1
Matched Filters and Ambiguity Functions for
RADAR SignalsPart 2
SOLO HERMELIN
Updated: 01.12.08http://www.solohermelin.com
2
SOLO Matched Filters and Ambiguity Functions for RADAR Signals
Table of Content
RADAR RF Signals
Maximization of Signal-to-Noise Ratio
The Matched FilterThe Matched Filter Approximations
1. Single RF Pulse
2. Linear FM Modulated Pulse (Chirp)
Continuous Linear Systems
Discrete Linear Systems
RADAR SignalsSignal Duration and BandwidthComplex Representation of Bandpass Signals
Matched Filter Response to a Band Limited Radar Signal
Matched Filter Response to Phase Coding
Matched Filter Response to its Doppler-Shifted Signal
MATCHED
FILTERS
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 Train
Ambiguity Function Examples (Rihaczek, A.W., “Principles of High Resolution Radar”)
References
4
SOLO Matched Filters and Ambiguity Functions for RADAR Signals
Continue fromMatched Filters
5
( ) ( ) ( ) ( )∫+∞
∞−
∗ −= dttfjtgtgfX DD πττ 2exp:,
SOLO
Definition of Ambiguity Function:
( ) ( ) ( ) ( ) ( ) ( )[ ]( ) ( )[ ]tjta
tjttatsjtstg QI
θθθ
exp
sincos:
=
+=+=
• Ambiguity Function is an analytic tool for investigating the effect of target motion on the matching filter response.
• It is a function of waveform only.
• It can be used to characterize:
- Range Resolution
- Doppler Resolution
- Range – Doppler coupling
- Loses due to mismatched Doppler
Ambiguity Function for RADAR Signals
Return to Table of Content
6
Ambiguity Function for RADAR Signals
( ) ( ) ( ) ( )∫+∞
∞−
∗ −= dttfjtgtgfX DD πττ 2exp:,
( ) ( ) ( ) ( ) ( ) ( )[ ]( ) ( )[ ]tjta
tjttatsjtstg QI
θθθ
exp
sincos:
=
+=+=
SOLO
Definition of Ambiguity Function:
Ambiguity Function has the following properties:
( ) ( ) 12
1 22 == ∫∫+∞
∞−
+∞
∞−
ωωπ
dGdttgAssume that the complex signal envelope has a unit energy:
( ) ( ) 10,0, =≤ XfX Dτ1
( ) 1, =∫ ∫+∞
∞−
+∞
∞−DD dfdfX ττ2
( ) ( )DD fXfX ,, ττ =−−3
4 ( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫+∞
∞−
+∞
∞−
∗ −=−= fdfjfGffGtdtfjtgtgfX DDD τππττ 2exp*2exp:,
5 ( ) ( )DD fXfX −=− ,, ττ
Ambiguity Function Properties
7
Ambiguity Function for RADAR Signals
( ) ( ) ( ) ( )∫+∞
∞−
∗ −= dttfjtgtgfX DD πττ 2exp:,
SOLO
Definition of Ambiguity Function:
Ambiguity Function properties (continue - 1):
( ) ( ) 10,0, =≤ XfX Dτ1
( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( ) 1
2exp
2exp,
1
2
1
2
22
2
2
=−=
−≤
−=
∫∫
∫∫
∫
∞+
∞−
∗∞+
∞−
∞+
∞−
∗∞+
∞−
∞+
∞−
∗
dttgdttg
dttfjtgdttg
dttfjtgtgfX
D
DD
τ
πτ
πττ
( ) ( ) 12
1:
22 === ∫∫+∞
∞−
+∞
∞−
ωωπ
dGdttgEs
( ) 1,2 ≤DfX τ ( ) ( ) 10,0, =≤ XfX Dτ
Proof:
Schwarzinequality
8
Ambiguity Function for RADAR Signals( ) ( ) ( ) ( )∫
+∞
∞−
∗ −= dttfjtgtgfX DD πττ 2exp:,
SOLO
Definition of Ambiguity Function:
Ambiguity Function properties (continue – 2):
( ) 1, =∫ ∫+∞
∞−
+∞
∞−DD dfdfX ττ2
( ) ( ) ( ) ( )∫+∞
∞−
∗ −= dttfjtgtgfX DD πττ 2exp:,
( )DfX ,τ is the Fourier Transform of ( ) ( )τ−∗ tgtg
Using Parseval’s Theorem: ( ) ( ) ( )∫∫+∞
∞−
+∞
∞−
∗ =− DD dffXdttgtg22
,ττ
Integrating both sides on τ we obtain: ( ) ( ) ( ) VddffXddttgtg DD ==− ∫ ∫∫ ∫+∞
∞−
+∞
∞−
+∞
∞−
+∞
∞−
∗ ττττ 22,
V is the volume under the Ambiguity Function.
( ) ( ) ( ) ( ) ( )∫ ∫∫ ∫+∞
∞−
+∞
∞−
∗=
=−
+∞
∞−
+∞
∞−
∗ =−= 2121
2
21
2,
1
2
dtdtttJtgtgddttgtgVtt
tt τττ ( ) 1
11
01
//
//,
22
1121 =
−=
∂∂∂∂∂∂∂∂
=ττ
ttt
tttttJ
( ) ( ) ( ) ( ) ( )∫ ∫∫∫∫ ∫+∞
∞−
+∞
∞−
+∞
∞−
∗+∞
∞−
+∞
∞−
+∞
∞−
∗ ==== DD dfdfXdttgdttgdtdttgtg ττ 2
1
2
2
2
1
1
2
121
2
2
2
1 ,1
Proof:
9
Ambiguity Function for RADAR Signals
( ) ( ) ( ) ( )∫+∞
∞−
∗ −= dttfjtgtgfX DD πττ 2exp:,
SOLO
Definition of Ambiguity Function:
Ambiguity Function properties (continue – 3):
( ) ( )DD fXfX ,, ττ =−−3
( ) ( ) ( ) ( )∫+∞
∞−
∗ −= dttfjtgtgfX DD πττ 2exp:,Proof:
( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )*
1111
1111
1111
2exp2exp
2exp2exp
2exp2exp:,1
−=
−−=
−−−=−+=−−
∫
∫
∫∫
∞+
∞−
∗
∞+
∞−
∗
+∞
∞−
∗=++∞
∞−
∗
dttfjtgtgfj
dttfjtgtgfj
dttfjtgtgdttfjtgtgfX
DD
DD
D
tt
DD
πττπ
πττπ
τπτπτττ
( ) ( ) ( )DDD fXfjfX ,2exp, * ττπτ =−−
( ) ( ) ( ) ( )DDDD fXfXfjfX ,,2exp, * τττπτ ==−−
10
Ambiguity Function for RADAR Signals
( ) ( ) ( ) ( )∫+∞
∞−
∗ −= dttfjtgtgfX DD πττ 2exp:,
( ) ( ){ } ( ) ( )∫+∞
∞−
−== dttfjtgtgfG π2exp:F
SOLO
Definition of Ambiguity Function:
Ambiguity Function properties (continue – 4):
4
Proof:
( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫+∞
∞−
+∞
∞−
∗ −=−= fdfjfGffGtdtfjtgtgfX DDD τππττ 2exp*2exp:,
( ) ( ){ } ( ) ( )∫+∞
∞−
== fdtfjfGGtg πω 2exp:1-F
( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∫∫+∞
∞−
∗+∞
∞−
+∞
∞−
∗ −
−=−= tdtgfdtfjffGtdtgtfjtgfX DDD τπτπτ 2exp2exp,
( )tg-1F
F ( )fG
( ) ( ) ( )[ ]∫+∞
∞−
−=− fdtfjfGtg τπτ 2exp ( )τ−tg -1FF ( ) ( )τπ fjfG 2exp −
( ) ( ) ( )[ ] ( ) ( ) ( )∫∫+∞
∞−
+∞
∞−
−−=−−= fdfjffGfjfdtffjffGtg DDDD πππ 2exp2exp2exp
( ) ( ) ( ) ( )∫+∞
∞−
−= fdfjffGfjtg DD ππ 2exp2exp ( ) ( )Dfjtg π2exp -1FF ( )DffG −
( ) ( ) ( )∫+∞
∞−
−= fdfjfGffG D τπ2exp*
( ) ( ) ( ) ( ) ( ) ( )[ ]∫∫ ∫∞+
∞−
∞+
∞−
∞+
∞−
−−=
−−−= fdfjfGffGfdtdtfjtgffG DD
*
*
2exp2exp τππτ
q.e.d.
