Pi - Chudnovsky
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Fixed point numbers are a simple and easy way to express fractional numbers, using a fixed number of bits. Systems without floating-point hardware support frequently use fixed-point numbers to represent fractional numbers. The term "Fixed-Point" refers to the position of the binary point. The binary point is analogous to the decimal point of a base-ten number, but since this is binary rather than decimal, a different term is used.
In binary, bits can be either 0 or 1 and there is no separate symbol to designate where the binary point lies. However, we imagine, or assume, that the binary point resides at a fixed location between designated bits in the number. For instance, in a bit number, we can assume that the binary point exists directly between bits 15 15 because the first bit is numbered 0, not 1 and 16, giving binare formel 1 bits for the whole number part and 16 bits for the fractional part.
Note that the most significant bit in the whole number field is generally designated as the sign bit leaving 15 bits for the whole number's magnitude. The width of a fixed-point number is the total number of bits assigned for storage for the fixed-point number.
If we are storing the whole part and the fractional part in different storage locations, the width would be the total amount of storage for the number. The range of a fixed-point number is the difference between the minimum number possible, and the maximum number possible. The precision of a fixed-point number is the total number of bits for the fractional part of the number.
Because we can define where we want the fixed binary point to be located, the precision can be any number up to and including the width of binare formel 1 number. Note however, that the more precision we have, the less total range we have. There are a number of standards, but in this book we will use n for the width of a fixed-point number, p for the precision, and R for the total range. Not all numbers can be represented exactly by a fixed-point number, and so the closest approximation is used.
The formula for calculating binare formel 1 integer representation X in a Qm. Some examples in the extremely common  Q7. Because the position of the binary point is entirely conceptual, the logic for adding and subtracting fixed-point numbers is identical to the binare formel 1 required for adding and subtracting integers.
Thus, when adding one half plus one half in Q3. Which is equal to one as we would expect. This applies equally binare formel 1 subtraction. In other words, when we add or subtract fixed-point numbers, the binary point in the sum or difference will be located in exactly the same place as in the two numbers upon which we binare formel 1 operating.
When multiplying two 8-bit fixed-point numbers we will need 16 bits binare formel 1 hold the product. Clearly, binare formel 1 there are a different number of bits in the result as compared to the inputs, the binary point should be expected to move. However, it works exactly the same way in binary as it does in decimal. When we multiply two numbers in decimal, the location of the binare formel 1 point is N digits to the left of the product's rightmost digit, where N is sum of the number of digits located to the right side of the decimal point in the multiplier and the multiplicand.
Thus, in decimal when we multiply 0. The multiplier has one digit to the binare formel 1 of the decimal point, and the multiplicand has two digits to the right of the decimal point.
Thus, the product has three digits to the right of the decimal point which is to say, the decimal point is located three digits to the left. From the addition example above, we know that the number one half binare formel 1 Q3. Since 0x8 times 0x8 in hex is 0x also in hexthe fixed-point result can also be binare formel 1 to be 0x - as long as we know where the binary point is located. Let's write the product out in binary:. Since both the multiplier and multiplicand have four bits to the right of the binary point, the location of the binary point in the product is eight bits to the left.
Thus, our answer is If we want the format of the output to be the same as the format of the input, we must restrict the range of the inputs to prevent overflow. To convert from Q7. Many embedded systems that produce sine waves, such as DTMF generators, store a "sine table" in program memory. It's used for approximating the mathematical sine and cosine functions. Since such systems often have very limited amounts of program memory, often fixed-point numbers are used two different ways when such tables are used: Typically one quadrant of the sine and cosine functions are stored in that table.
The values in such tables are usually stored as fixed point numbers—often bit numbers in unsigned Q0. There seems to be two popular ways to handle the fact that Q0. A few people draw fairly accurate circles and calculate fairly accurate sine and cosine with a Bezier spline. Many people prefer to represent rotation such as angles in terms of "turns". The integer part of the "turns" tells how many whole revolutions have happened.
The main advantage to storing angles as a fixed-point fraction of a turn is speed. Combining some "current position" angle with some positive or negative "incremental angle" to get the "new position" is very fast, even on slow 8-bit microcontrollers: Other formats for storing angles require the same addition, plus special cases to handle the edge cases of overflowing degrees or underflowing 0 degrees. Using a binary angle format in units of "turns" allows us to quickly using shift-and-mask, avoiding multiply binare formel 1 the bits into:.
Some systems use the low-order bits to linearly interpolate between binare formel 1 in the table. A few systems binare formel 1 even more accuracy using an binare formel 1 smaller table by sacrificing a few more cycles to use those low-order bits to calculate cubic interpolation. Many embedded systems store the angle, the fractional part of the "turns", binare formel 1 a single byte binary angle format. One full turn  is brads  is degrees.
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