Abstract:
An array multiplier using modified Booth encoding of multiplier input signals is formed on the surface of a monolithic integrated circuit using masks generated by a computer, in accordance with a silicon compiler program, by arranging an array of standard cells selected from a library of standard cell designs in a tessellation procedure. The array multiplier is laid out in accordance with one of particular tessellation patterns, which employ simpler and more regular patterns of interconnections between cells. Carry-save addition is used in combining partial product terms to avoid concatenating long ripple carry times.

Description:
RELATED APPLICATIONS 
     This application is a continuation-in-part of application Ser. No. 07/709,562 filed Jun. 6, 1991, now abandoned. 
    
    
     FIELD OF THE INVENTION 
     The invention relates to digital multipliers and, more paticularly, to digital multipliers using modified Booth encoding and having been constructed within the confines of monolithic integrated circuits in accordance with designs generated by silicon compilers. 
     BACKGROUND OF THE INVENTION 
     A digital multiplier embodying the invention is essentially a Booth multiplier. To understand the invention, it is necessary to have some understanding of the mode of operation of a Booth multiplier. Booth multiplication is discussed in the book Computer Arithmetic by Kai Hwang (Wiley, 1979). The method described in the detailed description is the so-called modified Booth encoding, but it will be called Booth encoding for brevity. 
     A primary goal of the invention is to have standardized layout procedures for integrated-circuit digital multipliers that accommodate their design by silicon compiler methods. It is desirable for &#34;custom&#34; integrated circuit design purposes to have an array multiplier in which a close-packed array by row and by column of multiplier cells can be extended in both the row and column dimensions to accommodate multiplier and multiplicand signals having words of any specified bit-width. Further aspects of the invention extend to &#34;semi-custom&#34; design procedures in which standardized &#34;chunks&#34; or close-packed arrays of multiplier cells by row and by column are disposed on a monolithic integrated circuit and are interconnected by discretionary top-level metallization to form digital multipliers with multiplier and multiplicand signals having respective words of differing specified bit-widths. A basic problem encountered in the design of multipliers that can accomodate multiplier and multiplicand signals of differing specified bit-widths is how best to combine partial product terms, in order to avoid concatenating long ripple carry times, which would slow the generation of digital products. The use of Booth encoding is known to be helpful in this regard since it halves the number of partial products that have to be combined. 
     Where the bit-width of the multiplier words is always the same, a procedure that has been generally used in the prior art to reduce the time required for combining partial products is to sum the partial products with a tree of adders rather than a chain of adders. This speed-up procedure is not used in digital multipliers constructed in accordance with the invention, because the tree connection of adders is not susceptible readily (if at all) to the regular tessellation and interconnection procedures preferred by the silicon compiler. 
     SUMMARY OF THE INVENTION 
     A Booth multiplier constructed within a monolithic integrated circuit in accordance with the invention is made up of: a column of Booth encoder cells, a plurality of decoder/adder cells arranged in M rows and N columns to form a close-packed array, and an additional column of 2M adder cells and an additional row of N adder cells together forming the final adder. The decoder/adder cells use carry-save addition to combine partial product terms, in order to avoid concatenating long ripple carry times for the complete addition of each successive partial product. Each decoder/adder cell may be considered as a gestalt by the silicon compiler or, alternatively, may be constructed by the silicon compiler from smaller cells such as individual decoder and adder cells. In embodiments of the invention favored for multiplying multiplicands having a large number of bits by multipliers that have a large number of bits, the adder cells in the final adder are configured to perform carry-select addition. 
    
    
     BRIEF DESCRIPTION OF THE DRAWING 
     FIG. 1 is a schematic diagram of an array multiplier using modified Booth encoding, which array multiplier is constructed in accordance with the invention from a tessellation of standard cells including modified Booth encoding cells, decoder/adder gestalt cells and final adder cells. 
     FIG. 2 is a schematic diagram of circuitry included in a decoder/adder gestalt cell used in the FIG. 1 array multiplier for decoding, in accordance with a bit of a multiplicand signal Y, the modified Booth encoding of a multiplier signal X. 
     FIG. 3 is a schematic diagram of circuitry included in a basic cell used in the FIG. 1 array multiplier for performing a portion of the modified Booth encoding. 
     FIG. 4 is a schematic diagram of a plurality (or &#34;chunk&#34;) of decoder/adder gestalt cells per FIG. 2 arranged by row and by column into a rectangular array as used for the core of the FIG. 1 array multiplier. 
