4 matched filters and ambiguity functions for radar signals-2

1
Matched Filters and
Ambiguity Functions for
RADAR Signals
Part 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 Filter
The Matched Filter Approximations
1.Single RF Pulse
2. Linear FM Modulated Pulse (Chirp)
Continuous Linear Systems
Discrete Linear Systems
RADAR Signals
Signal Duration and Bandwidth
Complex 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
M
A
T
C
H
E
D
F
I
L
T
E
R
S

3
SOLO
Table of Content (continue – 1)
Ambiguity Function for RADAR Signals
Definition of Ambiguity Function
Ambiguity Function Properties
Cuts Through the Ambiguity Function
Ambiguity 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
Continue from
Matched Filters

5
( ) ( ) ( ) ( )∫
+∞
∞−
∗
−= dttfjtgtgfX DD πττ 2exp:,
SOLO
Definition of Ambiguity Function:
( ) ( ) ( ) ( ) ( ) ( )[ ]
( ) ( )[ ]tjta
tjttatsjtstg QI
θ
θθ
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
Return to Table of Content

6
( ) ( ) ( ) ( )∫
+∞
∞−
∗
( ) ( ) ( ) ( ) ( ) ( )[ ]
( ) ( )[ ]tjta
tjttatsjtstg QI
θ
θθ
exp
sincos:
=
+=+=
SOLO
Ambiguity Function has the following properties:
( ) ( ) 1
2
1 22
== ∫∫
+∞
∞−
+∞
∞−
ωω
π
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
( ) ( ) ( ) ( )∫
+∞
∞−
∗
SOLO
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
( ) ( ) 1
2
1
:
22
=== ∫∫
+∞
∞−
+∞
∞−
ωω
π
dGdttgEs
( ) 1,
2
≤DfX τ ( ) ( ) 10,0, =≤ XfX Dτ
Proof:
Schwarz
inequality

8
( ) ( ) ( ) ( )∫
+∞
∞−
∗
SOLO
Ambiguity Function properties (continue – 2):
( ) 1, =∫ ∫
+∞
∞−
+∞
∞−
DD dfdfX ττ2
( ) ( ) ( ) ( )∫
+∞
∞−
∗
( )DfX ,τ is the Fourier Transform of ( ) ( )τ−∗
tgtg
Using Parseval’s Theorem: ( ) ( ) ( )∫∫
+∞
∞−
+∞
∞−
∗
=− DD dffXdttgtg
22
,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
dtdtttJtgtgddttgtgV
tt
tt τ
ττ ( ) 1
11
01
//
//
,
22
11
21 =
−
=
∂∂∂∂
∂∂∂∂
=
τ
τ
ttt
ttt
ttJ
( ) ( ) ( ) ( ) ( )∫ ∫∫∫∫ ∫
+∞
∞−
+∞
∞−
+∞
∞−
∗
+∞
∞−
+∞
∞−
+∞
∞−
∗
==== DD dfdfXdttgdttgdtdttgtg ττ
2
1
2
2
2
1
1
2
121
2
2
2
1 ,1
  
Proof:

9
( ) ( ) ( ) ( )∫
+∞
∞−
∗
SOLO
( ) ( )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
τπτπττ
τ
( ) ( ) ( )DDD fXfjfX ,2exp, *
ττπτ =−−
( ) ( ) ( ) ( )DDDD fXfXfjfX ,,2exp, *
τττπτ ==−−

10
( ) ( ) ( ) ( )∫
+∞
∞−
∗
( ) ( ){ } ( ) ( )∫
+∞
∞−
−== dttfjtgtgfG π2exp:F
SOLO
4
Proof:
( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫
+∞
∞−
+∞
∞−
∗
( ) ( ){ } ( ) ( )∫
+∞
∞−
== fdtfjfGGtg πω 2exp:1-
F
( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∫∫
+∞
∞−
∗
+∞
∞−
+∞
∞−
∗
−





−=−= tdtgfdtfjffGtdtgtfjtgfX DDD τπτπτ 2exp2exp,
( )tg -1
F
F
( )fG
( ) ( ) ( )[ ]∫
+∞
∞−
−=− fdtfjfGtg τπτ 2exp ( )τ−tg -1
F
F ( ) ( )τπ fjfG 2exp −
( ) ( ) ( )[ ] ( ) ( ) ( )∫∫
+∞
∞−
+∞
∞−
−−=−−= fdfjffGfjfdtffjffGtg DDDD πππ 2exp2exp2exp
( ) ( ) ( ) ( )∫
+∞
∞−
−= fdfjffGfjtg DD ππ 2exp2exp ( ) ( )Dfjtg π2exp -1
F
F
( )DffG −
( ) ( ) ( )∫
+∞
∞−
−= fdfjfGffG D τπ2exp*
( ) ( ) ( ) ( ) ( ) ( )[ ]∫∫ ∫
∞+
∞−
∞+
∞−
∞+
∞−
−−=





−−−= fdfjfGffGfdtdtfjtgffG DD
*
*
2exp2exp τππτ
q.e.d.

11
( ) ( ) ( ) ( )∫
+∞
∞−
∗
SOLO
( ) ( ) ( )DDD fXfjfX −−=− ,*2exp, ττπτ
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
  

12
( ) ( ) ( ) ( )∫
+∞
∞−
∗
SOLO
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
( ) ( ) ( ) ( )∫
+∞
∞−
∗
SOLO
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 is
independent of any phase or frequency modulation and is determined only by the
magnitude of the complex envelope of the signal – that is by amplitude modulation.
( ) ( ) ( ) ( ) ( ) ( )∫∫
+∞
∞−
+∞
∞−
∗
=== dttfjtgdttfjtgtgfX DDD ππτ 2exp2exp,0
2
( )DfX ,0=τ( ) 2
tg
F
F-1

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
( ) ( ) tfj
etgts 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
2
22
2
11
:
:
τπ
τπ
τ
τ
tffj
tffj
D
D
etgts
etgts
- Transmitted signal
- Received signals

15
Ambiguity as a Measure of Range and Doppler Resolution (continue – 1)
( ) ( ) ( ) ( ) ( )[ ] ( )[ ] ( )[ ]
( ) ( ) ( )[ ] ( )[ ] ( )[ ]∫
∫∫∫
∞+
∞−
+∞
∞−
+∞
∞−
+∞
∞−
−−−−−−−
−−−−−−−−−+=
dttfjtfjfjtgtg
DD
DD
221121021
221121021
222
2exp2exp2exp*
2exp2exp2exp*
τπτπττπττ
τπτπττπτττε
2121 :&: DDD fff −=∆−=∆ τττDefine
We found
( ) ( ) ( ) ( ) ( )[ ] ( )[ ] ( )[ ]
( ) ( ) ( )[ ] ( )[ ] ( )[ ]∫
∫∫∫
∞+
∞−
+∞
∞−
+∞
∞−
+∞
∞−
−+−−−+−
−+−−−−+−−+=
""2exp"2exp2exp""*
''2exp'2exp2exp'*'
212121012
212121021
222
dttfjtfjfjtgtg
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
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
( ) ( ) ( )
( )
( )[ ] ( ){ }
( )
( )
( )[ ] ( ){ }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 for
non-zero target delay Δτ and Doppler ΔfD.
( )DfX ∆∆ ,τ
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )




−=
−=
−+
−+
220
110
2
22
2
11
:
:
τπ
τπ
τ
τ
tffj
tffj
D
D
etgts
etgts
Received complex signal
( ) ( ) ( ) ( )∫
+∞
∞−
∗
−= dttfjtgtgfX DD πττ 2exp:,where ( ) ( ) ( )DDD fXfjfX −−=− ,*2exp, ττπτ

