Problems of Information Transmission, Vol. 39, No. 2, 2003, pp. 216–226. Translated from Problemy Peredachi Informatsii, No. 2, 2003, pp. 63–74.
Original Russian Text Copyright
2003 by Malozemov, Tsvetkov.
METHODS OF SIGNAL PROCESSING
On Optimal Signal–Filter Pairs
V. N. Malozemov
and K. Yu. Tsvetkov
Math. and Mech. Dept., St. Petersburg University
Mozhaisky Military Space Engineering University, St. Petersburg
Received September 16, 2002; in ﬁnal form, February 27, 2003
Abstract—We extend the notion of an optimal signal–ﬁlter pair to the case where n-con-
volution is used instead of cyclic convolution. We construct n-optimal n-adic signals for all
n =2, 3,....
Among discrete periodic signals of ﬁxed energy, one which corresponds to a side lobe ﬁlter (SLF)
with minimum (in energy) impulse response is considered to be optimal. An optimal signal together
with an optimal SLF form an optimal signal–ﬁlter pair.
If there are no restrictions on signal samples, an optimal signal–ﬁlter pair can be characterized
as follows : the optimal signal should be delta-correlated and the corresponding SLF should be
In the present paper, the notion of an optimal pair is extended to the case where n-convolution
[2,3] is used instead of cyclic convolution. We deﬁne delta-n-correlated signals and matched n-ﬁlters
and prove that, in an n-optimal pair, the signal should be delta-n-correlated and the n-SLF should
be matched. Based on the Vilenkin–Chrestenson functions and Frank construction, we construct
delta-n-correlated n-adic signals of periods N = n
for all n =2, 3,.... The latter result is of
signiﬁcance: note, for instance, that among binary signals of period N>4therearenodelta-
correlated ones  (but there exist delta-2-correlated signals of periods N =2
2.1. Denote by C
the set of complex-valued N -periodic integer-argument functions x = x(j),
j ∈ Z.Bysignals,wecallelementsofthisset. InC
, operations of multiplication by a complex
number and addition and multiplication of two signals are introduced in a standard way, namely,
y = cx ⇐⇒ y(j)=cx(j),j∈ Z,
y = x
y = x
becomes a commutative algebra with unity. The unity is signal
all of whose samples
are equal to one.
The signal x
inverse to x is deﬁned by the condition xx
. Clearly, a signal x has the
inverse (is invertible) if and only if all of its samples x(j) are nonzero. In this case, x
j ∈ Z.
Along with a signal x, signals Re x,Imx,¯x,and|x| are considered. They are deﬁned compo-
nentwise. For example, |x|(j)=|x(j)|, j ∈ Z. Clearly, x¯x = |x|
Supported in part by the Russian Foundation for Basic Research, project no. 01-01-00231.
2003 MAIK “Nauka/Interperiodica”