Page 88 - Electrical Engineering Dictionary

bilinear transformation (1) conformal matical operations for such codes are defined
mapping of the complex plane of the form over the finite or Galois field consisting of
az+b
f(z) = , where the real values a, b, two elements denoted by GF(2). The math-
cz+d
c, d satisfy ad − bc 6= 0. Also called linear ematical operations for such a Galois field
fractional transformation or M¨ obius transfor- are addition and multiplication. For addition
mation. over GF(2) one finds that
(2) a special case of (1) is a mapping from
1 + 1 = 0, 1 + 0 = 1 and 0 + 0 = 0.
the jω axis in the s-plane to the unit circle
2 1−z −1
|z|= 1inthe z- plane, given by x = ,
T 1+z −1 For multiplication over GF(2) one finds that
whereT isthetimeintervalbetweensamples.
Such bilinear transformations are used in 1 · 1 = 1, 1 · 0 = 0 and 0 · 0 = 0.
the design of recursive digital filters from
equivalent analogue filters in the following See also block coding, convolutional coding,
procedure: error control coding.
1. define characteristic digital frequencies
 i . binary-coded decimal (BCD) (1) a
2. prewarp these to analog frequencies ω i us- weighted code using patterns of four bits to
2  i T represent each decimal position of a number.
ing ω i = tan 1 ≤ i ≤ k.
T 2
3. design a suitable analog filter with frequen- (2) decimal digits 0 to 9, encoded by their
four-bit binary representation. Thus: 0 =
cies ω i .
0000, 1 = 0001, 2 = 0010, 3 = 0011, 4 = 0100,
4. use the bilinear transformation to replace
2 1−z −1 5 = 0101, 6 = 0110, 7 = 0111, 8 = 1000, 9 =
s in the analog filter with s = .
T 1+z −1 1001.
bimetal overload device an overload de- binary erase channel a channel where
vice that employs a bimetal strip as the actu- an error detecting circuit is used and the er-
ating element. The bimetal strip consists of roneous data is rejected as erasure asking for
two metals bonded together. When heated, retransmission. The inputs are binary and
the bimetal strip will bend due to the differ- the outputs are ternary, i.e., 0, 1 and erasure.
ent coefficients of linear expansion of the two Used for ARQ (automatic request for retrans-
metals. The bending operates a set of con- mission) type data communication.
tacts that automatically removes the affected
load from the source of electrical power. See binary hypothesis testing a special two-
also overload heater, overload relay. hypothesis case of the M-ary hypothesis
testing problem. The problem is to as-
bimodal histogram a histogram with sess the relative likelihoods of two hypothe-
two main groupings of values, such as the ses H 1 ,H 2 , normally given prior statistics
sum of two displaced Gaussians. See also P(H 1 ), P(H 2 ), and given observations y
histogram. whose dependence p(y|H 1 ), p(y|H 2 ) on the
hypotheses is known. The receiver operat-
binary (1) a signal or other information ing characteristic is an effective means to
item that has two possible states. visualize the possible decision rules. See
(2) representation of quantities in base 2. also m-ary hypothesis testing. See also
conditional statistic, prior statistics, pos-
binary code a code, usually for error con- terior statistics.
trol, in which the fundamental information
symbols which the codewords consist of are
two-valued or binary and these symbols are binary image an image whose pixels can
usually denoted by either “1” or “0,” Mathe- have only two values, 0 or 1 (i.e., “off” or

c
2000 by CRC Press LLC