[ 0001 ] Field of the Invention

[ 0002 ] The present invention relates to a multiplieraccumulator . In particular, the invention relates to an asynchronous multiplier operable to be cascaded and configured to operate as a multiplier-accumulator, which accepts digital inputs and generates a summed analog output which can be digiti zed to form a digital output representing a sum of multiplication products .

[ 0003 ] Background of the Invention

[ 0004 ] The expanded use of Arti ficial Intelligence

(Al ) software applications has created a need for scalable hardware multiplier-accumulators for acceleration of software algorithms used in machine learning (ML ) . An nxn multiplier increases in gate complexity as n ² , and large numbers of adders are further needed for multiply- accumulate operations . Additionally, prior art multipliers relied on synchronous , clocked stages to operate , and the clocked operation results in increased power dissipation .

[ 0005 ] For example , in machine learning applications , it is often desired to form dot products in the form of multiply-accumulate operations between a Ixn input row vector A and a nxm B coef ficient matrix to generate a Ixm result R, such as :

[ 0006 ] It is desired to provide a new low-power architecture for a multiplier and multiplier-accumulator which operates asynchronously and minimi zes displacement currents internal to the unit element multiplier by arrangement of internal circuits , speci fically through the use of an architecture which minimi zes internal state changes when B is a static weighting matrix, as is commonly used in ML calculations , thereby reducing power consumption per calculation . It is further desired to provide a simpli fied architecture for a cascaded analog multiplieraccumulator to form a product as an analog quantity which is converted to a digital form for subsequent processing .

[ 0007 ] Obj ects of the Invention

[ 0008 ] A first obj ect of the invention is a unit element accepting a digital A input and a digital B input , the unit element performing a single multiply operation and adding a corresponding charge to an analog charge bus , the unit element comprising an AND-group comprising a plurality of AND gates , each AND gate of an AND-group having one input coupled to a unique one of the A input bits and the other AND gate inputs commonly coupled to one of the B input bits , the output of each AND gate representing a product term and having a bit order equal to the sum of the A input bit position and B input bit position, each AND gate output coupled to an analog charge line associated with the bit order through a charge trans fer capacitor of value Cu, each analog charge line common to other unit elements , each analog charge line coupled to a charge summing unit comprising charge summing capacitors which provide a charge trans fer for each analog charge line of two to the power of the bit order, the charge summing unit having a first terminal of a respective charge summing capacitor coupled to a respective analog charge line according to the bit weight of the analog charge line , the charge summing capacitors having a second terminal which are coupled together and also coupled to the input of an analog to digital converter (ADC ) which provides the product of A and B for all unit elements at the output of the ADC output . [ 0009 ] A second obj ect of the invention is a multiplier-accumulator comprising a plurality of unit elements , each unit element accepting a digital A input and a digital B input , each unit element comprising a plurality of AND-groups , each AND-group comprising a plurality of AND gates , each AND gate having one input coupled to a unique one of the A input bits and the other AND gate input commonly coupled to one of the B input bits , the output of each AND gate of each AND-group having an associated bit order according to a sum of bit position of the associated A input bit and B input bit , each AND gate output coupled through a capacitor of value Cu to a particular analog charge line of an analog charge bus according to bit order, the plurality of unit elements sharing respective analog charge lines of the analog charge bus , the analog charge bus coupled to a charge summing unit comprising a plurality of charge summing capacitors which trans fer charge according to two to the power of bit order, each analog charge line coupled to a first terminal of a respective charge summing capacitor trans ferring charge according to two to the power of bit order, the charge summing capacitors having a second terminal which is coupled together and coupled to an analog to digital converter (ADC ) , the ADC thereby outputting a digital value representing the sum of products from the plurality of unit elements .