18
( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫
+∞
∞−
+∞
∞−
∗
==−== fdfjfGfGRdttgtgfX ggD τπτττ 2exp*:0,
SOLO
Range Resolution
( )
( ) 2
2
0,0
0,
:
X
dfX
T
D
res
∫
+∞
∞−
=
=
ττ
Assume the two signals have the same Doppler fD = 0. The range resolution is defined as:
( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫
+∞
∞−
+∞
∞−
∗
Using
we have
( )τggR -1
F
F ( ) 2
fG
( ) ( ) ( ) ( ) ( ) ( )∫∫
+∞
∞−
+∞
∞−
∗
====== fdfGfGRdttgtgfX ggD *0:0,0 ττ
( )
( )
( )
( )
2
2
4
2
2
0






==
∫
∫∫
∞+
∞−
+∞
∞−
+∞
∞−
fdfG
fdfG
R
dR
T
gg
gg
res
ττ
Parseval’s Theoremand

19
Doppler Resolution
( )
( ) 2
2
0,0
,0
:
X
fdfX
F
DD
res
∫
+∞
∞−
=
=
τ
Assume the two signals have the same range delay τ = 0. The Doppler resolution
is defined as:
( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫
+∞
∞−
+∞
∞−
∗
Using
we have
( ) 2
tg -1
F
F ( )DGG fR
( ) ( ) ( ) ( ) ( ) ( ) ( )0*0:0,0 ====== ∫∫
+∞
∞−
+∞
∞−
∗
fRfdfGfGRdttgtgX GGgg τ
( )
( )
( )
( )
2
2
4
2
2
0






==
∫
∫∫
∞+
∞−
+∞
∞−
+∞
∞−
tdtg
tdtg
R
fdfR
F
GG
GG
res
Parseval’s Theoremand
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )DGGDGGDDD fRfRfdfGffGtdtfjtgtgfX =−=−=== ∫∫
+∞
∞−
+∞
∞−
∗
*2exp:,0 πτ

20
Range – Doppler Resolution
( )
( )
( )
( )
2
2
4
2
2
0






==
∫
∫∫
∞+
∞−
+∞
∞−
+∞
∞−
tdtg
tdtg
R
fdfR
F
GG
GG
res
( )
( )
( )
( )
2
2
4
2
2
0






==
∫
∫∫
∞+
∞−
+∞
∞−
+∞
∞−
fdfG
fdfG
R
dR
T
gg
gg
res
ττ
From Schwarz Inequality: ( ) ( ) ( ) ( )∫∫∫
+∞
∞−
+∞
∞−
+∞
∞−
≤ dtthdttfdtthtf
22
Choose ( ) ( ) ( ) ( ) ( )tg
td
tgd
thtgttf ':& ===
( ) ( ) ( ) ( )∫∫∫
+∞
∞−
+∞
∞−
+∞
∞−
≤ dttsdttstdttstst
22
''we obtain

21
Good resolution is obtained when (Ambiguity Function) is minimum for
non-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.

22
( ) ( ) ( ) ( )∫
+∞
∞−
∗
SOLO
Ambiguity Function Close to Origin
( ) ( ) ( ) ( ) DfD
D
fDD ffX
f
fXXfX DD 0
02
0
0222
,,0,0, =
=
=
=
∂
∂
+
∂
∂
+= ττ
τττ
τ
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,, 0
02
0
02
=
∂
∂
=
∂
∂
=
=
=
=
DD fD
D
fD fX
f
fX ττ
ττ
τ
( ) ( ) ( ) +






∂
∂
+
∂
∂
∂
∂
+
∂
∂
+ =
=
=
=
=
= 2
0
02
2
2
0
022
0
02
2
2
,,2,
2
1
DfD
D
DfD
D
fD ffX
f
ffX
f
fX DDD
τττ
τττ
τ
ττ
τ
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ( )
( )
∫ ∫∫ ∫
∫ ∫
∞+
∞−
∞+
∞−
∗
∞+
∞−
∞+
∞−
∗
+∞
∞−
+∞
∞−
∗
=
−−
∂
∂
+−−
∂
∂
=






−−
∂
∂
=
∂
∂
    
ττ
ττ
τ
ττ
τ
ττ
τ
τ
τ
gggg
D
RR
fD
dttgtgdttgtgdttgtgdttgtg
dtdttgtgtgtgfX
111222
*
222111
2122110
2
**
*,
also
( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
00*0,
000
0
02
=
∂
∂
=






∂
∂
+
∂
∂
=
∂
∂
=≠
+∞
∞−
+∞
∞−
∗
≠
=
=
∫∫
τ
τ
τ
τ
τ
τ
gg
gggg
f
D
R
Rdttg
t
tgdttg
t
tgRfX D


23
( ) ( ) ( ) ( )∫
+∞
∞−
∗
( ) ( ) ( ) ( ) ( ) ( )  ,2,1,0,,2,1,0*
2
*
==




−
=− ∫∫
+∞
∞−
+∞
∞−
mnd
d
Sd
S
j
dt
td
tsd
tstj m
m
n
n
n
n
mm
ω
ω
ω
ωω
π
SOLO
Ambiguity Function Close to Origin (continue – 1)
( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫
+∞
∞−
+∞
∞−
∗
= −
∂
∂
+−
∂
∂
=
∂
∂
ττ
τ
ττ
τ
τ
τ
ggggfD RdttgtgRdttgtgfX D 2221110
2
**,
( ) ( ) ( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ( ) ( )
( )
∫∫
∫∫
∞+
∞−
∞+
∞−
+∞
∞−
∗
+∞
∞−
∗
=
∂
∂
−
∂
∂
+−
∂
∂
+
∂
∂
−
∂
∂
+−
∂
∂
=
∂
∂
τ
τ
τ
τ
ττ
τ
τ
τ
τ
τ
ττ
τ
τ
τ
gg
gg
gg
ggfD
R
dttgtgRdttgtg
R
dttgtgRdttgtgfX D
222222
2
2
111112
2
10
2
2
2
**
*
*,
Since is a maximum for τ=0, we have( ) ( ) sgggg ERR 20*0 ==
( ) ( )
0
0*0
=
∂
=∂
=
∂
=∂
τ
τ
τ
τ gggg RR
( ) ( ) ( ) ( ) ( ) ( )






∂
∂
+
∂
∂
=
∂
∂
∫∫
+∞
∞−
+∞
∞−
∗
=
=
dttgtgdttgtgRfX
s
D
E
gg
f
D 2
2
2
2
2
0
02
2
2
*0,
ττ
τ
τ
τ

n=2
m=0
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( )
  
sE
ggss
f
s
Parseval
dffGffEdffGfEdGG
E
2
222222
2
2
222:2222*
2
22
∫∫∫
+∞
∞−
+∞
∞−
=+∞
∞−
+∆−=−=−= πππωωωω
π
πω
Relationship
from Parseval’s
Theory

