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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63 1 Introduction to Digital Electronics

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 2 Learning Objectives To introduce analog and digital system Combinational circuit Identify the basic gates and describe the behavior of each Combine basic gates into circuits Adders Multiplexer and de multiplexer Encoder and decoder Sequential circuit Latch and flip flop Types of flip flop

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 3 To be transmitted, data must be transformed to electromagnetic signals. Analog and Digital Data can be analog or digital. The term analog data refers to information that is continuous; digital data refers to information that has discrete states.

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 4 Analog and Digital Data and Signal Analog and digital data Data can be analog or digital. Analog data are continuous and take continuous values. Digital data have discrete states and take discrete values. Signals can be analog or digital. Analog signals can have an infinite number of values in a range. Digital signals can have only a limited number of values.

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 5 Comparison of Analog and Digital Signals

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 6 Digital Electronics Digital Electronics represents information (0, 1) with only two discrete values. Ideally no voltage (e.g., 0v) represents a 0 and full source voltage (e.g., 5v) represents a 1 Realistically low voltage (e.g., 4v) represents a 1 We achieve these discrete values by using switches. We use transistor switches, which operates at high speed, electronically, a small in size.

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 7 Electronic Aspects of Digital Design How we represent digital information in electronic devices? By discrete voltages.

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 8 What is the Basic Digital Element in Electronics ? a Switch

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 9 Using Switch to Represent Digital Information

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 10 Digital Abstraction It is difficult to make ideal switches means a switch is completely ON or completely OFF. So, we impose some rules that allow analog behavior to be ignored in most cases, so circuits can be modeled as if they really did process 0s and 1s, known as digital abstraction. Digital abstraction allows us to associate a noise margin with each logic values (0 and 1).

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 11 Logic levels Undefined region is inherent digital, not analog Switching threshold varies with voltage, temp need noise margin Logic voltage levels decreasing with new processors. 5, 3.3, 2.5, 1.8 V

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 12 Analog versus Digital Analog systems process time-varying signals that can take on any value across a continuous range of voltages (in electrical/electronics systems). Digital systems process time-varying signals that can take on only one of two discrete values of voltages (in electrical/electronics systems). Discrete values are called 1 and 0 (ON and OFF, HIGH and LOW, TRUE and FALSE, etc.)

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 13 Representing Information Electronically Analog electronics deals with non-discrete values Digital electronics deals with discrete values

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 14 Benefits of Digital over Analog Reproducibility Not effected by noise means quality Ease of design Data protection Programmable Speed Economy

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 15 Digital Revolution Digital systems started back in 1940s. Digital systems cover all areas of life: still pictures digital video digital audio telephone traffic lights Animation

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 16 Basic terminology Gate A device that performs a basic operation on electrical signals Circuits Gates combined to perform more complicated tasks How do we describe the behavior of gates and circuits? Boolean expressions Uses Boolean algebra, a mathematical notation for expressing two- valued logic Logic diagrams A graphical representation of a circuit; each gate has its own symbol Truth tables A table showing all possible input value and the associated output values

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 17 Circuits Circuits can be Combinational or Sequential Combinational logic circuits produce a specified output (almost) at the instant when input values are applied. The addition of a memory device to a combinational circuit allows the output to be fed back into the input: Sequential circuit circuit memory Input(s) Output(s) Sequential circuit Combinational circuit

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 18 Digital Devices Combinational circuit Gates Multiplexer Demultiplexer Adders Encoder Decoder Sequential circuit Flip-Flops Registers Counters

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi Combinational Circuits

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 20 Overview Gates Iterative combinational circuits Binary adders Half and full adders Ripple carry Binary subtraction Binary adder-subtractors Signed binary numbers Signed binary addition and subtraction Overflow

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 21 Combinational Circuits Combinational logic circuits produce a specified output (almost) at the instant when input values are applied.

