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1
Matched Filters and Ambiguity Functions for
RADAR SignalsPart 1
SOLO HERMELIN
Updated: 01.12.08http://www.solohermelin.com
2
SOLO Matched Filters and Ambiguity Functions for RADAR Signals
Table of Content
RADAR RF SignalsMaximization of Signal-to-Noise Ratio
Continuous Linear Systems
The Matched FilterThe Matched Filter Approximations
1. Single RF Pulse2. Linear FM Modulated Pulse (Chirp)
Discrete Linear Systems
RADAR SignalsSignal Duration and Bandwidth
Complex Representation of Bandpass SignalsMatched Filter Response to a Band Limited Radar Signal
Matched Filter Response to Phase Coding
Matched Filter Response to its Doppler-Shifted Signal
3
SOLO Matched Filters and Ambiguity Functions for RADAR Signals
Table of Content (continue – 1)
Ambiguity Function for RADAR Signals
Definition of Ambiguity Function
Ambiguity Function Properties
Cuts Through the Ambiguity FunctionAmbiguity as a Measure of Range and Doppler Resolution
Ambiguity Function Close to Origin
Ambiguity Function for Single RF Pulse
Ambiguity Function for Linear FM Modulation Pulse
Ambiguity Function for a Coherent Pulse TrainAmbiguity Function Examples (Rihaczek, A.W.,
“Principles of High Resolution Radar”)
References
AMBIGUITY
FUNCTIONS
4
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 RF 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
Matched Filters in RADAR Systems
Doppler Effect
7
SOLO
The received signal model:
( ) ( ) ( ) ( ) ( )[ ] ( )tnoisettttffttEtE ddDdr +−+−⋅+−= ϕπα 00 2cos
Matched Filters in RADAR Systems
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
8
Matched Filters in RADAR SystemsSOLO
α MV R
EVTarget
Transmitter &Receiver
The transmitted RADAR RF Signal is:
( ) ( ) ( )[ ]ttftEtEt θπ += 00 2cos
( )c
tRtd
02≅
Since the received signal preserve the envelope shape of the known transmitted signalwe want to design a Matched Filter that will distinguish the signal from the receiver noise.
( ) ( )λ
λ0
/
00 22 0 tR
fc
tRf
fc
D
−=−≅
=
the received signal is: ( ) ( ) ( ) ( ) ( )[ ] ( )tnoisettttffttEtE ddDdr +−+−+−≈ θπα 00 2cos
Scaled DownIn Amplitude Two-Way
Delay
Possible Phase Modulation
DopplerFrequency
For R1 = R2 = R we obtain that
Return to Table of Content
9
Matched Filters for RADAR Signals
( ) ( ) ( )tntstv ii += Linear Filter
( )thopt
( ) ( ) ( )tntsty oo +=
SOLO
Maximization of Signal-to-Noise Ratio
Consider the problem of choosing a linear time-invariant filter hopt (t) that maximizesthe output signal-to-noise ratio at a predefined time t0.
The input waveform is: ( ) ( ) ( )tntstv ii +=
( )tsi - a known signal component
( )tni - noise (stationary random process) component
The output waveform is: ( ) ( ) ( )tntsty oo +=
Assume that the linear filter has a finite time memory T, then
( ) ( ) ( )∫ −=T
iopto dtshts0
00 τττ ( ) ( ) ( )∫ −=T
iopto dtnhtn0
00 τττ
The signal-to-noise ratio is defined as:( )( )02
02
tn
ts
N
S
o
o=
To find hopt (t) a variational technique is applied, by defining a non-optimal filter
( ) ( ) ( )tgthth opt ε+= ( ) ( ) 00
0 =−∫T
i dtsg τττwith: and ε any real.
( )02 tno - the mean square value of ( )0tno
Continuous Linear Systems
10
Matched Filters for RADAR Signals
( ) ( ) ( )tntstv ii += Linear Filter
( ) ( )tgthopt ε+( ) ( ) ( )tntsty oo ''' +=
SOLOMaximization of Signal-to-Noise Ratio
The output signal s’o (t) and noise n’o (t) at time t0 are:
( ) ( ) ( )[ ] ( )
( ) ( ) ( ) ( ) ( )00
0
0
0
0
0
00'
tsdtsgdtsh
dtsghts
o
T
i
T
iopt
T
iopto
=−+−=
−+=
∫∫
∫
τττετττ
τττετ
( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )∫
∫∫∫
−+=
−+−=−+=
T
io
T
i
T
iopt
T
iopto
dtngtn
dtngdtnhdtnghtn
0
00
0
0
0
0
0
00'
τττε
τττεττττττετ
( )[ ] ( )[ ] ( ) ( ) ( ) ( ) ( )2
0
02
0
002
02
0 2'
−+−+= ∫∫
T
i
T
iooo dtngdtngtntntn τττετττε
By the definition of the optimal filter ( )[ ] ( )[ ]202
0' tntn oo ≥
Therefore ( ) ( ) ( ) ( ) ( ) 02
2
0
02
0
00 ≥
−+− ∫∫
T
i
T
io dtngdtngtn τττετττε
Continuous Linear Systems (continue – 1(
11
Matched Filters for RADAR Signals
( ) ( ) ( )tntstv ii += Linear Filter
( ) ( )tgthopt ε+( ) ( ) ( )tntsty oo ''' +=
SOLOMaximization of Signal-to-Noise Ratio
This inequality is satisfied for all values of ε if and only if the first term vanishes
