Transcript
1
RADAR SignalProcessing
SOLO HERMELIN
Updated: 28.11.08http://www.solohermelin.com
2
SOLO RADAR Signal Processing
Table of Contents
RADAR Signals
Waveform Hierarchy
RADAR TypesRadar Generic Procedures
Fourier Transform
Waveforms
Quadrature Form
Spectrum
Energy
Complex and Analytic Signals
Signal Duration and BandwidthComplex Representation of Bandpass Signals
Autocorrelation
Sampling and z-Transform
Nyquist-Shannon Sampling Theorem
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SOLO RADAR Signal Processing
Table of Contents (continue – 1)
The Discrete Time Fourier Transform (DTFT)
The Discrete Fourier Transform (DFT)
Fast Fourier Transform (FFT)
Digital Filtering
Windowing
Doppler Frequency Shift
Coherent Pulse Doppler Radar
Signal Processing
Decision/Detection Theory
Search & Detect Mode
Acquisition Mode
References
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SOLO
The transmitted RADAR RF Signal is:
( ) ( ) ( )[ ]ttftEtEt 0000 2cos ϕπ +=E0 – amplitude of the signal
f0 – RF frequency of the signal φ0 –phase of the signal (possible modulated)
The returned signal is delayed by the time that takes to signal to reach the target and toreturn back to the receiver. Since the electromagnetic waves travel with the speed of lightc (much greater then RADAR andTarget velocities), the received signal is delayed by
c
RRtd
21 +≅
The received signal is: ( ) ( ) ( ) ( )[ ] ( )tnoisettttfttEtE dddr +−+−⋅−= ϕπα 00 2cos
To retrieve the range (and range-rate) information from the received signal thetransmitted signal must be modulated in Amplitude or/and Frequency or/and Phase.
ά < 1 represents the attenuation of the signal
RADAR Signal ProcessingRADAR Signals
5
SOLO
The received signal is:
( ) ( ) ( ) ( )[ ] ( )tnoisettttfttEtE dddr +−+−⋅−= ϕπα 00 2cos
( ) ( ) tRRtRtRRtR ⋅+=⋅+= 222111 &
We want to compute the delay time td due to the time td1 it takes the EM-wave to reachthe target at a distance R1 (at t=0), from the transmitter, and to the time td2 it takes the EM-wave to return to the receiver, at a distance R2 (at t=0) from the target. 21 ddd ttt +=
According to the Special Theory of Relativitythe EM wave will travel with a constant velocity c (independent of the relative velocities ).21 & RR
The EM wave that reached the target at time t was send at td1 ,therefore
( ) ( ) 111111 ddd tcttRRttR ⋅=−⋅+=− ( )1
111 Rc
tRRttd
+⋅+=
In the same way the EM wave received from the target at time t was reflected at td2 , therefore
( ) ( ) 222222 ddd tcttRRttR ⋅=−⋅+=− ( )2
222 Rc
tRRttd
+⋅+=
RADAR Signal Processing
6
SOLO
The received signal is:
( ) ( ) ( ) ( )[ ] ( )tnoisettttfttEtE dddr +−+−⋅−= ϕπα 00 2cos
21 ddd ttt += ( )1
111 Rc
tRRttd
+⋅+= ( )
2
222 Rc
tRRttd
+⋅+=
( ) ( )2
22
1
1121 Rc
tRR
Rc
tRRtttttttt ddd
+⋅+−
+⋅+−=−−=−
+
−+−+
+
−+−=−
2
2
2
2
1
1
1
1
2
1
2
1
Rc
Rt
Rc
Rc
Rc
Rt
Rc
Rctt d
From which:
or:
Since in most applications we canapproximate where they appear in the arguments of E0 (t-td), φ (t-td),however, because f0 is of order of 109 Hz=1 GHz, in radar applications, we must use:
cRR <<21,
1,2
2
1
1 ≈+−
+−
Rc
Rc
Rc
Rc
( )
−⋅
++
−⋅
+=
−⋅
−+
−⋅
−⋅≈− 2
.
201
.
1022
011
00 2
1
2
1
2
121
2
121
21
D
RalongFreqDoppler
DD
RalongFreqDoppler
Dd ttffttffc
Rt
c
Rf
c
Rt
c
Rfttf
( ) ( ) ( ) ( ) ( )[ ] ( )tnoisettttffttEtE ddDdr +−+−⋅+−= ˆˆˆ2cosˆ00 ϕπα
where 212
21
1212
021
01ˆˆˆ,,,ˆˆˆ,
2ˆ,2ˆ
dddddDDDDD tttc
Rt
c
Rtfff
c
Rff
c
Rff +=≈≈+=−≈−≈
Finally
RADAR Signal Processing
Doppler Effect
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SOLO
The received signal model:
( ) ( ) ( ) ( ) ( )[ ] ( )tnoisettttffttEtE ddDdr +−+−⋅+−≈ ϕπα 00 2cos
Delayed by two-way trip time
Scaled downAmplitude Possible phase
modulated
CorruptedBy noise
Dopplereffect
We want to estimate:
• delay td range c td/2
• amplitude reduction α
• Doppler frequency fD
• noise power n (relative to signal power)
• phase modulation φ
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SOLO Waveform Hierarchy
Radar Waveforms
CW Radars Pulsed Radars
FrequencyModulated CW
PhaseModulated CW
bi – phase & poly-phase
Linear FMCWSawtooth, or
Triangle
Nonlinear FMCWSinusoidal,
Multiple Frequency,Noise, Pseudorandom
Intra-pulse Modulation
Pulse-to-pulse Modulation,
Frequency AgilityStepped Frequency
FrequencyModulate Linear FM
Nonlinear FM
PhaseModulatedbi – phase poly-phase
Unmodulated CW
Multiple FrequencyFrequency
Shift Keying
Fixed Frequency
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SOLO
( )tf
2
τ2
τ−
A
∞→t
2
τ+T2
τ−T
A
2
τ+−T2
τ−−T
A
t←∞−
T T
NONCOHERENT PULSESCOHERENT PULSES
( )tf
t
A
2
τ2
τ−T
AA
T T
A
22
τ+T2
2τ−T
A
T T
A
2
τ− 2
τ+T
TN
PULSED (UNCODED)
A Partial List of the Family of RADAR Waveforms
PRI – Pulse Repetition Interval PRF – Pulse Repetition Frequency
τ – Pulse Width [μsec]
PRF = 1/PRI
Pulse Duty Cycle = DC = τ / PRI = τ * PRF
Paverrage = DC * Ppeak
Pulse Waveform Parameters
Continuous Waves (CW)
Pulses• Coherent – Phase is predictable from pulse-to-pulse• Non-coherent – Phase from pulse-to-pulse is not predictable
Waveform Hierarchy
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SOLO
( )tf
2
τ2
τ−
A
∞→t
2τ+T
2τ−T
A
2τ+−T
2τ−−T
A
t←∞−
T TA
t
A
t
A
LINEAR FM PULSECODED PULSE
T T
PULSED (INTRAPULSE CODING)
t
( )tf
A
2
τ2
τ−T
AA
T T
A
22
τ+T2
2τ−T
A
T T
A
2τ− 2
τ+T
TN
t
( )tf
A
2
τ2
τ−T
AA
T T
A
22
τ+T2
2τ−T
A
T T
A
2
τ− 2τ+T
TN
PHASE CODED PULSES HOPPED FREQUENCY PULSES
PULSED (INTERPULSE CODING)
t
( )tf
A
T
2/τ−
LOW PRFMEDIUM PRF
PULSED( )tf
T T T T
2/τ+
τ
HIGH PRF
TT T T
A Partial List of the Family of RADAR Waveforms (continue – 1)
Pulses
Waveform Hierarchy
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SOLO RADAR Types
Frequency Modulated CW Radar Multi-Frequencies CW Radar
Step Frequency Pulse Radar Coherent Pulse Radar
Examples of CW and Pulse Radars
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SOLO
Radar Generic Procedures: Matched Filters in RADAR Systems
• Transmits high frequency (f0) EM signal: ( ) ( ) ( )[ ]ttftEtEt 0000 2cos ϕπ +=
( ) ( ) ( ) ( ) ( )[ ] ( )tnoisettttffttEtE ddDdr +−+−⋅+−≈ ϕπα 00 2cos
• Receives low power reflected EM signal that contains doppler information (f0 + fD):
• Down-converts to Intermediate Frequency (IF) signal (fIF + fD), Amplifies at Low Noise, and Automatically Controls the Gain (AGC) of the receiver:
( ) ( ) ( ) ( ) ( )[ ] ( )tnoisettttffttEGtE IFddDIFdIFIF +−+−⋅+−≈ ϕπα 2cos0
• Down-converts to Video Frequency (V) signal (fV + fD), (often using a Synchronous I,Q configuration), samples the video (A/D) for Digital Signal Processing.
• The Digital Signal Processing (DSP) performs Fast Fourier Transforms (FFT), to produce the Data Cube (Range, Doppler, Receiving Channels). Using the data DSP detects the potential targets, and computes the receiving delay td (Range), Doppler frequency (closing velocity), angular target position. According to the Radar policy, he will acquire the targets of interest, and will track them. Doing this he prevents unwanted signal (Clutter, ECM, …) to interfere with the target of interest received signals.
This presentation deals with some aspects of the Radar Digital Processing.
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SOLO
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Fourier Transform
( ) ( ){ } ( ) ( )∫+∞
∞−
−== dttjtftfF ωω exp:F
SOLO
Jean Baptiste JosephFourier
1768 - 1830
F (ω) is known as Fourier Integral or Fourier Transformand is in general complex
( ) ( ) ( ) ( ) ( )[ ]ωφωωωω jAFjFF expImRe =+=
Using the identities
( ) ( )tdtj δ
πωω =∫
+∞
∞− 2exp
we can find the Inverse Fourier Transform ( ) ( ){ }ωFtf -1F=
( ) ( ) ( ) ( ) ( )
( ) ( )( ) ( ) ( ) ( ) ( )[ ]002
1
2exp
2expexp
2exp
++−=−=−=
−=
∫∫ ∫
∫ ∫∫∞+
∞−
∞+
∞−
∞+
∞−
+∞
∞−
+∞
∞−
+∞
∞−
tftfdtfdd
tjf
dtjdjf
dtjF
ττδττπωτωτ
πωωττωτ
πωωω
( ) ( ){ } ( ) ( )∫+∞
∞−
==πωωωω
2exp:
dtjFFtf -1F
( ) ( ) ( ) ( )[ ]002
1 ++−=−∫+∞
∞−
tftfdtf ττδτ
If f (t) is continuous at t, i.e. f (t-0) = f (t+0)
This is true if (sufficient not necessary)f (t) and f ’ (t) are piecewise continue in every finite interval1
2 and converge, i.e. f (t) is absolute integrable in (-∞,∞)( )∫+∞
∞−
dttf
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( )atf −-1F
F ( ) ( )ωω ajF −exp
Fourier TransformSOLO( )tf
-1FF ( )ωFProperties of Fourier Transform (Summary)
Linearity 1 ( ) ( ){ } ( ) ( )[ ] ( ) ( ) ( )ωαωαωαααα 221122112211 exp: FFdttjtftftftf +=−+=+ ∫+∞
∞−
F
Symmetry 2
( )tF-1F
F ( )ωπ −f2
Conjugate Functions3 ( )tf *
-1FF ( )ω−*F
Scaling4 ( )taf-1F
F
a
Fa
ω1
Derivatives5 ( ) ( )tftj n−-1F
F ( )ωω
Fd
dn
n
( )tftd
dn
n
-1FF ( ) ( )ωω Fj n
Convolution6
( ) ( )tftf 21-1F
F ( ) ( )ωω 21 * FF( ) ( ) ( ) ( )∫+∞
∞−
−= τττ dtfftftf 2121 :*-1F
F ( ) ( )ωω 21 FF
( ) ( ) ( ) ( )∫∫+∞
∞−
+∞
∞−
= ωωωπ
dFFdttftf 2*
12*
1 2
1
Parseval’s Formula7
Shifting: for any a real 8( ) ( )tajtf exp
-1FF ( )aF −ω
Modulation9 ( ) ttf 0cos ω-1F
F( ) ( )[ ]002
1 ωωωω −++ FF
( ) ( ) ( ) ( ) ( ) ( )∫∫∫+∞
∞−
+∞
∞−
+∞
∞−
−=−= ωωωπ
ωωωπ
dFFdFFdttftf 212121 2
1
2
1
16
( ) ( )∫+∞
∞−
−= ωωπ
ω dejFj
tf tj
2
1
Signal
(1) C.W.
