5 pulse compression waveform

Pulse Compression Waveform

SOLO HERMELIN

Updated: 27.10.08http://www.solohermelin.com

http://www.solohermelin.com/

Table of Content

SOLO Pulse Compression Waveform

Resolution

Pulse Range Resolution

Pulse Compression Waveform Introduction

Waveform Hierarchy

Linear FM Modulated Pulse (Chirp)

Barker Codes

Combined Barker Codes

Poly-Phase Codes

Phase Coded Waveforms

Matched Filter Response to Phase Coding

Bi-Phase Codes

Table of Content (continue – 1)

SOLO Pulse Compression Waveform

Poly-Phase Codes

Frank Codes

P1, P2, P3, P4 Poly-Phase Codes

Pseudo-Random Codes

Frequency Codes

Costas Codes

Complementary Pulse Codes

Summary of Pulse Compression Codes

References

Range & Doppler Measurements in RADAR SystemsSOLO

Resolution

Resolution is the spacing (in range, Doppler, angle, etc.) we must have in order todistinguish between two different targets.

first targetresponse

second targetresponse

compositetarget

response

greather then 3 db

DistinguishableTargets

first targetresponse

second targetresponse

compositetarget

response

UndistinguishableTargets

less then 3 db

The two targets are distinguishable ifthe composite (sum) of the received signal has a deep (between the twopicks) of at least 3 db.

Return to Table of Content

SOLOUnmodulated Pulse Range Resolution

Resolution is the spacing (in range, Doppler, angle, etc.) we must have in order todistinguish between two different targets.

Range Resolution

RADAR

τ

c

R

RR ∆+

Target # 1Target # 2

Assume two targets spaced by a range Δ R and a unmodulated radar pulse of τ seconds.

The echoes start to be receivedat the radar antenna at times: 2 R/c – first target 2 (R+Δ R)/c – second target

The echo of the first target endsat 2 R/c + τ

τ τ

time from pulsetransmission

c

R2 ( )c

RR ∆+2τ+

c

R2

ReceivedSignals

Target # 1 Target # 2

The two targets echoes can beresolved if:

c

RR

c

R ∆+=+ 22 τ2

τcR =∆ Pulse Range Resolution

( ) ( ) ≤≤+

=elsewhere

ttAts

0

0cos: 0 τϕω

Unmodulated Pulse

SOLO

Energy ( ) ( ) ( )∫∫∫+∞

∞−

+∞

∞−

+∞

∞−

=== ωωπ

dSdttsdttsEs222

2

1:

( ) ( ) ≤≤+

=elsewhere

ttAts

0

0cos: 0 τϕω

( ) ( )

( )[ ] ( ) ( )

−++=++=

+==

∫

∫∫+∞

∞−

τϕϕτωτϕω

ϕω

τ

τ

0002

0

00

2

0

2

002

2cos22cos1

222cos1

2

cos

Adtt

A

dttAdttsEs

Unmodulated Pulse

RADAR SignalsSOLO

( ) ( ) ≤≤+

=elsewhere

ttAts

0

0cos: 0 τϕω

Energy( ) ( )

2

2cos22cos1

2

2000

2 ττ

ϕϕτωτ AE

AE ss =⇒

−++=

2

τcR =∆ Pulse Range Resolution

Decreasing Pulse Width Increasing

Decreasing SNR, Radar Performance Increasing

Increasing Range Resolution Capability Decreasing

For the Unmodulated Pulse, there exists a coupling between Range Resolution andWaveform Energy. Return to Table of Content

Pulse Compression WaveformsSOLO

Pulse Compression Waveforms permit a decoupling between Range Resolution and Waveform Energy.

- An increased waveform bandwidth (BW) relative to that achievable with an unmodulated pulse of an equal duration

τ1>>BW

22

τcBW

cR <<=∆

- Waveform duration in excess of that achievable with unmodulated pulse of equivalent waveform bandwidth

BW

1>>τ

PCWF exhibit the following equivalent properties:

This is accomplished by modulating (or coding) the transmit waveform and compressing the resulting received waveform.

Pulse Compression Waveform Introduction


SOLO Waveform Hierarchy

Radar Waveforms

CW Radars Pulsed Radars

FrequencyModulated CW

PhaseModulated CW

bi – phase & poly-phase

Linear FMCWSawtooth, or

Triangle

Nonlinear FMCWSinusoidal,

Multiple Frequency,Noise, Pseudorandom

Intra-pulse Modulation

Pulse-to-pulse Modulation,

Frequency AgilityStepped Frequency

FrequencyModulate Linear FM

Nonlinear FM

PhaseModulatedbi – phase poly-phase

Unmodulated CW

Multiple FrequencyFrequency

Shift Keying

Fixed Frequency

SOLO Waveform Hierarchy

• Pulse Compression Techniques• Wave Coding

• Frequency Modulation (FM)

- Linear

• Phase Modulation (PM)]

- Non-linear

- Pseudo-Random Noise (PRN)

- Bi-phase (0º/180º)

- Quad-phase (0º/90º/180º/270º)

• Implementation

• Hardware

- Surface Acoustic Wave (SAW) expander/compressor

• Digital Control- Direct Digital Synthesizer (DDS)

- Software compression “filter”Return to Table of Content

SOLO

• Pulse Compression Techniques

SOLO Waveform Hierarchy• Pulse Compression Techniques

SOLO Coherent Pulse Doppler Radar


SOLO

Linear FM Modulated Pulse (Chirp)

( ) ( )2/cos 203 ttAtf ωω ∆+=

t

A

2/τ−2/τ ( )