     FIGS. 5A and 5B (referred to collectively as FIG. 5) are the left and right halves of a schematic diagram of a 4×4 chunk of array multiplier circuitry, which together with other such chunks, can be programmed to provide a variety of multiplier configurations. 
     FIG. 6 is a schematic diagram of a 6×6 array multiplier, which array multiplier except for using a carry-select final adder rather than a ripple-carry final adder is of the same general type as the FIG. 1 array multiplier. 
     FIGS. 7 and 8 are expanded portions of FIG. 6. 
     FIG. 9 is a schematic diagram of a carry-select final adder for a 10×10 array multiplier. 
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     FIG. 1 shows the layout for a Booth multiplier construction within a monolithic integrated circuit in accordance with the invention. The FIG. 1 Booth multiplier is made up of: a column M in number of Booth encoder cells B 0 , B 2 , . . . B 2  (M-1), a plurality M×N in number of decoder/adder gestalt cells arrayed by row and by column, and a row and a column of adder cells together forming the final adder. The decoder/adder gestalt cells are identified by call-outs of the form G 2m ,n, where 2m is the number of the row in which the decoder/adder gestalt cell is located and where n is the number of the column in which the decoder/adder gestalt cell is located. Row numbering is by consecutive even ordinal numbers zeroeth through 2(M-1) th , and column numbering is by consecutive ordinal numbers zeroeth through (N-1) th . In the final adder, the column of adder cells used to generate the minor product have call-outs of the form A m ,N with m ranging from zero to 2M-1; and the row of adder cells used to generate the major product have call-outs of the form A 2m ,n with n ranging from zero to N-1. 
     FIG. 1 shows the array composed of the Booth encoder cells B 2m , the decoder/adder gestalt cells G 2m ,n, the column of adder cells A m ,N and the row of adder cells A 2m ,n in somewhat exploded form with electrical connections from certain ones of the cells to certain others of the cells spanning gaps between the cells. This is done to facilitate comprehension of the interconnection relationships between the cells by one viewing the drawing and to leave room for the call-outs identifying the cells. Preferably, however, in actual array multipliers laid out on the surface of a monolithic integrated-circuit die, the cells are in close-packed array without gaps between adjoining cells. Segments of the electrical connections between cells are included within the cells themselves, so the electrical connections are completed as a part of the tiling or tesselation procedure carried forward by the silicon compiler, avoiding or substantially reducing the need for recourse to a subsequent routing procedure to establish point-to-point electrical connections. 
     FIG. 2 shows more particularly the form the decoder/adder gestalt cell takes. Each decoder/adder gestalt cell G 2m ,n comprises a respective decoder cell D 2m ,n and a respective adder cell A 2m ,n. Since the adder cells load their decoder cells identically, the drive power design for the decoder cells can be optimized. The decoder cell D 2m ,n comprises a respective two-input AND gate &amp; 2m ,n, a respective two-input exclusive-OR gate XOR 2m ,n, a respective two-input multiplexer MUX 2m ,n, and a respective logic inverter NOT 2m ,n. The SR0 output port of the decoder/adder gestalt cell G 2m ,n provides a sum bit output signal; the GR output port of the decoder/adder gestalt cell G 2m ,n provides a first-carry bit output signal with weight twice that of the sum bit output signal; and the CL output port of the decoder/adder gestalt cell G 2m ,n provides a second-carry bit output signal. 
     The cell designs for the silicon compiler library of basic cells preferably include to-edge wiring that obviates most of the need for separate wire-routing routines. As shown in FIG. 2, the decoder/adder gestalt cell includes segments of the electrical busses used to correctly apply the sum bit and first-carry bit output signals of the decoder/adder gestalt cells to its left, in accordance with the descriptions provided in the following two paragraphs. 
     The SR0 sum bit output signal of a decoder/adder gestalt cell G 2m ,n is the SL0 input signal of any decoder/adder gestalt cell G 2m ,(n-1) to its immediate right and thence its SR1 output signal. The SR1 output signal of any decoder/adder gestalt cell G 2m ,(n-1) is the SL1 input signal of any decoder/adder gestalt cell G 2m ,(n-2) to its immediate right and thence its SB output signal. Thus, the SR0 sum bit output signal of a decoder/adder gestalt cell G 2m ,n is furnished as a ST input signal to any decoder/adder gestalt cell G 2 (m+1),(n-2) located in a row next down, two columns to the right. Or, where 2(m+1)=2M, the SR0 sum bit output signal of a decoder/adder gestalt cell G 2m ,n is furnished as a ST input signal to an adder A 2m ,(n-2). 
