4 matched filters and ambiguity functions for radar signals-2

1

Matched Filters and Ambiguity Functions for

RADAR SignalsPart 2

SOLO HERMELIN

Updated: 01.12.08http://www.solohermelin.com

http://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


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


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


Return to Table of Content

6


( ) ( ) ( ) ( )∫+∞

∞−


( ) ( ) ( ) ( ) ( ) ( )[ ]( ) ( )[ ]tjta

tjttatsjtstg QI

θθθ

exp

sincos:

=

+=+=

SOLO


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


( ) ( ) ( ) ( )∫+∞

∞−


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

( ) ( ) 12

1:

22 === ∫∫+∞

∞−

+∞

∞−

ωωπ

dGdttgEs

( ) 1,2 ≤DfX τ ( ) ( ) 10,0, =≤ XfX Dτ

Proof:

Schwarzinequality

8

Ambiguity Function for RADAR Signals( ) ( ) ( ) ( )∫

+∞

∞−


SOLO


Ambiguity Function properties (continue – 2):

( ) 1, =∫ ∫+∞

∞−

+∞

∞−DD dfdfX ττ2

( ) ( ) ( ) ( )∫+∞

∞−


( )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


( ) ( ) ( ) ( )∫+∞

∞−


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

τπτπτττ

( ) ( ) ( )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-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


( ) ( ) ( ) ( )∫+∞

∞−


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 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


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 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


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



( ) ( ) ( )( )

( )[ ] ( ){ }

( )( )

( )[ ] ( ){ }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


( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫+∞

∞−

+∞

∞−

∗ ==−== fdfjfGfGRdttgtgfX ggD τπτττ 2exp*:0,

SOLO


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:

( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫+∞

∞−

+∞

∞−


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



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:

( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫+∞

∞−

+∞

∞−


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 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



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( ) ( ) ( ) ( )∫

+∞

∞−


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


( ) ( ) ( ) ( )∫+∞

∞−


( ) ( ) ( ) ( ) ( ) ( ) ,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


( ) ( ) ( ) ( )∫+∞

∞−


( ) ( ) ( )[ ]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


( ) ( ) ( ) ( )∫+∞

∞−


SOLO


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


( ) ( ) ( ) ( )∫+∞

∞−


SOLO


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


( ) ( ) ( ) ( )∫+∞

∞−


SOLO


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


( ) ( ) ( ) ( )∫+∞

∞−


SOLO


( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( )

( ) ( ) ( ) ( ) ( ) 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


( ) ( ) ( ) ( )∫+∞

∞−


SOLO


( ) ( ) ( ) ( ) ( ) +∂

∂+∂∂

∂+∂∂+=

==

==

==

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


( ) ( ) ( ) ( )∫+∞

∞−


SOLO


( )( )

( ) ( )( ) ( ) ( ) ( ) +∆−

∂∂+∆−= ∫

∞+

∞−==

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


( ) ( ) ( ) ( )∫+∞

∞−


SOLO


( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫ ∫

+∞

∞−

+∞

∞−

+∞

∞−

−=

∂∂− 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


( ) ( ) ( ) ( )∫+∞

∞−


( ) ( ) ( )

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


( ) ( ) ( ) ( ) ( ) ( ) ( )

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


( ) ( ) ( ) ( )∫+∞

∞−


SOLO


( ) ( ) ( ) ( ) ( ) ( ) ( )

∂∂=

−∂∂=

∂∂

∂∂=

∂∂

∂∂

∫∫+∞

∞−

∗+∞

∞−

∗=

==

= 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 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 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


( )

( ) 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 ==


38


( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( )∫∫+∞

∞−

∗+∞

∞−

∗ −−−=−= 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 Linear FM Modulation Pulse (continue – 1)

( ) ( ) ( ) ( )[ ]( ) ( )

( )ττ

τττπ

ττπτ

τ

kfX

elsewere

tttkf

ttkft

fX

DSP

pppD

ppDp

DFMSP

+=

≤

−+−+

−=

,

0

/1

/1sin/1

,

40



( ) ( ) ( ) ( )[ ]( ) ( )

( )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



( )( )

( )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


42


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 a Coherent Pulse Train (continue – 1)

( ) ( ) ( )∑ ∑−

=

−

=

−=−=1

0

1

0

:,2exp1

,N

n

N

mDRSPRDDPT mnpfTpXTnfj

NfX τπτ


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




( ) ( ) ( )( )

( ) ( ) ( )∑ ∑

∑ ∑−

=

−−

=

−−=

−−

=

−+

−=

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




( ) ( )[ ] ( ) ( )[ ]( )( )

∑−

−−=

−−+−=

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



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



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


52

SOLORihaczek, A.W., “Principles of High Resolution Radar”, McGraw Hill, 1969

53


54


55


56


57


58


59


60


61


62


63


64


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 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


0=z

planez

Poles of

( )zF

C


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


( ) ( ) ( )∑∑∞

=

−+∞

−∞=

=

+=

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


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


( )

( )

( )

( )

( )

( )

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

4 matched filters and ambiguity functions for radar signals-2

Science

ambiguity function properties

ambiguity functioncut

function of time

autocorrelation function

function of waveform

radar signalssoloambiguity

signal envelope energy

complex signal envelope