11
Ambiguity Function for RADAR Signals
( ) ( ) ( ) ( )∫+∞
∞−
∗ −= dttfjtgtgfX DD πττ 2exp:,
SOLO
Definition of Ambiguity Function:
( ) ( ) ( )DDD fXfjfX −−=− ,*2exp, ττπτ
Ambiguity Function properties (continue – 5):
5 ( ) ( )DD fXfX −=− ,, ττ
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )[ ] ( ) ( ) ( ) ( )**
'
*
''2exp''2exp''2exp''
2exp2exp:,
−−=
−−−=
−+=+=−
∫∫
∫∫∞+
∞−
∗∞+
∞−
∗−→
∞+
∞−
∗∞+
∞−
∗
dttfjtgtgfjdttfjtgtg
dttfjtgtgdttfjtgtgfX
DDD
tt
DDD
πττπτπτ
πτπττ
τ
Proof:
or
From which( ) ( ) ( ) ( )DDDD fXfXfjfX −=−−=− ,,*2exp,
1
τττπτ
Return to Table of Content
12
Ambiguity Function for RADAR Signals
( ) ( ) ( ) ( )∫+∞
∞−
∗ −= dttfjtgtgfX DD πττ 2exp:,
SOLO
Definition of Ambiguity Function:
Cuts Through the Ambiguity Function
( ) ( ) ( ) ( )τττ ggD RdttgtgfX =−== ∫+∞
∞−
∗0,Cut through the delay axis:
where Rgg (τ) is the autocorrelation function of the signal envelope.
The cut along the Ambiguity Function along the delay axis is the shape of the“range window” at zero Doppler. This is how the envelope of the Matched Filter will look as a function of time.
Linear FM pulse
Single pulse
13
Ambiguity Function for RADAR Signals
( ) ( ) ( ) ( )∫+∞
∞−
∗ −= dttfjtgtgfX DD πττ 2exp:,
SOLO
Definition of Ambiguity Function:
Cuts Through the Ambiguity Function (continue – 1)
Cut through the frequency axis:
This is the Fourier Transform of signal envelope energy, and the cut at τ = 0 isindependent of any phase or frequency modulation and is determined only by themagnitude of the complex envelope of the signal – that is by amplitude modulation.
( ) ( ) ( ) ( ) ( ) ( )∫∫+∞
∞−
+∞
∞−
∗ === dttfjtgdttfjtgtgfX DDD ππτ 2exp2exp,02
( )DfX ,0=τ( ) 2tgF
F -1
Return to Table of Content
14
Ambiguity Function for RADAR SignalsSOLO
Ambiguity as a Measure of Range and Doppler Resolution
Suppose that the transmitted signal s (t) is returned by two targets whose signals s1 (t) and s2 (t) differ only in range (delay time τ) and Doppler (frequency fD).
The Resolution of the Radar is related to how it can distinguish between the two signals. A tractable criteria of resolution is the integrated square difference magnitude, denoted by |ε|2, and defined by
( ) ( ) tfjetgts 02: π=
( ) ( ) ( ) ( )[ ] ( ) ( )[ ] ( ) ( ) ( ) ( )[ ] ( ) ( )[ ]{ }∫∫∫+∞
∞−
+∞
∞−
+∞
∞−
−−+=−−=−= dttstststststsdttstststsdttsts 2121
2
2
2
12121
2
21
2****ε
In order to obtain a difference in delay and Doppler we will define the complex signals:
( ) ( ) ( ) ( ) ( )[ ] ( )[ ] ( )[ ]
( ) ( ) ( )[ ] ( )[ ] ( )[ ]∫
∫∫∫∞+
∞−
+∞
∞−
+∞
∞−
+∞
∞−
−−−−−−−
−−−−−−−−−+=
dttfjtfjfjtgtg
dttfjtfjfjtgtgdttgdttg
DD
DD
221121021
221121021
222
2exp2exp2exp*
2exp2exp2exp*
τπτπττπττ
τπτπττπτττε
Note: The real signals are ( ) ( ) ( )[ ] ( ) ( ) ( )[ ] 2/*&2/* 222111 tstststststs +=+=
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
−=
−=−+
−+
220
110
222
211
:
:τπ
τπ
ττ
tffj
tffj
D
D
etgts
etgts
- Transmitted signal
- Received signals
15
Ambiguity Function for RADAR SignalsSOLO
Ambiguity as a Measure of Range and Doppler Resolution (continue – 1)
( ) ( ) ( ) ( ) ( )[ ] ( )[ ] ( )[ ]
( ) ( ) ( )[ ] ( )[ ] ( )[ ]∫
∫∫∫∞+
∞−
+∞
∞−
+∞
∞−
+∞
∞−
−−−−−−−
−−−−−−−−−+=
dttfjtfjfjtgtg
dttfjtfjfjtgtgdttgdttg
DD
DD
221121021
221121021
222
2exp2exp2exp*
2exp2exp2exp*
τπτπττπττ
τπτπττπτττε
2121 :&: DDD fff −=∆−=∆ τττDefine
We found
( ) ( ) ( ) ( ) ( )[ ] ( )[ ] ( )[ ]
( ) ( ) ( )[ ] ( )[ ] ( )[ ]∫
∫∫∫∞+
∞−
+∞
∞−
+∞
∞−
+∞
∞−
−+−−−+−
−+−−−−+−−+=
""2exp"2exp2exp""*
''2exp'2exp2exp'*'
212121012
212121021
222
dttfjtfjfjtgtg
dttfjtfjfjtgtgdttgdttg
DD
DD
πττπττπττ
ττππττπτττε
( ) ( )[ ] ( ) ( ) ( )[ ]
( )[ ] ( ) ( ) ( )[ ]∫
∫∫∞+
∞−
+∞
∞−
+∞
∞−
∆−∆−∆+−
∆∆+∆+−−=
""2exp""*2exp
''2exp'*'2exp2
10
20
22
dttfjtgtgffj
dttfjtgtgffjdttg
DD
DD
πττπ
πττπε
( ) ( ) ( ) ( )∫+∞
∞−
∗ −= dttfjtgtgfX DD πττ 2exp:,where ( ) ( ) ( )DDD fXfjfX −−=− ,2exp, * ττπτ
( ) ( ) ( ) ( )[ ] ( ) ( )[ ] ( )DDDD fXffjfXffjdttgdttsts ∆−∆∆+−∆∆−∆+−−=−= ∫∫+∞
∞−
+∞
∞−
,2exp,2exp2 1020
22
21
2 ττπττπε
16
Ambiguity Function for RADAR SignalsSOLO
Ambiguity as a Measure of Range and Doppler Resolution (continue – 2)We found
( ) ( ) ( ) ( )[ ] ( ) ( )[ ] ( )DDDD fXffjfXffjdttgdttsts ∆−∆∆+−∆∆−∆+−−=−= ∫∫+∞
∞−
+∞
∞−
,2exp,2exp2 1020
22
21
2 ττπττπε
( ) ( ) ( ) ( )∫+∞
∞−