( )2
cos00
0
tjtj eeAtAtf
ωω
ω−+==
0ω - carrier frequency
Frequency
( ) ( )∫+∞
∞−
= dtetfjF tjωωFourier Transform
( ) ( ) ( )00 22ωωδωωδω ++−= AA
jFFourier Transform
SOLO Fourier Transform of a Signal
17
( ) ( )∫+∞
∞−
−= ωωπ
ω dejFj
tf tj
2
1
Signal
(2) Single Pulse
( )
>≤≤−
=2/0
2/2/
τττ
t
tAtf
τ - pulse width
Frequency
( ) ( )∫+∞
∞−
= dtetfjF tjωωFourier Transform
( ) ( ) ( )( )2/
2/sin2/
2/ τωτωτω
τ
τ
ω AdteAjF tj == ∫−
Fourier Transform
SOLO Fourier Transform of a Signal
18
( ) ( )∫+∞
∞−
−= ωωπ
ω dejFj
tf tj
2
1
Signal
( ) ( )
>≤≤−
=2/0
2/2/cos 0
τττω
t
ttAtf
τ - pulse width
Frequency
( ) ( )∫+∞
∞−
= dtetfjF tjωωFourier Transform
( ) ( )
( )
( )
( )
( )
−
−
++
+
=
= ∫−
2
2sin
2
2sin
2
cos
0
0
0
0
2/
2/
0
τωω
τωω
τωω
τωωτ
ωωτ
τ
ω
A
dtetAjF tjFourier Transform
0ω - carrier frequency
(3) Single Pulse Modulated at a frequency
0ω
ω
( )ωjF
0
τπω 2
0 +
2
τA
0ω
τπω 2
0 −τπω 2
0 +−
2
τA
0ω−
τπω 2
0 −−
τπω 2
20 +τπω 2
20 −
SOLO Fourier Transform of a Signal
19
( ) ( )∫+∞
∞−
−= ωωπ
ω dejFj
tf tj
2
1
Signal
( ) ( )
±±=>−≤−≤−+
=,2,1,0,2/0
2/2/cos 0
kkkTt
kTttAtf
rand
τττϕω
τ - pulse width
Frequency
( ) ( )∫+∞
∞−
= dtetfjF tjωωFourier Transform
( ) ( )
( )
( )
( )
( )
−
−
++
+
=
= ∫−
2
2sin
2
2sin
2
cos
0
0
0
0
2/
2/
0
τωω
τωω
τωω
τωωτ
ωωτ
τ
ω
A
dtetAjF tj
Fourier Transform
0ω - carrier frequency
(4) Train of Noncoherent Pulses (random starting pulses), modulated at a frequency 0ω
T - Pulse repetition interval (PRI)
SOLO Fourier Transform of a Signal
20
( ) ( )∫+∞
∞−
−= ωωπ
ω dejFj
tf tj
2
1
Signal
( ) ( )
( ) ( )( ) ( )( )[ ]
−++
+=
±±=>−≤−≤−
=
∑∞
=1000
0
coscos
2
2sin
cos
,2,1,0,2/0
2/2/cos
nPRPR
PR
PRseriesFourier
tntnn
n
tT
A
kkkTt
kTttAtf
ωωωωτω
τω
ωτ
τττω
τ - pulse width
Frequency
( ) ( )∫+∞
∞−
= dtetfjF tjωωFourier Transform
Fourier Transform
0ω - carrier frequency
(5) Train of Coherent Pulses, of infinite length, modulated at a frequency 0ω
T - Pulse repetition interval (PRI)
( ) ( ) ( ){
( ) ( ) ( ) ( )[ ]
+−+−+−−++
+
−+=
∑∞
=10000
00
2
2sin
2
nPRPRPRPR
PR
PR
nnnnn
n
T
AjF
ωωδωωδωωδωωδτω
τω
ωδωδτω
T/1 - Pulse repetition frequency (PRF)TPR /2πω =
SOLO Fourier Transform of a Signal
21
( ) ( )∫+∞
∞−
−= ωωπ
ω dejFj
tf tj
2
1
Signal
( ) ( )
( ) ( )( ) ( )( )[ ]
−++
+=
±±=>−≤−≤−
=
∑∞
=
≤≤−
1000
22
0
coscos
2
2sin
cos
2/,,2,1,0,2/0
2/2/cos
nPRPR
PR
PRNTt
NT
tntnn
n
tT
A
NkkkTt
kTttAtf
ωωωωτω
τω
ωτ
τττω
τ - pulse width
Frequency
( ) ( )∫+∞
∞−
= dtetfjF tjωωFourier Transform
Fourier Transform
0ω - carrier frequency
(6) Train of Coherent Pulses, of finite length N T, modulated at a frequency 0ω
T - Pulse repetition interval (PRI)
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
−−
−−
++−
+−
++
+
+
−+
−+
+++
++
++
+
=
∑
∑
∞
=
∞
=
10
0
0
0
0
0
10
0
0
0
0
0
2
2sin
2
2sin
2
2sin
2
2sin
2
2sin
2
2sin
2
2sin
2
2sin
2
nPR
PR
PR
PR
PR
PR
nPR
PR
PR
PR
PR
PR
TNn
TNn
TNn
TNn
n
n
TN
TN
TNn
TNn
TNn
TNn
n
n
TN
TN
T
AjF
ωωω
ωωω
ωωω
ωωω
τω
τω
ωω
ωω
ωωω
ωωω
ωωω
ωωω
τω
τω
ωω
ωωτω
T/1 - Pulse repetition frequency (PRF)TPR /2πω =
SOLO Fourier Transform of a Signal
22
Signal
( ) ( )
+=
±±=>−≤−≤−
= ∑∞
=11 cos
2
2sin
21,2,1,0,2/0
2/2/
nPR
PR
PRSeriesFourier
tnn
n
T
AkkkTt
kTtAtf ω
τω
τωτ
τττ
τ - pulse width0ω - carrier frequency
(6) Train of Coherent Pulses, of finite length N T, modulated at a frequency 0ω
T - Pulse repetition interval (PRI)
T/1 - Pulse repetition frequency (PRF)TPR /2πω =
( ) ( )tAtf 03 cos ω=
t
A A
( )tf1
t
2
τ2
τ−T
A
T T
22
τ+T
22
τ−T
T T
2
τ− 2
τ+T
( )tf 2
t
TN
2/TN2/TN−
( ) ( ) ( ) ( )tftftftf 321 ⋅⋅=
( ) ( ) ( ) ( ) ( )
( ) ( )( ) ( )( )[ ]
−++
+=
±±=>−≤−≤−
=⋅⋅=
∑∞
=
≤≤−
1000
22
0
321
coscos
2
2sin
cos
2/,,2,1,0,2/0
2/2/cos
nPRPR
PR
PRNTt
NT
tntnn
n
tT
A
NkkkTt
kTttAtftftftf
ωωωωτω
τω
ωτ
τττω
( )
>≤≤−
=2/0
2/2/12 TNt
TNtTNtf ( ) ( )ttf 03 cos ω=
SOLO Fourier Transform of a Signal
23
Range & Doppler Measurements in RADAR SystemsSOLORadar Waveforms and their Fourier Transforms
24
Range & Doppler Measurements in RADAR SystemsSOLORadar Waveforms and their Fourier Transforms
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25
RADAR SignalsSOLO
Waveforms
( ) ( ) ( )[ ]tttats θω += 0cos
a (t) – nonnegative function that represents any amplitude modulation (AM)
θ (t) – phase angle associated with any frequency modulation (FM)
ω0 – nominal carrier angular frequency ω0 = 2 π f0
f0 – nominal carrier frequency
Transmitted Signal
( ) ( ) ( )[ ]{ }ttjtats θω += 0exp
Phasor (complex) Transmitted Signal
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26
RADAR SignalsSOLO
Quadrature Form( ) ( ) ( )[ ]
( ) ( )[ ] ( ) ( ) ( )[ ] ( )tttattta
tttats
00
0
sinsincoscos
cos
ωθωθθω
−=+=
where: ( ) ( ) ( )[ ]( ) ( ) ( )[ ]ttats
ttats
Q
I
θθ
sin
cos
==
( ) ( ) ( ) ( ) ( )ttsttsts QI 00 sincos ωω −=
One other form: ( ) ( ) ( )[ ] ( ) ( ) ( )[ ]tjtjtjtj eeta
tttats θωθωθω −−+ +=+= 00
2cos 0
( ) ( ) ( )[ ]tjtj etgetgts 00 *
2
1 ωω −+= ( ) ( ) ( ) ( ) ( )tjQI etatsjtstg θ=+=:
Envelope of the signal
( ) ( ) tjetgts 0ω=
Phasor (complex) Transmitted Signal
Transmitted Signal
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27
RADAR SignalsSOLO
Spectrum
Define the Fourier Transfer F
( ) ( ){ } ( ) ( )∫+∞
∞−
−== dttjtstsS ωω exp:F ( ) ( ){ } ( ) ( )∫+∞
∞−
==πωωωω
2exp:
dtjSSts 1-F
( ) ( ) ( )[ ]tjtj etgetgts 00 *
2
1 ωω −+= ( ) ( ) ( )[ ]0*
02
1 ωωωωω −−+−= GGS-1FF
-1FF
( ) ( ) ( ) ( ) ( )tjQI etatsjtstg θ=+=:
( ) ( ) ( )[ ]tttats θω += 0cosInverse Fourier Transfer F -1
Envelope of the signalWe defined:
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28
RADAR SignalsSOLO
Energy ( ) ( ) ( )[ ]tttats θω += 0cos
( ) ( ) ( )[ ]{ } ( )∫∫∫+∞
∞−
+∞
∞−
+∞
∞−
≈++== dttadttttadttsEs2
022
2
122cos1
2
1: θω
Parseval’s Formula
Proof:
( ) ( ) ( ) ( )∫∫+∞
∞−
+∞
∞−
= ωωωπ
dFFdttftf 2*
12*
1 2
1
( ) ( ) ( )∫+∞
∞−
−= dttjtfF ωω exp11
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫ ∫∫ ∫∫+∞
∞−
+∞
∞−
+∞
∞−
+∞
∞−
+∞
∞−
+∞
∞−
=−=−=πωωω
πωωω
πωωω
22exp
2exp 2
*
112*
2*
12*
1
dFF
ddttjtfFdt
dtjFtfdttftf
( ) ( ) ( )∫+∞
∞−
−=πωωω
2exp*
2
*
2
dtjFtf
If s (t) is real, than s (t) = s*(t) and
( ) ( ) ( )∫∫∫+∞
∞−
+∞
∞−
+∞
∞−
=== ωωπ
dSdttsdttsEs
222
2
1:
29
RADAR SignalsSOLO
Energy (continue – 1) ( ) ( ) ( )[ ]tttats θω += 0cos
( ) ( ) ( )∫∫∫+∞
∞−
+∞
∞−
+∞
∞−
=== ωωπ
dSdttsdttsEs
222
2
1:
( ) ( ) ( ) ( )[ ] ( ) ( )[ ]( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
−−−+−−−+
−−−−+−−=
−−+−−−+−=
−
−−
00
0000
0
*
0
*2
00
0
*
00
*
0
00
*
0
*
0
*
4
1
4
1
ϕϕ
ϕϕϕϕ
ωωωωωωωωωωωωωωωω
ωωωωωωωωωω
jj
jjjj
eGGeGG
GGGG
eGeGeGeGSS
For finite band (W << ω0 ) signals (see Figure)
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )∫∫∫
∫∫∞+
∞−
∞+
∞−
∞+
∞−
+∞
∞−
−+∞
∞−
=−−−−=−−
≈−−−=−−−
ωωωωωωωωωωωωω
ωωωωωωωωωω ϕϕ
dGGdGGdGG
deGGdeGG jj
*
0
*
00
*
0
2
0
*
0
*2
00 000
( ) ( ) gs EdGdSE2
1
2
1
2
1
2
1:
22 =≈= ∫∫+∞
∞−
+∞
∞−
ωωπ
ωωπ
Return to Table of Content
30
RADAR SignalsSOLO
Complex and Analytic Signals
( ) ( ) ( )[ ]tttats θω += 0cosWe have the following definitions:
Real signal
( ) ( ) ( )[ ]tjtj etgetgts 00 *
2
1 ωω −+=
( ) ( ) ( ) ( ) ( )tjQI etatsjtstg θ=+=: Envelope of the signal
( ) ( ) ( )[ ] ( ) tjetgtjtjtats 00exp: ωθω =+= Complex Signal
( ) ( ) ( )[ ]tjtj etgetgts 00 *
2
1 ωω −+= ( ) ( ) ( )[ ]0*
02
1 ωωωωω −−+−= GGS-1F
F
( ) ( ){ } ( ){ } ( )00 ωωω ω −=== GetgtsS tjFF
( ) ( ) ( ) ( ) ( )ωωωωω
ωωω SUS
GS 200
020 =
<>
≈−=
For Band limited signals
31
RADAR SignalsSOLO
Complex and Analytic Signals (continue – 1)
( ) ( ) ( )[ ] ( ) tjetgtjtjtats 00exp: ωθω =+=
Complex Signal
( ) ( ){ } ( ){ } ( )00 ωωω ω −=== GetgtsS tjFF
( ) ( ) ( ) ( ) ( )ωωωωω
ωωω SUS
GS 200
020 =
<>
≈−=
For Band limited signals
Analytic Signal
The Analytic Signal is a Complex Signal chosen that its spectrum if forced to be zero for ω<0.
( ) ( ) ( ) ( )[ ] ( )ωωωωω SsignSUS +== 12:~
( )
<−=>+
=01
00
01
:
ωωω
ωsign
( )[ ] ( )[ ] ( )t
jtsignU
πδωω +=+= −− 12 11 FF
The time function corresponding to the product of the spectrums of two time functions isgiven by the time convolution of the two functions
( ) ( )[ ] ( ) ( )[ ] ( ) ( ) ( ) ( ) ( )∫∫
+∞
∞−
+∞
∞−
−−
−+=
−
+−=== ξξ
ξπ
ξξπ
ξδξωωω dt
sjtsd
t
jtsSUS 2
~~ 11 FFts
32
RADAR SignalsSOLO
Analytic Signal
The Analytic Signal is a Complex Signal chosen that its spectrum if forced to be zero for ω<0.