222cos

2

0

ττµω ≤≤−

+= tt

tAtsi

Pulse Compression Waveforms

Linear Frequency Modulation is a technique used to increase the waveform bandwidthBW while maintaining pulse duration τ, such that

BW

1>>τ 1>>⋅BWτ

222 0

2

0

ττµωµωω ≤≤−+=

+= tt

tt

td

d

Matched Filters for RADAR Signals

( ) ( )( ) ( )

≤≤−== −∗

Ttttsth

eSH

i

tji

00

0ωωω

SOLO

The Matched Filter (Summary(

si (t) - Signal waveform

Si (ω) - Signal spectral density

h (t) - Filter impulse response

H (ω) - Filter transfer function

t0 - Time filter output is sampled

n (t) - noise

N (ω) - Noise spectral density

Matched Filter is a linear time-invariant filter hopt (t) that maximizesthe output signal-to-noise ratio at a predefined time t0, for a given signal si (t(.

The Matched Filter output is:

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) 00

00

tjiii

iii

eSSHSS

dttssdthsts

ωωωωωω

ξξξξξξ

−∗

+∞

∞−

+∞

∞−

⋅=⋅=

+−=−= ∫∫

SOLO

Linear FM Modulated Pulse (continue – 1)


Concept of Group Delay

BW

1>>τ

τ

BW

1

( )222

cos2

0

ττµω ≤≤−

+= tt

tAtsi

( ) ( )222

cos2

0

00 ττµω ≤≤−

−=−=

=

tt

tAtsth i

t

MF

Matched Filter

( )tsi ( )tso

( ) ( )tsth i

t

MF −==00 ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )ωωωωω

ξξξξξξ

∗=

+∞

∞−

=+∞

∞−

⋅=⋅=

−=−= ∫∫

ii

t

i

ii

t

i

SSHSS

dtssdthsts

0

0

0

0

0

0

SOLO



Concept of Group Delay (continue -1)

BW

1>>τ

τ

BW

1

( )222

cos2

0

ττµω ≤≤−

+= tt

tAtsi

Matched Filter

( )tsi ( )tso

( ) ( )tsth i

t

MF −==00 ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )ωωωωω

ξξξξξξ

∗=

+∞

∞−

=+∞

∞−

⋅=⋅=

−=−= ∫∫

ii

t

i

ii

t

i

SSHSS

dtssdthsts

0

0

0

0

0

0

( ) ( ) ( ) ( )

>≤≤+−

<+≤≤−

−+−

+=−

022

022

2cos

2cos

2

0

2

02

tt

ttt

tAtss ii τξτ

τξτξµξωξµξωξξ

( ) ( ) ( ) ( ) ( )∫∫

+

+−

>∞+

∞−

>=

−+−

+=−=

2/

2/

2

0

2

02

000

0 2cos

2cos

0 τ

τ

ξξµξωξµξωξξξt

t

ii

tt

dt

tAdtssts

( ) ( )222

cos2

0

00 ττµω ≤≤−

−=−=

=

tt

tAtsth i

t

MF

Ignoring terms of 2 ω0

( ) ( ) ( )

( ) ( )t

tttA

t

tttA

t

tttAdttt

Ats

tt

tt

µµτµω

µµτµω

µµξµωξµξµω

τ

τ

τ

τ

2/2/sin

2

2/2/sin

2

2/sin

22/cos

22

022

02

2/

2/

20

22/

2/

20

200

0

0

+−−−+=

−+=−+≅+

+−

+

+−

>=

∫

( ) ( ) βαβαβα coscos2coscos =−++( )[ ] ( )∫∫+

+−

+

+−

−+++−+−=2/

2/

20

22/

2/

220

2

2/cos2

2/2/2cos2

τ

τ

τ

τ

ξµξµωξµξµξµξωtt

dttA

dttA

SOLO



Concept of Group Delay (continue -2)

BW

1>>τ

τ

BW

1

( )222

cos2

0

ττµω ≤≤−

+= tt

tAtsi

Matched Filter

( )tsi ( )tso

( ) ( )tsth i

t

MF −==00 ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )ωωωωω

ξξξξξξ

∗=

+∞

∞−

=+∞

∞−

⋅=⋅=

−=−= ∫∫

ii

t

i

ii

t

i

SSHSS

dtssdthsts

0

0

0

0

0

0

( ) ( )222

cos2

0

00 ττµω ≤≤−

−=−=

=

tt

tAtsth i

t

MF

Ignoring terms of 2 ω0 ( ) ( ) ( )t

tttA

t

tttAts

tt

µµτµω

µµτµω 2/2/sin

2

2/2/sin

2

20

220

200

0

0

+−−−+≅>=

( ) ( )ttt

tt

tAts

tt

0

200

0 cos1

2

12

sin

12

0

ω

ττµ

ττµ

ττ

−

−

−≅

>= ( ) ( ) βαβαβα sincos2sinsin =−−+

−==

ττµβωα tt

t 12

,0

If we re-due for t < 0 and combine, we obtain

( ) ( )ttt

tt

tAts

t

0

20

0 cos

12

12

sin

12

0

ω

ττµ

ττµ

ττ

−

−

−≅

=

SOLO

( ) ( )2/cos 203 ttAtf ωω ∆+=

t

A

2/τ−2/τ


( )222

cos2

0

ττµω ≤≤−

+= tt

tAtsi

The Fourier Transform is:

( ) [ ]