∗ −= dttfjtgtgfX DD πττ 2exp:,where ( ) ( ) ( )DDD fXfjfX −−=− ,2exp, * ττπτ
( ) ( ) ( ) ( )[ ] ( ) ( ) ( )[ ] ( )
( ) ( )[ ] ( ) ( )[ ] ( )DDDD
DDD
f
DDD
fXffjfXffjdttg
fXffjfXffjffjdttgdttsts
D
∆−∆∆+−∆−∆∆+−−=
∆−∆∆+−∆−∆
∆−−∆+−−=−=
∫
∫∫∞+
∞−
∆
∞+
∞−
∞+
∞−
,2exp,*2exp2
,2exp,*2exp2exp2
1010
2
102120
22
21
2
ττπττπ
ττπττπτπε
( ) ( ) ( ) ( )[ ] ( ){ }DD fXffjdttgdttsts ∆−∆∆+−=−= ∫∫+∞
∞−
+∞
∞−
,2expRe22 10
22
21
2 ττπε
( ) ( ) ( ) ( )[ ] ( ) ( )[ ] ( )
( ) ( )[ ] ( ) ( )[ ] ( )[ ] ( )DDDDDD
DDDD
fXffjffjfXffjdttg
fXffjfXffjdttgdttsts
∆∆−∆−∆+−∆∆−∆+−−=
∆−∆∆+−∆∆−∆+−−=−=
∫
∫∫∞+
∞−
+∞
∞−
+∞
∞−
,*2exp2exp,2exp2
,2exp,2exp2
211020
2
1020
22
21
2
ττπτπττπ
ττπττπε
( ) ( ) ( ) ( )[ ] ( ){ }DD fXffjdttgdttsts ∆∆−∆+−−=−= ∫∫+∞
∞−
+∞
∞−
,2expRe22 20
22
21
2 ττπε
17
Ambiguity Function for RADAR SignalsSOLO
Ambiguity as a Measure of Range and Doppler Resolution (continue – 3)
( ) ( ) ( )( )
( )[ ] ( ){ }
( )( )
( )[ ] ( ){ }DD
tgofEnergy
DD
tgofEnergy
fXffjdttg
fXffjdttgdttsts
∆∆−∆+−−=
∆−∆∆+−=−=
∫
∫∫
∞+
∞−
+∞
∞−
+∞
∞−
,2expRe22
,2expRe22
20
2
10
22
21
2
ττπ
ττπε
2121 :&: DDD fff −=∆−=∆ τττDefine
Good Resolution requires that |ε|2 be large for any delay Δτ ≠0 and Doppler ΔfD ≠0. The first term is the energies (positive) of the complex envelopes of the two signals. The second term has a minus sign, hence |ε|2 will be increased when the second term will decrease.
( )[ ] ( ){ } ( ) ( )[ ] ( ){ }( )DDDDD fXffjfXfXffj ∆∆∆+−∆∆=∆∆∆+− ,2expargcos,,2expRe 2020 ττπτττπ
Good resolution is obtained when (Ambiguity Function) is minimum fornon-zero target delay Δτ and Doppler ΔfD.
( )DfX ∆∆ ,τ
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
−=
−=−+
−+
220
110
222
211
:
:τπ
τπ
ττ
tffj
tffj
D
D
etgts
etgtsReceived complex signal
( ) ( ) ( ) ( )∫+∞
∞−
∗ −= dttfjtgtgfX DD πττ 2exp:,where ( ) ( ) ( )DDD fXfjfX −−=− ,*2exp, ττπτ
18
Ambiguity Function for RADAR Signals
( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫+∞
∞−
+∞
∞−
∗ ==−== fdfjfGfGRdttgtgfX ggD τπτττ 2exp*:0,
SOLO
Ambiguity as a Measure of Range and Doppler Resolution (continue – 4)
Range Resolution
( )
( ) 2
2
0,0
0,
:X
dfX
TD
res
∫+∞
∞−
==
ττ
Assume the two signals have the same Doppler fD = 0. The range resolution is defined as:
( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫+∞
∞−
+∞
∞−
∗ −=−= fdfjfGffGtdtfjtgtgfX DDD τππττ 2exp*2exp:,
Using
we have
( )τggR -1FF ( ) 2fG
( ) ( ) ( ) ( ) ( ) ( )∫∫+∞
∞−
+∞
∞−
∗ ====== fdfGfGRdttgtgfX ggD *0:0,0 ττ ( )
( )
( )
( )2
2
4
2
2
0
==
∫
∫∫∞+
∞−
+∞
∞−
+∞
∞−
fdfG
fdfG
R
dR
Tgg
gg
res
ττ
Parseval’s Theoremand
19
Ambiguity Function for RADAR SignalsSOLO
Ambiguity as a Measure of Range and Doppler Resolution (continue – 5)
Doppler Resolution
( )
( ) 2
2
0,0
,0
:X
fdfX
FDD
res
∫+∞
∞−
==
τ
Assume the two signals have the same range delay τ = 0. The Doppler resolution is defined as:
( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫+∞
∞−
+∞
∞−
∗ −=−= fdfjfGffGtdtfjtgtgfX DDD τππττ 2exp*2exp:,
Using
we have
( ) 2tg -1FF ( )DGG fR
( ) ( ) ( ) ( ) ( ) ( ) ( )0*0:0,0 ====== ∫∫+∞
∞−
+∞
∞−
∗ fRfdfGfGRdttgtgX GGgg τ ( )
( )
( )
( )2
2
4
2
2
0
==
∫
∫∫∞+
∞−
+∞
∞−
+∞
∞−
tdtg
tdtg
R
fdfR
FGG
GG
res
Parseval’s Theoremand
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )DGGDGGDDD fRfRfdfGffGtdtfjtgtgfX =−=−=== ∫∫+∞
∞−
+∞
∞−
∗ *2exp:,0 πτ
20
Ambiguity Function for RADAR SignalsSOLO
Ambiguity as a Measure of Range and Doppler Resolution (continue – 6)Range – Doppler Resolution
( )
( )
( )
( )2
2
4
2
2
0
==
∫
∫∫∞+
∞−
+∞
∞−
+∞
∞−
tdtg
tdtg
R
fdfR
FGG
GG
res
( )
( )
( )
( )2
2
4
2
2
0
==
∫
∫∫∞+
∞−
+∞
∞−
+∞
∞−
fdfG
fdfG
R
dR
Tgg
gg
res
ττ
From Schwarz Inequality: ( ) ( ) ( ) ( )∫∫∫+∞
∞−
+∞
∞−
+∞
∞−
≤ dtthdttfdtthtf22
Choose ( ) ( ) ( ) ( ) ( )tgtd
tgdthtgttf ':& ===
( ) ( ) ( ) ( )∫∫∫+∞
∞−
+∞
∞−
+∞
∞−
≤ dttsdttstdttstst22
''we obtain
21
Ambiguity Function for RADAR SignalsSOLO
Ambiguity as a Measure of Range and Doppler Resolution (continue – 7)
Good resolution is obtained when (Ambiguity Function) is minimum fornon-zero target delay τ and Doppler fD.
( )DfX ,τ
A waveform has an Ideal Ambiguity Function if it has a “Thumbtack” shape:• No response unless the echo is closely matched to the Doppler for which the filter
is designed.• And a very narrow peak in range, yielding good range resolution.