( ) ( )[ ] ( ) ( )[ ] ( ) ( ) ( ) ( ) ( )∫∫
+∞
∞−
+∞
∞−
−−
−+=
−
+−=== ξξ
ξπ
ξξπ
ξδξωωω dt
sjtsd
t
jtsSUS 2
~~ 11 FFts
( ) ( ) ( ) ( ) ( )tsjtsdt
sjts ˆ~ +=
−+= ∫
+∞
∞−
ξξ
ξπ
ts
or
Complex and Analytic Signals (continue – 2)
From ( ) ( )[ ] ( ) ( ) ( )ωωωωω SjSSsignS ˆ1:~ +=+=
we have
( ) ( ) ( )( )
( )
<+=>−
=−=0
00
0ˆ
ωωωωω
ωωωSj
Sj
SsignjS
Assuming a Band Limited signal we can assume that
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )tsjtststsSUSS ˆ~2~ +=≈⇒=≈ ωωωω
where is the Hilbert Transform of s (t)( ) ( )∫
+∞
∞− −= ξ
ξξ
πd
t
sts
1:ˆ
(see “Hilbert Transformation” Presentation)
Return to Table of Content
33
Signals
( ) ( )∫+∞
∞−
= fdefSts tfi π2
SOLO
Signal Duration and Bandwidth
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )∫∫ ∫
∫ ∫∫ ∫∫∞+
∞−
∞+
∞−
∞+
∞−
−
∞+
∞−
∞+
∞−
−∞+
∞−
∞+
∞−
∞+
∞−
=
=
=
=
dffSfSdfdesfS
dfdefSsdfdefSsdss
tfi
tfitfi
ττ
τττττττ
π
ππ
2
22
( ) ( )∫+∞
∞−
= fdefSts tfi π2 ( ) ( ) ( )∫+∞
∞−
== fdefSfitd
tsdts tfi ππ 22'
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )∫∫ ∫
∫ ∫∫ ∫∫∞+
∞−
∞+
∞−
∞+
∞−
−
+∞
∞−
+∞
∞−
−+∞
∞−
+∞
∞−
−+∞
∞−
=
−=
−=
−=
dffSfSfdfdesfSfi
dfdesfSfidfdefSfsidss
tfi
tfitfi
222
22
2'2
'2'2''
πττπ
ττπττπτττ
π
ππ
( ) ( )∫∫+∞
∞−
+∞
∞−
= dffSds 22 ττ
Parseval Theorem
From
From
( ) ( )∫∫+∞
∞−
+∞
∞−
= dffSfdtts2222
4' π
34
Signals
( )
( )
( ) ( )
( )
( ) ( )
( )
( ) ( )
( )
( ) ( )
( )∫
∫
∫
∫ ∫
∫
∫ ∫
∫
∫
∫
∫∞+
∞−
+∞
∞−∞+
∞−
+∞
∞−
+∞
∞−
−
∞+
∞−
+∞
∞−
+∞
∞−
−
∞+
∞−
+∞
∞−∞+
∞−
+∞
∞− =====dffS
fdfdfSd
fSi
dffS
fdtdetstfS
dffS
tdfdefStst
dffS
tdtstst
tdts
tdtst
t
fifi
22
2
2
2
22
2
2:
πππ
SOLO
Signal Duration and Bandwidth (continue – 1)
( ) ( )∫+∞
∞−
−= tdetsfS tfi π2 ( ) ( )∫+∞
∞−
= fdefSts tfi π2Fourier
( ) ( )∫+∞
∞−
−−= tdetstifd
fSd tfi ππ 22( ) ( )∫
+∞
∞−
= fdefSfitd
tsd tfi ππ 22
( )
( )
( ) ( )
( )
( ) ( )
( )
( ) ( )
( )
( ) ( )
( )∫
∫
∫
∫ ∫
∫
∫ ∫
∫
∫
∫
∫∞+
∞−
+∞
∞−∞+
∞−
+∞
∞−
+∞
∞−∞+
∞−
+∞
∞−
+∞
∞−∞+
∞−
+∞
∞−∞+
∞−
+∞
∞−
−=
====tdts
tdtd
tsdtsi
tdts
tdfdefSfts
tdts
fdtdetsfSf
tdts
fdfSfSf
fdfS
fdfSf
f
fifi
22
2
2
2
22
2 2222
:
ππ ππππ
35
Signals
( ) ( ) ( ) ( ) ( )∫∫∫∫∫+∞
∞−
+∞
∞−
+∞
∞−
+∞
∞−
+∞
∞−
=≤
dffSfdttstdttsdttstdtts
222222
2
2 4'4
1 π
( ) ( )∫∫+∞
∞−
+∞
∞−
= dffSdts22 τ
SOLO
Signal Duration and Bandwidth (continue – 2)
0&0 == ftChange time and frequency scale to get
From Schwarz Inequality: ( ) ( ) ( ) ( )∫∫∫+∞
∞−
+∞
∞−
+∞
∞−
≤ dttgdttfdttgtf22
Choose ( ) ( ) ( ) ( ) ( )tstd
tsdtgtsttf ':& ===
( ) ( ) ( ) ( )∫∫∫+∞
∞−
+∞
∞−
+∞
∞−
≤ dttsdttstdttstst22
''we obtain
( ) ( )∫+∞
∞−
dttstst 'Integrate by parts( )
=+=
→
==
sv
dtstsdu
dtsdv
stu '
'
( ) ( ) ( ) ( ) ( )∫∫∫+∞
∞−
+∞
∞−
∞+
∞−
+∞
∞−
−−= dttststdttsstdttstst '' 2
0
2
( ) ( ) ( )∫∫
+∞
∞−
+∞
∞−
−= dttsdttstst 2
2
1'
( ) ( )∫∫+∞
∞−
+ ∞
∞−
= dffSfdtts2222
4' π
( )
( )
( )
( )
( )
( )
( )
( )∫
∫
∫
∫
∫
∫
∫
∫∞+
∞−
+∞
∞−∞+
∞−
+∞
∞−∞+
∞−
+∞
∞−∞+
∞−
+∞
∞− =≤dffS
dffSf
dtts
dttst
dtts
dffSf
dtts
dttst
2
222
2
2
2
222
2
244
4
1ππ
assume ( ) 0lim =→∞
tstt
36
SignalsSOLO
Signal Duration and Bandwidth (continue – 3)
( )
( )
( )
( )
( )
( )
22
2
222
2
24
4
1
ft
dffS
dffSf
dtts
dttst
∆
∞+
∞−
+∞
∞−
∆
∞+
∞−
+∞
∞−
≤
∫
∫
∫
∫ π
Finally we obtain ( ) ( )ft ∆∆≤2
1
0&0 == ftChange time and frequency scale to get
Since Schwarz Inequality: becomes an equalityif and only if g (t) = k f (t), then for:
( ) ( ) ( ) ( )∫∫∫+∞
∞−
+∞
∞−
+∞
∞−
≤ dttgdttfdttgtf22
( ) ( ) ( ) ( )tftsteAttd
sdtgeAts tt ααα αα 222:
22
−=−=−==⇒= −−
we have ( ) ( )ft ∆∆=2
1
37
Signals
t
t∆2
t
( ) 2ts
ff
f∆2
( ) 2fS
SOLO
Signal Duration and Bandwidth – Summary
then
( ) ( )∫+∞
∞−
−= tdetsfS tfi π2 ( ) ( )∫+∞
∞−
= fdefSts tfi π2
( ) ( )
( )
2/1
2
22
:
−
=∆
∫
∫∞+
∞−
+∞
∞−
tdts
tdtstt
t
( )
( )∫
∫∞+
∞−
+ ∞
∞−=tdts
tdtst
t2
2
:
Signal Duration Signal Median
( ) ( )
( )
2/1
2
2224
:
−
=∆
∫
∫∞+
∞−
+∞
∞−
fdfS
fdfSff
f
π ( )
( )∫
∫∞+
∞−
+ ∞
∞−=fdfS
fdfSf
f2
22
:
π
Signal Bandwidth Frequency Median
Fourier
( ) ( )ft ∆∆≤2
1
38
Signal Duration and BandwidthSOLO
( )tf-1F
F ( )ωFRelationships from Parseval’s Formula
( ) ( ) ( ) ( )∫∫+∞
∞−
+∞
∞−
= ωωωπ
dFFdttftf 2*
12*
1 2
1Parseval’s Formula7
Choose ( ) ( ) ( ) ( )tstjtftf m−== 21
( ) ( ),2,1,0
2
12
22 == ∫∫∞+
∞−
∞+
∞−
ndd
Sddttst
m
mm ω
ωω
π
( ) ( )tftj n−-1F
F ( )ωω
Fd
dn
n
and use 5a
Choose ( ) ( ) ( )n
n
td
tsdtftf == 21 and use 5b ( )tf
td
dn
n
-1FF ( ) ( )ωω Fj n
( ) ( ) ,2,1,02
1 22
2
== ∫∫∞+
∞−
∞+
∞−
ndSdttd
tsd mn
n
ωωωπ
Choosec
( ) ( ) ( ) ( ) ( ) ( ) ,2,1,0,,2,1,0
2* ==
= ∫∫
+∞
∞−
+∞
∞−
mndd
SdS
jdt
td
tsdtstj
m
mn
n
n
nmm ω
ωωωω
π
( ) ( )n
n
td
tsdtf =1
( ) ( ) ( )tstjtf m−=2
Return to Table of Content
39
( ) ( ) ( )[ ]tttats θω += 0cos
SOLOComplex Representation of Bandpass Signals The majority of radar signals are narrow band signals, whose Fourier transform islimited to an angular-frequency bandwidth of W centered about a carrier angularfrequency of ±ω0.
Another form of s (t) is
( ) ( ) ( )( )
( ) ( ) ( )( )
( )
( ) ( ) ( ) ( )ttstts
tttatttats
QI
tsts QI
00
00
sincos
sinsincoscos
ωω
ωθωθ
−=
−=
sI (t) – in phase component sQ (t) – quadrature component
1
2
Define the signal complex envelope: ( ) ( ) ( ) ( ) ( ) ( )[ ]( ) ( )[ ]tjta
tjttatsjtstg QI
θθθ
exp
sincos:
=
+=+=
Therefore:
( ) ( ) ( )[ ] ( )[ ]tstjtgts ReexpRe 0 == ω
( ) ( ) ( ) ( ) ( ) ( ) ( )tststjtgtjtgts *2
1
2
1exp
2
1exp
2
100 +=−+= ∗ ωω
or:
3
4
( ) ( ) ( )[ ]tjtjtats θω += 0expAnalytic (complex) signal
Return to Table of Content
40
( ) ( ) ( )[ ]tttats θω += 0cos
SOLOAutocorrelation The Autocorrelation Function is extensively used in Radar Signal Processing
( ) ( ) ( )∫+∞
∞−
−= tdtstsRss ττ :
Real signal For
The Autocorrelation Function is defined as:
Properties of the Autocorrelation Function:
2 ( ) ( )ττ ssss RR =−
( ) ( ) ( ) ( ) ( ) ( )τττττ
ss
tt
ss RtdtststdtstsR =−=+=− ∫∫+∞
∞−
+=+∞
∞−
''''
1 ( ) ( ) ( ) ( ) ( ) sss EfdfSfStdtstsR === ∫∫+∞
∞−
+∞
∞−
*0 Es – signal energy
3
( ) ( ) ( ) ( ) ( ) ( ) 2222
2
20sss
EE
InequalitySchwarz
ss REtdtstdtstdtstsR
ss
==−≤−= ∫∫∫∞+
∞−
∞+
∞−
∞+
∞−
τττ
( ) ( )0ssss RR ≤τ
Autocorrelation is a mathematical tool for finding specific patterns, such as the presence of a known signal which has been buried under noise.
41
SOLOAutocorrelation (continue – 1(
The Autocorrelation Function is extensively used in Radar Signal Processing
( ) ( ) ( )∫+∞
∞−
−= tdtgtgRgg ττ *:
Signal complex envelope For
The Autocorrelation Function is defined as:
Properties of the Autocorrelation Function:
2 ( ) ( )ττ *gggg RR =−
( ) ( ) ( ) ( ) ( ) ( )τττττ
*''*'*'
gg
tt
gg RtdtgtgtdtgtgR =−=+=− ∫∫+∞
∞−
+=+∞
∞−
1 ( ) ( ) ( ) ( ) ( ) sgg EfdfGfGtdtgtgR 2**0 === ∫∫+∞
∞−
+∞
∞−
Es – signal energy
3
( ) ( ) ( ) ( ) ( ) ( ) 22
2
2
2
2
22
04** ggs
EE
InequalitySchwarz
gg REtdtgtdtgtdtgtgR
ss
==−≤−= ∫∫∫∞+
∞−
∞+
∞−
∞+
∞−
τττ
( ) ( )0gggg RR ≤τ
( ) ( ) ( )[ ]tjtatg θexp:=
42
SOLOAutocorrelation (continue – 2(
The Autocorrelation Function is extensively used in Radar Signal Processing
( ) ( ) ( )∫+∞
∞−
−= tdtgtgRgg ττ *:
Signal complex envelope For
The Autocorrelation Function is defined as:
3
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )( )
( ) ( ) ( ) ( )( )
∫ ∫∫ ∫
∫ ∫∞+
∞−
∞+
∞−
∞+
∞−
∞+
∞−
=
+∞
∞−
+∞
∞−
∂∂+
∂∂=
−−∂∂==
∂∂=
0
111222
2
0
222111
1
0
212211
2
****
**00
gggg RR
gg
tdtgtgtdtgt
tgtdtgtgtdtgt
tg
tdtdtgtgtgtgRτ
τττ
ττ
( ) ( )0gggg RR ≤τ
( ) ( ) ( )[ ]tjtatg θexp:=
(continue – 1)Since Rgg (0) is a maximum of a continuous function at τ=0, we must have
( ) 002
==∂∂ ττ ggR
Therefore ( ) ( ) ( ) ( ) 0** =∂∂+
∂∂
∫∫+∞
∞−
+∞
∞−
tdtgt
tgtdtgt
tg
Return to Table of Content
43
Fourier Transform
( )tf
( ) ( )∑∞
=
−=0n
T Tntt δδ
( ) ( ) ( ) ( ) ( )∑∞
=
−==0
*
n
T TntTnfttftf δδ
( )tf *
( )tfT t
( ) ( ){ } ( ) σσ <==+∫
∞
−f
ts dtetftfsF0
L
SOLO
Sampling and z-Transform
( ) ( ){ } ( ) σδδ <−
==
−==−
∞
=
−∞
=∑∑ 0
1
1
00sT
n
sTn
n
T eeTnttsS LL
( ) ( ){ }( ) ( ) ( )
( ) ( ){ } ( ) ( )
<<−
=
=
−
==
−
∞+
∞−−−
∞
=
−∞
=
+∫
∑∑
0
00**
1
1
2
1 σσσξξπ
δ
δ
ξ
σ
σξ f
j
j
tsT
n
sTn
n
de
Fj
ttf
eTnfTntTnf
tfsF
L
LL
( )
( ) ( )( )
( )( )
( )
( )
( )( )
( )( )
( )
−=
−
−=
−=
∑∫
∑∫
∑
−−−
−−
Γ
−−
−−
Γ
−−
∞
=
−
tse
ofPoleststs
FofPoles
tsts
n
nsT
e
FResd
e
F
j
e
FResd
e
F
j
eTnf
sF
ξ
ξξ
ξ
ξξ
ξξξπ
ξξξπ
1
1
0
*
112
1
112
1
2
1
Poles of
( ) Tse ξ−−−1
1
Poles of
( )ξF
planes
Tnsn
πξ 2+=
ωj
ωσ j+
0=s
Laplace Transforms
The signal f (t) is sampled at a time period T.
1Γ2Γ
∞→R
∞→R
Poles of
( ) Tse ξ−−−1
1
Poles of
( )ξF
planeξ
Tnsn
πξ 2+=
ωj
ωσ j+
0=s
44
Fourier Transform
( )tf
( ) ( )∑∞
=
−=0n
T Tntt δδ
( ) ( ) ( ) ( ) ( )∑∞
=
−==0
*
n
T TntTnfttftf δδ
( )tf *
( )tfT t
SOLO
Sampling and z-Transform (continue – 1)
( ) ( )( )
( )
( )
( ) ( ) ∑∑
∑∑
∞+
−∞=
∞+
−∞=−−→
∞+
−∞=−−
+→
+=−
−−
+=
−
+
−=
+
−
−−−=
−−=
−−
−−
nnTse
nts
T
njs
T
njs
e
ofPolests
T
njsF
TeT
Tn
jsF
T
njsF
eT
njs
e
FRessF
ts
n
ts
ππ
ππξξ
ξ
ξπξ
πξ
ξ
ξ
ξ
212
lim
2
1
2
lim1
1
2
21
1
*
Poles of
( )ξF
ωj
σ0=s
T
π2
T
π2
T
π2
Poles of
( )ξ*F plane
js ωσ +=
The signal f (t) is sampled at a time period T.