24
Signal Duration and BandwidthSOLO
( )tf
-1
F
F
( )ωFRelationships from Parseval’s Formula
( ) ( ) ( ) ( )∫∫
+∞
∞−
+∞
∞−
= ωωω
π
dFFdttftf 2
*
12
*
1
2
1
Parseval’s Formula7
Choose ( ) ( ) ( ) ( )tstjtftf
m
−== 21
( ) ( ) ,2,1,0
2
1
2
22
== ∫∫
∞+
∞−
∞+
∞−
nd
d
Sd
dttst m
m
m
ω
ω
ω
π
( ) ( )tftj
n
−
-1
F
F ( )ω
ω
F
d
d
n
n
and use 5a
Choose ( ) ( ) ( )
n
n
td
tsd
tftf == 21 and use 5b
( )tf
td
d
n
n
-1
F
F
( ) ( )ωω Fj
n
( ) ( ) ,2,1,0
2
1 22
2
== ∫∫
∞+
∞−
∞+
∞−
ndSdt
td
tsd m
n
n
ωωω
π
Choose or the oppositec
( ) ( ) ( ) ( ) ( ) ( )  ,2,1,0,,2,1,0
2
*
*
==





= ∫∫
∞+
∞−
∞+
∞−
mnd
d
Sd
S
j
dt
td
tsd
tstj m
m
n
n
n
n
mm
ω
ω
ω
ωω
π
( ) ( )
n
n
td
tsd
tf =1
( ) ( ) ( )tstjtf
m
−=2
( ) ( ) ( ) ( ) ( ) ( )  ,2,1,0,,2,1,0*
2
*
==




−
=− ∫∫
+∞
∞−
+∞
∞−
mnd
d
Sd
S
j
dt
td
tsd
tstj m
m
n
n
n
n
mm
ω
ω
ω
ωω
π
c1
c2

25
( ) ( ) ( ) ( )∫
+∞
∞−
∗
( ) ( ) ( )[ ]222
0
02
2
2
22, ggsfD ffEfX D
+∆−=
∂
∂
=
=τ
τ
τ
( )
( ) ( ) ( )
( )∫
∫
∞+
∞−
+∞
∞−
−
=∆
dffG
dffGff
f
g
g
2
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 ( ) ( ) ( )∫∫∫
+∞
∞−
+∞
∞−
+∞
∞−
=== tdtgfdfGtdtsEs
222
2
1
2
2
1
: π
fg – is signal envelope frequency median
( ) ( )
( )∫
∫
∞+
∞−
+∞
∞−
=
dffG
dffGf
fg
2
2
2
22
:
π
ππ
( )
( ) ( )
( )∫
∫
∞+
∞−
+∞
∞−
=+∆
dffG
dffGf
ff gg
2
222
22
2
22
π
ππ

26
( ) ( ) ( ) ( )∫
+∞
∞−
∗
SOLO
In the same way:
( ) ( ) ( ) ( ) ( ) ( )[ ]
( ) ( ) ( ) ( )[ ]∫ ∫
∫ ∫
∞+
∞−
∞+
∞−
+∞
∞−
+∞
∞−
∗∗
−−=






−
∂
∂
=
∂
∂
2121
2
2
2
121
21212211
2
2exp2
2exp,0
dtdtttfjtgtgttj
dtdtttfjtgtgtgtg
f
fX
f
D
D
D
D
D
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
0
0
2
≡−=
−=
∂
∂
⇔
∞+
∞−
∞+
∞−
∞+
∞−
∞+
∞−
+∞
∞−
+∞
∞−=
=
∫ ∫∫ ∫
∫ ∫
tt
f
D
D
dttgdttgtjdttgdttgtj
dtdttgtgttjfX
f D
ππ
πτ
τ

27
( ) ( ) ( ) ( )∫
+∞
∞−
∗
SOLO
Return to:
( ) ( ) ( ) ( ) ( )[ ]∫ ∫
+∞
∞−
+∞
∞−
−−=
∂
∂
2121
2
2
2
121
2
2exp2,0 dtdtttfjtgtgttjfX
f
DD
D
ππ
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( )












+−−=
−−=
∂
∂
∫∫∫ ∫∫ ∫
∫ ∫
∞+
∞−
∞+
∞−
∞+
∞−
∞+
∞−
∞+
∞−
∞+
∞−
+∞
∞−
+∞
∞−
=
=
2
2
2
2
2
2
1
2
12
2
221
2
11
2
2
2
21
2
1
2
1
2
21
2
2
2
1
2
21
2
0
0
2
2
2
22
2,
dttgtdttgdttgtdttgtdttgdttgt
dtdttgtgttfX
f
ss
D
EE
f
D
D

π
πτ
τ
Define:
( )
( )
( )
( ) ( )
( )
( )
( )
( )∫
∫
∫
∫
∫
∫
∞+
∞−
+∞
∞−
∞+
∞−
+∞
∞−
∞+
∞−
+∞
∞−
=+∆⇒
−
=∆=
dttg
dttgt
tt
dttg
dttgtt
t
dttg
dttgt
t gg
g
gg
2
22
22
2
22
2
2
2
::
( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) 2222222222
0
0
2
2
2
222222, gsgggggs
f
D
D
tEtttttEfX
f D
∆−=+∆+−+∆−=
∂
∂
=
=
ππτ
τ

28
( ) ( ) ( ) ( )∫
+∞
∞−
∗
SOLO
In the same way:
( ) ( ) ( ) ( ) ( ) ( )[ ]
( ) ( ) ( ) ( ) ( ) ( )[ ]∫ ∫
∫ ∫
∞+
∞−
∞+
∞−
∗
+∞
∞−
+∞
∞−
∗
−−−−=






−−−
∂
∂
=
∂
∂
2121221121
21212211
2
2exp*2
2exp*,
dtdtttfjtgtgtgtgttj
dtdtttfjtgtgtgtg
f
fX
f
D
D
D
D
D
πττπ
πτττ
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )∫ ∫
∫ ∫
∞+
∞−
∞+
∞−
+∞
∞−
+∞
∞−=
=
∂
∂
−−
∂
∂
−−=
∂∂
∂
212
2
21121
21221
1
121
0
0
2
2
**2
**2,
dtdttg
t
tgtgtgttj
dtdttgtgtg
t
tgttjfX
f Df
D
D
π
πτ
τ
τ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∫∫ ∫
∫ ∫∫ ∫
∞+
∞−
∞+
∞−
∞+
∞−
∞+
∞−
+∞
∞−
+∞
∞−
+∞
∞−
+∞
∞−
∂
∂
+
∂
∂
−
∂
∂
+
∂
∂
−=
22
2
2211122
2
21111
222211
1
122211
1
11
**2**2
**2**2
dttg
t
tgtdttgtgjdttg
t
tgdttgtgtj
dttgtgtdttg
t
tgjdttgtgdttg
t
tgtj
ππ
ππ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫∫ ∫
+∞
∞−
+∞
∞−
+∞
∞−
+∞
∞−






∂
∂
−
∂
∂
−





∂
∂
−
∂
∂
+= dttg
t
tgtg
t
tgdttgtgtjdttg
t
tgtg
t
tgtdttgtgj ***2***2 ππ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )






∂
∂






−






∂
∂






=
∂∂
∂
∫∫∫∫
+∞
∞−
+∞
∞−
+∞
∞−
+∞
∞−=
=
dttg
t
tgdttgtgtdttg
t
tgtdttgtgfX
f Df
D
D
*Im*4*Im*4,0
0
0
2
2
ππ
τ
τ

29
( ) ( ) ( ) ( )∫
+∞
∞−
∗
SOLO
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( )
( ) ( ) ( ) ( ) ( ) gs fEdffGfGf
dGGdGGGGdttg
t
tgtg
t
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
*
*
==





= ∫∫
∞+
∞−
∞+
∞−
mnd
d
Sd
S
j
dt
td
tsd
tstj m
m
n
n
n
n
mm
ω
ω
ω
ωω
π
( ) ( ) ( ) ( ) ( ) ( )  ,2,1,0,,2,1,0*
2
*
==