( ) ( ) ( ) ( ) ( ) 02
2
0
02
0
00 ≥
−+− ∫∫
T
i
T
io dtngdtngtn τττετττε
( ) ( ) 00
0 =−∫T
i dtsg τττ
( ) ( ) ( ) 020
00 =−∫T
io dtngtn τττ
Using we obtain:( ) ( ) ( )∫ −=T
iopto dtnhtn0
00 τττ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 00 0
00
0 0
00 =−−=−− ∫ ∫∫ ∫T T
iiopt
T T
iiopt ddtntnhgddtntnhg στστστστστστ
where is the Autocorrelation Function of the input noise.( ) ( ) ( )στστ −−=− 00: tntnR iinn ii
Continuous Linear Systems (continue – 2(
12
Matched Filters for RADAR Signals
( ) ( ) ( )tntstv ii += Linear Filter
( ) ( )tgthopt ε+( ) ( ) ( )tntsty oo ''' +=
SOLOMaximization of Signal-to-Noise Ratio
Therefore the optimality condition is:
( ) ( ) ( ) 00 0
=
−∫ ∫ τσστστ ddRhg
T T
nnopt ii
( ) ( ) 00
0 =−∫T
i dtsg τττComparing with the condition:
we obtain:
( ) ( ) ( ) TtskdRh i
T
nnopt ii≤≤−=−∫ ττσστσ 00
0
k is obtained using:
( ) ( ) ( ) ( ) ( ) ( ) ( )k
tnddRhh
kdtshts o
T T
nnoptopt
T
iopto ii
02
0 00
00
1 =−=−= ∫ ∫∫ στστσττττ( )
( )00
2
ts
tnk
o
o=
For T → ∞ we can take the Fourier Transfer of the result:
( ) ( ) ( )( ) ( )[ ]τσστσ −=
−∫∞→ 0
0
02
0
lim tsts
tndRh i
o
oT
nnoptT ii
FF
Continuous Linear Systems (continue – 3(
13
Matched Filters for RADAR SignalsSOLO
Maximization of Signal-to-Noise Ratio
( ) ( ) ( )( ) ( )[ ]τσστσ −=
−∫
∞
00
02
0
tsts
tndRh i
o
onnopt ii
FF
( ) ( ) ( )tntstv ii += Linear Filter
( )thopt
( ) ( ) ( )tntsty oo +=
( ) ( ) ( )( ) ( ) 0*
0
02
tji
o
onnopt eS
ts
tnH
ii
ωωωω −=Φ ( ) ( )( )
( )( )ω
ωωω
iinn
tji
o
oopt
eS
ts
tnH
Φ=
− 0*
0
02
Continuous Linear Systems (continue – 4(
Return to Table of Content
14
Matched Filters for RADAR Signals
( )tsi
t
T0mt
SOLO
The Matched Filter
Assume that the two-sided noise spectrum density is of a white noise, i.e.
( ) ( )στδστ −=−20N
Riinn
( ) ( )[ ]20N
Riiii nnnn ==Φ τω F
then
( ) ( ) 0*2 tji
oopt eS
N
kH ωωω −= ( ) ( ) Tttts
N
kth i
oopt ≤≤−= 0
20
( )tsi
t
t
( )tsi −T0
0T−
mt
( )tsi
t
t
t
( )tsi −
( ) ( ) TttttsN
kth mmiopt ≤≤−= ,0
2
0
T0
0T−
0 mtmtT −
mt
The optimal filter, that maximizes the Signal-to-Noise Ratio for a white noise iscalled a Matched Filter of the knownSignal si (t).
We can see that for a known inputsignal of finite duration T the optimal Matched Filter is also of finite duration T.
15
Matched Filters for RADAR Signals
( ) ( ) ( ) ( ) 02
00
2 tjiiopt eS
N
kSHS ωωωωω −==
SOLO
The Matched Filter
The signal and the noise at the output of the matched filter are found as follows:
then ( ) ( ) ( ) ( ) ( ) ( )∫ ∫∫+∞
∞−
+∞
∞−
−−+∞
∞−
−
==
πωω
πωω ωωω
2
2
2
20*2 d
dvevseSN
kdeS
N
kts vj
ittj
io
ttji
oo
m
( ) ( ) 0*2 tji
oopt eS
N
kH ωωω −=
( ) ( ) ( ) ( ) ( )∫∫ ∫+∞
∞−
+∞
∞−
+∞
∞−
−− +−== dvttvsvsN
kdv
deSvs
N
kii
o
vttjii
o0
* 2
2
20
πωω ω
The Autocorrelation Function of the input signal is defined as: ( ) ( ) ( )∫+∞
∞−
−= dvvsvsR iiss iiττ :
therefore: ( ) ( )02
ttRN
kts
iisso
o −=
( ) ( ) ( ) ( ) ( )∫∫+∞
∞−
+∞
∞−
∗
=Φ= ωω
πωωωω
πd
NS
N
kdHHtn o
io
optnnopto ii 2
2
2
1
2
1 22
2( ) ( ) mtj
i
o
opt eSN
kH ωωω −= *2
( ) ( ) ( )[ ]∫+∞
∞−
== dvvsN
kR
N
kts i
oss
oo ii
20
20
2
16
Matched Filters for RADAR SignalsSOLO
The Matched Filter
therefore:
( ) ( ) ( ) ( ) ( )∫∫+∞
∞−
+∞
∞−
∗
=Φ= ωω
πωωωω
πd
NS
N
kdHHtn o
io
optnnopto ii 2
2
2
1
2
1 22
2
( ) ( ) ( )[ ]∫+∞
∞−
== dvvsN
kR
N
kts i
oss
oo ii
20
20
2
( )[ ]( )
( )[ ]
( )
( )[ ]
( )∫
∫
∫
∫∞+
∞−
∞+
∞−
∞+
∞−
∞+
∞−
=
==
πωω
πωω 2222
2
2
2
2
2
22
2
2
2
20
dS
N
dvvs
dNS
Nk
dvvsNk
tn
ts
N
S
io
i
oi
o
io
o
o
Max
Since by Parseval’s relation: (E – input signal energy)( )[ ] ( ) Ed
Sdvvs ii == ∫∫+∞
∞−
+∞
∞−πωω2
22
( )[ ]( ) oo
o
Max N
E
tn
ts
N
S 22
20 ==
We have: ( ) ( ) ( )∫+∞
∞−
+−= dvttvsvsN
kts ii
oto 0
20
Independent of signal waveform
17
Matched Filters for RADAR Signals
( ) ( )( ) ( )
≤≤−== −∗
Ttttsth
eSH tj
00
0ωωω
SOLO
The Matched Filter (Summary(
s (t) - Signal waveform
S (ω) - Signal spectral density
h (t) - Filter impulse response
H (ω) - Filter transfer function
t0 - Time filter output is sampled (for Radar signals this is the time the received returned signal is expected to arrive)
n (t) - noise
N (ω) - Noise spectral density
Matched Filter is a linear time-invariant filter hopt (t) that maximizesthe output signal-to-noise ratio at a predefined time t0, for a given signal s (t(.