( ) ( )∫∫

∫

−−

−

++−+

+−=

−

+=

2/

2/

2

0

2/

2/

2

0

2/

2/

2

0

2exp

2

1

2exp

2

1

exp2

cos

τ

τ

τ

τ

τ

τ

µωωµωω

ωµωω

dtt

tjAdtt

tjA

dttjt

tAS i

∫∫−−

++−

++

−−

−−=

2/

2/

2

0

2

0

2/

2/

2

0

2

0

2exp

2exp

22exp

2exp

2

τ

τ

τ

τµ

ωωµµωω

µωωµ

µωω

dttjjA

dttjjA

Change variables: xt =

−−

µωω

πµ 0 yt =

++

µωω

πµ 0

( ) ∫∫−−

−

++

−−=

2

1

2

1

2exp

2exp

22exp

2exp

2

22

02

2

0

Y

Y

X

X

i dty

jjA

dtx

jjA

Sπ

µωωπ

µωωω

−−=

−+=

µωωτ

πµ

µωωτ

πµ 0

20

1 2&

2XX

+−=

++=

µωωτ

πµ

µωωτ

πµ 0

20

1 2&

2YY

Define: ( )fnf ∆=−=∆ πωωτµ

π2

2&

2

1: 0


SOLO

( ) ( )2/cos 203 ttAtf ωω ∆+=

t

A

2/τ−2/τ


( )222

cos2

0

ττµω ≤≤−

+= tt

tAtsi


( ) ( ) ( ) ∫∫−−

−

++

−−=

2

1

2

1

2exp

2exp

22exp

2exp

2

22

022

0

Y

Y

X

X

i dty

jjA

dtx

jjA

Sπ

µωωπ

µωωω

The first part gives the spectrum around ω = ω0, and the second part around ω = -ω0 :

where: are Fresnel Integrals,

which have the properties:

( ) ( ) ∫∫ ==UU

dzz

USdzz

UC0

2

0

2

2sin&

2cos

ππ

( ) ( ) ( ) ( )USUSUCUC −=−−=− &

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

( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) −+ ++−=−+−

−+

+++

−−=

ωωωωµωω

µπ

µωω

µπω

002211

2

0

2211

2

0

2exp

2

2exp

2

ii

i

SSYSjYCYSjYCjA

XSjXCXSjXCjA

S

ωωωπωτµπ

∆=−∆=∆=∆2

:&2:2

1: 0

nff


SOLO Fresnel Integrals

Augustin Jean Fresnel1788-1827

Define Fresnel Integrals

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )∫ ∑

∑∫∞

=

+

∞

=

+

+−=

=

++−=

=

α

α

ααπα

ααπα

0 0

142

0

34

0

2

!2141

2sin:

!12341

2cos:

n

nn

n

nn

nn

xdS

nn

xdC

( ) ( )ααααπα

SjCdj +=

∫

0

2

2exp

( ) ( ) 5.0±=∞±=∞± SC

( ) ( ) ( ) ( )USUSUCUC −=−−=− &

The Cornu Spiral is defined as the plot of S (u) versus C (u)

duuSd

duuCd

=

=

2

2

2sin

2cos

π

π

( ) ( ) duSdCd =+ 22

Therefore u may be thought as measuring arc length along the spiral.

SOLO

( ) ( )2/cos 203 ttAtf ωω ∆+=

t

A

2/τ−2/τ


( )222

cos2

0

ττµω ≤≤−

+= tt

tAtsi


ωωωπωτµπ

∆=−∆=∆=∆2

:&2:2

1: 0

nff

Define:

( ) ( ) ( )[ ] ( ) ( )[ ]{ }221

2

210 2XSXSXCXC

AS i +++=−

+ µπωωAmplitude Term:

Square Law Phase Term: ( ) ( )µωωω2

2

01

−−=Φ

Residual Phase Term: ( ) ( ) ( )( ) ( ) 4

1tan5.05.0

5.05.0tantan 111

21

211

2

πωτ

==++→

++=Φ −−

>>∆−

f

XCXC

XSXS

( ) ( )nfXnfX −∆=+∆= 12

&12 21

ττ

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

( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) −+ ++−=−+−

−+

+++

−−=

ωωωωµωω

µπ

µωω

µπω

002211

2

0

2211

2

0

2exp

2

2exp

2

ii

i

SSYSjYCYSjYCjA

XSjXCXSjXCjA

S( )ω2Φ( )

+− ωω 0iS


SOLO



Linear FM Modulated Pulse (Chirp) Summary

• Chirp is one of the most common type of pulse compression code

• Chirp is simple to generate and compress using IF analog techniques, for example, surface acoustic waves (SAW) devices.

• Large pulse compression ratios can be achieved (50 – 300).

• Chirp is relative insensitive to uncompressed Doppler shifts and can be easily weighted for side-lobe reduction.

• The analog nature of chirp sometimes limits its flexibility.

• The very predictibility of chirp mades it asa poor choice for ECCM purpose.


SOLO Pulse Compression Techniques

Phase Coded Waveforms

• Phase Coded Waveforms consists of N contiguous sub-pulses where the phase of each pulse is chosen to shape the range sidelobe response at the output of the matched filter.

- sub-pulse length = τ

- total length = N τ Poly-phase codes allow for any of M phase shifts on a sub-pulse basis, where Mis called the order of the code and the possible phase states are

φi = (2π/M) i, for i = 1,…,M

SOLOPulse Compression Techniques

Phase CodingA transmitted radar pulse of duration τ is divided in N sub-pulses of equal durationτ’ = τ /N, and each sub-pulse is phase coded in terms of the phase of the carrier.