−
=− ∫∫
+∞
∞−
+∞
∞−
mnd
d
Sd
S
j
dt
td
tsd
tstj m
m
n
n
n
n
mm
ω
ω
ω
ωω
π
c1
c2
Relationships
from Parseval’s
Theorem
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
∫∫
+∞
∞−
+∞
∞− 









−



−=





∂
∂
−
∂
∂
− ω
ω
ω
ω
ω
ω
ωωπ d
d
Sd
S
d
Sd
Sjdttg
t
tgttg
t
tgtj
*
***2
( ) ( ) ( ) ( ) ( ) ( ) ( ) ggss
f
D
D
ftEdttg
t
tgtEfX
f D
22
0
0
2
2
222*Im222,0 ππ
τ
τ
+






∂
∂
=
∂∂
∂
∫
+∞
∞−=
=
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫∫ ∫
+∞
∞−
+∞
∞−
+∞
∞−
+∞
∞−=
=






∂
∂
−
∂
∂
−





∂
∂
−
∂
∂
+=
∂∂
∂
dttg
t
tgtg
t
tgdttgtgtjdttg
t
tgtg
t
tgtdttgtgjfX
f Df
D
D
***2***2,0
0
0
2
2
ππ
τ
τ

30
( ) ( ) ( ) ( )∫
+∞
∞−
∗
SOLO
( ) ( ) ( ) ( ) ( ) +
∂
∂
+
∂∂
∂
+
∂
∂
+=
=
=
=
=
=
=
2
0
0
2
2
2
0
0
2
2
2
0
0
2
2
2
22
,
2
1
,,
2
1
0,0, D
f
D
D
D
f
D
Df
DD ffX
f
ffX
f
fXXfX
DDD
τττ
τττ
τ
ττ
τ
τ
( ) ( ) ( )[ ]( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) +∆−








+





∂
∂
++∆−= ∫
+∞
∞−
2222
22222222
22
222*Im22222,
Dgs
DggsssggsD
ftE
fftEdttg
t
tgtEEffEfX
π
τππττ
( )
( )
( )[ ] ( )
( )
( ) ( ) ( ) ( ) ( ) +∆−








+





∂
∂
++∆−= ∫
∞+
∞−
222222
2
2
222*Im
2
2
21
0,0
,
DgDggs
s
gg
D
ftfftEdttg
t
tgt
E
ff
X
fX
πτπ
π
τ
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
0
0
2
2
2*Im
2
2
21
0,0
,
DgD
s
g
t
f
D
ftfdttg
t
tgt
E
f
X
fX
g
g
πτ
π
τ
τ

31
( ) ( ) ( ) ( )∫
+∞
∞−
∗
SOLO
( )
( )
( ) ( )
( )
( ) ( ) ( ) +∆−





∂
∂
+∆−= ∫
∞+
∞−
=
=
22222
0
0
2
2
2*Im
2
2
21
0,0
,
DgD
s
g
t
f
D
ftfdttg
t
tgt
E
f
X
fX
g
g
πτ
π
τ
τ
Helstrom’s Uncertainty Ellipse
The curve resulting from the interception of a plane parallel to the τ, fD plane and the
Normalized 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
3
2*Im
2
2
21
0,0
, 22222
0
0
2
2
=+∆−





∂
∂
+∆−= ∫
∞+
∞−
=
=
DgD
s
g
t
f
D
ftfdttg
t
tgt
E
f
X
fX
g
g
πτ
π
τ
τ
( ) ( )
( )
( ) ( ) ( )
4
1
2*Im
2
2
2
22222
=∆+





∂
∂
−∆ ∫
+∞
∞−
DgD
s
g ftfdttg
t
tgt
E
f πτ
π
τ

32
( ) ( ) ( ) ( )∫
+∞
∞−
∗
SOLO
( ) ( ) ( ) ( ) ( ) ( ) ( )
∫∫ ∫
+∞
∞−
+∞
∞−
+∞
∞−






−=
∂
∂
− ω
ω
ω
ωωπ d
d
Gd
GEjdttgtgdttg
t
tgtj s *2**2 22211
1
11
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
∫∫∫ ∫
+∞
∞−
+∞
∞−
+∞
∞−
+∞
∞−





−
=
∂
∂
+ ω
ω
ω
ωωωωω
π
π d
d
Gd
GdGG
j
dttgtgtdttg
t
tgj
*
222211
1
1 *
2
**2
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫∫ ∫
+∞
∞−
+∞
∞−
+∞
∞−
+∞
∞−




=
∂
∂
− ωωωω
π
ω
ω
ω
ωπ dGG
j
d
d
Gd
Gdttg
t
tgdttgtgtj *
2
***2 22
2
21111
( ) ( ) ( ) ( ) ( ) ( ) ( )
∫∫ ∫
+∞
∞−
+∞
∞−
+∞
∞−






=
∂
∂
+ ω
ω
ω
ωωπ d
d
Gd
GEjdttg
t
tgtdttgtgj s
*
22
2
22111 2**2
c2 m=n=1
c2 m=0
n=1
c1
m=1
n=0
c2 m=1
n=0
c1
m=0
n=1
c1 m=n=1
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
∫∫
∫
∞+
∞−
∞+
∞−
∞+
∞−=
=












−



−












−



=
∂∂
∂
ω
ω
ω
ω
ω
ω
ωωωωω
π
ω
ω
ω
ω
ω
ω
ωω
τ
τ
d
d
Gd
G
d
Gd
GjdGG
d
d
Gd
G
d
Gd
GjEfX
f
s
f
D
D D
**
2
1
*2,0
*
*
0
0
2
2
( ) ( ) ( ) ( ) ( ) ( )  ,2,1,0,,2,1,0
2
*
*
==





= ∫∫
∞+
∞−
∞+
∞−
mnd
d
Sd
S
j
dt
td
tsd
tstj m
m
n
n
n
n
mm
ω
ω
ω
ωω
π
( ) ( ) ( ) ( ) ( ) ( )  ,2,1,0,,2,1,0*
2
*
==




−
=− ∫∫
+∞
∞−
+∞
∞−
mnd
d
Sd
S
j
dt
td
tsd
tstj m
m
n
n
n
n
mm
ω
ω
ω
ωω
π
c1
c2
Relationships
from Parseval’s
Theorem

33
( ) ( ) ( ) ( )∫
+∞
∞−
∗
( ) ( ) ( )
  
s
D
E
g
f
D dffGffX
2
22
0
0
2
2
22, ∫
+∞
∞−
=
= ∆−=
∂
∂
πτ
τ
τ
( )
( ) ( )
( )∫
∫
∞+
∞−
+∞
∞−
=∆
dffG
dffGf
fg
2
222
2
2
22
:
π
ππ
SOLO
( ) ( ) ( ) ( ) ( ) ( ) ( )
  
s
D
E
g
f
D
D
dttgtgtdttgtgtfX
f
2
222
0
0
2
2
22, ∫∫
+∞
∞−
∗
+∞
∞−
∗
=
= ∆−=−=
∂
∂
ππτ τ
We found:
where:
Δfg – is signal envelope bandwidth
Es – is signal energy ( ) ( ) ( )∫∫∫
+∞
∞−
+∞
∞−
+∞
∞−
=== tdtgfdfGtdtsEs
222
2
1
2
2
1
: π
or ( ) ( ) ( ) ( )
  
s
D
E
g
f
D
D
dttgtgtfX
f
2
2
0
0
2
2
2, ∫
+∞
∞−
∗
=
= ∆−=
∂
∂
πτ τ
Δtg – is signal envelope duration ( )
( )
( )∫
∫
∞+
∞−
+∞
∞−
=∆
tdtg
tdtgt
tg
2
22
2
:

34
( ) ( ) ( ) ( )∫
+∞
∞−
∗
SOLO
( ) ( ) ( ) ( ) ( ) ( ) ( )






∂
∂
=






−
∂
∂
=






∂
∂
∂
∂
=
∂
∂
∂
∂
∫∫
+∞
∞−
∗
+∞
∞−
∗
=
=
=
= dttg
t
tgtdttfjtgtgtjfX
f
fX
f
DfD
D
f
D
D
DD
ππτ
τ
πτ
τ
τ
τ
ττ 2Im2exp2Re,Re, 0
0
0
0
Define
( ) ( )
( ) ( )
( )
( ) ( )






∂
∂
∆∆
−=





∂
∂
∆∆
−= ∫
∫
∫ ∞+
∞−
∗
∞+
∞−
∗
+∞
∞−
∗
dttg
t
tgt
Eft
dttgtg
dttg
t
tgt
ft sgggg 2
1
Im
1
:ρ
Error Coupling
Coefficient
We obtain
( ) ( ) ( ) ( ) ggs
f
D
D
ftEdttg
t
tgtfX
f D
∆∆−=






∂
∂
=
∂
∂
∂
∂
∫
+∞
∞−
∗
=
= ρππτ
τ
τ 222Im,
0
0

35
Ambiguity Function for Single RF Pulse
( )
( )




>
≤≤−
=
2/0
2/2/cos 0
p
pp
SPi
tt
ttttA
ts
ω
The complex envelope is
( )





>
≤≤−
=
2/0
2/2/
1
p
pp
pSP
tt
ttt
ttg
( ) ( ) ( ) ( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )









<
<
=








<
<
=−= ++
−
+
+−
+
−
+
+−
∞+
∞−
∗
∫
∫
∫
02exp
2
1
02exp
2
1
02exp
1
02exp
1
2exp:, 2/
2/
2/
2/
2/
2/
2/
2/
τπ
π
τπ
π
τπ
τπ
πττ τ
τ
τ
τ
p
p
p
p
p
p
p
p
t
t
D
pD
t
t
D
pD
t
t
D
p
t
t
D
p
DDSP
tfj
tfj
tfj
tfj
tdtfj
t
tdtfj
t
tdtfjtgtgfX











<





 +
−−




 +
<













 −
−−




 −






=











<






−−













+
<














+−−





=
0
2
2
2exp
2
2exp
0
2
2
2exp
2
2exp
2
2exp
0
2
2
2exp
2
2exp
0
2
2
2exp
2
2exp
τ
π
τ
π
τ
π
τ
π
τ
τ
π
τ
π
τ
π
τ
π
πτπ
τ
π
τππ
pD
p
D
p
D
pD
p
D
p
D
D
pD
p
D
p
D
pD
p
D
p
D
tfj
t
fj
t
fj
tfj
t
fj
t
fj
fj
tfj
t
fj
t
fj
tfj
t
fj
t
fj

36
Ambiguity Function for Single RF Pulse (continue – 1)
( ) ( )
( )[ ] ( ) ( ) ( )[ ]
( ) p
ppD
ppD
pD
pD
pD
DDSP t
ttf
ttf
tfj
tf
tf
fjfX ≤
−
−
−=
−
= τ
τπ
τπ
ττπ
π
τπ
τπτ
/1
/1sin
/1exp
sin
exp,
Therefore:
( )
( ) ( )[ ]
( )





≤
−
−
−
=
elsewere
t
ttf
ttf
t
fX
p
ppD
ppD
p
DSP
0
/1
/1sin
/1
,
τ
τπ
τπ
τ
τ
( ) ( ) ppSP ttX ≤−= τττ /10,
( )
[ ]
pD
pD
DSP
tf
tf
fX
π
πsin
,0 =

37
( )
( )
p
t
p
t
pSP
DSP
res t
t
d
tX
dfX
T
pp
=








−=








−=
=
= ∫
∫
+∞
∞−
0
2
0
2
2
2
212
0,0
0,
:
τ
ττ
τ
ττ
( )
( )
( )
( )
p
t
p
SP
SP
SP
DDSP
res
t
td
t
tdtg
tdtg
X
fdfX
F
p
12
0,0
,0
:
2/
0
2
1
2
2
4
1
2
2
==






=
=
= ∫
∫
∫∫
∞+
∞−
+∞
∞−
+∞
∞−
  

τ
( )





>
≤≤−
=
2/0
2/2/
1
p
pp
pSP
tt
ttt
ttg
SOLO
Ambiguity Function for Single RF Pulse (continue – 2)
( )
( ) ( )[ ]
( )





≤
−
−
−
=
elsewere
t
ttf
ttf
t
fX
p
ppD
ppD
p
DSP
0
/1
/1sin
/1
,
τ
τπ
τπ
τ
τ
( ) ( ) ppSP ttX ≤−= τττ /10,
Range Resolution
( ) 10,0 =SPX
Doppler Resolution
p
resres
t
FV
22
λλ
==

38
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( )∫∫
+∞
∞−
∗
+∞
∞−
∗
−−−=−= tdtfjtkjtgtkjtgtdtfjtgtgfX DSPSPDFMSPFMSPDFMSP πτπτππττ 2expexpexp2exp:,
22
SOLO
Ambiguity Function for Linear FM Modulation Pulse
( )







>
≤





+
=
2
0
22
cos
2
0
τ
τπ
ω
t
t
tk
tA
ts FMSPi
( )
[ ]
( ) [ ]2
2
exp
2
0
2
exp
1
tkjtg
t
t
t
ttkj
t
tg SP
p
p
p
FMSP π
π
=







>
≤
=
The signal
of Single Pulse
Frequency Modulated
The complex envelope
of Single Pulse
Frequency Modulated
( )tgSP
- the complex envelope of Single RF Pulse
( ) ( ) ( ) ( ) ( )[ ] ( ) ( )τττπτπττπτ kfXkjtdtkfjtgtgkjfX DSPDSPSPDFMSP +−=+−−= ∫
+∞
∞−
∗
,exp2expexp, 22
( ) ( ) ( )[ ]
( ) p
ppD
ppD
pDSP t
ttf
ttf
tfX ≤
−
−
−= τ
τπ
τπ
ττ
/1
/1sin
/1,where Ambiguity Function of the Single
Frequency Pulse
( )
( ) ( ) ( )[ ]
( ) ( )





≤
−+
−+
−
=
elsewere
t
ttkf
ttkf
t
fX
p
ppD
ppD
p
DFMSP
0
/1
/1sin
/1
,
τ
ττπ
ττπ
τ
τ

39
Ambiguity Function for Linear FM Modulation Pulse (continue – 1)
( )
( ) ( ) ( )[ ]
( ) ( )
( )ττ
τ
ττπ
ττπ
τ
τ
kfX
elsewere
t
ttkf
ttkf
t
fX
DSP
p
ppD
ppD
p
DFMSP
+=





≤
−+
−+
−
=
,
0
/1
/1sin
/1
,

40
( )
( ) ( ) ( )[ ]
( ) ( )
( )ττ
τ
ττπ
ττπ
τ
τ
kfX
elsewere
t
ttkf
ttkf
t
fX
DSP
p
ppD
ppD
p
DFMSP
+=