     The GR first-carry bit output signal of the decoder/adder gestalt cell G 2m ,n is the GL input signal of any decoder/adder gestalt cell G 2m ,(n-1) to its immediate right and thence its GB output signal. Thus, the GR first-carry bit output signal of a decoder/adder gestalt cell G 2m ,n is furnished as a GT input signal to any decoder/adder gestalt cell G 2 (m+l),(n-1) located in a row next down, one column to the right. Or, where 2(m+1)=2M, the GR first-carry bit output signal of the decoder/adder gestalt cell G 2m ,n is furnished as a GT input signal to an adder A 2m ,(n-1). 
     In Booth encoding, a binary twos&#39; complement number is recoded in a redundant radix-four data format, in which numbers are represented using digit values -2, -1, 0, 1 or 2. Thus, in encoded form, a number &lt;A n  A.sub.(n-1) . . . A 0  &gt;, where each A i  is one of the digits above, represents the numerical value ##EQU1## The determination of the digit A i  is done by examining the bits Y 2j-1 , Y 2j  and Y 2j+1  of the uncoded twos&#39; complement value. Thus, the input value Y is divided into overlapping sections of three bits. For the determination of the coded value Y 0 , a dummy value of ZERO for the non-existent bit Y -1  is assumed. However, a dummy value of ONE for the nonexistent bit Y -1  can instead be used, where it is desired to calculate X(Y+1) instead of X*Y. The encoded digit values are represented by three bit values, C0, C1 and C2 representing the allowable digit values -2, -1, 0, 1, 2. Each encoder cell takes three bits Y 2j-1 , Y 2j  and Y 2j+1  of the multiplier input, Y, and encodes them in three bits C0, C1 and C2. There is some flexibility in the representation of the digit value by the three code bits C0, C1 and C2; it need not be a twos&#39; complement representation. 
     The following table shows a Booth encoding of three bits of input. 
     
         ______________________________________Y.sub.2j+1    Y.sub.2j          Y.sub.2j-1 A.sub.i                           C0     C1  C2______________________________________0        0     0          0     X      X   00        0     1          1     1      0   10        1     0          1     1      0   10        1     1          2     1      1   11        0     0          -2    0      1   11        0     1          -1    0      0   11        1     0          -1    0      0   11        1     1          0     X      X   0______________________________________ 
    
     The meanings of the three Booth-code bits, C0, C1 and C2 are as follows: 
     C2 if 0 set to 0 
     C1 if 1 multiply by 2 (shift left) 
     C0 if 0 multiply by -1 (invert) 
     FIG. 3 shows a cell containing one arrangement of logic elements that generates the desired modified Booth encoding. Logic inverters NOT 2m ,N, NOT 2m ,(N+1) and NOT 2m ,(N+2) complement repective ones of the three bits Y 2j-1 , Y 2j  and Y 2j+1 . The logic inverter NOT 2m ,(N+2) output signal corresponds to C0 Booth code value. 
     An AND gate &amp; 2m ,N responds to the output signals of the logic inverters NOT 2m ,N and NOT 2m ,(N+1) to generate a ONE only when both of the bits Y 2j-1  and Y 2j  are ZEROs. An AND gate &amp; 2m ,(N+1) responds to the bits Y 2j-1  and Y 2j  to generate a ONE only when both of them are ONEs. An OR gate OR 2m  responds to the output signals of AND gate &amp; 2m ,N and AND gate &amp; 2m ,(N+1) to generate C1 Booth code value, which is a ONE when the bits Y 2j-1  and Y 2j  are alike and is a ZERO when the bits Y 2j-1  and Y 2j  differ from each other. 
     An AND gate &amp; 2m ,(N+2) responds to the output signals of the logic inverters NOT 2m ,N, NOT 2m ,(N+1) and NOT 2m ,(N+2) to generate a ONE only when all three bits Y 2j-1 ,  Y   2j  and Y 2j+1  are ZEROs. An AND gate &amp; 2m ,(N+3) responds to bits Y 2j-1 , Y 2j  and Y 2j+1  to generate a ONE only when all three are ONEs. A NOR gate NOR 2m  responds to the output signals of AND gate &amp; 2m ,(N+2) and AND gate &amp; 2m ,(N+3) to generate C2 Booth code value, which is a ONE except when the three bits Y 2j-1 , Y 2j  and Y 2j+1  are all ZEROs or are all ONEs. 