The complex envelope of the phase codedsignal is given by:

( ) ( ) ( )∑−

=

−=1

02/1 '

'

1 N

nn ntu

Ntg τ

τ where:

( ) ( ) ≤≤

=elsewhere

tjtu n

n 0

'0exp τϕ


Matched Filters for RADAR SignalsSOLO

Matched Filter Response to Phase Coding

( ) ( ) ( ) ( ) ( )tjtgtjtgts 00 exp2

1exp

2

1 ωω −+= ∗

( ) ( ) ( ) ∆<<

=∆−=∑−

= elsewhere

tttftptfctg

M

pp 0

011

0

Let the signal be a phase-modulated carrier, in which the modulation is in discrete and equal steps Δt. The complex envelope of the signal can be described by a sequence of complex numbers , such thatkc

( ) [ ] ( ) ( )∫+∞

∞−

∗ +−−= dtttgtgtjgo 000exp2

1 τωτ

Constant Phase

Matched Filter output envelope (change t ↔τ):

( )ttk ∆<≤+∆→ τττ 0

tMt ∆=0

( ) [ ] ( ) ( )[ ]

[ ] ( )[ ]( )

∑ ∫

∫ ∑−

=

∆+

∆

∗

+∞

∞−

∗−

=

∆−+−∆−=

∆−+−∆−∆−=+∆

1

0

1

0

1

00

exp2

1

exp2

1

M

p

tp

tp

p

M

ppo

dttkMtgctMj

dttkMtgtptfctMjtkg

τω

τωτ

Change variable of integration to t1 = t – τ + (M - k) Δt

( ) [ ] ( )( )

( )

∑ ∫−

=

−∆+−+

−∆−+

∗∆−=+∆1

0

1

110exp2

1 M

p

tkMp

tkMp

po dttgctMjtkgτ

τ

ωτ


Matched Filter Response to Phase Coding (continue – 1(

Matched Filter output envelope for a Phase Coding is:

( ) [ ] ( )[ ]( )

∑ ∫−

=

∆+

∆

∗ ∆−+−∆−=+∆1

0

1

0exp2

1 M

p

tp

tp

po dttkMtgctMjtkg τωτ

Change variable of integration to t1 = t – τ + (M - k) Δt

( ) [ ] ( )( )

( )[ ] ( )

( )

( )( )

( )

( )

∑ ∫∫∑ ∫−

=

−∆+−+

∆−+

∗∆−+

−∆−+

∗−

=

−∆+−+

−∆−+

∗

+∆−=∆−=+∆

1

0

1

11110

1

0

1

110 exp2

1exp

2

1 M

p

tkMp

tkMp

tkMp

tkMp

p

M

p

tkMp

tkMp

po dttgdttgctMjdttgctMjtkgτ

τ

τ

τ

ωωτ

( ) ( ) ( )( ) ( ) ( ) τ

τ−∆+−+<<∆−+=

∆−+<<−∆−+=

−+∗

−−+∗

tkMpttkMpctg

tkMpttkMpctg

kMp

kMp

11*

1

11*

1

( ) [ ] ∑ ∫∫−

=

−∆

−+

−

−−+

+∆−=+∆

1

0 0

1*

0

11*

0exp2

1 M

p

t

kMpkMppo dtcdtcctMjtkgτ

τ

ωτ

( ) [ ] ∑−

=−+−−+

∆

−+

∆

∆−∆

=+∆1

0

*1

*0 1exp

2

1 M

p

kMpkMppo tc

tcctMj

ttkg

ττωτ

This equation describes straight lines in the complex plane, that can have corners only atτ = 0. At those corners

( ) [ ] ∑−

=−+∆−

∆=∆

1

0

*0exp

2

1 M

p

kMppo cctMjt

tkg ω

Constant Phase



Matched Filter output envelope for a Phase Coding is:

( ) [ ] ∑−

=−+−−+

∆

−+

∆

∆−∆

=+∆1

0

*1

*0 1exp

2

1 M

p

kMpkMppo tc

tcctMj

ttkg

ττωτ

This equation describes straight lines in the complex plane, that can have corners only atτ = 0. At those corners

( ) [ ] ∑−

=−+∆−

∆=∆

1

0

*0exp

2

1 M

p

kMppo cctMjt

tkg ω

Constant Phase

We can see that is the discrete autocorrelation function for the observation time t0 = M Δt (the time the received Radar signal return is expected)

∑−

=−+

1

0

*M

p

kMpp cc



Example: Pulse poly-phase coded of length 4

Given the sequence: { } 1,,,1 −−++= jjck

which corresponds to the sequence of phases 0◦, 90◦, 270◦ and 180◦, the matched filter is given in Figure bellow.

{ } 1,,,1* −+−+= jjck

Pulse poly-phase coded of length 4

At the Receiver the coded pulse enters a 4 cells delay lane (from left to right), a bin at each clock.The signals in the cells are multiplied by -1,+j,-j or +1 and summed.

clock

SOLOPoly-Phase Modulation

-1 = -11 1+

-j +j = 02 1+j+

+j -1-j = -13 1+j+j−

+1 +1+1+1 = 44 1+j+j−1−

-j-1+j = -15 j+j−1−

+j - j = 06

j−1−7

1− -1 = -1

8 0

Σ

{ } 1,,,1 −−++= jjck

1− 1+j+ j− {ck*}

0 = 00

0

1

2

3

4

5

6

7

{ } 1,,,1* −+−+= jjck



Bi-Phase Codes

• easy to implement

• significant range sidelobe reduction possible

• Doppler intolerant

A bi-phase code switches the absolute phase of the RF carrier between two states180º out of phase.

