10. Bit Operations and Radix Conversion

Transcription

1 10. Bit Operations and Radix Conversion by Douglas Lyon and Maynard Marquis One hopes to achieve the zero option, but in the absence of that we must achieve balanced numbers. - Margaret Thatcher, Prime Minister of Great Britain The study of numbering systems and bit operations are an important part of the software engineering body of knowledge. Bitwise operations enable data communications of all types. For example, in the transmission of an attachment, we often find that binary data is blocked by a mailer. As a result, a base-64 encoding is used to allow the data to pass unmolested by the mail processing software. Exercise 1 shows how a point-of-sale data transaction system might be coded, using a bar code reader to track inventory. Most modern coding systems, for error correction, detection, compression and protection, require bit manipulations. In fact, without bit manipulations, message authentication would be highly improbable. Bit-operations and base-conversions are elementary operations performed automatically by many subsystems in the computer. To write numbers, humans have devised several numbering systems, which have evolved over time. The ancient Egyptians had a numbering system, so did the ancient Romans, and so did many other civilizations. The current or modern numbering system comes from many sources, but in particular from the Hindus and Arabians. The system was further developed in Europe, and has become

2 142 Java for Programmers almost universal in its use. This system uses the ten Arabic numerals or digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 in a dual role of number and position. For example, the number 2 is associated with a particular unit or quantity, but that quantity depends on its position in the three numbers 275, 325, 572. In this chapter you will learn: How to convert from one base to another How to perform bit manipuations on binary numbers Form of Numbering Systems The form or structure of the modern numbering system is given by its dual role. The number 2 in 572 means two units, in 325 twenty units, and in 275 two hundred units. Its value depends on its position within the total number. In 572, its value is 2 times 1 = 2, in 325: 2 times 10 = 20, in 275: 2 times 100 = 200. So as one moves from right to left within the number, a particular number increases by a multiple of 10. In the first position, the numeral, 2, is multiplied by 10 0 ; in the second position by 10 1 ; and in the third by So the number consists of a summation of numerals which have been multiplied by 10 taken to power specified by the position of the numeral. N number = a N i (10) N i i =0 Where number = base 10 or decimal value, N = number of digits - 1, and a i = the symbol at position i

3 Java for Programmers 143 Using position notation, we can represent the number (572) base 10 as: 5(10) 2 + 7(10) 1 + 2(10) 0 = 5(100) + 7(10) + 2(1) The base 10 number system is also called decimal. Radix is another term used for the base. The base refers to the quantity that is raised to the exponent and is equal to the number of symbols in the system. The decimal system consists of symbols (0,1,2,3,4,5,6,7,8,9). For an arbitrary radix, R we write: N number = a N i (R) N i i =0 For systems with a radix greater than 10, letters are used as the symbols. The minimum value for the radix is 2. The radix is always an integer and there is no maximum value Common Computer Numbering Systems Several numbering systems are common in computing. Typically the radix is increased by a power of two. As a result, base 2, 8, 10, and 16 are the most common. However, larger bases (like 64) are used for transmitting textual codes on channels that cannot handle binary data. For example MIME (Multi-purpose Internet Mail Extensions) attachements will often be base 64 encoded. The hexadecimal system is called base 16 and so uses 16 symbols (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E and F). Binary (called base 2) uses only two symbols (0 and 1). Octal is called base 8 and uses 8 symbols (0, 1, 2, 3, 4, 5, 6, 7). The higher the base the more bits are represented by each symbol. For example, binary symbols represent one bit of information. Octal symbols represent 3 bits of information. Hexadecimal symbols represent 4 bits of information. Consequently, the octal and hexadecimal systems are used to write numbers with significantly