[ 00010 ] A third obj ect of the invention is a multiplier for n input bits on at least one multiplicand, where the n input bits of the at least one multiplicand are subdivided into at least one plurality of m bits , each of the plurality of m bits coupled to a sub-product multiplier generating a digital sub-product value , each sub-product multiplier input coupled to a plurality of unit elements , each unit element accepting a digital A input of m bits and a digital B input of m bits , each sub-product multiplier comprising a plurality of AND-groups , each AND-group comprising a plurality of AND gates , each AND gate having one input coupled to a unique one of the A input bits and the other AND gate inputs commonly coupled to one of the B input bits , the output of each AND gate of each AND-group associated with a bit order according to the sum of an A input bit position and B input bit position, each AND gate coupled to a particular analog charge line through a charge trans fer capacitor of value Cu, the unit element charge lines coupled to a charge summing unit having a charge summing capacitor for each analog charge line , each analog charge line coupled to a first terminal of a respective summing capacitor trans ferring charge from a respective analog charge line in an amount proportional to two to the power of the bit order, each summing capacitor having a second terminal which is coupled to the second terminal of other charge summing capacitors and also coupled to an analog to digital converter generating a digital value as the output of the sub-product multiplier, each of the subproduct multiplier digital outputs shi fted by a number of bits representative of the magnitude of the sub-product multiplier output , and where the shi fted sub-product multiplier outputs are summed together to form a multiplier output .

[ 0011 ] A fourth obj ect of the invention is a plurality of unit elements , each unit element multiplying a unique A digital input and a B digital input , the A digital input and B digital input and trans ferring charge values according to a multiplication result , the multiplication result provided as a plurality of charges trans ferred to respective analog charge lines of a shared analog charge bus , each unit element of the plurality of unit elements contributing a trans ferred charge to a respective analog charge line according to a bit order determined by a sum of a bit position of each A digital input bit and each B digital input bit , the shared analog charge bus coupled to a charge summing unit comprising a plurality of summing capacitors , each summing capacitor having a value which trans fers charge according to two to the power of the bit order, each summing capacitor having a first terminal connected to one of the analog charge lines , the other terminal of the binary weighted capacitors coupled together and to an analog to digital converter for providing an accumulated multiplication result of the A and B digital inputs .

[ 0012 ] A fi fth obj ect of the invention is a di f ferential unit element (UE ) for multiplying an A digital value and a B digital value and trans ferring the result to an analog charge bus , the di f ferential unit element having a positive unit element part and a negative unit element part ; the positive unit element part comprising a plurality of AND-groups , each AND-group having a plurality of AND gates having one input coupled to a unique one of the A bits and an input coupled to one of the B bits , each AND gate coupled to a positive analog charge line according to a bit order determined by a sum of the bit position of a corresponding A bit and the bit position of a corresponding B bit ; the negative unit element part also comprising a plurality of AND-groups , each AND-group having a plurality of AND gates having one input coupled to a unique one of the A bits and an input coupled to one of the B bits , each AND gate coupled a negative analog charge line according to a bit order determined by a sum of the bit position of a corresponding A bit and a bit position of the corresponding B bit ; the positive charge lines and negative charge lines being coupled to a positive charge summing unit and a negative charge summing unit , each positive charge summing unit and negative charge summing unit comprising a plurality of charge summing capacitors , each charge summing capacitor having a value which trans fers charge according to a binary weight of the respective bit order of each associated charge line , each summing capacitor having a first terminal connected to a respective positive charge line or negative charge line , the positive charge summing unit and negative charge summing unit having a respective positive output terminal and a respective negative output terminal connected respectively to an analog to digital converter (ADC ) positive input and negative input , the ADC having an output providing a digital output value , the positive analog charge bus and negative analog charge bus optionally coupled to other unit elements . [ 0013 ] A sixth obj ect of the invention is a di f ferential unit element (UE ) for multiplying an A digital value and a signed B digital value and trans ferring the result to analog charge lines of an analog charge bus , the di f ferential unit element having a positive unit element part and a negative unit element part ; the positive unit element part enabled when the B sign is positive and comprising a plurality of AND-groups , each AND-group having a plurality of AND gates having one input coupled to a unique one of the A bits and an input coupled to one of the B bits , each AND gate coupled through a capacitor having a value Cu to a positive analog charge line , each AND gate having a bit order according to the sum of an associated A input bit and B input bit ; the negative unit element part enabled when the B sign is negative , the negative unit element part comprising a plurality of AND-groups , each AND-group having a plurality of AND gates having one input coupled to a unique one of the A bits and an input coupled to one of the B bits , each AND gate coupled through a capacitor having a value Cu to a negative analog charge line , each AND gate having a bit order according to the sum of an associated A input bit and

B input bit ; the positive charge lines and negative charge lines being coupled to a positive charge summing unit and a negative charge summing unit , respectively, each positive charge summing unit and negative charge summing unit comprising a plurality of summing capacitors , each summing capacitor trans ferring a charge equal to two to the power of the order of a respective charge line , each summing capacitor having a first terminal connected to a respective positive charge line or negative charge line , the positive charge summing unit and negative charge summing unit having a respective positive output terminal and a respective negative output terminal connected respectively to an analog to digital positive input and negative input , the ADC having an output providing a digital output value , the positive analog charge bus and negative analog charge bus optionally coupled to other unit elements .

[ 0014 ] Summary of the Invention

[ 0015 ] In a first example of the invention, a unit element for multiplication of two binary multiplicands A and B, each having m bits , comprises a plurality of AND- groups , each AND-group comprising a plurality m of AND gates , each AND gate of an AND-group receiving individual bits of A and a unique one of the bits of B . The output of each AND gate of each unit cell is coupled to a respective charge trans fer capacitor with value Cu to an analog charge line arranged according to a bit order determined by a sum of the A input bit number and B input bit number coupled to the respective AND gate . The analog charge lines form an analog charge bus which is coupled to a charge summing unit which performs a charge trans fer according to a respective bit order of an associated analog charge line . Each of the analog charge lines is coupled to a first terminal of a charge summing capacitor, each charge summing capacitors trans ferring a charge proportional to two to the power of the bit order of the associated analog charge line , the charge summing capacitors having a second terminal which are connected together and connected to the input of analog to digital converter which provides the multiplication product as a digital value .

[ 0016 ] In a second example of the invention providing an accumulating multiplier, a plurality of the unit elements of the first example of the invention are coupled to a common set of analog charge lines , the analog charge lines coupled to charge summing capacitors and to the analog to digital converter to generate an accumulated multiplication result for the plurality of unit elements . [ 0017 ] In a third example of the invention, the invention provides for a multiplier for n bits of each multiplicand, the n bits subdivided into multiple pluralities , the subdivided bits provided to separate multipliers to generate sub-results which are shi fted and summed together to form a product output .

[ 0018 ] In a fourth example of the invention, a di f ferential unit element has a positive unit element part and a negative unit element part ; the positive unit element part comprising a plurality of AND-groups , each AND-group comprising a plurality of AND gates performing bit-wise multiplications having a respective bit order, each positive unit element part AND gate output coupled through a charge trans fer capacitor to a positive charge trans fer line on a charge trans fer line according to a binary weight of its bit order, the positive charge trans fer lines coupled to a positive charge summing unit , the charge summing unit comprising charge summing capacitors , each charge summing capacitor having a first terminal connected to a respective positive charge trans fer line and trans ferring a charge proportional to the charge trans fer line binary weight , the capacitors of the positive charge summing unit having a second terminal connected together and to a positive input of an analog to digital converter (ADC ) ; the negative unit element part comprising a plurality of AND-groups , each AND-group comprising a plurality of AND gates performing bit-wise multiplications having a respective bit order, each negative unit element part AND gate output coupled through a charge trans fer capacitor to a negative charge trans fer line on a charge trans fer line according to a binary weight of its bit order, the negative charge trans fer bus lines coupled to a negative charge summing unit , the negative charge summing unit comprising charge summing capacitors , each charge summing capacitor having a first terminal connected to a respective negative charge trans fer line and trans ferring a charge proportional to the charge trans fer line binary weight , the capacitors of the negative charge summing unit having a second terminal connected together and to a negative input of an analog to digital converter (ADC ) , the ADC generating a product output value . In one example of the invention, a plurality of unit elements share a common positive analog charge bus and a common negative analog charge bus . [0019] Brief Description of the Drawings

[0020] Figure 1A shows an example multiplication of two 3 bit multiplicands.

[0021] Figure IB shows an expansion of the multiplication of figure 1A identifying individual terms.

[0022] Figure 1C shows a block diagram for an accumulating multiplier performing dot product operations.

[0023] Figure 2A shows a schematic diagram of a unit element with analog charge bus gain balancing capacitors.

[0024] Figure 2B shows a schematic diagram of the unit element of figure 2A with analog bus gain balancing capacitors incorporated into the charge summing capacitors.

[0025] Figure 3A shows the multiplication of two 4 bit binary numbers .

[0026] Figures 3B shows an expansion of the multiplication of figure 3A.

[0027] Figure 4A shows a schematic diagram for a 4x4 unit element coupled to binary weighted charge summing capacitors which incorporate gain balancing.

[0028] Figure 4B shows a block diagram for an example

4x4 multiplier-accumulator for use with figure 4A. [0029] Figure 5 shows an example expansion for multiplication of two 8 bit binary numbers.