Bandwidth ~ 1/τ

Transmitted Pulse

Received Pulse

• Peak Sidelobe Level

PSL = 10 log (maximum side-lobe power/ peak response power)

• Integrated Side-lobe Level

ISL = 10 log (total power in the side-lobe/ peak response power)

Bi-Phase Codes Properties

The most known are the Barker Codes sequence of length N, with sidelobes levels, atzero Doppler, not higher than 1/N. Return to Table of Content


Bi-Phase Codes

LengthN

Barker Code PSL(db)

ISL(db)

2 + - - 6.0 - 3.0

2 + + - 6.0 - 3.0

3 + + - - 9.5 - 6.5

3 + - + - 9.5 - 6.5

4 + + - + - 12.0 - 6.0

4 + + + - - 12.0 - 6.0

5 + + + - + - 14.0 - 8.0

7 + + + - - + - - 16.9 - 9.1

11 + + + - - - + - - + - - 20.8 - 10.8

13 + + + + + - - + + - + - + - 22.3 - 11.5

Barker Codes-Perfect codes –Lowest side-lobes forthe values of N listed in the Table.

-1

Pulse bi-phase Barker coded of length 3

Digital Correlation At the Receiver the coded pulse enters a 3 cells delay lane (from left to right), a bin at each clock.The signals in the cells are multiplied according to ck* sign and summed.

clock

-1 = -11

+1 -1 = 02

-( +1) = -15

0 = 06

+1 +1-( -1) = 33

+1-( +1) = 04


1

2

3

4

5

6

0

+1+1

0 = 00


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


+1-1+1+1+1 { }*kc

+1 = +11

+1 = 19

0 = 010

2 -1 +1 = 0

+1 +1 -1-( +1) = 04

+1 +1 +1 –(-1)+1 = 55

0 = 0 0

3 +1-1 +1 = 1

+1 +1 -(+1) -1 = 06

+1-( +1) +1 = 17

–(+1) +1 = 08



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


-1-1 -1+1+1+1+1 { }*kc

Pulse bi-phase coded of length 8


clock


+1 = 11

-1-1 -1+1+1+1+1 { }*kc+1

-1 +1 = 02-1 -1 +1 = -13

+1 -1 -1-( +1) =-24

-1 +1 -1 –(-1)+1= 15+1 -1 +1-(-1) -1–(+1)= 06

1+1 -1-( +1)-1 –(-1)-(+1)=- 17+1+1+1 –(-1)+1-( -1) -( -1)+1= 88

+1+1 –(+1) -1-( +1)-(-1) -1= -19+1-(+1)+1-(-1) -( +1)-1= 010

-(+1)+1-(+1)-(-1)+1 = 111+1-(+1)-(+1)-1 = -212

-(+1)-(+1)+1 = -113-(+1)+1 = 014

+1 = 115

SOLO Pulse Compression TechniquesBi-Phase Codes

Combined Barker CodesOne scheme of generating codes longer than 13 bits is the method of forming combinedBarker codes using the known Barker codes. For example to obtain a 20:1 pulsecompression rate, one may use eithera 5x4 or a 4x5 codes.

The 5x4 Barker code (see Figure) consists of the 5 Barker code, each bit of which is the 4-bit Barker code. The 5x4 combined code is the 20-bit code.

• Barker Code 4

• Barker Code 5


Binary Phase Codes Summary

• Binary phase codes (Barker, Combined Barker) are used in most radar applications.

• Binary phase codes can be digitally implemented. It is applied separately to I and Q channels.

• Binary phase codes are Doppler frequency shift sensitive.

• Barker codes have good side-lobe for low compression ratios.

• At Higher PRFs Doppler frequency shift sensitivity may pose a problem.


SOLO Pulse Compression TechniquesPoly-Phase Codes

Frank Codes

In this case the pulse of width τ is divided in N equal groups; each group issubsequently divided into other N sub-pulses each of width τ’. Therefore thetotal number of sub-pulses is N2, and the compression ratio is also N2.

A Frank code of N2 sub-pulses is called a N-phase Frank code. The fundamentalphase increment of the N-phase Frank code is: N/360=∆ ϕ

For N-phase Frank code the phase of each sub-pulse is computed from:

( )

( ) ( ) ( ) ( )

ϕ∆

−−−−

−−

21131210

126420

13210

00000

NNNN

N

N

Each row represents the phases of the sub-pulses of a group


Frank Codes (continue – 1)

Example: For N=4 Frank code. The fundamental phase increment of the 4-phase Frank code is: 904/360 ==∆ ϕ

We have:

−−−−−−

⇒

→

jj

jjj

formcomplex

11

1111

11

1111

901802700

18001800

270180900

0000

90

Therefore the N = 4 Frank code has the following N2 = 16 elements

{ }jjjjF 11111111111116 −−−−−−=

The phase increments within each row represent a stepwise approximation of an up-chirp LFM waveform.



Example: For N=4 Frank code (continue – 1).

If we add 2π phase to the third N=4 Frank phase row and 4π phase to the forth(adding a phase that is a multiply of 2π doesn’t change the signal) we obtain aanalogy to the discrete FM signal.

If we use then the phases of the discrete linear FM and the Frank-coded signals are identical at all multipliers of τ’.

'/1 τ=∆ f



Fig. 8.8 Levanon pg.157



Fig. 8.8 Levanon pg.158,159



The phase-code pulses envelope is given by: ( ) ( ) ( )∑=

−−=

N

mm

mtrecj

NTtg

1 '

1exp

1

τϕ

The phases φm are chosen such that the autocorrelation function has the smallest Peak-to-sidelobe ratio (PSR), for a certain code length. PSR is bounded from bellowby the code length N ( )NPSR log20=

Binary phase codes use only φm=0 or π. The main drawback of binary codes, such as Barker codes or m-sequences is their sensitivity to Doppler shift.