     A signal CL which is equal to CO is generated, and is to be passed to the CR input of the decoder/adder gestalt cell G 2m ,O. Furthermore, CL is inverted by inverter NOT 2m ,(N+3) and connected to output pin CC, which in turn is connected by an exterior connection to a first summand bit for a respective adder cell A 2m ,N. It is desirable to locate the column of adders A 2m ,N on the right hand side of the rectangular array of decoder/adder gestalt cells, as shown in FIG. 1, to reduce the length of the connections from the array of G 2m ,O and G 2m ,1 decoder/adder cells to the column of adders A 2m ,N, m being a variable ranging over zero to (M-1). It is desirable to keep as short as possible the connection for applying complemented CL signal from each Booth encoder cell B 2m  to the respective adder cell A 2m ,N. It is also desirable to keep as short as possible the connection for CL signal from each Booth encoder cell B 2m  to the decoder/adder gestalt cell G 2m ,O in the same row. Accordingly, it is generally preferable to locate the column of Booth encoder cells B 0 , B 2  . . . B 2 (M-1) on the right hand side of the rectangular array of decoder/adder gestalt cells to avoid the need for running two additional lines per row across the array multiplier. However, where the array multiplier is located together with other digital circuitry on a monolithic integrated circuit, connections to which other circuitry are better arranged such that the array multiplier minor product is better supplied from the side of the array multiplier opposite the side across which multiplier input signal is supplied, it is as well to locate the column of Booth encoder cells B 0 , B 2  . . . B 2 (m-1) on the left hand side of the rectangular array of decoder/adder gestalt cells. 
     FIG. 4 shows a plurality of FIG. 2 decoder/adder gestalt cells arranged in a rectangular array of size M×N, as provides the core of the FIG. 1 array multiplier, where 2M is the number of bits in the multiplier Y and N is the number of bits in the multiplicand X. The decoder/adder gestalt cells in each row have as inputs the multiplicand input, X, one bit per cell, and the three bits, C0, C1 and C2 representing the encoded digit for that row. The purpose of the decoder/adder gestalt cells is to generate the correct multiple, -2X, -X, 0, X or 2X of the multiplicand. To allow values 2X and -2X to be generated by a one place shift left, where possible, each decoder/adder gestalt cell G 2m ,n first generates a value -X, X or 0, which is passed down and right to the cell G 2 (m+1),(n-1). The C1 bit of the encoded digit determines whether to select the shifted or unshifted value. At the right hand end of the array, in each row, the value of CL from the Booth encoder cell is passed as a shift-in signal to the CR input of the rightmost decoder/adder gestalt cell. At the left hand end of the array, in each row, the final shift out passes out from the CL output of the leftmost decoder/adder gestalt cell, to be used as will be described further on in this specification. To generate -X and -2X, the bits of X are inverted, which means that a correction must be made in the least significant bit to get the correct negative. This correction is done later in the final adder by passing the value CC as a summand input to one of the two adder cells associated with the same row as the Booth encoder cell. The partial product generated by the decoder portions of the gestalt cells is added to the accumulated product by the adder portions of the gestalt cells in the row. A carry-save addition is used, meaning that both sum and carry bits are saved. Each array position has a binary weight 2 k  associated with it. This weight is defined such that the position (i,j) counting from (O,O) at the top right has weight 2 i+2j , where i counts vertical columns and j counts horizontal rows. What this means is that a 1 input to an adder at a position with weight 2 k  really represents the number 2 k . Since each Booth-encoded digit represents two bits of the 2M-bit multiplier, Y, there are only M rows, as already stated. Each row has binary weight equal to four times the previous one. Consequently, the sum output from one row of the array of decoder/adder gestalt cells must be passed two places right and down into the next row, whereas the first carry output (which has double the weight) must be passed one place right and down. The second carry output is to the adder cell portion of the decoder/adder gestalt cell to the left. 
