IEEE TRANSACTIONS ON SONICS AND ULTRASONICS, VOL. SU-21, NO. 1, JANUARY 1974 7

A Numerical Fourier Transform Technique and Its Application to Acoustic-Surface- Wave

Bandpass Filter Synthesis and Design CARMINE F. VASILE

Abstract-This paper describes a numerical Fourier trans- form technique that leads to a closed form synthesis procedure for acoustic surface wave bandpass filters. The procedure is direct, requiring no computer iteration process, and leads to an apodization function of finite extent, thereby eliminating truncation errors. A numerical example is given wherein the interdigital electrode apodization function is derived for a bandpass filter having a 22% flat bandwidth and 4% transition width. Sources of distortion and an alternative electrode design are discussed.

Y S Io 'S'

Fig. 1 . Symmetrical bandpass function.

INTRODUCTION L,', T i

This paper describes a numerical Fourier transform tech- nique that leads to a closed form synthesis procedure for acoustic surface wave bandpass filters. The procedure is direct, requiring no computer iteration process, and leads to an apodization function of finite extent, thereby eliminating / \ \

truncation errors. f

/I \ ,

// \ \

2 1 ~ = T T

L e T T

TRANSFORM TECHNIQUE Fig. 2. Eigenfunctions.

The bandpass characteristics illustrated in Fig. 1 will be sufficient to illustrate the technique. H m will be approxi- their respective 6 dB points as illustrated with the dashed mated by triangles in Fig. 2. This requires

N fn + 1 - f n = 2 4 T . (4) '(0 = C ~ C f r z ) ~ C f - f n , 73, f > 0. ( 1 )

n = I For the bandpass characteristic of Fig. 1 we choose to span

Such that B wi thN translated eigenfunctions. It, therefore, follows that

Equations (2) and (3) will be satisfied if H(f , ) , E and f, are chosen properly.

that is normalized to unity at f = 0 and is nearly triangular in shape with its half amplitude points occurring at I f l = a/T. It is required to have a sufficiently small amplitude for I f 1 Z 2a lT as illustrated in Fig. 2. The parameters T and

CY will be defined below. The sampling points or eigenfre- quencies fn are chosen to be equally spaced at intervals Af = 2a/Tsuch that the adjacent eigenfunctions cross at

The function E(f , 73 is defined as a frequency eigenfunction

Manuscript received March 9, 1973; revised May 24, 1973. The author was with Motorola, Inc., Scottsdale, Ariz. He is now

with the Research Laboratories, Hazeltine Corporation, Greenlawn, N.Y. 11740.

where

N 2 B/S + 1 (6 )

and

2a /T= B/(N - 1). (7 )

Thus once the bandwidth B, the transition width S, and the current frequencyf, are specified, (6) gives the number of sampling points required, (7) gives the ratio a / T and ( 5 ) gives the equally spaced sampling frequenciesf, . The procedure outlined is equivalent to connecting the sample points H(&) by a series of chords within the passband B and using the natural roll off of the two eigenfunctions centered at the band edges to satisfy the rejection requirements. Clearly the eigen-

8 IEEE TRANSACTIONS ON SONlCS AND ULTRASONICS, JANUARY 1974

functions are required to have low sidelobes if a high degree of rejection and a small in-band ripple is required.

I f E were a perfect triangle, its inverse Fourier transform would be infinite in extent. This is undesirable for it is re- quired to deal with finite duration time functions in order to eliminate truncation errors. It might seem plausible to select a sin x/x function for E. However, the large sidelobes of this function leads to a large in-band ripple and poor off- band rejection. Schroeder [ l ] gives a number of antenna distributions and their corresponding Fourier transforms [2]. The combination cosine squared weighting function, whose spectrum has the lowest sidelobes, appears to be best suited for our purpose. It is defined by:

0.44 + 0.5 cos 2nt/T + 0.07 cos4nt/T, I tl< TI2 W(t, T) =

It1 > T/2.

Fig. 3 (a) is a plot of W(t, T). The Fourier transform of W(t, T) is given in closed form by:

sin (x - n) sin (x + n) + x - 71 X + n

sin (x - 2n) sin (x + 2n) + x + 2n Fig. 3. Combination cosinesquared weighting and its Fourier

transform. where x = nf T. Letting E(f, T) = W ( t , 7')/0.44 T, it can be seen from Fig. 3 (b), where E ( f , T) is plotted on a linear and dB scale, that E(f, T) is a very good approximation to the desired triangular spectrum if we set (Y = 1 .OS, so that adjacent eigenfunctions cross at their - 6dB points. For I f 1 = 2.1/T E(f, T) is down 24 dB and for I f 1 > 2.5/T it is below 40 dB. The sidelobes rise to a maximum of 64 dB nearf= 7.5/T.

