Signal Processing - January 2017 - 49
fact, calculation is a type of counting. In
many calculations, finding the solutions
of equations is always one of the most
important tasks. Among finding the
solutions of equations, finding the roots
of polynomials is probably the most
important. The fundamental theorem
of algebra tells us that any nonconstant
single-variable polynomial has at least
one complex root, which means that any
single-variable polynomial equation can
be solved with possibly complex numbers as solutions/roots. We know that
roots of a polynomial of a degree lower
than five have closed forms in terms
of the coefficients of the polynomial.
However, for a polynomial of a degree
of five or higher, its roots may not have
closed forms in terms of its coefficients,
which was first mathematically proven
by Galois and is, therefore, called the
Galois theory. To do so, Galois invented
the concepts of group, ring, and field,
which led to modern mathematics. The
smallest field is the binary field {0, 1},
and the largest is the complex field C
that is the set of all complex numbers.
The reason why C is the largest field is
because every polynomial equation over
the complex field can be solved already
by the fundamental theorem of algebra. There are many kinds of subfields
and extended fields, such as algebraic
number fields, by including, e.g., some
roots of unity, i.e., exp (- 2r j/m), for
some positive integer m, in the middle
of {0, 1} and C. After the complex field,
mathematicians generalized C to quaternionic numbers that form, in fact, a
domain as well as octonionic numbers.
For example, a quaternionic number
can be equivalently written as
c
x y
m,
- y) x)
where x and y are two complex numbers. With these generalizations, mathematicians found that the most important
property from all of these structures is
the norm identity
x:y = x : y
the real multiplication, and
stands
for the norm used in the domain. In
other words, the norm of the product of
any two elements is equal to the product
of the norms of the two elements. This
is clear when x and y are two complex
numbers but is less obvious for other
cases. A general design satisfying (1),
as generalizations of complex numbers,
quaternionic numbers, and octonionic
numbers, is called compositions of quadratic forms [1]. A 6k, n, p@ Hermitian
composition formula is
^ x 1 2 + g + x k 2h ^ y 1 2 + g + y n 2h
= z1 2 + g + z p
2
(2)
where | | stands for the absolute value,
X = ^ x 1, f, x k h and Y = ^ y 1, f, y n h
are systems of indeterminates, and
z i = z i (X, Y) is a bilinear form of X and
Y. As an example, let k = n = p = 2 and
z 1 = x 1 y 1 - x 2 y 2, z 2 = x 1 y 2 + x 2 y 1 .
This corresponds to the following case.
The product of the absolute values of
two complex numbers is equal to the
absolute value of the product of the two
complex numbers, i.e., if x = x 1 + jx 2
and y = y 1 + jy 2 for real-valued
x 1, x 2, y 1, y 2 and z = z 1 + jz 2 = xy,
then z = xy = x y . More designs
on the compositions of quadratic forms
can be found in [2], which has found
applications as space-time coding in
wireless communications with multiple
transmit antennas.
With this in mind, I would say that
algebra is with the norm identity, where
you are able to count precisely (the same
as the first apple example mentioned previously), where 2 $ 4 = 8 = 2 $ 4 and
500 + 400 = 500 + 400 , when the
dot sign in (1) is the real multiplication
and the real addition, respectively. This,
in my opinion, corresponds to small data.
Mid data and analysis
In most cases, the norm identity (1) does
not hold. Instead, it is the following
inequality:
(1)
x:y # x : y
for any two elements x and y in the
domain of interest, where the dot stands
for the multiplication in the domain or
(3)
for any two elements x and y in a set
called space. This leads to the concept
IEEE Signal Processing Magazine
|
January 2017
|
of a norm space, i.e., if there is an operation
on a set that satisfies (3) for any
two elements x and y in the set, this set
with some additional scaling property
is called a norm space. It is the key for
functional analysis or analysis, including measure theory and/or probability
theory and statistics. In this case, in (3),
the dot sign is the addition +, and (3)
is correspondingly called the triangular
inequality. In my opinion, the difference between algebra and analysis is the
difference between the norm equality
and the norm inequality shown in (1)
and (3), respectively. It is the same as the
second apple example mentioned previously, where
{400 apples in one family}
, {500 apples in another family}
# {400 apples in one family}
+ {500 apples in another family}
= 400 + 500 = 900,
where the dot sign in (3) corresponds to
the union of two sets and the real addition, respectively. I feel that this corresponds to mid data.
Another observation about the above
norm inequality is that the dot operation in (3) for two elements x and y can
be thought of as a general operation as
we have seen above for different cases
of the dot sign. The norm inequality (3)
becomes the triangular inequality when
the dot is +, as mentioned previously.
