Mathematical Fundamentals

135

binary digits

unsigned integer

bits

least significant bit

most significant bit

signed integers

two's complement

representation

and it is convergent for values of z that are larger than e

(z

0

)

Similarly to what we saw for continuous-variable functions, the Fourier trans-

form for discrete-variable functions can be obtained as a specialization of the

Z transform where the values of the complex variable are restricted to the unit

circumference.

Y () = Y

Z

(e

j

) ,

(58)

or, in detail,

Y () =

+

n=-

y(n)e

-jn

.

(59)

In this book, we use the symbol for the radian frequency in the case of discrete-

variable functions, leaving for the continuous-variable functions.

###

A.9

Computer Arithmetics

A.9.1

Integer Numbers

In order to fully understand the behavior of several hardware and software tools

for sound processing, it is important to know something about the internal

representation of numbers within computer systems. Numbers are represented

as strings of binary digits (0 and 1), but the specific meaning of the string

depends on the conventions used. The first convention is that of unsigned integer

numbers, whose value is computed, in the case of 16 bits, by the following

formula

x =

15

i=0

x

i

× 2

i

,

(60)

where x

i

is the i-th binary digit starting from the right. The binary digits are

called bits, the rightmost digit is called least significant bit (LSB), and the

leftmost digit is called the most significant bit (MSB). For instance, we have

0100001100100110

2

= 2

1

+ 2

2

+ 2

5

+ 2

8

+ 2

9

+ 2

14

= 17190 ,

(61)

where the subscript 2 indicates the binary representation, being the usual deci-

mal representation indicated with no subscript.

The leftmost bit is often interpreted as a sign bit: if it is set to one it means

that the sign is minus and the absolute value is given by the bits that follow.

However, this is not the representation that is used for the signed integers. For

these numbers the two's complement representation is used, where the leftmost

bit is still a sign bit, but the absolute value of a negative number is recovered

by bitwise complementation of the following bits, interpretation of the result as

a positive integer, and addition of one. For instance, with four bits we have

1010

2

= -(0101

2

+ 1) = -(5 + 1) = -6 .

(62)

The two's complement representation has the following advantages:

18

(x) is the real part of the complex number x