Restriction to bandpass characteristics such that f= 0 lies in the stop band, negative frequency terms can be neglected and it follows from (1)-(9) that the inverse Fourier transform of A ( f ) is given simply by:

3-1 A ( f ) = c H(fn)cos(2nf,t) (10) N

0.22 T ,=, with

2.1 B T 2

f, =fo +-(n - 1)- -

In cases where the negative frequency terms are not negli- gible the exact transform pair is defined by (10) and:

N

A(f)= C H ( f n ) E ( f - f n , T)+H(- fn )E( f+fn ,T) . n=1

(1 1)

It is obvious that a corresponding transform pair also exists:

N

where t, is analogous to f, and F is analogous to T. Here we represent a time function g(t) by a series of chords which are expressed by sums of translated eigenfunctions E(t - t,, F). Thus the Fourier transform ofg(t), so approximated, is given exactly by the band limited function (1 2b).

It may be noted that the summation in (12b) is merely the discrete Fourier transform and that of (10) the discrete inverse Fourier transform [3]. The use of a "triangular" sampling function instead of isolated samples allows the approxima- tion of a function by a series of chords without introducing a periodicity into the corresponding transform. If E were a perfect triangle in (1 2a), equatinn (1 2b) would become;

C G(t,) (13) n= 1

where A t = tn+ - t, is less than or equal to the Nyquist sampling rate used in obtaining the fast Fourier transform [3] . Equation (13) is no longer a band limited function as is (12b), nor is it periodic as is the discrete Fourier transform.

SURFACE WAVE BANDPASS FILTER SYNTHESIS If H( f) is required to be symmetrical about fo, (1) can

be rewritten as:

+M

For an even number of N samples

M=N/2 , fn=f ,+- - - (nFO.5) , n > O f n . (14b) B

N - 1

VASILE: NUMERICAL FOURIER TRANSFORM TECHNIQUE 9

For N odd:

M = ( N - 1)/2, f, =fo + n B / ( N - 1). (1 4c)

Employing the identity cos (x y ) = cos x cosy sin x sin y , equation (10) simplifies to:

if H(' has even symmetry about fo as in Fig. 1, or to:

if H(' has odd symmetry about fo . Since any function may be written as the sum of an even

and odd function, (14) and (15) may be used in place of (1) and (1 0). The importance of (15) is that is shows that the inverse transform ofA(f) is given simply by a carrier term amplitude modulated by an apodization function. For the even symmetry case, whbh is of interest here:

W( t , T) O . l l T

Ae(t) = ___ 2 H ( f n ) cos 2n(f, - fo)t, N even (16a)

or

N odd (16b)

Ao(t) in 15(b) is given by 16 with cos 2n(f, - fo)t replaced by -sin 2n(fn - f o ) t .

with uniformly spaced electrodes having equal widths and spaces, can be used to excite acoustic surface waves whose impulse response h(t) is of the form (15). Since the corre- sponding Fourier transform of h(t) is given by (l), the frequency response of such a transducer will be'the desired bandpass characteristic given by (1).

The preceding theory leads to a closed form expression for the apodization required to realize a prescribed surface wave filter characteristic. The value of this approach may best be judged with the aid of the numerical example to be discussed below.

Fig. 4 illustrates the simplest interdigital electrode pattern that may be used to realize a surface wave filter characteristic. Using an unapodized transducer in combination with an apodized transducer minimizes diffraction effects and the overall transfer function is given simply by the product of the Fourier transforms of the impulse response of each transducer. The unapodized transducer consisting ofN, electrode pairs introduces a frequency dependence of the form:

It is well known that apodized interdigital electrode patterns,

Hu(f) = sin xix, x = @Tu - N u ) (17)

where Nu =fo Tu and Tu is the physical length of the Nu pairs divided the surface wave velocity.

An important feature of the synthesis procedure described above is that a flat (or shaped) overall transfer function is realizable because it is possible to shape the frequency response

Fig. 4. Typical acoustic surface wave filter configuration.

of the apodized structure by adjusting the apodization func- tion in a very simple manner, i.e. by varying the coefficients H&) in (1 6). Arbitrarily let:

Nu = 0.885 f o / B (18)

so that the unapodized response is 3 dB down at the band edges. Two cases will be considered. In the first, the apodized transducer has a flat bandpass that leads to an overall response having a 3-dB bandwidth equal to B. In the second, the apodization is adjusted to flatten the overall response.