Can’t get rid of the pedestal because of the “constant volume” property.Return to Table of Content
22
Ambiguity Function for RADAR Signals( ) ( ) ( ) ( )∫
+∞
∞−
∗ −= dttfjtgtgfX DD πττ 2exp:,
SOLO
Ambiguity Function Close to Origin
( ) ( ) ( ) ( ) DfDD
fDD ffXf
fXXfXDD 002
00222
,,0,0, ==
==
∂∂+
∂∂+= ττ ττττ
τ
Let develop the Square of the Ambiguity Function in a Taylor series around the origin τ=0, fD=0
Since |X (0,0)|2 is the maximum of the continuous |X (τ,fD)|2 we must have
( ) ( ) 0,, 002
002 =
∂∂=
∂∂
==
==
DD fDD
fD fXf
fX ττ τττ
( ) ( ) ( ) +
∂∂+
∂∂
∂∂+
∂∂+ =
==
==
= 2002
2
2
0022
002
2
2
,,2,2
1DfD
D
DfDD
fD ffXf
ffXf
fXDDD
τττ ττττ
τττ
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )( )
( ) ( ) ( ) ( )( )
∫ ∫∫ ∫
∫ ∫∞+
∞−
∞+
∞−
∗∞+
∞−
∞+
∞−
∗
+∞
∞−
+∞
∞−
∗=
−−∂∂+−−
∂∂=
−−∂∂=
∂∂
ττ
τττ
τττ
τττ
ττ
gggg
D
RR
fD
dttgtgdttgtgdttgtgdttgtg
dtdttgtgtgtgfX
111222
*
222111
21221102
**
*,
also
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )00*0,
000
002 =
∂∂
=
∂∂+
∂∂=
∂∂
=≠
+∞
∞−
+∞
∞−
∗
≠
== ∫∫
ττ τ
ττ
τgg
ggggf
D
RRdttg
ttgdttg
ttgRfX D
23
Ambiguity Function for RADAR Signals
( ) ( ) ( ) ( )∫+∞
∞−
∗ −= dttfjtgtgfX DD πττ 2exp:,
( ) ( ) ( ) ( ) ( ) ( ) ,2,1,0,,2,1,0*
2
* ==
−=− ∫∫+∞
∞−
+∞
∞−
mndd
SdS
jdt
td
tsdtstj
m
mn
n
n
nmm ω
ωωωω
π
SOLO
Ambiguity Function Close to Origin (continue – 1)
( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫+∞
∞−
+∞
∞−
∗= −
∂∂+−
∂∂=
∂∂ ττ
τττ
ττ
τ ggggfD RdttgtgRdttgtgfX D 22211102
**,
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )∫∫
∫∫∞+
∞−
∞+
∞−
+∞
∞−
∗+∞
∞−
∗=
∂∂
−∂∂+−
∂∂+
∂∂
−∂∂+−
∂∂=
∂∂
ττ
ττ
τττ
ττ
ττ
τττ
ττ
gggg
ggggfD
RdttgtgRdttgtg
RdttgtgRdttgtgfX D
222222
2
2
111112
2
102
2
2
**
**,
Since is a maximum for τ=0, we have( ) ( ) sgggg ERR 20*0 ==( ) ( )
00*0
=∂
=∂=
∂=∂
ττ
ττ gggg RR
( ) ( ) ( ) ( ) ( ) ( )
∂∂+
∂∂=
∂∂
∫∫+∞
∞−
+∞
∞−
∗=
= dttgtgdttgtgRfX
s
D
E
ggf
D 2
2
2
2
2
002
2
2
*0,ττ
ττ τ
n=2m=0
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( )
sE
ggss
fs
Parseval
dffGffEdffGfEdGGE
2
2222222
2 222:2222*2
22∫∫∫
+∞
∞−
+∞
∞−
=+∞
∞−
+∆−=−=−= πππωωωωπ
πω
Relationshipfrom Parseval’s
Theory
24
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
Choose or the oppositec
( ) ( ) ( ) ( ) ( ) ( ) ,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
( ) ( ) ( ) ( ) ( ) ( ) ,2,1,0,,2,1,0*
2
* ==
−=− ∫∫+∞
∞−
+∞
∞−
mndd
SdS
jdt
td
tsdtstj
m
mn
n
n
nmm ω
ωωωω
π
c1
c2
25
Ambiguity Function for RADAR Signals
( ) ( ) ( ) ( )∫+∞
∞−
∗ −= dttfjtgtgfX DD πττ 2exp:,
( ) ( ) ( )[ ]222002
2
2
22, ggsfD ffEfXD
+∆−=∂∂
==ττ
τ
( )( ) ( ) ( )
( )∫
∫∞+
∞−
+∞
∞−
−=∆
dffG
dffGff
fg
g2
222
2
2
22
:
π
ππ
SOLO
Ambiguity Function Close to Origin (continue -2)
We found:
where:
Δfg – is signal envelope bandwidth
Es – is signal energy ( ) ( ) ( )∫∫∫+∞
∞−
+∞
∞−
+∞
∞−
=== tdtgfdfGtdtsEs222
2
12
2
1: π
fg – is signal envelope frequency median
( ) ( )
( )∫
∫∞+
∞−
+∞
∞−=dffG
dffGf
f g2
2
2
22
:
π
ππ
( )( ) ( )
( )∫
∫∞+
∞−
+∞
∞−=+∆dffG
dffGf
ff gg2
222
22
2
22
π
ππ
26
Ambiguity Function for RADAR Signals
( ) ( ) ( ) ( )∫+∞
∞−
∗ −= dttfjtgtgfX DD πττ 2exp:,
SOLO
Ambiguity Function Close to Origin (continue -3)
In the same way:
( ) ( ) ( ) ( ) ( ) ( )[ ]
( ) ( ) ( ) ( )[ ]∫ ∫
∫ ∫∞+
∞−
∞+
∞−
+∞
∞−
+∞
∞−
∗∗
−−=
−∂∂=
∂∂
2121
2
2
2
121
21212211
2
2exp2
2exp,0
dtdtttfjtgtgttj
dtdtttfjtgtgtgtgf
fXf
D
DD
DD
ππ
π
Since |X (0,0)|2 is the maximum of the continuous |X (τ,fD)|2 we must have
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) 022
2,
21
1
2
12
2
222
2
21
2
11
21
2
2
2
121
00
2
≡−=
−=∂∂
⇔
∞+
∞−
∞+
∞−
∞+
∞−
∞+
∞−
+∞
∞−
+∞
∞−==
∫ ∫∫ ∫
∫ ∫
tt
fD
D
dttgdttgtjdttgdttgtj
dtdttgtgttjfXf D
ππ
πττ
27
Ambiguity Function for RADAR Signals
( ) ( ) ( ) ( )∫+∞
∞−
∗ −= dttfjtgtgfX DD πττ 2exp:,
SOLO
Ambiguity Function Close to Origin (continue -4)
Return to:
( ) ( ) ( ) ( ) ( )[ ]∫ ∫+∞
∞−
+∞
∞−
−−=∂∂
2121
2
2
2
121
22exp2,0 dtdtttfjtgtgttjfX
f DDD
ππ
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( )
+−−=
−−=∂
∂
∫∫∫ ∫∫ ∫
∫ ∫
∞+
∞−
∞+
∞−
∞+
∞−
∞+
∞−
∞+
∞−
∞+
∞−
+∞
∞−
+∞
∞−==
2
2
222
2
1
2
12
2
221
2
11
2
2
2
21
2
121
2
21
2
2
2
12
212
00
2
2
2
22
2,
dttgtdttgdttgtdttgtdttgdttgt
dtdttgtgttfXf
ss
D
EE
fD
D
π
πττ
Define:( )
( )( )
( ) ( )
( )( )
( )
( )∫
∫
∫
∫
∫
∫∞+
∞−
+ ∞
∞−∞+
∞−
+∞
∞−∞+
∞−
+∞
∞− =+∆⇒−
=∆=dttg
dttgt
tt
dttg
dttgtt
t
dttg
dttgt
t gg
g
gg2
22
22
2
22
2
2
2
::
( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) 2222222222
00
2
2
2
222222, gsgggggsf
D
D
tEtttttEfXf D
∆−=+∆+−+∆−=∂
∂
==
ππττ
28
Ambiguity Function for RADAR Signals
( ) ( ) ( ) ( )∫+∞
∞−
∗ −= dttfjtgtgfX DD πττ 2exp:,
SOLO
Ambiguity Function Close to Origin (continue -5)
In the same way:
( ) ( ) ( ) ( ) ( ) ( )[ ]
( ) ( ) ( ) ( ) ( ) ( )[ ]∫ ∫
∫ ∫∞+
∞−
∞+
∞−
∗
+∞
∞−
+∞
∞−
∗
−−−−=
−−−∂∂=
∂∂
2121221121
21212211
2
2exp*2
2exp*,
dtdtttfjtgtgtgtgttj
dtdtttfjtgtgtgtgf
fXf
D
DD
DD
πττπ
πτττ
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )∫ ∫
∫ ∫∞+
∞−
∞+
∞−
+∞
∞−
+∞
∞−==
∂∂−−
∂∂−−=
∂∂∂
2122
21121
212211
121
00
22
**2
**2,
dtdttgt
tgtgtgttj
dtdttgtgtgt
tgttjfXf Df
DD
π
πττ
τ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∫∫ ∫
∫ ∫∫ ∫∞+
∞−
∞+
∞−
∞+
∞−
∞+
∞−
+∞
∞−
+∞
∞−
+∞
∞−
+∞
∞−
∂∂+
∂∂−
∂∂+
∂∂−=
222
22111222
21111
2222111
1222111
11
**2**2
**2**2
dttgt
tgtdttgtgjdttgt
tgdttgtgtj
dttgtgtdttgt
tgjdttgtgdttgt
tgtj
ππ
ππ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫∫ ∫+∞
∞−
+∞
∞−
+∞
∞−
+∞
∞−
∂∂−
∂∂−
∂∂−
∂∂+= dttg
ttgtg
ttgdttgtgtjdttg
ttgtg
ttgtdttgtgj ***2***2 ππ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
∂∂
−
∂∂
=∂∂
∂∫∫∫∫
+∞
∞−
+∞
∞−
+∞
∞−
+∞
∞−==
dttgt
tgdttgtgtdttgt
tgtdttgtgfXf Df
DD
*Im*4*Im*4,000
22
ππτ
τ
29
Ambiguity Function for RADAR Signals
( ) ( ) ( ) ( )∫+∞
∞−
∗ −= dttfjtgtgfX DD πττ 2exp:,
SOLO
Ambiguity Function Close to Origin (continue -6)
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( )
( ) ( ) ( ) ( ) ( ) gs fEdffGfGf
dGGdGGGGdttgt
tgtgt
tgj
22222*22
*2****2
22 ππππ
ωωωωωωωωωωωπ
==
=+=
∂∂−
∂∂
∫
∫∫∫∞+
∞−
+∞
∞−
+∞
∞−
+∞
∞−
( ) ( ) sEdttgtg 2* =∫+∞
∞−( ) ( ) ( )
( )
( )( ) gs tE
dttg
dttgt
dttgdttgtgt 2*2
2
2 ==
∫
∫∫∫ ∞+
∞−
+∞
∞−∞+
∞−
∞+
∞−
( ) ( ) ( ) ( ) ( ) ( ) ,2,1,0,,2,1,0
2*
*
==
= ∫∫
∞+
∞−
∞+
∞−
mndd
SdS
jdt
td
tsdtstj
m
mn
n
n
nmm ω
ωωωω
π
( ) ( ) ( ) ( ) ( ) ( ) ,2,1,0,,2,1,0*
2
* ==
−=− ∫∫+∞
∞−
+∞
∞−
mndd
SdS
jdt
td
tsdtstj
m
mn
n
n
nmm ω
ωωωω
π
c1
c2
Relationshipsfrom Parseval’sTheorem
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫
+∞
∞−
+∞
∞−
−
−=
∂∂−
∂∂− ω
ωωω
ωωωωπ d
d
SdS
d
SdSjdttg
ttgttg
ttgtj
****2
( ) ( ) ( ) ( ) ( ) ( ) ( ) ggssf
DD
ftEdttgt
tgtEfXf D
22
00
22
222*Im222,0 ππτ
τ
+
∂∂=
∂∂∂
∫+∞
∞−==
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫∫ ∫+ ∞
∞−
+ ∞
∞−
+ ∞
∞−
+ ∞
∞−==
∂∂−
∂∂−
∂∂−
∂∂+=
∂∂∂
dttgt
tgtgt
tgdttgtgtjdttgt
tgtgt
tgtdttgtgjfXf Df
DD
***2***2,000
22
ππτ
τ
30
Ambiguity Function for RADAR Signals
( ) ( ) ( ) ( )∫+∞
∞−
∗ −= dttfjtgtgfX DD πττ 2exp:,
SOLO
Ambiguity Function Close to Origin (continue -7)
( ) ( ) ( ) ( ) ( ) +∂
∂+∂∂
∂+∂∂+=
==
==
==
2
00
2
2
2
00
22
2
00
2
2
222
,2
1,,
2
10,0, D
fD
D
Df
DDf
DD ffXf
ffXf
fXXfXDDD
τττ
ττττ
τττ
τ
( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) +∆−
+
∂∂++∆−= ∫
+∞
∞−
2222
22222222
22
222*Im22222,
Dgs
DggsssggsD
ftE
fftEdttgt
tgtEEffEfX
π
τππττ
( )( )
( )[ ] ( )( ) ( ) ( ) ( ) ( ) ( ) +∆−
+
∂∂++∆−= ∫
∞+
∞−
222222
2
2
222*Im2
221
0,0
,DgDggs
sgg
D ftfftEdttgt
tgtE
ffX
fXπτππτ
τ
If we choose the time and frequency origins such that
( )
( )
( ) ( )
( )0
2
22
:&0:2
2
2
2
====
∫
∫
∫
∫∞+
∞−
+∞
∞−∞+
∞−
+ ∞
∞−
dffG
dffGf
f
dttg
dttgt
t gg
π
ππ
( )( )
( ) ( )( ) ( ) ( ) ( ) +∆−
∂∂+∆−= ∫
∞+
∞−==
22222
00
2
2
2*Im2
221
0,0
,DgD
sg
tf
D ftfdttgt
tgtE
fX
fX
g
g
πτπττ
31
Ambiguity Function for RADAR Signals
( ) ( ) ( ) ( )∫+∞
∞−
∗ −= dttfjtgtgfX DD πττ 2exp:,
SOLO
Ambiguity Function Close to Origin (continue -8)
( )( )
( ) ( )( ) ( ) ( ) ( ) +∆−
∂∂+∆−= ∫
∞+
∞−==
22222
00
2
2
2*Im2
221
0,0
,DgD
sg
tf
D ftfdttgt
tgtE
fX
fX
g
g
πτπττ
Helstrom’s Uncertainty Ellipse
The curve resulting from the interception of a plane parallel to the τ, fD plane and theNormalized Ambiguity Function is an ellipse. The ellipse computed when the plane is at a height of 0.75 is referred to as Helstrom’s Uncertainty Ellipse.
