Page 373 - Analog and Digital Filter Design

370 Analog and Digital Filter Design

Subtracting Two’s Complement Numbers
The principle of a carry still applies (where 1 is being subtracted from 0). Con-
sider a straightforward equation with a smaller number subtracted from a larger
one.

Number 0011,1110,1101,1001 (3ED9h or 16089)
Subtract 0010,0101,1101,0101 (25D5h or 9685)
Result 0001,1001,0000,0100 (1904h or 6404)

Now consider what happens when the number being subtracted from is
negative.
Number 1011,1110,1111,1000 (BEF8h or -4108h or -16648)

Subtract 0010,0101,1101,0101 (25D5h or 9685)
Result 1001,1001,0010,0011 (99231.1 or -66DDh or -26333)

Two’s complement numbers are well behaved in both addition and subtraction
operations. The exception to this is when the number range is exceeded.

Multiplication
Consider a multiplication of two simple binary numbers.

Number #1 101 1 (Bh or 11)
Number #2 0111 (7h or 7)

Product 101 1
+ 1,0110
+ 10,1100
Result 0100,1101 (4Dh or 77)

So binary multiplication gives us the correct result, but in two’s complement
notation there are some complications. Multiplying two positive numbers gives
the correct result. However, the product of a positive and negative number, or
the product of two negative numbers, gives the wrong answer.

As an example, I will multiply two positive numbers 3Fh x 55 h = 14EBh (or
63 x 85 = 5355). This is shown by a series of binary additions.

001 1,111 1 (3Fh)

X 0101,0101 (55W