# HDL Butterworth Filter

This example illustrates how to generate HDL code for a 5th order Butterworth filter. The cutoff-frequency for this filter is very low relative to the sample rate, leading to a filter that is difficult to make practical. Also, small input (8-bit) and output (9-bit) word sizes cause the quantized filter to require scaling to be realizable.

## Contents

## Design the Filter

Use the CD sampling rate of 44.1 kHz and a cut-off frequency of 500 Hz. First, create the filter design object, then create the DF1 Biquad Filter System object. Finally, examine the response in log frequency using fvtool.

Fs = 44100; F3db = 500; filtdes = fdesign.lowpass('n,f3db', 5, F3db, Fs); butterFilter = design(filtdes,'butter',... 'SystemObject',true,'FilterStructure','df1sos'); fvtool(butterFilter, 'Fs', Fs, 'FrequencyScale', 'log');

## Create the Quantized Filter

Apply the fixed point settings to the filter object. Assume 9-bit fixed-point output data with 12-bit coefficients, 20-bit states, full precision products, and 32-bit adders. Check the response with fvtool.

butterFilter.NumeratorCoefficientsDataType = 'Custom'; butterFilter.CustomNumeratorCoefficientsDataType = numerictype([],12); butterFilter.CustomDenominatorCoefficientsDataType = numerictype([],12); butterFilter.CustomScaleValuesDataType = numerictype([],12); butterFilter.SectionInputDataType = 'Custom'; butterFilter.CustomSectionInputDataType = numerictype([],20,15); butterFilter.SectionOutputDataType = 'Custom'; butterFilter.CustomSectionOutputDataType = numerictype([],20,15); butterFilter.NumeratorProductDataType = 'Full precision'; butterFilter.DenominatorProductDataType = 'Full precision'; butterFilter.NumeratorAccumulatorDataType = 'Custom'; butterFilter.CustomNumeratorAccumulatorDataType = numerictype([],32,24); butterFilter.DenominatorAccumulatorDataType = 'Custom'; butterFilter.CustomDenominatorAccumulatorDataType = numerictype([],32,25); butterFilter.OutputDataType = 'Custom'; butterFilter.CustomOutputDataType = numerictype([],9,7); butterFilter.RoundingMethod = 'nearest'; butterFilter.OverflowAction = 'wrap'; fvtool(butterFilter, 'Fs', Fs, 'FrequencyScale', 'log','Arithmetic','fixed');

## Requantize the Filter

In the plot above, fvtool shows that the quantized passband is approximately 2 dB lower than the desired response. Adjust the coefficient word length from 12 to 16 to get the quantized response closer to the reference double-precision response and zoom in on the passband response. The quantized filter is now just over 0.1 dB lower than the reference filter.

butterFilter.CustomNumeratorCoefficientsDataType = numerictype([],16); butterFilter.CustomDenominatorCoefficientsDataType = numerictype([],16); butterFilter.CustomScaleValuesDataType = numerictype([],16); fvtool(butterFilter, 'Fs', Fs, 'FrequencyScale', 'log','Arithmetic','fixed'); axis([0 1.0 -1 1]);

## Examine the Scale Values

A key step for hardware realization of the filter design is to check whether the scale values are reasonable and adjust the scale value if needed. First, examine the quantized scale values relative to the input specification--an 8-bit value with fraction length of 7 bits. Since the first two scale values are smaller than the input settings, most of the input values are quantized away. To correct this, the filter needs to be scaled.

scaless = butterFilter.ScaleValues .* 2^7; disp(scaless);

Now scale the filter using the frequency domain infinity norm. After scaling, the scale value are all one in this case.

```
scale(butterFilter,'Linf');
scaless = butterFilter.ScaleValues;
disp(scaless);
```

## Generate HDL Code and Test Bench from the Quantized Filter

Starting with the correctly quantized filter, generate VHDL or Verilog code. You have the option of generating a VHDL or Verilog test bench to verify that the HDL design matches the MATLAB® filter.

To generate Verilog instead, change the value of the property 'TargetLanguage', from 'VHDL' to 'Verilog'.

Since the passband of this filter is so low relative to the sampling rate, a custom input stimulus is a better way to test the filter implementation. Build the test input with one cycle of each of 50 to 300 Hz in 50 Hz steps.

Assume 8-bit signed fixed-point input with 7 bits of fraction.

Generate a VHDL test bench to verify that the results match the MATLAB results exactly.

Create a temporary work directory. Generate VHDL code for the filter and a VHDL test bench to verify that the results match the MATLAB results exactly.

Open the generated VHDL file for the filter in the editor.

workingdir = tempname; userstim = []; for n = [50, 100, 150, 200, 250, 300] userstim = [userstim, sin(2*pi*n/Fs*(0:Fs/n))]; %#ok end generatehdl(butterFilter, 'Name', 'hdlbutter',... 'TargetLanguage', 'VHDL',... 'TargetDirectory', workingdir, ... 'GenerateHDLTestbench', 'on', ... 'TestBenchUserStimulus', userstim, ... 'InputDataType',numerictype(1,8,7)); edit(fullfile(workingdir, 'hdlbutter.vhd')); % Open the generated VHDL test bench in the editor. edit(fullfile(workingdir, 'hdlbutter_tb.vhd'));

## Generate HDL Code and Test Bench Using FDHDLTool

HDL code and test bench can optionally be generated using the FDHDLTOOL command that opens the dialog which lets you customize and generate Verilog or VHDL code and test benches for the quantized filter.

The GUI is customized to 'butterFilter' in such a way that only the relevant widgets are available to set. To generate HDL code and test bench you should first go to the working directory and then call the FDHDLTOOL command.

fdhdltool(butterFilter, numerictype(1,8,7));

You can modify the default settings and click Generate to generate HDL and/or test bench.

## ModelSim® Simulation Results

The following display shows the ModelSim HDL simulator after running the VHDL test bench. Compare the ModelSim result with the MATLAB result below.

xrange = (0:length(userstim) - 1); y = butterFilter(fi(userstim.',1,8,7)); subplot(2,1,1); plot(xrange, userstim); axis([0 length(userstim) -1.1 1.1]); title('HDL Butterworth filter in Stimulus.'); xlabel('Sample #'); subplot(2,1,2); plot(xrange, y); axis([0 length(userstim) -1.1 1.1]); title('HDL Butterworth filter out Response.'); xlabel('Sample #');

## Conclusion

You designed a Butterworth filter to meet the given specification. You then quantized the filter and discovered that the passband requirement was not met. Requantizing the coefficients and scaling the filter fixed this issue. You then generated VHDL code for the filter and a VHDL test bench.

You can use the ModelSim HDL Simulator, to verify these results. You can also experiment with VHDL and Verilog for both filters and test benches.