SURFACE ACOUSTIC WAVE DISPERSIVE FILTER WITH VARIABLE, LINEAR, FREQUENCY - TIME SLOPE

C 0 Newton and E G S Paige ERE, Malvern, Worcs., U.K.

ABSTRACT A wide var ie ty of appl ica t ions has been found f o r SAW dispersive f i l t e r s of f ixed dispers ive slope. There are, however, appl ica t ions f o r which a var iab le dispers ive s lope would be advantagebus. described which a c t s as a dispersive f i l t e r with a slope which can be e lec t ronica l ly varied. The scheme is similar t o that used i n a delay l i n e with var iab le time delay except that the dispers ive f i l t e r s have a cubic var ia t ion of phase v i t h frequency. s u i t a b l e choice of dispers ive f i l t e r s , the frequency-time slope of the syatem may be maintained l i n e a r but var ied over an extensive range of time-bandwidth product simply by varying a l o c a l o s c i l l a t o r frequency. P r a c t i c a l aspec ts and l i m i t a t i o n s are discussed.

I n t h i s paper a new SAW device is

By

INTRODUCTION DDL 1 DDLZ F

This paper descr ibes a scheme f o r the implementation of var iab le , l i n e a r , chirp. A f i l t e r is described i n d e t a i l whose t ime bandwidth product can be var ied by a f a c t o r of two using a vol tage control led o s c i l l a t o r . It is shown i n out l ine tha t a v a r i a t i o n by a f a c t o r of ten i n useful time bandwidth product i s avai lable .

ch i rp system has the following a t t r a c t i v e fea tures : For s igna l processing appl ica t ions , t h i s var iab le + operat ion is asynchronous; the system works a t f ixed bandwidth, the chirp rate var ia t ion being provided by a chirp length var ia t ion ; amplitude weighting can be b u i l t d i r e c t l y i n t o the f i l t e r f o r pulse compression appl ica t ions ; a pulse compression loop using a p a i r of var iab le ch i rp f i l t e r s gives a pu lse delay t i m e j i t t e r which is automatically cancelled i n the receiver ; i n u r a c t i c a l terms the system operates i n the

DDL 1,OULZ

\'C 0

c

SAW d ispers ive delay lines P i t h cub i c phase charoct er i st i c

v o l t a g e controtled o s c i l l a t o r c o r i r r o l 1 i r ig d I spers i ve sl oFe

Filter : o reject mixer sideband prefer red condition with- the mixers sa tura ted by t h e voltage cont ro l led o s c i l l a t o r ; t h e chirp r a t e is l i n e a r l y control led by the o s c i l l a t o r frequency so t h a t p r o g r a d n g i s easy; d i r e c t p ropor t iona l i ty can RRRRNGEMENT F O R VRRIRBLE SLDPE DISPERSION be chosen o r t h e ch i rp rate can be swung through p o s i t i v e and negative values. pig. 1

Speci f ica t ion c a l l s f o r a p a i r of accurately demigned nonl inear chirp f i l t e r s . these w i l l be surface acous t ic wave devices because surface wave devices can be designed t o meet the d e t a i l e d spec i f ica t ion requiremmts f o r nonl inear chirp f i l t e r s ( l ) . Development e f f o r t has s h m t h a t SAW devices s u b s t a n t i a l l y meet t h e o r e t i c a l perform- ance predict ions; therefore i t is f e l t j u s t i f i a b l e t o present signal processing r e s u l t s simulated i n the conputer ahead of ac tua l device performance.

It is assumed here t h a t

FILTER DESIGN

Design Scheme

The b a s i c arrangement of components is shown i n P ig 1. It is s i m i l a r t o t h a t used i n the var iab le t i m e delayC2) except t h a t the d ispers ive f i l t e r s DDL1, DDLZ have nonlinear c h a r a c t e r i s t i c s . We w i l l now show t h a t a var iab le l i n e a r dispers ion can be obtained f o r a p a i r of f i l t e r s with cubic variations of phase with frequency; domain, aiming towards l i n e a r dispers ion i n the frequency domain as opposed t o the t ime domain.

