120
D. Rocchesso: Sound Processing
vectors
-1
-0.5
0
0.5
1
-1
0
1
Figure 5: Roots of the polynomial 1 + 2x
2
+ 5x
5
in the complex plane
coefficients of the operands. The product is done by application of the usual dis-
tributive and associative properties to the product of sums of powers. The order
of the product is given by the sum of the orders of the polynomial operands,
and the k-th coefficient of the product is obtained by the coefficients a
i
and b
j
of the operands by the formula
c
k
=
i+j=k
a
i
b
j
,
(9)
where this notation indicates a sum whose addenda are characterized by a couple
of indices i, j that sum up to k.
As it can be seen from sec. 1.4, the polynomial multiplication is formally
identical to the convolution of discrete signals, and this latter operation is fun-
damental in digital signal processing.
A.4
Vectors and Matrices
Physicists use arrows to indicate physical quantities having both an intensity and
a direction (e.g., forces or velocities). These arrows, sometimes called vectors,
are oriented according to the direction of the physical quantity and their length
is proportional to the intensity. These vectors can be located in the plane (or
the 3D space) as if they were departing from the origin. In this way, they can be
represented by the couple (or triple) of coordinates of their second extremity.
This representation allows to perform the sum of vectors and the multiplication
of a vector by a constant as the usual algebraic operations done with each
separate coordinate:
(x
1
, y
1
, z
1
) + (x
2
, y
2
, z
2
) = (x
1
+ x
2
, y
1
+ y
2
, z
1
+ z
2
)
(x
1
, y
1
, z
1
) = (x
1
, y
1
, z
1
)
(10)
More generally, an n-coordinate vector is defined in a field F as the ordered