Post on 17-Sep-2015
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Chapter 5
Radar Waveforms Analysis
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Ch 5. Radar Waveforms Analysis
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- Choosing a waveform type and a signal processing technique in a radar system
depends on the radars specific mission and role
- Radar systems can use
Continuous Waveforms(CW)
Pulse waveforms with modulation
Pulse waveforms without modulation
- Range and Doppler resolutions are directly related to the specific waveform
frequency characteristics. power spectrum density of waveform is very critical
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5.1. LP, BP Signals and Quadrature Components
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Low Pass, Band Pass Signals and Quadrature Components
- Low Pass(LP) signals : contained significant frequency composition at a low
frequency band that includes DC
- Band Pass(BP) signals : have significant frequency away from the origin
Real BP Signal - Real BP signal represented by )(tx
frequencycarrier:
modulationphase:)(
envelopeormodulationamplitude:)(where
)1.5())(2cos()()(
0
0
f
t
tr
ttftrtx
x
x
* Both and have frequency components significantly smaller than . )(tr )(tx 0f
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Real BP Signal
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)2.5()(2
1)( t
dt
dtf xm
- Frequency modulation is
- Instantaneous frequency is
)3.5()()(22
1)( 00 tffttf
dt
dtf mxi
- Signal bandwidth B
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< Extraction of quadrature componentcs>
Real BP Signal
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5.2 CW and Pulsed Waveforms
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CW and Pulsed Waveforms
- Energy signal (finite energy): haracterized by its Energy Spectrum Density
(ESD) function (Joules/Hz)
- Power signal (finite power): characterized by its Power Spectrum Density
(PSD) function (Watts/Hz)
- Signal bandwidth: range of freq over which the signal has a nonzero spectrum.
- Signal defined using its duration (time domain) and bandwidth (freq. domain)
- Finite bandwidth band-limited
- Signals that have
finite duration (time-limited) have infinite bandwidth
finite bandwidth (band-limited) have infinite duration
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CW and Pulsed Waveforms
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- A time domain signal has )(tf
- The signal autocorrelation function is )(fR
)8.5()()()( *
dttftfR f
dtetfF tj )()(
deFtf tj)(2
1)(
Fourier Transform (FT) :
Inverse FT (IFT) :
(5.6)
(5.7)
Signal Autocorrelation Function
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CW and Pulsed Waveforms
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)9.5()()(
deRS jff
- Signal amplitude spectrum is .
if were an energy signal, then its ESD is .
if were a power signal, then its PSD is which is the FT of the
autocorrelation function
)(F
)(F)(tf
)(tf )(fS
CW Waveform - CW waveform given by tAtf 01 cos)(
- The FT of is )(1 tf )()()( 001 AF
00 2where f
(5.10)
(5.11)
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CW Waveform
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- Signal has infinitesimal bandwidth, located at . )(1 tf 0f
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Time domain signal
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- Time domain signal given by )(2 tf
)14.5()sin(
)(where
)13.5(2
)(
)12.5(
0
22)(
2
2
x
xxSinc
SincAF
otherwise
tAtRectAtf
Time Domain Signal
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Time domain signal
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- Bandwidth is infinite. Since infinite
bandwidth
cannot be physically implemented.
- Signal bandwidth is approximated by
radians per second or Hertz.
accounts for most of the signal energy
/2
/1
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Coherent gated CW waveform
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Coherent gated CW waveform Signal
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Coherent gated CW waveform
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- Coherent gated CW waveform given by )(3 tf
n
nTtftf )15.5()()( 23
is periodic, where is the period( is the PRF)
Using the complex exponential Fourier series
)(3 tf T Tfr /1
)17.5(
)16.5()(2
3
T
nSinc
T
AF
eFtf
n
n
Tntj
n
- The FT of is )(3 tf
)18.5()2(2)(3
n
rn fnFF
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Amplitude spectrum
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Amplitude spectrum for a coherent pulse train of finite length
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Amplitude spectrum for a coherent pulse train of finite length
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4) Function as )(4 tf
N
n
nTtftf0
24 )19.5()()(
Note that is a limited . The FT of is )(4 tf )(4 tf)(3 tf
n.convolutioindicatesoperatorthewhere
)20.5()2()(2
)(4
n
rr fnfnSincNT
SincANF
- The envelope is still a sinx/x which corresponds to the pulse width. But
the spectral lines are replaced by sinx/x spectra that correspond to the
duration NT
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5.3 LFM Waveform
Linear Frequency Modulation Waveforms
- Frequency or phase modulated waveforms can be used to achieve much wider
operating bandwidths. Linear Freq. Modulation(LFM) is commonly used
- LFM : freq. is swept linearly across the pulse width, either upward(up-chirp)
or downward(down-chirp).
- The matched filter bandwidth is proportional to the sweep bandwidth, and is
independent of the pulse width.
- The LFM up-chirp instantaneous phase can be expressed by
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LFM Waveform
tcoefficienLFM:/)2(
frequencycenterradar:where
)21.5(222
2)(
0
2
0
B
f
tttft
- the instantaneous frequency is
)22.5(22
)(2
1)( 0
ttft
dt
dtf
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LFM Waveform
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)(a )(b
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LFM Waveform
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- The down-chirp instantaneous phase and frequency are given by
)24.5(22
)(2
1)(
)23.5(222
2)(
0
2
0
ttftdt
dtf
tttft
- A typical LFM waveform can be expressed in complex notation by
widthofpulserrectangulaadenotes)/(where
)25.5()(
20
22
1
tRect
et
Recttsttfj
)26.5()()( 02
1 tsetstfj
))(offunctionenvelopecomplex()(where 12
tset
Rectts tj
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LFM Waveform
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- The spectrum of the signal is determined from its complex
envelope . taking the FT of yields
)(1 ts
)(ts )(ts
)29.5(
variableofchangetheperformand,/22Let
)28.5(2
2exp)(
2
2
22
dt
;dx
t
x
B
dtetj
dteet
RectS tjtjtj
)31.5()(
)30.5()S(
12
222
2
1
22
0
2/
0
2/2/
2/2/
x
xj
x
xjj
x
x
xjj
dxedxeeS
dxee
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LFM Waveform
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)35.5(2
sin)(
)34.5(2
cos)(
bydefinedare),(and)(denotedintegrals,FresnelThe
)33.5(2/
122
)32.5(2/
122
where
0
2
0
2
2
1
x
x
dxS
dxC
xSxCby
B
fBx
B
fBx
- Fresnel integrals are approximated by
)37.5(1;2
cos1
2
1)(
)36.5(1;2
sin1
2
1)(
2
2
xxx
xS
xxx
xC
)()()()(, xSxSandxCxCthatNote
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LFM Waveform
C(x) and S(x) for 0 x 10
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Fresnel Spectrum
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)38.5(
2
)()()()(1)(
)31.5()(
)35.5(2
sin)(
)34.5(2
cos)(
1212)4/(
0
2/
0
2/2/
0
2
0
2
2
12
222
xSxSjxCxCe
BS
dxedxeeS
dxS
dxC
Bj
x
xj
x
xjj
x
x
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Fresnel Spectrum
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Typical plot for the amplitude spectrum of an LFM waveform. The square-like spectrum is widely known as the Fresnel spectrum