16:1 MUX Function Implementation

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1. Function Definition & MUX Basics Function: $F(A,B,C,D,E)= \sum m(2,4,5,7,10,14,15,16,17,25,26,30,31)$ Variables: $A, B, C, D, E$ (5 variables) MUX: 16:1 Multiplexer. A 16:1 MUX has 4 select lines ($S_3, S_2, S_1, S_0$) and 16 data inputs ($I_0$ to $I_{15}$). The output of the MUX is determined by the binary value on the select lines. 2. MUX Input Assignment Strategy To implement a 5-variable function with a 16:1 MUX (4 select lines), we need to use 4 variables as select lines and the remaining variable as the data input controller. Select Lines: Let's choose $B, C, D, E$ as the select lines ($S_3=B, S_2=C, S_1=D, S_0=E$). Data Input Variable: The remaining variable is $A$. The data inputs ($I_0$ to $I_{15}$) will be functions of $A$ (i.e., $0, 1, A, \bar{A}$). 3. K-Map for MUX Input Derivation We'll create a K-map where $B, C, D, E$ form the rows/columns, and $A$ is used to determine the cell values. The minterms are for $ABCDE$. The decimal values correspond to $16A + 8B + 4C + 2D + E$. The MUX input $I_k$ corresponds to the row where $BCDE$ is the binary representation of $k$. 3.1 Minterm Mapping to MUX Inputs For each $I_k$ (where $k$ is $BCDE$ in binary), we check the function $F$ for $A=0$ and $A=1$: If $F=0$ for both $A=0$ and $A=1$, then $I_k = 0$. If $F=1$ for both $A=0$ and $A=1$, then $I_k = 1$. If $F=0$ for $A=0$ and $F=1$ for $A=1$, then $I_k = A$. If $F=1$ for $A=0$ and $F=0$ for $A=1$, then $I_k = \bar{A}$. 3.2 Minterm List: $m(2) \Rightarrow 00010 \quad (A=0, BCDE=0010 \Rightarrow I_2)$ $m(4) \Rightarrow 00100 \quad (A=0, BCDE=0100 \Rightarrow I_4)$ $m(5) \Rightarrow 00101 \quad (A=0, BCDE=0101 \Rightarrow I_5)$ $m(7) \Rightarrow 00111 \quad (A=0, BCDE=0111 \Rightarrow I_7)$ $m(10) \Rightarrow 01010 \quad (A=0, BCDE=1010 \Rightarrow I_{10})$ $m(14) \Rightarrow 01110 \quad (A=0, BCDE=1110 \Rightarrow I_{14})$ $m(15) \Rightarrow 01111 \quad (A=0, BCDE=1111 \Rightarrow I_{15})$ $m(16) \Rightarrow 10000 \quad (A=1, BCDE=0000 \Rightarrow I_0)$ $m(17) \Rightarrow 10001 \quad (A=1, BCDE=0001 \Rightarrow I_1)$ $m(25) \Rightarrow 11001 \quad (A=1, BCDE=1001 \Rightarrow I_9)$ $m(26) \Rightarrow 11010 \quad (A=1, BCDE=1010 \Rightarrow I_{10})$ $m(30) \Rightarrow 11110 \quad (A=1, BCDE=1110 \Rightarrow I_{14})$ $m(31) \Rightarrow 11111 \quad (A=1, BCDE=1111 \Rightarrow I_{15})$ 4. MUX Input Table Let $S_3=B, S_2=C, S_1=D, S_0=E$. Decimal ($BCDE$) $I_k$ $A=0$ ($m_k$) $A=1$ ($m_{16+k}$) MUX Input ($I_k$) 0 (0000) $I_0$ 0 $m_{16}$ (1) $A$ 1 (0001) $I_1$ 0 $m_{17}$ (1) $A$ 2 (0010) $I_2$ $m_2$ (1) 0 $\bar{A}$ 3 (0011) $I_3$ 0 0 0 4 (0100) $I_4$ $m_4$ (1) 0 $\bar{A}$ 5 (0101) $I_5$ $m_5$ (1) 0 $\bar{A}$ 6 (0110) $I_6$ 0 0 0 7 (0111) $I_7$ $m_7$ (1) 0 $\bar{A}$ 8 (1000) $I_8$ 0 0 0 9 (1001) $I_9$ 0 $m_{25}$ (1) $A$ 10 (1010) $I_{10}$ $m_{10}$ (1) $m_{26}$ (1) 1 11 (1011) $I_{11}$ 0 0 0 12 (1100) $I_{12}$ 0 0 0 13 (1101) $I_{13}$ 0 0 0 14 (1110) $I_{14}$ $m_{14}$ (1) $m_{30}$ (1) 1 15 (1111) $I_{15}$ $m_{15}$ (1) $m_{31}$ (1) 1 5. Final MUX Connections Connect the 16:1 MUX as follows: Select Lines: $S_3 = B$ $S_2 = C$ $S_1 = D$ $S_0 = E$ Data Inputs: $I_0 = A$ $I_1 = A$ $I_2 = \bar{A}$ $I_3 = 0$ $I_4 = \bar{A}$ $I_5 = \bar{A}$ $I_6 = 0$ $I_7 = \bar{A}$ $I_8 = 0$ $I_9 = A$ $I_{10} = 1$ $I_{11} = 0$ $I_{12} = 0$ $I_{13} = 0$ $I_{14} = 1$ $I_{15} = 1$ 6. Summary Diagram (Conceptual) 16:1 MUX B (S3) C (S2) D (S1) E (S0) F I0=A I1=A I2=A' I3=0 I4=A' I5=A' I6=0 I7=A' I8=0 I9=A I10=1 I11=0 I12=0 I13=0 I14=1 I15=1