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MulDiv, Constant Multiplication and Division
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When you get into advanced mathematics (especially if you continue your academic career into electrical engineering), you will learn to appreciate the power of arithmetic series. With a modest amount of computing power, quite impressive results can be
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MULDIV, CONSTANT MULTIPLICATION AND DIVISION
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produced in terms of calculating data values. A good example of this is using arithmetic series to calculate a sine, cosine, or logarithmic function value for a given parameter. Arithmetic series can be used in analog electronics to prove that summing a number of simple sine waves can result in squarewave, sawtooth, or other arbitrary repeating waveforms. An arithmetic series has the form
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Result = + P1X1 + P2X2 + P3X3 + P4X4 + . . .
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where the pre x value (P#) is calculated to provide the function value. X# is the parameter value that is modi ed for each value in the series. The parameter change can be a number of different operations, including squaring, square rooting, multiplying by the power of a negative number, and so on. For the multiplication and division operations shown here, I will be shifting the parameter by 1 bit for each series element. The theory and mathematics of calculating the pre x and parameter for arithmetic series can be quite complex but it can be used in cases such as producing the pre x values for simple multiplication or division operations, as I am going to show in this section. To demonstrate the operations, I have created the multiply and divide macros that can be found in the muldiv.inc le in the Macros\ MulDiv folder. The two macros provide the multiplication and division functions using MPLAB assembler capabilities that I haven t explained yet (although I do in Chap. 10). To avoid confusion, I will explain how the macros work from the perspective of a high level language before presenting the actual code. To further help explain how the macros work, I will present them from the perspective of implementing the function in straight PIC microcontroller assembler. Multiplication (and division) can be represented by a number of different methods. When you were taught basic arithmetic, multiplication was repeated addition. If you had to program it, you would use the high level code
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Product = 0; for (i = 0; i < Multiplier; i++ ) Product = Product + Multiplicand;
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This method works very well but will take a differing amount of time based on the multiplier (i.e., eight times something takes four times longer than two times something). This is not a problem for single-digit multiplication, but when multiplication gets more complex, the operations become signi cantly longer, which can have a negative impact on operation of the application code. Ideally, a multiplication method (or algorithm) that does not have such extreme ranges should be used. As you would expect, this is where the arithmetic series is involved. As you know, every number consists of constants multiplied by exponents of the number s base. For example, 123 decimal is actually
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123 = 1 * Hundreds + 2 * tens + 3 * ones
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This also works for binary numbers and is used to convert constants between numbering systems. 123 decimal is 1111011 binary (0x07B). This can be represented like 123 decimal above as
123 = 1 * sixty-four + 1 * thirty-two + 1 * sixteen + 1 * eight + 0 * four + 1 * two + 1 * one
In this binary sequence, I also have included any digits that are zero (which is 4 in the case of 123) because they will be used when multiplying two numbers together. This binary sequence can be used as the pre x of a multiplication arithmetic series if each value is used to add the multiplicand that has been shifted up by the number of the bit. The shifted-up multiplicand can be thought of as the parameter of the series. This series can be written out as
A * B = ((A ((A ((A ((A & (1 << & (1 << 1)) & (1 << 2)) & (1 << 7)) 0)) != 0)*(B << 0) + != 0)*(B << 1) + != 0)*(B << 2) + + != 0)*(B << 7)
This series can be converted to high level code very easily:
int Multiply(int A, int B) // Multiply two eight bit values // together and return the result { int Product = 0; int i; for (i = 0; i < 8; i++) { if ((A & 1) != 0) Product = Product + B; // Repeat for each bit // // // // // Bit of Multiplier is Set, Add Multiplicand to Product Shift down the Multiplier Shift up the Multiplicand
A = A >> 1; B = B << 1; } return Product;
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