The poles of are given by( ) tse ξ−−−1
1
( ) ( )T
njsnjTsee n
njTs πξπξπξ 221 2 +=⇒=−−⇒==−−
( ) ∑+∞
−∞=
+=
n T
njsF
TsF
π21*
45
Fourier TransformSOLO
F F -1
frequency-B/2 B/2B
F F -1
-B/2 B/2
B
1/Ts-1/Ts frequency
Sample
Sampling a function at an interval Ts (in time domain)
Anti-aliasing filters is used to enforce band-limited assumption.
causes it to be replicated at 1/ Ts intervals in the other (frequency) domain.
Sampling and z-Transform (continue – 2)
46
Fourier Transform
( )tf
( ) ( )∑∞
=
−=0n
T Tntt δδ
( ) ( ) ( ) ( ) ( )∑∞
=
−==0
*
n
T TntTnfttftf δδ
( )tf *
( )tfT t
SOLO
Sampling and z-Transform (continue – 3)
0=z
planez
Poles of
( )zF
C
The signal f (t) is sampled at a time period T.
The z-Transform is defined as:
( ){ } ( ) ( )( )
( ) ( )( )
−
−===
∑
∑
=
−
→
∞
=
−
=
iF
iF
iiF
Ts
FofPoles
T
F
n
n
ze
ze
F
zTnf
zFsFtf
ξξξ
ξ
ξξξξξ
1
0*
1
lim:Z
( ) ( )
<
>≥= ∫ −
00
,02
1 1
n
RzndzzzFjTnf
fCC
n
π
47
Fourier TransformSOLO
Sampling and z-Transform (continue – 4)
( ) ( ) ( )∑∑∞
=
−+∞
−∞=
=
+=
0
* 21
n
nsT
n
eTnfT
njsF
TsF
πWe found
For the δ (t) function we have:
( ) 1=∫+∞
∞−
dttδ ( ) ( ) ( )τδτ fdtttf =−∫+∞
∞−
The following series is a periodic function: ( ) ( )∑ −=n
Tnttd δ:
therefore it can be developed in a Fourier series:
( ) ( ) ∑∑
−=−=
n
n
n T
tnjCTnttd πδ 2exp:
where: ( )T
dtT
tnjt
TC
T
T
n
12exp
12/
2/
=
= ∫
+
−
πδ
Therefore we obtain the following identity:
( )∑∑ −=
−
nn
TntTT
tnj δπ2exp
Second Way
48
Fourier Transform
( ) ( ){ } ( ) ( )∫+∞
∞−
−== dttjtftfF νπνπ 2exp:2 F
( ) ( ) ( )∑∑∞
=
−+∞
−∞=
=
+=
0
* 21
n
nsT
n
eTnfT
njsF
TsF
π
( ) ( ){ } ( ) ( )∫+∞
∞−
== ννπνπνπ dtjFFtf 2exp2:2-1F
SOLOSampling and z-Transform (continue – 5)
We found
Using the definition of the Fourier Transform and it’s inverse:
we obtain ( ) ( ) ( )∫+∞
∞−
= ννπνπ dTnjFTnf 2exp2
( ) ( ) ( ) ( ) ( ) ( )∑∫∑∞
=
+∞
∞−
∞
=
−=−=0
111
0
* exp2exp2expnn
n sTndTnjFsTTnfsF ννπνπ
( ) ( ) ( )[ ]∫ ∑+∞
∞−
+∞
−∞=
−−== 111
* 2exp22 νννπνπνπ dTnjFjsFn
( ) ( ) ∑∫ ∑+∞
−∞=
+∞
∞−
+∞
−∞=
−=
−−==
nn T
nF
Td
T
n
TFjsF νπνννδνπνπ 2
1122 111
*
We recovered (with –n instead of n) ( ) ∑+∞
−∞=
+=
n T
njsF
TsF
π21*
Second Way (continue)
Making use of the identity: with 1/T instead of T
and ν - ν 1 instead of t we obtain: ( )[ ] ∑∑
−−=−−
nn T
n
TTnj 11
12exp ννδννπ
( )∑∑ −=
−
nn
TntTT
tnj δπ2exp
Return to Table of Content
49
Fourier TransformSOLO
Henry Nyquist1889 - 1976
http://en.wikipedia.org/wiki/Harry_Nyquist
Nyquist-Shannon Sampling Theorem
The sampling theorem was implied by the work of Harry Nyquist in 1928 ("Certain topics in telegraph transmission theory"), in which he showed that up to 2B independent pulse samples could be sent through a system of bandwidth B; but he did not explicitly consider the problem of sampling and reconstruction of continuous signals. About the same time, Karl Küpfmüller showed a similar result, and discussed the sinc-function impulse response of a band-limiting filter, via its integral, the step response Integralsinus; this band-limiting and reconstruction filter that is so central to the sampling theorem is sometimes referred to as a Küpfmüller filter (but seldom so in English).
http://en.wikipedia.org/wiki/Nyquist-Shannon_sampling_theorem
Karl Küpfmüller 1887-1977
http://www.iec.ch/cgi-bin/tl_to_htm.pl?section=person&item=71
50
Claude Elwood Shannon 1916 – 2001
http://en.wikipedia.org/wiki/Claude_E._Shannon
Fourier TransformSOLONyquist-Shannon Sampling Theorem
The sampling theorem, essentially a dual of Nyquist's result, was proved by Claude E. Shannon in 1949 ("Communication in the presence of noise"). V. A. Kotelnikov published similar results in 1933 ("On the transmission capacity of the 'ether' and of cables in electrical communications", translation from the Russian), as did the mathematician E. T. Whittaker in 1915 ("Expansions of the Interpolation-Theory", "Theorie der Kardinalfunktionen"), J. M. Whittaker in 1935 ("Interpolatory function theory"), and Gabor in 1946 ("Theory of communication").
http://en.wikipedia.org/wiki/Nyquist-Shannon_sampling_theorem
Edmund Taylor Whittaker
1873 - 1956
Dennis Gabor 1900 - 1979
Vladimir Aleksandrovich Kotelnikov 1908 - 2005
John Macnaughten Whittaker
1905 - 1985
51
Fourier TransformSOLONyquist-Shannon Sampling Theorem (continue – 1)
• Signal can be recovered if Fourier spectrum of the sampling signal do not overlap.
• Start with a band limited signal s (t) ( )2
0 fBfforfS >≡
• Sample s (t) at a time period Ts, replicates spectrum every 1/Ts Hz.
( ) ∑∞+
−∞=
−=
k sTkfjSfS
12* π
fjs π2=
( ) ( ) ( )
−= ∑
+∞
−∞=nsTnttsts δ* ( )
−= ∑
∞+
−∞=k sTjksSsS
π2*
L -1
L
FF -1
52
Fourier Transform
2
1
2
B
T
B
s
−<
SOLONyquist-Shannon Sampling Theorem (continue – 2)
• Signal can be recovered if Fourier spectrum of the sampling signal do not overlap.
BB
Ts
=
>
22
1
(Nyquist Sampling Rate)
• Complex signal band-limited to B/2 Hz requires B complex samples/second, or 2 B real samples/seconds (twice the highest frequency)
• Start with a band-limited signal f (t) ( )2
0 fBfforfF >≡ • Sample f (t) at a time period Ts,
replicates spectrum every 1/Ts Hz.
Nyquist-Shannon Sampling Theorem:
Return to Table of Content
53
Fourier TransformSOLOThe Discrete Time Fourier Transform (DTFT)
• Start with a band limited signal s (t) ( )2
0 fBfforfS >≡
• Sample s (t) at a time period Ts, replicates spectrum every 1/Ts Hz.
( )
−= ∑
∞+
−∞=k sTkfSfS
1*
( ) ( ) ( )
( ) ( )∑
∑∞+
−∞=
+∞
−∞=
−=
−=
nss
ns
TntTns
Tnttsts
δ
δ*
( ) ( )∫+∞
∞−
−= tdetsfS tfj π2 ( ) ( )∫+∞
∞−
= fdefSts tfj π2F
F -1
Continuous Fourier Transform
F
F -1
Discretization of a Continuous Signal ( ) ( )∫+∞
∞−
== fdefSTnts sTnfjs
π2
( ) ( ) ( )∑∑∞+
−∞=
−
=∞+
−∞=
− ==n
nf
fj
s
Tf
n
TnfjsDTFT
ss
s
s eTnseTnsfSπ
π2
1
2:
DTFT provides an approximation of the continuous-time Fourier transform.
Discrete Time Fourier Transform (DTFT)Define
54
Fourier TransformSOLO
The Discrete Time Fourier Transform (DTFT) (continue-1)
• Signal can be recovered if Fourier spectrum of the sampling signal do not overlap.
Discretization of a Continuous Signal ( ) ( )∫+∞
∞−
== fdefSTnts sTnfjs
π2
DTFT -1
DTFT
Discrete Time Fourier Transform (DTFT)
( ) ( ) ( )∑∑∞+
−∞=
−
=∞+
−∞=
− ==n
nf
fj
s
Tf
n
TnfjsDTFT
ss
s
s eTnseTnsfSπ
π2
1
2:
We can see that
( ) ( ) ( ) ( )∑∑∞+
−∞=
−
−∞+
−∞=
+−
===+n
DTFTnkj
nf
fj
sn
nf
fkfj
ssDTFT fSeeTnseTnsfkfS ss
s
1
222
πππ
The Discrete Time Fourier Transform SDTFT (fs) is periodic with period fs.Let compute
( ) ( )( )
( )( )
( )( )
( ) ( ) ( )[ ]( ) ( )∑ ∑
∑ ∫∫ ∑∫
∞+
−∞=
∞+
−∞=
=←≠←
+
−
−
∞+
−∞=
+
−
−
+
−
∞+
−∞=
−
+
−
=−
−=−=
==
ns
sn
nmnm
ss
f
fs
nmf
fj
s
n
f
f
nmf
fj
s
f
f n
nmf
fj
s
f
f
mf
fj
DTFT
TmsTnm
nmfTns
fnm
j
eTns
fdeTnsdfeTnsdfefS
s
s
s
s
s
s
s
s
s
s
s
s
1sin
2
10
2/
2/
2
2/
2/
22/
2/
22/
2/
2
π
π
π
π
πππ
( ) ( )∑+∞
−∞=
−=n
TnfjsDTFT
seTnsfS π2: ( ) ( )( )
( )
∫+
−
=s
s
s
T
T
nTfjDTFTss dfefSTTns
2/1
2/1
2π
55
Fourier TransformSOLO
The Discrete Time Fourier Transform (DTFT) (continue-2)
Normalization of the frequency
DTFT -1
DTFT( ) ( )∑
+∞
−∞=
−=n
TnfjsDTFT
seTnsfS π2: ( ) ( )( )
( )
∫+
−
=s
s
s
T
T
nTfjDTFTss dfefSTTns
2/1
2/1
2π
( ) ( )[ ][ ]2/1,2/1
2/1,2/1
:
*
*
+−∈
+−∈=
f
TTf
Tff
ss
s
( ) ( )∑+∞
−∞=
−=n
nfjDTFT ensfS *2* : π
DTFT -1
DTFT ( ) ( )∫+
−
=2/1
2/1
*2 ** dfefSns nfjDTFT
π
Example ( ) 1,,1,002 −== − NneAns nfj π
( ) ( )( )
( )
( ) ( )
( ) ( )
( )
( )
( )[ ]( )[ ]
( ) ( )1*
0
0
*
*
**
**
*2
*21
0
*2*
0
0
0
00
00
0
0
0
*sin
*sin
1
1
−−−
−−
−−
−−−
−−−
−−
−−−
=
−−
−−=
−−=
−−== ∑
Nffj
ffj
Nffj
ffjffj
NffjNffj
ffj
NffjN
n
nffjDTFT
eff
NffA
e
e
ee
eeA
e
eAeAfS
π
π
π
ππ
ππ
π
ππ
ππ
|SDTFT(f*)|
Normalized Frequency
56
Fourier TransformSOLO
The Discrete Time Fourier Transform (DTFT) (continue-3)
( ) ( )∑+∞
−∞=
−=n
nfjDTFT ensfS *2* : π
DTFT -1
DTFT ( ) ( )∫+
−
=2/1
2/1
*2 ** dfefSns nfjDTFT
π
Example ( )
≥==
=−
22&8,,00
21,,10,902
nn
nens
nfj
π
( )
≥==
=−
27&4,,00
26,,10,302
nn
nens
nfj
π
Frequency Resolution Increases with Observation Time N Ts
DTFT
DTFT
Return to Table of Content
57
Fourier Transform
( ) ( )∑−
=
−=
1
0
2
:N
n
nkN
j
sDFT eTnskSπ
SOLOThe Discrete Fourier Transform (DFT)
Assume a periodic sequence, sampled at a time period Ts, such that s (n Ts) = s [(n+kN) Ts]
The Discrete Fourier Transform (DFT) requires an input function that is discrete and whose non-zero values have a limited (finite) duration.