     At the end of the partial product accumulation, a sum word and a carry word obtain, since the accumulation of partial products is done saving both sum and carry bits in each adder cell A 2m ,n. These two words are added together to obtain the final product. To obtain a (2M+N)-bit product, the summation forming the minor product is done in the adder cells A 0 ,N through A 2M-1 ,N and the summation forming the major product is done in the adder cells A 2M ,0 through A 2M ,(N-1). In different embodiments of the invention different types of adder are used for the final addition. A ripple-carry final adder is suitable only for multipliers in which 2M+N is small and speed requirements are not demanding. In multipliers in which 2M+N is larger or in which speed requirements are more demanding, it is preferable to use a final adder of carry-select type, as will be described further on with reference to FIGS. 6-9. 
     The FIG. 1 array multiplier is provided with the correct connections for the hanging inputs of the FIG. 4 array of decoder/adder gestalt cells that forms its core. To get correct multiplication results, it is necessary to connect these inputs up carefully. The correct connections shown in FIG. 1 for these inputs are as follows. At the top of the array, besides the bits X i  there are two inputs per bit position in order to make up to three the number of inputs to the top adder row. Generally, one of these two inputs per bit position (identified as GT and ST in the FIG. 2 schematic diagram of circuitry included in a decoder/adder gestalt cell) must be connected to receive a ONE value and the other to receive a ZERO value. Since the two inputs are equivalent, it does not matter which is connected to receive a ONE value and which is connected to receive a ZERO value. (However, if multiplication/accumulation is being done, one of the two inputs GT and ST per bit position will be connected always to receive a ONE value, and the other will be connected to receive a respective bit of the number being accumulated). On the left hand side of the array there are three inputs (S0L, S1L, GL) for each row of the array. These values are passed to the adders in the first two columns from the left of the array. In addition, the shift left output (CL) from the leftmost decoder/adder gestalt cell issues from the left of the add/decode array. The correct and necessary connection is to return this shift out value (CL) to the carry-in input (S0L) which connects to the cell one row down in the second column from the left. The other two values on the left both connect to adders in the leftmost column of the array. One of these connections should be set to a ONE value and one should be set to a ZERO. The carry in Yin to the low-order Booth encoder B 0  should be a ZERO. Finally, the carry in Cin to the first adder cell in the final addition chain should be a ONE, not a ZERO as might be expected. 
     Here is a brief justification of these input connections. The purpose of the input of ONE values at selected input locations is to ensure the correct results when some of the partial products are negative. Consider first the case of all partial products being positive. In this case, the shift out value (CL) from each decoder row is a ONE (since CL is inverted from the correct bit value). Along with the ONE inputs to the top row and left column of adders in the array, there will also be a ONE value input to each of the adders in the second from left column in the array. Taking account of the weights of these array locations, the total sum contributed by these 
     ONE values is ##EQU2## Along with the ONE carry input to the Cin position, this makes a total contribution of 2 2M+N  which lies beyond the range represented by the 2M+N bits of the result and so has no effect. Consider now the case of a partial product being negative, say in row 0. Let the bits of the partial product be &lt;a N-1  . . . a i  . . . a 0  &gt;. Since the partial product is negative, the shift left output (CL) will be a ZERO, and hence, the ONE adder input will be missing from array position (N-2,1) which has weight 2 N . The net effect will be the same as if 2 N  had been subtracted from the final result. The contribution of the partial sum is then ##EQU3## which is equal to the twos complement numerical value of &lt;a.sub.(n-1) . . . a o  &gt; since a.sub.(N-1) =1. 
     By setting to ZERO the most significant bits of the multiplicand X and of the multiplier Y, one can achieve signed-magnitude multiplication. Mixed-mode multiplication wherein one of the operands is twos&#39; complement and the other is signed-magnitude is also possible. When the array multiplier is not just a chunk of a larger array multiplier, the leftmost adder stage A 2m ,(N-1) in its final adder, such as adder stage A 2m ,3 in the FIG. 1 array multiplier, may merely provide a one-bit sign extension allowing it to be dispensed with, shortening the product by one bit from N+2M bits. 
