Computer words consist of a finite numbers of bits. This means that the binary encoding of variables is only an approximation of an arbitrarily precise real-world value. Therefore, the limitations of the binary representation automatically introduce limitations on the precision of the value.
The precision of a fixed-point word depends on the word size and radix point location. Extending the precision of a word can always be accomplished with more bits although you face practical limitations with this approach. Instead, you must carefully select the data type, word size, and scaling such that numbers are accurately represented. Rounding and padding with trailing zeros are typical methods implemented on processors to deal with the precision of binary words.
Rounding
The result of any operation on a fixed-point number is typically stored in a register that is longer than the number's original format. When the result is put back into the original format, the extra bits must be disposed of. That is, the result must be rounded. Rounding involves going from high precision to lower precision and produces quantization errors and computational noise.
The blockset provides four rounding modes, which are shown below.
The Fixed-Point Blockset rounding modes are discussed below. The data is generated using Simulink's Signal Generator block and doubles are converted to signed 8-bit numbers with radix point-only scaling of 2-2.
Round Toward Zero
The computationally simplest rounding mode is to drop all digits beyond the number required. This mode is referred to as rounding toward zero, and it results in a number whose magnitude is always less than or equal to the more precise original value. In MATLAB, you can round to zero using the fix function.
Rounding toward zero introduces a cumulative downward bias in the result for positive numbers and a cumulative upward bias in the result for negative numbers. That is, all positive numbers are rounded to smaller positive numbers, while all negative numbers are rounded to smaller negative numbers. Rounding toward zero is shown below.
An example comparing rounding to zero and truncation for unsigned and two's complement numbers is given in Example: Rounding to Zero Versus Truncation.
Round Toward Nearest
When rounding toward nearest, the number is rounded to the nearest representable value. This mode has the smallest errors associated with it and these errors are symmetric. As a result, rounding toward nearest is the most useful approach for most applications.
In MATLAB, you can round to nearest using the round function. Rounding toward nearest is shown below.
Round Toward Ceiling
When rounding toward ceiling, both positive and negative numbers are rounded toward positive infinity. As a result, a positive cumulative bias is introduced in the number.
In MATLAB, you can round to ceiling using the ceil function. Rounding toward ceiling is shown below.
Round Toward Floor
When rounding toward floor, both positive and negative numbers are rounded to negative infinity. As a result, a negative cumulative bias is introduced in the number.
In MATLAB, you can round to floor using the floor function. Rounding toward floor is shown below.
Rounding toward ceiling and rounding toward floor are sometimes useful for diagnostic purposes. For example, after a series of arithmetic operations, you may not know the exact answer because of word-size limitations, which introduce rounding. If every operation in the series is performed twice, once rounding to positive infinity and once rounding to negative infinity, you obtain an upper limit and a lower limit on the correct answer. You can then decide if the result is sufficiently accurate or if additional analysis is required.
Example: Rounding to Zero Versus Truncation
Rounding to zero and truncation or chopping are sometimes thought to mean the same thing. However, the results produced by rounding to zero and truncation are different for unsigned and two's complement numbers.
To illustrate this point, consider rounding a 5-bit unsigned number to zero by dropping (truncating) the two least significant bits. For example, the unsigned number 100.01 = 4.25 is truncated to 100 = 4. Therefore, truncating an unsigned number is equivalent to rounding to zero or rounding to floor.
Now consider rounding a 5-bit two's complement number by dropping the two least significant bits. At first glance, you may think truncating a two's complement number is the same as rounding to zero. For example, dropping the last two digits of -3.75 yields -3.00. However, digital hardware performing two's complement arithmetic yields a different result. Specifically, the number 100.01 = -3.75 truncates to 100 = -4, which is rounding to floor.
As you can see, rounding to zero for a two's complement number is not the same as truncation when the original value is negative. For this reason, the ambiguous term "truncation" is not used in this guide, and four explicit rounding modes are used instead.
Padding with Trailing Zeros
Padding with trailing zeros involves extending the least significant bit (LSB) of a number with extra bits. This method involves going from low precision to higher precision.
For example, suppose two numbers are subtracted from each other. First, the exponents must be aligned, which typically involves a right shift of the number with the smaller value. In performing this shift, significant digits can "fall off" to the right. However, when the appropriate number of extra bits is appended, the precision of the result is maximized. Consider two 8-bit fixed-point numbers that are close in value and subtracted from each other
where q is an integer. To perform this operation, the exponents must be equal.
If the top number is padded by two zeros and the bottom number is padded with one zero, then the above equation becomes
which produces a more precise result. An example of padding with trailing zeros using the Fixed-Point Blockset is illustrated in Digital Controller Realization.
Example: Limitations on Precision and Errors
Fixed-point variables have a limited precision because digital systems represent numbers with a finite number of bits. For example, suppose you must represent the real-world number 35.375 with a fixed-point number. Using the encoding scheme described in Scaling, the representation is
The two closest approximations to the real-world value are Q = 13 and Q = 14.
In either case, the absolute error is the same.
For fixed-point values within the limited range, this represents the worst-case error if round-to-nearest is used. If other rounding modes are used, the worst-case error can be twice as large.
Example: Maximizing Precision
Precision is limited by slope. To achieve maximum precision, the slope should be made as small as possible while keeping the range adequately large. The bias will be adjusted in coordination with the slope.
Assume the maximum and minimum real-world value is given by max(V) and min(V), respectively. These limits may be known based on physical principles or engineering considerations. To maximize the precision, you must decide upon a rounding scheme and whether overflows saturate or wrap. To simplify matters, this example assumes the minimum real-world value corresponds to the minimum encoded value, and the maximum real-world value corresponds to the maximum encoded value. Using the encoding scheme described in Scaling, these values are given by
Solving for the slope, you get
This formula is independent of rounding and overflow issues, and depends only on the word size, ws.