Unlike the Discrete-time Fourier transform (DTFT), it only evaluates enough frequency components to reconstruct the finite segment that was analyzed. Its inverse transform cannot reproduce the entire time domain, unless the input happens to be periodic (forever). Therefore it is often said that the DFT is a transform for Fourier analysis of finite-domain discrete-time functions
For the sequence s (0), s (Ts),…,s [(N-1) Ts] we define the Discrete Fourier Transform:
58
Fourier Transform
( ) ( ) ( )∑∑−
=
−
=
−==
1
0
1
0
2
:N
n
nks
N
n
nkN
j
sDFT WTnseTnskSπ
SOLOThe Discrete Fourier Transform (DFT) (continue – 1)
For the sequence s (0), s (Ts),…,s [(N-1) Ts] we define the Discrete Fourier Transform:
where is a primitive N'th root of unityand is periodic
Nj
eWπ2
:−
=
n
Nm
Nj
n
Nj
Nmn
Nj
Nmn WeeeW =
=
=
−−+
−+
1
222 πππ
( )( )( )
( )( )
( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )
[ ]
( )( )( )
( )[ ]( )[ ]
N
N
N s
s
s
s
s
s
W
NNNNNNN
NNNNNNN
NN
NN
NN
S
DFT
DFT
DFT
DFT
DFT
TNs
TNs
Ts
Ts
Ts
WWWWW
WWWWW
WWWWW
WWWWW
WWWWW
NS
NS
S
S
S
⋅−⋅−
⋅⋅⋅
=
−−
−−−−−−−
−−−−−−−
−−
−−
−−
1
2
2
1
0
1
2
2
1
0
1121211101
1222221202
1222221202
1121211101
1020201000
[ ] NNN sWS = [ ]NW is a Vandermonde type of Matrix
59
Fourier TransformSOLOThe Discrete Fourier Transform (DFT) (continue – 2)
nNmn WW =+
[ ] [ ] NH
NN IN
WW1=
Nj
eWπ2−
= 12
* −== WeW Nj
π
[ ]
( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )
=
−−−−−−−
−−−−−−−
−−
−−
−−
1121211101
1222221202
1222221202
1121211101
1020201000
NNNNNNN
NNNNNNN
NN
NN
NN
N
WWWWW
WWWWW
WWWWW
WWWWW
WWWWW
W
[ ] [ ]
( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )
==
−+−−+−−−−−−
−+−−+−−−−−−
+−+−−−
+−+−−−
+−+−−−
1112121110
2122222120
2122222120
1112121110
0102020100
*
NNNNNNN
NNNNNNN
NN
NN
NN
TN
HN
WWWWW
WWWWW
WWWWW
WWWWW
WWWWW
WW
Let multiply those two matrices
[ ] [ ]( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )( )
( )( ) ( )
( )
=
≠=−
−=−−
==
+++++=
−
−
−
−−
=
−
+−−−−
∑mkN
mkW
W
W
WW
WWWWWWWWWW
mk
mk
N
mk
NmkN
j
jmk
mNNkmjjkmkmkmk
HNN
01
1
1
1
1
1
0
111100
,
Where IN is the N x N identity matrix
60
Fourier Transform
( ) ( ) ( )∑∑−
=
−
=
−==
1
0
1
0
2
:N
n
nks
N
n
nkN
j
sDFT WTnseTnskSπ
SOLOThe Discrete Fourier Transform (DFT) (continue – 3)
For the sequence s (0), s (Ts),…,s [(N-1) Ts] we defined the Discrete Fourier Transform:
[ ] NNN sWS = [ ]NW is a Vandermonde type of Matrix
We found that
[ ] [ ] NH
NN IN
WW1= Where IN is the NxN identity matrix
Therefore the Inverse Discrete Fourier Transform (IDFT) is
[ ] NH
NN SWN
s1=
( ) ( ) ( )∑∑−
=
−
=
− ==1
0
21
0
11 N
n
nkN
j
DFT
N
k
nkDFTs ekS
NWkS
NTns
π
D.F.T.
I.D.F.T.
61
Fourier TransformSOLOThe Discrete Fourier Transform (DFT) (continue – 4)
Second way to find the Inverse Discrete Fourier Transform (IDFT). Let compute:
( ) ( ) ( ) ( ) ( )∑ ∑∑∑∑
−
=
−
=
−−−
=
−
=
−−−
=
+==
1
0
1
0
21
0
1
0
21
0
2 N
n
N
k
rnkN
j
s
N
k
N
n
rnkN
j
s
N
k
rkN
j
DFT eTnseTnsekSπππ
( )
( )
( )
( )
( )
( )[ ] ( )[ ]( ) ( )
( )[ ]( )
( )[ ] ( )[ ]( ) ( )
( )[ ]( )
( )
( )( )[ ] ( )[ ]( ) ( )
≠−=−
=
−+
−
−+−
−
−
−
−=
−+
−
−+−
−
−=
−+
−−
−+−−=−
−=−
−
=−−
−−
−−
−−
−
=
−−
∑
Nmrn
NmrnN
rnN
jrnN
rnjrn
rnN
rnN
rn
rnN
rnN
jrnN
rnjrn
rnN
rn
rnN
jrnN
rnjrn
e
e
e
e
ern
Nj
rnj
rnN
j
Nrn
Nj
N
k
rnkN
j
0cossin
cossin
sin
sin
cossin
cossin
sin
sin
2sin
2cos1
2sin2cos1
1
1
1
1
2
2
2
2
1
0
2
ππππ
π
π
ππ
ππππ
ππ
ππππ
π
π
π
π
π
( ) ( )[ ] ,2,1,01
0
2
±±=+=∑−
=
+mTmNrsNekS s
N
k
rkN
j
DFT
π
62
Fourier Transform
( ) ( ) ( )∑∑−
=
−
=
−==
1
0
1
0
2
:N
n
nks
N
n
nkN
j
sDFT WTnseTnskSπ
SOLOThe Discrete Fourier Transform (DFT) (continue – 5)
For the sequence s (0), s (Ts),…,s [(N-1) Ts] we define the Discrete Fourier Transform:
where is a primitive N'th root of unityand is periodic
Nj
eWπ2
:−
=
n
Nm
Nj
n
Nj
Nmn
Nj
Nmn WeeeW =
=
=
−−+
−+
1
222 πππ
( )( )( )
( )( )
( )( )( )
( )[ ]( )[ ]
⋅−⋅−
⋅⋅⋅
=
−−
−−
−−
−−
−−
s
s
s
s
s
NN
NN
NN
NN
DFT
DFT
DFT
DFT
DFT
TNs
TNs
Ts
Ts
Ts
WWWWW
WWWWW
WWWWW
WWWWW
WWWWW
NS
NS
S
S
S
1
2
2
1
0
1
2
2
1
0
12210
23320
23420
12210
00000
63
Fourier TransformSOLOThe Discrete Fourier Transform (DFT) (continue – 6)
The DFT ant Inverse DFT (IDFT) are given by
( ) ( )∑−
=
+=
1
0
21 N
k
nkN
j
DFTs ekSN
Tnsπ
( ) ( )∑−
=
−=
1
0
2
:N
n
nkN
j
sDFT eTnskSπ
IDFT
DFT
with the periodic properties
( )[ ] ( ),2,1,0 ±±=
=+m
TnsTmNns ss( ) ( )
,2,1,0 ±±==+
m
kSNmkS DFTDFT
The sequence s (0), s (Ts),…,s [(N-1) Ts] can be interpreted to be a sequence of finitelength, given for r = 0, 1,…,N-1, and zero otherwise or a periodic sequence, defined for all r.
64
Fourier Transform
( ) ( )∑−
=
−=
1
0
2
:N
n
nkN
j
sDFT eTnskSπ
SOLOThe Discrete Fourier Transform (DFT) (continue – 7)
The DFT ant Inverse DFT (IDFT) are given by
( ) ( )∑−
=
+=
1
0
21 N
k
nkN
j
DFTs ekSN
Tnsπ
IDFT
DFT
( ) ( )∑+∞
−∞=
−=n
nfjDTFT ensfS *2* : π( ) ( )∫
+
−
=2/1
2/1
*2 ** dfefSns nfjDTFT
π
IDTFT
DTFT
The DTFT ant Inverse DTFT (IDTFT) where given by
We can see that DFT is a sampled version of DTFT by tacking:
( ) ( )[ ][ ]2/1,2/1
2/1,2/1
1,,1,0
*
*
+−∈
+−∈
−==⇒==
f
TTf
NkTN
kf
N
kfTf
ss
ss
( ) ( ) ( ) 1,,1,0:1
0
2
−====
−
=
−
∑ NkfSeTnskSsTN
kfDTFT
N
n
nkN
j
sDFT π
65
Fourier TransformSOLOThe Discrete Fourier Transform (DFT) (continue –8)
We can see that DFT is a sampled version of DTFT :
( ) ( ) ( ) 1,,1,0:1
0
2
−====
−
=
−
∑ NkfSeTnskSsTN
kfDTFT
N
n
nkN
j
sDFT π
By changing f0 from 0.25 to 0.275 we move |SDTFT (f)| to the right, and since the samplingpoints didn’t change, we obtain different |SDFT (k)| values.
66
Fourier TransformSOLOThe Discrete Fourier Transform (DFT) (continue – 9)
We can see that DFT is a sampled version of DTFT :
( ) ( ) ( ) 1,,1,0:1
0
2
−====
−
=
−
∑ NkfSeTnskSsTN
kfDTFT
N
n
nkN
j
sDFT π
Increase sampling density from N=20 to N=60.
67
Fourier TransformSOLOThe Discrete Fourier Transform (DFT) (continue – 10)
Zero Padding
( ) ( )∑−
=
−=
1
0
2
:N
n
nkN
j
sDFT eTnskSπ
The DFT ant Inverse DFT (IDFT) are given by
( ) ( )∑−
=
+=
1
0
21 N
k
nkN
j
DFTs ekSN
Tnsπ
IDFT
DFT
Let add to the signal L-N zeros (L > N) for k = N, N+1,…,L-1, to obtain:
( ) ( )
−+=−==
=1,,1,0
1,,1,0,
LNNk
NknkTnsTks ss
( ) ( )∑−
=
−=
1
0
2
:'L
m
mkL
j
sDFT eTksmSπ
Define: ( )∑−
=
−=
1
0
2N
n
mnL
j
s eTnsπ
( )∑−
=
−
=1
0
2N
n
L
Nmn
Nj
s eTnsπ
=
L
NmSDFT
68
Fourier TransformSOLOThe Discrete Fourier Transform (DFT) (continue – 11)
Zero Padding (continue – 1)
We added to the signal L-N zeros (L > N) for k = N, N+1,…,L-1, to obtain:
( ) ( )
−+=−==
=1,,1,0
1,,1,0,
LNNk
NknkTnsTks ss
( ) ( ) ( )∑∑−
=
−−
=
−==
1
0
21
0
2
:'N
n
L
Nmn
Nj
s
L
m
mkL
j
sDFT eTnseTksmSππ
Define:
( )( )
( )∑ ∑∑ ∑−
=
−
=
−+−
=
−−
=
+==
1
0
1
0
21
0
21
0
2 11 N
k
N
k
L
Nmkn
Nj
DFT
N
n
L
Nmn
Nj
Tks
N
k
knN
j
DFT ekSN
eekSN
s
πππ
≠≠=
=
−
−
=
−
−=−
−=
−
−+
−−
−+
−−
−+
−+
−+
−+
−+
−
=
−+
∑
integer/0
integer/&/0
integer/&/1
sin
sin
1
1
1
2
2
1
0
2
notNLk
NLkNLkm
NLkNLkm
LN
mkN
LN
mk
e
ee
ee
e
e
e
ee
L
Nmk
N
Nj
L
Nmk
Nj
L
Nmk
Nj
L
Nmkj
L
Nmkj
L
Nmk
Nj
L
Nmkj
L
Nmk
Nj
L
NmkN
Nj
N
n
L
Nmkn
Nj
π
ππ
ππ
ππ
π
π
π
ππ
( ) ( )∑−
=
−
−+
−
−
=1
0
1
sin
sin1
'N
k
L
Nmk
N
Nj
DFTDFT
LN
mkN
LN
mk
ekSN
mSπ
ππ
69
Fourier TransformSOLOThe Discrete Fourier Transform (DFT) (continue – 12)
Zero Padding (continue – 11)
We added to the signal L-N zeros (L > N) for k = N, N+1,…,L-1, to obtain:
( ) ( )
−+=−==
=1,,1,0
1,,1,0,
LNNk
NknkTnsTks ss
Define:
( ) ( ) ( )∑∑−
=
−
−+−
=
−
−
−
=
==
1
0
11
0
2
sin
sin1
:'N
k
LN
mkNN
j
DFTDFT
L
m
mkL
j
sDFT
L
Nmk
N
LN
mk
ekSNL
NmSeTksmS
π
πππ
We can see that S’ DFT has more points that S DFT by a factor of L/N, but it containsno more information because it uses only the N values s (nTs). If L/N is an integer then for the m=n L/N S’ DFT (m) = S DFT (n). Between thosepoints S’ DFT (m) is an interpolation of S DFT points, with the weight
≠≠=
=
−
−
−
−+
integer/0
integer/&/0
integer/&/1
sin
sin1
notNLk
NLkNLkm
NLkNLkm
LN
mkN
LN
mk
e L
Nmk
N
Nj
π
ππ
70
Fourier TransformSOLO
The Discrete Fourier Transform (DFT) (continue – 13)
Increase sampling density from N=20 to N=60.
0
0.5
1
0
60 - SAMPLE PULSE
Signal sample
Sig
nal a
mpl
itud
e
5 10 15 20 25 30 35 40 45 50 55 60
Zero Padding from n=21 to L=60.
DFT
DFT
DFT
71
SOLO
72
SOLOProperties of The Discrete Fourier Transform (DFT) (continue – 14)
( )mns − ( ) mkN
j
DFT ekSπ2−
Linearity 1 ( ) ( )nsns 2211 αα +
Shift of a Sequence2
3
4
5
Periodic Convolution
6
7
Conjugate
8
9
IDFTDFT ( ) ( )∑
−
=
−=
1
0
2
:N
n
nkN
j
DFT enskSπ
( ) ( )∑−
=
+=
1
0
21 N
k
nkN
j
DFT ekSN
nsπ
( ) ( )kSkS DFTDFT 2211 αα +
( ) ( )nsns 21 , Periodic Sequence(Period N)
( ) ( )kSkS DFTDFT 21 , DFT(Period N)
( ) nlN
jens
π2− ( )lkSDFT −
( ) ( )∑−
=−⋅
1
021
N
m
mnsms( ) ( )kSkS DFTDFT 21 ⋅
( ) ( )nsns 21 ⋅ ( ) ( )∑−
=−⋅
1
021
1 N
lDFTDFT lkSlS
N
( )ns∗ ( )kSDFT −∗
( )ns −∗ ( )kSDFT∗
Real & Imaginary ( )[ ]nsRe
( )[ ]nsImj
( ) ( ) ( )[ ] 2/kSkSkS DFTDFTeven −+= ∗
( ) ( ) ( )[ ] 2/kSkSkS DFTDFTodd −−= ∗
73
SOLOProperties of The Discrete Fourier Transform (DFT) (continue – 15)
( ) ( ) ( )[ ] 2/: nsnsnseven −+= ∗ ( )kSDFTReEven Part10
11
12 Symmetric Proprties(only when s (n) is real)
IDFTDFT ( ) ( )∑
−
=
−=
1
0
2
:N
n
nkN
j
DFT enskSπ
( ) ( )∑−
=
+=
1
0
21 N
k
nkN
j
DFT ekSN
nsπ
( ) ( )nsns 21 , Periodic Sequence(Period N)
( ) ( )kSkS DFTDFT 21 , DFT(Period N)
( )lkSDFT −
( ) ( )( )[ ] ( )[ ]( )[ ] ( )[ ]( ) ( )( ) ( )
−−∠=∠
−=
−−=
−=
−= ∗
kSkS
kSkS
kSmkSm
kSkS
kSkS
DFTDFT
DFTDFT
DFTDFT
DFTDFT
DFTDFT
II
ReRe
Odd Part ( ) ( ) ( )[ ] 2/: nsnsnsodd −−= ∗
Return to Table of Content
74
Fourier TransformSOLOFast Fourier Transform (FFT)
John Wilder Tukey 1915 – 2000
http://en.wikipedia.org/wiki/John_Tukey
James W. Cooley1926 -
http://www.ieee.org/portal/pages/about/awards/bios/2002kilby.html
The Cooley-Tukey algorithm, is the most common fast Fourier transform (FFT) algorithm. It re-expresses the discrete Fourier transform (DFT) of an arbitrary composite size N = N1N2 in terms of smaller DFTs of sizes N1 and N2, recursively, in order to reduce the computation time to O(N log N) for highly-composite N (smooth numbers).