[0030] Figure 6 shows an example block diagram for an 8x8 multiplier.

[0031] Figure 7A shows a schematic diagram for a differential unit element.

[0032] Figure 7B shows a schematic diagram for a differential unit element with a signed multiplicand and gain balancing.

[0033] Figure 7C shows a schematic diagram for a differential charge summing unit with binary weighting.

[0034] Figure 8A shows a schematic diagram for an analog charge bus biasing.

[0035] Figure 8B shows a schematic diagram for performing a charge state reset.

[0036] Figure 9 shows a unit element with individual charge transfer lines coupled to a charge summing unit incorporating gain balancing capacitance.

[0037] Detailed Description of the Invention

[0038] By way of convention, in the present application, similar reference numbers on different figures indicate the same element or function. Where a function is performed by individual elements, the suffix a, b, c etc may be appended as appears in the drawings, whereas the elements taken as a whole are understood to be without suffix, so for example unit element 201 or analog charge bus 210 are understood to refer to the entire structure when a suffix a, b, c, etc are not present.

[0039] Figures 1A and IB show an example expansion for multiplication of two 3 bit binary numbers. This may also be described as the partial product expansion: p0[2:0] = {a[0] &b[2J , a [ 0 ] &b [ 1 ] , a [ 0 ] &b [ 0 ] } pl [2:0] = {a[l] &b[2J , a [ 1 ] &b [ 1 ] , a [ 1 ] &b [ 0 ] } p2 [2:0] = {a[2] &b[2J , a [ 2 ] &b [ 1 ] , a [ 2 ] &b [ 0 ] } which can be rearranged as:

P = l*p0[0] + 2 * (pO [ 1 ] +pl [ 0 ] ) + 4 * (pO [ 2 ] +pl [ 1 ] +p2 [ 0 ] ) + 8* (pl [2] +p2 [1] ) + 16* (p2 [2] )

In one example embodiment, the binary charge summing may be performed by selection of relative capacitor values in the charge summing unit. If each product term has its own individual capacitors, this would result in

( 1 + 2 + 2 + 4 + 4 + 4 + 8 + 8 + 16 ) =49 individual capacitors. [0040] Figure 1C shows a block diagram example dot product for implementing the dot product:

[0041] Unit Elements (UE) 102, 104, ...,106 perform the computation for element R1 of the dot product, UE 108, 110, ..., 112 perform the computation for element R2 of the dot product, and UE 114, 116, ..., 118 perform the computation for element Rn. Accordingly, the architecture of the present invention provides for any number of UEs to be arranged in rows and columns as shown to provide an expandable dot matrix computation for an arbitrary size of the A and B matrices. Additionally, the architecture provides flexibility in being reconfigured for a larger or smaller number of A and B matrices.

[0042] Figure 2A shows a schematic diagram of a unit element 200 comprising a plurality of AND-groups 203a, 203b, and 203c, where each AND-group such as 203a comprises AND gates 202a, 202b, and 202c and charge transfer capacitors 208a coupling each AND gate output to an analog charge line according to bit order of the multiplication, as described by the 2 ⁿ terms of figure IB. Each of the multiplicand bits for A is applied to the inputs of AND gates 202a, 202b, 202c ; 204a, 204b, 204c ; and 206a, 206b, 206c for each respective AND-groups 203a, 203b, and 203c . Each of the multiplicand bits for B is separately input as a term for each of the AND gates 202a, 202b, 202c ; 204a, 204b, 204c ; and 206a, 205b, 206c, of the respective AND- groups 203a, 203b, and 203c, as shown . The output of each AND gates of AND-groups 203a, 203b, 203c is provided through respective charge trans fer capacitors 208a, 208b, and 208c with capacitance value Cu to the analog charge bus 210 which comprises analog charge trans fer lines 210a, 210b, 210c, 210d, and 210e , which are ordered by corresponding binary weighted charge conferred, with a most signi ficant bit (MSB ) trans ferring greater charge than a least signi ficant bit ( LSB ) , each charge trans fer line having a power of 2 greater charge trans fer than a lower signi ficant charge trans fer line . Additionally, each of the analog charge trans fer lines has gain balancing capacitors 209a and 209b such that the capacitance associated with the number of AND gate charge trans fer capacitors connected to a particular charge line of the analog charge bus 210 is equali zed, such that each charge trans fer line 210a, 210b, 210c, 210d, and 210e have a total capacitance equal to the AND gate charge trans fer capacitance of other charge trans fer lines . The binary weighted charge summing capacitors in charge summing unit