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 22 Gates The most basic digital devices are called gates. Gates got their name from their function of allowing or blocking (gating) the flow of digital information. A gate has one or more inputs and produces an output depending on the input(s). A gate is called a combinational circuit. Three most important gates are: AND, OR, NOT

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 23 Digital Logic Binary system -- 0 & 1, LOW & HIGH, negated and asserted. Basic building blocks -- AND, OR, NOT

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 24 NOT Gate A NOT gate accepts one input signal (0 or 1) and returns the o pposite signal as output

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 25 AND Gate An AND gate accepts two input signals If both are 1, the output is 1; otherwise the output is 0

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 26 OR Gate An OR gate accepts two input signals If both are 0, the output is 0; otherwise, the output is 1

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 27 XOR Gate An XOR gate accepts two input signals If both are the same, the output is 0; otherwise, the output is 1

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 28 XOR Gate Note the difference between the XOR gate and the OR gate; they differ only in one input situation When both input signals are 1, the OR gate produces a 1 and the XOR produces a 0 XOR is called the exclusive OR

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 29 NAND Gate The NAND gate accepts two input signals If both are 1, the output is 0; otherwise, the output is 1

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 30 NOR Gate The NOR gate accepts two input signals If both are 0, the output is 1; otherwise, the output is 0

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 31 De Morgan again A NAND gate: Y = A.B = A + B is the same as an OR gate with two NOT gates Similarly a NOR gate is the same as an AND gate with two inverters Y = A + B = A.B

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 32 Dual gates

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 33 Truth Tables and Boolean Notation NAND Gate Representation It is possible to implement any boolean expression using only NAND gates XX NOT AND A B A.B OR A A+B B

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 34 Truth Tables and Boolean Notation NAND Gate representation Implement the following circuit using only NAND gates x3 x2 x4

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 35 Solution Dual the gates, remember two nots together can be removed. x3 x2 x4 A B A.B A A+B B AND feeding OR

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 36 Exercise Implement NOT, AND and OR using NOR gates Example AND gate dual circuit:

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 37 Solution Similar pattern to using NAND gates (not surprising) NOT AND OR XX A B A.B A A+B B X X A B A.B A+B A A.B B

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 38 Logic Gates NAND and NOR are known as universal gates because they are inexpensive to manufacture and any Boolean function can be constructed using only NAND or only NOR gates.

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 39 Truth Tables and Boolean Notation NOR Gate representation It is also possible to implement any boolean expression using only NOR gates Implement the following circuit using only NOR gates X4 X3 X2

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 40 Solution Two NOR gates in sequence acting as NOTs can be eliminated: X4 X3 X2

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 41 Logic Gates Gates can have multiple inputs and more than one output. A second output can be provided for the complement of the operation.

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 42 Computers are implementations of Boolean logic. Boolean functions are completely described by truth tables. Logic gates are small circuits that implement Boolean operators. The basic gates are AND, OR, and NOT. The XOR gate is very useful in parity checkers and adders. The universal gates are NOR, and NAND. Conclusion

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 43 Implementation F=X.Y.+X.Y.Z

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 44 Iterative Combinational Circuits Arithmetic functions Operate on binary vectors Use the same subfunction in each bit position Can design functional block for subfunction and repeat to obtain functional block for overall function Cell - subfunction block Iterative array - a array of interconnected cells An iterative array can be in a single dimension (1D) or multiple dimensions

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 45 Block Diagram of a 1D Iterative Array Example: n = 32 Number of inputs = ? Truth table rows = ? Equations with up to ? input variables Equations with huge number of terms Design impractical! Iterative array takes advantage of the regularity to make design feasible

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 46 Functional Blocks: Addition Binary addition used frequently Addition Development: Half-Adder (HA), a 2-input bit-wise addition functional block, Full-Adder (FA), a 3-input bit-wise addition functional block, Ripple Carry Adder, an iterative array to perform binary addition, and Carry-Look-Ahead Adder (CLA), a hierarchical structure to improve performance. *(Details not required)

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 47 Functional Block: Half-Adder A 2-input, 1-bit width binary adder that performs the following computations: A half adder adds two bits to produce a two-bit sum The sum is expressed as a sum bit, S and a carry bit, C The half adder can be specified as a truth table for S and C X Y C S X Y C S

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 48 Logic Simplification: Half-Adder The K-Map for S, C is: This is a pretty trivial map! By inspection: and These equations lead to several implementations. Y X S Y X C )YX()YX(S YXYXYXS )(C YXC )YX(

