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CHAPTER 5. DIGITAL SIGNAL PROCESSING
CPU Memory I/O Interface
Keyboard
CRT
Disks
Network
Figure 5.1: Generic computer hardware organization.
The generic computer contains input devices (keyboard, mouse, A/D (analog-to-digital) converter, etc.), a computational unit, and output devices (monitors, printers, D/A converters). The computational unit is the computer’s heart, and usually consists of a central processing unit (CPU), a memory, and an input/output (I/O) interface. What I/O devices might be present on a given computer vary greatly. • A simple computer operates fundamentally in discrete time. Computers are clocked devices, in which computational steps occur periodically according to ticks of a clock. This description belies clock speed: When you say “I have a 1 GHz computer,” you mean that your computer takes 1 nanosecond to perform each step. That is incredibly fast! A “step” does not, unfortunately, necessarily mean a computation like an addition; computers break such computations down into several stages, which means that the clock speed need not express the computational speed. Computational speed is expressed in units of millions of instructions/second (Mips). Your 1 GHz computer (clock speed) may have a computational speed of 200 Mips. • Computers perform integer (discrete-valued) computations. Computer calculations can be numeric (obeying the laws of arithmetic), logical (obeying the laws of an algebra), or symbolic (obeying any law you like).4 Each computer instruction that performs an elementary numeric calculation — an addition, a multiplication, or a division — does so only for integers. The sum or product of two integers is also an integer, but the quotient of two integers is likely to not be an integer. How does a computer deal with numbers that have digits to the right of the decimal point? This problem is addressed by using the so-called floating-point representation of real numbers. At its heart, however, this representation relies on integer-valued computations.
5.2.2 Representing Numbers Focusing on numbers, all numbers can represented by the positional notation system.5 The b-ary positional representation system uses the position of digits ranging from 0 to b-1 to denote a number. The quantity b is known as the base of the number system. Mathematically, positional systems represent the positive integer n as ∞ X n= dk bk , dk ∈ {0, . . . , b − 1} (5.1) k=0
and we succinctly express n in base-b as nb = dN dN −1 . . . d0 . The number 25 in base-10 equals 2×101 +5×100 , so that the digits representing this number are d0 = 5, d1 = 2, and all other dk equal zero. This same number in binary (base-2) equals 11001 (1 × 24 + 1 × 23 + 0 × 22 + 0 × 21 + 1 × 20 ) and 19 in hexadecimal 4 An
example of a symbolic computation is sorting a list of names. number representation systems exist. For example, we could use stick figure counting or Roman numerals. These were useful in ancient times, but very limiting when it comes to arithmetic calculations: ever tried to divide two Roman numerals? 5 Alternative