Poly-phase codes are not restricted on code elements and are generated from phasehistory of frequency-modulated pulse. The Frank code and the P1 and P2 codes,The modified version of Frank code, are derived from the linear stepped frequency Modulation. These three codes are only applicable for perfect square length (N = L2),and can be expressed as: ( ) ( )

( ) ( )[ ] ( )[ ]

( )[ ] ( )[ ]jLiLL

P

jLiLjL

P

jiL

Frank

ji

ji

ji

−+−+=

−−−+−=

−−=

2/12/12

:2

12112

:1

112

:

,

,

,

πϕ

πϕ

πϕ


( )

( )

( ) ( )NiiN

P

oddNNiiiN

evenNNiiNP

i

i

−−−=

=−

=−=

11:4

;,,2,1;1

;,,2,1;1:3

2

πϕ

π

π

ϕ

Another two well known poly-phase codes are P3 and P4 derived from linear frequencymodulated pulse. Unlike Frank, P1 and P2 codes, P3 and P4 code lengths are arbitrary.P3 and P4 codes can be expressed as:

It is known that Frank, P1 and P2 codes are more Doppler shift insensitive thanbinary codes, but P3 and P4 are even better.




P4, N = 25 Elements


Frank, P1, P2, P3, P4 Codes Summary

• Frank, P1, P2, P3, P4 Codes are digital versions of the chirp

• They are insensitive to Doppler frequency shift provided that fmax . τ’ < 0.3 but more sensitive then chirp.• They can have very long length..

P1, P2, P3, P4, P(n,k) Poly-Phase Codes


SOLO

Pseudo-Random Codes

Pseudo-Random Codes are binary-valued sequences similar to Barker codes.

The name pseudo-random (pseudo-noise) stems from the fact that they resemblea random like sequence.

The pseudo-random codes can be easily generated using feedback shift-registers.

It can be shown that for N shift-registers we can obtain a maximum length sequenceof length 2N-1.

0 1 0 0 1 1 123-1=7

Register# 1

Register# 2

Register# 3

XOR

clock

A

B

Input A Input B Output XOR

0 0 0 0 1 1 1 0 1 1 1 0

Register # 1

Register # 2

Register # 3

0 1 0 sequence

I.C.

0 0 1 1

1 0 0 2

1 1 0 3

1 1 1 4

0 1 1 5

1 0 1 60 1 0 7

clock

0 0 1 8

0

Pulse Compression Techniques

SOLO

Pseudo-Random Codes (continue – 1)To ensure that the output sequence from a shift register with feedback is maximal length, the biths used in the feedback path like in Figure bellow, must be determined by the 1 coefficients of primitive, irreducible polynomials modulo 2. As an example for N = 4, length 2N-1=15, can be written in binary notation as 1 0 0 1 1.

The primitive, irreductible polynomial that this denotes is (1)x4 + (0)x3 + (0)x2 + (1)x1 + (1)x0

1 0 0 1 0 0 0 1 1 1 1 0 1 0 1

24-1=15

sequence

1 0 0 1 I.C.0

The constant (last) 1 term in every such polynomial corresponds to the closing of the loop to the first bit in the register.

Register# 1

Register# 2

Register# 3

XOR

clock

AB


0 0 0 0 1 1 1 0 1 1 1 0

Register# 4

Register # 1

Register # 2

Register # 3clock

Register # 4

1 0 1 0 0

0 0 1 0 2

0 0 0 1 3

1 0 0 0 4

1 1 0 0 5

1 1 1 0 6

1 1 1 1 7

0 1 1 1 8

1 0 1 1 9

0 1 0 1 10

1 0 1 0 11

1 1 0 1 12

0 1 1 0 13

0 0 1 1 14

1 0 0 1 15

0 1 0 0 16

0 0 1 017


SOLO

Pseudo-Random Codes (continue – 2)



0 0 0 0 1 1 1 0 1 1 1 0

Register# 1

Register# 2

Register# n

XOR

clock

AB

Register# (n-1)

Register# m

. . .. . .

2 3 1 2 ,1

3 7 2 3 ,2

4 15 2 4 ,3

5 31 6 5 ,3

6 63 6 6 ,5

7 127 18 7 ,6

8 255 16 8 ,6 ,5 ,4

9 511 48 9 ,5

10 1,023 60 10 ,7

11 2,047 176 11 ,9

12 4,095 144 12 ,11 ,8 ,6

13 8,191 630 13 ,12 ,10 ,9

14 16,383 756 14 ,13 ,8 ,4

15 32,767 1,800 15 ,14

16 65,535 2,048 16 ,15 ,13 ,4

17 131,071 7,710 17 ,4

18 262,143 7,776 18 ,11

19 524,287 27,594 19 ,18 ,17 ,14

20 1,048,575 24,000 20 ,17

Number ofStages n

Length ofMaximal Sequence N

Number ofMaximal Sequence M

Feedback stageconnections

Maximum Length Sequence

n – stage generator

N – length of maximum sequence

12 −= nNM – the total number of maximal-length sequences that may be obtained from a n-stage generator

∏

−=

ipN

nM

11

where pi are the prime factors of N.

SOLO

Pseudo-Random Codes (continue – 3)


Pseudo-Random Codes Summary

• Longer codes can be generated and side-lobes eventually reduced.

• Low sensitivity to side-lobe degradation in the presence of Doppler frequency shift.

• Pseudo-random codes resemble a noise like sequence.

• They can be easily generated using shift registers.

• The main drawback of pseudo-random codes is that their compression ratio is not large enough.


SOLO Pulse Compression TechniquesFrequency Codes

Costas CodesIn this case a pulse of duration T is divided in N equal sub-pulses of duration NT /1 =τIn Linear Stepped Frequency Modulation (LSFM) the frequency of each sub-pulse is increased linearly according to: Nififf i ,,2,10 =+= δ

where f0 is a constant frequency and f0 >> δ f.

The maximum change in frequency is Δ f = N δ f during the time τ.

The pulse has a time-bandwidth of: ( ) 2

1

12

1 NfNNfNTf ≈⋅=⋅=⋅∆≈τδτδ

0 1 2 3 4 5 6 7 8 9

12

345678

910

0 1 2 3 4 5 6 7 8 9

12

345678

910

Column number, j (time) Column number, j (time)

Row

nu

mbe

r, i

(fre

quen

cy)

Row

nu

mbe

r, i

(fre

quen

cy)

Frequency-time array for LSFM Frequency-time array for Costas code

Costas codes are similar to LSFM, only the frequency steps are chosen randomly.


Costas Codes (continue – 1)

The normalized complex envelope of a Costas signal is given by:

( ) ( ) ( ) ( ) ≤≤

=−= ∑−

= elsewhere

ttfjtgltg

Ntg l

l

N

ll 0

02exp&

1 11

01

1

τπτ

τ

Costas showed that the output of the matched filter is given by:

( ) ( ) ( ) ( )( )∑ ∑−

=

−

≠=

−−Φ+Φ=1

0

1

01,,2exp

1,

N

l

N

lqq

DlqDlllD fqlftfjN

f τττπτχ

( ) ( ) 11

1 2expsin

, τττπβα

αττ

ττ ≤−−

−=Φ qq

q

qDlq fjjf

( ) ( )( ) ( )ττπβ

ττπα

+−−=

−−−=

1

1

Dqlq

Dqlq

fff

fff



0 1 2 3 4 5 6 7 8 9

12

345678

910

Column number, j (time)

Row

nu

mbe

r, i

(fre

quen

cy)

Frequency-time array for Costas code

Fig. 8.3, 8.4 Levanon pg.150,151



• All side-lobes, except for few around the origin, have amplitude 1/N. Few side-lobes close to the origin have amplitude 2/N, which is typical to Costas codes.

• The compression ratio of Costas codes is approximately N.

• The ambiguity function of Costas codes is approaching the ideal thumbtack shape..

• Costas codes have low sensitivity to coherence requirements.


SOLO Pulse Compression TechniquesComplementary Pulse Codes

Complementary codes consist of a pair of codes with complementaryside-lobes, that is, their side-lobes are equal and opposite Golay, 1961).

Bogler, P.L., “Radar Principles with Applications to Tracking Systems”, John Wiley & Sons, 1989

References


Levanon, N., “Radar Principles”, John Wiley & Sons, 1988

Mahafza, B.R., “Radar System Analysis and Design Using MATLAB”, Chapman & Hall/CRC, 2000

Nathanson, F.E., “Radar Design Principles”, McGraw-Hill, 1969

Morris, G.V., “Airborne Pulse Radar”, Artech House, 1988

Berkowitz, R.S. Ed., “Modern Radar – Analysis, Evaluation, and System Design”, John Wiley & Sons, 1965

Richards, M.A., “Fundamentals of Radar Signal Processing”, Georgia Tech Course ECE 6272, Spring 2000

“Principles of Modern Radar” Georgia Tech, 2004, Jim Scheer,

“Advanced Radar Waveforms”Hermelin, S., “Matched Filters and Ambiguity Functions for RADAR Signals”,

Power Point PresentationReturn to Table of Content

January 20, 2015 62

SOLO

TechnionIsraeli Institute of Technology

1964 – 1968 BSc EE1968 – 1971 MSc EE

Israeli Air Force1970 – 1974

RAFAELIsraeli Armament Development Authority

1974 – 2013

Stanford University1983 – 1986 PhD AA

SOLO Fourier Transform of a Signal

The Fourier transform of a signal f (t) can be written as:

A sufficient (but not necessary) condition for theexistence of the Fourier Transform is:

( ) ( ) ∞<= ∫∫∞

∞−

∞

∞−

ωωπ

djFdttf22

2

1

JEAN FOURIER1768-1830

( ) ( )∫+∞

∞−

−= ωωπ

ω dejFj

tf tj

2

1

The Inverse Fourier transform of F (j ω) is given by:

( ) ( )∫+∞

∞−

= dtetfjF tjωω

( ) ( )∫+∞

∞−

−= ωωπ

ω dejFj

tf tj

2

1

Signal

(1) C.W.