The Matched Filter output is:
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) 0
00
tjo eSSHSS
dttssdthsts
ωωωωωω
ττττττ
−∗
+∞
∞−
+∞
∞−
⋅=⋅=
+−=−= ∫∫
Return to Table of Content
18
Matched Filters for RADAR SignalsSOLO
Matched Filter Output for White Noise Spectrum
s (t) - Signal waveform with energy E
S (ω) - Signal spectral density
h (t) - Filter impulse response
H (ω) - Filter transfer function
( ) ( ) ( ) ( ) ( ) ( ) ( )0* 0
2
1
2
1ttRdeSSdeSHts ss
ttjtjo −=== ∫∫
+∞
∞−
−+∞
∞−
ωωωπ
ωωωπ
ωω
( ) ( ) ( ) ( )0*'
0
2
ss
TheoremsParsevalT
RdSSdttsE === ∫∫+∞
∞−
ωωω
so (t) - Filter output signal
N (ω) - Noise spectral density η/2
Rnn (τ) - Noise Autocorrelation Function η/2 δ (τ) ( ) ( ) ( )∫−
∞→+=
T
TT
nn dttntnT
R ττ 1lim
Rss (τ) - Signal Autocorrelation Function ( ) ( ) ( )∫−
∞→+=
T
TT
ss dttstsT
R ττ 1lim
S/N - Output Power signal-to-noise ratio E/(η/2)
t0 - Time filter output is sampled (for Radar signals this is the time the received returned signal is expected to arrive)
Return to Table of Content
19
Matched Filters for RADAR SignalsSOLO
The Matched Filter Approximations
1. Single RF Pulse
( )( )
>
≤≤−=
2/0
2/2/cos 0
p
pp
itt
ttttAts
ω
pt - pulse width
( ) ( )
( )
( )
( )
( )
−
−
++
+
=
= ∫−
−
2
2sin
2
2sin
2
cos
0
0
0
0
2/
2/
0
p
p
p
p
p
t
t
tji
t
t
t
t
tA
dtetAjSp
p
ωω
ωω
ωω
ωω
ωω ω
Fourier Transform
0ω - carrier frequency
We found: ( ) ( ) ( ) ( )∫+∞
∞−
−=−= dvtvsvsN
kttR
N
kts ii
otss
oto ii
2200
0
therefore:
( ) ( )( )
( )
( )[ ]
( )[ ]
( ) ( )[ ]
( ) ( )[ ]
( )( ) ( ) ttt
N
tkAttt
N
kA
tttvttt
tttvttt
N
kA
ttdvtvt
ttdvtvt
N
kA
ttdvtvAvA
ttdvtvAvA
N
ktR
N
kts
po
pp
o
tt
ptt
tp
pt
ttp
o
p
tt
t
p
t
tt
ott
t
p
p
t
tt
oss
oto
p
p
p
p
p
p
p
p
p
p
p
p
p
ii
0
2
0
21
1
2/
2/00
0
2/
2/00
02
2/
2/
00
2/
2/
002
2/
2/
00
2/
2/
00
0
cos/1cos
02sin2
1cos
02sin2
1cos
02coscos
02coscos
0coscos
0coscos22
02
0
ωωω
ωω
ωω
ω
ωω
ωω
ωω
ωω
ω
−=−≈
<<−−++
<<−+−=
<<−−+
<<−+
=
<<−−
<<−
==
<<−
+−
−
+
−
−
+
−
−
=
∫
∫
∫
∫
( ) ( ) ( )tntstv ii += Linear Filter
( )thopt
( ) ( ) ( )tntsty oo +=
20
Matched Filters for RADAR SignalsSOLO
The Matched Filter Approximations
1. Single RF Pulse (continue – 1(
( )( )
>
≤≤−=
2/0
2/2/cos 0
p
pp
itt
ttttAts
ω
pt - pulse width
( ) ( )( )
( )
( )
( )0
0
2
2sin
2
2sin
2 0
0
0
0
*
tj
p
p
p
p
p
tjiMF
et
t
t
t
tA
ejSjS
ω
ω
ωω
ωω
ωω
ωωωω
−
−
−
−
++
+
=
=0ω - carrier frequency
We obtained:
( ) ( )
≥
<−=
=
p
ppo
p
to
tt
tttttN
tkA
ts
0
cos/1 0
2
00
ω
( ) ( ) ( )tntstv ii += Linear Filter
( )thopt
( ) ( ) ( )tntsty oo +=
t2
τ2
τ−
( )tso
0
2
N
Ak τ
ττ−
0=mt
Return to Table of Content
21
Matched Filters for RADAR SignalsSOLO
The Matched Filter Approximations
1. Single RF Pulse (continue – 2(
( )( )
>
≤≤−=
2/0
2/2/cos 0
p
pp
itt
ttttAts
ω
pt - pulse width
( ) ( )( )
( )
( )
( )0
0
2
2sin
2
2sin
2 0
0
0
0
*
tj
p
p
p
p
p
tjiMF
et
t
t
t
tA
ejSjS
ω
ω
ωω
ωω
ωω
ωωωω
−
−
−
−
++
+
=
=
0ω - carrier frequency
We obtained:
Return to Table of Content
22
SOLO
2. Linear FM Modulated Pulse (Chirp)
( )222
cos2
0pp
i
tt
tttAts ≤≤−
+= µω
The Fourier Transform is:
( ) [ ]
( ) ( )∫∫
∫
−−
−
++−+
+−=
−
+=
2/
2/
2
0
2/
2/
2
0
2/
2/
2
0
2exp
2
1
2exp
2
1
exp2
cos
p
p
p
p
p
p
t
t
t
t
t
t
i
dtt
tjAdtt
tjA
dttjt
tAS
µωωµωω
ωµωω
∫∫−−
++−
++
−−
−−=2/
2/
2
0
2
0
2/
2/
2
0
2
0
2exp
2exp
22exp
2exp
2
p
p
p
p
t
t
t
t
dttjjA
dttjjA
µωωµ
µωω
µωωµ
µωω
Change variables: xt =
−−
µωω
πµ 0 yt =
++
µωω
πµ 0
( ) ∫∫−−
−
++
−−=
2
1
2
1
2exp
2exp
22exp
2exp
2
22
02
2
0
Y
Y
X
X
i dty
jjA
dtx
jjA
Sπ
µωωπ
µωωω
−−=
−+=µ
ωωπµ
µωω
πµ 0
20
1 2&
2pp t
Xt
X
+−=
++=µ
ωωπµ
µωω
πµ 0
20
1 2&
2pp t
Yt
Y
Define: ( )fntf p ∆=−=∆ πωωµ
π2
2&
2
1: 0
Matched Filters for RADAR Signals
23
SOLO
2. Linear FM Modulated Pulse (continue – 1)
The Fourier Transform is:
( ) ( ) ( )∫∫
−−
−
++
−−=2
1
2
12
exp2
exp22
exp2
exp2
220
220
