A curiosum concerning discrete time convolution

It is shown that the discrete time convolution of two absolutely summable nowhere zero sequences may be identically equal to zero. DEVELOPMENT Consider two absolutely summable sequences a = {a,,: n E Z } and b = { b,,: n E Z } of real numbers. Further, assume that the sequences a and b are nowhere zero. Does it follow that the discrete time convolution a * b is nowhere zero? Does it follow that a * b is nonzero on some nonempty subset of Z? From a linear systems viewpoint, does an absolutely summable nowhere zero input to a discrete time linear time-invariant system described via discrete time convolution with a fixed absolutely summable nowhere zero sequence result in an output which is nonzero somewhere? 'The following development, inspired by [I . pp. 354-3561. addresses these questions. To begin, we will use the following notation. For an absolutely summable sequence of real numbers a = { a,,: n E Z 1, let T, map absolutely summable sequences of real numbers into absolutely sumrnable sequences of real numbers via where a = {a, , : n E Z } is any absolutely summable sequence of real numbers. For any two absolutely summable sequences of real Manuscript received May 25. 1989; revised July 20. 1989. This work was supported in part by the Office of Naval Research under Grant NOOO1490-J17 12. and in pan by the Air Force Office of kientific Research under Grant AFOSR-86-0026. E. B. Hall is with the Department of Electrical Engineering. Southern Methodist University. Dallas. TX 75275. G. L. Wise is with the Department of Statistics. University of California. Berkeley. CA 94720. IEEE Log Number 9034983. 0096-35 18/90/06001058$01 .OO O 1990 IEEE lEEE TRANSACTIONS ON ACOUSTICS. SPEECH. AND SIGNAL PROCESSING. VOL. 38. NO. 6. JUNE I W ) 1 059 numbers a = ( a , , : n E Z } and f l = { fl,,: t l E Z ). it follows that I m where we define X, = E , apflq. Finally, note that for any two absolutely summable sequences of real numbers a and b . i t follows via Fubini's theorem that T , ( a ) * T p ( b ) = (T,, 0 T @ ) ( N * 6). Theorem I : Letting the above paragraph set notation. there exist two nonidentically zero absolutely summable sequences of real numbers a and fl such that for any absolutely summable sequences of real numbers a and b , T , ( u ) * T @ ( b ) = 0 . Proof: Recall that the function I cos(x) I is expressible as a Fourier series given by C,",-, c , exp (inx) where i denotes the imaginary unit and where it follotvs easily that c,, = 0 if n is odd and c,, = ( 2 / + ) [ ( 1 ) " I 2 / ( 1 n 2 ) 1 if n is even. Further, if we define fi (x) = ( I COS (x ) I + cos (x)) and