( )2

cos00

0

tjtj eeAtAtf

ωω

ω−+==

0ω - carrier frequency

Frequency

( ) ( )∫+∞

∞−

= dtetfjF tjωωFourier Transform

( ) ( ) ( )00 22ωωδωωδω ++−= AA

jFFourier Transform


( ) ( )∫+∞

∞−

−= ωωπ

ω dejFj

tf tj

2

1

Signal

(2) Single Pulse

( )

>≤≤−

=2/0

2/2/

τττ

t

tAtf

τ - pulse width

Frequency

( ) ( )∫+∞

∞−


( ) ( ) ( )( )2/

2/sin2/

2/ τωτωτω

τ

τ

ω AdteAjF tj == ∫−

Fourier Transform


( ) ( )∫+∞

∞−

−= ωωπ

ω dejFj

tf tj

2

1

Signal

( ) ( )

>≤≤−

=2/0

2/2/cos 0

τττω

t

ttAtf

τ - pulse width

Frequency

( ) ( )∫+∞

∞−


( ) ( )

( )

( )

( )

( )

−

−

++

+

=

= ∫−

2

2sin

2

2sin

2

cos

0

0

0

0

2/

2/

0

τωω

τωω

τωω

τωωτ

ωωτ

τ

ω

A

dtetAjF tjFourier Transform


(3) Single Pulse Modulated at a frequency

0ω

ω

( )ωjF

0

τπω 2

0 +

2

τA

0ω

τπω 2

0 −τπω 2

0 +−

2

τA

0ω−

τπω 2

0 −−

τπω 2

20 +τπω 2

20 −


( ) ( )∫+∞

∞−

−= ωωπ

ω dejFj

tf tj

2

1

Signal

( ) ( )

±±=>−≤−≤−+

=,2,1,0,2/0

2/2/cos 0

kkkTt

kTttAtf

rand

τττϕω

τ - pulse width

Frequency

( ) ( )∫+∞

∞−


( ) ( )

( )

( )

( )

( )

−

−

++

+

=

= ∫−

2

2sin

2

2sin

2

cos

0

0

0

0

2/

2/

0

τωω

τωω

τωω

τωωτ

ωωτ

τ

ω

A

dtetAjF tj

Fourier Transform


(4) Train of Noncoherent Pulses (random starting pulses), modulated at a frequency 0ω

T - Pulse repetition interval (PRI)


( ) ( )∫+∞

∞−

−= ωωπ

ω dejFj

tf tj

2

1

Signal

( ) ( )

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

−++

+=

±±=>−≤−≤−

=

∑∞

=1000

0

coscos

2

2sin

cos

,2,1,0,2/0

2/2/cos

nPRPR

PR

PRseriesFourier

tntnn

n

tT

A

kkkTt

kTttAtf

ωωωωτω

τω

ωτ

τττω

τ - pulse width

Frequency

( ) ( )∫+∞

∞−


Fourier Transform


(5) Train of Coherent Pulses, of infinite length, modulated at a frequency 0ω


( ) ( ) ( ){

( ) ( ) ( ) ( )[ ]

+−+−+−−++

+

−+=

∑∞

= 10000

00

2

2sin

2

nPRPRPRPR

PR

PR

nnnnn

n

T

AjF

ωωδωωδωωδωωδτω

τω

ωδωδτω

T/1 - Pulse repetition frequency (PRF)TPR /2πω =


( ) ( )∫+∞

∞−

−= ωωπ

ω dejFj

tf tj

2

1

Signal

( ) ( )

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

−++

+=

±±=>−≤−≤−

=

∑∞

=

≤≤−

1000

22

0

coscos

2

2sin

cos

2/,,2,1,0,2/0

2/2/cos

nPRPR

PR

PRNTt

NT

tntnn

n

tT

A

NkkkTt

kTttAtf

ωωωωτω

τω

ωτ

τττω

τ - pulse width

Frequency

( ) ( )∫+∞

∞−


Fourier Transform


(6) Train of Coherent Pulses, of finite length N T, modulated at a frequency 0ω


( )( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

−−

−−

++−

+−

++

+

+

−+

−+

+++

++

++

+

=

∑

∑

∞

=

∞

=

10

0

0

0

0

0

10

0

0

0

0

0

2

2sin

2

2sin

2

2sin

2

2sin

2

2sin

2

2sin

2

2sin

2

2sin

2

nPR

PR

PR

PR

PR

PR

nPR

PR

PR

PR

PR

PR

TNn

TNn

TNn

TNn

n

n

TN

TN

TNn

TNn

TNn

TNn

n

n

TN

TN

T

AjF

ωωω

ωωω

ωωω

ωωω

τω

τω

ωω

ωω

ωωω

ωωω

ωωω

ωωω

τω

τω

ωω

ωωτω



Signal

( ) ( )

+=

±±=>−≤−≤−

= ∑∞

=11 cos

2

2sin

21,2,1,0,2/0

2/2/

nPR

PR

PRSeriesFourier

tnn

n

T

AkkkTt

kTtAtf ω

τω

τωτ

τττ

τ - pulse width0ω - carrier frequency

(6) Train of Coherent Pulses, of finite length N T, modulated at a frequency 0ω



( ) ( )tAtf 03 cos ω=

t

A A

( )tf1

t

2

τ2

τ−T

A

T T

22

τ+T2

2τ−T

T T

2

τ− 2

τ+T

( )tf 2

t

TN

2/TN2/TN−

( ) ( ) ( ) ( )tftftftf 321 ⋅⋅=

( ) ( ) ( ) ( ) ( )

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

−++

+=

±±=>−≤−≤−

=⋅⋅=

∑∞

=

≤≤−

1000

22

0

321

coscos

2

2sin

cos

2/,,2,1,0,2/0

2/2/cos

nPRPR

PR

PRNTt

NT

tntnn

n

tT

A

NkkkTt

kTttAtftftftf

ωωωωτω

τω

ωτ

τττω

( )

>≤≤−

=2/0

2/2/12 TNt

TNtTNtf ( ) ( )ttf 03 cos ω=


5 pulse compression waveform

Science

coherent pulse radar

resulting received waveform

radar antenna

different targets

targets echoes

order todistinguish

received signal

frf fifblock diagram