FFTs became popular after J. W. Cooley of IBM and John W. Tukey of Princeton published a paper in 1965 reinventing the algorithm (first invented by Gauss) and describing how to perform it conveniently on a computer
75
Fourier TransformSOLOFast Fourier Transform (FFT)
The radix-2 DIT Algorithm
The radix-2 decimation-in-time (DIT) FFT is the simplest and most common form of the Cooley-Tukey algorithm, although highly optimized Cooley-Tukey implementations typically use other forms of the algorithm as described below. Radix-2 DIT divides a DFT of size N into two interleaved DFTs (hence the name "radix-2") of size N/2 with each recursive stage.
( ) ( ) ( )∑∑−
=
−
=
−==
1
0
1
0
2
:N
n
nks
N
n
nkN
j
sDFT WTnseTnskSπ
For the sequence s (0), s (Ts),…,s [(N-1) Ts] we define the Discrete Fourier Transform:
1,1, 22/12
*2
+==−====→= −−−− ππππ
jNjevenN
NNj
Nj
eWeWWeWeW
Suppose N is a power of 2; i.e. N=2L (L is integer). Since N is a even integer, let compute SDFT (k) by separate s (nTs) into two (N/2)-point sequences consisting of the even-numberedpoints (n=2r) and odd numbered points (n=2r+1).
( ) ( )( )
( ) ( )( )
( ) ( )( )( ) ( )( )
∑∑
∑∑−
=
−
=
−
=
+−
=
++=
++=
12/
0
212/
0
2
12/
0
1212/
0
2
122
122
N
n
kr
Nk
N
N
n
kr
N
N
n
krN
N
n
krNDFT
WrsWWrs
WrsWrskS
76
Fourier TransformSOLO
Fast Fourier Transform (FFT)
The radix-2 DIT Algorithm (continue – 1)
2/2/
2222
NN
jN
j
N WeeW ==
=
−− ππ
We divided the N-point DFT into two N/2-points DFTs.
( ) ( ) ( )( )( ) ( )( )
( )( )
( )
( )( )
( )
kH
N
n
krN
kN
kG
N
n
krN
N
n
krN
kN
N
n
krNDFT
WrsWWrs
WrsWWrskS
∑∑
∑∑−
=
−
=
−
=
−
=
++=
++=
12/
02/
12/
02/
12/
0
212/
0
2
122
122
Since
77
Fourier TransformSOLO
Fast Fourier Transform (FFT)
The radix-2 DIT Algorithm (continue – 2)
We divided the N-point DFT into two N/2-points DFTs.
Reduction of an 8-points FFT to two4-points FFTs
A 2-points FFT(Butterfly)
Reduction of an 4-points FFT to two2-points FFTs
78
Fourier TransformSOLO
Fast Fourier Transform (FFT)
The radix-2 DIT Algorithm (continue – 3)
Flow Diagram for an 8-points FFT
79
Fourier TransformSOLO
Fast Fourier Transform (FFT)
The radix-2 DIT Algorithm (continue – 2) ( ) ( ) kkj
kN
N
jNk
N eeW 12
2/ −==
= −− π
π
We divided the N-point DFT into two N/2-points DFTs.
( ) ( ) ( ) ( )[ ]( )( ) ( )
( )
( )
∑∑−
=−
−
=
+
++=++=
12/
01
2/12/
0
2/ 2/2/N
n
knN
NkN
N
n
NnkN
knNDFT WWNnsnsWNnsWnskS
k
Since N/2 is an even integer (N=2L)
( ) ( ) ( )[ ]( )
( )( )
( )
( )
tgofFFTN
N
n
nlN
WW
N
N
n
nlN
ng
DFT WngWNnsnslkSNN
L
2/
12/
02/
2
12/
0
2 2/2
2/2 ∑∑−
=
=
=
−
=
=++==
( ) ( ) ( )[ ]( )
( )( )
( )
( )
thofFFTN
N
n
nlN
WW
N
N
n
nlN
nh
nNDFT WnhWWNnsnslkS
NN
L
2/
12/
02/
2
12/
0
2 2/2
2/12 ∑∑−
=
=
=
−
=
=+−=+=
80
Fourier TransformSOLO
Fast Fourier Transform (FFT)
The radix-2 DIT Algorithm (continue – 3)
We divided the N-point DFT into two N/2-points DFTs.
Reduction of an 8-points FFT to two4-points FFTs
Reduction of an 4-points FFT to two2-points FFTs
A 2-points FFT(Butterfly)
81
Fourier TransformSOLO
Fast Fourier Transform (FFT)
The radix-2 DIT Algorithm (continue – 4)
Flow Diagram for an 8-points FFT
82
Fourier Transform
( ) ( ) 1,,1,0:1
0
2
−== ∑−
=
−NkeTnskS
N
n
nkN
j
sDFT π
8 64 24 64 8
16 256 64 256 24
32 1024 160 1024 64
64 4096 384 4096 160
128 16384 896 16384 384
SOLOFast Fourier Transform (FFT)
Arithmetic Operations for a Radix FFT versus DFT
For N = 2L we have L stages of Radix FFT and:
For N-point DFT we have:
For each row we have N complex additions and N complex multiplications, therefore for the N rows we have
Number of complex additions DFT = Number of complex multiplications DFT = NxN=N2
Number of complex additions FFT =N L=N log2 N
Number of complex additions FFT =N/2 (multiplications per stage) x L -1 =N/2 log2 (N/2)
Operation
Complex additions Complex multiplications
DFT DFTFFT FFTN=2L
Approximate number of Complex Arithmetic Operations Required for 2L-point DFT and FFT computations
Return to Table of Content
83
Fourier TransformSOLODigital Filtering
Digital Filters can be partitioned in two distinct classes:
• Finite Impulse Response (FIR) filters that have an impulse response h (nT) of finite duration
( ) ( )
≥<−=
=Nnn
NnnhTnh
&,00
1,,1,0
• Infinite Impulse Response (IIR) filters that have an impulse response h (nT) of infinite duration
If s (n) is an input signal to the digital filter, then the output of the digital filter y (k) is related to the input by a relation of the type:
( ) ( ) ( )[ ] ( )[ ]( )[ ] ( )[ ]TMkybTkyb
TNksaTksaTksaTky
M
N
−−−−−−++−+=
1
1
1
10
If all the coefficients ai, bi are constants we can use the z transform to obtain:
( ) ( ) ( ) ( )zSzHzSzbzb
zazaazY
MM
NN ⋅=⋅
++++++= −−
−−
1
1
110
1
For a causal filter N ≤ M.
84
Fourier TransformSOLODigital Filtering (continue – 1)
( ) ( ) ( )zSzHzY ⋅=
( )N
N
NN
zbzb
zazaazH −−
−−
++++++=
1
1
110
1
If b1 = b2= … =bN =0 ( ) NN zazaazH −− +++= 1
10
This is Finite Impulse Filter (FIR) with ( ) −=
=otherways
Nnanh n
0
1,,1,0
If this is not the case we obtain the Infinite Impulse Filter (IIR) with
( ) Hn
nNN
NN rzzczcazbzb
zazaazH <++++=
++++++= −−
−−
−−
1101
1
110
1
where
( ) ( )zCzrzdzzHzj
c H
C
nn ∈∀<= ∫π2
1
85
Fourier TransformSOLODigital Filtering (continue – 2)
( ) ( ) ( )zSzHzY ⋅= ( )N
N
NN
zbzb
zazaazH −−
−−
++++++=
1
1
110
1Rewrite this as:
( ) ( ) ( ) ( ) ( ) 00111
111 =−+−++−+− −
−−−−− YSaYbSazYbSazYbSaz NN
NNN
N
( ) ( ) ( ) ( )YbSazYbSazYbSazSaY NNN
NNN −+−++−+= −
−−−−−
111
111
0
Finally:
Transformation from Transfer Function to State-Space (Method 1)
86
Fourier TransformSOLODigital Filtering (continue – 3)
( ) ( ) ( )zSzHzY ⋅= ( )N
N
NN
zbzb
zazaazH −−
−−
++++++=
1
1
110
1Rewrite this as:
( )( )( )( ) SazazazaW
WzbzbzbYN
NN
N
NN
NN
01
11
1
11
11 1
++++=
++++=−−−
−−
−−−−
−
Transformation from Transfer Function to State-Space (Method 2)
87
Fourier TransformSOLODigital Filtering (continue – 4)
88
Fourier TransformSOLODigital Filtering (continue – 5)
89
Fourier TransformSOLODigital Filtering (continue – 6)
90
Fourier TransformSOLODigital Filtering (continue – 7)
Return to Table of Content
91
Fourier TransformSOLOWindowing
• Windowing is used for DFT data to reduce Doppler side lobes
• Windowing widen main lobe and this decreases Doppler resolution
• Windowing reduces the peak of the DFT producing a processing loss, PL
• Windowing causes a modest signal to noise (S/N) loss, called loss in peak gain, or LPG.
Windows are an overlay applied to a given time series to improve the spectral qualityof the data base.
92
Fourier TransformSOLOWindowing
Rectangular [ ] ≤≤
=otherwise
Mnnw
,0
0,1
Bartlett(triangular) [ ]
≤<−≤≤
=otherwise
MnMMn
MnMn
nw
,0
2/,/22
2/0,/2
Hanning
Hammming
[ ] ( ) ≤≤−
=otherwise
MnMnnw
,0
0,/2cos5.05.0 π
[ ] ( ) ≤≤−
=otherwise
MnMnnw
,0
0,/2cos46.054.0 π
Blackman [ ] ( ) ( ) ≤≤+−
=otherwise
MnMnMnnw
,0
0,/4sin08.0/2cos5.042.0 ππ
Julius Ferdinand von Hann (1839 -1921)
Richard Wesley Hamming (1915 –1998)
93
Fourier TransformSOLOWindowing (continue – 1)
cosine
[ ]
≤≤<
−−=
otherwise
MnM
Mnnw
,0
0&5.02/
2/
2
1exp
2
σσ
Lanczos[ ]
≤≤
−
=otherwise
MnM
nnw
,0
0,12
sinc
Gauss
[ ]
≤≤
=
−
=otherwise
MnM
n
M
nnw
,0
0,sin2
cosπππ
[ ]( )
≤≤
−−
=
otherwise
MnI
Mn
I
nw
,0
0,
12
1
0
2
0
α
αKaiser
α=2π
α=3π
94
Fourier TransformSOLOWindowing (continue – 2)
Bartlett–Hann window
( )
38.0;42,0;62.0
1
2cos
2
1
1
210
210
===
−−−
−−=
aaa
N
na
N
naanw
π
Bartlett–Hann window; B=1.46 Low-resolution (high-dynamic-range) windows
Nuttall window, continuous first derivative
( )
012604.0;144232.0;487396,0;355768.0
1
6cos
1
4cos
1
2cos
3210
3210
====
−−
−+
−−=
aaaa
N
na
N
na
N
naanw
πππ
Nuttall window, continuous first derivative; B=2.02
Blackman–Harris window
( )
01168.0;14128.0;48829,0;35875.0
1
6cos
1
4cos
1
2cos
3210
3210
====
−−
−+
−−=
aaaa
N
na
N
na
N
naanw
πππ
Blackman–Nuttall window Blackman–Harris window, B=1.98
Blackman–Nuttall window, B=3.77
( )
0106411.0;1365995.0;4891775,0;3635819.0
1
6cos
1
4cos
1
2cos
3210
3210
====
−−
−+
−−=
aaaa
N
na
N
na
N
naanw
πππ
95
Fourier TransformSOLOWindowing (continue – 3)
Dolph-Chebyshev window
( ) ( )[ ]
( ) ( )[ ]( ) ( )4,3,2,10cosh
1cosh
1,,2,1,0,coshcosh
coscoscos
1
1
1
≈
=
−=
=
=
−
−
−
αβ
β
πβω
ω
α
N
NkN
Nk
N
W
WIDFTnw
k
k
The α parameter controls the side-lobe level via the formula:
Side-Lobe Level in dB = - 20 α
The Dolph-Chebyshev Window (or Dolph window) minimizes the Chebyshev norm of the side lobes for a given main lobe width 2 ωc:
( ) ( ){ }ωωω WWsidelobescwwww >=∞= ∑=∑ maxmin:min
1,1,
The Chebyshev norm is also called the L - infinity norm, uniform norm, minimax norm, or simply the maximum absolute value.