     In high-density interconnect (HDI) technology a desire is to configure chunks of multiplier cells within a monolithic integrated circuit, so as to facilitate the formation of multipliers of selected sizes. The basic chunk consists of an M×N array of decoder/adder gestalt cells like the one shown in FIG. 1 sufficient to carry out a 2M×N multiplication. The requirement is to be able to connect an a×b matrix of chunks together together to form 2aM×bN multipliers. This is done by connecting an a×b matrix of chunks together. It is required that the chunks used to form the multipliers be physically adjacent so that communications between them can be made on the chip, rather than through HDI wires. For instance, a chip may contain a 4×4 array of 8×8 multiplier chunks. These chunks may be configured into a single 32×32 bit multiplier, or four separate 16×16 multipliers, sixteen separate 8×8 multipliers or some mixture of different sized multipliers which may be formed by partitioning the array. Since in the case where the chip is partitioned into multipliers which consist of a single chunk, each chunk will require its own set of Booth encoders and adders, the Booth encoders and adders are included with every chunk. Those adders which fall between chunks of the same multiplier will not be used. On the other hand, the Booth encoders will be used in each chunk. The Y multiplier inputs will be fed to each chunk and decoded locally. Hence, the C0, C1 and C2 inputs of the multiplier array will always be connected to the outputs of the local Booth encoding cell. 
     In forming arrays of multiplier chunks, the connections to the pins of the multiplier chunk shown in FIG. 2 vary depending on whether the chunk is on the edge of or in the middle of an array of chunks forming a single multiplier. Consider for example making a 16×16 multiplier by taking a 4×4 array of 4×4 multipliers as shown in FIG. 1 and connecting them together edge to edge, ignoring superfluous encoders and final adders. The following connections are necessary for every row or column (as appropriate) of each FIG. 4 chunk of the decode/encode array. 
     CR If the chunk is at the right of the array, connect to C0 output of the Booth encoder cell. Otherwise connect to CL of the chunk to the right. 
     GT If chunk is on the top, then connect to 1. Otherwise connect to GB of the chunk above. 
     ST If chunk is on the top, then connect to 0. Otherwise connect to SB of the chunk above. 
     S0L If chunk is on the left, then connect to CL of the same chunk. Otherwise, connect to the S0R of the chunk to the left. 
     S1L If chunk is on the left, then connect to 1. Otherwise, connect to the S1R of the chunk to the right. 
     GL If chunk is on the left, then connect to 0. Otherwise connect to GR of the chunk to the right. 
     X If chunk is on the top, then connect to the input mini-pad. Otherwise, connect to Xout of the chunk above. 
     Certain connections to the Booth encoder cells and the adder cells in the final adder must also be made taking into consideration the position of the chunk in the multiplier configuration. Study of the FIG. 1 array multiplier will make clear the nature of these modified connections, which are as follows. 
     Cin to bit 0. If chunk is on the top, then connect to 1. Otherwise, connect to Cout of bit 2M-1 of chunk above. 
     Cin to bit 2M. If chunk is on the right, then connect to Cout of Bit 2M-1 of the same chunk. Otherwise, connect to Cout of bit 2M+N-1 of the chunk to the right. 
     Yin If chunk is on the top, then connect to 0. Otherwise, connect to Yout of the chunk above. 
     To configure a chip containing an array of multiplier chunks into several independent multipliers it is only necessary to make the correct connections, as described above, to neighboring chunks, to fixed signals or to internal signals. These connections can be by using logical gates and multiplexers controlled by certain configuration pins. 
     FIGS. 5A and 5B together show a single 4 ×4 bit multiplier chunk. The input/output to the chunk are of two types. The square pins marked with a cross, such as X 0 , are mini-pads used for HDI connections (call these &#34;hard input/output&#34;). The pentagonal arrow-shaped pins are connected by top metallization directly to the correspondingly placed pin in the abutting chunk (&#34;soft input/output&#34;). The pentagonal arrow-shaped pins along the edges of the chip are not connected and tend to receive arbitrary values of signal. The summands to the adder cells are connected through the adder cells to respective soft outputs and connect to corresponding soft inputs in multiplier chunks to the right and to the bottom of the FIG. 5 chunk, presuming such other multiplier chunks exist. The second carry-out bits of the (N-1) th  decoder/adder cells in each row are applied to respective soft outputs and connect to corresponding soft inputs in any multiplier chunks to the left of the FIG. 5 chunk. Apart from the mini-pad X and Y inputs and the SUM outputs, there are three configuration mini-pads, RIGHT, LEFT and TOP to which signals are applied indicating to each chunk whether it lies on the right, left or top border of a complete multiplier configured as an array of chunks. These signals control the values of the soft inputs to the chunk according to the rules given above. For instance, on the left side of the chunk shown in FIG. 5, the OR gate on the S1L line sets S1L to 1 if LEFT signal is 1, otherwise it passes the signal through from the adjacent chunk. The AND gate on the GL line, receptive of the logical complement of the LEFT signal, sets GL to 0 if LEFT signal is 1, otherwise it passes the signal through from the adjacent chunk. The other multiplexers, OR gates and AND gates on the soft inputs similarly implement the correct connections for these pins. 
