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<TITLE> 2.4.4 Exclusive-OR Cell</TITLE>
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<H2>2.4.4 Exclusive-OR Cell</H2>
<P><P CLASS="BodyAfterHead"><A NAME="pgfId=53548"></A>The two-input <B>exclusive-OR</B>
(<B> XOR</B> , EXOR, not-equivalence, ring-OR) function is</P>
<P><P CLASS="EqnNmbrdAlign"><A NAME="pgfId=54093"></A> A1<SPAN CLASS="White"> </SPAN><SPAN CLASS="Symbol">
</SPAN> <SPAN CLASS="White"> </SPAN>A2<SPAN CLASS="White"> </SPAN>=<SPAN CLASS="White"> </SPAN>XOR(A1, A2)<SPAN CLASS="White"> </SPAN>=<SPAN CLASS="White"> </SPAN>A1<SPAN CLASS="White"> </SPAN>·<SPAN CLASS="White"> </SPAN>A2'<SPAN CLASS="White"> </SPAN>+<SPAN CLASS="White"> </SPAN>A1'<SPAN CLASS="White"> </SPAN>·<SPAN CLASS="White"> </SPAN>A2.(2.32)</P>
<P><P CLASS="Body"><A NAME="pgfId=54107"></A>We are now using multiletter
symbols, but there should be no doubt that A1' means anything other than
NOT(A1). We can implement a two-input XOR using a MUX and an inverter as
follows (2 gates):</P>
<P><P CLASS="EqnNmbrdAlign"><A NAME="pgfId=54153"></A> XOR(A1, A2)<SPAN CLASS="White"> </SPAN>=<SPAN CLASS="White"> </SPAN>MUX[NOT(A1), A1, A2],(2.33)</P>
<P><P CLASS="BodyAfterHead"><A NAME="pgfId=54158"></A>where</P>
<P><P CLASS="EqnNmbrdAlign"><A NAME="pgfId=54179"></A> MUX(A,
B, S)<SPAN CLASS="White"> </SPAN>=<SPAN CLASS="White"> </SPAN>A<SPAN CLASS="White"> </SPAN>·<SPAN CLASS="White"> </SPAN>S<SPAN CLASS="White"> </SPAN>+<SPAN CLASS="White"> </SPAN>B<SPAN CLASS="White"> </SPAN>·<SPAN CLASS="White"> </SPAN>S<SPAN CLASS="White"> </SPAN>'.(2.34)</P>
<P><P CLASS="BodyAfterHead"><A NAME="pgfId=54148"></A>This implementation
only buffers one input and does not buffer the MUX output. We can use inverter
buffers (3.5 gates total) or an inverting MUX so that the XOR cell does
not have any external connections to source/drain diffusions as follows
(3 gates total):</P>
<P><P CLASS="EqnNmbrdAlign"><A NAME="pgfId=56615"></A> XOR(A1, A2)<SPAN CLASS="White"> </SPAN>=<SPAN CLASS="White"> </SPAN>NOT[MUX(NOT[NOT(A1)], NOT(A1), A2)].(2.35)</P>
<P><P CLASS="Body"><A NAME="pgfId=54279"></A>We can also implement a two-input
XOR using an AOI21 (and a NOR cell), since</P>
<P><P CLASS="EquationAlign"><A NAME="pgfId=54280"></A> XOR(A1, A2)<SPAN CLASS="White"> </SPAN>=<SPAN CLASS="White"> </SPAN>A1<SPAN CLASS="White"> </SPAN>·<SPAN CLASS="White"> </SPAN>A2'<SPAN CLASS="White"> </SPAN>+<SPAN CLASS="White"> </SPAN>A1'<SPAN CLASS="White"> </SPAN>·<SPAN CLASS="White"> </SPAN>A2<SPAN CLASS="White"> </SPAN>=<SPAN CLASS="White"> </SPAN>[<SPAN CLASS="White"> </SPAN>(A1<SPAN CLASS="White"> </SPAN>·<SPAN CLASS="White"> </SPAN>A2)<SPAN CLASS="White"> </SPAN>+<SPAN CLASS="White"> </SPAN>(A1<SPAN CLASS="White"> </SPAN>+<SPAN CLASS="White"> </SPAN>A2)'
]'</P>
<P><P CLASS="EqnNmbrdAlign"><A NAME="pgfId=197508"></A> =<SPAN CLASS="White"> </SPAN>AOI21[A1,
A2, NOR(A1, A2)],(2.36)</P>
<P><P CLASS="BodyAfterHead"><A NAME="pgfId=197509"></A>(2.5 gates). Similarly
we can implement an exclusive-NOR (XNOR, equivalence) logic cell using an
inverting MUX (and two inverters, total 3.5 gates) or an OAI21 logic cell
(and a NAND cell, total 2.5 gates) as follows (using the MUX function of
Eq. 2.34):</P>
<P><P CLASS="EquationAlign"><A NAME="pgfId=54094"></A> XNOR(A1,
A2)<SPAN CLASS="White"> </SPAN>=<SPAN CLASS="White"> </SPAN>A1<SPAN CLASS="White"> </SPAN>·<SPAN CLASS="White"> </SPAN>A2<SPAN CLASS="White"> </SPAN>+<SPAN CLASS="White"> </SPAN>NOT(A1)<SPAN CLASS="White"> </SPAN>·<SPAN CLASS="White"> </SPAN>NOT(A2)</P>
<P><P CLASS="EquationAlign"><A NAME="pgfId=54453"></A> =<SPAN CLASS="White"> </SPAN>NOT[NOT[MUX(A1,
NOT (A1), A2]]</P>
<P><P CLASS="EqnNmbrdAlign"><A NAME="pgfId=54458"></A> =<SPAN CLASS="White"> </SPAN>OAI21[A1, A2, NAND(A1, A2)](2.37)</P>
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