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cw and pulse radr
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Prof. Y Kwag@ RSP LabKorea Aerospace Univ.
Chapter 3
Continuous Wave and Pulsed Radars
Prof. Y Kwag@ RSP LabKorea Aerospace Univ.
3.1. Functional Block Diagram
< CW radar block diagram >
Prof. Y Kwag@ RSP LabKorea Aerospace Univ.
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
Prof. Y Kwag@ RSP LabKorea Aerospace Univ.
: 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),
Prof. Y Kwag@ RSP LabKorea Aerospace Univ.
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
Prof. Y Kwag@ RSP LabKorea Aerospace Univ.
3.3 Frequency Modulation
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
Prof. Y Kwag@ RSP LabKorea Aerospace Univ.
3.3 Frequency Modulation
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
Prof. Y Kwag@ RSP LabKorea Aerospace Univ.
3.3 Frequency Modulation
- 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
)(
Prof. Y Kwag@ RSP LabKorea Aerospace Univ.
3.3 Frequency Modulation
- 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
Prof. Y Kwag@ RSP LabKorea Aerospace Univ.
3.3 Frequency Modulation
- 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
Prof. Y Kwag@ RSP LabKorea Aerospace Univ.
3.3 Frequency Modulation
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)
Prof. Y Kwag@ RSP LabKorea Aerospace Univ.
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
Prof. Y Kwag@ RSP LabKorea Aerospace Univ.
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
Prof. Y Kwag@ RSP LabKorea Aerospace Univ.
3.4 Linear FM (LFM) CW Radar
(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)
Prof. Y Kwag@ RSP LabKorea Aerospace Univ.
3.4 Linear FM (LFM) CW Radar
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
Prof. Y Kwag@ RSP LabKorea Aerospace Univ.
3.4 Linear FM (LFM) CW Radar
-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
Prof. Y Kwag@ RSP LabKorea Aerospace Univ.
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
Prof. Y Kwag@ RSP LabKorea Aerospace Univ.
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
Prof. Y Kwag@ RSP LabKorea Aerospace Univ.
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
Prof. Y Kwag@ RSP LabKorea Aerospace Univ.
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
Prof. Y Kwag@ RSP LabKorea Aerospace Univ.
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
Prof. Y Kwag@ RSP LabKorea Aerospace Univ.
3.7 Range and Doppler Ambiguities
Range and Doppler Ambiguities
Fig 3.8. Spectra of transmitted and received wavwforms and Doppler bank.
Prof. Y Kwag@ RSP LabKorea Aerospace Univ.
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
Prof. Y Kwag@ RSP LabKorea Aerospace Univ.
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.
Prof. Y Kwag@ RSP LabKorea Aerospace Univ.
Resolving Range Ambiguity
Prof. Y Kwag@ RSP LabKorea Aerospace Univ.
3.8 Resolving Range Ambiguity
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
Prof. Y Kwag@ RSP LabKorea Aerospace Univ.
Resolving Range Ambiguity
)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
Prof. Y Kwag@ RSP LabKorea Aerospace Univ.
Resolving Range Ambiguity
)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
Prof. Y Kwag@ RSP LabKorea Aerospace Univ.
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.
Prof. Y Kwag@ RSP LabKorea Aerospace Univ.
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
Prof. Y Kwag@ RSP LabKorea Aerospace Univ.
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
Prof. Y Kwag@ RSP LabKorea Aerospace Univ.
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
Prof. Y Kwag@ RSP LabKorea Aerospace Univ.
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
Prof. Y Kwag@ RSP LabKorea Aerospace Univ.
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