Y
Y
X
X
i dty
jjA
dtx
jjA
Sπ
µωωπ
µωωω
The first part gives the spectrum around ω = ω0, and the second part around ω = -ω0 :
where: are Fresnel Integrals,
which have the properties:
( ) ( ) ∫∫ ==UU
dzz
USdzz
UC0
2
0
2
2sin&
2cos
ππ
( ) ( ) ( ) ( )USUSUCUC −=−−=− &
( ) ( ) ( ) ( ) ( ) ( )[ ]
( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) −+ ++−=−+−
−+
+++
−−=
ωωωωµωω
µπ
µωω
µπω
002211
2
0
2211
2
0
2exp
2
2exp
2
ii
i
SSYSjYCYSjYCjA
XSjXCXSjXCjA
S
Matched Filters for RADAR Signals
( )222
cos2
0pp
i
tt
tttAts ≤≤−
+= µω
ωωωπωµπ
∆=−∆=∆=∆2
:&2:2
1: 0
nftf p
24
SOLO Fresnel Integrals
Augustin Jean Fresnel1788-1827
Define Fresnel Integrals
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )∫ ∑
∑∫∞
=
+
∞
=
+
+−=
=
++−=
=
α
α
ααπα
ααπα
0 0
142
0
34
0
2
!2141
2sin:
!12341
2cos:
n
nn
n
nn
nn
xdS
nn
xdC
( ) ( )ααααπα
SjCdj +=
∫
0
2
2exp
( ) ( ) 5.0±=∞±=∞± SC
( ) ( ) ( ) ( )USUSUCUC −=−−=− &
The Cornu Spiral is defined as the plot of S (u) versus C (u)
duuSd
duuCd
=
=
2
2
2sin
2cos
π
π
( ) ( ) duSdCd =+ 22
Therefore u may be thought as measuring arc length along the spiral.
25
SOLO
2. Linear FM Modulated Pulse (continue – 2)
The Fourier Transform is:
Define:
( ) ( ) ( )[ ] ( ) ( )[ ]{ }221
2
210 2XSXSXCXC
AS i +++=−
+ µπωωAmplitude Term:
Square Law Phase Term: ( ) ( )µωωω2
2
01
−−=Φ
Residual Phase Term: ( ) ( ) ( )( ) ( ) 4
1tan5.05.0
5.05.0tantan 111
21
211
2
πωτ
==++→
++=Φ −−
>>∆−
f
XCXC
XSXS
( ) ( )ntfXn
tfX pp −∆=+∆= 1
2&1
2 21
( )ω2Φ( )+
− ωω 0iS( ) ( ) ( ) ( ) ( ) ( )[ ]
( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) −+ ++−=−+−
−+
+++
−−=
ωωωωµωω
µπ
µωω
µπω
002211
2
0
2211
2
0
2exp
2
2exp
2
ii
i
SSYSjYCYSjYCjA
XSjXCXSjXCjA
S
Matched Filters for RADAR Signals
( )222
cos2
0pp
i
tt
tttAts ≤≤−
+= µω
ωωωπωµπ
∆=−∆=∆=∆2
:&2:2
1: 0
nftf p
26
SOLO
2. Linear FM Modulated Pulse (continue – 3)
Matched Filters for RADAR Signals
( )222
cos2
0pp
i
tt
tttAts ≤≤−
+= µω
ωωωπωµπ
∆=−∆=∆=∆2
:&2:2
1: 0
nftf p
( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]
( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) −+−
−−
++−=−+−
−+
+++
−−==
ωωωωµωω
µπ
µωω
µπωω
ω
ωω
002211
20
2211
20*
0
00
2exp
2
2exp
2
MFMFtj
tjtjiMF
SSeYSjYCYSjYCjA
eXSjXCXSjXCjA
eSS
27
SOLO
2. Linear FM Modulated Pulse (continue – 4)
Matched Filters for RADAR Signals
( )222
cos2
0pp
i
tt
tttAts ≤≤−
+= µω ωωωπωµ
π∆=−∆=∆=∆2
:&2:2
1: 0
nftf p
The Matched Filter output is given by: ( ) ( ) ( ) ( )∫+∞
∞−
−=−= dvtvsvsN
kttR
N
kts ii
otss
oto ii
2200
0
( )( ) ( )
( ) ( )
<<−
−+−
+
<<
−+−
+
=
∫
∫+
−
+
−
2/
2/
2
0
2
0
2/
2/
2
0
2
0
02
cos2
cos
02
cos2
cos2
0 p
p
p
p
tt
t
p
t
tt
p
oto
ttdvtv
tvv
v
ttdvtv
tvv
v
N
kts
µωµω
µωµω
We discard the double frequency term, whose contribution to the value of integral is small for large ω0,
( )( )
( )
<<−
−++−+
−+
<<
−++−+
−+
=
∫
∫+
−
+
−
2/
2/
22
0
2
0
2/
2/
22
0
2
02
02
222cos
2cos
02
222cos
2cos
0 p
p
p
p
tt
t
p
t
tt
p
oto
ttdvtvtv
tvt
tvt
ttdvtvtv
tvt
tvt
N
Akts
µµµωµµω
µµµωµµω
( )
( )
<<−−+
−++−+
+−
<<−+
−++−+
+−
=+
−
+
−
+
−
+
−
0242
222
2sin2
sin
0242
2
222sin
2sin
2/
2/
0
22
0
2/
2/
2
0
2/
2/
0
22
0
2/
2/
2
0
2
tttv
tvtvtv
t
tvt
t
tttv
tvtvtv
t
tvt
t
N
Ak
p
tt
t
tt
t
p
t
tt
t
tt
o p
p
p
p
p
p
p
p
µµω
µµµω
µ
µµω
µµω
µµµω
µ
µµω
Expanding the integrand trigonometrically
28
SOLO
2. Linear FM Modulated Pulse (continue – 5)
Matched Filters for RADAR Signals
Return to Table of Content
( )222
cos2
0pp
i
tt
tttAts ≤≤−
+= µω
ωωωπωµπ
∆=−∆=∆=∆2
:&2:2
1: 0
nftf p
The Matched Filter output is given by:
( )
<<−
+−
<<
+−
≈+
−
+
−
02
sin
02
sin
2/
2/
2
0
2/
2/
2
02
0
tttvt
t
tttvt
t
tN
Akts
p
tt
t
p
t
tt
oto
p
p
p
p
µµω
µµω
µ
( )
( )
<<−
−−−
++−
<<
−+−−
+−
=0
22sin2/
2sin
02/2
sin22
sin
2
0
2
0
2
0
2
02
ttttt
ttttt
t
tttttt
tttt
t
tN
Ak
pp
p
ppp
o µµωµµω
µµωµµω
µ
( ) ( )
( ) ( )