212 each have individual capacitance values which increase by powers of two as Cs=2 ^m , where m is the magnitude order, and each charge summing capacitor Cs of charge summing unit 212 has a value which is much smaller than the aggregate capacitance value of the individual charge trans fer capacitors Cu coupled to each associated analog charge line such as 210a . It is preferred that the aggregate capacitance of Cu on each analog charge line exceed Cs by a factor of 8 or more , to avoid systematic errors . As was observed for analog charge buses 210- 1 , 210-2 , and 210-m of figure 1C, the analog charge lines are connected to many UE , which makes the requirement of the Cs to be smaller in value than the total number of charge trans fer capacitors Cu from the many UEs on a charge trans fer line of 210- 1 , 210-2 , etc attached to the analog charge lines easier to meet . One terminal of each charge summing capacitor of charge summing unit 212 is coupled to an associated analog charge line 210a through 210e , and the other terminal of the charge summing capacitors of charge summing unit 212 are coupled together and coupled to the input of an analog to digital converter 214 for conversion to a digital output value P representing the product of A and B . In one example of the invention, Gain balancing capacitors , also referred to as "dummy capacitors" 209a, and 209b are added to equalize the loading of capacitance on each analog charge line of bus 210, in this example, to equalize to a total of 3C per charge transfer line per unit element connected to each charge transfer line. In this example of the invention, the charge summing capacitors have a value Cs * (sum of charge transfer capacitors coupled to the charge transfer line) * 2 ^{bit order} , where bit_order is the sum of the bit positions of the multiplicand input bits. For example, the bit_order of AND gate 206c multiplying bO and aO is 0+0=0, with binary weight 2°=1. The bit order of AND gate 202a multiplying a2 and b2 is 2+2=4, with binary weight 2 ⁴ =16. In these and other examples, each AND gate is coupled through a charge transfer capacitor to a respective charge transfer line with a bit order and binary weight (two to the power of the respective bit_order) . As will be seen, the binary weight provides a comparative power of two transfer of charge contributed by each AND gate output compared to other AND gate outputs, although the charge balancing capacitors and values of individual charge summing capacitors take different values depending on how gain balancing is performed - either using gain balancing capacitors 209a and 209b to equalize charge, or by incorporating the gain balancing capacitors into the charge summing capacitors .

[ 0043 ] The term " gain balancing capacitor" is understood to describe capacitors such as 209a and 209b that are added to individual analog charge lines 210 to equali ze the capacitance of each analog charge line so that the charge trans ferred from each Cu 208a and 208b has equal ef fect . However, each analog charge line has an associated binary weight which depends on a bit weight of the multiplication result coupled to each particular analog charge line , and the capacitance of each particular summing capacitor for each associated analog charge line provides that bit weight in the amount of charge trans ferred to the output of the summing unit for each line according to bit weight of the charge trans fer line . The summing unit 212 thereby has capacitors with values on each charge trans fer line which provide the required binary analog line weight as shown in figure 2A.

[ 0044 ] Figure 2B shows the unit element 200 of figure 2A with gain balancing capacitors incorporated in charge summing unit 212 . In this example , the gain balancing capacitors 209a and 209b of figure 2A are trans ferred to the values of the charge summing capacitors of 212 associated with each analog charge line of analog charge bus 210 . Each charge summing capacitor of 212 has a respective value for each nth charge trans fer line Cs = num_charge_trans f er_caps* 2 ⁿ , where num_charge_trans f er_caps is the number of charge trans fer caps on analog charge line n . As with figure 2A, each analog charge line is operative to trans fer a charge according to a binary weight based on the bit order of the analog charge line , which determines the si ze of each summing capacitor coupled to each analog charge line .

[ 0045 ] Figures 3A and 3B are similar to figures 2A and 2B, respectively, showing the individual bits and organi zation into associated terms for multiplication of two 4 bit binary numbers rather than the 3 bit examples of figure 2A and 2B .

[ 0046 ] Figure 4A shows an example 4x4 unit element 401 , which has an expanded architecture for four bits similar to the three bit example of figure 2B, and includes the gain balancing capacitors incorporated into charge summing capacitors Cs of binary weighted charge summing unit 412 .

