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To completely understand the results generated by fixed-point Simulink^{®} blocks, you must be aware of these issues:
When numerical block parameters are converted from doubles to Fixed-Point Designer™ data types
When input signals are converted from one Fixed-Point Designer data type to another (if at all)
When arithmetic operations on input signals and parameters are performed
For example, suppose a fixed-point Simulink block performs an arithmetic operation on its input signal and a parameter, and then generates output having characteristics that are specified by the block. The following diagram illustrates how these issues are related.
The sections that follow describe parameter and signal conversions. Rules for Arithmetic Operations discusses arithmetic operations.
Parameters of fixed-point blocks that accept numerical values are always converted from double to a fixed-point data type. Parameters can be converted to the input data type, the output data type, or to a data type explicitly specified by the block. For example, the Discrete FIR Filter block converts its Initial states parameter to the input data type, and converts its Numerator coefficient parameter to a data type you explicitly specify via the block dialog box.
Parameters are always converted before any arithmetic operations are performed. Additionally, parameters are always converted offline using round-to-nearest and saturation. Offline conversions are discussed below.
An offline conversion is a conversion performed by your development platform (for example, the processor on your PC), and not by the fixed-point processor you are targeting. For example, suppose you are using a PC to develop a program to run on a fixed-point processor, and you need the fixed-point processor to compute
$$y=\left(\frac{ab}{c}\right)u=Cu$$
over and over again. If a, b, and c are constant parameters, it is inefficient for the fixed-point processor to compute ab/c every time. Instead, the PC's processor should compute ab/c offline one time, and the fixed-point processor computes only C·u. This eliminates two costly fixed-point arithmetic operations.
Consider the conversion of a real-world value from one fixed-point data type to another. Ideally, the values before and after the conversion are equal.
$${V}_{a}={V}_{b},$$
where V_{b} is the input value and V_{a} is the output value. To see how the conversion is implemented, the two ideal values are replaced by the general [Slope Bias] encoding scheme described in Scaling:
$${V}_{i}={F}_{i}{2}^{{E}_{i}}{Q}_{i}+{B}_{i}.$$
Solving for the output data type's stored integer value, Q_{a} is obtained:
$$\begin{array}{c}{Q}_{a}=\frac{{F}_{b}}{{F}_{a}}{2}^{{E}_{b}-{E}_{a}}{Q}_{b}+\frac{{B}_{b}-{B}_{a}}{{F}_{a}}{2}^{-{E}_{a}}\\ ={F}_{s}{2}^{{E}_{b}-{E}_{a}}{Q}_{b}+{B}_{net},\end{array}$$
where F_{s} is the adjusted fractional slope and B_{net} is the net bias. The offline conversions and online conversions and operations are discussed below.
Both F_{s} and B_{net} are computed offline using round-to-nearest and saturation. B_{net} is then stored using the output data type and F_{s} is stored using an automatically selected data type.
The remaining conversions and operations are performed online by the fixed-point processor, and depend on the slopes and biases for the input and output data types. The conversions and operations are given by these steps:
The initial value for Q_{a} is given by the net bias, B_{net}:
$${Q}_{a}={B}_{net}.$$
The input integer value, Q_{b}, is multiplied by the adjusted slope, F_{s}:
$${Q}_{RawProduct}={F}_{s}{Q}_{b}.$$
The result of step 2 is converted to the modified output data type where the slope is one and bias is zero:
$${Q}_{Temp}=convert\left({Q}_{RawProduct}\right).$$
This conversion includes any necessary bit shifting, rounding, or overflow handling.
The summation operation is performed:
$${Q}_{a}={Q}_{Temp}+{Q}_{a}.$$
This summation includes any necessary overflow handling.
Note that the maximum number of conversions and operations is performed when the slopes and biases of the input signal and output signal differ (are mismatched). If the scaling of these signals is identical (matched), the number of operations is reduced from the worst (most inefficient) case. For example, when an input has the same fractional slope and bias as the output, only step 3 is required:
$${Q}_{a}=convert\left({Q}_{b}\right).$$
Exclusive use of binary-point-only scaling for both input signals and output signals is a common way to eliminate mismatched slopes and biases, and results in the most efficient simulations and generated code.