( )( )
( )( )
>
<
−
−
−=
<<−
+
<<
−
=
p
p
pp
pp
po
p
pp
pp
o
tt
ttt
tttt
tttt
ttN
tAk
tttttt
tttttt
tN
Ak
0
cos
/12
/12
sin
/12
0cos2
sin2
0cos2
sin20
2
0
02 ωµ
µ
ωµ
ωµ
µ
29
SOLO
2. Linear FM Modulated Pulse (continue – 6)
Matched Filters for RADAR Signals
Return to Table of Content
( )222
cos2
0pp
i
tt
tttAts ≤≤−
+= µω
ωωωπωµπ
∆=−∆=∆=∆2
:&2:2
1: 0
nftf p
The Matched Filter output is given by:
( )( )
( )
( )( )
>
<
−
−
−≈
p
p
pp
pp
po
p
to
tt
ttttt
tt
tttt
ttN
tAk
ts
0
cos/1
2
/12
sin
/1 0
2
0
ωµ
µ
o
p
N
tAk 2
ptt
µπ2=∆
1>>ptµ
30
SOLO
2. Linear FM Modulated Pulse (continue – 6)
Matched Filters for RADAR Signals
Return to Table of Content
( )222
cos2
0pp
i
tt
tttAts ≤≤−
+= µω
ωωωπωµπ
∆=−∆=∆=∆2
:&2:2
1: 0
nftf p
31
Matched Filters for RADAR Signals
( ) ( ) ( )tntstv ii +=Linear Filter
( )Tnhopt
( ) ( ) ( )tntsty oo +=
( ) ( ) ( )TnnTnsTnv ii +=
T T
( ) ( ) ( )TnnTnsTny oo +=
SOLOMaximization of Signal-to-Noise Ratio
Consider the problem of choosing a discrete linear time-invariant filter hopt (n T) that Maximizes the discrete output signal-to-noise ratio at a predefined time mT.
The input waveform is: ( ) ( ) ( )tntstv ii +=
( )tsi - a known signal component
( )tni - noise (stationary random process) component
The output waveform is: ( ) ( ) ( )TnnTnsTny oo +=
The signal-to-noise ratio at discrete time mT is defined as:( )( )Tmn
Tms
N
S
o
o
2
2
=
( )Tmno2
- the mean square value of ( )Tmno
Discrete Linear Systems
The input and output of the discrete linear filter are synchronous discretized witha constant time period T. S (z) is the Z -transform of the discrete signal input si (nT)
We have: ( ) ( ) ( )∫+
−
=σ
σ
ωωω ωσ
deeeTns TjTjTj
o HS2
1
( ){ } ( ) ( )∫+
−
=σ
σ
ω ωωσ
deTnnE Tjo
22
2
1HN
32
Matched Filters for RADAR Signals
( ) ( ) ( )tntstv ii +=Linear Filter
( )Tnhopt
( ) ( ) ( )tntsty oo +=
( ) ( ) ( )TnnTnsTnv ii +=
T T
( ) ( ) ( )TnnTnsTny oo +=
SOLOMaximization of Signal-to-Noise Ratio
Consider the problem of choosing a discrete linear time-invariant filter hopt (n T) that Maximizes the discrete output signal-to-noise ratio at a predefined time mT.
Like in the continuous case the optimal H (z) is:
( )( )
( ) ( )
( ) ( )∫
∫+
−
+
−==
σ
σ
ω
σ
σ
ω
ωωσ
ωωσ
de
de
Tmn
Tms
N
S
Tj
Tj
i
o
o
2
2
2
2
2
1
2
1
HN
HS
Discrete Linear Systems (continue – 1(
If N (ω) = N0 we have:
( ) ( ) ( )∫+
−
=σ
σ
ωωω ωσ
deeeTns TjTjTj
o HS i2
1
( ){ } ( ) ( )∫+
−
=σ
σ
ω ωωσ
deTnnE Tj
o
22
2
1HN
( ) ( )( )
mTjTj
iTj ee
ke ωω
ω
ω−=
N
SH
( ) [ ] [ ]nmsN
knhz
zN
kz i
m
i −=⇔
= −
00
1SH
Return to Table of Content
33
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
34
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
35
RADAR SignalsSOLO
Spectrum
Define the Fourier Transfer F
( ) ( ){ } ( ) ( )∫+∞
∞−
−== dttjtstsS ωω exp:F ( ) ( ){ } ( ) ( )∫+∞
∞−
==πωωωω2
exp:d
tjSSts -1F
( ) ( ) ( )[ ]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:
36
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
( ) ( ) ( )∫+∞
∞−
−=πωωω2
exp*
2
*
2
dtjFtf
If s (t) is real, than s (t) = s*(t) and
( ) ( ) ( )∫∫∫+∞
∞−
+∞
∞−
+∞
∞−
=== ωωπ
dSdttsdttsEs
222
2
1:
37
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 EdGdSE 22
1
2
1
2
1:
22 =≈= ∫∫+∞
∞−
+∞
∞−
ωωπ
ωωπ
Return to Table of Content
38
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
39
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
40
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
41
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
42
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
Return to Table of Content
43
Matched Filters for RADAR Signals
( ) ( ) ( )[ ]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
44
Matched Filters for RADAR Signals
( ) ( ) ( )[ ]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.
