OTHER ARITHMETIC OPERATORS in VS .NET

Encoding QR-Code in VS .NET OTHER ARITHMETIC OPERATORS
OTHER ARITHMETIC OPERATORS
QR Code Decoder In VS .NET
Using Barcode Control SDK for Visual Studio .NET Control to generate, create, read, scan barcode image in .NET applications.
degree-63 polynomial using the degree-16 ones as primitives. The respective cells to be implemented correspond to the following polynomials: First stage cells (Figure 14.14a) Ci4 (x) c4i 3 :x3 c4i 2 :x2 c4i 1 :x c4i (c4i 3 :x c4i 2 ):x c4i 1 ):x c4i , Second stage cells (Figure 14.15a)
Painting Denso QR Bar Code In Visual Studio .NET
Using Barcode printer for .NET framework Control to generate, create QR Code 2d barcode image in Visual Studio .NET applications.
4 4 C 16 (x) (C4j 3 (x):x4 C4j 2 (x)):x4 j 4 4 C4j 1 (x)):x4 C4j (x),
QR Code JIS X 0510 Decoder In .NET
Using Barcode reader for VS .NET Control to read, scan read, scan image in .NET applications.
i 0, 1, . . . , 15:
Barcode Creation In Visual Studio .NET
Using Barcode printer for .NET Control to generate, create barcode image in Visual Studio .NET applications.
(14:27)
Barcode Scanner In VS .NET
Using Barcode scanner for .NET Control to read, scan read, scan image in .NET applications.
j 0, 1, 2, 3:
Generate QR In Visual C#
Using Barcode printer for .NET Control to generate, create QR image in VS .NET applications.
(14:28)
Encode QR Code In .NET
Using Barcode printer for ASP.NET Control to generate, create QR Code image in ASP.NET applications.
x c4i+3 a.x+b c4i+2 c4i+3 c4i+2 c4i+1 c4i x x
Denso QR Bar Code Maker In VB.NET
Using Barcode creation for .NET framework Control to generate, create QR Code 2d barcode image in .NET applications.
c4i+1
USS-128 Printer In Visual Studio .NET
Using Barcode generation for VS .NET Control to generate, create GTIN - 128 image in .NET framework applications.
C4i(x)
DataMatrix Printer In .NET
Using Barcode creation for Visual Studio .NET Control to generate, create Data Matrix 2d barcode image in Visual Studio .NET applications.
(a) First-stage degree-3 GHE cell C4 (x) i
EAN / UCC - 13 Creator In .NET
Using Barcode creator for Visual Studio .NET Control to generate, create EAN-13 Supplement 5 image in .NET framework applications.
c63 c62 c61 c60
Painting Leitcode In .NET Framework
Using Barcode drawer for Visual Studio .NET Control to generate, create Leitcode image in Visual Studio .NET applications.
c7 c6 c5 c 4
Code 128 Code Set C Printer In VB.NET
Using Barcode drawer for .NET Control to generate, create Code 128 image in .NET applications.
c3 c2 c1 c0
Barcode Printer In Visual Studio .NET
Using Barcode printer for ASP.NET Control to generate, create barcode image in ASP.NET applications.
C4 (x) 15 C4 (x) 1 (b) First-stage GHE: 16 degree-3 cells C4 (x) 0
GS1 - 13 Creation In .NET Framework
Using Barcode printer for ASP.NET Control to generate, create EAN / UCC - 13 image in ASP.NET applications.
GHE degree-63 polynomial: rst stage.
Code 128C Creator In Java
Using Barcode printer for Java Control to generate, create Code 128C image in Java applications.
14.2 POLYNOMIAL COMPUTATION CIRCUITS
Bar Code Creator In Java
Using Barcode creator for Java Control to generate, create bar code image in Java applications.
x4 C4 4j+3 a.x4+b C4 4j+2
Making Code 3 Of 9 In Java
Using Barcode printer for Java Control to generate, create Code 39 Full ASCII image in Java applications.
4 4 C4 4j+3 C 4j+2 C 4j+1
EAN-13 Supplement 5 Printer In VB.NET
Using Barcode generator for .NET framework Control to generate, create UPC - 13 image in Visual Studio .NET applications.
C4 4j
Generate EAN 128 In Java
Using Barcode creation for Java Control to generate, create GS1-128 image in Java applications.
C4 4j+1
C4 4j
C16 (x) j
C16 (x) j
(a) Second-stage degree-15 GHE cell
C4 15
C4 14
C4 13
C4 12
C4 11
C4 10
C4 9
C4 8
C4 7
C4 6
C4 5
C4 4
C4 3
C4 2
C4 1
C4 0
C16 (x) 3
C16 (x) 2
C16 (x) 1
C16 (x) 0
(b) Second-stage GHE: 4 degree-15 cells
Figure 14.15 GHE degree-63 polynomial: second stage.