NUMERICAL EXAMPLE The filter parameters t o be used in this example are similar

to those used by Tancrell [4] and are symmetrical about fo.

fo = 60 MHz

B=0.22f , = 13.2 MHz

S = 0.04 fo = 2.4 MHz.

From (6), N > B/S + 1 = 6.5, thus N = 7 eigenfunctions are required in the synthesis. From (7),

T=2(1.05)(N- 1)/B=O.955 p.

Hence there are N A = fo T = 57 carrier cycles required in the apodized transducer. A conventional interdigital design would therefore require 57 finger pairs. From (18), the unapodized transducer must have Nu = 4 finger pairs. The unapodized transducer is therefore Tu = 4/60 = 0.0667-ps long. The eigenfrequencies are determined from (14c) as:

ffl=60+---B=60+2.2n, n = - 3 t o + 3 (19) n

N - l

For a flat overall response the end transducer response is sampled at the 7 eigenfrequencies f, . The 7 samples Q, as obtained from (17) are:

Q ,=s inx , /x , , x ,=n[ fnTu-Nu] (20)

withf, given by (19),Nu = 4 , and Tu = 0.0667 ps. From (16b) the required apodization is:

A&) = [ 1 + 2 cos 4.4nt W(t , 0.955)

0.22 T

+ 2 cos 8.87rt + 2 cos 13.27rtI (21)

for the first case and

W ( t , 0.955) Ae(t) = 0.22 T

[ 1 + 2.06 cos 4.4nt

+ 2.31 cos 8.8nt + 2.83 cos 13.2ntI (22)

10 IEEE TRANSACTIONS ON SONICS AND ULTRASONICS, JANUARY 1974

Fig. 5 . Apodization functions.

for the second case in which the overall frequency response is flat. In using (21) and (22), W ( t , 0.955) is given by (8) with T = 0.955 P S .

The overall filter frequency response corresponding to the preceding apodization functions and the unapodized end transducer is given simply by the product of (17) and (14a) as:

for the first case and

for the second case in which the overall frequency response is flat. In (23) and (24), Cl and C2 are constants, E is given by (9) with T = 0.955 ,f, is given by (1 9), Q,, is given by (20), and x by (17).

ization functions, given by (21) and (22), which have even symmetry about t = 0. It should be noted that there are 7 lobes on each side of 6 = 0 corresponding to the 7 eigenfunc- tions used in the synthesis. The compensated apodization shown in Fig. 5(b) is seen to be very similar to that of the uncompensated design of Fig. S(a). This is to be expected since only a 3 dB roll off is being compensated for.

Fig. 6 is a normalized plot of (24) with and without the sin x/x factor. Figure 7 is a similar plot of (23). The overall

Figs. S (a) and (b) are graphs showing one half of the apod-

Fig. 6. Frequency response of compensated apodized transducer and overall filter response.

filter response shown in Fig. 6 exhibits an in-band ripple on the order of k0.2 dB and remarkably good sidelobes. The phase response of both filters is linear with a slope that cor- responds to a time delay of OS(T + Tu) in addition to a fixed delay corresponding to the physical separation of the elec- trode patterns shown in Fig. 4. It should be noted that the compensated design exhibits a -32 dB rejection at the edge of the transition band. This fall off is slightly slower than that shown by Tancrell. To increase the rate of fall off simply increase the number of eigenfunctions from 7 to 8. The corresponding apodization functions (2 l ) and (22) would then have 8 lobes on each side of the origin as does Tancrell's.

Several comments bearing upon the physical realization of the desirable filter characteristics illustrated in Figs. 6 and 7, are in order. If, for example, a specification calls for a flatness across the passband of kO.5 dB it must be ensured that every spurious signal, including triple transit, contained in the impulse response of the surface wave device is below a given level. The ripple shown in Fig. 6 is about k0.2 dB. Assuming a worst case addition of a single spurious signal, the overall flatness of kO.5 dB implies a time spurious level of less than 25 dB. An overall flatness of k0.3 dB requires greater than 30 dB spurious rejection. The realization of 30 dB rejection is not practical with the simple two-transducer configuration illustrated in Fig. 4 (because of the triple transit signal) and one requires a more sophisticated electrode structure as will be discussed later.

The high off-band rejection level in excess of 60 dB illus- trated in Fig. 6 is also difficult to achieve in practice. To

VASILE: NUMERICAL FOURIER TRANSFORM TECHNIQUE 11

l I l / \\

- 6 7

Fig. 7. Frequency response of uncompensated apodized transduce1 and overall filter response.

approach this level, good bulk wave suppression is required as well as good electrical shielding. These effects will place a limit on off-band rejection close to the passband, while the response of the electrical matching network and harmonic response of the interdigital transducer will contribute to the rejection away from the passband. hleasurements made at Motorola GED indicate 45-50 dB off-band rejection is readily achieved in a single, miniaturized device. It is not possible to set a value on ultimate off-band rejection as it is dependent upon many factors peculiar to a given design.