( )( )
( ) ( )( ) ( ) ( ) ( )
4
32*Im
2
221
0,0
, 22222
00
2
2
=+∆−
∂∂+∆−= ∫
∞+
∞−==
DgDs
g
tf
D ftfdttgt
tgtE
fX
fX
g
g
πτπττ
( ) ( )( ) ( ) ( ) ( )
4
12*Im
2
22 22222 =∆+
∂∂−∆ ∫
+∞
∞−DgD
sg ftfdttg
ttgt
Ef πτπτ
32
Ambiguity Function for RADAR Signals
( ) ( ) ( ) ( )∫+∞
∞−
∗ −= dttfjtgtgfX DD πττ 2exp:,
SOLO
Ambiguity Function Close to Origin (continue -4)
( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫ ∫
+∞
∞−
+∞
∞−
+∞
∞−
−=
∂∂− ω
ωωωωπ d
d
GdGEjdttgtgdttg
ttgtj s *2**2 22211
111
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫∫ ∫
+∞
∞−
+∞
∞−
+∞
∞−
+∞
∞−
−=
∂∂+ ω
ωωωωωωω
ππ d
d
GdGdGG
jdttgtgtdttg
ttgj
*
2222111
1 *2
**2
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫∫ ∫+∞
∞−
+∞
∞−
+∞
∞−
+∞
∞−
=
∂∂− ωωωω
πω
ωωωπ dGG
jd
d
GdGdttg
ttgdttgtgtj *
2***2 22
221111
( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫ ∫
+∞
∞−
+∞
∞−
+∞
∞−
=
∂∂+ ω
ωωωωπ d
d
GdGEjdttg
ttgtdttgtgj s
*
222
22111 2**2
c2 m=n=1
c2 m=0n=1
c1m=1n=0
c2 m=1n=0
c1m=0n=1
c1 m=n=1
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )∫∫
∫∞+
∞−
∞+
∞−
∞+
∞−==
−
−
−
=
∂∂∂
ωωωω
ωωωωωωω
π
ωωωω
ωωωω
ττ
dd
GdG
d
GdGjdGG
dd
GdG
d
GdGjEfX
f sf
DD D
**2
1
*2,0
*
*
00
22
( ) ( ) ( ) ( ) ( ) ( ) ,2,1,0,,2,1,0
2*
*
==
= ∫∫
∞+
∞−
∞+
∞−
mndd
SdS
jdt
td
tsdtstj
m
mn
n
n
nmm ω
ωωωω
π
( ) ( ) ( ) ( ) ( ) ( ) ,2,1,0,,2,1,0*
2
* ==
−=− ∫∫+∞
∞−
+∞
∞−
mndd
SdS
jdt
td
tsdtstj
m
mn
n
n
nmm ω
ωωωω
π
c1
c2
Relationshipsfrom Parseval’sTheorem
33
Ambiguity Function for RADAR Signals
( ) ( ) ( ) ( )∫+∞
∞−
∗ −= dttfjtgtgfX DD πττ 2exp:,
( ) ( ) ( )
s
D
E
gf
D dffGffX
2
22
00
2
2
22, ∫+∞
∞−=
= ∆−=∂∂ πττ
τ
( )( ) ( )
( )∫
∫∞+
∞−
+∞
∞−=∆dffG
dffGf
f g2
222
2
2
22
:
π
ππ
SOLO
Ambiguity Function Close to Origin (continue -1)
( ) ( ) ( ) ( ) ( ) ( ) ( )
s
D
E
gf
D
D
dttgtgtdttgtgtfXf
2
222
002
2
22, ∫∫+∞
∞−
∗+∞
∞−
∗
== ∆−=−=
∂∂ ππτ τ
We found:
where:
Δfg – is signal envelope bandwidth
Es – is signal energy ( ) ( ) ( )∫∫∫+∞
∞−
+∞
∞−
+∞
∞−
=== tdtgfdfGtdtsEs222
2
12
2
1: π
or ( ) ( ) ( ) ( )
s
D
E
gf
D
D
dttgtgtfXf
2
2
002
2
2, ∫+∞
∞−
∗
== ∆−=
∂∂ πτ τ
Δtg – is signal envelope duration ( )( )
( )∫
∫∞+
∞−
+∞
∞−=∆tdtg
tdtgt
tg2
22
2 :
34
Ambiguity Function for RADAR Signals
( ) ( ) ( ) ( )∫+∞
∞−
∗ −= dttfjtgtgfX DD πττ 2exp:,
SOLO
Ambiguity Function Close to Origin (continue -2)
( ) ( ) ( ) ( ) ( ) ( ) ( )
∂∂=
−∂∂=
∂∂
∂∂=
∂∂
∂∂
∫∫+∞
∞−
∗+∞
∞−
∗=
==
= dttgt
tgtdttfjtgtgtjfXf
fXf DfD
Df
DD
DD
ππττ
πττ
ττ
ττ 2Im2exp2Re,Re,00
00
Define( ) ( )
( ) ( ) ( ) ( ) ( )
∂∂
∆∆−=
∂∂
∆∆−= ∫
∫
∫ ∞+
∞−
∗∞+
∞−
∗
+∞
∞−
∗
dttgt
tgtEft
dttgtg
dttgt
tgt
ft sgggg 2
1Im
1:ρ
Error Coupling Coefficient
We obtain( ) ( ) ( ) ( ) ggs
fD
D
ftEdttgt
tgtfXf D
∆∆−=
∂∂=
∂∂
∂∂
∫+∞
∞−
∗
== ρππτ
ττ 222Im,
00
35
Ambiguity Function for RADAR SignalsSOLO
Ambiguity Function for Single RF Pulse
( )( )
>
≤≤−=
2/0
2/2/cos 0
p
pp
SPitt
ttttAts
ω
The complex envelope is
( )
>
≤≤−=
2/0
2/2/1
p
pp
pSP
tt
tttttg
( ) ( ) ( ) ( )( )
( )
( )( )
( )
( )( )
( )( )
( )
<
<
=
<
<
=−=++
−
+
+−
+
−
+
+−∞+
∞−
∗
∫
∫∫
02exp2
1
02exp2
1
02exp1
02exp1
2exp:,2/
2/
2/
2/
2/
2/
2/
2/
τππ
τππ
τπ
τπ
πτττ
τ
τ
τ
p
p
p
p
p
p
p
p
t
t
DpD
t
t
DpD
t
t
Dp
t
t
Dp
DDSP
tfjtfj
tfjtfj
tdtfjt
tdtfjt
tdtfjtgtgfX
<
+−−
+
<
−−−
−
=
<
−−
+
<
+−−
=
02
22exp
22exp
02
22exp
22exp
22exp
02
22exp
22exp
02
22exp
22exp
τπ
τπ
τπ
τπ
ττ
πτ
π
τπ
τπ
πτπ
τπ
τππ
pD
pD
pD
pD
pD
pD
D
pD
pD
pD
pD
pD
pD
tfj
tfj
tfj
tfj
tfj
tfj
fj
tfj
tfj
tfj
tfj
tfj
tfj
36
Ambiguity Function for RADAR SignalsSOLO
Ambiguity Function for Single RF Pulse (continue – 1)
( ) ( ) ( )[ ] ( ) ( ) ( )[ ]( ) p
ppD
ppDpD
pD
pDDDSP t
ttf
ttftfj
tf
tffjfX ≤
−−
−=−
= ττπ
τπττπ
πτπ
τπτ/1
/1sin/1exp
sinexp,
Therefore:
( ) ( ) ( )[ ]( )
≤
−−
−=
elsewere
tttf
ttft
fX pppD
ppDp
DSP
0
/1
/1sin/1
,τ
τπτπ
ττ
( ) ( ) ppSP ttX ≤−= τττ /10,
( ) [ ]pD
pDDSP tf
tffX
ππsin
,0 =
37
Ambiguity Function for RADAR Signals
( )
( ) p
t
p
t
pSP
DSP
res tt
dtX
dfX
Tpp
=
−=
−=
== ∫
∫+∞
∞−
0
2
02
2
2212
0,0
0,
:ττττ
ττ
( )
( )
( )
( ) p
t
p
SP
SP
SP
DDSP
res ttd
ttdtg
tdtg
X
fdfX
Fp 12
0,0
,0
:2/
02
1
2
2
4
1
2
2
==
=
== ∫
∫
∫∫∞+
∞−
+∞
∞−
+∞
∞−
τ
( )
>
≤≤−=
2/0
2/2/1
p
pp
pSP
tt
tttttg
SOLO
Ambiguity Function for Single RF Pulse (continue – 2)
( ) ( ) ( )[ ]( )
≤
−−
−=
elsewere
tttf
ttft
fX pppD
ppDp
DSP
0
/1
/1sin/1
,τ
τπτπ
ττ
( ) ( ) ppSP ttX ≤−= τττ /10,
Range Resolution
( ) 10,0 =SPX
Doppler Resolution
presres tFV
22
λλ ==