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 49 Five Implementations: Half-Adder We can derive following sets of equations for a half-adder: (a), (b), and (e) are SOP, POS, and XOR implementations for S. In (c), the C function is used as a term in the AND-NOR implementation of S, and in (d), the function is used in a POS term for S. YXC )(S)c( YXC )YX()YX(S)b( YXC YXYXS)a( YX C YXC YXS)e( )YX(C C)YX(S)d( C

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 50 Implementations: Half-Adder The most common half adder implementation is: (e) A NAND only implementation is: YXC YXS )(C C)YX(S )YX( X Y C S X Y C S

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 51 Functional Block: Full-Adder A full adder is similar to a half adder, but includes a carry- in bit from lower stages. Like the half-adder, it computes a sum bit, S and a carry bit, C. For a carry-in (Z) of 0, it is the same as the half-adder: For a carry- in (Z) of 1: Z0000 X Y C S Z1111 X Y C S

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 52 Logic Optimization: Full-Adder Full-Adder Truth Table: Full-Adder K-Map: XYZCS X Y Z S X Y Z C

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 53 Equations: Full-Adder From the K-Map, we get: The S function is the three-bit XOR function (Odd Function): The Carry bit C is 1 if both X and Y are 1 (the sum is 2), or if the sum is 1 and a carry-in (Z) occurs. Thus C can be re-written as: The term X·Y is carry generate. The term X Y is carry propagate. ZYZXYXC ZYXZYXZYXZYXS ZYXS Z)YX(YXC

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 54 Full Adder Circuit X Y Z=Cin Cout Sum

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 55 Implementation: Full Adder Full Adder Schematic Here X, Y, and Z, and C (from the previous pages) are A, B, C i and C o, respectively. Also, G = generate and P = propagate. Note: This is really a combination of a 3-bit odd function (for S)) and Carry logic (for C o ): (G = Generate) OR (P =Propagate AND C i = Carry In) Co = G + P · Ci AiAi BiBi CiCi C i+1 GiGi PiPi SiSi

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 56 Binary Adders To add multiple operands, we bundle logical signals together into vectors and use functional blocks that operate on the vectors Example: 4-bit ripple carry adder: Adds input vectors A(3:0) and B(3:0) to get a sum vector S(3:0) Note: carry out of cell i becomes carry in of cell i + 1 Description Subscript Name Carry In CiCi Augend AiAi Addend BiBi Sum SiSi Carry out C i+1

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 57 Combinational Circuits Just as we combined half adders to make a full adder, full adders can connected in series. The carry bit ripples from one adder to the next; hence, this configuration is called a ripple-carry adder. Todays systems employ more efficient adders.

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 58 4-bit Ripple-Carry Binary Adder A four-bit Ripple Carry Adder made from four 1-bit Full Adders:

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 59 Signed Integer Representations Signed-Magnitude – here the n – 1 digits are interpreted as a positive magnitude. Signed-Complement – here the digits are interpreted as the rest of the complement of the number. There are two possibilities here: Signed 1's Complement Uses 1's Complement Arithmetic Signed 2's Complement Uses 2's Complement Arithmetic

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 60 Signed Integers Positive numbers and zero can be represented by unsigned n-digit, radix r numbers. We need a representation for negative numbers. To represent a sign (+ or –) we need exactly one more bit of information (1 binary digit gives 2 1 = 2 elements which is exactly what is needed). Since computers use binary numbers, by convention, the most significant bit is interpreted as a sign bit: s a n–2 a 2 a 1 a 0 where: s = 0 for Positive numbers s = 1 for Negative numbers and a i = 0 or 1 represent the magnitude in some form.

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 61 Signed-Magnitude Arithmetic I f the parity of the three signs is 0: 1. Add the magnitudes. 2. Check for overflow (a carry out of the MSB) 3. The sign of the result is the same as the sign of the first operand. If the parity of the three signs is 1: 1. Subtract the second magnitude from the first. 2. If a borrow occurs: take the twos complement of result and make the result sign the complement of the sign of the first operand. 3. Overflow will never occur.