≤
−+
−+
−
=
,
0
/1
/1sin
/1
,
( ) ( )
( )0,
,,
τ
τττττ
SP
SPFMSP
X
kkXkX
=
+−=−

41
( )
( )
( )
p
p
p
p
p
p
DFMSP t
t
tk
t
tk
t
fX ≤








−
















−








−== τ
τ
τπ
τ
τπ
τ
τ
1
1sin
10,
tp
τ1’st null
( ) π
τ
τπ =








−
p
p
t
tk 1
p
tk
pp
nullst
tkk
tt p
11
42
42
'1
2
>>
≈−−=τ
k tp = Δf is the total frequency
deviation during the pulse.
p
nullst
p
p
tk
nullst tf
t
DrationCompressio
ftk
p
∆===
∆
=≈
>>
'1
4
'1
11
2
τ
τ

42
The envelope of each pulse is of unit energy and the
coherence is maintained from pulse to pulse.
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )∑ ∑ ∫
∫∑ ∑∫
−
=
−
=
∞+
∞−
+∞
∞−
−
=
−
=
+∞
∞−
−−−=
−−−=−=
1
0
1
0
1
0
1
0
*
2exp*
1
2exp*
1
2exp,
N
n
N
m
DRSPRSP
D
N
n
N
m
RSPRSPDPTPTDPT
tdtfjTmtgTntg
N
tdtfjTmtgTntg
N
tdtfjtgtgfX
πτ
πτπττ
( ) ( ) ( ) ( )[ ] ( )∑ ∑ ∫
−
=
−
=
+∞
∞−
−=
−−−=
1
0
1
0
1111 2exp*2exp
1
,
1 N
n
N
m
DRSPSPRD
Tntt
DPT tdtfjTnmtgtgTnfj
N
fX
R
πτπτ
( ) [ ] ( ) ( )
( ) ( )[ ]
( ) ( )





≤
−
−
−
==−∫
∞+
∞−
elsewere
tfj
ttf
ttf
t
fXtdtfjtgtg
pD
ppD
ppD
p
DSPDSPSP
0
2exp
/1
/1sin
/1
,2exp* 1111
ττπ
τπ
τπ
τ
τπτ
( )





>
≤≤−
=
2/0
2/2/
1
p
pp
pSP
tt
ttt
ttg Envelope of
Single Pulse
( ) ( )∑
−
=
−=
1
0
1 N
n
RSPPT Tntg
N
tg
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
(continue – 1)
( ) ( ) ( )∑ ∑
−
=
−
=
−=−=
1
0
1
0
:,2exp
1
,
N
n
N
m
DRSPRDDPT mnpfTpXTnfj
N
fX τπτ
Construction Table for the
Double Sum with p=n-m
n
m 0 1 2 … N-1
0 0 1 2 … N-1
1 -1 0 1 … N-2
2 -2 -1 0 … N-3
… … … … … …
N-1 -N-1 -N-2 -N-3 … 0
p=n-m
( )
    
BlockTriangularRight
pmn
N
p
pN
m
DiagonalBlockTriangularLow
pnm
Np
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,2exp
1
2exp,
1
,
N
p
pN
m
RDDRSPRD
Np
pN
n
RDDRSPDPT
TmfjfTpXTpfj
N
TnfjfTpX
N
fX
πτπ
πττ

44
(continue – 2)
( ) ( ) ( )
( )
( ) ( ) ( )∑ ∑
∑ ∑
−
=
−−
=
−−=
−−
=
−+
−=
1
1
1
0
0
1
1
0
2exp,2exp
1
2exp,
1
,
N
p
pN
m
RDDRSPRD
Np
pN
n
RDDRSPDPT
TmfjfTpXTpfj
N
TnfjfTpX
N
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
y
y
pNpNpNpNpN
n
n
−
−
=
−
−
= −
−−−−−−−
=
∑
take: ( )RD Tfjy π2exp=
( ) ( )[ ] ( )[ ]
( )RD
RD
RD
pN
n
RD
Tf
TpNf
TpNfjTnfj
π
π
ππ
sin
sin
1exp2exp
1
0
−
−−=∑
−−
=
Using this result we obtain:
( ) ( )[ ] ( )
( )[ ]
( )( )
∑
−
−−=
−
−+−=
1
1 sin
sin
,1exp
1
,
N
Np RD
RD
DRSPRDDPT
Tf
TpNf
fTpXTpNfj
N
fX
π
π
τπτ

45
(continue – 3)
( ) ( )[ ] ( )
( )[ ]
( )( )
∑
−
−−=
−
−+−=
1
1 sin
sin
,1exp
1
,
N
Np RD
RD
DRSPRDDPT
Tf
TpNf
fTpXTpNfj
N
fX
π
π
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
pRpD
pR
N
Np RD
RD
DRSPDPT
Tt
Tf
TpNf
tTptf
tTptf
tTp
N
Tf
TpNf
fTpX
N
fX
<
−
−−
−−
−−=
−
−=
∑
∑
−
−−=
−
−−=
π
π
τπ
τπ
τ
π
π
ττ
( )
( ) ( )[ ]
( ) ( )





≤
−
−
−
=
elsewere
tfj
ttf
ttf
t
fX
pD
ppD
ppD
p
DSP
0
2exp
/1
/1sin
/1
,
ττπ
τπ
τπ
τ
τ

46
(continue – 4)
The Ambiguity Function for a Coherent Pulse Train:
Setting fD = 0 we obtain:
( ) ( ) ( )[ ]
( )
( )[ ]
( )( )
2/
sin
sin
/1
/1sin
/1
1
,
1
1
Rp
N
Np RD
RD
pRpD
pRpD
pRDPT Tt
Tf
TpNf
tTptf
tTptf
tTp
N
fX <
−
−−
−−
−−= ∑
−
−−= π
π
τπ
τπ
ττ
( )
( )[ ]
( )( ) ( )
( )pN
Tf
Tf
TpNf
TpNf
t
Tp
N
fX
DD
fRD
RD
N
Np
fRD
RD
p
R
DPT −
−
−







 −
−=
=
−
−−=
=
∑
    
1
0
1
1
1
0
sin
sin
1
1
,
π
π
π
πτ
τ
( )
( )
pR
N
Np p
R
DPT tTp
N
p
t
Tp
fX <−







−







 −
−== ∑
−
−−=
τ
τ
τ 110,
1
1
or

47
(continue – 5)
The Ambiguity Function for a Coherent Pulse Train:
( )
( ) ( )[ ]
( )
( )[ ]
( )( )
2/
sin
sin
/1
/1sin
/1
1
,
1
1
Rp
N
Np RD
RD
pRpD
pRpD
pR
DPT
Tt
Tf
TpNf
tTptf
tTptf
tTp
N
fX
<
−
−−
−−
−−= ∑
−
−−= π
π
τπ
τπ
τ
τ

48
Pulse bi-phase Barker coded of length 7
Digital Correlation
At the Receiver the coded pulse enters a
7 cells delay lane (from left to right),
a bin at each clock.
The signals in the cells are multiplied
by 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) = -1
13
0 = 014
SOLO Pulse Compression Techniques
-1-1 -1+1+1+1+1 { }*
kc

50
SOLO

52
SOLO
Rihaczek, A.W., “Principles of High Resolution Radar”, McGraw Hill, 1969

53
SOLO

54
SOLO

55
SOLO

56
SOLO

57
SOLO

58
SOLO

59
SOLO

60
SOLO

61
SOLO

62
SOLO

63
SOLO

64
SOLO

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 Presentation

January 19, 2015 67
SOLO
Technion
Israeli Institute of Technology
1964 – 1968 BSc EE
1968 – 1971 MSc EE
Israeli Air Force
1970 – 1974
RAFAEL
Israeli Armament Development Authority
1974 –2013
Stanford University
1983 – 1986 PhD AA
Vector Analysis

