Revisiting the Basics - Digital Signal Processing: A Practitioner's Approach - page 38

to various inputs. This is a fairly simple problem known as synthesis. Given
the input and output, finding the system is more difficult; plenty of methods
are available and they depend on the specific nature of the problem. This section
tries to lay foundations for understanding parameters in more than one or two
dimensions.
It is the limitation of the mind that makes it difficult to perceive variables beyond
three dimensions. Even a school student can understand two dimensions with ease. A
point in the Cartesian frame is represented by an ordered pair with reference to x 1 and
x 2 axes, which are orthogonal to each other (Figure 2.15). Our mind is so well
trained it can understand and perceive all the concepts in two dimensions. The
difficulty comes once we go beyond two dimensions, and this puts the mind under
strain.

Figure 2.15 Two-dimensional representation

2.6.1 Vectors of More Than Two Dimensions
This section presents a novel way of visualising multidimensional variables. The
traditional approach is to imagine a hyperspace and treat it as a point. But we will
present a simple way of looking at the multidimensional vector, yet without losing
any generality. Consider a four-dimensional vector. x t x 4 has been

represented as a polygon of 5 sides in Figure 2.16. This provides more insight into
understanding parameters of many dimensions. A multidimensional vector is a
figure but not a point. Visualising an n-dimensional vector as a polygon of order
n þ 1 is an important aspect of understanding multivariate systems. This shows how
difficult it is to compare two figures or perform any standard binary operations on
them. Many times we need to find out maxima or minima in this domain. This will
help us to understand these concepts better. Comparing numbers or performing the
familiar binary operations such as þ; ; ; is easy but it is more difficult with
multidimensional vectors as they are polygons. It doesn't mean that we cannot
compare two polygons; there are methods for doing this.

¼½ x 1 ; x 2 ; x 3 ;

2.6.2 Functions of Several Variables
Multivariate functions are so common in reality, we cannot avoid handling them.
They could be representing a weather system or a digital filter; the motion
parameters of a space shuttle, a missile or an aircraft; the performance of a person