I zyxwvutsrqponmlkjihgfedcbaZYXW I 1 zyxwvuts p(km) zyxwvutsrqpon Fig. 4. Signal-to-noise ratio; lateral waves near air-sea boundary. possible to enhance the unit dipole moment zyxwvutsrqp (/Ad= 1 A . m) to a more practical value, then useful values of zyxwvutsrq S/N will result. A typical lateral-wave system could consist of /Ad= 100 X loo0 = zyxwvu IO5 A . m. The curves for ElP in Fig. 2 would, in this case, be raised 100 dB. This result is shown in Fig. 3. Thus at a frequency as high as 1 MHz, useful zyxwvutsrqponm S/N ratios would result. The resulting signal-to-noiseratios have been plotted as func- tions oftheradialrange zyxwvutsrqp p in Fig. 4. Even more optimistic values would obtain at f = lo7 Hz (not shown) since there is an optimum at that frequency for the lateral wave. V. CONCLUSION Useful signal-to-noise ratios are obtainable with lateral waves between submerged horizontal dipoles for an air-sea model. These results infer useful communication to submarines using lateral waves limited by attenuated atmospheric noise, for shallow depths. For greater depths, lower frequencies should give more optimum signal-to-noise ratios; this will be reported on in a future paper. REFERENCES [l] R. W. P. King and M. F. Brown, “Lateral electromagnetic waves along plane boundaries: A summarizing approach,” Proc. I€€€, vol. [2] T. T. Wu and R. W. P. King, “Lateral waves: A new formula and 72. no. 5, pp. 595-611, May 1984. interference patterns,” Radio Sci., vol. 17, pp. 521-531, May-June 1982. [3] CClR Report No. 322, “World distribution and characteristics of atmospheric radio noise,” Documents of the Xth Plenary Assembly, [4] J. T. deBettencourt, “Theory of attenuation of atmospheric noise Geneva, 1963. Geneva, Switzerland, ITU, 1964, Figs. 19a. b. and interference by the overburden,” in J. T. deBettencourt and AFCRL Contract AF19(604)-8359. Raytheon Co., Oct. 1962, appen- R. A. Sutcliffe, “Studies in deep strata communications,” Final Rep., dix G. A Simple Deductive Proof of a Stability Test for Two-Dimensional Digital Filters P. K. RAJAN AND H. C. REDDY A simple proof based on a continuity argument is proposed for d theorem on the stability tests of two-dimensional (2-0) digital filters, independently proposed by Strintzis and by DeCarlo, Murray, and Saeks. This work was supported in part by the National Science Foundation Manuscript received November 23, 1963; revised February 24, zyxwvuts 1984. under Grant ECS-8307541 and in part under a Tennessee Technological University Faculty Research Grant. nessee Technological University, Cookeville, TN 38505, USA. The authors are with the Department of Electrical Engineering, Ten- I. INTRODUCTION To test the stability of two-dimensional (2-D) digital filters a powerful theorem [ I ] has been proposed independently by Strintzis [2] and by DeCarlo, Murray, and Saeks [3]. Strintzis proved the theorem using Cauchy’s principal value formula and DeCarlo et a/. proved it using homotopy arguments. In this letter, we prove this theorem using a continuity argument, which we believe is simpler than the earlier proofs. First, we state the continuity argument in a form suitable for this application. II. CONTINUITY ARGUMENT Let B(z,,z,) be a two-variable polynomial of degree M in z, and N in z2. Let a be a point in 2,-plane. Then, B(u,z~) is a one-vari- able polynomial in z2 and has in general N zeros in the finite region of the 2,-plane. If zyxw a is such that some coefficients of B(a,z,) vanish and B(a,z,) becomes a polynomial of degree less than N, say N,, then we can treat the disappeared N - N1 zeros to be at infinity (the infinite distant points of the z,-plane are as- sumed to converge to a single point at infinity) canceling with an equal number of poles there so that the resulting polynomial has N - N1 poles at infinity. For the discussion below, we ignore the cancellatlon in such cases and assume that B(a,r,) has N, zeros in the finite region, N - N1 zeros at infinity, and N poles at infinity. Consider a zero 2, of 6(a,z2), When a is moved on a continuous line in the 2,-plane, 2, either moves continuously or remains stationary. The latter case occurs when a factor of the form (z, - i2) independent of z1 is present in 6(z1,z2). Further, when (k d N) zeros are present at a point among which k’ zerosare stationary, there will be (k - k’) incoming and (k - k’) outgoing loci at that point. We may associate an incoming locus with an outgoing locus arbitrarily on a one-to-one basis. With these remarks, we can state the continuity argument as follows: The locus of a zero 2, of B(u,z,) generated by the movement of a on a continuous line in the 2,-plane is a continuous line or a fixed point. The proof of this argument can be given based on the properties of algebraic func- tions discussed by Bliss [4]. Similar argument also applies to the loci of the zeros of B(z,, a) as a is moved on a continuous line in the 2,-plane. 111. THE THEOREM AND A SIMPLE PROOF For i = 1,2 let q = u, + T, = (2;l l2;l d I}. The symbol ”+” is used to denote the set theoretic union and the symbol “X” is used to denote the set theoretic product. Theorem [Z], [3]: Let B(.z~,z~) be a polynomial in z1 and 2,. Then 6(2,,z2) # 0, V(z,,z,) E a, X zyxw G if and only if i) B(z,,zz) f 0, V(zl,z2) E T, X T2 (1 1 ii) B(a,.?,) # 0, for some a,la( = 1 and Vz, E 4 (2) iii) B(z,,b) # 0, forsome b,lb( d 1 and VZ, E U,. (3) Proof: The ngessity part is obvious as the specified regions are subsets of u, X 4. We will prove the sufficiency part. a) i) implies B(z z has no zero sets in T, X T,. b) In ii) as la1 = 1, a-ls a point on T,, so i) and ii) jointly imply B(~,z,) has no zeros in &. Now, consider any other point a‘ such that a’ E T,; a can be connected to a’ by a continuous line ad’ such that all the points of ad’ are in Tl (Fig. 1). Let a” be a point on ad‘. The locus of any zero of B(~”,z,) as a” is moved from a to a’ cannot enter & without crossing T2. However, no zero can be present on T, so long as a” is on (condition i)). As this is true for all a’ on T,, we have l’ ,A . 6(z,,z2) f 0, V(zl,z2) E TI X G. (4) c) -Now consider iii) in conjunction with i) and ii). Let at b E &, 6(z1, b) # 0, Vz, E-Ulii.e., B(z,, b) has no zeros i n U,. Now consider a point b’ in Q; b can be connected t_o b ’ by a continuous line btf such that all the points of btf are in 4 (Fig. 2). Let b“ be a point on btf. As all the zeros of B(z~, b) are outside the 001 8-921 9/&1/09iX)-1221 M .oO sl984 I EEE PROCEEDINGS OF THE IEEE. VOL. 72, NO. 9, SEPTEMBER 1984 1221