[ 0047 ] A key advantage of the present architecture is the ability to scale to larger and larger multiply- accumulation operations by addition of UEs, and figure 4B shows an example using several unit element 401 of figure 4A to form the multiply/accumulate product P = (A1*B1+A2*B2+ . . . +An*Bn) . In some examples of the invention for Al purposes, an input-independent bias term Bias= (En . . E0 ) may be applied, such as P = (A1*B1+A2*B2+ . . ,+An*Bn + Bias) .

[0048] One difficulty of the present invention is that the terms expansion requires larger and larger numbers of AND gates. For example, multiplication of an 8x8 by continued expansion of the architecture of figure 4A would require 64 AND gates. Figures 5 and 6 show an expansion of the present multiplier to perform 8x8 multiply operations without the geometric expansion of AND gates and used in the example block diagram of figure 6. The 8 bit A value [a7..a0] is separated into two 4 bit values, Al and A0, and similarly the 8 bit B value [b7..b0] is separated into two four bit Bl and B0 values and applied to the respective figure 6 4x4 sub-multiplier blocks 612, 614, 616, 618 which then generate a binary output from each sub-multiplier, which may comprise a four bit unit element such as 401 coupled to a charge summing unit 412 and ADC 414 generating each sub-multiplier digital output. This separation then provides that the multiplication operation may be performed as four bit multiplies using the sub-product 4-bit multipliers 401, 412, and 414 of figure 4A in the 4x4 submultiplier blocks 612, 614, 616, 618 of figure 6, with each digital output sub-result from sub-multipliers 612, 614, 616, and 618 bit shifted according to order (which accomplishes multiplication by 2 for each shift) by shift blocks 602, 604, and 606, and summing of the result which is output at 610 as a complete digital result. The number of bits assigned to each sub-multiplier 612, 614, 616, or 618 by an input splitter separating those bits to each submultiplier may be any number of bits, preferably in the range of three to five bits.

[0049] Figure 7A shows a differential unit element 700 for multiplication of [a2..a0] with [b2..b0] . The differential unit element has several advantages over the single ended unit elements 200 and 401. A differential unit element provides greater isolation from disturbances from the supply voltage, as it tends to be self-cancelling as present on both inputs of the combining ADC, whereas the same voltage disturbance in a single ended UE 200 and 401 propagates through to the ADC input. Additionally, the differential UE provides a factor of 2 greater signal swing for a given UE, providing greater noise immunity and improved signal to noise ratio. Further, Machine Learning (ML ) algorithms which make use of the recti fied linear unit

(ReLU) function require operation with negative coef ficients . An additional advantage of a di f ferential UE is the ability to enable separately the positive and negative parts of the unit element to support ReLU signed weighting coef ficients for the B input .

[ 0050 ] Figure 7A shows a first example of a di f ferential unit element which does not support negative ( signed) coef ficients . A plurality of positive AND-groups 203aP, 203bP, and 203cP comprising AND gates and charge trans fer capacitors with the suf fix P is generating positive ( P ) charge displacements to positive analog charge bus 706P, and a companion plurality of negative AND-groups 203aN, 203bN, and 203cN comprising AND gates and trans fer capacitors with the suf fix N is generating negative (N) charge displacement to respective analog charge busses 706N . Positive analog charge bus 706P is functionally equivalent to 210 of figure 3B, with the respective elements carrying the P suf fix . Similarly, the negative analog charge bus 706N is coupled to respective analog charge lines 210aN to 210eN . A plurality of di f ferential unit elements 700 may be cascaded, sharing positive charge buss 706P and 706N, with the charges summing with the binary weighted charge summing unit 212P and 212N, which sum charges into differential ADC 702 and forming the dot product output 704, as before. The individual charge summing capacitors of 212P and 212N are weighted as was described for figure 3B, and can incorporate gain balancing capacitors in the summing capacitors as previously described for figure 2B or with gain balancing capacitors as described in figure 2A. The illustration of figure 7A using the structures of figure 3B are shown as a simple example, however it is understood that figures 3A, 4A, or 4B could be similarly expanded into differential unit element configurations.