Phase s h i f t s on passing through DDLl, DDL2 are respec t ive ly character ised as follows:

design i s car r ied out i n the frequency

Frequency w o s c i l l a t o r VCO t o the two mixers. The frequency range of w is chosen so t h a t the f i l ters DDLl, DDL2, F pass oneC complete frequency sideband without frequency shift.Then the t o t a l phase s h i f t across DDLl and DDL2 is:

is supplied from the voltage cont ro l led

2 $T - +1 + +* = olo + +20 + alw + a2Cw+wc) + blw

+ b2(w+wC)* + c1w3 + c2(w+wc) 3

In order to achieve the l i n e a r dispers ion which we seek it is necessary t o set c2 - -c1:

2 OT * +lo + $20 + a2wc + b2wc - c p C 3 +

n

w(al + a2 + 2b2wc - 3c1wc') +

w 2 (bl + b2 - 3c1wC)

The d ispers ive delay t(v) given by dgT/dw is:

'I = al + a2 + 2b2wc - 3c1wc2 + 2w(bl + b2 - 3ClWC)

+,(w+wC) - #20 + a2(W+VC) + b2Cw+vc)2 +

3

The dispers ion rate 6 given by dr/dw is:

= 2bl + 2b2 - 6c1q ( 2 ) C2(W4=)

(4)

I976 Ultrasonics symposium Proceedings,

IEEE Cat. g 76 ca1120-5sn 424

C 0 Newton & E G S Paige

Thus we have a new 'component', a component whose l i n e a r d i spers ion rate is l i n e a r l y cont ro l led by the frequency of a s i n g l e ex terna l o s c i l l a t o r . onent i s f u l l y asynchronous. the prefer red condi t ion of amplitude s a t u r a t i o n a t the o s c i l l a t o r port . i s given by equation ( 5 ) ;

The COW- The mixers operate i n

The v a r i a t i o n i n signal delay time can i n p r i n c i p l e be

removed by var iab le delay ti) ; The d ispers ive f i l t e r s DDL1, DDL2 have been character- i sed above i n the frequency domain; i n a surface wave implementation, the f i l t e r i s of necess i ty constructed by reference t o i ts behaviour i n the time domain. The SAW f i l t e r design is obtained by Fourier transforma- t i o n of the frequency c h a r a c t e r i s t i c . Truncation i n t i m e is necessary, introducing response imperfection i n t o the frequency domain. The e f f e c t of t h i s w i l l become apparent i n the simulation.

Parameter Constraints f o r Linear Chirp

Reciprocity ensures t h a t f i l t e r a c t i o n i s i d e n t i c a l f o r s igna ls en ter ing a t e i t h e r p o r t . However i t i s advantageous t o consider the system of Fig 1 a s a l i n e a r chirp generator f o r an impulse en ter ing on the l e f t hand s i d e as indicated. The requirement f o r s i n g l e sideband f i l t e r i n g ac t ion has already been mentioned. Correct parameter choice ensures t h a t the bandwidth of the output chirp i s exac t ly the band- width A of DDLl; thus var iab le d ispers ion i s achieved by varying the s igna l t i m e durat ion a t f ixed bandwidth.

var ia t ion , w e can charac te r i se the o v e r a l l f i l t e r by i t s r a t i o of maximum t o minimum dispers ive s lope R, with R - T ~ ~ / fmin.

Further cons t ra in ts a r e imposed by SAW device e ra t ions . F i r s t l y , SAW f i l t e r bandwidths a r e l imi ted t o around an octave due t o bulk wave generation by surface wave transducers; the l a r g e s t acceptable d ispers ive delay l i n e frequency range i s here chara- c t e r i s e d by the c o e f f i c i e n t F where F = f /f *

F S&onde: min' S b device manufacturing circumstances s e t a maximum, Dm, on the t i m e bandwidth product which can reasonably be achieved f o r individual delay l ines . shown how optimum s y s t e m parameters may be deduced f o r given values of A, R, Fm, Dm.