Return to Table of Content
38
Ambiguity Function for RADAR Signals
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( )∫∫+∞
∞−
∗+∞
∞−
∗ −−−=−= tdtfjtkjtgtkjtgtdtfjtgtgfX DSPSPDFMSPFMSPDFMSP πτπτππττ 2expexpexp2exp:, 22
SOLO
Ambiguity Function for Linear FM Modulation Pulse
( )
>
≤
+
=
20
22cos
2
0
τ
τπω
t
ttk
tAts FMSPi
( )[ ]
( ) [ ]22
exp
20
2exp
1
tkjtgt
t
tttkj
ttg SP
p
p
pFMSP π
π=
>
≤=
The signalof Single PulseFrequency Modulated
The complex envelopeof Single PulseFrequency Modulated
( )tgSP - the complex envelope of Single RF Pulse
( ) ( ) ( ) ( ) ( )[ ] ( ) ( )τττπτπττπτ kfXkjtdtkfjtgtgkjfX DSPDSPSPDFMSP +−=+−−= ∫+∞
∞−
∗ ,exp2expexp, 22
( ) ( ) ( )[ ]( ) p
ppD
ppDpDSP t
ttf
ttftfX ≤
−−
−= ττπ
τπττ
/1
/1sin/1,where Ambiguity Function of the Single
Frequency Pulse
( ) ( ) ( ) ( )[ ]( ) ( )
≤
−+−+
−=
elsewere
tttkf
ttkft
fX pppD
ppDp
DFMSP
0
/1
/1sin/1
,τ
ττπττπ
ττ
39
Ambiguity Function for RADAR SignalsSOLO
Ambiguity Function for Linear FM Modulation Pulse (continue – 1)
( ) ( ) ( ) ( )[ ]( ) ( )
( )ττ
τττπ
ττπτ
τ
kfX
elsewere
tttkf
ttkft
fX
DSP
pppD
ppDp
DFMSP
+=
≤
−+−+
−=
,
0
/1
/1sin/1
,
40
Ambiguity Function for RADAR SignalsSOLO
Ambiguity Function for Linear FM Modulation Pulse (continue – 2)
( ) ( ) ( ) ( )[ ]( ) ( )
( )ττ
τττπ
ττπτ
τ
kfX
elsewere
tttkf
ttkft
fX
DSP
pppD
ppDp
DFMSP
+=
≤
−+−+
−=
,
0
/1
/1sin/1
,
( ) ( )( )0,,,
τ
τττττ
SP
SPFMSP
X
kkXkX
=
+−=−
41
Ambiguity Function for RADAR SignalsSOLO
Ambiguity Function for Linear FM Modulation Pulse (continue – 3)
( )( )
( )p
pp
pp
pDFMSP t
ttk
ttk
tfX ≤
−
−
−== τ
ττπ
ττπτ
τ1
1sin
10,
tpτ1’st null
( ) πτ
τπ =
−
pp ttk 1
p
tkpp
nullst tkk
tt p 11
42
42
'1
2>>
≈−−=τ
k tp = Δf is the total frequencydeviation during the pulse.
pnullst
p
p
tk
nullst tft
DrationCompressioftk
p
∆===∆
=≈>>
'1
4
'1
112
ττ
Return to Table of Content
42
Ambiguity Function for RADAR SignalsSOLO
Ambiguity Function for a Coherent Pulse Train
The envelope of each pulse is of unit energy and thecoherence is maintained from pulse to pulse.
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )∑ ∑ ∫
∫ ∑ ∑∫−
=
−
=
∞+
∞−
+∞
∞−
−
=
−
=
+∞
∞−
−−−=
−−−=−=
1
0
1
0
1
0
1
0
*
2exp*1
2exp*1
2exp,
N
n
N
mDRSPRSP
D
N
n
N
mRSPRSPDPTPTDPT
tdtfjTmtgTntgN
tdtfjTmtgTntgN
tdtfjtgtgfX
πτ
πτπττ
( ) ( ) ( ) ( )[ ] ( )∑ ∑ ∫−
=
−
=
+∞
∞−
−=
−−−=1
0
1
01111 2exp*2exp
1,
1 N
n
N
mDRSPSPRD
Tntt
DPT tdtfjTnmtgtgTnfjN
fXR
πτπτ
( ) [ ] ( ) ( ) ( ) ( )[ ]( ) ( )
≤
−−
−==−∫
∞+
∞−elsewere
tfjttf
ttft
fXtdtfjtgtg pDppD
ppDp
DSPDSPSP
0
2exp/1
/1sin/1
,2exp* 1111
ττπτπ
τπτ
τπτ
( )
>
≤≤−=
2/0
2/2/1
p
pp
pSP
tt
tttttg Envelope of
Single Pulse( ) ( )∑
−
=
−=1
0
1 N
nRSPPT Tntg
Ntg
Envelope of a Pulse Train
( ) ( ) ( ){ }tfjtgts PT 02expRe π= Pulse Train Signal
For a Coherent Pulse Train:
where for a Single Pulse, we found:
implies coherency
43
Ambiguity Function for RADAR SignalsSOLO
Ambiguity Function for a Coherent Pulse Train (continue – 1)
( ) ( ) ( )∑ ∑−
=
−
=
−=−=1
0
1
0
:,2exp1
,N
n
N
mDRSPRDDPT mnpfTpXTnfj
NfX τπτ
For a Coherent Pulse Train:
Construction Table for the Double Sum with p=n-m
nm 0 1 2 … N-1
0 0 1 2 … N-11 -1 0 1 … N-22 -2 -1 0 … N-3
… … … … … …
N-1 -N-1 -N-2 -N-3 … 0
p=n-m
( ) BlockTriangularRight
pmn
N
p
pN
m
DiagonalBlockTriangularLow
pnmNp
pN
n
N
n
N
m +=
−
=
−−
=−=−−=
−−
=
−
=
−
=∑ ∑∑ ∑∑ ∑ +=1
1
1
0
&
0
1
1
0
1
0
1
0
( ) ( ) ( )( )
( ) ( ) ( )∑ ∑
∑ ∑−
=
−−
=
−−=
−−
=
−+
−=
1
1
1
1
0
1
1
0
2exp,2exp1
2exp,1
,
N
p
pN
mRDDRSPRD
Np
pN
nRDDRSPDPT
TmfjfTpXTpfjN
TnfjfTpXN
fX
πτπ
πττ
44
Ambiguity Function for RADAR SignalsSOLO
Ambiguity Function for a Coherent Pulse Train (continue – 2)
For a Coherent Pulse Train:
( ) ( ) ( )( )
( ) ( ) ( )∑ ∑
∑ ∑−
=
−−
=
−−=
−−
=
−+
−=
1
1
1
0
0
1
1
0
2exp,2exp1
2exp,1
,
N
p
pN
mRDDRSPRD
Np
pN
nRDDRSPDPT
TmfjfTpXTpfjN
TnfjfTpXN
fX
πτπ
πττ
To compute the sums of the exponents, we use:
( ) ( ) ( )
2/12/1
2/2/
2/1
2/1
0 1
1
yy
yy
y
y
y
yy
pNpNpNpNpN
n
n
−−=
−−= −
−−−−−−−
=∑
take: ( )RD Tfjy π2exp=
( ) ( )[ ] ( )[ ]( )RD
RDRD
pN
nRD Tf
TpNfTpNfjTnfj
ππ
ππsin
sin1exp2exp
1
0
−−−=∑
−−
=
Using this result we obtain:
( ) ( )[ ] ( ) ( )[ ]( )( )
∑−
−−=
−−+−=
1
1 sin
sin,1exp
1,
N
Np RD
RDDRSPRDDPT Tf
TpNffTpXTpNfj
NfX
ππ
τπτ
45
Ambiguity Function for RADAR SignalsSOLO
Ambiguity Function for a Coherent Pulse Train (continue – 3)
For a Coherent Pulse Train:
( ) ( )[ ] ( ) ( )[ ]( )( )
∑−