[0051] In another example of the invention supporting the ReLU function with signed weighting coefficients, the A input (such as a [2:0] ) is always a positive quantity, and does not have a sign bit, and the weighting function W (such as b[2:0] may include a sign bit Bsgn. It was previously seen from figure 4A that the UE complexity increases with additional multiplication bits. An available simplification which allows for additional resolution is where the generally static weight terms W are provided as B coefficients comprising three bits of coefficient value b[2:0] and a sign bit Bsgn, and the A coefficients have three bits of value a [2:0] which are unsigned positive values. In a second example of a differential UE shown in figure 7B, unit element 700B is operative with signed B weight coefficients which use the Bsgn bit 710 to alternately enable the positive unit element (P suffix) or enable the negative unit element (N suffix) of figure 7B. In this embodiment, unit element 700B has a positive unit element part comprising a plurality of positive AND-groups 203aP, 203bP, 203cP, each positive AND-group having AND gates 202aP, 202bP, 202cP with charge coupling capacitors 208aP, 208bP, 208cP, respectively, coupled to positive analog charges lines 706P. Unit element 700B also has a negative unit element part comprising a plurality of negative AND-groups 203aN, 203bN, 203cN, each negative AND- group having AND gates 202aN, 202bN, 202cN with charge coupling capacitors 208aN, 208bN, 208cN, respectively, coupled to negative analog charges lines 706N. In the embodiment of figure 7B, a positive sign bit Bsgn 710 associated with the weight coefficients b[2:0] is applied to enable the upper P-suffixed AND-groups 203aP, 203bP, and 203cP to enable charge transfers to the P suffixed charge transfer bus 210[a-e]P. A negative sign bit Bsgn 710 results in the assertion of /Sign 714, which enables the lower N suffixed AND-groups 203aN, 203bN, and 203cN, with the positive and negative multiplication results being transferred to differential ADC 702 of figure 7C, as was described for figure 7A. An additional advantage of the architecture of figure 7B over the single ended UE of figures 2 and 4 is that a single CLR 711 input can be asserted to return all of the capacitors to an initial state . The AND-group functions are shown as all NAND in figure 7B compared with the mixture of AND and NAND in figure 7A for understanding the invention, the inversion sense from the AND/NAND gate outputs can be done by appropriate selection of the inversion/non-inversion for each P and N input to the ADC 702 of figure 7C .

[ 0052 ] In many deep learning applications , a Recti fied Linear Unit (ReLU) function may be used, which provides functionality similar to a diode . In the present dot product architecture , the ReLU function may be implemented using the SGN bit to disable multiplication for negative coef ficients in addition to having the ADC output 0 , thereby saving power in both stages .

Additionally, it may be desired to provide a bias term as an of fset input to the analog charge bus 210/ 420 . Figure 8A shows an example charge bias circuit attached to an example analog charge bus 210 or 420 , where the capacitors 806 are weighed to provide binary power of 2 weight from the bias input bits of 802 to the charge transferred to the associated line of the analog charge bus 210/420. In this manner, a binary bias value Bbias expressed as a binary input [E8..E0] may be applied at input 802 which is transferred through drivers 804 and through charge transfer capacitors 806 to transfer charge to the analog charge bus 420. The number of bits of the bias input 802 may be provided as any number of bits which are equal to or less than the number of bits in the analog charge bus 210 or 420.

[0053] It may also be desired to reset the charge transfer capacitors 208 of figures 2 and charge summing unit 212/412 capacitor charges to 0 to initialize the circuit, or to clear previous accumulated values or stored results. Figure 8B shows an example charge state reset, whereby at least one of the [a3..a0] or [b3..O] inputs is set to 0 so that the AND gate 408a, 408b, 408c all generate a zero output, and then reset circuit 820 with input 822 shorts the input lines and output line of the charge storage 412 to ground, so that all capacitors are initialized to a 0V state. Other methods for performing charge reset or initialization may be provided, these are given only as examples for understanding the invention and its capabilities. [ 0054 ] An important consideration for accuracy of the analog multiplication result is accurate charge trans fer weights , which relates to the ratio of charge trans fer capacitor 208 value Cu of 410 or 208 to the value of capacitance Cs , where each value of summing capacitor of charge summing unit 212 is binary weighted as previously described . Accordingly, the charge which is trans ferred from the unit elements 203 via all of the individual charge trans fer capacitors 208 / 410 should be large compared to nominal value Cs of binary weighted charge summing unit 212 / 412 to avoid systematic errors . Speci fically, the total value Cu contributed by all charge trans fer capacitors for a particular analog charge bus line 210 ( such as those associated with unit element 401 of figure 4B ) coupled to a line of the analog charge bus 210 or 420 is on the order of 8x or more greater than the largest summing capacitor such as 16Cs of 212 in figure 2B, 64Cs of 412 in figure 4A or 4B, or 16C of 212P and 212N of figure 7 .

