12. waveforms, design and diversity_2014

12. Waveforms, Design and Diversity

Waveforms, Design and Diversity

Barker codes!Polyphase codes!Nonlinear FM!Linear period modulation!Step-CW and step-chirp!

Phase Coded Pulse Compression

In contrast to analogue linear-FM pulse compression, it is equally possible to generate almost any waveform by digital means, and to perform the pulse compression processing digitally, with all of the advantages (high performance, reproduceability, flexibility, ) of digital processing.!

Many types of waveform, and especially phase coded waveforms have been studied and used. Phase coded waveforms can be biphase (simplest) or polyphase.!

One of the simplest types are the Barker codes.!

Barker Codes

Barker codes are a set of binary (biphase) codes with perfect autocorrelation properties (in the sense that the sidelobes are either zero or + 1.!The codes are as follows :!

Code length Code elements Sidelobe level, dB

2 + , + + 6.0 3 + + 9.5 4 + + +, + + + 12.0 5 + + + + 14.0 7 + + + + 16.9 11 + + + + + 20.8 13 + + + + + + + + + 22.3

We can evaluate the autocorrelation function of any of these codes (for example, of length 5) by considering the output from a tapped delay-line matched filter :!

+ + + +

+ + + +

Barker Codes

+ + + +

+ + + +

Barker Codes

+ + + +

+ + + +

Barker codes

+ + + +

+ + + +

Barker Codes

Peak side-lobe level = 20 log10 n!!Processing gain = 10 log10 n!!where n is the code length.!!!!The side-lobe performance degrades quite badly away from zero Doppler.!!No Barker codes of length greater than 13 have ever been discovered. But it is possible to build concatenated codes which have greater length.!

Barker Codes

Barker Codes

Digital Pulse Compression

Time Domain vs Frequency Domain Processing

compressedwaveform

W1 W2 W3

WNf. . . . .

input

input forwardFTinverseFT compressedsignal

replicaspectrum

time domain!!!!!!!!!!frequency domain!

( ) ( )( ) ( )( ) ( )( ) g t h t g t h t = F F F

( ) ( ) ( )( ) ( )( )( ) g t h t g t h t = - 1F F F

Concatenated Barker Codes

For example take a 5-bit Barker code and a 4-bit Barker code :!!!!!!The matched filter is built up by cascading the matched filters for the individual codes :!

!"#!"#!"#!"#!"# ++++

+++

++++

++++

++++

+ 0 0 0 - 0 0 0 + 0 0 0 + 0 0 0 +

+ - + +

Polyphase Codes

Greater flexibility over biphase codes is provided by polyphase codes, though the processing involved in generation and compression is more complicated than for biphase codes. Several of these have been proposed and studied; most are based on approximations to linear FM.!!Examples are Frank codes, Huffman codes, and the P-codes developed by Kretschmer and Lewis. !

Frank Codes

Greater flexibility over biphase codes is provided by polyphase codes, though the processing involved in generation and compression is more complicated than for biphase codes. Several of these have been proposed and studied; most are based on approximations to linear FM.!!Examples considered here are Frank codes, Huffman codes, and the P-codes developed by Kretschmer and Lewis. !

Kretschmer & Lewis P-Codes

Costas Codes

NUMBER AND DENSITY OF COSTAS ARRAYSN 3 4 5 6 7 8 9 10 11 12M(N) 4 12 40 116 200 444 760 2,160 4,368 7,852M(N)|N! 0.67 0.5 0.33 0.16 0.04 0.011 210-3 610-4 110-4 1.610-5

fc+f6fc+f5fc+f4fc+f3fc+f2fc+f1

fcf / t t0 t1 t2 t3 t4 t5 t6

*

*

*

*

*

**

The permutation matrix shown is one of 200 possible Costas arrays for N = 7.

Costas codes

Ambiguity diagrams

Barker Code!

Pseudo Random Binary Code!

Costas Code!

Ultra low side-lobe waveform

We had a requirement for a waveform with very low range sidelobes (better than -60 dB) for a satellite-borne rain radar.!

v

h

D

r

d

Welti Codes

frequency

time

The waveform can be written as!!!!where is the central linear FM component and corresponds to the additional higher FM rate portions!!The higher FM rate can be thought of as a way of reducing the energy per unit bandwidth (i.e. amplitude taper) whilst keeping the amplitude constant.!

)(~ )(~ 21 tsts +

)(~1 ts )(~2 ts

Griffiths, H.D. and Vinagre, L., Design of low-sidelobe pulse compression waveforms, Electronics Letters, Vol.30, pp1004-1005, 1994.


The point target response is the sum of the different crosscorrelation terms :!

{ }

= dttgtstss *110 )()(~)(~ )(

{ }

+ dttgtsts *12 )()(~)(~

{ }

+ dttgtsts *21 )()(~)(~

{ }

+ dttgtsts *22 )()(~)(~

)()()()( 4321 RRRR +++=

R3(t) and R4(t) are negligible.

(i) R1(t) ; (ii) R2(t) ; (iii) s0(t) for optimised waveform


Improved DDFC waveform

1

, 1 2

2

. 0

. 2

2

i pos

M t t t

f M t t t tT t

t t T

= + <

<

,

,

2 0

0 2

i pos

i

i pos

f T tf

f t T

12. waveforms, design and diversity_2014

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