45
Matched Filters for RADAR SignalsSOLOAutocorrelation (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:=
46
Matched Filters for RADAR SignalsSOLOAutocorrelation (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
47
Matched Filters for RADAR Signals
( ) ( ) ( )[ ]tttats θω += 0cos
SOLO
Matched Filter for Received Radar 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.
The received signal will be:
1
• attenuated by a factor α• retarded by a time t0 = 2 R/c
• affected by the Doppler effectc
RRc
f
c
D
222 0
2
00
ωλ
πωωπλ
−=−===
( ) ( ) ( ) ( ) ( )[ ]0000 cos ttttttats Dr −+−+−= θωωα2
Since the range and range-rate (t0, ωD) are not known exactly in advance, the matched filter is designed to match the received signal at any time t0
assuming zero Doppler ωD=0.Return to Table of Content
48
Matched Filters for RADAR Signals
( ) ( ) ( ) ( ) ( )tjtgtjtgts 00 exp2
1exp
2
1 ωω −+= ∗
( ) ( )( ) ( )
≤≤−== −∗
Ttttsth
eSH tj
00
0ωωω
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( )[ ]∫
∫∫∞+
∞−
∗∗
+∞
∞−
+∞
∞−
+−−+−++−+−
−+=
+−=−=
00000000
0
exp2
1exp
2
1exp
2
1exp
2
1ttjttgttjttgjgjg
dttssdthstso
τωττωττωττωτ
ττττττ
SOLO
The Matched Filter is a linear time-invariant filter hopt (t) that maximizesthe output signal-to-noise ratio at a predefined time t0, for a known transmitted signal s (t(.
Assuming no Doppler let find the Matched Filter for the received radar signal at a time t0:
( ) ( ) ( ) ( ) ( )tjtgtjtgts 00 exp2
1exp
2
1 ωω −+= ∗
( )[ ] ( ) ( ) ( )[ ] ( ) ( )∫∫+∞
∞−
∗+∞
∞−
∗ +−−−++−−= τττωτττω dttggttjdttggttj 000000 exp4
1exp
4
1
( )[ ] ( ) ( ) ( ) ( )[ ] ( ) ( ) ( )∫∫+∞
∞−
∗+∞
∞−
∗ +−−−+−+−−+ τωττωτωττω dtjttggttjdtjttggttj 00000000 2expexp4
12expexp
4
1
Matched Filter Response to a Band Limited Radar Signal
49
Matched Filter output envelope
Matched Filters for RADAR Signals
( ) ( ) ( ) ( ) ( )∫∫+∞
∞−
+∞
∞−
+−=−= ττττττ dttssdthstso 0
SOLO
Matched Filter Response to a Band Limited Radar Signal (continue – 1(
The transmitted radar signal:
( ) ( ) ( ) ( ) ( )tjtgtjtgts 00 exp2
1exp
2
1 ωω −+= ∗
( )[ ] ( ) ( ) ( )[ ] ( ) ( ) ( )
−+−−+
+−−= ∫∫+∞
∞−
∗+∞
∞−
∗ τωττωτττω dtjttggttjdttggttj 0000000 2expexpRe2
1expRe
2
1
The integral in the second term on the r.h.s. is the Fourier transform ofevaluated at ω = 2 ω0. Since the spectrum of is limited by ω = W << ω0, thissecond term can be neglected, therefore:
( ) ( )[ ]0ttgg +−∗ ττ( )τg
( ) ( )[ ] ( ) ( ) ( ) ( )[ ]tjtgdttggttjts o
filtermatchedsignal
o 0000 expRe2
1expRe
2
1 ωτττω =
+−−≈ ∫∞+
∞−
∗
( ) [ ] ( ) ( ) [ ] ( )000000 exp2
1exp
2
1ttRtjdttggtjtg gg
filtermatchedsignal
o −−=+−−= ∫+∞
∞−
∗ ωτττω
( ) ( ) ( ) ( ) ( ) ( )[ ]( ) ( )[ ]tjta
tjttatsjtstg QI
θθθ
exp
sincos:
=
+=+=
Constant Phase
Matched Filter (for time t0) output is:
Autocorrelation Function of ( )tgReturn to Table of Content
50
Matched Filters for RADAR SignalsSOLO
Matched Filter Response to Phase Coding
( ) ( ) ( ) ∆<<
=∆−=∑−
= elsewhere
tttftptfctg
M
pp 0
011
0
Let the signal be a phase-modulated carrier, in which the modulation is in discrete and equal steps Δt. The complex envelope of the signal can be described by a sequence of complex numbers , such thatkc
( ) [ ] ( ) ( )∫+∞
∞−
∗ +−−= dtttgtgtjgo 000exp2
1 τωτ
Constant Phase
Matched Filter output envelope (change t ↔τ):
( )ttk ∆<≤+∆→ τττ 0
( ) [ ] ( ) ( )[ ]
[ ] ( )[ ]( )
∑ ∫
∫ ∑−
=
∆+
∆
∗
+∞
∞−
∗−
=
∆−+−∆−=
∆−+−∆−∆−=+∆
1
0
1
0
1
00
exp2
1
exp2
1
M
p
tp
tp
p
M
ppo
dttkMtgctMj
dttkMtgtptfctMjtkg
τω
τωτ
Change variable of integration to t1 = t – τ + (M - k) Δt
( ) [ ] ( )( )
( )
∑ ∫−
=
−∆+−+
−∆−+
∗∆−=+∆1
0
1
110exp2
1 M
p
tkMp
tkMp
po dttgctMjtkgτ
τ
ωτ
tMt ∆=0 (expected receiving time)
( ) ( ) ( ) ( ) ( )tjtgtjtgts 00 exp2
1exp
2
1 ωω −+= ∗The signal:
51
Matched Filters for RADAR SignalsSOLO
Matched Filter Response to Phase Coding (continue – 1(
Matched Filter output envelope for a Phase Coding is:
( ) [ ] ( )[ ]( )
∑ ∫−
=
∆+
∆
∗ ∆−+−∆−=+∆1
0
1
0exp2
1 M
p
tp
tp
po dttkMtgctMjtkg τωτ
Change variable of integration to t1 = t – τ + (M - k) Δt
( ) [ ] ( )( )
( )[ ] ( )
( )
( )( )
( )
( )
∑ ∫∫∑ ∫−
=
−∆+−+
∆−+
∗∆−+
−∆−+
∗−
=
−∆+−+
−∆−+
∗
+∆−=∆−=+∆
1
0
1
11110
1
0
