Error Analysis in the Walsh Sampling Theorem

After presenting some results on the Walsh Fourier transform of dyadic continuous functions, a short proof of the Walsh sampling theorem is given. Using the dyadic modulus of continuity esti­ mates of the main errors arising in sampling series approximation are built up, namely: the aliasing error, jitter error, and round-off error. The con­ cept of dyadic (Walsh-) differentiability and Lipschitz classes then leads to the evaluation of rates of convergence. This finally enables us to reconstruct non-sequeney-limited signal functions by a jittered and quantized Walsh sampling series within a given scope of accuracy. PRELIMIHARIES IM WALSH ANALYSIS For each t,u 6 R+ = {x,x>0} Fine's generalized Walsh functions (Ref. 6) are given by *w(t) exp N(to)+1 iri J u, .t. j-N(t) 1_J J where t = 7 t.2~ j=-N(t) J u = y ai.2 -H(oi) is valid. More generally, one proves © Lemma 1. Any f £ C can be represented by 2-n f(t) = lim 2n / f (t ©u)du (2) n-*» o uniformly in t £R+ . Indeed, one has 2-n |f(t) -2n / f(t ©u)du| O 2 ~ n < 2n / |f(t) -f(t©u)|du o < sup I f ( • ) f (• ©u)ll , 0<u<2~n © which tends to zero for n-*" since f £ C . In order to deduce the inversion formula, Eq.1, recall that the 2n-th partial inversion integral is given by 2n / F(oi)i|i (t)doi = f * J(2n ;.)(t), (3) are the dyadic expansions of t and oi; N(t) £ 2 : = {0,±1,±2, ...} is the largest integer j with t_.;s£:0 the same for H(oi). For functions f€L, ** + namely those that are absolutely integrable on IR in Lebesgue's sense, the+Walsh Fourier transform is defined for each o>£R by F(oi) = / f(t)if) (t )dt. If furthermore f belongs to C , the class of boun­ ded and uniformly dyadic continuous functions, i.e., functions which satisfy I f(» © h) f(») I ■+ 0 (h ->0+), where I fll = I f (• )I is the usual sup-norm and © de­ notes dyadic (termwise modulo 2) addition, then the Walsh Fourier inversion formula f(t) = f F(u)i|) (t)dd i. Ii\ (1) where P J(p ;t) := / >l> (t) du o is the Walsh Dirichlet kernel, and where the con­ volution f * g of two functions f, g£ L is defined by OO f*g(t) := / f(u)g(t ©u)du. o From the particular numerical values J(2 ;u) = 2n , u£[ 0,2 n) W 0, elsewhere, one easily verifies that 2n 2_n / F(u)il) (t)du = 2n / f (t ©u)du, ‘ oi 1 o o so that Eq. 1 follows directly from Eq. 2 in case CH1538-8/80/0000-0366$00.75 © 1980 IEEE 366 © f £ LDC . Note that Eqs. 1-2 are still valid at each point of dyadic continuity if f does not be­ long to C® (see also Refs. kr5)• Even a result in­ verse to that of Lemma 1 holds: Lemma 2. If Eq. 2 holds uniformly in t £IR+ for an essentially bounded function f, then f belongs to Proof. If u£[2 nk,2 n(k+l)) for some k £ P = {0 ,1 ,2 ,...}, then u © h belongs to the same interval if h£[0,2_n). Therefore, one deduces s e q u e n c y 2 , n £ £ , t h e n