[ 0055 ] Figure 9 shows another example of the invention 900 for performing gain balancing and accumulation of charge across an expandable plurality of unit element (UE ) 910a, 910b, 910c . . . etc, where each UE 910 includes three AND-groups 201a, 201b, 201c for each UE , and each AND gate of each AND-group such as 202a, 202b, 202c ; 204a, 204b, 204c; 206a, 206b, 206c is coupled to a respective charge transfer capacitor group comprising 208a, 208b, and 208c, respectively. Each charge transfer capacitor of 208a, 208b, and 208c is coupled to a separate charge transfer line 908a, 908b, 908c, 908d, 908e, 908f, 908g, 908h, and 908i, as shown, which are common across the other UE 901b, 901c, etc (not shown) . A computation for large number N of UE (specifically N=766) results in charge summing unit 912 having corresponding capacitance weights 8Cs, 16Cs, 33Cs, 16Cs, 33Cs, 69Cs, 33Cs, 69Cs, and 152Cs, each charge summing capacitor having a first terminal coupled to a respective charge transfer line 908a, 908b, 908c, 908d, 908e, 908f, 908g, 908h, and 908i, respectively. The second terminals of the charge summing capacitors of 912 are connected together and applied to the Analog to Digital Converter ADC 214 which provides an accumulated product output A1*B1+A2*B2+A3+B3+ . . . , as before. The advantage of the individual charge line connection is that the resulting chip layout provides a highly symmetrical layout geometry, minimizing parasitic capacitances compared to the alternative embodiment described in figures 2A or 2B. For the configuration of figure 9, each summing capacitor can be determined from the formula: where :

Cu is the value of each charge transfer capacitor of 208a, 208b, 208c;

C[l] is the value of the charge summing capacitor of 912 coupled to charge transfer line 908a;

C[k] is the value of each higher order charge summing capacitor of 912. k=l corresponds to the value of the summing capacitor associated with the lowest order charge transfer line 908a, with k incrementing for each charge transfer line 908b. . .908i .

[0056] As an example, C[l] is selected to be 8x the value of Cu. In general, C[l] should be larger than the total capacitance attributable to Cu associated with the bit order 0 charge transfer line such as by a factor of 8 or more to minimize summing errors, and the values of 912 are for N=766 (number of UEs sharing an analog charge bus.

[0057] Figure 9 shows nine charge transfer lines for clarity in understanding the invention, and the example does not limit the general scope of the invention to the number of charge trans fer lines , si ze of unit element 910 , or number of Unit Elements 901a, 901b, . . . , 901n .

[ 0058 ] The present multiplier architectures have certain advantages . In the prior art , multi-stage multipliers are synchronous devices with a running clock, which requires energy for displacement currents associated with each clock edge transition . In the various examples of the invention, the multiplication is operative asynchronously, and without any clocks , the multiplication value changing and being updated asynchronously when a multiplicand input changes value . Additionally, the present invention has the advantage of scalability, in that additional unit elements 200 , 401 , or 901 may be chained together on a common charge trans fer but leading to a charge summing unit as shown in the figures , such that each additional unit element may be flexibly added or isolated from the analog charge bus , and the accumulation of each multiplication result occurs on a respective analog charge bus 210 , 420 908 . In an example use case , the invention may be used where the B multiplicands are static weight coef ficients and the A multiplicands are dynamic for dot product computations in arti ficial intelligence applications . [ 0059 ] The proceeding has been a description of the various embodiments of the invention . It will be appreciated that deviations and modi fications can be made without departing from the scope of the invention, which is defined by appended claims . For example , the present invention is described in terms of AND-groups using AND gates for understanding the invention in terms of additive charge trans fers for 0 to 1 output transitions , however it is understood by one skilled in the art that NAND gates may be used throughout all of the AND-groups , as NAND gates have the advantage of requiring fewer transistors per gate for reali zation . In this case , the charge trans fer from 1 to 0 for the same transition in a NAND-group would be subtractive , and can be accounted for by inversion of the summed result . Various approximations may be used in the speci fication of the patent application, the approximations are understood to refer to ranges from a nominal value . A value which is " substantially" a nominal value is understood to be in the range of a factor of 1 /2 to 2 times the nominal value . A value which is "on the order of" a nominal value or " an order of magnitude" of a nominal value is understood to be in the range l / 10th to l Ox the nominal value .