Revisiting the Basics - Digital Signal Processing: A Practitioner's Approach - page 41

2

1

0

−1

−2

−3

−4

1 2 3 4 5 6 7 8

Figure 2.18 Innovations e k

least-squares sense. The other way is that the rows of A þ corresponds to MA
processes or FIR filters. Figure 2.18 depicts the error e k (in simple language it is the
deviation from the straight line fit). Sometimes k is called an innovation process [4]

and is given as e k ¼ AA þ y y (Figure 2.18).

2.7 Number Systems
A good understanding of binary number systems provide a proper insight into digital
filters and their implementations. Consider a binary number b n ¼ x n x n 1 ... x 3 x 2 x 1 ,

where the symbol x i , called a bit of the word b n , takes value 0 or 1. The string of
binary symbols has a value given as

b n ¼ x n x n 1 ... x i ... x 3 x 2 x 1 ¼ X

n

x i 2 i 1

: ð 2 : 36 Þ

i ¼ 1

This representation is an unsigned representation of integer numbers. For a given
value of n we can generate 2 n 1 patterns. If we fix the value n, normally known as

word length, to accommodate the positive and negative numbers, the usable
numbers gets partitioned into half each. There are three common ways of doing
it. For illustration we choose the value n ¼ 3.

2.7.1 Representation of Numbers

2.7.1.1 Sign Magnitude Form

Let us consider a set of 8 numbers f 0 ; 1 ; 2 ; 3 ; 4 ; 5 ; 6 ; 7 g representing a 3-bit binary

numbers and mapped as fþ 0

; þ 1 ; þ 2 ; þ 3 ; 0 ; 1 ; 2 ; 3 g:

Here we have divided
the given set into two partitions of positive numbers and negative numbers. Human
beings understand decimal numbers the best, so we are trying to explain with
equivalent decimal numbers whose binary bit patterns are the same. The basic
problem is that the hardware is relatively complex and the non-uniqueness of þ 0