We make the a r b i t r a r y choice b2 * -b so t h a t the dispers ion r a t e i s t o the frequency of the vol tage cont ro l led o s c i l l a t o r . Taking the frequency l i m i t s of DDLl as f and taking the frequency range of the VCO as f s t o Rfs, t h e above cons t ra in ts requi re :

With T~~ and T~~~ a s t h e extremes of time

consid-

w i l l l a t e r be given the vayue 2 .

It i s now

i n equation ( 6 ) d i r e c t l y proport ional 1

and f x + A X

f s = A/2 (7)

f x = {(R+l)/(Fm-l) -11 A / 2 ( 8)

Put t ing Fm = 2, we obtain:

f x = M/Z

I n designing cubic phase law d ispers ion f i l t e r s it i s essential t o keep i n mind which sec t ion of the chara- c t e r i s t i c i s being employed. the quadra t ic curve for group delay aga ins t frequency (Fig 2).

We focus a t t e n t i o n on

3 $(U) = + a w + b w2 + clw 1 1

2 ~ ( w ) = al + 2b w + 3c w 1 1

UELflY I I TIME

'I

"I i i I 3 0

FREQUENCY w

Fig. 2 QUADRATIC DELRY FUNCTIBN

Referring t o Fig 2 , i t is physical ly allowable t o construct devices which operate over a frequency range w1 t o w3 spanning the frequency w2 a t which the group

delay has a turning poin t ; t h i s case i s discussed l a t e r . l i m i t e d t o the range w1 t o w

Then i t can be shown t h a t the cubic phase law para- meters a r e given by:

Meanwhile w e take operat ion f o r DDL2 t o be

2'

bl = 41(ii(R+1)L).Dm/AL (12)

(13) 2 3 3 c = -2/(3n (R+1) ).Dm/A 1

The l i n e a r output ch i rp has a dispers ion time var ia- b i l i t y from

3 ( 1 4 ) 8/ (R+1) '. Dm/A t o 8R/ (R+1) . Dm/A

and the time bandwidth product v a r i a b i l i t y i s from

(15) ~ / ( R + I ) ~ . D , t o ~ R / ( R + I ) 3 . D ~

We take as a convenient lower l i m i t t o time bandwidth product a value DF which provides a useful processing gain and which avoids the m r e extreme departures from l i n e a r i t y associated with Fresnel r i p p l e i n t h e t rans- formation between time and frequency domains. Then the l a r g e s t value of the range f o r dispers ion var ia- b i l i t y R which is achieved above is:

R = 2?J(Dm/Df) -1 (16)

= H) and s e t t i n g the SAW device construc- Put t ing D t i o n limi! Dm a r b i t r a r i l y a t 2000, the l a r g e s t achievable f a c t o r f o r dispers ion v a r i a b i l i t y i s given as R = 5.8. of Fig 2 and operat ing between equivalent frequencies w1 and w , w e can obta in R = 10 as w i l l be shown l a t e r . Designs 3based on equations (12), (13) require oper- a t i o n over the f u l l frequency range wl, t o w2 without

any margin between sidebands. follows t h i s s i t u a t i o n is avoided by s e t t i n g the working bandwidth t o be 75% of A.

By spanning the peak of the quadrat ic

In the example which

425

C 0 Newton & E G S Paige omponent Parts

Table 1

PARAMETER VALUES

Overall Filter

I DDLl Voltage Controlled Oscillator

Centre Frequency Bandwidth Dispersion Time Time Bandwidth Product Signal Delay at Centre Frequency

60 30

13.3 400

15.1

DEVICE SIMULATION

Choice of Parameters

The aim is a fairly modest implementation for an octave variation in dispersion (R = 2). The dis- persive bandwidth is chosen as 30 MHz; in order to give an acceptable margin between sidebands, A is set as 40 MHz. b = - 7 . 1 -14 sec -1 and c = 3.1610-23 sec-'. The range for the voltage controlled oscillator is 20 to 40 MHz; the corresponding range in dispersion time is 4.6 to 9 .2 psec and in time bandwidth product 136 to 2 7 2 .