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 62 2s Complement Method Given: an n-bit binary number, beginning at the least significant bit and proceeding upward: Copy all least significant 0s Copy the first 1 Complement all bits thereafter. 2s Complement Example: Copy underlined bits: 100 and complement bits to the left:

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 63 Signed Integer Representation Example

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 64 Signed-Complement Arithmetic Addition: 1. Add the numbers including the sign bits, discarding a carry out of the sign bits (2's Complement), or using an end-around carry (1's Complement). 2. If the sign bits were the same for both numbers and the sign of the result is different, an overflow has occurred. 3. The sign of the result is computed in step 1. Subtraction: Form the complement of the number you are subtracting and follow the rules for addition.

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 65 2s Complement Adder/Subtractor Subtraction can be done by addition of the 2's Complement. 1. Complement each bit (1's Complement.) 2. Add 1 to the result. The circuit shown computes A + B and A – B: For S = 1, subtract, the 2s complement of B is formed by using XORs to form the 1s comp and adding the 1 applied to C 0. For S = 0, add, B is passed through unchanged

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 66 Overflow Detection Overflow occurs if n + 1 bits are required to contain the result from an n-bit addition or subtraction Overflow can occur for: Addition of two operands with the same sign Subtraction of operands with different signs 1Signed number overflow cases with correct result sign Detection can be performed by examining the result signs which should match the signs of the top operand

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 67 Overflow Detection Signed number cases with carries C n and C n-1 shown for correct result signs: Signed number cases with carries shown for erroneous result signs (indicating overflow): Simplest way to implement overflow V = C n + C n - 1 This works correctly only if 1s complement and the addition of the carry in of 1 is used to implement the complementation! Otherwise fails for

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 68 Other Arithmetic Functions Convenient to design the functional blocks by contraction - removal of redundancy from circuit to which input fixing has been applied Functions Incrementing Decrementing Multiplication by Constant Division by Constant

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 69 Incrementing & Decrementing Incrementing Adding a fixed value to an arithmetic variable Fixed value is often 1, called counting (up) Examples: A + 1, B + 4 Functional block is called incrementer Decrementing Subtracting a fixed value from an arithmetic variable Fixed value is often 1, called counting (down) Examples: A - 1, B - 4 Functional block is called decrementer

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 70 Multiplication/Division by 2n (a) Multiplication by 100 Shift left by 2 (b) Division by 100 Shift right by 2 Remainder preserved B 0 B 1 B 2 B 3 C 0 C 1 00 C 2 C 3 C 4 C 5 (a) B 0 B 1 B 2 B 3 C 0 C 2 1 C 2 2 C 1 C 2 00 C 3 (b)

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 71 Combinational Circuits A multiplexer does just the opposite of a decoder. It selects a single output from several inputs. The particular input chosen for output is determined by the value of the multiplexers control lines. To be able to select among n inputs, log 2 n control lines are needed. This is a block diagram for a multiplexer.

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 72 Example of a Combinatorial Circuit: A Multiplexer (MUX) Consider an integer m, which is constrained by the following relation: m = 2 n, where m and n are both integers. A m-to-1 Multiplexer has m Inputs: I 0, I 1, I 2, I (m-1) one Output: Y n Control inputs: S 0, S 1, S 2, S (n-1) One (or more) Enable input(s) such that Y may be equal to one of the inputs, depending upon the control inputs.

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 73 Examples The Multiplexer Selects one of 2 n inputs and copies it to a single output The selected line is determined from the bit combination (address) on the n selection lines e.g. 1 from 2 mutiplexer selabout sel ab out = a b sel out n = 1 0 1

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 74 K map for 2:1 Multiplexer AB sel output = sel.a + sel.b Principal can be extended to 4:1 – 2 select lines and 4 data lines 8:1 – 3 select lines and 8 data lines and so on… data sel out

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 75 2:1 Multiplexer selabout selabout 00?0 01?1 1?00 1?11 if a is selected, dont care about b. AB sel

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 76 Combinational Circuits This is what a 4-to-1 multiplexer looks like on the inside. If S 0 = 1 and S 1 = 0, which input is transferred to the output?

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 77 Demultiplexer (DMUX)/ Decoder A 1-to-m DMUX, with ACTIVE HIGH Outputs, has 1 Input: I ( also called as the Enable input when the device is called a Decoder) m ACTIVE HIGH Outputs: Y 0, Y 1, Y 2, …………….Y (m-1) n Control inputs: S 0, S 1, S 2, S (m-1)

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 78 Characteristic table of the 1-to-4 DMUX with ACTIVE HIGH Outputs: Table 1

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 79 Characteristic Table of a 1-to-4 DMUX, with ACTIVE LOW Outputs: Table 2

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 80 The diagram below shows the relation between a multiplexer and a Demultiplexer. I0 I1 I2 I3 S1 S0 Y out Y0 Y1 Y2 Y4 S1 S0 Input 4 to 1 MUX 1 to 4 DEMUX

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 81 Combinational Circuits Decoders are another important type of combinational circuit. Among other things, they are useful in selecting a memory location according a binary value placed on the address lines of a memory bus. Address decoders with n inputs can select any of 2 n locations. This is a block diagram for a decoder.

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 82 Combinational Circuits This is what a 2-to-4 decoder looks like on the inside. If x = 0 and y = 1, which output line is enabled?

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 83 A Decoder is a Demultiplexer with a change in the name of the inputs : Y0 Y1 Y2 Y4 S1 S0 ENABLE INPUT 2 to 4 Decoder When the IC is used as a Decoder, the input I is called an Enable input

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 84 DECODER The unexcited state of an Output is 0 for an IC with ACTIVE HIGH Outputs. The unexcited state of an Output is 1 for an IC with ACTIVE LOW Outputs. Enable Input: In a Decoder, the Enable Input can be ACTIVE LOW or ACTIVE HIGH.

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 85 Characteristic Table of a 2-to-4 DECODER, with ACTIVE LOW Outputs and with ACTIVE LOW Enable Input: Logic expressions for the outputs of the Decoder of Table: Y0 = E + S1 + S0 Y1 = E + S1+ S0 Y2 = E + S1 + S0 Y3 = E + S1 + S0

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 86 Encoders Multiple-input/multiple-output device. Performs the inverse function of a Decoder. Outputs ( m ) are less than inputs ( n ). Converts input code words into output code words. input code output code ENCODER

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 87 Encoders vs. Decoders DecoderEncoder 2^n-to-n encoder Input code : 1-out-of-2^n. Output code : Binary Code n-to-2^n Input code : Binary Code Output code :1-out-of-2^n. Binary decoders/encoders

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 88 Encoder/Decoder Vocabulary ENCODER- a digital circuit that produces a binary output code depending on which of its inputs are activated. DECODER- a digital circuit that converts an input binary code into a single numeric output.

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 89 ENCODERS AND DECODERS ONLY ONE INPUT ACTIVATED AT A TIME BINARY CODE OUTPUT BINARY CODE INPUT ONLY ONE OUTPUT ACTIVATED AT A TIME

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 90 Binary Encoder 2^n-to-n encoder : 2^n inputs and n outputs. Input code : 1-out-of-2^n. Output code : Binary Code Example : n=3, 8-to-3 encoder Inputs Outputs I0 I1 I2 I3 I4 I5 I6 I7 Y2 Y1 Y I1 I2 I3Y1 Y2I4 I5 I6 I0 Y0 I7 Binary encoder

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 91 8-to-3 encoder Implementation Simplified implementation: - From the truth table Y0 = I1 + I3 + I5 + I7 Y1 = I2 + I3 + I6 + I7 Y2 = I4 + I5 + I6 + I7 Limitations : - I0 has no effect on the output - Only one input can be activated Application: Handling multiple devices requests But, no simultaneous requests Establishing priorities solve the problem of multiple requests I1 I2 I3 I4 I5 I6 I0 I7 Y1 Y0 Y2

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 92 What you should be able to do: Change circuits using one set of gates (e.g. AND, OR, NOT) to their equivalent using NAND or NOR gates only (and vice versa). Be familiar with half-, full- adders and multiplexer, de multiplexer, encoder and decoder circuits.

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 93 TEST ANSWER THE FOLLOWING QUESTIONS WITH ONE OR MORE OF THESE WORDS: MUX, DEMUX, ENCODER, DECODER. A. Has more inputs than outputs.ENCODER, MUX B. Uses select inputs.MUX, DEMUX C. Can be used in parallel-to-serial conversion.MUX D. Produces a binary code at its output.ENCODER E. Only one of its outputs is activated at one time.DEMUX, DECODER F. Used to route input signals to one of several outputs.MUX G. Used to generate arbitrary logic functions.MUX, DEMUX H. 3 line-to-8 line or binary to octal.DECODER I. Data Selectors are alsoMUX.

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi Sequential Logic

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 95 Sequential Logic Circuits So far we have only considered circuits where the output is purely a function of the inputs With sequential circuits the output is a function of the values of past and present inputs This particular example is not very useful X = X + A

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 96 Sequential Circuits Combinational logic circuits are perfect for situations when we require the immediate application of a Boolean function to a set of inputs. There are other times, however, when we need a circuit to change its value with consideration to its current state as well as its inputs. These circuits have to remember their current state. Sequential logic circuits provide this functionality for us.

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 97 Integrated Circuits A collection of one or more gates fabricated on a single silicon chip is called an integrated circuit (IC). ICs were classified by size: SSI - small scale integration - 1~20 gates MSI - medium scale integration - 20~200 gates LSI - large scale integration - 200~200,000 gates VLSI - very large scale integration - over 1M transistors Pentium-III - 40 million transistors

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 98 Sequential circuit concepts The addition of a memory device to a combinational circuit allows the output to be fed back into the input: To retain their state values, sequential circuits rely on feedback. Feedback in digital circuits occurs when an output is looped back to the input. circuit memory Input(s) Output(s)

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 99 Synchronous and Asynchronous With synchronous circuits a clock pulse is used to regulate the feedback, input signal only enabled when clock pulse is high – acts like a gate being opened. circuit memory Input(s) Output(s) Clock pulse

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 100 Sequential Circuits As the name implies, sequential logic circuits require a means by which events can be sequenced. State changes are controlled by clocks. A clock is a special circuit that sends electrical pulses through a circuit. Clocks produce electrical waveforms such as the one shown below.

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 101 Sequential Circuits State changes occur in sequential circuits only when the clock ticks. Circuits can change state on the rising edge, falling edge, or when the clock pulse reaches its highest voltage.

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 102 Sequential Circuits Circuits that change state on the rising edge, or falling edge of the clock pulse are called edge-triggered. Level-triggered circuits change state when the clock voltage reaches its highest or lowest level.

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 103 Clock Pulse Definition Edges can also be referred to as leading and trailing. Positive Pulse Positive Edge Negative Edge Negative Pulse Positive Edge Negative Edge

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 104 Flip-flops A device that stores either a 0 or 1. Stored value can be changed only at certain times determined by a clock input. New value depend on the current state and its control inputs A digital circuit that contains filp-flops is called a sequential circuit

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 105 Flip-flops S-R latch symbolsD flip-flop J-K flip-flops

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 106 Two cross-coupled NOR gates form an SR (set and reset) latch Latches

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 107 Latches The SR Latch Consider the following circuit S R Q Q Circuit Function Table RSQ n+1 00Q n ? n+1 represents output at some future time n represents current output. R S Q Q Symbol Q Q S R

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 108 SR Latch operation Assume some previous operation has Q as a 1 Assume R and S are initially inactive S = 0 R = 0 Q = 1 Q = 0 Circuit RSQ n+1 00Q n ? Indicates a stable state at some future time (n+ = now plus) ~Q = Q, ie is the complement of Q. Now assume R goes first to 1 then returns to 0, what happens:

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 109 Reset goes active When R goes active 1, the output from the first gate must be 0. S = 0 R = 1 Q = 0 ~Q = 1 This 0 feeds back to gate 2 Since both inputs are 0 the output is forced to 1 The output ~Q is fed back to gate 1, both inputs being 1 the output Q stays at 0. S = 0 R = 1 Q = 0 ~Q = 1

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 110 Reset goes in-active When R now goes in-active 0, the feedback from ~Q (still 1), holds Q at 0. S = 0 R = 0 Q = 0 ~Q = 1 The pulse in R has changed the output as shown in the function table: RSQ n+1 00Q n ? We went from here To here And back again In that process, Q changed from 1 to 0. Further signals on R will have no effect.