4 144 Java for Programmers fewer digits than the binary system. The symbols used in the various numbering systems are shown in Figure Number System Base or Radix Numerals Binary Octal Decimal Hexadecimal A B C D E F Figure Symbols in Various Numbering Systems For example, the decimal number ( 427) 10 converts to the 9-digit binary number ( ) 2. This can be written as the 3-digit octal number, ( 653) 8, or the 3- digit hexadecimal number, ( 1AB) 16. To write octal numbers into program code, the character zero precedes the number. So 0653 would appear in the code. For hexadecimal numbers the character zero followed by an x precedes the number. So 0x1AB signifies a hexadecimal number. The octal and hexadecimal numbers can be operated upon just like decimal numbers. The code statements shown below all yield the decimal value, x = x = 3*427; x = 3*0653; x = 3*0x1AB; 10.3 Converting from Binary, Octal, and Hexadecimal to Decimal Conversion of a number from the binary, octal, and hexadecimal systems to the decimal system is accomplished by using positional notation. For example, the 9- digit binary number ( ) 2 is equal to the decimal number ( 427) 10. 1(2) 8 +1(2) 7 +0(2) 6 +1(2) 5 +0(2) 4 +1(2) 3 +0(2) 2 +1(2) 1 +1(2) 0 = = ( 427) 10

5 Java for Programmers 145 The 3-digit octal number ( 653) 8 is also ( 427) 10 in the decimal system. As another example, ( 1AB) 16 = ( 427) 10. 6(8) 2 +5(8) 1 +3(8) 0 = = ( 427) 10 1 (16) 2 +10(16) 1 +11(16) 0 = = ( 427) Converting from Binary to Octal and Hexadecimal To convert binary numbers to octal, group the binary number in groups of 3. The grouping starts from the right. Then each group is converted to octal. For example: Binary: ( ) = = = = = = 3 Octal: ( 653) 8 To convert binary to hexadecimal, form groups of 4 bits, starting from the right. For example: Binary: ( ) 2 1 = = 1(8)+0(4)+1(2)+0(1) = 10 = A

6 146 Java for Programmers 1011 = 1(8)+0(4)+1(2)+1(1) = 11 = B Hexadecimal: ( 1AB) Converting from Decimal to Binary, Octal, and Hexadecimal Converting decimal numbers to binary, octal, and hexadecimal numbers can be done by a number of methods, each of which requires successive operations. The first method is called successive division. With successive division, we take a base 10 number and divide by 10. The remainder is used to form the symbol of the new base R number, from least to most significant (generally written from right to left). The quotient is used for the next division, until the quotient is zero. An example of successive division is shown in Figure radix quotient remainder Figure Successive Division converts decimal to binary in Excel In Java the quotient formula is i/2 (in Excel we write; =TRUNC(i/2)). The remainder formula is written in Java as i % 2 (in Excel we write; =MOD(i,2)). The same technique works for converting a decimal number to its octal equivalent. This is shown in Figure

7 Java for Programmers 147 radix quotient remainder Figure Successive Division converts decimal to octal in Excel In Java the quotient formula is i/8 (in Excel we write; =TRUNC(i/8)). The remainder formula is written in Java as i % 8 (in Excel we write; =MOD(i,8)). For small numbers, this is easy to to by hand. The octal number is written from the botton up as 653 ( ) 8. ( ) 16 as the ( ) 10. This is shown in Figure Repeating the same technique for the hexadecimal produces 1AB equivalence for decimal 427 radix quotient remaindersymbol b a Figure Successive Division converts decimal to hexadecimal in Excel In Java the quotient formula is i/16 (in Excel we write; =TRUNC(i/16)). The remainder formula is written in Java as i % 16 (in Excel we write; =MOD(i,16)). Using repeated division, you can convert from decimal into any base. Because humans are able to do binary division with ease, it is sometime easier to perform a decimal to binary conversion, using successive division, then convert to the target base using a grouping of the binary symbols. Either process can yield a correct result.

8 148 Java for Programmers radix quotient remainder symbol b a Figure A Demonstration of Repeated Division for Radix 16 positional notation R=8 regroup R=10 ungroup positional notation group by 3 group by 4 R=16 ungroup R=2 repeated division Figure Operations that Transform From One Base to Another Figure shows serveral operations that can be used to transform from one base to another Bitwise Operators Figure lists the Bitwise Operators.

