1. Digital Electronics and Computer Interfacing Tim Mewes 3. Digital Electronics

2. Digital Electronics and Computer Interfacing2 3.1 Digital Information Information is stored in two distinct physical states –Charge state of a capacitor (DRAM) –Magnetization direction (Hard disk, MRAM) –… The two states are referred to as –TRUE/FALSE (Boolean) –1/0 –On/Off –High/Low

3. Digital Electronics and Computer Interfacing3 3.1 Digital Information Information can be transmitted using: discrete Voltage levels –TTL: 1: 2.0 V or greater 0: 0.8 V or less –CMOS: 1: 3.7 V or greater 0: 1.3 V or less –… Light –typical wavelengths: 850, 1310, or 1550 nm Radio frequencies –Bluetooth 2.4GHz –…

4. Digital Electronics and Computer Interfacing4 3.2 Digital number representation 3.2.1 Unsigned integers use base 2 (binary) representation: 167 10 =1  2 7 +0  2 6+ 1  2 5 +0  2 4 +0  2 3 +1  2 2 +1  2 1 +1  2 0 =1010 0111 2 Each digit in the binary representation is called a bit Eight bits are called a byte The largest possible number that can be represented by n-bits is 2 n -1 (255 in case of a byte) The leftmost bit is also called most significant bit The rightmost bit is also called least significant bit

5. Digital Electronics and Computer Interfacing5 3.2.1 Unsigned integers How to convert from base 2 to base 10? Example: 11001001, n=8 bits Most significant bit 1  2 n-1 =1  2 7 =128  128 1  2 n-2 =1  2 6 =64  64 0  2 n-3 =0  2 5 =0  0 0  2 n-4 =0  2 4 =0  0 1  2 n-5 =1  2 3 =8  8 0  2 n-6 =0  2 2 =0  0 0  2 n-7 =0  2 1 =0  0 1  2 n-8 =1  2 0 =1  1 Sum: 201

6. Digital Electronics and Computer Interfacing6 3.2.1 Unsigned integers How to convert from base 10 to base 2? Example: 100 with n=8 bits Most significant bit 2 n-1 =2 7 =128 > 100  0 2 n-2 =2 6 =64 < 100  1 100-64 = 362 n-3 =2 5 =32 < 36  1 36-32 = 42 n-4 =2 4 =16 > 4  0 2 n-5 =2 3 =8 > 4  0 2 n-6 =2 2 =4 = 4  1 4-4 =02 n-7 =2 1 =2> 0  0 2 n-8 =2 0 =1> 0  0 100 10 =01100100 2

7. Digital Electronics and Computer Interfacing7 3.2.2 Signed integers Sign-and-magnitude Use the most significant bit to represent the sign 0 represents ‘+’, 1 represents ‘-’ For n-bits numbers from -2 n-1 +1 to 2 n-1 -1 can be represented Advantage: similar to the way we usually indicate the sign of a number Disadvantage: arithmetic calculations tricky Zero has two representations: -0 10 =1000 0000 2 +0 10 =0000 0000 2

8. Digital Electronics and Computer Interfacing8 3.2.2 Signed integers Ones’ complement Negative numbers are represented by complementing all the bits (1  0) of the binary representation of the magnitude of the number 42 10 = 0010 1010 2 -42 10 = 1101 0101 2 Zero still has two representations

9. Digital Electronics and Computer Interfacing9 3.2.2 Signed integers Two’s complement For negative numbers calculate the ones’ complement and add 1 to the result: 42 10 = 0010 1010 2 Ones’ complement: -42 10 = 1101 0101 2 Two’s complement: -42 10 = 1101 0110 2 Zero has only one representation Range for n bits: -2 n-1 to 2 n-1 -1 (-128 to +127 for a byte) Advantage: convenient for computer arithmetic

10. Digital Electronics and Computer Interfacing10 3.2.3 Comparison (4-Bit) Base 10Unsigned IntegerSign-and-magnitudeOnes’ complementTwo’s comlement +81000--- +70111 +60110 +50101 +40100 +30011 +20010 +10001 +00000 -0-10001111- -100111101111 -2-101011011110 -3-101111001101 -4-110010111100 -5-110110101011 -6-111010011010 -7-111110001001 -8---1000

11. Digital Electronics and Computer Interfacing11 3.3 Gates A logic gate is an arrangement of switches to calculate operations using Boolean logic in digital circuits The output of a gate only depends on its inputs and not its history

12. Digital Electronics and Computer Interfacing12 3.3 Gates A Q A B Q A B Q

13. Digital Electronics and Computer Interfacing13 3.3 Gates A B Q A B Q A B Q

14. Digital Electronics and Computer Interfacing14 3.4 Boolean algebra associativity: commutativity: absorption: complements: distributivity: De Morgan’s theorem:

15. Digital Electronics and Computer Interfacing15 3.4 Boolean algebra How many gates do we really need? Just one: either NAND or NOR (universal gates)! One can build all other gates using for example only NAND: AND: NOT: OR: XOR: