The Boolean function F = Σ (0,3,5,6,7) with inputs A, B, C (where A is MSB and C is LSB) can be implemented using a Decoder, a Multiplexer, and a Programmable Logic Array (PLA) as follows:

The minterms for which F is true are:
m0 (000)
m3 (011)
m5 (101)
m6 (110)
m7 (111)

1. Implementation using Decoder

A 3-to-8 line decoder generates all 8 minterms (m0 through m7) of the three input variables A, B, and C. To implement the function F, an OR gate is used to combine the specific minterms where F is true.

• Components:
  • One 3-to-8 Decoder with inputs A, B, C.
  • One 5-input OR gate.
• Connections:
  • The inputs A, B, and C are connected to the select lines of the 3-to-8 decoder.
  • The outputs of the decoder corresponding to minterms m0, m3, m5, m6, and m7 are connected to the inputs of the 5-input OR gate.
  • The output of the OR gate represents the function F.

<<<GRAPHVIZ_START>>>
digraph DecoderImplementation {
rankdir=LR;
node [shape=box];

// Inputs
A_in [label="A", shape=none];
B_in [label="B", shape=none];
C_in [label="C", shape=none];

// Decoder module
decoder [label="{<p_A>A|<p_B>B|<p_C>C}|{<p_m0>m0|<p_m1>m1|<p_m2>m2|<p_m3>m3|<p_m4>m4|<p_m5>m5|<p_m6>m6|<p_m7>m7}}", shape=record, fillcolor=lightblue, style=filled, width=1.5];

// OR Gate
or_gate [label="OR", shape=oval, fixedsize=true, width=0.8, height=0.8];

// Output
F_out [label="F", shape=none];

// Connections from inputs to decoder
A_in -> decoder:p_A;
B_in -> decoder:p_B;
C_in -> decoder:p_C;

// Decoder outputs to OR gate for minterms (0,3,5,6,7)
decoder:p_m0 -> or_gate;
decoder:p_m3 -> or_gate;
decoder:p_m5 -> or_gate;
decoder:p_m6 -> or_gate;
decoder:p_m7 -> or_gate;

// OR gate to output
or_gate -> F_out;
}
<<<GRAPHVIZ_END>>>

2. Implementation using Multiplexer

A 2^n-to-1 multiplexer can implement any n-variable Boolean function. For a 3-variable function (A, B, C), an 8-to-1 multiplexer is required.

• Components:
  • One 8-to-1 Multiplexer.
  • Constant logic '1' (VCC) and '0' (GND) sources.
• Connections:
  • The input variables A, B, C are connected to the select lines S2, S1, S0 respectively (A as MSB, C as LSB).
  • The data inputs (I0 to I7) are set to '1' or '0' based on the function's truth table for each minterm:
    • I0 = 1 (for m0)
    • I1 = 0 (for m1)
    • I2 = 0 (for m2)
    • I3 = 1 (for m3)
    • I4 = 0 (for m4)
    • I5 = 1 (for m5)
    • I6 = 1 (for m6)
    • I7 = 1 (for m7)
  • The output of the multiplexer directly provides the function F.

<<<GRAPHVIZ_START>>>
digraph MultiplexerImplementation {
rankdir=LR;
node [shape=box];

// Inputs (Select Lines)
A_sel [label="A", shape=none];
B_sel [label="B", shape=none];
C_sel [label="C", shape=none];

// Constants
VCC [label="1 (VCC)", shape=ellipse, fixedsize=true, width=0.5, height=0.5, fillcolor=green, style=filled];
GND [label="0 (GND)", shape=ellipse, fixedsize=true, width=0.5, height=0.5, fillcolor=red, style=filled];

// 8-to-1 Multiplexer
mux [label="{S2|S1|S0}|{I0|I1|I2|I3|I4|I5|I6|I7}|F_mux", shape=record, fillcolor=lightblue, style=filled];

// Output
F_out [label="F", shape=none];

// Connections for Select Lines
A_sel -> mux:s2 [headlabel="A (MSB)"];
B_sel -> mux:s1 [headlabel="B"];
C_sel -> mux:s0 [headlabel="C (LSB)"];

// Connections for Data Inputs based on F = Σ (0,3,5,6,7)
VCC -> mux:d0; // I0 = 1 (m0)
GND -> mux:d1; // I1 = 0 (m1)
GND -> mux:d2; // I2 = 0 (m2)
VCC -> mux:d3; // I3 = 1 (m3)
GND -> mux:d4; // I4 = 0 (m4)
VCC -> mux:d5; // I5 = 1 (m5)
VCC -> mux:d6; // I6 = 1 (m6)
VCC -> mux:d7; // I7 = 1 (m7)

// Mux output to Function Output
mux:out -> F_out;
}
<<<GRAPHVIZ_END>>>

3. Implementation using Programmable Logic Array (PLA)

A PLA consists of a programmable AND array and a programmable OR array. First, the function needs to be simplified to its Sum-of-Products (SOP) form.

• K-map Simplification for F = Σ (0,3,5,6,7):

  A\BC	00	01	11	10
  0	1	0	1	0
  1	0	1	1	1

  Grouping terms:

  1. Group (m6, m7): ABC' + ABC = AB
  2. Group (m3, m7): A'BC + ABC = BC
  3. Group (m5, m7): AB'C + ABC = AC
  4. Minterm (m0): A'B'C'

  Therefore, the simplified SOP expression is: F = A'B'C' + AB + BC + AC

• PLA Structure:

  • Inputs: A, B, C (and their complements A', B', C').
  • AND Plane: Programmed to generate the four product terms: P1 = A'B'C', P2 = AB, P3 = BC, P4 = AC.
  • OR Plane: Programmed to sum these product terms: F = P1 + P2 + P3 + P4.

• Connections:

  • Input lines A, A', B, B', C, C' feed into the AND plane.
  • Each AND gate in the AND plane is configured to produce one product term:
    • AND gate 1 for P1 (A'B'C'): connected to A', B', C'.
    • AND gate 2 for P2 (AB): connected to A, B.
    • AND gate 3 for P3 (BC): connected to B, C.
    • AND gate 4 for P4 (AC): connected to A, C.
  • The outputs of these four AND gates feed into the OR gate in the OR plane.
  • The output of the OR gate is the function F.

<<<GRAPHVIZ_START>>>
digraph PLA_Implementation {
rankdir=LR;
node [shape=box, style=filled, fillcolor=lightyellow];

// Input Buffers/Inverters
subgraph cluster_inputs {
label="Inputs & Inverters";
A_in [label="A", shape=none];
B_in [label="B", shape=none];
C_in [label="C", shape=none];
inv_A [label="", shape=invtriangle, width=0.4, height=0.4];
inv_B [label="", shape=invtriangle, width=0.4, height=0.4];
inv_C [label="", shape=invtriangle, width=0.4, height=0.4];

A_in -> inv_A [label="A'"];
B_in -> inv_B [label="B'"];
C_in -> inv_C [label="C'"];

}

// AND Plane (Programmable)
subgraph cluster_and_plane {
label="AND Plane";
node [shape=and_gate, width=0.8, height=0.8, fillcolor=orange, style=filled];
P1 [label="P1\n(A'B'C')"];
P2 [label="P2\n(AB)"];
P3 [label="P3\n(BC)"];
P4 [label="P4\n(AC)"];

// Connections to AND gates
inv_A -> P1; inv_B -> P1; inv_C -> P1;
A_in -> P2; B_in -> P2;
B_in -> P3; C_in -> P3;
A_in -> P4; C_in -> P4;

}

// OR Plane (Programmable)
subgraph cluster_or_plane {
label="OR Plane";
node [shape=or_gate, width=0.8, height=0.8, fillcolor=lightgreen, style=filled];
F_out_OR [label="F"];

// Connections from AND gates to OR gate
P1 -> F_out_OR;
P2 -> F_out_OR;
P3 -> F_out_OR;
P4 -> F_out_OR;

}

// Final Output
F_final [label="F", shape=none];
F_out_OR -> F_final;
}
<<<GRAPHVIZ_END>>>

Differentiation between PAL and PLA

Programmable Array Logic (PAL) and Programmable Logic Array (PLA) are types of programmable logic devices (PLDs) used to implement combinational and sequential logic circuits. They differ primarily in their internal architecture, flexibility, and complexity.

• AND/OR Array Structure:

  • PAL: Features a programmable AND array and a fixed OR array. The AND array generates product terms, which are then fed into the fixed OR gates to produce the sum-of-products form for each output.
  • PLA: Consists of both a programmable AND array and a programmable OR array. This allows for greater flexibility as both the product terms and their combinations into sum-of-products can be customized.

• Flexibility and Logic Implementation:

  • PAL: Less flexible. The number of product terms that can be connected to each OR gate (output) is fixed. This limits the complexity of the logic function that can be implemented at each output.
  • PLA: More flexible. Any product term generated by the AND array can be connected to any OR gate. This allows for efficient sharing of product terms among multiple output functions, making it suitable for implementing highly complex sum-of-products expressions.

• Complexity and Size:

  • PAL: Simpler internal structure with fewer programmable connections (fuses) compared to a PLA for a given number of inputs and outputs. This generally leads to smaller chip size.
  • PLA: More complex internal structure due to two programmable arrays, requiring a larger number of programmable connections (fuses). This typically results in a larger chip size.

• Cost:

  • PAL: Generally lower cost due to its simpler architecture and fewer fuses, making it more cost-effective for designs where its fixed OR array structure is sufficient.
  • PLA: Typically higher cost due to the increased complexity and larger number of fuses required for two programmable arrays.

• Ease of Design and Programming:

  • PAL: Simpler to design and program because only the AND array needs to be programmed.
  • PLA: More challenging to design and program, especially in terms of minimizing the number of product terms (to reduce the size of the programmable arrays), due to the flexibility of both AND and OR arrays.

• Applications:

  • PAL: Often used for implementing logic functions where the number of product terms required per output is well-defined and within the device's fixed OR gate capacity, such as address decoders, simple state machines, and combinational glue logic.
  • PLA: Ideal for complex logic functions requiring extensive product term sharing, such as sophisticated state machines, control logic units, and designs where the number of product terms per output varies significantly.

---

Design of a Counter based on the State Diagram

The counter design uses three JK flip-flops (Q2, Q1, Q0) for the 3-bit states.

1. State Table and JK Flip-Flop Excitation:

Present State (Q2 Q1 Q0)	Next State (Q2+ Q1+ Q0+)	J2	K2	J1	K1	J0	K0
000	X X X	X	X	X	X	X	X
001	0 1 1	0	X	1	X	X	0
010	1 1 1	1	X	X	0	1	X
011	0 1 0	0	X	X	0	X	1
100	X X X	X	X	X	X	X	X
101	0 0 1	X	1	0	X	X	0
110	0 0 1	X	1	X	1	1	X
111	1 1 0	X	0	X	0	X	1

Note: 'X' in the Next State column for unused states (000, 100) and in J/K columns represents a "don't care" condition, which can be 0 or 1 to simplify logic.

2. Derivation of Flip-Flop Input Equations (using K-Maps):

• J2:

  Q2\Q1Q0	00	01	11	10
  0	X	0	0	1
  1	X	X	X	X
  J2 = Q2'Q1Q0' (from state 010)

• K2:

  Q2\Q1Q0	00	01	11	10
  0	X	X	X	X
  1	X	1	0	1
  K2 = Q2Q1' + Q2Q0' = Q2(Q1' + Q0') (from states 101 and 110, using 100 as don't care)

• J1:

  Q2\Q1Q0	00	01	11	10
  0	X	1	X	X
  1	X	0	X	X
  J1 = Q2'Q1'Q0 (from state 001, simplifies to Q2'Q0 using don't cares if other Q1 values are X)
  Re-evaluation: Only Q2'Q1'Q0 (001) has J1=1. No other '1's or useful 'X's to simplify.
  J1 = Q2'Q1'Q0

• K1:

  Q2\Q1Q0	00	01	11	10
  0	X	X	0	0
  1	X	X	0	1
  K1 = Q2Q1Q0' (from state 110)

• J0:

  Q2\Q1Q0	00	01	11	10
  0	X	X	X	1
  1	X	X	X	1
  J0 = Q1Q0' (from states 010 and 110)

• K0:

  Q2\Q1Q0	00	01	11	10
  0	X	0	1	X
  1	X	0	1	X
  K0 = Q1Q0 (from states 011 and 111)

Final Derived Equations:

• J2 = Q2'Q1Q0'
• K2 = Q2(Q1' + Q0')
• J1 = Q2'Q1'Q0
• K1 = Q2Q1Q0'
• J0 = Q1Q0'
• K0 = Q1Q0

3. Logic Diagram:

The circuit consists of three JK flip-flops (FF2, FF1, FF0) and combinational logic gates to generate the J and K inputs based on the current state outputs (Q2, Q1, Q0). All flip-flops are synchronously clocked by a common clock signal.

<<<GRAPHVIZ_START>>>
digraph CounterDesign {
rankdir=LR;
node [shape=box style=filled fillcolor=lightgrey fontname="Arial"];

// Flip-flops
subgraph cluster_ffs {
    label = "JK Flip-Flops";
    color=blue;
    FF2 [label="JK FF (Q2)"];
    FF1 [label="JK FF (Q1)"];
    FF0 [label="JK FF (Q0)"];
}

// Input/Output nodes for Q and Q_bar
Q2 [shape=circle label="Q2" style=solid fillcolor=white];
Q1 [shape=circle label="Q1" style=solid fillcolor=white];
Q0 [shape=circle label="Q0" style=solid fillcolor=white];
Q2_bar [shape=circle label="Q2'" style=solid fillcolor=white];
Q1_bar [shape=circle label="Q1'" style=solid fillcolor=white];
Q0_bar [shape=circle label="Q0'" style=solid fillcolor=white];

// Clock
CLK [label="CLK", shape=octagon style=filled fillcolor=lightgreen];

// Connections from FF outputs to Q/Q_bar nodes
FF2:Q -> Q2;
FF2:Q_bar -> Q2_bar;
FF1:Q -> Q1;
FF1:Q_bar -> Q1_bar;
FF0:Q -> Q0;
FF0:Q_bar -> Q0_bar;

// Combinational Logic for J2, K2
node [shape=Mrecord style=solid fillcolor=white];
J2_AND [label="{AND| J2 = Q2'Q1Q0'}"];
K2_OR [label="{OR| Q1'+Q0'}"];
K2_AND [label="{AND| K2 = Q2(Q1'+Q0')}"];

Q2_bar -> J2_AND:p1 [headport=w];
Q1 -> J2_AND:p2 [headport=w];
Q0_bar -> J2_AND:p3 [headport=w];
J2_AND -> FF2:J;

Q1_bar -> K2_OR:p1 [headport=w];
Q0_bar -> K2_OR:p2 [headport=w];
K2_OR -> K2_AND:p1 [headport=w];
Q2 -> K2_AND:p2 [headport=w];
K2_AND -> FF2:K;

// Combinational Logic for J1, K1
J1_AND [label="{AND| J1 = Q2'Q1'Q0}"];
K1_AND [label="{AND| K1 = Q2Q1Q0'}"];

Q2_bar -> J1_AND:p1 [headport=w];
Q1_bar -> J1_AND:p2 [headport=w];
Q0 -> J1_AND:p3 [headport=w];
J1_AND -> FF1:J;

Q2 -> K1_AND:p1 [headport=w];
Q1 -> K1_AND:p2 [headport=w];
Q0_bar -> K1_AND:p3 [headport=w];
K1_AND -> FF1:K;

// Combinational Logic for J0, K0
J0_AND [label="{AND| J0 = Q1Q0'}"];
K0_AND [label="{AND| K0 = Q1Q0}"];

Q1 -> J0_AND:p1 [headport=w];
Q0_bar -> J0_AND:p2 [headport=w];
J0_AND -> FF0:J;

Q1 -> K0_AND:p1 [headport=w];
Q0 -> K0_AND:p2 [headport=w];
K0_AND -> FF0:K;

// Clock connections
CLK -> FF0:CLK;
CLK -> FF1:CLK;
CLK -> FF2:CLK;

// Arrange nodes
{rank=same; Q2_bar; Q1_bar; Q0_bar;}
{rank=same; Q2; Q1; Q0;}
{rank=same; J2_AND; K2_OR; K2_AND;}
{rank=same; J1_AND; K1_AND;}
{rank=same; J0_AND; K0_AND;}
{rank=same; FF2; FF1; FF0;}

}
<<<GRAPHVIZ_END>>>

Block Diagram

<<<GRAPHVIZ_START>>>
digraph BlockDiagram {
rankdir=LR;
node [shape=rect, style=filled, fillcolor=lightblue, fontname="Arial"];

// Inputs
D_in [label="Data_in", shape=point, style=invis];
EN [label="Enable", shape=point, style=invis];
S3 [label="S3", shape=point, style=invis];
S2 [label="S2", shape=point, style=invis];
S1 [label="S1", shape=point, style=invis];
S0 [label="S0", shape=point, style=invis];

// Demultiplexer Block
Demux [label="1x16 Demultiplexer\n(DEMUX)", width=2, height=3];

// Outputs
Y0 [label="Y0", shape=point, style=invis];
Y1 [label="Y1", shape=point, style=invis];
Y_dots [label="...", shape=none, fontcolor=darkgray, fontsize=18];
Y15 [label="Y15", shape=point, style=invis];

// Visible labels for inputs
label_Din [label="Data Input (D_in)", shape=plaintext];
label_EN [label="Enable (EN)", shape=plaintext];
label_S [label="Select Lines\n(S3, S2, S1, S0)", shape=plaintext];

// Visible labels for outputs
label_Y0 [label="Y0", shape=plaintext];
label_Y1 [label="Y1", shape=plaintext];
label_Y15 [label="Y15", shape=plaintext];

// Connections to Demux
label_Din -> Demux [arrowhead=vee, label="", headport=w, tailport=e];
label_EN -> Demux [arrowhead=vee, label="", headport=w, tailport=e];
label_S -> Demux [arrowhead=vee, label="", headport=w, tailport=e];

// Connections from Demux
Demux -> label_Y0 [arrowhead=vee, label="", headport=e, tailport=w];
Demux -> label_Y1 [arrowhead=vee, label="", headport=e, tailport=w];
Demux -> Y_dots [style=invis, headport=e, tailport=w]; // Connect for spacing
Demux -> label_Y15 [arrowhead=vee, label="", headport=e, tailport=w];

// Arrange nodes
{rank=same; label_Din; label_EN; label_S;}
{rank=same; Demux;}
{rank=same; label_Y0; label_Y1; Y_dots; label_Y15;}

// Invisible edges for alignment and spacing
label_Din -> Demux [minlen=1, style=invis];
label_EN -> Demux [minlen=1, style=invis];
label_S -> Demux [minlen=1, style=invis];
Demux -> label_Y0 [minlen=1, style=invis];
Demux -> label_Y1 [minlen=1, style=invis];
Demux -> label_Y15 [minlen=1, style=invis];

label_S -> Demux [style=invis, dir=none, constraint=false];
label_S -> label_Y0 [style=invis, dir=none, constraint=false];
label_S -> label_Y15 [style=invis, dir=none, constraint=false];

// Vertical alignment
{rank=sink; label_Y0; label_Y1; Y_dots; label_Y15;}
{rank=source; label_Din; label_EN; label_S;}

// Explicit positioning for a cleaner look
label_Din -> Demux [arrowhead=vee, ltail=label_Din, lhead=Demux, style=solid, penwidth=1.5];
label_EN -> Demux [arrowhead=vee, ltail=label_EN, lhead=Demux, style=solid, penwidth=1.5];
label_S -> Demux [arrowhead=vee, ltail=label_S, lhead=Demux, style=solid, penwidth=1.5];

Demux -> label_Y0 [arrowhead=vee, ltail=Demux, lhead=label_Y0, style=solid, penwidth=1.5];
Demux -> label_Y1 [arrowhead=vee, ltail=Demux, lhead=label_Y1, style=solid, penwidth=1.5];
Demux -> Y_dots [arrowhead=none, ltail=Demux, lhead=Y_dots, style=dotted, penwidth=1.5];
Demux -> label_Y15 [arrowhead=vee, ltail=Demux, lhead=label_Y15, style=solid, penwidth=1.5];

// Align the text labels for inputs vertically
{ rank=same; label_Din; label_EN; label_S; }
{ rank=same; label_Y0; label_Y1; Y_dots; label_Y15; }

label_Din -> Demux [arrowhead=vee, dir=forward, label="", tailport=e, headport=w];
label_EN -> Demux [arrowhead=vee, dir=forward, label="", tailport=e, headport=w];
label_S -> Demux [arrowhead=vee, dir=forward, label="", tailport=e, headport=w];

Demux -> label_Y0 [arrowhead=vee, dir=forward, label="", tailport=e, headport=w];
Demux -> label_Y1 [arrowhead=vee, dir=forward, label="", tailport=e, headport=w];
Demux -> label_Y15 [arrowhead=vee, dir=forward, label="", tailport=e, headport=w];

}
<<<GRAPHVIZ_END>>>

Truth Table

EN	S3	S2	S1	S0	D_in	Y0	Y1	...	Y15
0	X	X	X	X	X	0	0	...	0
1	0	0	0	0	0	0	0	...	0
1	0	0	0	0	1	1	0	...	0
1	0	0	0	1	0	0	0	...	0
1	0	0	0	1	1	0	1	...	0
...	...	...	...	...	...	...	...	...	...
1	1	1	1	1	0	0	0	...	0
1	1	1	1	1	1	0	0	...	1

• X denotes a "don't care" condition.
• When EN = 0, all outputs (Y0-Y15) are forced to 0, regardless of other inputs.
• When EN = 1, the select lines (S3 S2 S1 S0) determine which output channel mirrors the D_in signal. All other unselected outputs remain 0.

Logic Circuit

<<<GRAPHVIZ_START>>>
digraph LogicCircuit {
rankdir=LR;
node [shape=rect, style=filled, fillcolor=lightgray, fontname="Arial"];

// Inputs
D_in_node [label="Data Input (D_in)"];
EN_node [label="Enable (EN)"];
S3_node [label="Select S3"];
S2_node [label="Select S2"];
S1_node [label="Select S1"];
S0_node [label="Select S0"];

// Intermediate AND for Data & Enable
node [shape=circle, label="&", fillcolor=white, fontname="Arial"];
AND_EN_D [label="AND Gate"];

// Decoder Block
node [shape=component, label="4-to-16\nDecoder", fillcolor=lightblue, fontname="Arial"];
Decoder;

// Output AND gates (showing first two and last one, others implied)
node [shape=circle, label="&", fillcolor=white, fontname="Arial"];
AND0 [label="AND Gate"];
AND1 [label="AND Gate"];
AND_dots [label="...", shape=none, fontcolor=darkgray, fontsize=18];
AND15 [label="AND Gate"];

// Outputs
node [shape=rect, style=filled, fillcolor=lightgray, fontname="Arial"];
Y0_node [label="Output Y0"];
Y1_node [label="Output Y1"];
Y_dots_out [label="...", shape=none, fontcolor=darkgray, fontsize=18];
Y15_node [label="Output Y15"];

// Connections
D_in_node -> AND_EN_D;
EN_node -> AND_EN_D;

S3_node -> Decoder;
S2_node -> Decoder;
S1_node -> Decoder;
S0_node -> Decoder;

// Connect Data_Enabled to all output AND gates
AND_EN_D -> AND0 [label="Data_Enabled"];
AND_EN_D -> AND1 [label="Data_Enabled"];
AND_EN_D -> AND_dots [style=dotted, label=""]; // Implied connection to all
AND_EN_D -> AND15 [label="Data_Enabled"];

// Connect Decoder outputs (minterms) to respective AND gates
Decoder -> AND0 [label="M0"];
Decoder -> AND1 [label="M1"];
Decoder -> AND_dots [style=dotted, label="M2-M14"]; // Implied connections
Decoder -> AND15 [label="M15"];

// Connect AND gates to final outputs
AND0 -> Y0_node;
AND1 -> Y1_node;
AND_dots -> Y_dots_out [style=invis]; // Link for visual continuity
AND15 -> Y15_node;

{rank=same; D_in_node; EN_node; S3_node; S2_node; S1_node; S0_node;}
{rank=same; AND_EN_D; Decoder;}
{rank=same; AND0; AND1; AND_dots; AND15;}
{rank=same; Y0_node; Y1_node; Y_dots_out; Y15_node;}

}
<<<GRAPHVIZ_END>>>

Working Principle

A 1x16 Demultiplexer (DEMUX) functions as a data distributor, taking a single data input and routing it to one of 16 possible output lines, determined by a set of select inputs.

1. Inputs:

   • Data Input (D_in): This is the single data bit (0 or 1) that needs to be transmitted to one of the output lines.
   • Select Lines (S3, S2, S1, S0): These four lines form a 4-bit binary code (2^4 = 16) that specifies which of the 16 output lines should receive the data input.
   • Enable Input (EN): This control input enables or disables the entire demultiplexer. When EN is inactive (e.g., LOW), all outputs are held in a default inactive state (e.g., LOW) regardless of the data and select inputs. When EN is active (e.g., HIGH), the demultiplexer operates normally.

2. Operation:

   • When the Enable (EN) input is active, the demultiplexer becomes operational.
   • The 4-bit binary code on the select lines (S3 S2 S1 S0) is interpreted to identify one specific output line out of the 16 available. For example, if S3S2S1S0 = "0000", output Y0 is selected; if S3S2S1S0 = "1111", output Y15 is selected.
   • The value present at the Data Input (D_in) is then transferred to the currently selected output line.
   • All other unselected output lines remain in their inactive state (typically LOW).

3. Internal Logic:

   • The core of a 1x16 Demultiplexer is often a 4-to-16 decoder. This decoder takes the four select lines as input and generates 16 unique active signals (minterms), each corresponding to one of the 16 possible binary combinations on the select lines.
   • These 16 minterm outputs from the decoder are then combined with the Data Input (D_in) using 16 separate 2-input AND gates.
   • Specifically, the enable input (EN) is usually ANDed with the Data Input (D_in) first to create an "enabled data" signal. This "enabled data" signal then serves as one input to all 16 AND gates. Each corresponding minterm from the decoder serves as the second input to its respective AND gate.
   • Only the AND gate whose decoder input (minterm) is active (due to the select lines) will pass the "enabled data" signal to its output, thus making the selected output Y_i = D_in, while all other outputs remain 0.

In essence, a 1x16 demultiplexer acts as a switch, directing a single stream of input data to one of 16 distinct output channels based on a binary address provided by its select lines, with an overall control provided by the enable input.

The operations are performed using 7-bit signed 2's complement representation. The range for 7-bit signed 2's complement is -64 to +63, which accommodates the given numbers.

1. Binary Representation:

• +42:
  • Decimal 42 = 0101010₂
  • In 7-bit signed 2's complement: 0101010
• -13:
  • Decimal 13 = 0001101₂
  • 1's complement of +13 (0001101): 1110010
  • Add 1 for 2's complement: 1110010 + 1 = 1110011
• -42:
  • Decimal 42 = 0101010₂
  • 1's complement of +42 (0101010): 1010101
  • Add 1 for 2's complement: 1010101 + 1 = 1010110

2. Arithmetic Operations:

(a) (+42) + (-13)

• Operation: Add the 2's complement representations.
• Calculation:

    0101010  (+42)
  + 1110011  (-13)
  ---------
  (1)0011101  (Carry out ignored for 7-bit result)
  

• Result: 0011101
  • Verification: 0011101₂ = 16 + 8 + 4 + 1 = 29₁₀.
  • (+42) + (-13) = +29.

(b) (-42) - (-13)

• This operation is equivalent to (-42) + (+13) in 2's complement.
• We need the 2's complement of -13, which is +13.
  • +13 in 7-bit signed 2's complement: 0001101
• Calculation:

    1010110  (-42)
  + 0001101  (+13)
  ---------
    1100011
  

• Result: 1100011
  • Verification: The MSB is 1, so it's a negative number.
    • Take 2's complement of 1100011 to find its magnitude:
      • 1's complement: 0011100
      • Add 1: 0011100 + 1 = 0011101
    • 0011101₂ = 29₁₀.
    • Therefore, 1100011 represents -29₁₀.
  • (-42) - (-13) = -42 + 13 = -29.

The function F(A, B, C, D) is defined by its min-terms:
F(A, B, C, D) = Σ(0, 2, 6, 11, 13, 14)

For a 4-variable function (A, B, C, D), there are 2^4 = 16 possible min-terms, ranging from m0 to m15.
The complement of the function, F'(A, B, C, D), consists of all min-terms that are not present in F.

The complete set of min-terms is {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}.
The min-terms in F are {0, 2, 6, 11, 13, 14}.

To find the min-terms for F', subtract the min-terms of F from the complete set:
F' = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} - {0, 2, 6, 11, 13, 14}

The resulting set of min-terms for F' is:
{1, 3, 4, 5, 7, 8, 9, 10, 12, 15}

Therefore, the complement of the function in sum of min-terms is:
F'(A, B, C, D) = Σ(1, 3, 4, 5, 7, 8, 9, 10, 12, 15)

To reduce the function F = wxy + yz + xy’z + x’y using a K-map:

1. Convert to Minterms (Optional, but helps with K-map population):

• wxy: w=1, x=1, y=1 (z can be 0 or 1) -> m14 (1110), m15 (1111)
• yz: y=1, z=1 (w, x can be any) -> m3 (0011), m7 (0111), m11 (1011), m15 (1111)
• xy’z: x=1, y=0, z=1 (w can be 0 or 1) -> m5 (0101), m13 (1101)
• x’y: x=0, y=1 (w, z can be any) -> m2 (0010), m3 (0011), m10 (1010), m11 (1011)

Combined minterms for F: {m2, m3, m5, m7, m10, m11, m13, m14, m15}

2. Draw the 4-variable K-map and fill '1's for the minterms:
<<<GRAPHVIZ_START>>>
digraph Kmap {
graph [splines=false overlap=false]
node [shape=none fontname="monospace" fontsize=10]

kmap [label=<

	yz
	00	01	11	10
wx	00	0	0	1	1
01	0	1	1	0
11	0	1	1	1
10	0	0	1	1

]
}
<<<GRAPHVIZ_END>>>

3. Group the '1's for Prime Implicants:

The goal is to cover all '1's with the largest possible groups (octets, quads, pairs) and the minimum number of groups.

• Group 1 (Quad - yz): Covers minterms m3, m7, m11, m15. (Column 11 across all rows).
• Group 2 (Quad - x’y): Covers minterms m2, m3, m10, m11. (Rows 00 and 10, columns 10 and 11 - this group wraps around the K-map horizontally).
• Group 3 (Pair - xy’z): Covers minterms m5, m13. (Column 01, rows 01 and 11).
• Group 4 (Pair - wxy): Covers minterms m14, m15. (Row 11, columns 10 and 11).

4. Identify Essential Prime Implicants:

• m2 is uniquely covered by x'y. Therefore, x'y is an Essential Prime Implicant.
• m5 is uniquely covered by xy'z. Therefore, xy'z is an Essential Prime Implicant.
• m7 is uniquely covered by yz. Therefore, yz is an Essential Prime Implicant.
• m10 is uniquely covered by x'y.
• m13 is uniquely covered by xy'z.
• m14 is uniquely covered by wxy. Therefore, wxy is an Essential Prime Implicant.

Since all '1's are covered by these four essential prime implicants, they form the minimal sum-of-products expression.

5. Write the Reduced Function:

F = x’y + yz + xy’z + wxy

Therefore, the given function is already in its minimal Sum-of-Products form.

Combinational Circuit Design: 3-bit Squarer

The circuit takes a 3-bit binary number as input and produces a 6-bit binary number as output, representing the square of the input.

1. Input and Output Specification

• Inputs: Let the 3-bit input be $I_2 I_1 I_0$, where $I_2$ is the MSB and $I_0$ is the LSB. This represents decimal numbers from 0 to 7.
• Outputs: Let the 6-bit output be $O_5 O_4 O_3 O_2 O_1 O_0$, where $O_5$ is the MSB and $O_0$ is the LSB. The maximum output is $7^2 = 49$, which requires 6 bits ($2^5 = 32, 2^6 = 64$).

2. Truth Table

The truth table maps all possible 3-bit inputs to their corresponding 6-bit squared outputs.

Decimal Input	$I_2 I_1 I_0$	Decimal Output	$O_5 O_4 O_3 O_2 O_1 O_0$
0	000	0	000000
1	001	1	000001
2	010	4	000100
3	011	9	001001
4	100	16	010000
5	101	25	011001
6	110	36	100100
7	111	49	110001

3. Boolean Expressions for Each Output

Deriving simplified Sum-of-Products (SOP) expressions for each output bit from the truth table (using K-maps or algebraic simplification):

• $O_0 = I_0$
• $O_1 = I_0(I_1 + I_2)$
• $O_2 = I_1\overline{I_0}$
• $O_3 = I_0(I_1 \oplus I_2)$
• $O_4 = I_2(\overline{I_1} + I_0)$
• $O_5 = I_2I_1$

4. Circuit Diagram

The combinational circuit can be implemented using basic logic gates (AND, OR, NOT, XOR) according to the derived Boolean expressions.

<<<GRAPHVIZ_START>>>
digraph G {
rankdir=LR;
node [shape=box, style=filled, fillcolor=lightgrey];

// Inputs
I2 [label="I2"];
I1 [label="I1"];
I0 [label="I0"];

// Inverters
not_I0 [shape=circle, label="NOT"];
not_I1 [shape=circle, label="NOT"];

I0 -> not_I0;
I1 -> not_I1;

// Output O0
O0 [label="O0", shape=rpromoter, style=filled, fillcolor=lightgreen];
I0 -> O0;

// Output O1
or_O1 [shape=ellipse, label="OR"];
and_O1 [shape=box, label="AND"];
I1 -> or_O1;
I2 -> or_O1;
I0 -> and_O1;
or_O1 -> and_O1;
O1 [label="O1", shape=rpromoter, style=filled, fillcolor=lightgreen];
and_O1 -> O1;

// Output O2
and_O2 [shape=box, label="AND"];
I1 -> and_O2;
not_I0 -> and_O2;
O2 [label="O2", shape=rpromoter, style=filled, fillcolor=lightgreen];
and_O2 -> O2;

// Output O3
xor_O3 [shape=doublecircle, label="XOR"];
and_O3 [shape=box, label="AND"];
I1 -> xor_O3;
I2 -> xor_O3;
I0 -> and_O3;
xor_O3 -> and_O3;
O3 [label="O3", shape=rpromoter, style=filled, fillcolor=lightgreen];
and_O3 -> O3;

// Output O4
or_O4 [shape=ellipse, label="OR"];
and_O4 [shape=box, label="AND"];
not_I1 -> or_O4;
I0 -> or_O4;
I2 -> and_O4;
or_O4 -> and_O4;
O4 [label="O4", shape=rpromoter, style=filled, fillcolor=lightgreen];
and_O4 -> O4;

// Output O5
and_O5 [shape=box, label="AND"];
I2 -> and_O5;
I1 -> and_O5;
O5 [label="O5", shape=rpromoter, style=filled, fillcolor=lightgreen];
and_O5 -> O5;

// Invisible edges for consistent input spacing
{rank=same; I2; I1; I0;}
{rank=same; O5; O4; O3; O2; O1; O0;}
}
<<<GRAPHVIZ_END>>>

<<<GRAPHVIZ_START>>>
digraph G {
rankdir=LR;
node [shape=box, style="filled", fillcolor="#e6e6fa"];
edge [dir=forward];

// Input signals
A4 [label="A4 (MSB)", shape=circle, fixedsize=true, width=0.3];
A3 [label="A3", shape=circle, fixedsize=true, width=0.3];
A2 [label="A2", shape=circle, fixedsize=true, width=0.3];
A1 [label="A1", shape=circle, fixedsize=true, width=0.3];
A0 [label="A0 (LSB)", shape=circle, fixedsize=true, width=0.3];

// 2x4 Decoder block
dec2x4 [label="{ A4 | A3 | 2x4 Decoder | E0 | E1 | E2 | E3 }", shape=record];

// 3x8 Decoder blocks
dec3x8_0 [label="{ A2 | A1 | A0 | Enable | 3x8 Decoder (0) | D0-D7}", shape=record];
dec3x8_1 [label="{ A2 | A1 | A0 | Enable | 3x8 Decoder (1) | D8-D15}", shape=record];
dec3x8_2 [label="{ A2 | A1 | A0 | Enable | 3x8 Decoder (2) | D16-D23}", shape=record];
dec3x8_3 [label="{ A2 | A1 | A0 | Enable | 3x8 Decoder (3) | D24-D31}", shape=record];

// Connect inputs A4, A3 to 2x4 Decoder
A4 -> dec2x4:A4;
A3 -> dec2x4:A3;

// Connect outputs of 2x4 Decoder to Enable inputs of 3x8 Decoders
dec2x4:E0 -> dec3x8_0:EN;
dec2x4:E1 -> dec3x8_1:EN;
dec2x4:E2 -> dec3x8_2:EN;
dec2x4:E3 -> dec3x8_3:EN;

// Connect common inputs A2, A1, A0 to all 3x8 Decoders
A2 -> dec3x8_0:A2;
A1 -> dec3x8_0:A1;
A0 -> dec3x8_0:A0;

A2 -> dec3x8_1:A2;
A1 -> dec3x8_1:A1;
A0 -> dec3x8_1:A0;

A2 -> dec3x8_2:A2;
A1 -> dec3x8_2:A1;
A0 -> dec3x8_2:A0;

A2 -> dec3x8_3:A2;
A1 -> dec3x8_3:A1;
A0 -> dec3x8_3:A0;

// Output labels
output_D0_D7 [label="D0-D7", shape=none];
output_D8_D15 [label="D8-D15", shape=none];
output_D16_D23 [label="D16-D23", shape=none];
output_D24_D31 [label="D24-D31", shape=none];

dec3x8_0:OUT -> output_D0_D7;
dec3x8_1:OUT -> output_D8_D15;
dec3x8_2:OUT -> output_D16_D23;
dec3x8_3:OUT -> output_D24_D31;

// Arrange decoders vertically
{ rank=same; dec2x4; }
{ rank=same; A2; A1; A0; } // Keep A2,A1,A0 aligned horizontally
{ rank=same; dec3x8_0; dec3x8_1; dec3x8_2; dec3x8_3; }
{ rank=max; output_D0_D7; output_D8_D15; output_D16_D23; output_D24_D31; }

// Invisible edges for layout control
dec2x4 -> dec3x8_0 [style=invis, weight=10];
dec3x8_0 -> dec3x8_1 [style=invis];
dec3x8_1 -> dec3x8_2 [style=invis];
dec3x8_2 -> dec3x8_3 [style=invis];
}
<<<GRAPHVIZ_END>>>

A Decimal (BCD) adder is a digital circuit that adds two 4-bit Binary-Coded Decimal (BCD) numbers, producing a 4-bit BCD sum and a carry-out. Unlike standard binary adders, BCD adders require special correction logic because a 4-bit binary sum can represent values up to 15, while BCD digits only range from 0 to 9.

Design Explanation:

A BCD adder typically consists of two main stages:

1. Initial Binary Addition:

   • Two 4-bit BCD digits (A and B) and an input carry (Cin) are added using a standard 4-bit binary adder.
   • The output is an intermediate 4-bit binary sum (S3S2S1S0) and an initial carry-out (Cout1).

2. Correction Logic:

   • If the intermediate sum (S3S2S1S0) is less than or equal to 9, and Cout1 is 0, the sum is already a valid BCD digit. No correction is needed, and the final carry-out (Cout_final) is 0.
   • If the intermediate sum is greater than 9 (i.e., 10 to 15) OR if Cout1 is 1, the sum is an invalid BCD digit. To correct this, binary 6 (0110) is added to the intermediate sum. This correction forces the sum to roll over to the next BCD decade, and the Cout_final is set to 1. The result of this addition (S_corrected) becomes the final BCD sum.

The condition for correction (K) is determined by:
K = Cout1 OR (S3 AND S2) OR (S3 AND S1)
Where S3S2S1S0 is the output of the first 4-bit adder.

Truth Table (Conceptual for Correction Logic):

The correction logic effectively adjusts the binary sum to its BCD equivalent.

Initial Sum (S3S2S1S0)	Initial Carry (Cout1)	Correction Condition (K)	Action	Final BCD Sum (R3R2R1R0)	Final Carry (Cout_final)
0000-1001 (0-9)	0	0	No Correction	Same as Initial Sum	0
1010-1111 (10-15)	0	1	Add 0110 (binary 6)	Initial Sum + 0110	1
X	1	1	Add 0110 (binary 6)	Initial Sum + 0110	1

*(X indicates 'don't care' since Cout1=1 already triggers correction)*

Suitable Diagram (Block Diagram of a Single-Digit BCD Adder):

<<<GRAPHVIZ_START>>>
digraph BCD_Adder {
rankdir=LR;
node [shape=box, style=filled, fillcolor=lightgoldenrodyellow];

// Inputs
A [label="A (BCD Digit)\n(A3 A2 A1 A0)"];
B [label="B (BCD Digit)\n(B3 B2 B1 B0)"];
Cin [label="Cin"];

// First 4-bit Binary Adder
Adder1 [label="4-bit Binary Adder"];

// Outputs of first adder
Sum_initial [label="Sum_initial\n(S3 S2 S1 S0)"];
Cout1 [label="Cout1"];

// Correction Logic
Correction_Logic [label="Correction Logic\n(K = Cout1 OR (S3S2) OR (S3S1))", shape=octagon, style="filled", fillcolor=lightblue];

// Output of Correction Logic (Correction Flag / Final Carry)
K [label="K (Correction Flag\n & Final Carry)", shape=box, style="filled", fillcolor=lightgreen];

// Second 4-bit Binary Adder for correction
Adder2 [label="4-bit Binary Adder"];
Add6 [label="0110 (Binary 6)", shape=plaintext]; // Constant input for adding 6

// Outputs
Sum_final [label="Sum_final\n(R3 R2 R1 R0)", style="filled", fillcolor=lightgray];
Cout_final [label="Cout_final", style="filled", fillcolor=lightgray];

// Connections
A -> Adder1;
B -> Adder1;
Cin -> Adder1;

Adder1 -> Sum_initial;
Adder1 -> Cout1;

Sum_initial -> Correction_Logic;
Cout1 -> Correction_Logic;

Correction_Logic -> K;

Sum_initial -> Adder2 [label="Input 1"];
Add6 -> Adder2 [label="Input 2"];
K -> Adder2 [label="Carry-in"];

Adder2 -> Sum_final;
K -> Cout_final [label=" (Same as K)"]; // The correction flag K also acts as the final carry-out
}
<<<GRAPHVIZ_END>>>

A shift register with parallel load is a digital circuit capable of performing both serial shifting of data and simultaneous parallel loading of data into its internal flip-flops. It typically includes control inputs to select between these operational modes.

Operation:

• Parallel Load Mode: When the control signal (e.g., Load) is active, data present on the parallel input lines is loaded into all flip-flops of the register simultaneously on the clock edge.
• Shift Mode: When the control signal (e.g., Shift) is active, data is shifted serially from one flip-flop to the next on each clock edge, similar to a standard shift register. Data can be shifted in or out serially.

Practical Implications:

• Data Conversion: Facilitates both Serial-to-Parallel (SIPO) and Parallel-to-Serial (PISO) data conversion, essential in communication systems and data processing.
• Data Manipulation: Useful for temporary data storage, data alignment, and reordering of bit sequences.
• Bus Interface: Can act as an interface between parallel data buses and serial data lines, commonly found in microcontrollers and peripheral devices.
• Pattern Generation: Can be configured to generate specific bit patterns or sequences.

A master-slave J-K flip-flop is a cascaded connection of two J-K flip-flops, where the first is designated as the 'master' and the second as the 'slave'. This configuration is designed to overcome the race-around condition present in single J-K flip-flops when operating in toggle mode with a positive level-triggered clock.

Key characteristics:

• Structure: Consists of two stages:
  • Master Flip-Flop: A level-triggered J-K flip-flop that operates when the clock pulse is HIGH.
  • Slave Flip-Flop: A level-triggered J-K flip-flop that operates when the clock pulse is LOW.
• Clock Inversion: The clock signal applied to the slave flip-flop is an inverted version of the clock signal applied to the master flip-flop.
• Operation:
  1. When the clock pulse is HIGH, the master flip-flop is enabled. Its outputs (Q_M and Q̄_M) respond to the J and K inputs. The slave flip-flop is disabled during this phase.
  2. When the clock pulse transitions from HIGH to LOW, the master flip-flop becomes disabled, latching its current output state.
  3. Simultaneously, the slave flip-flop becomes enabled (due to the inverted clock). It then reads the latched outputs of the master flip-flop (Q_M and Q̄_M) and transfers them to its own outputs (Q and Q̄).
  4. When the clock pulse is LOW, the slave is enabled and holds its state. The master is disabled.
• Race-around Prevention: By using the master to sample inputs during the first half of the clock pulse and the slave to output during the second half, the master-slave configuration ensures that the flip-flop output changes only after the active clock edge (typically the falling edge) has passed, effectively isolating the input from the output during the critical period and preventing the race-around condition.

De-Morgan’s Theorem

De-Morgan’s theorems are a pair of transformation rules in Boolean algebra that are fundamental for simplifying Boolean expressions and for converting between different forms of logic gates (e.g., NAND to OR, NOR to AND). They state that:

• The complement of a product of variables is equal to the sum of the complements of the variables.
  • $\overline{A \cdot B} = \overline{A} + \overline{B}$
  • This implies that a NAND gate behaves like an OR gate with inverted inputs.
• The complement of a sum of variables is equal to the product of the complements of the variables.
  • $\overline{A + B} = \overline{A} \cdot \overline{B}$
  • This implies that a NOR gate behaves like an AND gate with inverted inputs.

These theorems are extensively used in digital circuit design for logic minimization and for implementing circuits using universal gates (NAND or NOR gates exclusively).

TTL (Transistor-Transistor Logic)

Transistor-Transistor Logic (TTL) is a family of digital integrated circuits constructed using bipolar junction transistors (BJTs). It was a dominant technology for digital logic circuits for several decades.

Key characteristics of TTL:

• Construction: Employs multiple-emitter transistors at the input stage, followed by a phase splitter and a totem-pole output stage.
• Voltage Levels: Operates with specific voltage ranges for logic HIGH and LOW states, typically:
  • Output HIGH (V_OH_): > 2.4V (min)
  • Output LOW (V_OL_): < 0.4V (max)
  • Input HIGH (V_IH_): > 2V (min)
  • Input LOW (V_IL_): < 0.8V (max)
• Power Dissipation: Generally higher compared to CMOS, and increases with operating frequency.
• Speed: Offers good switching speeds, with propagation delays in the range of nanoseconds. Various sub-families (e.g., Schottky TTL, Low-Power Schottky TTL) were developed to optimize speed and power consumption.
• Noise Margin: Typically good noise immunity due to distinct voltage thresholds.

TTL devices are widely used in general-purpose digital applications, interfacing with microprocessors, and building combinational and sequential logic circuits.