When the dot is a true product of two
elements, such as the matrix multiplication of two matrices, the inequality (3)
is the conventional norm inequality. The
norm inequality (3) becomes the Cauchy-Schwarz inequality when the dot is
the inner product
#a b f (t) g (t) dt
# 8#
a
b
f (t) 2 dtB 8 #
1/ 2
a
b
g (t) 2 dtB ,
(4)
1/ 2
where the equality holds if, and only
if, functions f (t) and g (t) are linearly dependent, i.e., f (t) = cg (t) or
g (t) = cf (t) for some constant c. From
this observation, almost all inequalities
can be derived from the norm inequality
49
Table of Contents for the Digital Edition of Signal Processing - January 2017
Signal Processing - January 2017 - Cover1
Signal Processing - January 2017 - Cover2
Signal Processing - January 2017 - 1
Signal Processing - January 2017 - 2
Signal Processing - January 2017 - 3
Signal Processing - January 2017 - 4
Signal Processing - January 2017 - 5
Signal Processing - January 2017 - 6
Signal Processing - January 2017 - 7
Signal Processing - January 2017 - 8
Signal Processing - January 2017 - 9
Signal Processing - January 2017 - 10
Signal Processing - January 2017 - 11
Signal Processing - January 2017 - 12
Signal Processing - January 2017 - 13
Signal Processing - January 2017 - 14
Signal Processing - January 2017 - 15
Signal Processing - January 2017 - 16
Signal Processing - January 2017 - 17
Signal Processing - January 2017 - 18
Signal Processing - January 2017 - 19
Signal Processing - January 2017 - 20
Signal Processing - January 2017 - 21
Signal Processing - January 2017 - 22
Signal Processing - January 2017 - 23
Signal Processing - January 2017 - 24
Signal Processing - January 2017 - 25
Signal Processing - January 2017 - 26
Signal Processing - January 2017 - 27
Signal Processing - January 2017 - 28
Signal Processing - January 2017 - 29
Signal Processing - January 2017 - 30
Signal Processing - January 2017 - 31
Signal Processing - January 2017 - 32
Signal Processing - January 2017 - 33
Signal Processing - January 2017 - 34
Signal Processing - January 2017 - 35
Signal Processing - January 2017 - 36
Signal Processing - January 2017 - 37
Signal Processing - January 2017 - 38
Signal Processing - January 2017 - 39
Signal Processing - January 2017 - 40
Signal Processing - January 2017 - 41
Signal Processing - January 2017 - 42
Signal Processing - January 2017 - 43
Signal Processing - January 2017 - 44
Signal Processing - January 2017 - 45
Signal Processing - January 2017 - 46
Signal Processing - January 2017 - 47
Signal Processing - January 2017 - 48
Signal Processing - January 2017 - 49
Signal Processing - January 2017 - 50
Signal Processing - January 2017 - 51
Signal Processing - January 2017 - 52
Signal Processing - January 2017 - 53
Signal Processing - January 2017 - 54
Signal Processing - January 2017 - 55
Signal Processing - January 2017 - 56
Signal Processing - January 2017 - 57
Signal Processing - January 2017 - 58
Signal Processing - January 2017 - 59
Signal Processing - January 2017 - 60
Signal Processing - January 2017 - 61
Signal Processing - January 2017 - 62
Signal Processing - January 2017 - 63
Signal Processing - January 2017 - 64
Signal Processing - January 2017 - 65
Signal Processing - January 2017 - 66
Signal Processing - January 2017 - 67
Signal Processing - January 2017 - 68
Signal Processing - January 2017 - 69
Signal Processing - January 2017 - 70
Signal Processing - January 2017 - 71
Signal Processing - January 2017 - 72
Signal Processing - January 2017 - 73
Signal Processing - January 2017 - 74
Signal Processing - January 2017 - 75
Signal Processing - January 2017 - 76
Signal Processing - January 2017 - 77
Signal Processing - January 2017 - 78
Signal Processing - January 2017 - 79
Signal Processing - January 2017 - 80
Signal Processing - January 2017 - 81
Signal Processing - January 2017 - 82
Signal Processing - January 2017 - 83
Signal Processing - January 2017 - 84
Signal Processing - January 2017 - 85
Signal Processing - January 2017 - 86
Signal Processing - January 2017 - 87
Signal Processing - January 2017 - 88
Signal Processing - January 2017 - 89
Signal Processing - January 2017 - 90
Signal Processing - January 2017 - 91
Signal Processing - January 2017 - 92
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Signal Processing - January 2017 - 94
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Signal Processing - January 2017 - 96
Signal Processing - January 2017 - 97
Signal Processing - January 2017 - 98
Signal Processing - January 2017 - 99
Signal Processing - January 2017 - 100
Signal Processing - January 2017 - 101
Signal Processing - January 2017 - 102
Signal Processing - January 2017 - 103
Signal Processing - January 2017 - 104
Signal Processing - January 2017 - 105
Signal Processing - January 2017 - 106
Signal Processing - January 2017 - 107
Signal Processing - January 2017 - 108
Signal Processing - January 2017 - 109
Signal Processing - January 2017 - 110
Signal Processing - January 2017 - 111
Signal Processing - January 2017 - 112
Signal Processing - January 2017 - 113
Signal Processing - January 2017 - 114
Signal Processing - January 2017 - 115
Signal Processing - January 2017 - 116
Signal Processing - January 2017 - Cover3
Signal Processing - January 2017 - Cover4
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