Third stage cell (Figure 14.16a)
16 16 16 16 C 64 (x) (C3 (x):x16 C2 (x)):x16 C1 (x)):x16 C0 (x):
(14:29)
The 16 rst-stage cells are represented in gure 14.14b; the 4 second-stage cells are represented in Figure 14.15b, and the full 3-stage circuit is shown in Figure 14.16b. Each cell of Figure 14.16 implements recursively the function a.xk b: three times in this example. Observe that the sizes of the operands increase with the stage level. The practical implementation of the basic cells may depart from the direct application of the Horner scheme. Nevertheless, the synthesis problem may become quickly unmanageable, for example, dealing with integer numbers with a signi cant precision; observe that the inputs of the output cell of Figure 14.16 are made up of four degree-15 polynomials and one power of x (x16). Actually, as far as a suf cient precision is desired for the intermediate polynomial results, the number of binary variables to handle may become prohibitive for hardware implementation (LUT or circuits). Firmware approaches can be
OTHER ARITHMETIC OPERATORS
x16 C16 3 a.x16+b C16 2 C16 3 C16 2 C16 1 C16 0 x16 x16
C16 1
C16 0
C64(x)
C64(x)
(a) Third-stage degree-63 GHE cell
c(59..56) c(63..60)
c(51..48)
c(43..40)
c(35..32)
c(27..24)
c(19..16)
c(11..8)
c(3..0) c(7..4)
c(55..52)
c(47..44)
c(39..36)
c(31..28)
c(23..20)
c(15..12)
C4 15
C4 14
C4 13
C4 12
x 4 C4 11
C4 10
C4 9
C4 8
C4 7
C4 6
C4 5
C4 4
C4 3
C4 2
C4 1
C4 0
C16 (x) 3
C16 (x) 2
C16 (x) 1
C16 (x) 0 x16
C64(x)
(b) Full GHE circuit degree-63 polynomial
GHE degree-63 polynomial: full GHE 3-stage circuit.
suitable alternatives. Special applications of particular nite elds (e.g., GF(2)) look more realistic for a full hardware implementation. The overall circuit complexity will depend on the cell cost and computation time. According to the degree of the polynomial to synthesize, several cell sizes can be foreseen. Whenever this size is selected, the value k (power of x) is set for the next stage level. Cell sizes and number of stages are optimization parameters to be considered by the designer according to the time/cost constraints. As an alternative to the synthesis of degree-63 polynomials, one could have considered a 6-level tree using cells C2(x), C4(x), C8(x), . . . , C64 (x).
14.3 LOGARITHM OPERATOR
The cost and computation time of an s-level tree using cells Ct(x) (t rs2i; 0 ! i ! s 2 1) to synthesize polynomials of degree n rs 2 1, are given by C(n) S0 T(n) S0
i s 1 r i
:C(r s i ),
i s 1 T(r
(14:30)
where C(rs2i) and T(rs2i) stand for the cost and computation time of computing cell Ct(x), t rs2i. 14.3 LOGARITHM OPERATOR
This section presents an implementation for binary logarithms computation using multiplicative normalization. As shown in Section 7.3.3.1, the main (logarithm) sequence may be computed in another base different from the auxiliary sequence, built up in binary. The implementation, displayed in Figure 14.17, handles all data and results in binary. Algorithm 7.8 (logarithm computation by multiplicative normalization) assumes that the numerical values of (1 22i), (1 2 22i), ln (1 22i), and ln (1 2 22i) are available. In practical implementations, those values are read out from a look-up table to be preset by the designer. As the precision of the result is linear (1 bit-result per step), then, for p-bit precision, 2p logarithms ln (1 + 22i) and 2p values (1 + 22i) have to be precomputed and loaded. On the other hand, the precision of the stored values has to be de ned too: if p is the required precision for the result, at least p bits are needed per LUT entry. Actually, to cope with the errors generated by rounding and error propagation, some more bits have to be included. Nevertheless, 4p2 is a fair order of magnitude of the LUT cost. The argument X is in [1/2, 2]. A counter, not represented in the circuit displayed in gure 14.17, may be used to increment step number i. LUT outputs are thus updated while, at the same time, a combinational circuit computes x2i(i) and x2i(i).not x2i21(i). During the second phase of step i, the registers X and acc are loaded with X(i 1) and acc(i 1), respectively. The nal result is stored in the register acc after step p: ln x acc(p): Observe that the stop condition test (x(i) 1 ), which is optional, is not represented in Figure 14.17. The cost and computation time are given by C(p) 4:CLUT (p2 ) 2:Cmux4 (p) Cmux2 Cmultiplier (p) Csubtractor (p) Ccomb:circ: (p) Cacc (p) Creg (p), T(p) p:(max (TLUT (p2 ), (Tcomb:circ: (p) Tmux2 )) Tmux4 (p) Tmultiplier (p)), (14:31)