Dispersive Performance

The impulse responses of the pair of delay lines DDLl, DDL2 are shown in Fig 3, with amplitude and instant- aneous frequency plotted against delay time. problems need be anticipated in the physical construc- tion of devices with such impulse responses.

The chirp performance of the total system is shown in Fig 4. uency domain and in the time domain. show the performance of the real filter truncated in time; the filter is thus not perfect in either domain since the corresponding perfect chirp in either domain would have a rectangular amplitude profile and a linear frequency slope. Fig 4 is plotted for the mid-value of dispersion (f = 30 MHz); the performance is not significantly different at the dispersion extremes (fs = 2 0 , 40 MHz).

Table 1 shows the parameter used, with

1 10

No

The output chirp is shown in both the freq- The diagrams

AMPL I TUDE INSTANTANEOUS FREQUENCY

0 5 10 15 20 0 5 10 15 20 T I M E MUSEC

Fig. 3 COMPONENT I M P U L S E RESPONSES

120 MH Z

100

80

bO

40

Pulse Compression Loop Performance

A sensitive and relevant test of the performance of a dispersive filter is its performance in a pulse com- pression loop. chosen the variable chirp filter is considered as the receiving filter; we consider three alternative schemes for generating the expanded signal, namely a variable chirp filter, a "perfect" frequency chirp and a "perfect" time chirp. "perfect" chirp as one with a rectangular amplitude envelope and a quadratic phase characteristic.

In the first instance an expanded signal generated by a variable chirp filter is compressed by the equiva- lent variable chirp filter with sidelobe suppression weighting. The result is shown in Fig 5 . The expanded signal has been amplitude limited in the time domain to simulate the action of a radar trans- mitter value; the variable chirp receiving filter has -40dB Taylor weighting imposed directly onto the amplitudeprofile ofDDL1. All sidelobes in Fig 5 are below -40dB. Thus imperfections introduced by SAW implementation requirements do not compromise -40dB performance for the above parameters. Fig 5 shows the operation at the mid-value of dispersion; -4OdB performance is achieved over the full range of dispersion values.

In the second instance an expanded signal generated as a "perfect" frequency chirp is compressed by a -40dB Taylor weighted variable chirp receiver. In this arrangement the sidelobe performance is superior to that of Fig 5.

In the pulse compression loop

Here we define the

INSTANTANEQUS AMPLITUDE GROUP DELAY FREQUENCY [MHZ) (MUSEC)

U 0 - ' 1 1

1 l o o

15

10

5

0 0 5 10 15 U 0 bO 80

DELAY (MUSECI FREQUENCY (MHZ)

T I M E DOMAIN FREQUENCY D O M A I N

~ i ~ . 4 OVERALL F I L T E R PERFORMANCE

426

c o Newton ik E G s Paige I n the t h i r d instance an expanded s i g n a l generated a s a "perfect" time chirp is compressed by the -40dB Taylor weighted var iab le chirp receiver . shows the c e n t r a l sidelobes of the compressed pulse on an expanded t ime sca le . I n t h i s case -40dB side- lobes are not achieved a t the loves t value of dis- pers ion, but from mid-range t o the upper l i m i t of the var iab le dispers ion -40dB performance i s achieved. This sets a performance l i m i t f o r the parameters of Table 1.

Maximum Dispersion V a r i a b i l i t y

The exanple given above with the parameters of Table 1 has an octave range of d i spers ion v a r i a t i o n (R-2). It is now shown t h a t an order of magnitude v a r i a t i o n (R=10) i s possible . Referring back t o Fig 3, we now operate between equivalent points w1 and w3. SAW implementation there a r e advantages i n construct- ing the device i n two halves, having equal band- widths w2 - w1 and w - w2.

maximum TB Dm.