1
110 exp2
1exp
2
1 M
p
tkMp
tkMp
tkMp
tkMp
p
M
p
tkMp
tkMp
po dttgdttgctMjdttgctMjtkgτ
τ
τ
τ
ωωτ
( ) ( ) ( )( ) ( ) ( ) τ
τ−∆+−+<<∆−+=
∆−+<<−∆−+=
−+∗
−−+∗
tkMpttkMpctg
tkMpttkMpctg
kMp
kMp
11*
1
11*
1
( ) [ ] ∑ ∫∫−
=
−∆
−+
−
−−+
+∆−=+∆
1
0 0
1*
0
11*
0exp2
1 M
p
t
kMpkMppo dtcdtcctMjtkgτ
τ
ωτ
( ) [ ] ∑−
=−+−−+
∆
−+
∆
∆−∆
=+∆1
0
*1
*0 1exp
2
1 M
p
kMpkMppo tc
tcctMj
ttkg
ττωτ
This equation describes straight lines in the complex plane, that can have corners only atτ = 0. At those corners
( ) [ ] ∑−
=−+∆−
∆=∆
1
0
*0exp
2
1 M
p
kMppo cctMjt
tkg ω
Constant Phase
52
Matched Filters for RADAR SignalsSOLO
Matched Filter Response to Phase Coding (continue – 2(
Matched Filter output envelope for a Phase Coding is:
( ) [ ] ∑−
=−+−−+
∆
−+
∆
∆−∆
=+∆1
0
*1
*0 1exp
2
1 M
p
kMpkMppo tc
tcctMj
ttkg
ττωτ
This equation describes straight lines in the complex plane, that can have corners only atτ = 0. At those corners
( ) [ ] ∑−
=−+∆−
∆=∆
1
0
*0exp
2
1 M
p
kMppo cctMjt
tkg ω
Constant Phase
We can see that is the Discrete Autocorrelation Function for the observation time t0 = M Δt (the time the received Radar signal return is expected)
∑−
=−+
1
0
*M
p
kMpp cc
53
Matched Filters for RADAR SignalsSOLO
Matched Filter Response to Phase Coding (continue – 3(
Example: Pulse poly-phase coded of length 4
Given the sequence: { } 1,,,1 −−++= jjck
which corresponds to the sequence of phases 0◦, 90◦, 270◦ and 180◦, the matched filter is given in Figure bellow.
{ } 1,,,1* −+−+= jjck
54
Pulse poly-phase coded of length 4
At the Receiver the coded pulse enters a 4 cells delay lane (from left to right), a bin at each clock.The signals in the cells are multiplied by -1,+j,-j or +1 and summed.
clock
SOLOPoly-Phase Modulation
-1 = -11 1+
-j +j = 02 1+j+
+j -1-j = -13 1+j+j−
+1 +1+1+1 = 44 1+j+j−1−
-j-1+j = -15 j+j−1−
+j - j = 06
j−1−7
1− -1 = -1
8 0
Σ
{ } 1,,,1 −−++= jjck
1− 1+j+ j− {ck*}
0 = 00
0
1
2
3
4
5
6
7
{ } 1,,,1* −+−+= jjck
55
-1
Pulse bi-phase Barker coded of length 3
Digital Correlation At the Receiver the coded pulse enters a 3 cells delay lane (from left to right), a bin at each clock.The signals in the cells are multiplied according to ck* sign and summed.
clock
-1 = -11
+1 -1 = 02
-( +1) = -15
0 = 06
+1 +1-( -1) = 33
+1-( +1) = 04
SOLO Pulse Compression Techniques
1
2
3
4
5
6
0
+1+1
0 = 00
56
Pulse bi-phase Barker coded of length 5
Digital CorrelationAt the Receiver the coded pulse enters a7 cells delay lane (from left to right),a bin at each clock.The signals in the cells are multipliedby ck* and summed.
clock
SOLO Pulse Compression Techniques
+1-1+1+1+1 { }*kc
+1 = +11
+1 = 19
0 = 010
2 -1 +1 = 0
+1 +1 -1-( +1) = 04
+1 +1 +1 –(-1)+1 = 55
0 = 0 0
3 +1-1 +1 = 1
+1 +1 -(+1) -1 = 06
+1-( +1) +1 = 17
–(+1) +1 = 08
Return to Table of Content
57
Matched Filters for RADAR Signals
( ) ( ) ( ) ( ) ( ) ( )[ ]( ) ( )[ ]tjta
tjttatsjtstg QI
θθθ
exp
sincos:
=
+=+=
SOLO
Matched Filter Response to its Doppler-Shifted Signal
Matched Filter for the transmitted radar signal:
The received radar signal has the form:
( ) ( ) ( )[ ]( ) ( ) ( ) ( )tjtgtjtg
tttats
00
0
exp2
1exp
2
1
cos
ωω
θω
−+=
+=
∗
( ) ( ) ( ) ( )[ ]000 cos tttttakts Dr −++−= θωω
( ) ( ) ( ) ( )[ ]
( ) ( )[ ] ( ) ( ) ( )[ ] ( )tjtjtgtjtjtgk
tttakts
DD
Dtr
0*
0
00
expexp2
1expexp
2
cos0
ωωωω
θωω
−+=
++==
( ) ( ) ( )∫+∞
∞−
∗=
−= τττ dtggtgfiltersignal
to 2
100
Matched Filter output envelope (designed under zero Doppler assumption) was found to be:
( ) ( ) ( ) ( )∫+∞
∞−
∗=
−= τττωτω dtgjgtgfiltersignal
DtDo exp
2
1,
00
For a nonzero Doppler (ωD ≠ 0) the Matched Filter output envelope is:
58
Matched Filters for RADAR Signals
( ) ( ) ( ) ( ) ( ) ( )[ ]( ) ( )[ ]tjta
tjttatsjtstg QI
θθθ
exp
sincos:
=
+=+=
SOLO
Matched Filter Response to its Doppler-Shifted Signal (continue – 1(
For a nonzero Doppler (ωD ≠ 0) the Matched Filter output complex envelope is:
( ) ( ) ( ) ( )∫+∞
∞−
∗=
−= τττωτω dtgjgtgfiltersignal
DtDo exp
2
1,
00
Change between t and τ and define:
( ) ( ) ( ) ( )∫+∞
∞−
∗ −= dttfjtgtgfX DD πττ 2exp:,
The magnitude of the complex envelope ,is called the Ambiguity Function. ( )DfX ,τ
