ch02.6.htm

介绍asci设计的一本书

NA</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableLeft">
<A NAME="pgfId=162410">
 </A>
Z = A;</P>
<P CLASS="TableLeft">
<A NAME="pgfId=162411">
 </A>
SG(Z) = NOT(SG(A))</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableLeft">
<A NAME="pgfId=162413">
 </A>
Z = NOT(A)</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableLeft">
<A NAME="pgfId=162415">
 </A>
Z = NOT(A) + 1</P>
</TD>
</TR>
</TABLE>
</DIV>
<DIV>
<H3 CLASS="Heading2">
<A NAME="pgfId=162537">
 </A>
2.6.2&nbsp;Adders</H3>
<P CLASS="BodyAfterHead">
<A NAME="pgfId=194409">
 </A>
We can view addition in terms of <SPAN CLASS="Definition">
generate</SPAN>
, G[<SPAN CLASS="EquationVariables">
i</SPAN>
], and <SPAN CLASS="Definition">
propagate</SPAN>
, P[<SPAN CLASS="EquationVariables">
i</SPAN>
], signals. </P>
<TABLE>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=380769">
 </A>
<SPAN CLASS="BodyText">
method 1</SPAN>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=380771">
 </A>
<SPAN CLASS="BodyText">
method 2</SPAN>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=380773">
 </A>
&nbsp;</P>
</TD>
</TR>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=380775">
 </A>
G[i] = A[i] &#183; B[i]</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=380777">
 </A>
G[<SPAN CLASS="EquationVariables">
i</SPAN>
] = A[<SPAN CLASS="EquationVariables">
i</SPAN>
] &#183; B[<SPAN CLASS="EquationVariables">
i</SPAN>
]</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnNumber">
<A NAME="pgfId=380779">
 </A>
(2.42)</P>
</TD>
</TR>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=380781">
 </A>
P[<SPAN CLASS="EquationVariables">
i</SPAN>
] = A[<SPAN CLASS="EquationVariables">
i</SPAN>
] <SPAN CLASS="Symbol">
&#8853; </SPAN>
B[<SPAN CLASS="EquationVariables">
i</SPAN>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=380783">
 </A>
P[<SPAN CLASS="EquationVariables">
i</SPAN>
] = A[<SPAN CLASS="EquationVariables">
i</SPAN>
] + B[<SPAN CLASS="EquationVariables">
i</SPAN>
]</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnNumber">
<A NAME="pgfId=380785">
 </A>
(2.43)</P>
</TD>
</TR>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=380787">
 </A>
C[<SPAN CLASS="EquationVariables">
i</SPAN>
] = G[<SPAN CLASS="EquationVariables">
i</SPAN>
] + P[<SPAN CLASS="EquationVariables">
i</SPAN>
] &#183; C[<SPAN CLASS="EquationVariables">
i</SPAN>
&#8211;1]</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=380789">
 </A>
C[<SPAN CLASS="EquationVariables">
i</SPAN>
] = G[<SPAN CLASS="EquationVariables">
i</SPAN>
] + P[<SPAN CLASS="EquationVariables">
i</SPAN>
] &#183; C[<SPAN CLASS="EquationVariables">
i</SPAN>
&#8211;1]</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnNumber">
<A NAME="pgfId=380791">
 </A>
(2.44)</P>
</TD>
</TR>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=380793">
 </A>
S[<SPAN CLASS="EquationVariables">
i</SPAN>
] = P[<SPAN CLASS="EquationVariables">
i</SPAN>
] <SPAN CLASS="Symbol">
&#8853; </SPAN>
C[<SPAN CLASS="EquationVariables">
i</SPAN>
&#8211;1]</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=380795">
 </A>
S[<SPAN CLASS="EquationVariables">
i</SPAN>
] = A[<SPAN CLASS="EquationVariables">
i</SPAN>
] <SPAN CLASS="Symbol">
&#8853; </SPAN>
B[<SPAN CLASS="EquationVariables">
i</SPAN>
] <SPAN CLASS="Symbol">
&#8853; </SPAN>
C[<SPAN CLASS="EquationVariables">
i</SPAN>
&#8211;1]</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnNumber">
<A NAME="pgfId=380797">
 </A>
(2.45)</P>
</TD>
</TR>
</TABLE>
<P CLASS="BodyAfterHead">
<A NAME="pgfId=195387">
 </A>
where C[<SPAN CLASS="EquationVariables">
i</SPAN>
] is the carry-out signal from stage <SPAN CLASS="EquationVariables">
i</SPAN>
, equal to the carry in of stage (<SPAN CLASS="EquationVariables">
i</SPAN>
 + 1). Thus, C[<SPAN CLASS="EquationVariables">
i</SPAN>
] = COUT[<SPAN CLASS="EquationVariables">
i</SPAN>
] = CIN[<SPAN CLASS="EquationVariables">
i</SPAN>
 + 1]. We need to be careful because C[0] might represent either the carry in or the carry out of the LSB stage. For an adder we set the carry in to the first stage (stage zero), C[&#8211;1] or CIN[0], to '0'. Some people use <SPAN CLASS="Definition">
delete</SPAN>
 (D) or <SPAN CLASS="Definition">
kill</SPAN>
 (K) in various ways for the complements of G[i] and P[i], but unfortunately others use C for COUT and D for CIN&#8212;so I avoid using any of these. Do not confuse the two different methods (both of which are used) in Eqs.&nbsp;&nbsp;2.42&#8211;2.45 when forming the sum, since the propagate signal, P[<SPAN CLASS="EquationVariables">
i</SPAN>
] , is different for each method. </P>
<P CLASS="Body">