−−=
−−+−=
1
1 sin
sin,1exp
1,
N
Np RD
RDDRSPRDDPT Tf
TpNffTpXTpNfj
NfX
ππ
τπτ
where
The expression |XPT (τ,fD)| can be simplified if the separation between pulses is larger than the duration of individual pulses.
( ) ( ) ( )[ ]( )( )
( ) ( )[ ]( )
( )[ ]( )( )
2/sin
sin
/1
/1sin/1
1
sin
sin,
1,
1
1
1
1
Rp
N
Np RD
RD
pRpD
pRpDpR
N
Np RD
RDDRSPDPT
TtTf
TpNf
tTptf
tTptftTp
N
Tf
TpNffTpX
NfX
<−
−−−−
−−=
−−=
∑
∑−
−−=
−
−−=
ππ
τπτπ
τ
ππ
ττ
( ) ( ) ( )[ ]( ) ( )
≤
−−
−=
elsewere
tfjttf
ttft
fX pDppD
ppDp
DSP
0
2exp/1
/1sin/1
,ττπ
τπτπ
ττ
46
Ambiguity Function for RADAR SignalsSOLO
Ambiguity Function for a Coherent Pulse Train (continue – 4)
The Ambiguity Function for a Coherent Pulse Train:
Setting fD = 0 we obtain:
( ) ( ) ( )[ ]( )
( )[ ]( )( )
2/sin
sin
/1
/1sin/1
1,
1
1Rp
N
Np RD
RD
pRpD
pRpDpRDPT Tt
Tf
TpNf
tTptf
tTptftTp
NfX <
−−−
−−−−= ∑
−
−−= ππ
τπτπ
ττ
( ) ( )[ ]( )( ) ( ) ( )pN
Tf
Tf
TpNf
TpNf
t
Tp
NfX
DDfRD
RDN
Np fRD
RD
p
RDPT −
−−
−−=
=
−
−−= =
∑
1
0
1
1
1
0sin
sin1
1,
ππ
ππτ
τ
( )( )
pR
N
Np p
RDPT tTp
N
p
t
TpfX <−
−
−−== ∑
−
−−=
ττ
τ 110,1
1
or
47
Ambiguity Function for RADAR SignalsSOLO
Ambiguity Function for a Coherent Pulse Train (continue – 5)
The Ambiguity Function for a Coherent Pulse Train:
( )
( ) ( )[ ]( )
( )[ ]( )( )
2/sin
sin
/1
/1sin/1
1
,
1
1Rp
N
Np RD
RD
pRpD
pRpDpR
DPT
TtTf
TpNf
tTptf
tTptftTp
N
fX
<−
−−−−
−−= ∑−
−−= ππ
τπτπ
τ
τ
48
Pulse bi-phase Barker coded of length 7
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
-1 = -11
+1 -1 = 02
-1 +1 -1 = -13
-1 -1 +1-( -1) = 04
+1 -1 -1 –(+1)-( -1) = -15
+1 +1 -1-(-1) –(+1)-1= 06
+1+1 +1-( -1)-(-1) +1-(-1)= 77
+1+1 –(+1)-( -1) -1-( +1)= 08
+1-(+1) –(+1) -1-( -1)= -19
-(+1)-(+1) +1 -( -1)= 010
-(+1)+1-(+1) = -111
+1-(+1) = 012-(+1) = -113
0 = 014
SOLO Pulse Compression Techniques
-1-1 -1+1+1+1+1 { }*kc
49
SOLO
50
SOLO
Return to Table of Content
51
52
SOLORihaczek, A.W., “Principles of High Resolution Radar”, McGraw Hill, 1969
53
SOLORihaczek, A.W., “Principles of High Resolution Radar”, McGraw Hill, 1969
54
SOLORihaczek, A.W., “Principles of High Resolution Radar”, McGraw Hill, 1969
55
SOLORihaczek, A.W., “Principles of High Resolution Radar”, McGraw Hill, 1969
56
SOLORihaczek, A.W., “Principles of High Resolution Radar”, McGraw Hill, 1969
57
SOLORihaczek, A.W., “Principles of High Resolution Radar”, McGraw Hill, 1969
58
SOLORihaczek, A.W., “Principles of High Resolution Radar”, McGraw Hill, 1969
59
SOLORihaczek, A.W., “Principles of High Resolution Radar”, McGraw Hill, 1969
60
SOLORihaczek, A.W., “Principles of High Resolution Radar”, McGraw Hill, 1969
61
SOLORihaczek, A.W., “Principles of High Resolution Radar”, McGraw Hill, 1969
62
SOLORihaczek, A.W., “Principles of High Resolution Radar”, McGraw Hill, 1969
63
SOLORihaczek, A.W., “Principles of High Resolution Radar”, McGraw Hill, 1969
64
SOLORihaczek, A.W., “Principles of High Resolution Radar”, McGraw Hill, 1969
65
Ambiguity function for a square pulse
Ambiguity function for an LFM pulse Return to Table of Content
66
Matched Filters for RADAR SignalsSOLO
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
N. Levanon, “Waveform Analysis and Design”, 2008 IEEE Radar Conference, Tutorial, MA2, May 26 – 30, 2008, Rome, Italy
Hermelin, S., “Pulse Compression Techniques”, Power Point PresentationReturn to Table of Content
January 19, 2015 67
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
Vector Analysis
68
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
( ) ( )td
tj δπωω =∫
+∞
∞− 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
69
( )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
aF
a
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
70
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
71
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*
72
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
73
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
74
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
75
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
76
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
77
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
78
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
79
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
80
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
81
SOLO
82
SOLO
83
SOLO