68
Fourier Transform
( ) ( ){ } ( ) ( )∫
+∞
∞−
−== dttjtftfF ωω exp:F
SOLO
Jean Baptiste Joseph
Fourier
1768-1830
F (ω) is known as Fourier Integral or Fourier Transform
and is in general complex
( ) ( ) ( ) ( ) ( )[ ]ωφωωωω jAFjFF expImRe =+=
Using the identities
( ) ( )t
d
tj δ
π
ω
ω =∫
+∞
∞− 2
exp
we can find the Inverse Fourier Transform ( ) ( ){ }ωFtf -1
F=
( ) ( ) ( ) ( ) ( )
( ) ( )( ) ( ) ( ) ( ) ( )[ ]00
2
1
2
exp
2
expexp
2
exp
++−=−=−=




−=
∫∫ ∫
∫ ∫∫
∞+
∞−
∞+
∞−
∞+
∞−
+∞
∞−
+∞
∞−
+∞
∞−
tftfdtfd
d
tjf
d
tjdjf
d
tjF
ττδττ
π
ω
τωτ
π
ω
ωττωτ
π
ω
ωω
( ) ( ){ } ( ) ( )∫
+∞
∞−
==
π
ω
ωωω
2
exp:
d
tjFFtf -1
F
( ) ( ) ( ) ( )[ ]00
2
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 −
-1
F
F ( ) ( )ωω ajF −exp
Fourier TransformSOLO
( )tf
-1
F
F
( )ωFProperties of Fourier Transform (Summary)
Linearity1
( ) ( ){ } ( ) ( )[ ] ( ) ( ) ( )ωαωαωαααα 221122112211 exp: FFdttjtftftftf +=−+=+ ∫
+∞
∞−
F
Symmetry2
( )tF
-1
F
F
( )ωπ −f2
Conjugate Functions3 ( )tf *
-1
F
F
( )ω−*
F
Scaling4 ( )taf
-1
F
F






a
F
a
ω1
Derivatives5 ( ) ( )tftj
n
−
-1
F
F ( )ω
ω
F
d
d
n
n
( )tf
td
d
n
n
-1
F
F
( ) ( )ωω Fj
n
Convolution6
( ) ( )tftf 21
-1
F
F ( ) ( )ωω 21
* FF( ) ( ) ( ) ( )∫
+∞
∞−
−= τττ dtfftftf 2121
:*
-1
F
F ( ) ( )ωω 21
FF
( ) ( ) ( ) ( )∫∫
+∞
∞−
+∞
∞−
= ωωω dFFdttftf 2
*
12
*
1
Parseval’s Formula7
Shifting: for any a real8
( ) ( )tajtf exp
-1
F
F ( )aF −ω
Modulation9 ( ) ttf 0
cos ω -1
F
F
( ) ( )[ ]00
2
1
ωωωω −++ FF
( ) ( ) ( ) ( ) ( ) ( )∫∫∫
+∞
∞−
+∞
∞−
+∞
∞−
−=−= ωωω
π
ωωω
π
dFFdFFdttftf 212121
2
1
2
1

70
Fourier Transform
( )tf
( ) ( )∑
∞
=
−=
0n
T Tntt δδ
( ) ( ) ( ) ( ) ( )∑
∞
=
−==
0
*
n
T
TntTnfttftf δδ
( )tf *
( )tf
T t
( ) ( ){ } ( ) σσ <== +∫
∞
−
f
ts
dtetftfsF
0
L
SOLO
Sampling and z-Transform
( ) ( ){ } ( ) σδδ <
−
==






−== −
∞
=
−
∞
=
∑∑ 0
1
1
00
sT
n
sTn
n
T
e
eTnttsS LL
( ) ( ){ }
( ) ( ) ( )
( ) ( ){ } ( ) ( )






<<
−
=
=






−
==
−
∞+
∞−
−−
∞
=
−
∞
=
+∫
∑∑
0
00
**
1
1
2
1
σσσξξ
π
δ
δ
ξ
σ
σ
ξ f
j
j
tsT
n
sTn
n
d
e
F
j
ttf
eTnfTntTnf
tfsF
L
L
L
( )
( ) ( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )













−
=
−
−
=
−
=
∑∫
∑∫
∑
−−
−
−−
Γ
−−
−−
Γ
−−
∞
=
−
ts
e
ofPoles
tsts
F
ofPoles
tsts
n
nsT
e
F
Resd
e
F
j
e
F
Resd
e
F
j
eTnf
sF
ξ
ξξ
ξ
ξξ
ξ
ξ
ξ
π
ξ
ξ
ξ
π
1
1
0
*
112
1
112
1
2
1
Poles of
( ) Ts
e ξ−−
−1
1
Poles of
( )ξF
planes
T
nsn
π
ξ
2
+=
ωj
ωσ j+
0=s
Laplace Transforms
The signal f (t) is sampled at a time period T.
1Γ
2
Γ
∞→R
∞→R
Poles of
( ) Ts
e ξ−−
−1
1
Poles of
( )ξF
planeξ
T
nsn
π
ξ
2
+=
ωj
ωσ j+
0=s

71
Fourier Transform
( )tf
( ) ( )∑
∞
=
−=
0n
T Tntt δδ
( ) ( ) ( ) ( ) ( )∑
∞
=
−==
0
*
n
T
TntTnfttftf δδ
( )tf *
( )tf
T t
SOLO
Sampling and z-Transform (continue – 1)
( ) ( )
( )
( )
( )
( ) ( ) ∑∑
∑∑
∞+
−∞=
∞+
−∞=
−−→
∞+
−∞=
−−
+→
+=
−
−−






+=
−






+
−=






+












−
−−
−=
−
−=
−−
−−
nn
Tse
n
ts
T
n
js
T
n
js
e
ofPoles
ts
T
n
jsF
TeT
T
n
jsF
T
n
jsF
e
T
n
js
e
F
RessF
ts
n
ts
π
π
π
π
ξ
ξ
ξ
ξπ
ξ
π
ξ
ξ
ξ
ξ
21
2
lim
2
1
2
lim
1
1
2
2
1
1
*
Poles of
( )ξF
ωj
σ
0=s
T
π2
T
π2
T
π2
Poles of
( )ξ*
F plane
js ωσ +=
The poles of are given by( )ts
e ξ−−
−1
1
( )
( )
T
n
jsnjTsee n
njTs π
ξπξπξ 2
21 2
+=⇒=−−⇒==−−
( ) ∑
+∞
−∞=






+=
n T
n
jsF
T
sF
π21*

72
Fourier Transform
( )tf
( ) ( )∑
∞
=
−=
0n
T Tntt δδ
( ) ( ) ( ) ( ) ( )∑
∞
=
−==
0
*
n
T
TntTnfttftf δδ
( )tf *
( )tf
T t
SOLO
0=z
planez
Poles of
( )zF
C
The z-Transform is defined as:
( ){ } ( ) ( )
( )
( ) ( )
( )








−
−===
∑
∑
=
−
→
∞
=
−
=
iF
iF
i
iF
Ts
FofPoles
T
F
n
n
ze
ze
F
zTnf
zFsFtf
ξξ
ξ
ξ
ξξ
ξξξ
1
0
*
1
lim:Z
( )
( )





