Fixed-point Representation of Numbers

Transcription

1 Fixed-point Representation of Numbers

2 Fixed Point Representation of Numbers Sign-and-magnitude representation Two s complement representation Two s complement binary arithmetic Excess code representation Binary fractions Fixed point representation of fractional numbers

3 Sign-and-magnitude representation magnitude of a number the numerical value of the no. excluding its sign the sign-and-magnitude method the leftmost bit represent the sign of the no. the remaining bits represent the binary equivalent of the magnitude of the no. sign bit - the bit represents the sign of a no.

4 Example Express and in 8-bit binary codes using the sign-and-magnitude method convert the magnitude of the nos. into binary = = if necessary, add 0s at the left until 7 binary digits = = = = add a 0 or 1 at the left according to the sign of the no = =

5 Pros & Cons of sign-and-magnitude method Advantages simple and easy to understand Disadvantages the no. zero can be represented in two ways: and the sign bit and the magnitude part have to be handled separately complicates the design of the circuit used for addition particularly

6 Two s complement representation counters of some cassette recorders and most video cassette recorders ( ) reset the counter to 0000 rewind a tape the counter changes to 9999 & decreases gradually similar to the design of the memory of a computer ( )

7 Two s Compl. the leftmost bit of the binary code a+veno. 0 a-veno. 1 the range of nos. represented is -128 to = =

8 The relation between a no. & its negative The codes (+126) & (-126) are called the two s complement of each other it can be obtained by changing all 0s to 1s and all 1s to 0s adding the resulting binary code by 1

9 Example Obtain the two s complement of and change to = change to =

10 To convert a denary no. into an n-bit binary code using the two s complement notation convert the magnitude of the denary no. into a binary no. add 0s at the left until it contains n digits if the original no. is negative, obtain the two s complement of the binary code (Note that we need not find the two s complement if the original no. is non-negative)

11 Example represent and in 8-bit two s complement code = = = = = =

12 To convert a 8-bit binary code back to its denary equivalent if the leftmost bit is 0 (i.e. the no. is +ve) treat it as a simple binary no. and convert it into denary if the leftmost bit is 1 (i.e. the no. is -ve) obtain its magnitude by finding its two s complement change it to a denary no. and add the sign - in front of it

13 Advantages of two s complement representation electronic circuit used is very simple in design no separate circuit is needed for subtraction no separate circuit is needed to handle the sign bit and the magnitude part each no. is represented by a unique code avoid ambiguity and widens the range of no. represented

14 Two s complement binary arithmetic Addition Subtraction Overflow all nos. are represented in two s complement notation, and the no. of bits of each binary code is fixed

15 Addition addition is done in usual binary nos. if a carry occurs beyond the leftmost bit, it is discarded Example: perform the following additions in two s complement binary arithmetic using 8-bit binary codes (-60) + (-41) 60 + (-41)

16 Check: Denary equivalent Carry ) ) Ans

17 (-60) + (-41)( carry ) The extra digit is discarded Check: Denary equivalent -60 +) ans

18 60 + (-41)( Check: Denary equivalent Carry ) ) The extra digit is discarded ans

19 Subtraction Since x - y = x + (-y), x - y is found by adding x to the two s complement of y. Example: perform the following additions in two s complement binary arithmetic using 8-bit binary codes = 60 + (-41) 60 - (-41) = (-60) - (-41) = (-60) + 41

20 Overflow overflow is said to occur when at some stage during processing binary arithmetic, a no. outside the finite range is generated for 8-bit binary codes in two s complement form, the finite range is to concentrate on the possible occurrence of overflow during addition only when two nos. to be added are of the same sign, overflow may occur

21 To find Carry ) Check: Denary equivalent 86 +) The answer is , whose denary equivalent is -110 i.e, overflow has occurred.

22 Conclusion of Overflow In most microcomputers, 16-bit (or even 32-bit) binary codes are used the range becomes to inclusively However, no matter how many bits we use, it is still possible to find nos. outside the finite range

23 Binary fractions denary nos place value decimal point binary fraction place value binary point

24 = = 2 + 1/4 + 1/8 = = = /8 = /2 + 1/8 = =

25 Fixed point representation of fractional numbers *P