The purpose of this example is to show you how to simulate a continuous real- world signal using a generalized fixed-point data type. The model used is the simplest possible model and employs only two fixed-point blocks. Although simple in design, the model gives you the opportunity to explore many of the important features of the Fixed-Point Blockset including:
Data types
Scaling
Rounding
Logging minimum and maximum simulation values to the workspace
Overflow handling
The model used in this example is given by the fxpdemo_dbl2fix demo. You can launch this demo by typing its name at the MATLAB command line.
fxpdemo_dbl2fix
The model is shown below.
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Block Descriptions
The Signal Generator block is configured to output a sine wave with an amplitude defined on the interval [-5 5]. It always outputs double-precision numbers.
The FixPt Gateway In block is used as the interface between Simulink and the Fixed-Point Blockset. Its function is to convert the double-precision numbers from the Signal Generator block into one of the Fixed-Point Blockset data types. For simplicity, its output signal is limited to 5 bits in this example.
The FixPt Gateway Out block is used as the interface between the Fixed-Point Blockset and Simulink. Its function is to convert one of the Fixed-Point Blockset data types into a Simulink data type. In this example, it outputs double-precision numbers.
The FixPt GUI block launches the Fixed-Point Blockset Interface tool, fxptdlg. This tool provides convenient access to the global override and logging parameters, the logged minimum and maximum simulation data, the automatic scaling script, and the plot interface tool. It is not used in this example. However, if you have many fixed-point blocks whose scaling must be optimized, you should use this tool. Refer to Chapter 6, Tutorial: Feedback Controller Simulation for more information.
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As described in Compatibility with Simulink Blocks, you can eliminate the gateway blocks from your fixed-point model if all signals use built-in data types. |
Simulation Results
The results of two simulation trials are given below. The first trial uses radix point-only scaling while the second trial uses slope/bias scaling.
Trial 1: Radix Point-Only Scaling
When using radix point-only scaling, your goal is to find the optimal power-of-two exponent E, as defined in Selecting the Output Scaling. For this scaling mode, the fractional slope F is set to 1 and no bias is required.
The FixPt Gateway In block is configured in this way:
Output data type - The output data type is given by
sfix(5). This creates a MATLAB structure that is a 5-bit, signed generalized fixed-point number.Output scaling - The output scaling is given by
2^-2, which puts the radix point two places to the left of the rightmost bit. This gives a maximum value of 011.11 = 3.75, a minimum value of 100.00 = -4.00, and a precision of (1/2)2 = 0.25.Rounding - The rounding mode is given by
Nearest. This rounds the fixed-point result to the nearest representable number, with the exact midpoint rounded towards positive infinity.Overflows - Fixed-point values that overflow will saturate to the maximum or minimum value represented by the word.
The resulting real-world and fixed-point simulation results are shown below.
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The simulation clearly demonstrates the quantization effects of fixed-point arithmetic. The combination of using a 5-bit word with a precision of (1/2)2 = 0.25 produces a discretized output that does not span the full range of the input signal.
If you want to span the complete range of the input signal with 5 bits using radix point-only scaling, then your only option is to sacrifice precision. Hence, the output scaling would be given by 2^-1, which puts the radix point one place to the left of the rightmost bit. This scaling gives a maximum value of 0111.1 = 7.5, a minimum value of 1000.0 = -8.0, and a precision of (1/2)1 = 0.5.
Trial 2: Slope/Bias Scaling
When using slope/bias scaling, your goal is to find the optimal fractional slope F and fixed power-of-two exponent E, as defined in Selecting the Output Scaling. No bias is required for this example since the sine wave is defined on the interval [-5 5]. The FixPt Gateway In block configuration is the same as that of the previous trial except for the scaling.
To arrive at a value for the slope, you can begin by assuming a fixed power-of-two exponent of -2. In the previous trial, this value defined the radix point-only scaling and resulted in a precision of 0.25. To find the fractional slope, you divide the maximum value of the sine wave by the maximum value of the scaled 5-bit number. The result is 5.00/3.75 = 1.3333. The slope (and precision) is 1.3333.(0.25) = 0.3333. You specify this value as [0.3333] for the Output scaling parameter.
Of course, you could have specified a fixed power-of-two exponent of -1 and a corresponding fractional slope of 0.6667. Naturally, the resulting slope is the same since E was reduced by one bit but F was increased by one bit. In this case, the blockset would automatically store F as 1.3332 and E as -2 due to the normalization condition of 1 ⋜ F < 2.
The resulting real-world and fixed-point simulation results are shown below.
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This somewhat cumbersome process used to find the slope is not really necessary. All that is required is the range of the data you are simulating and the size of the fixed-point word used in the simulation. In general, you can achieve reasonable simulation results by selecting your scaling based on the formula
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where:
max is the maximum value to be simulated.
min is the minimum value to be simulated.
ws is the word size in bits.
2ws - 1 is the largest value of a word with whose size is given by ws.
For this example, the formula produces a slope of 0.32258.