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CHAPTER 5. DIGITAL SIGNAL PROCESSING
the only floating point number having a zero fraction. The sign of the mantissa represents the sign of the number and the exponent can be a signed integer. A computer’s representation of integers is either perfect or only approximate, the latter situation occurring when the integer exceeds the range of numbers that a limited set of bytes can represent. Floating point representations have similar representation problems: if the number x can be multiplied/divided by enough powers of two to yield a fraction lying between 1/2 and 1 that has a finite binary-fraction representation, the number is represented exactly in floating point. Otherwise, we can only represent the number approximately, not catastrophically in error as with integers. For example, the number 2.5 equals 0.625 × 22 , the fractional part of which has an exact binary representation.8 However, the number 2.6 does not have an exact binary representation, and only be represented approximately in floating point. In single precision floating point numbers, which require 32 bits (one byte for the exponent and the remaining 24 bits for the mantissa), the number “2.6” will be represented as 2.600000079 . . .. Note that this approximation has a much longer decimal expansion. This level of accuracy may not suffice in numerical calculations. Double precision floating point numbers consume 8 bytes, and quadruple precision 16 bytes. The more bits used in the mantissa, the greater the accuracy. This increasing accuracy means that more numbers can be represented exactly, but there are always some that cannot. Such inexact numbers have an infinite binary representation.9 Realizing that real numbers can be only represented approximately is quite important, and underlies the entire field of numerical analysis, which seeks to predict the numerical accuracy of any computation. Exercise 5.2 (Solution on p. 191.) What are the largest and smallest numbers that can be represented in 32-bit floating point? in 64-bit floating point that has sixteen bits allocated to the exponent? Note that both exponent and mantissa require a sign bit. So long as the integers aren’t too large, they can be represented exactly in a computer using the binary positional notation. Electronic circuits that make up the physical computer can add and subtract integers without error. (This statement isn’t quite true; when does addition cause problems?)
5.2.3 Computer Arithmetic and Logic The binary addition and multiplication tables are 0+0=0
0×0=0
0+1=1
0×1=0
1 + 1 = 10
1×1=1
1+0=1
1×0=0
(5.4)
Note that if carries are ignored,10 subtraction of two single-digit binary numbers yields the same bit as addition. Computers use high and low voltage values to express a bit, and an array of such voltages express numbers akin to positional notation. Logic circuits perform arithmetic operations. Exercise 5.3 (Solution on p. 191.) Add twenty-five and seven in base-2. Note the carries that might occur. Why is the result “nice?” T The variables of logic indicate truth or falsehood. A B, the AND of A and B, represents a statement that both A and B must be true for the statement to be true. You use this kind of statement S to tell search engines that you want to restrict hits to cases where both of the events A and B occur. A B, the OR of A and B, yields a value of truth if either is true. Note that if we represent truth by a “1” and falsehood by a “0,” binary multiplication corresponds to AND (ignoring carries) to XOR. S and addition T XOR, the exclusive or operator, equals the union of A B and A B. The Irish mathematician George 8 See
if you can find this representation. that there will always be numbers that have an infinite representation in any chosen positional system. The choice of base defines which do and which don’t. If you were thinking that base-10 numbers would solve this inaccuracy, note that 1/3 = 0.333333.... has an infinite representation in decimal (and binary for that matter), but has finite representation in base-3. 10 A carry means that a computation performed at a given position affects other positions as well. Here, 1 + 1 = 10 is an example of a computation that involves a carry. 9 Note