96
Fourier TransformSOLOWindowing (continue – 3)
Comparison of Windows
97
Fourier Transform
SOLOWindowing (continue – 4)
Comparison of Windows
WindowType
Peak Sidelobe
Amplitude (Relative)
Approximate Width of Mainlobe
Peak Approximation
Error20 log10δ
(dB)
Equivalent Kaiser
Windowβ
Transition Width
of EquivalentKaiser
Window
Rectangular -13 4π/(M+1) -21 0 1.81π/M
Bartlett -25 8π/M -25 1.33 2.37π/M
Hanning -31 8π/M -44 3.86 5.01π/M
Hamming -41 8π/M -53 4.86 6.27π/M
Blackman -57 12π/M -74 7.04 9.19π/M
98
Fourier TransformSOLOWindowing (continue – 5)
Comparison of Windows
99
Fourier TransformSOLOWindowing (continue – 6)
Effect of Window in the Fourier Transform
• Good Effects
- Reduction of sidelobes
- Reduction of straddle loss
• Bad Effects
- Reduction in peak
- Widening of mainlobe
- Reduction in SNR
No Window
Hamming Window
∑−
=
1
0
21 N
nnw
N
21
0
1
0
2
1
∑
∑−
=
−
=
N
nn
N
nn
w
w
N
100
Fourier TransformSOLO
Return to Table of Content
101
102
103
104
105
106Return to Table of Content
107
SOLO Doppler Frequency Shift
( )ωjF
2
NAτ
ω
TNπω 2
0 +
0ω−
TNπω 2
0 −
PRωω +− 0PRωω −− 0
TPR
πω 2=TPR
πω 2=
ω0
TNπω 2
0 +
0ω
TNπω 2
0 −
PRωω +0PRωω −0
TPR
πω 2=TPR
πω 2=
2
2sin
2 τω
τωτ
n
n
NA
PR
PR
( )
( )2
2sin
0
0
NT
NT
ωω
ωω
−
−
( )
( )2
2sin
2
2s in
20
0
NTn
NTn
n
n
NA
RP
RP
PR
PR
ωωω
ωωω
τω
τωτ
−−
−−
( )ωjF
( )02ωωδτ −NA
ω
0ω−PRωω +− 0PRωω −− 0
TPR
πω 2=TPR
πω 2=
ω0
PRωω +0PRωω −0
TPR
πω 2=TPR
πω 2=
2
2sin
2 τω
τωτ
n
n
NA
PR
PR
2
2sin
2 τω
τωτ
n
nNA
PR
PR
0ω PRωω 20 +PRωω 20 −PRωω 20 −−
PRωω 30 −−PRωω 40 −−
PRωω 20 +−
PRωω 30 +− PRωω 40 +−
Fourier Transform of an Infinite Train Pulses
Fourier Transform of an Finite Train Pulses of Lenght N
( )PR
PR
PR
NA ωωωδτω
τωτ
−−
0
2
2sin
2
( ) ( )tAtf 03 cos ω=
t
A A
( )tf1
t
2
τ2
τ−T
A
T T
22
τ+T
22
τ−T
T T
2
τ− 2
τ+T
( )tf 2
t
TN
2/TN2/TN−
( ) ( ) ( ) ( )tftftftf 321 ⋅⋅=Train of Coherent Pulses,of finite length N T,modulated at a frequency 0ω
The pulse coherency is a necessary conditionto preserve the frequency information andto retrieve the Doppler of the returned signal.
Transmitted Train of Coherent Pulses
108
SOLO Doppler Frequency Shift
Fourier Transform of an Finite Train Pulses of Lenght N
2
NAτ
ω
TN
πω 20 +
0ω
TN
πω 20 −
PRωω+0PRωω−0
TPR
πω 2=TPR
πω 2=
2NAτ
ω
TN
πω 20 +
0ω
TN
πω 20 −
PRωω+0PRωω−0
TPR
πω 2=TPR
πω 2=
2
2sin
2 τω
τωτ
n
n
NA
PR
PR
( )
( )2
2sin
0
0
NT
NT
ωω
ωω
−
−
2NAτ
ω
TN
πω 20 +
0ω
TN
πω 20 −
PRωω+0PRωω−0
TPR
πω 2=TPR
πω 2=
πω
λ 2&
2P R
DopplerDopple r ftdRd
f <
−=
πω
λ 2&
2PR
Dopple rDopple r ftdRd
f >
−=
Fourier Transform of theTransmitted Signal
Fourier Transform of theReceiveded Signal
with Unambiguous Doppler
Fourier Transform of theReceiveded Signal
with Ambiguous Doppler
Received Train of Coherent Pulses
The bandwidth of a single pulse is usually several order of magnitude greater than theexpected doppler frequency shift 1/τ >> f doppler. To extract the Doppler frequency shift,the returns from many pulses over an observation time T must be frequency analyzed sothat the single pulse spectrum will separate into individual PRF lines with bandwidthsapproximately given by 1/T.
From the Figure we can seethat to obtain an unambiguousDoppler the following conditionmust be satisfied:
PRFc
td
Rdf
td
Rd
f PRMaxMaxdoppler =≤==
πω
λ 2
22 0
or02 f
PRFc
td
Rd
Max
≤
Return to Table of Content
109
SOLO Coherent Pulse Doppler Radar
An idealized target doppler response will provide at IF Amplifier output the signal:
( ) ( )[ ] ( ) ( )[ ]tjtjdIFIF
dIFdIF eeA
tAts ωωωωωω +−+ +=+=2
cos
that has the spectrum:f
fIF+fd-fIF-fd
-fIF fIF
A2/4A2/4 |s|2
0
Because we used N coherent pulses ofwidth τ and with Pulse Repetition Time Tthe spectrum at the IF Amplifier output
f
-fd fd
A2/4A2/4|s|2
0
After the mixer and base-band filter:
( ) ( ) [ ]tjtjdd
dd eeA
tAts ωωω −+==2
cos
We can not distinguish between positive to negative doppler!!!
and after the mixer :
110
SOLO Coherent Pulse Doppler Radar
We can not distinguish between positive to negative doppler!!!
Split IF Signal:
( ) ( )[ ] ( ) ( )[ ]tjtjdIFIF
dIFdIF eeA
tAts ωωωωωω +−+ +=+=2
cos
( ) ( )[ ]
( ) ( )[ ]tAts
tA
ts
dIFQ
dIFI
ωω
ωω
+=
+=
sin2
cos2
Define a New Complex Signal:
( ) ( ) ( ) ( )[ ]tjQI
dIFeA
tsjtstg ωω +=+=2
ffIF+fd
fIF
A2/2|g|2
0
f
fd
A2/2|s|2
0
Combining the signals after the mixers
( ) tjd
deA
tg ω
2=
We now can distinguish between positive to negative doppler!!!
111
SOLO Coherent Pulse Doppler Radar
Split IF Signal:
( ) ( )[ ]
( ) ( )[ ]tAts
tA
ts
dIFQ
dIFI
ωω
ωω
+=
+=
sin2
cos2
Define a New Complex Signal:
( ) ( ) ( ) ( )[ ]tjQI
dIFeA
tsjtstg ωω +=+=2
ffd
A2/2|s|2
0
Combining the signals after the mixers
( ) tjd
deA
tg ω
2=
We now can distinguish between positive to negative doppler!!!
From the Figure we can see that in this case the doppler is unambiguous only if:
Tff PRd
1=<
Because we used N coherent pulses ofwidth τ and with Pulse Repetition Time Tthe spectrum after the mixer output is
Return to Table of Content
112
SOLO Signal Processing
Collecting Pulsed Radar Data: 1 Pulse, Multiple Range-Gates Samples
• when using a coherent receiver, each range sample comprises one “I” sample and one “Q” sample, forming one complex number I+j Q.• Each range cells contains an echo from a different range interval.
• Also called Range-Bins, Range-Gates, Fast-Time Samples.
113
SOLO Signal Processing
Collecting Pulsed Radar Data: Multiple Pulses
• when using a coherent receiver, each range sample comprises one “I” sample and one “Q” sample, forming one complex number I+j Q.• Repeat for multiple pulses in a “coherent processing interval” (CPI) or “dwell”
Sequence of samples for a fixed range bin represents echoes from same range interval over a period of time.
114
SOLO Signal Processing
Perform FFT in Each Range Gate
After FFT a Range-DopplerMap is obtained for SignalProcessing
FFT
Run This
115
SOLO Signal Processing
Perform FFT in Each Range Gate
Data-cube for Signal Processing
Repeat the Operation for each Receiver Channel (Σ,ΔAz,ΔEl,Γ for monopulse antenna or Σi,j for each element in an Electronic Scanned Antenna)
Range – Doppler Cells in Σ and ΔAz, ΔEl
FFT
FFT
FFT
FFT
Run This
116
SOLO Signal Processing
Adaptive algorithms use additional data from the cube for weight estimation.
Datacube for Signal Processing
Standard radar signal processing algorithms correspond to operating in 1- or 2-D alongvarious axes of the data-cube
Space-Time Adaptive Processing:2-D joint adaptive weighting acrossantenna element and pulse number
Beamforming:1-D weighting acrossElectrical Scan Antennaelement number
Pulse Compression:1-D convolution alongthe range axis(“fast time”)
Synthetic Aperture Imaging:2-D matched filtering in slowand fast time
Doppler Processing:1-D filtering or spectralanalysis along the pulse axis(“slow time”)
Run This
117
SOLO Signal Processing
Range – Doppler Cells in Σ and ΔAz, ΔEl
118
SOLO Signal Processing
Generation of Σ , ΔAz, ΔEl Range – Doppler Maps
The Parameters defining the Range – Doppler Maps are:
Δ R – Map Range Resolution
Δ f – Map Doppler Resolution
RUnambiguous – Unambiguous Range
fUnambiguous – Unambiguous Doppler
Range – DopplerCell
Range – DopplerMap
f
fM
R
RN sunambiguousunambiguou
∆=
∆= &
Range Gates are therefore i = 1, 2, …, NNumber of Range-Doppler Cells = N x M
Doppler Gates are therefore j = 1, 2, …, M
Note: The Map Range & Doppler resolution (Δ R, Δ f) may change as function of Radar task (Search, Detection, Acquisition, Track). This is done by choosingthe Pulse Repetition Interval (PRI) and the number of pulses in a batch.
resolutionresolution ffRR ≥∆≥∆ &
119
SOLO Signal Processing Generation of Σ , ΔAz, ΔEl Range – Doppler Maps (continue – 1)
The received signal from the scatter k is:
( ) ( )[ ] ( ) ( )ttTktttTkttfCts ddkdkr
kr
k ++≤≤++−= τθπ2cos
Ckr – amplitude of received signal
td (t) – round trip delay time given by ( )2/c
tRRtt kk
d
+=
θk – relative phase
The received signal is down-converted to base-band in order to extract the quadrature components. More precisely sk
r (t) is mixed with: ( ) [ ] τθπ +≤≤+= TktTktfCty kkk 2cos
After Low-Pass filtering the quadrature components of Σk, ΔAz k or ΔEl k signals are:
( ) ( )( ) ( )
==
tAtx
tAtx
kkQk
kkIk
ψψ
sin
cos
( ) ( )
+−≅−=
c
tR
c
Rfttft kkkdkk
2222 ππψ
The quadrature samples are given by:( ) ( )
+−≅=
c
tR
c
RfjAjAtX kkkkkkk
222expexp πψ
Ak - amplitude of Σk, ΔAz k or ΔEl k signals ψk - phase of Σk, ΔAz k or ΔEl k signals
( )
+−
+≅+=
c
tR
c
RfAj
c
tR
c
RfAxjxtX kk
kkkk
kkQkIkk
222sin
222cos ππ
120
SOLO Signal Processing Generation of Σ , ΔAz, ΔEl Range – Doppler Maps (continue – 2)
The received signal from the scatter k is:
The energy of the received signal is given by: ( ) ( ) 2kkkk AtXtXP == ∗
( )
+−
+≅+=
c
tR
c
RfAj
c
tR
c
RfAxjxtX kk
kkkk
kkQkIkk
222sin
222cos ππ
where * is the complex conjugate.
Therefore:kk PA =
Return to Table of Content
121
Decision/Detection TheorySOLO
Hypotheses
H0 – target is not present
H1 – target is present
Binary Detection
( )0Hp - probability that target is not present
( )1Hp - probability that target is present
( )zHp |0 - probability that target is not present and not declared (correct decision)
( )zHp |1 - probability that target is present and declared (correct decision)
Using Bayes’ rule: ( ) ( ) ( )∫=Z
dzzpzHpHp |00( ) ( ) ( )∫=
Z
dzzpzHpHp |11
( )zp - probability of the event Zz ⊂
Since p (z) > 0 the Decision rules are:
( ) ( )zHpzHp || 01 < - target is not declared (H0)
( ) ( )zHpzHp || 01 > - target is declared (H1) ( ) ( )zHpzHpH
H
|| 01
0
1
<>
122
Decision/Detection TheorySOLO
Hypotheses H0 – target is not present H1 – target is present
Binary Detection
( )zHp |0 - probability that target is not present and not declared (correct decision)
( )zHp |1 - probability that target is present and declared (correct decision)
( )zp - probability of the event Zz ⊂
Decision rules are: ( ) ( )zHpzHpH
H
|| 01
0
1
<>
Using again Bayes’ rule:
( ) ( ) ( )( ) ( ) ( ) ( )
( )zp
HpHzpzHp
zp
HpHzpzHp
H
H
00
011
1
||
||
0
1
=<>=
( )0| Hzp - a priori probability that target is not present (H0)
( )1| Hzp - a priori probability that target is present (H1)
Since all probabilities arenon-negative
( )( )
( )( )1
0
0
1
0
1
|
|
Hp
Hp
Hzp
Hzp
H
H
<>
123
Decision/Detection TheorySOLO
Hypotheses
( )1| Hzp - a priori probability density that target is present (likelihood of H1)
( )0| Hzp - a priori probability density that target is absent (likelihood of H0)
Detection Probabilities
( ) M
z
D PdzHzpPT
−== ∫∞
1| 1
( )∫∞
=Tz
FA dzHzpP 0|
( ) D
z
M PdzHzpPT
−== ∫∞−
1| 1
PD - probability of detection = probability that the target is present and declared
PFA - probability of false alarm = probability that the target is absent but declared
PM - probability of miss = probability that the target is present but not declared
T - detection threshold
DP
FAP
( )1| Hzp( )0| Hzp
MPz
Tz
( )( ) T
Hzp
Hzp
T
T =0
1
|
|
H0 – target is not present H1 – target is present
Binary Detection
( )( )
( )( ) THp
Hp
Hzp
HzpLR
H
H
=<>=
1
0
0
1
0
1
|
|:Likelihood Ratio Test (LTR)
124
Decision/Detection TheorySOLO
Hypotheses
Decision Criteria on Definition of the Threshold T
1. Bayes Criterion
DP
FAP
( )1| Hzp( )0| Hzp
MPz
Tz
( )( ) T
Hzp
Hzp
T
T =0
1
|
|
H0 – target is not present H1 – target is present
Binary Detection
( )( )
( )( ) THp
Hp
Hzp
HzpLR
H
H
=<>=
1
0
0
1
0
1
|
|:Likelihood Ratio Test (LTR)
The optimal choice that optimizes the Likelihood Ratio is ( )( )1
0
Hp
HpTBayes =
This choose assume knowledge of p (H0) and P (H1), that in general are not known a priori.