Fig 6

I n a

Each sec t ion i s of

Then we deduce:

bl = 24/(a(R+1) 2 ) .Dm/P2

-30 1 t

-15 -10 -5 0 5 10 15

OB MUSEC

Fig. 5 ~ ' u - S E COi.'P;>iSSIClN L O C P

T X V A R I S E L E C H I R P R X VARIRBLE CHIRP

L

VCD 20 MHZ

-20

-30

- U 0

-50

-60

-70 1 c -U -3 -2 - 1 0 1 2 3 U

DB MUSEC

Fig. 6 PULSE COMPRESSION L O O P

T X T I M E C W I R P R X V ~ ~ R I A B L E C H I R P

(18)

The l i n e a r output chirp has a dispers ion time var ia- b i l i t y from

3 64/(R+1I3.Dm/A t o 64R/(R+l) .Dm/A

and the time bandwidth product v a r i a b i l i t y is from

6 4 / ( ~ + 1 ) ~ . D ~ t o 64R/(R+1) 3 .Dm

Thus R = 4?j(Dm/DF) -1 (21)

Put t ing Dm = 2000 and DF = 50 as before , we achieve

R - 12.7. This demonstrates a very s u b s t a n t i a l dispers ion v a r i a b i l i t y f o r a p a r t i c u l a r scheme of parameter choice.

Variat ion i n Pulse Delay Time

From equation (5) it i s apparent t h a t var iab le delay time accompanies var iab le dispers ion i n t h e config- ura t ion shown. For a pu lse compression loop, there i s automatic cance l la t ion of the delay v a r i a t i o n when compressor and expander have b 1, c1 equal and opposi te i n value. Thus a pulse compression loop with var iab le d ispersers o f f e r s pulse j i t t e r as wel l as var iab le s lope, with automatic compensation t o provide a c e n t r a l compressed pulse.

CONCLUSION

This paper has demonstrated a design f o r a d i spers ive f i l t e r whose l i n e a r t i m e frequency s lope i s l i n e a r l y control led by a s i n g l e vol tage control led o s c i l l a t o r . Analysis has shown t h a t an octave range of disper- s i v e s lopes can be achieved i n a modest implementa- t i o n and a decade range without unreasonable opt- i m i s m . components; nonlinear surface acoust ic wave d ispers ive delay l i n e s of moderate time bandwidth product a r e by now well t e s t e d devices; the mixers operate a t t h e i r sa tura t ion amplitude t o the osc i l - l a t o r por t . d i spers ive slope i s re la ted l i n e a r l y t o the o s c i l l - a t o r frequency. We have shown t h a t f o r pulse com- pression purposes amplitude weighting can be b u i l t d i r e c t l y i n t o the f i l t e r design, t h a t -40dB side- lobes a r e achievable and t h a t a pulse compression loop comprised of two var iab le delay l i n e s o f f e r s a pulse j i t t e r which i s automatically cancelled a t the receiver .

Further design opt ions a r e ava i lab le . of b o ther than b2 = -b

e l e c t r o n i c a l l y between p o s i t i v e and negative chirps . For d ispers ive delay l i n e s with phase c h a r a c t e r i s t i c s o ther than cubic i t i s possible

t o generate a v a r i e t y of nonlinear chirps and t o transform one waveform i n t o another. Emphasis i n t h i s paper has been concentrated on l i n e a r chirp and on a p a r t i c u l a r scheme of parameter choice i n order t o demonstrate the working p r i n c i p l e s of the e lec t ronica l ly var iab le d ispers ive f i l t e r .

The design scheme uses orthodox and simple

Operation is asynchronous and the

For choices it is poss ib le t o swing 1

REFERENCES

1. C 0 Neuton "Nonlinear Chirp Radar Signal Waveforms f o r Surface Acoustic Wave Pulse Compression F i l t e r s " Wave Electronics 1, pp 387-401 (1974-76)

2. J Burnsweig, W T Gosser and S H Arneson %lectronically Control lable Time Delay" Presented at IEEE G-MTT I n t . Symp. Boulder, Colorado 1973

427