division and multiplication by 2 · lesson02 page 3 . multiplication: signed (2's complement)...
TRANSCRIPT
Division and multiplication by 2m
Division: Unsigned Integers
Let a be an n‐bit unsigned integer and let m <= n‐1
a = an‐1 2n‐1 + … + am 2m + … + a0 20
Division by 2m
a/2m = [ an‐1 2n‐1 + … + am 2m + … + a0 20] 2‐m
a/2m = [ an‐1 2n‐1‐m + … + am 20 + … + a0 2‐m]
Integer part Fractional part
Discard the fractional part
a/2m = [ an‐1 2n‐1‐m + … + am 20 ]
a/2m (after discarding the fractional part) is the same as shifting the bits of a by mbits to the right
In bit notation:
a an‐1 an‐2 … am am‐1 … a0
a/2m 0 0 … 0 0 0 an‐1 an‐2 … am
discard
pad
n‐m bitsm bits
lsd (least significant digit)
You can safely use an n‐m bit bus for a/2m . Padding to the left is needed only if a wider bus is used
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Multiplication: Unsigned Integersa = an‐1 2n‐1 + … + am 2m + … + a0 20
Multiplication by 2m
a x 2m = [ an‐1 2n‐1 + … + a0 20] 2m
a x 2m = [ an‐1 2n+m‐1 + … + … + a0 2m] = [ an‐1 2n+m‐1 + … + … + a0 2m + 0 x 2m‐1+ … + 0 x 20]a x 2m is the same as shifting all the bits of a by m bits to the left
In bit notation:
a an‐1 an‐2 … am am‐1 … a
a x2m an‐1 an‐2 … am am‐1 … a0 0 0 … 0
n bits m bits
n+m bits
You can safely use an n+m bit bus for a x 2m . Padding to the left by 0's is needed if a wider bus is used
Insert 0's
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Division: Signed (2's complement) Integers
For signed integers stored in 2's complement notation the procedure is exactly the same as that for unsigned integers except that we need to be carful about the sign bit. This restricts m to m <= n‐2. Also we want to make sure that after shifting right by m bits the padding on the left hand side is by 0's if the sign of a is positive and by 1's if the sign is negative. The trick is to pad with the highest significant digit of a, i.e., an‐1 which is 0 for positive integers and 1 for negative integers.
In bit notation:
a an‐1 an‐2 … am am‐1 … a0
a/2m an‐1 … an‐1 an‐1 an‐2 … am
discard
pad
n‐m bitsm bits
lsd (least significant digit)
You can safely use an n‐m bit bus for a/2m . Padding to the left is needed only if a wider bus is used
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Multiplication: Signed (2's complement) Integers
Just as in the case of signed division we need to make sure that if we require padding on the left then we pad with an‐1, otherwise signed multiplication by 2m is similar to unsigned multiplication.
In bit notation:a an‐1 an‐2 … am am‐1 … a0
a 2m an‐1 an‐2 … am am‐1 … a0 0 0 … 0
n bits m bits
n+m bits
You can safely use an n+m bit bus for a x 2m . Padding to the left by an‐1 is needed if a wider bus is used
Insert 0's
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MSP430G2553
Address Bus
Data Bus
MSP430G2553
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