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 111 Set the latch Similar sequences can be followed to show that setting S to 1 then 0 – activating S – will set Q to a 1 stable state. When R and S are activated simultaneously both outputs will go to a 0 S = 1 R = 1 Q = 0 ~Q = 0 When R and S now go inactive 0, both inputs at both gates are 0 and both gates output a 1. This 1 fedback to the inputs drives the outputs to 0, again both inputs are 0 and so on and so on and so on and so on.

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 112 Metastable state In a perfect world of perfect electronic circuits the oscillation continues indefinitely. However, delays will not be consistent in both gates so the circuit will collapse into one stable state or another. RSQ n+1 00Q n ? This collapse is unpredictable. Thus our function table: Future output = present output Set the latch Reset the latch Dont know

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 113 Sequential Switching Elements R-S Latch Revisited Truth Table: Next State = F(S, R, Current State) S R Q R-S Latch Q+ Derived K-Map: Characteristic Equation: Q+ = S + R Q t S(t)R(t)Q(t) Q(t+ ) HOLD RESET SET X Not Allowed X

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 114 Application of the SR Latch An important application of SR latches is for recording short lived events e.g. pressing an alarm bell in a hospital

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 115 Clocks and synchronization A clock is a special device that whose output continuously alternates between 0 and 1. The time it takes the clock to change from 1 to 0 and back to 1 is called the clock period, or clock cycle time. The clock frequency is the inverse of the clock period. The unit of measurement for frequency is the hertz. Clocks are often used to synchronize circuits. They generate a repeating, predictable pattern of 0s and 1s that can trigger certain events in a circuit, such as writing to a latch. If several circuits share a common clock signal, they can coordinate their actions with respect to one another. This is similar to how humans use real clocks for synchronization. clock period

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 116 The Clocked SR Latch In some cases it is necessary to disable the inputs to a latch This can be achieved by adding a control or clock input to the latch When C = 0 R and S inputs cannot reach the latch Holds its stored value When C = 1 R and S inputs connected to the latch Functions as before S R Q Q C

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 117 Clocked SR Latch RSCQ n+1 XX0Q n Hold 001Q n Hold 0111Set 1010Reset 111?Unused S R Q Q Q Q R S C C

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 118 Clocked D Latch Simplest clocked latch of practical importance is the Clocked D latch S R Q Q C D It means that both active 1 inputs at R and S cant occur. Notice weve reversed S and R so when D is 1 Q is 1.

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 119 D Latch It removes the undefined behaviour of the SR latch Often used as a basic memory element for the short term storage of a binary digit applied to its input Symbols are often labeled data and enable/clock (D and C) DCQ n+1 X0Q n Hold 010Reset 111Set Circuit D C Q Q SymbolFunction Table S R Q Q Q Q C C D

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 120 Transparency The devices that we have looked so far are transparent That is when C = 1 the output follows the input There will be a slight lag between them C D Q t t t When the clock gate opens, changes in input take effect at outputs – transparency. Also known as level-triggered.

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 121 Latches - Summary Two cross-coupled NOR gates form an SR (set and reset) latch A clocked SR latch has an additional input that controls when setting and resetting can take place A D latch has a single data input the output is held when the clock input is a zero the input is copied to the output when the clock input is a one The output of the clocked latches is transparent The output of the clocked D latch can be represented by the following behavior DCQ n+1 X0Q n Hold 010Reset 111Set

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 122 Latches and Flip Flops Terms are sometimes used confusingly: A latch is not clocked whereas a flip-flop is clocked. A clocked latch can therefore equally be referred to as a flip flop (SR flip flop, D flip flop). However, as we shall see, all practical flip flops are edge- triggered on the clock pulse. Sometimes latches are included within flip flops as a sub-type. Clocked latches are level triggered. While the clock is high, inputs and thus outputs can change. This is not always desirable. A Flip Flop is edge-triggered – either by the leading or falling edge of the clock pulse. Ideally, it responds to the inputs only at a particular instant in time.

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© Bharati Vidyapeeths Institute of Computer Applications and Management, New Delhi-63, by Deepali Kamthania 123

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