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2-Mod5_LS_2011

2-Mod5_LS_2011 - ECE 270 Introduction to Digital System...

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ECE 270 Introduction to Digital System Design © 2011 by D. G. Meyer 1 Lecture Summary – Module 5-A Unsigned Number Base Conversion Reference: Text (4 th Ed.), pp. 25-34; (3 rd Ed.), pp. 25-34 overview o d n – digits of base R number o c n – converted corresponding digits in base 10 o dealing with unsigned numbers only at this point leading zeroes don’t matter o table of correspondence integer conversion: base R to base 10 o method: iterative multiply and add o based on nested expression of a number integer conversion: base 10 to base R o method: iterative division o remainders become digits of converted number o quotient of zero indicates conversion is complete (d 3 d 2 d 1 d 0 ) R = (N) 10 = c 3 xR 3 + c 2 xR 2 + c 1 xR 1 + c 0 xR 0 = (((c 3 x R + c 2 ) x R + c 1 ) x R + c 0 )
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ECE 270 Introduction to Digital System Design © 2011 by D. G. Meyer 2 short cut for conversion among powers of 2 (from base “A” to base “B”) o method: size log 2 R groupings o write an n-digit binary number for each base A digit in the original number, where n = log 2 A o starting at the least significant position, form m-digit groups, where m = log 2 B
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ECE 270 Introduction to Digital System Design © 2011 by D. G. Meyer 3 Lecture Summary – Module 5-B Signed Number Notation Reference: Text (4 th Ed.), pp. 34-39; (3 rd Ed.), pp. 34-39 overview – signed number notations o sign and magnitude (SM) o diminished radix (DR) o radix (R) o only negative numbers are different – positive numbers are the same in all 3 notations sign and magnitude o vacuum tube vintage o left-most (“most significant”) digit is sign bit square4 0 positive square4 R-1 negative (where R is radix or base of number) o positive-negative pairs are called sign and magnitude complements of each other o negation method: replace sign digit (n s ) with R-1-n s diminished radix o most significant digit is still sign bit o positive-negative pairs are called diminished radix complements of each other o negation method: subtract each digit (including n s ) from R-1, i.e. -(N) R = (R n -1) R – (N) R radix o most significant digit is still sign bit o positive-negative pairs are called radix complements of each other o negation method: add one to the DR complement of (N) R , i.e. -(N) R = (R n ) R – (N) R comparison (3-bit signed numbers, each notation): simplifications for binary (base 2) o SM: complement sign position (0 1) o DR (also called 1’s complement): complement each bit o R (also called 2’s complement): square4 add 1 to DR complement -or- square4 scan number from right to left and complement each bit to the left of the first “1” encountered sign extension: SM – pad magnitude with leading zeroes; R and DR – replicate the sign digit Observations : 1. SM and DR have a balanced set of positive and negative numbers (as well as +0 and -0) 2. R notation has a single representation for zero, which results in an “extra negative number” – this unbalanced set of positive and negative numbers can lead to round-off errors in numeric computations 3. Virtually all computers in service today use R notation
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