2. Maximum Likelihood Criterion
Since p (H0) and P (H1) are not known a priori, we choose TML = 1
( )1| Hzp( )0| Hzp
MP z
Tz
( )( ) 1
|
|
0
1 == ML
T
T THzp
Hzp
DP
FAP
125
Decision/Detection TheorySOLO
Hypotheses
Decision Criteria on Definition of the Threshold T (continue)
3. Neyman-Pearson Criterion
DP
γ=FAP
( )1| Hzp( )0| Hzp
MPz
Tz
( )( ) PN
T
T THzp
Hzp−=
0
1
|
|
H0 – target is not present H1 – target is present
Binary Detection
( )( )
( )( ) THp
Hp
Hzp
HzpLR
H
H
=<>=
1
0
0
1
0
1
|
|:Likelihood Ratio Test (LTR)
Neyman and Pearson choose to optimizes the probability of detection PD
keeping the probability of false alarm PFA constant.
Egon Sharpe Pearson1895 - 1980
Jerzy Neyman1894 - 1981
( )∫∞
=T
TT
zzDz
dzHzpP 1|maxmax ( ) γ== ∫∞
Tz
FA dzHzpP 0|constrained to
Let use the Lagrange’s multiplier λ to add the constraint
( ) ( )
−+= ∫∫
∞∞
TT
TT
zzzz
dzHzpdzHzpG 01 ||maxmax γλ
Maximum is obtained for:
( ) ( ) 0|| 01 =+−=∂∂
HzpHzpz
GTT
T
λ( )( ) PN
T
T THzp
Hzp−==
0
1
|
|λ
zT is define by requiring that: ( ) γ== ∫∞
Tz
FA dzHzpP 0|
Return to Table of Content
126
SOLO SEARCH & DETECT MODE During Search Mode the RADAR Seeker performs the following tasks:
• Slaves the Seeker Gimbals to the Designation Target direction (like in Slave Mode).
• Transmits the RF (by choosing the best waveform).
• Receives the returning RF.
• Compute the Σ Range-Doppler Map, chooses the Detection Threshold and policy.
• Perform Detections Clustering and compute Range and Doppler spread.
Note: Here is important to simulate the number of Batches that are needed to obtain the predefined probability of detection, the False Alarm Rate (FAR) and to resolve the differentdetections, i.e. the time necessary to perform this task.
• If a Detection is in the Target Designation (Uncertainty) Window we go to Acquisition Mode.
127
Target returns are the summation of signals (amplitude and phase) from all of the scattering centers within the radar resolution cell.
SOLO Target RCS
where
Nsc – number of scatters in the volume VResol
σk– Radar Cross Section of scatter k
Rk– Range to scatter k
The equivalent Radar Cross Section σTarget of the target in the resolution cell of volume VResol is:
2Nscatter i4
Target Resol 4i 1 iR
gV R
σσ η Σ
=
= = ∑24 N
scatter i
4i 1Resol iR
gR
V
ση Σ
=
= ∑ ( )2/4
2Resol τϕϕπ
cRV elaz=
gΣ (εAz,εEl) – antenna sum pattern ( gΣ(0,0)=1 )
R – Range to the center of the volume VResol
( ) ( )( )( )
∑=
Σ
+−=Σ
jiN
k
kkk
kElkAzproc
trver
RcvrXmtr
sc
cc
RRR
jgG
L
GGPji
,
12
k
kscatter
proc
Targ
3
20
2
Targ 22
2expR
,
L4,
πσεε
πλ
In the same way:
gΔ (εAz,εEl) – antenna difference pattern ( gΔ(0,0)=0 )
R G A AN TG EE S
DOPPLERFILTERS
Range-Doppler S cells
Detections
According to Range and Doppler of each scatter determine theRange-Doppler cell (i,j) for the scatter.
( ) ( )( )( )
∑=
∆
+−=∆
jiN
k
kkk
kElkAzElAzproc
trver
RcvrXmtr
sc
cc
RRR
jgG
L
GGPji
,
12
k
kscatter,
proc
Targ
3
20
2
Az/ElTarg 22
2expR
,
L4,
πσεε
πλ
128
SOLO SEARCH & DETECT MODE
According to the position of Target Uncertainty Window (TUW) versus Clutter chose the Range – Doppler magnitude (Runambiguous and funambiguous) by defining the Pulse Repetition Frequency (PRF) and the number of pulses in the batch, and choose resolution Δ R and Δ f.
Improvements
1. Change Range-Doppler cells indexes i,j tobring the Target Uncertainty Window inthe middle of the Range-Doppler Map
2. Choose on the Range-Doppler Map aarea that includes the Target UncertaintyWindow and perform Ground Cluttercomputations only for this area (we may addGround Clutter computations in Main Lobeand Altitude Line: Rk = hI).
Transmits the RF (by choosing the best waveform).
Computation of the Σ Range-Doppler Map, chooses the Detection Threshold and policy
129
SOLO SEARCH & DETECT MODE
Computation of the Σ Range-Doppler Map, chooses the Detection Threshold and policy (continue – 1)
• Computation of Noise Threshold in each cell: ( ) ( ) ( ) BFTkjijijiN NoiseNoise 0,,, =Σ⋅Σ= ∗
• Computation of Clutter Power in CFAR Window cells (Cells in area around Target Uncertainty Window):
( ) ( ) ( ) ∗Σ⋅Σ= jijijiCCFAR
,,,
• Computation of Signal Power in Target Uncertainty Window cells:
( ) ( ) ( )∗Σ⋅Σ= jijijiS ,,,Window
yUncertaintTarget
• For each Range-Doppler Cell (i,j) perform the summation of complex signals for all the scatters in this cell:
∑∑∑===
∆=∆∆=∆Σ=Σjijiji N
kkEljiEl
N
kkAzjiAz
N
kkji
,,,
1,
1,
1, ,,
130
SOLO SEARCH & DETECT MODEComputation of the Σ Range-Doppler Map, chooses the Detection Threshold and policy (continue – 2).
DOPPLERWINDOW
R W A IN NG DE O W
R G A AN TG EE S
DOPPLERFILTERS
S cells
CFARWindow
R∆
f∆
Target Uncertainty
Window
( ) ( ) ( )[ ]∑ ∗+ Σ⋅Σ=
n
j WindowCFARNoiseClutter jiji
niC ,,
1
Guard(Gap)
Window
• Computation of Clutter + Noise Threshold
• Coherent Detection:
( ) ( )( ) ( ) ClutterThjiNiCIf
ClutternoThjiNiCIf
NoiseClutter
NoiseClutter
⇒+>⇒+≤
+
+
1,
1,
( ) NoiseThNjiS +≥Window
yUncertaintTarget,
( ) ( ) ( )[ ]∑ ∗+ Σ⋅Σ=
n
j WindowCFARNoiseClutter jiji
niC ,,
1
1. If no Clutter declare a Detection in the (i,j) cell of the Target Window if
ThNoise is chosen to assure a predefinedProbability of Detection pd and of False Alarm pFA
( ) NoiseClutterNoiseClutter ThCjiS ++ +≥Window
yUncertaintTarget,
2. If Clutter declare a Detection in the (i,j) cell of the Target Window if
ThNoise is chosen to assure a predefinedProbability of Detection pd and of False Alarm pFA
131
SOLO SEARCH & DETECT MODEComputation of the Σ Range-Doppler Map, chooses the Detection Threshold and policy (continue – 3).
• Coherent Detection (M-out-of-N):How to Increase Probability of Detection and Reduce Probability of False Alarm:
Suppose that by Coherent Detection using one Range – Doppler Map we haveProbability of Detection pd and Probability of False Alarm pfa.
To Increase Probability of Detection to pD and Reduce Probability of False Alarmto pFA we use N consecutive batches (at different PRFs) , in each of them performing the Coherent Detection procedure. We declare a detection in the if we have at least M Detections for corresponding resolved Range-Doppler cells. In this way:
( ) ( )∑=
−−−
=N
Ml
lNd
ldD pp
lNl
NP 1
!!
!
( ) ( )∑=
−−−
=N
Ml
lNfa
lfaFA pp
lNl
NP 1
!!
!
Example: pd = 0.6, pfa = 10-3, N = 4, M = 2 gives pD = 0.82, pFA = 6 x10-6
Since we use different PRFs,to obtain correlation betweenDetections we must resolve theRange-Doppler ambiguities.
132
SOLO SEARCH & DETECT MODEComputation of the Σ Range-Doppler Map, chooses the Detection Threshold and policy (continue – 4).
How to Increase Probability of Detection and Reduce Probability of False Alarm:
• Non-Coherent Detection:
To Increase Probability of Detection we use N consecutive batches, we compute thepower of each (i,j) cell, , in each Range-Doppler Map and we add (non-coherently) the powers of each corresponding (i,j) cell to obtaina non-coherent Range-Doppler Map. Now we perform the detection procedureas described before to declare a Detection.
( ) ( ) ( )∗Σ⋅Σ= jijijiS ,,,
133
SOLO SEARCH & DETECT MODEPerform Detections Clustering and compute Range and Doppler spread.
• Clustering
The Target signal may be spread in more then one Σ Range-Doppler cell. Clustering Process is to group the detections in the Σ Range-Doppler Map.
Group l parameters are mean and spread:
( )( )
( )( )∑
∑∑
∑==
il
ill
l
il
ill
l jiS
jiSii
jiS
jiSii
,
,&
,
, 2
2
( )( )
( )( )∑
∑∑
∑==
il
ill
l
il
ill
l jiS
jiSjj
jiS
jiSjj
,
,&
,
, 2
2
Range
Doppler
integer=∆+= mRiRmRlsunambiguoul
RiillRl
∆−= 22σ
integer=∆+= nfifnflsunambiguoul
fjjllf l
∆−= 22σ
If the spread of Target Range/Doppler spread σRl/ σRl are too high, we may remove theTarget detection assumption and declare the group l as Clutter.
lRadar
lf
f
cR
2=
ll fRadar
R f
c σσ2
=
134
SOLO SEARCH & DETECT MODEPerform Detections Clustering and compute Range and Doppler spread.
• Altitude Line and Main Lobe Clutter
The Interceptor altitude above ground hI is unknown (for simulation purposes we assume that the SeekerProcessor uses an estimation ĥI of hI). Thereforeis necessary to search for Altitude Line (Zero Doppler)and the Main Lobe Clutter in order to properly choosethe PRFs and the Σ Range-Doppler Map.
clutterdf _
( )RangeR
( )RangeR
Clutter
No Clutter
ClutterPower
ClutterPower
Main LobeClutter(MLC)
Altitude Return
λMV2
pMV θ
λcos
2
AAM e
Vcoscos
2 ψλ
pMV θ
λsin
2
pMV θ
λcos
2−
TargetRange
TargetDoppler
( ) ApA
I
e
h
ψθ cossin +
12
N1 2 M
Range-Doppler Map
• Check that the detection are from returns in the Main Lobe by comparing the signal power with the antenna Γuard power.
( ) ( ) ( ) ∗∗ Γ⋅Γ>Σ⋅Σ= jijijiS ,,,Window
yUncertaintTarget
If true the received signal is in the Main Lobe If not the received signal is in the Side Lobe and therefore rejected.
Return to Table of Content
135
SOLO ACQUISITION MODE
During Acquisition Mode the RADAR Seeker performs the following tasks:
• Slaves the Seeker Gimbals to the Designated Target direction.• The Angular Tracker is initialized.
• Confirms that the Detection is steady and in the Designated Zone by solving the ambiguities in Range and Doppler by using a number of Batches with different PRFs (Pulse Repetition Frequency).
• The Angular Tracker uses the Δ Elevation and Δ Azimuth Maps, computes the Radar Errors in the Detected Range-Doppler cells, and controls the gimbals in the Track Mode, by closing the track loops.
• Compute the Σ and Δ Range-Doppler Maps.
136
SOLO ACQUISITION MODE
In the Acquisition Mode the RADAR Seeker Signal Processor continue toPerform Detection in the Target Uncertainty Window of the Σ Range-Doppler Map asin Detection Mode, performing Detection cells Clustering. The Δ Elevation and Δ Azimuth Maps, are used to compute the Angular Radar Errors in the Detected Range-Doppler cells. For a cluster of l cells:
( ) ( )( ) ( )∑
Σ⋅Σ∆⋅Σ
= ∗
∗
lCluster ll
AzlldbAzAz
jiji
jiji
,,
,,Re
23θ
ε( ) ( )
( ) ( )∑
Σ⋅Σ∆⋅Σ
= ∗
∗
lCluster ll
EllldbElEl
jiji
jiji
,,
,,Re
23θ
ε
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137
SOLO
References
J.V. DiFranco, W.I. Rubin, “RADAR Detection”, Artech House, 1981, Ch.5, pp.143-201
C.E. Cook, M. Bernfeld, “RADAR Signals An Introduction to Theory and Application”, Artech House, 1993
D. C. Schleher, “MTI and Pulsed Doppler RADAR”, Artech House, 1991, Appendix B
J. Minkoff, “Signals, Noise, and Active Sensors”, John Wiley & Sons, 1992, Ch.5
M.A. Richards, ECE 6272, “Fundamentals of Signal Processing”, Georgia Institute of Technology, Spring 2000, Appendix A, Optimum and Sub-optimum Filters
W.B. Davenport,Jr., W.L. Root,”An Introduction to the Theory of Random Signals and Noise”, McGraw Hill, 1958, pp. 244-246
N. Levanon, “Radar Principles”, John Wiley & Sons, 1988, Ch.5 & 6
Peeble, P.Z. Jr, “Radar Principles”, John Wiley & Sons, 1998
RADAR Signal Processing
N. Levanon, E. Mozeson, “Radar Signals”, John Wiley & Sons, 2004
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January 17, 2015 138
SOLO
TechnionIsraeli Institute of Technology
1964 – 1968 BSc EE1968 – 1971 MSc EE
Israeli Air Force1970 – 1974
RAFAELIsraeli Armament Development Authority
1974 – 2013
Stanford University1983 – 1986 PhD AA
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