Thus to approach the characteristics of Fig. 6 one must deal effectively with spurious effects such as bulk wave excitation, electrode reflections, direct electromagnetic feed-through, multiple transit echos, diffraction, and reflections from crystal boundaries. Fortunately, a number of techniques have been developed that deal with all of the above problems. Means are available for the suppression of bulk waves over a large fractional bandwidth [ S ] . Electrode reflections are minimized by the use of the “double electrode transducer” [ 6 ] . Electrical feed-through is minimized by careful shielding and control of ground plane currents. The multistrip coupler [7 ,8] may be used to suppress bulk waves as well as the in- band ripple caused by a triple transit signal. It is also possible to eliminate a triple transit signal by using a three transducer configuration consisting of a center transducer symmetrically excited by two outer transducers [ g ] . Power matching the center electrode suppresses triple transit reflections. By coherently adding the output signals of the two outer trans- ducers, one can also drop the bi-direction loss from 6 to 3 dB. As it is difficult t o include diffraction effects in a synthesis procedure, it is better to initially neglect diffraction. If

necessary, the apodized electrode structure may be modified to compensate for undesirable phase and amplitude variations caused by diffraction. Reflections from crystal boundaries may be eliminated by the introduction of damping by means of surface roughening or surface loading with an absorbing medium.

CONCLUSIONS It has been shown that exceedingly good filter character-

istics can be synthesized in a very simple manner. The syn- thesis procedure is straightforward and does not rely on a computer search or iteration. Only the bandwidth, transition region width and center frequency must be specified. It should be obvious that filters having a shaped response, including SO-dB notches, can be synthesized in an equally simple fashion.

tive filter realization, based upon the mathematics of the synthesis procedure, is possible in cases where it is not desir- able to employ the multilobed apodization illustrated in Fig. 5 . Observing the filter impulse response given by (1 0) is merely a sum of N different carrier terms weighted by the Combination cosine squared apodization, it becomes apparent that the filter can be implemented as a bank of N channels. The impulse response of each channel must be made to cor- respond to each of the N terms in (10) and the output signals must be added coherently. While this realization uses more substrate area, it may be desirable to minimize undesirable diffraction effects or provide a more convenient impedance level. In cases where the fractional bandwidth is high, the channelized implementation provides more carrier cycles per apodization lobe.

Finally, to approach the theoretical response synthesized, spurious effects which are readily observed in poorly designed acoustic surface wave devices must be eliminated.

A point that may not be quite so obvious is that an alterna-

ACKNOWLEDGMENT The author wishes to thank Dr. F. S. Hickernell for reading

the manuscript and making helpful suggestions.

REFERENCES [ l ] K. G. Schroeder, “Beam Patterns For Phased hlonopulse Arrays,”

121 The author wishes to acknowledge John F. Crush of Hazeltine Microwaves, Vol. 2, No. 3, March 1963, pp. 18-27.

[ 3 ] W. T. Cochran et al. , “What is the Fast Fourier Transform?,” Corp. for calling his attention to the microwaves article, Ref. (1).

[4 ] R. H . Tancrell, “Analytic Design of Surface Wave Bandpass Proc. IEEE, Vol. 55, No. 10, pp. 1664-1674, October 1967.

Filters,” 1972 Ultrasonics Symposium Proceedings, IEEE Group on Sonics and Ultrasonics, pp. 215-217.

[S] R. LaRosa and C. F. Vasile, “Broadband Bulkwave Cancellation in Acoustic Surface-Wave Devices,” Electronics Letters, Vol. 8, No. 19, September 21, 1972, pp. 478-479.

[6 ] T. W. Bristol et al., “Double Electrodes in Acoustic Surface Wave Device Design,” 1972 Ultrasonics Symposium Proceedings, IEEE Group on Sonics and Ultrasonics, pp. 343-345.

[7 j F, G. Marshall and E. G, S. Paige, “Novel Acoustic-Surface Wave Directional Coupler with Diverse Applications,” Electron Letters,

[8] F. G. Marshall et al., “New Unidirectional Fansducer and Broad- Vol. 7, pp. 460462,1971.

band Reflector of Acoustic Surface Waves, Electronic Letters, Vol. 7 , pp. 638640, 1979.

[9 ] A. J. DeVries et al., “Characteristics of Surface Wave Interdigital Filters (SWIFS),” IEEE Trans BTR-17, No. 1, February 1971, pp. 16-22.