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Lecture no 6

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Two's Complement Given a negative number (N), represented using the Two's Complement representation (N*), the magnitude of the number (P) can be determined as follows: P = (2 n ) – N* or P = bit-wise complement of N* + 1 ECE 301 - Digital Electronics 2

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Signed Binary Numbers ECE 301 - Digital Electronics 3

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4 Binary Arithmetic of Signed Binary Numbers

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Two's Complement Addition Addition of n-bit signed numbers using Two's Complement addition is straightforward. Addition is carried out in the same way as the addition of n-bit positive numbers. Carry from the sign position (MSB) is ignored. ECE 301 - Digital Electronics 5

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Two's Complement Addition Implement the addition of the following signed numbers using Two's Complement Addition: 32 + 45 -17 + 63 82 + (-29) ECE 301 - Digital Electronics 6

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Overflow General rule for detecting overflow when adding two n-bit numbers using either One's Complement or Two's Complement Addition An overflow occurs when the addition of two positive numbers results in a negative value or the addition of two negative numbers results in a positive value. Cannot occur when adding a positive number and a negative number. ECE 301 - Digital Electronics 7

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8 Binary Codes

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Weighted and Unweighted Codes A weighted code is one in which each position in the code has a specific weight An unweighted code is one in which the positions in the code do not have a specific weight A 4-bit weighted code Weights: w 3, w 2, w 1, w 0 Code: a 3 a 2 a 1 a 0 Decimal: D = a 3 x w 3 + a 2 x w 2 + a 1 x w 1 + a 0 x w 0 ECE 301 - Digital Electronics 9

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Binary Codes Binary Coded Decimal (BCD) 4-bit binary number used to represent each decimal digit Weighted code: 8-4-2-1 The binary values 0000.. 1001 are used to represent the decimal digits 0.. 9 The binary values 1010.. 1111 are not used. How do we interpret these unused codes? ECE 301 - Digital Electronics 10

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Binary Codes 2-4-2-1 Code Weighted code with w 3 = 2, w 2 = 4, w 1 = 2, w 0 = 1 Excess-3 Code Obtained from the 8-4-2-1 (weighted code). Add 3 (0011 2 ) to each of the codes. ECE 301 - Digital Electronics 11

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Binary Codes ECE 301 - Digital Electronics 12

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1 Binary Numbers Again Recall that N binary digits (N bits) can represent unsigned integers from 0 to 2 N -1. 4 bits = 0 to 15 8 bits = 0 to 255 16 bits.

1 Binary Numbers Again Recall that N binary digits (N bits) can represent unsigned integers from 0 to 2 N -1. 4 bits = 0 to 15 8 bits = 0 to 255 16 bits.

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