The name is sometimes used for , and sometimes even for . ( )DfX ,τ ( ) 2, DfX τ
59
Matched Filters for RADAR SignalsSOLO
Matched Filter Response to its Doppler-Shifted Signal (continue – 2(
Properties of: ( ) ( ) ( ) ( )∫+∞
∞−
∗ −= dttfjtgtgfX DD πττ 2exp:,
( ) ( ) ( ) ( ) ( ) ( )∫∫∫+∞
∞−
∗+∞
∞−
+∞
∞−
∗ === dffGfGdttgdttgtgX2
:0,01
2 ( ) ( ) ( )DDD fXfjfX ,*2exp, ττπτ =−−( ) ( ) ( ) ( )
( ) ( ) ( ) ( )[ ]
( ) ( ) ( ) [ ] ( ) ( )DDDD
tt
DD
DD
fXfjdttfjtgtgfj
dttfjtgtgfj
dttfjtgtgfX
,*2exp''2exp''*2exp
2exp2exp
2exp,
*'
ττππττπ
τπττπ
πττ
τ=
−=
+−+=
=−+=−−
∫
∫
∫
∞+
∞−
+=
∞+
∞−
∗
+∞
∞−
∗
( ) ( ) ( )DDD fXfjfX −−=− ,*2exp, ττπτ3
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )[ ]
( ) ( ) ( ) ( )[ ] ( ) ( )DDDD
tt
DD
DD
fXfjdttfjtgtgfj
dttfjtgtgfj
dttfjtgtgfX
−−=
−−−=
++−=
=+=−
∫
∫
∫
∞+
∞−
+=
∞+
∞−
∗
+∞
∞−
∗
,*2exp''2exp''*2exp
2exp2exp
2exp,
*'
ττππττπ
τπττπ
πττ
τ
60
Matched Filters for RADAR SignalsSOLO
Matched Filter Response to its Doppler-Shifted Signal (continue – 3(
Properties of: ( ) ( ) ( ) ( )∫+∞
∞−
∗ −= dttfjtgtgfX DD πττ 2exp:,
4
5
( ) ( ) ( ) ( )τττ ggRdttgtgX =−= ∫+∞
∞−
∗ :0,
( ) ( ) ( ) ( ) ( ) ( ) ( )fRdfffGfGdttfjtgtgfX GGDDD =+== ∫∫+∞
∞−
+∞
∞−
∗ *2exp,0 π
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )[ ] ( ) ( ) ( )[ ]
( ) ( ) ( )fRdfffGfG
dfdttffjtgfGdfdttffjtgfG
dttfjtgdftfjfGdttfjtgtgfX
GGD
DD
DDD
=+=
+−=+=
==
∫
∫ ∫∫ ∫
∫ ∫∫
∞+
∞−
∞+
∞−
∞+
∞−
∞+
∞−
∞+
∞−
∗
+∞
∞−
∗+∞
∞−
+∞
∞−
∗
*
*
2exp2exp
2exp2exp2exp,0
ππ
πππ
Return to Table of Content
( ) [ ] ( ) ( ) [ ] ( ) [ ] ( )0,exp2
1exp
2
1exp
2
1000000000 ttXtjttRtjdttggtjtg gg
filtermatchedsignal
o −−=−−=+−−= ∫+∞
∞−
∗ ωωτττω
Autocorrelation Function of theSignal Complex Envelope ( )tg
We found that the Matched Filter Output Complex Envelope is:( )tgo
61
SOLO
Continue toAmbiguity Functions
Matched Filters and Ambiguity Functions for RADAR Signals
January 18, 2015 62
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
63
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
64
( )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
( ) ( ) ( ) ( )∫∫+∞
∞−
+∞
∞−
= ωωω dFFdttftf 2*
12*
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
65
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
66
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*
67
Fourier Transform
( )tf
( ) ( )∑∞
=
−=0n
T Tntt δδ
( ) ( ) ( ) ( ) ( )∑∞
=
−==0
*
n
T TntTnfttftf δδ
( )tf *
( )tfT t
SOLO
Sampling and z-Transform (continue – 2)
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
68
Fourier TransformSOLO
Sampling and z-Transform (continue – 3)
( ) ( ) ( )∑∑∞
=
−+∞
−∞=
=
+=
0
* 21
n
nsT
n
eTnfT
njsF
TsF
πWe found
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
69
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 – 4)
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
70
Fourier TransformSOLO
Henry Nyquist1889 - 1976
http://en.wikipedia.org/wiki/Harry_Nyquist
Nyquist-Shannon Sampling Theorem
Claude Elwood Shannon 1916 – 2001
http://en.wikipedia.org/wiki/Claude_E._Shannon
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).
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
71
SignalsSOLO
Signal Duration and Bandwidth
then
( ) ( )∫+∞
∞−
−= tdetsfS tfi π2 ( ) ( )∫+∞
∞−
= fdefSts tfi π2
t
t∆2
t
( ) 2ts
ff
f∆2
( ) 2fS
( ) ( )
( )
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
72
Signals
( ) ( )∫+∞
∞−
= fdefSts tfi π2
SOLO
Signal Duration and Bandwidth (continue – 1)
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )∫∫ ∫
∫ ∫∫ ∫∫∞+
∞−
∞+
∞−
∞+
∞−
−
∞+
∞−
∞+
∞−
−∞+
∞−
∞+
∞−
∞+
∞−
=
=
=
=
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
73
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
( ) ( )∫+∞
∞−
−= 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
74
Signals
( ) ( ) ( ) ( ) ( )∫∫∫∫∫+∞
∞−
+∞
∞−
+∞
∞−
+∞
∞−
+∞
∞−
=≤
dffSfdttstdttsdttstdtts
222222
2
2 4'4
1 π
( ) ( )∫∫+∞
∞−
+∞
∞−
= dffSdts22 τ
SOLO
Signal Duration and Bandwidth (continue – 1)
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
75
SignalsSOLO
Signal Duration and Bandwidth (continue – 2)
( )
( )
( )
( )
( )
( )
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
76
SOLO
77
SOLO
78
SOLO