<A NAME="pgfId=194423">
 </A>
Figure&nbsp;2.22(a) shows a conventional RCA. The delay of an <SPAN CLASS="EmphasisPrefix">
n</SPAN>
-bit RCA is proportional to <SPAN CLASS="EquationVariables">
n</SPAN>
 and is limited by the propagation of the carry signal through all of the stages. We can reduce delay by using pairs of &#8220;go-faster&#8221; bubbles to change AND and OR gates to fast two-input NAND gates as shown in Figure&nbsp;2.22(a). Alternatively, we can write the equations for the carry signal in two different ways: </P>
<TABLE>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=380815">
 </A>
either</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=380817">
 </A>
C[<SPAN CLASS="EquationVariables">
i</SPAN>
] = A[<SPAN CLASS="EquationVariables">
i</SPAN>
] &#183; B[<SPAN CLASS="EquationVariables">
i</SPAN>
] + P[<SPAN CLASS="EquationVariables">
i</SPAN>
] &#183; C[<SPAN CLASS="EquationVariables">
i</SPAN>
 &#8211; 1]</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnNumber">
<A NAME="pgfId=380819">
 </A>
(2.46)</P>
</TD>
</TR>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=380821">
 </A>
or</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=380823">
 </A>
C[<SPAN CLASS="EquationVariables">
i</SPAN>
] = (A[<SPAN CLASS="EquationVariables">
i</SPAN>
] + B[<SPAN CLASS="EquationVariables">
i</SPAN>
] ) &#183; (P[<SPAN CLASS="EquationVariables">
i</SPAN>
]' + C[<SPAN CLASS="EquationVariables">
i</SPAN>
 &#8211; 1]),</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnNumber">
<A NAME="pgfId=380825">
 </A>
(2.47)</P>
</TD>
</TR>
</TABLE>
<P CLASS="BodyAfterHead">
<A NAME="pgfId=306711">
 </A>
where P[<SPAN CLASS="EquationVariables">
i</SPAN>
]'= NOT(P[<SPAN CLASS="EquationVariables">
i</SPAN>
]). Equations 2.46 and 2.47 allow us to build the carry chain from two-input NAND gates, one per cell, using different logic in even and odd stages (Figure&nbsp;2.22b): </P>
<TABLE>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=380845">
 </A>
<SPAN CLASS="BodyText">
even stages</SPAN>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=380847">
 </A>
<SPAN CLASS="BodyText">
odd stages</SPAN>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=380849">
 </A>
&nbsp;</P>
</TD>
</TR>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=380851">
 </A>
<SPAN CLASS="BodyText">
C1[i]' = P[i ] &#183; C3[i &#8211; 1] &#183; C4[i &#8211; 1]</SPAN>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=380853">
 </A>
<SPAN CLASS="BodyText">
C3[i]' = P[i ] &#183; C1[i &#8211; 1] &#183; C2[i &#8211; 1]</SPAN>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnNumber">
<A NAME="pgfId=380855">
 </A>
(2.48)</P>
</TD>
</TR>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=380857">
 </A>
<SPAN CLASS="BodyText">
C2[i] = A[i ] + B[i ]</SPAN>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=380859">
 </A>
<SPAN CLASS="BodyText">
C4[i]' = A[i ] &#183; B[i ]</SPAN>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnNumber">
<A NAME="pgfId=380861">
 </A>
(2.49)</P>
</TD>
</TR>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=380863">
 </A>
<SPAN CLASS="BodyText">
C[i] = C1[i ] &#183; C2[i ]</SPAN>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=380865">
 </A>
<SPAN CLASS="BodyText">
C[i] = C3[i ] ' + C4[i ]'</SPAN>
</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnNumber">
<A NAME="pgfId=380867">
 </A>
(2.50)</P>
</TD>
</TR>
</TABLE>
<P CLASS="BodyAfterHead">
<A NAME="pgfId=306839">
 </A>
(the carry inputs to stage zero are C3[&#8211;1] = C4[&#8211;1] = '0'). We can use the RCA of Figure&nbsp;2.22(b) in a datapath, with standard cells, or on a gate array. </P>
<P CLASS="Body">
<A NAME="pgfId=195091">
 </A>
Instead of propagating the carries through each stage of an RCA, Figure&nbsp;2.23 shows a different approach. A <SPAN CLASS="Definition">
carry-save adder</SPAN>
 (<SPAN CLASS="Definition">
CSA</SPAN>
) cell CSA(A1[<SPAN CLASS="EquationVariables">
i</SPAN>
], A2[<SPAN CLASS="EquationVariables">
i</SPAN>
], A3[<SPAN CLASS="EquationVariables">
i</SPAN>
 ], CIN, S1[<SPAN CLASS="EquationVariables">
i</SPAN>
], S2[<SPAN CLASS="EquationVariables">
i</SPAN>
], COUT) has three outputs: &nbsp;</P>
<TABLE>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnLeft">
<A NAME="pgfId=380999">
 </A>
S1[<SPAN CLASS="EquationVariables">
i</SPAN>
] = CIN ,</P>
</TD>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="TableEqnNumber">
<A NAME="pgfId=381001">
 </A>
(2.51)</P>
</TD>
</TR>
<TR>
<TD ROWSPAN="1" COLSPAN="1">
<P CLASS="EquationAlign">
<A NAME="pgfId=381003">
 </A>
S2[<SPAN CLASS="EquationVariables">
i</SPAN>
] = A1[<SPAN CLASS="EquationVariables">