multiple PRFs.

Multiple PRF schemes;

incorporated sequentially within each dwell interval (scan or integration frame)

use a single PRF in one scan and resolve ambiguity in the next.

may have problems due to changing target dynamics from

one scan to the next.


Resolving Range Ambiguity


3.8 Resolving Range Ambiguity


- Radar uses two PRFs and , to resolve range ambiguity

- Desired PRF that corresponds to as

- One choice is to select and for some integer

- Within one period of the desired PRI( ) the two PRFs and coincide

only at one location true unambiguous range.

- M1(M2) : number of PRF1(PRF2) intervals between transmit of a pulse and receipt

of the true target return.

- Over the interval 0 to , the only possible result are M1=M2=M or M1+1=M2.

- Time delay t1 and t2 correspond to the time between the transmit of a pulse on

each PRF and receipt of a target return due to the same pulse.

)( 11 ur Rf )( 22 ur Rf

)(, 21 rangesunambiguouradardesiredRRR uuu

uR rdf

rdr Nff 1 rdr fNf )1(2 N

rdd fT 1 1rf 2rf

dT



)52.3(2

israngetargettrue

)51.3(

islocationtargettruethetotimetripround

1,1

)50.3(

)49.3(

:Icase.1

22

11

22

11

21

12

2

2

1

1

21

r

r

r

rr

rr

ctR

tMTt

tMTt

fT

fTwhere

TT

ttM

f

Mt

f

Mt

tt

)56.3(2

israngetargettrue

)55.3(

islocationtargettruethetotimetripround

1,1

)54.3()(

)53.3(1

:IIcase.2

1

111

22

11

21

212

2

2

1

1

21

r

r

rr

rr

ctR

tMTt

fT

fTwhere

TT

TttM

f

Mt

f

Mt

tt



)58.3(2

israngetargettrue

)57.3(

ambiguityfirsttheinistargetThe

:IIIcase.3

2

212

21

r

r

ctR

ttt

tt

- Blind range : pulse cannot be received while the following pulse is being

transmitted, these time correspond to blind range.

resolved by using a thired PRF

rdr

rdr

rdr

fNNf

fNNf

fNNf

)2)(1(

)2(

)1(

3

2

1


3.9 Resolving Doppler Resolution

Resolving Doppler Ambiguity

-The Doppler ambiguity problem is analogous to that of range ambiguity.

same methodology can be used to resolve Doppler ambiguity.

- Measure the Doppler frequency and instead of and .1df 2df 1t 2t

)62.3(

:.3

)61.3(

)60.3(

:.2

)59.3()(

:.1

21

21

2211

21

12

21

21

212

21

ddd

dd

drddrd

rr

dd

dd

rr

rdd

dd

fff

ffIIIcase

fMfforfMff

isDopplertrueand

ff

ffM

ffIIcase

ff

fffM

ffIcase

- Blind Doppler can occur can be resolved using a third PRF.


Example 3.3

kmf

cR

kmf

cR

kHzfNf

kHzNff

kHzR

cf

fPRF

r

u

r

u

rdr

rdr

u

rd

rd

667.110902

103

2

695.1105.882

103

2

90)1500)(159()1(

5.88)1500)(59(

thatfollowsIt

5.110200

103

2

,desired,first:solution*

3

8

2

2

3

8

1

1

2

1

3

8

.,,,f compute .59 Choose . 100 is range sunambiguou

desired The s.ambiguitie range resolve to PRFs two usesradar certain A

212r1 uuru RandRf NkmR


Example 3.4 (1)

PRF.eachfor17and2

8atappearingtargetanotherforfreq.)(Doppler Calculate . 550m/s isvelocity

whose target afor PRFeach ofposition frequency theCalculate .9 Assume

21 and 1815 PRFs; h threeradar wit aConsider

0

321

kHz,kHz

,kHzf

GHzf

kHz.fkHz,kHz, ff

d

rrr


Example 3.4 (1)

3321

3318

3315

321where Eq.(3.61) using

33103

10955022f

is frequency Doppler : *

33333

22222

11111

8

9

0d

ddr

ddr

ddr

ddirii

fnffn

fnffn

fnffn

,,ifffn

kHzc

vf

solution

.12and152 are freq.Doppler apparentThus,

.11way samevalue.acceptable:32Choose

.sincevalueacceptablenot:18and331and0 Choose

321

3211

1111

kHzf,khzfkHz,f

n,nkHzfn

ffkHzkHz,f,n

ddd

d

rdd


Example 3.4 (2)

1721

218

815

Eq.(3.61) using.problemtheofpartSecond

3333

2222

1111

nfffn

nfffn

nfffn

ddr

ddr

ddr

0 1 2 3 4

8 23 38 53 68

2 20 38 56

17 38 39

n

1from rd ff

2from rd ff

3from rd ff

smv

kHzfnnn

r

d

/7.6232

0333.038000

38isDopplertargettruetheand1and,2,Thus 321

isrelationsthreeabovethesatisfythatintegersSmallest 321 n,n,n


range_calc.m

3.10. MATLAB program range_calc.m

- The program range_calc.m solves the radar range equation of the form

)63.3()()4(

4

1

0

3

2

SNRLFTk

GGTfPR

e

rtirt

Peak transmitted power Boltzmans constant

Pulse width Effective noise figure

PRF System noise figure

Transmitting antenna gain Total system losses

Receiving antenna gain Dwell interval (time on target)

Wavelength Minimum SNR required for detection

Target cross section

tP

rf

tG

rG

k

eT

F

L

iT

0)(SNR

- This equation applies for both CW and pulsed radar.

- In the case of CW radars, the terms must be replaced by the average CW

power .

rt fP

CWP

MS_3장 CW and Pulse Radar

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