9 Java for Programmers 149 Operator Operation Java Meaning Expression ~ bit negation a = ~ b reverses the bits in b & bitwise AND a is equal to the bit-wise a = b & c anding of b and c ^ bitwise XOR x^y x is exclusive or'd with y bitwise OR a b a OR B 4 places, excluding the >> shift right a = a >> 4 SIGN bit >>> shift right a = a >>> 4 places, including the SIGN bit << shift left a = a << 4 a has its bits shifted left 4 places Figure Bitwise Operators The unary operator, ~, performs a bitwise complement. Java users a two s complement represent of negative numbers. In fact, this is true of many modern computer languages, like C and C++. Two s complement is discussed in Section Two of the bitwise operators, ^ and, are overloaded operators as logical boolean operators and as bitwise operators. The operator, <<, shifts bits to the left, and >> is a bitwise shift to the right with a sign extension. The operator, >>>, is a bitwise right shift with a zero extension. These operators are binary, and their associativity is from left-to-right. Six of the Bitwise Operators, &, ^,, <<, >>, and >>>, are combined with the assignment operator, =, to perform assignments of their operations to a specified variable.

10 150 Java for Programmers 10.7 Complement Operator Four of the fundamental operations in mathematics are: add, subtract, multiply, and divide, where multiply and divide can even be thought of as subsets of add and subtract. Subtraction can be performed using addition with negative numbers. Formulating the negative number is a process that varies depending on the numbering system used. One numbering system that represents negative numbers is called two s complement. Two s complement numbers are formed by negating the bits and then adding one. In binary we write the two s complement of 6 as: (0110) +1 = (1001). Two s complement, then, negates a number. So the two s complement of a number represents its negative value. The relationship given by the two s complement is, -x = ~ x + 1 where x is the two s complement and ~ x is the one s complement. The decimal number ( 427) 10 is written below as a sixteen-bit number with its one s complement ( a bit is a binary digit). A 32-bit version would have 16 more zeros to the left for the number and 16 more ones for the complement. number one s complement two s complement

11 Java for Programmers AND, OR, XOR Operators The AND operator, &, does a bit-wise computation between two fixed-point primative data types. For example, the following code statement, written using decimal number prints the decimal number 8. System.out.println ( 13&8 = + Integer.toString(13 & 8)); To understand the result, the numbers need to be converted to binaries & Then each position in the binaries is compared. Where they both contain a one, a one is placed in that position for the result, otherwise a zero is placed in the position. So in this example, only the third position (starting with zero) in both numbers contains a one, resulting in the binary 1000, or decimal 8. The inclusive OR operator,, operates in the same fashion. Except this operator places a one for the positions where a one exists in either number. So (13 8) yields = 1101 OR 8 = = 13 A B A B Figure Truth-table for the Inclusive OR

12 152 Java for Programmers The exclusive OR operator, ^, also operates in a similar way. This operator places a one in the result for those positions where only one of the numbers contains a one. If both numbers have a one in the same position, the result contains a zero. So (13 ^ 8) yields = 1101 XOR 8 = = 5 A B A^B Figure Truth-table for the Exclusive OR 10.9 Shift Operators The shift operators, <<, >>, and >>>, shift the positions of binary numbers by the amount specified. The left shift operator shifts the positions of the binary numbers to the left by the amount specified. The impact is to add zeros to the left of the number. The expression (12 << 2) yields 48. It is the same as multiplying 12 by 2 2. System.out.println ( 12<<2 = + Integer.toString( 12 << 2 )); 12 = << 2 = = 48 The right shift operator with a sign extension eliminates the number of digits on the right specified by the shift. So ( 12 >> 2) yields 3. It is the same as integer division, 12 / 2 2 = = 1100

13 Java for Programmers >> 2 = 1100 = 11 = 3 13 = >>2 = 1101 = 11 = 3 This operator also takes negative numbers. But in this case it is only the same as integer division when the real number division has no remainder. Thus 12 >> 2 yields -3 just as 12/4 does. But 13>>2 yields 4 and not 3. So in this latter case the shift is not the same as integer division. The reason for this can be explained using two s complement. The two s complements for 3, 4, 12, and 13 are shown below. Only 8 bits are shown, but there are actually more ones to the left of each number. Two s complement for 3 = = -3 Two s complement for 4 = = -4 Two s complement for 12 = = -12 Two s complement for 13 = = >> 2 = >> 2 = Shifting 2 positions on 12 eliminates the 2 zeros on the end. Now the binary number is the same as that for 3. Doing the same to 13, however, produces the binary number for 4. So more care needs to be exercised for shifts when dealing with a negative number. The right shift operator with a zero extension works the same as the previous operator, but only for positive numbers. The >>> operator handles only unsigned integers. 12>>>2 = 1100 = 1100 = 3 Combining the assignment operator with the shift operators assigns the result of the shift to the specified variable, operating the same as the += and -= operators.

14 154 Java for Programmers i >>= 3; // is the same as: i = i >> 3; // a right shift with sign in the extension i >>>= 9; // is the same as: i = i >>> 9; // a right shift with zeros in the extension The following code example will reverse the bits in an int. Bit reversal code is used to perform transforms such as the Fast Hartley Transform (FHT) and the Fast Fourier Transform (FFT). These are described in [Lyon 1999]: int bitr(int j) { int ans = 0; for (int i = 0; i< nu; i++) { ans = (ans <<1) + (j&1); j = j>>1; } return ans; } The variable nu is declared as an int and is equal to the number of bits to be processed. It is common to replace bit shifting code above with look-up tables that group the bits into 8 or 16 bit groups. An example of some code that demonstrates the bit-operations follows: public class BitTest { public static void printlnbits(string prompt, int x) { String s = Integer.toString(x,2); prompt = padstring(prompt, 16, ' '); s = padstring(s,16,'0'); System.out.println( prompt+ "= "+ s ); } /** padstring - pads string s with char c so that it has k total characters. */ public static String padstring(string s, int k, char c) { int n = k - s.length();

15 Java for Programmers 155 for (int i=0; i < n; i++) s = c+s; return s; } } public static void main(string args[]) { int a = 960; int b = 1<<8 ; printlnbits("a\t",a); printlnbits("b\t",b); printlnbits("a&b\t",a&b); printlnbits("a b\t",a b); printlnbits("a^b\t",a^b); } What follows is the output of BitTest : a = b = a&b = a b = a^b = Summary This section described some of the more popular bases and how to convert between them. Position notation was introduced as a means of representing decimal, binary octal and hexadecimal numbers. By extension, other bases may be represented in this way too. Grouping was used to map from bases 2, 8 and 16. The mechanics of the most elementary of bit-operations was also examined. These were used to shows how computers subtract numbers (via two s complement) Exercises 1. Bitwise operations enable the communication to embedded systems. For example, a popular bar-code scanner, called CueCat (available for free from Radio Shack) outputs an encoded bit-stream. Bitwise operations can convert the bit stream into UPC (Universal Price Code) or ISBN (International Standard Book

16 156 Java for Programmers Numbers). Go to a Radio Shack and get your free bar-code scanner. Then write a Java program to decode the output. See < for more information on CueCat 2. The number of available symbols limits the base that we can use. Sample symbols are: ABCDEFGHIJKLMONPQRSTUVWXYZ. These 36 symbols are what keeps us from using bases greater than 36. But suppose you wanted to express numbers in a higher base. An alternative symbol set might be: ABCDEFGHIJKLMONPQRSTUVWXYZ!@#$%^&*()`~;:<>,./?[] {}\ =_. This gives us 64 symbols, which enables base 64 numbers. Base 64 encoding is used for mail attachments, because mail processing software (called agents) tends to alter binary data, treating all data as if it were text. Write a base 64 encoder that will read a base 10 integer and write it out in base Write a base 64 decoder, able to return integers given the base 64 numbers generated by the answer to question What to the following Integer methods return? Hint: some throw a NumberFormatException. A. parseint("0", 10) B. parseint("473", 10) C. parseint("-0", 10) D. parseint("-ff", 16) E. parseint(" ", 2) F. parseint(" ", 10) G. parseint(" ", 10) H. parseint(" ", 10) I. parseint("99", 8) J. parseint("kona", 10) K. parseint("kona", 27) 5. Conversion to base 10: Convert the following numbers to base 10: A. (DEAD) 16 B. (757) 8 C. (FED) 16 D. (FEED) 16

17 Java for Programmers 157 E. (1000) 16 F. (1000) 8 G. (1000) 2 6. Convert the given base 10 numbers to the indicated base: A. to base 2 B. to base 16 C. to base 2 D. to base 8 E. to base 7