     The row of multiplexers that select X i  bits of the multiplicand signals to the G 0 ,n decoder/adder cells avoids the need for making separate HDI connections for these bits to each chunk in a complete array multiplier. Note how the multiplexer MUX 2m  (here MUX 4 ) at the bottom right of FIG. 5B causes the final addition carry to ripple round the corner or to ripple across from the chunk to the right according to whether the RIGHT signal is high or not. 
     Variants of the circuitry for implementing the correct connections for the soft inputs are possible, of course. If the LEFT signal is 0 to indicate that the multiplier chunk is in a leftmost position in an array of multiplier chunks, the inverter for complementing the LEFT signal is in the OR gate inputs, rather than the AND gate inputs, for the gates at the left of FIG. 5A; and the input connections for the multiplexers at the left of FIG. 5A are reversed. Similarly, if the TOP signal is 0 to indicate that the multiplier chunk is in a topmost position in an array of multiplier chunks, the inverter for complementing the TOP signal is in the OR gate inputs, rather than the AND gate inputs, for the gates at the top of FIGS. 5A and 5B; and the input connections for the multiplexers at the top of FIG. 5A are reversed. If the RIGHT signal is 0 to indicate that the multiplier chunk is in a rightmost position in an array of multiplier chunks, the input connections for the multiplexers at the left of FIG. 5B are reversed. The individual logic complementation of the LEFT, TOP or RIGHT signal at individual gate inputs as thusfar described may be supplanted by a single logic inverter driving a complementary signal line supplying those individual gate inputs. The claims following this specification should be construed to extend their scope of protection over these variants, considered to be equivalents of the circuitry for connecting soft inputs shown in FIGS. 5A and 5B. 
     A further feature of the FIG. 5 multiplier chunk is the capability of carrying out pipelined multiplication. The box marked ROW OF LATCH CELLS near the bottom of the chunk contains a battery of optional single-clock-cycle delays, one for each of the signals crossing it except the RIGHT configuration signal. The hard input signal LATCH controls whether the signals are delayed for one clock cycle or are passed through immediately. In the case where the signals are delayed, there will be a time shift of one clock cycle from the computations above the delay to those below. This means, that all inputs must be delayed one clock cycle, and all outputs will be delayed one clock cycle. For instance, suppose that chunks are made up of 4×4 bits and that they are configured into a 16×16 bit multiplier by connecting a 4×4 array of chunks together. Suppose that the LATCH signal is high for every chunk. For correct operation of the multiplier, the successive groups of 4 bits of the Y input must be delayed by one cycle with respect to the previous one. The output SUM values will be similarly delayed. 
     Separate multiplications can be pipelined so that they overlap in time. The timing is 
     
         ______________________________________X.sub.0 . . . X.sub.15 t = 0Y.sub.0 . . . Y.sub.3  t = 0Y.sub.4 . . . Y.sub.7  t = 1Y.sub.8 . . . Y.sub.11 t = 2Y.sub.12 . . . Y.sub.15                  t = 3SUM0 . . . SUM3        t = 0SUM4 . . . SUM7        t = 1SUM8 . . . SUM11       t = 2SUM12 . . . SUM15      t = 3SUM16 . . . SUM31      t = 4______________________________________ 
    
     This may be an impractical timing arrangement. A more usual case would be that the LATCH signal is high in the final row of chunks only. In this case, the final addition of the major product bits is delayed. There is only a two stage pipeline with input/output timing as follows. 
     
         ______________________________________X.sub.0 . . . X.sub.15 t = 0Y.sub.0 . . . Y.sub.15 t = 0SUM0 . . . SUM15       t = 0SUM16 . . . SUM31      t = 1______________________________________ 
    
     The purpose of this pipelining is to achieve a higher clock speed than would be possible with a single combinational multiplier. 
     A variant of the FIG. 5 multiplier chunk relocates the respective latch cells for the X i  bits after the respective input multiplexers for selecting these bits and previous to the respective columns of decoder/adder cells supplied by those bits. The latch cells can then serve as input latches for the topmost rank of multiplier chunks in the complete array multiplier. 