<
>≥
= ∫
−
00
0
2
1 1
n
RzndzzzF
jTnf
fC
C
n
π

73
( ) ( ) ( )∑∑
∞
=
−
+∞
−∞=
=





+=
0
* 21
n
nsT
n
eTnf
T
n
jsF
T
sF
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
tn
jCTnttd πδ 2exp:
where: ( )
T
dt
T
tn
jt
T
C
T
T
n
1
2exp
1
2/
2/
=





= ∫
+
−
πδ
Therefore we obtain the following identity:
( )∑∑ −=





−
nn
TntT
T
tn
j δπ2exp
Second Way

74
Fourier Transform
( ) ( ){ } ( ) ( )∫
+∞
∞−
−== dttjtftfF νπνπ 2exp:2 F
( ) ( ) ( )∑∑
∞
=
−
+∞
−∞=
=





+=
0
* 21
n
nsT
n
eTnf
T
n
jsF
T
sF
π
( ) ( ){ } ( ) ( )∫
+∞
∞−
== ννπνπνπ dtjFFtf 2exp2:2-1
F
SOLO
We found
Using the definition of the Fourier Transform and it’s inverse:
we obtain ( ) ( ) ( )∫
+∞
∞−
= ννπνπ dTnjFTnf 2exp2
( ) ( ) ( ) ( ) ( ) ( )∑∫∑
∞
=
+∞
∞−
∞
=
−=−=
0
111
0
*
exp2exp2exp
nn
n
sTndTnjFsTTnfsF ννπνπ
( ) ( ) ( )[ ]∫ ∑
+∞
∞−
+∞
−∞=
−−== 111
*
2exp22 νννπνπνπ dTnjFjsF
n
( ) ( ) ∑∫ ∑
+∞
−∞=
+∞
∞−
+∞
−∞=












−=





−−==
nn T
n
F
T
d
T
n
T
FjsF νπνννδνπνπ 2
11
22 111
*
We recovered (with –n instead of n) ( ) ∑
+∞
−∞=






+=
n T
n
jsF
T
sF
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
T
Tnj 11
1
2exp ννδννπ
( )∑∑ −=





−
nn
TntT
T
tn
j δπ2exp

75
Henry Nyquist
1889 - 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
then
( ) ( )∫
+∞
∞−
−
= tdetsfS tfi π2
( ) ( )∫
+∞
∞−
= fdefSts tfi π2
t
t∆2
t
( ) 2
ts
f
f
f∆2
( ) 2
fS
( ) ( )
( )
2/1
2
22
:














−
=∆
∫
∫
∞+
∞−
+∞
∞−
tdts
tdtstt
t
( )
( )∫
∫
∞+
∞−
+∞
∞−
=
tdts
tdtst
t
2
2
:
Signal Duration Signal Median
( ) ( )
( )
2/1
2
22
2
4
:














−
=∆
∫
∫
∞+
∞−
+∞
∞−
fdfS
fdfSff
f
π ( )
( )∫
∫
∞+
∞−
+∞
∞−
=
fdfS
fdfSf
f
2
2
2
:
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 ( ) ( ) ( )∫
+∞
∞−
== fdefSfi
td
tsd
ts tfi π
π 2
2'
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )∫∫ ∫
∫ ∫∫ ∫∫
∞+
∞−
∞+
∞−
∞+
∞−
−
+∞
∞−
+∞
∞−
−
+∞
∞−
+∞
∞−
−
+∞
∞−
=







−=








−=







−=
dffSfSfdfdesfSfi
dfdesfSfidfdefSfsidss
tfi
tfitfi
222
22
2'2
'2'2''
πττπ
ττπττπτττ
π
ππ
( ) ( )∫∫
+∞
∞−
+∞
∞−
= dffSds
22
ττ
Parseval Theorem
From
From
( ) ( )∫∫
+∞
∞−
+∞
∞−
= dffSfdtts
2222
4' π

78
Signals
( )
( )
( ) ( )
( )
( ) ( )
( )
( ) ( )
( )
( ) ( )
( )∫
∫
∫
∫ ∫
∫
∫ ∫
∫
∫
∫
∫
∞+
∞−
+∞
∞−
∞+
∞−
+∞
∞−
+∞
∞−
−
∞+
∞−
+∞
∞−
+∞
∞−
−
∞+
∞−
+∞
∞−
∞+
∞−
+∞
∞−
=====
dffS
fd
fd
fSd
fS
i
dffS
fdtdetstfS
dffS
tdfdefStst
dffS
tdtstst
tdts
tdtst
t
fifi
22
2
2
2
22
2
2
:
π
ππ
SOLO
( ) ( )∫
+∞
∞−
−
= tdetsfS tfi π2
( ) ( )∫
+∞
∞−
= fdefSts tfi π2
Fourier
( ) ( )∫
+∞
∞−
−
−= tdetsti
fd
fSd tfi π
π 2
2
( ) ( )∫
+∞
∞−
= fdefSfi
td
tsd tfi π
π 2
2
( )
( )
( ) ( )
( )
( ) ( )
( )
( ) ( )
( )
( ) ( )
( )∫
∫
∫
∫ ∫
∫
∫ ∫
∫
∫
∫
∫
∞+
∞−
+∞
∞−
∞+
∞−
+∞
∞−
+∞
∞−
∞+
∞−
+∞
∞−
+∞
∞−
∞+
∞−
+∞
∞−
∞+
∞−
+∞
∞−
−
=








====
tdts
td
td
tsd
tsi
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
( ) ( )∫∫
+∞
∞−
+∞
∞−
= dffSdts
22
τ
SOLO
0&0 == ftChange time and frequency scale to get
From Schwarz Inequality: ( ) ( ) ( ) ( )∫∫∫
+∞
∞−
+∞
∞−
+∞
∞−
≤ dttgdttfdttgtf
22
Choose ( ) ( ) ( ) ( ) ( )ts
td
tsd
tgtsttf ':& ===
( ) ( ) ( ) ( )∫∫∫
+∞
∞−
+∞
∞−
+∞
∞−
≤ dttsdttstdttstst
22
''we obtain
( ) ( )∫
+∞
∞−
dttstst 'Integrate by parts
( )



=
+=
→



=
=
sv
dtstsdu
dtsdv
stu '
'
( ) ( ) ( ) ( ) ( )∫∫∫
+∞
∞−
+∞
∞−
∞+
∞−
+∞
∞−
−−= dttststdttsstdttstst '' 2
0
2

( ) ( ) ( )∫∫
+∞
∞−
+∞
∞−
−= dttsdttstst 2
2
1
'
( ) ( )∫∫
+∞
∞−
+∞
∞−
= dffSfdtts
2222
4' π
( )
( )
( )
( )
( )
( )
( )
( )∫
∫
∫
∫
∫
∫
∫
∫
∞+
∞−
+∞
∞−
∞+
∞−
+∞
∞−
∞+
∞−
+∞
∞−
∞+
∞−
+∞
∞−
=≤
dffS
dffSf
dtts
dttst
dtts
dffSf
dtts
dttst
2
222
2
2
2
222
2
2
44
4
1
ππ
assume ( ) 0lim =
→∞
tst
t

80
SignalsSOLO
( )
( )
( )
( )
( )
( )
    
22
2
222
2
2
4
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 equality
if and only if g (t) = k f (t), then for:
( ) ( ) ( ) ( )∫∫∫
+∞
∞−
+∞
∞−
+∞
∞−
≤ dttgdttfdttgtf
22
( ) ( ) ( ) ( )tftsteAt
td
sd
tgeAts tt
ααα αα
222:
22
−=−=−==⇒= −−
we have ( ) ( )ft ∆∆=
2
1

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