93 lines
1.9 KiB
TeX
Executable File
93 lines
1.9 KiB
TeX
Executable File
\documentclass[
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nosolutions,
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singlenumbering,
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nopagenumber
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]{../../resources/ormc_handout}
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\usepackage{../../resources/macros}
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\title{Warm-Up: Adders}
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\uptitler{\smallurl{}}
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\subtitle{Prepared by Mark on \today}
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\begin{document}
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\maketitle
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\problem{}
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Fill the following binary addition table. \par
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\hint{s is \say{sum,} c is \say{carry}}
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\begin{center}
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\begin{tabular}{ c c || c c }
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$a$ & $b$ & s & c \\
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\hline
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0 & 0 & ? & ? \\
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0 & 1 & ? & ? \\
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1 & 0 & ? & ? \\
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1 & 1 & ? & ?
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\end{tabular}
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\end{center}
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\vfill
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\problem{}
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Draw a logic circuit that atisfies the above table. \par
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This is called a \textit{half adder}. \par
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\hint{You should need exactly two gates.}
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\begin{solution}
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$s = a \texttt{ xor } b$ \par
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$c = a \texttt{ and } b$
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\end{solution}
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\vfill
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\definition{}
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A \textit{full adder} is similar to a half adder, but it has an extra input: \par
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a full adder takes $a$, $b$, and $c_\text{in}$, and produces $s$ and $c_\text{out}$. \par
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\hint{$c_\text{in}$ is \say{carry in}}
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\problem{}
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Use two half adders to construct a full adder.
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\begin{solution}
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$s_1, c_1 = \texttt{HA}(a, b)$ \par
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$s_2, c_2 = \texttt{HA}(s_1, c_\text{in})$ \par
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$s_\text{out} = s_2$ \par
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$c_\text{out} = \texttt{OR}(c_1, c_2)$
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\vspace{2mm}
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Of course, the class should just draw the circuit.
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\end{solution}
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\vfill
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\pagebreak
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\problem{}<rippleadder>
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How can we add two four-bit binary numbers using the full adder? \par
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We want a four-bit output sum and a one-bit $c_\text{out}$.
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\vfill
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\problem{}
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Say that all basic logic gates need $1u$ of time to fully switch states. \par
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\note[Note]{This is called \textit{gate delay}}
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\vspace{2mm}
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How much time does a full adder need to fully switch states? \par
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How about your circuit from \ref{rippleadder}?
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\vfill
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\problem{Bonus}
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Design a faster solution to \ref{rippleadder}.
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\vfill
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\pagebreak
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\end{document} |