     Multiplier chunks that accept multiplicand signal segments with more than four bits, that accept multiplier signal segments with more than four bits, and that are constructed in accordance with the same precepts as the FIG. 5 4×4 multiplier chunk may be arrayed on a monolithic integrated circuit die. The same precepts may be used to make a very large array multiplier, chunks of which occupy respective monolithic integrated circuit dies. 
     FIG. 6 shows a 6×6 array multiplier which, except for using a carry-select final adder rather than a ripple-carry final adder, is of the same general type as the FIG. 1 array multiplier. FIGS. 7 and 8 show in greater detail respective portions of the carry-select adder that is the final adder for a 6×6 array multiplier. In an adder of carry-select type, the chain of adders is segmented into subchains and each subchain is duplicated. In each duplicate pair, one subchain adds as if its carry-in were a ZERO and the other subchain adds as if its carry-in were a ONE. When the actual carry-in for the chain segment is actually available, it is then used to select the one of the outputs of the duplicate pair of subchains that is the appropriate output for the chain segment. It is also used to select the proper carry-out from the two adders at the ends of the subchain and its duplicate, to be supplied as carry-in to the following subchain and its duplicate. Each succeeding pair of subchains can contain (and preferably does contain) one or a few more adder cells than the immediately preceding pair, since the propagation time for the successive carry-select signals through the multiplexers in the final adder affords progressively more time for summing before carry-select signal is available for the succeeding pair of subchains. The final pair of subchains may contain a reduced number of adder cells, of course. In FIGS. 7-8 each succeeding pair of subchains is shown as containing two more adder cells than the immediately preceding pair. 
     FIGS. 7-8 show each of the ripple-carry adder cells A m ,N in the final adder of a FIG. 1 type of array multiplier being replaced by a respective &#34;even&#34; adder cell EA m ,N included in a subchain having a ZERO applied thereto as carry-in, a respective &#34;odd&#34; adder cell OA m ,N included in a subchain having a ONE applied thereto as carry-in, and a multiplexer MUX m ,N for selecting one of the sum bits from the &#34;even&#34; adder cell EA m ,N and the &#34;odd&#34; adder cell OA m ,N. Each of the ripple-carry adder cells A 2M ,n in the final adder of a FIG. 1 type of array multiplier is replaced by a respective &#34;even&#34; adder cell EA 2M ,n included in a subchain having a ZERO applied thereto as carry-in, a respective &#34;odd&#34; adder cell OA 2M ,n included in a subchain having a ONE applied thereto as carry-in, and a multiplexer MUX 2M ,n for selecting one of the sum bits from the &#34;even&#34; adder cell EA 2M ,n and the &#34;odd&#34; adder cell OA 2M ,n. A respective two-cell adder chain constitutes each of the first pair of subchains, and a multiplexer MUX 1   selects one of their carries-out to the second pair of subchains. A respective four-cell adder chain constitutes each of the second pair of subchains, and a multiplexer MUX 2  selects one of their carries-out to the third pair of subchains. A respective six-cell adder chain constitutes each of the third pair of subchains, and a multiplexer MUX 3  for selecting one of their carries-out as the product overflow C out  from the array multiplier. Cells CON 1 , CON 2  and CON 3  provide connections across gaps in the final adder, so the silicon compiler does not have to perform connection routing procedures in laying out the array multiplier. In a 6×6 array multiplier, the corner connection between the minor-product-generating and major-product-generating portions of the final adder occurs where only a single-wire connection needs to be carried forward for carry-select signal. This single-wire corner connection can be made using a wire routing routine of the silicon compiler, or a corner cell can be added to the silicon compiler cell library so the connection can be provided as part of a tiling procedure. In certain other-size array multipliers the corner connection between the minor-product-generating and major-product-generating portions of the final adder occurs where a three-wire connection needs to be carried forward. 
     FIG. 9 shows a carry-select adder that is the final adder for a 10×10 array multiplier. The 10×10 array multiplier, except for using a carry-select final adder rather than a ripple-carry final adder, is presumed to be of the same general type as the FIG. 1 array multiplier. In this 10×10 array multiplier the corner connection between the minor-product-generating and major-product-generating portions of the final adder occurs is perforce a three-wire connection as shown in FIG. 9. 
     The foregoing disclosure will enable one skilled in the art to design a number of variations of the array multipliers as thusfar described, which variations also embody the invention; and the claims which follow should be construed so as to include such embodiments of the invention within the scope of protection these claims collectively afford. For example, chunks of digital multiplier using carry-select final adders are such embodiments of the invention.