## Digital Filter Implementations

### Using Digital Filter Blocks

DSP System Toolbox™ provides several blocks implementing digital filters, such as Discrete FIR Filter and Biquad Filter.

Use these blocks if you have already performed the design and analysis and know your desired filter coefficients. You can use these blocks to filter single-channel and multichannel signals, and to simulate floating-point and fixed-point filters. Then, you can use the Simulink® Coder™ product to generate highly optimized C code from your filters.

To implement a filter, you must provide the following basic information about the filter:

• The desired filter structure

• The filter coefficients

Note

Use the Digital Filter Design block to design and implement a filter. Use the Discrete FIR Filter and Biquad Filter blocks to implement a pre-designed filter. Both methods implement a filter in the same manner and have the same behavior during simulation and code generation.

### Implement a Lowpass Filter in Simulink

Use the Discrete FIR Filter (Simulink) block to implement a lowpass filter:

1. Define the lowpass filter coefficients in the MATLAB® workspace by typing

```lopassNum = [-0.0021 -0.0108 -0.0274 -0.0409 -0.0266 0.0374 0.1435 0.2465 0.2896 0.2465 0.1435 0.0374 -0.0266 -0.0409 -0.0274 -0.0108 -0.0021];```

2. Open Simulink and create a new model file.

3. From the DSP System Toolbox Filtering>Filter Implementations library, click-and-drag a Discrete FIR Filter block into your model.

4. Double-click the Discrete FIR Filter block. Set the block parameters as follows, and then click :

• Coefficient source = ```Dialog parameters```

• Filter structure = ```Direct form transposed```

• Coefficients = `lopassNum`

• Input processing = ```Columns as channels (frame based)```

• Initial states = `0`

Note that you can provide the filter coefficients in several ways:

• Type in a variable name from the MATLAB workspace, such as `lopassNum`.

• Type in filter design commands from Signal Processing Toolbox™ software or DSP System Toolbox software, such as `fir1(5, 0.2, 'low')`.

• Type in a vector of the filter coefficient values.

5. Rename your block Digital Filter - Lowpass.

The Discrete FIR Filter block in your model now represents a lowpass filter. In the next topic, Implement a Highpass Filter in Simulink, you use a Discrete FIR Filter block to implement a highpass filter. For more information about the Discrete FIR Filter block, see the Discrete FIR Filter (Simulink) block reference page. For more information about designing and implementing a new filter, see Using Digital Filter Design Block.

### Implement a Highpass Filter in Simulink

In this topic, you implement a highpass filter using the Discrete FIR Filter (Simulink) block:

1. If the model you created in Implement a Lowpass Filter in Simulink is not open on your desktop, you can open an equivalent model by typing

` ex_filter_ex1`

at the MATLAB command prompt.

2. Define the highpass filter coefficients in the MATLAB workspace by typing

```hipassNum = [-0.0051 0.0181 -0.0069 -0.0283 -0.0061 ... 0.0549 0.0579 -0.0826 -0.2992 0.5946 -0.2992 -0.0826 ... 0.0579 0.0549 -0.0061 -0.0283 -0.0069 0.0181 -0.0051];```
3. From the DSP System Toolbox Filtering library, and then from the Filter Implementations library, click-and-drag a Discrete FIR Filter block into your model.

4. Double-click the Discrete FIR Filter block. Set the block parameters as follows, and then click :

• Coefficient source = ```Dialog parameters```

• Filter structure = ```Direct form transposed```

• Coefficients = `hipassNum`

• Input processing = ```Columns as channels (frame based)```

• Initial states = `0`

You can provide the filter coefficients in several ways:

• Type in a variable name from the MATLAB workspace, such as `hipassNum`.

• Type in filter design commands from Signal Processing Toolbox software or DSP System Toolbox software, such as `fir1(5, 0.2, 'low')`.

• Type in a vector of the filter coefficient values.

5. Rename your block Digital Filter - Highpass.

You have now successfully implemented a highpass filter. In the next topic, Filter High-Frequency Noise in Simulink, you use these Discrete FIR Filter blocks to create a model capable of removing high frequency noise from a signal. For more information about designing and implementing a new filter, see Using Digital Filter Design Block.

### Filter High-Frequency Noise in Simulink

In the previous topics, you used Discrete FIR Filter (Simulink) blocks to implement lowpass and highpass filters. In this topic, you use these blocks to build a model that removes high frequency noise from a signal. In this model, you use the highpass filter, which is excited using a uniform random signal, to create high-frequency noise. After you add this noise to a sine wave, you use the lowpass filter to filter out the high-frequency noise:

1. If the model you created in Implement a Highpass Filter in Simulink is not open on your desktop, you can open an equivalent model by typing

`ex_filter_ex2 `

at the MATLAB command prompt.

2. If you have not already done so, define the lowpass and highpass filter coefficients in the MATLAB workspace by typing

```lopassNum = [-0.0021 -0.0108 -0.0274 -0.0409 -0.0266 ... 0.0374 0.1435 0.2465 0.2896 0.2465 0.1435 0.0374 ... -0.0266 -0.0409 -0.0274 -0.0108 -0.0021]; hipassNum = [-0.0051 0.0181 -0.0069 -0.0283 -0.0061 ... 0.0549 0.0579 -0.0826 -0.2992 0.5946 -0.2992 -0.0826 ... 0.0579 0.0549 -0.0061 -0.0283 -0.0069 0.0181 -0.0051];```
3. Click-and-drag the following blocks into your model file.

BlockLibraryQuantity

1

Random Source

Sources

1

Sine Wave

Sources

1

Time Scope

Sinks

1

4. Set the parameters for the rest of the blocks as indicated in the following table. For any parameters not listed in the table, leave them at their default settings.

BlockParameter Setting

• Icon shape  = `rectangular`

• List of signs = `++`

Random Source

• Source type = `Uniform`

• Minimum = `0`

• Maximum = `4`

• Sample mode = `Discrete`

• Sample time = `1/1000`

• Samples per frame = `50`

Sine Wave

• Frequency (Hz) = `75`

• Sample time = `1/1000`

• Samples per frame = `50`

Time Scope

• File > Number of Input Ports > 3

• File > Configuration ...

• Open the Visuals:Time Domain Options dialog and set Time span = ```One frame period```

5. Connect the blocks as shown in the following figure. You may need to resize some of your blocks to accomplish this task. 6. In the Modeling tab, click . The Configuration Parameters dialog opens.

7. In the Solver pane, set the parameters as follows, and then click :

• Start time = `0`

• Stop time = `5`

• Type = `Fixed-step`

• Solver = ```discrete (no continuous states)```

8. In the Simulation tab of the model toolstrip, click Run.

The model simulation begins and the Scope displays the three input signals.

9. After simulation is complete, select View > Legend from the Time Scope menu. The legend appears in the Time Scope window. You can click-and-drag it anywhere on the scope display. To change the channel names, double-click inside the legend and replace the current numbered channel names with the following:

• Add = `Noisy Sine Wave`

• Digital Filter – Lowpass = ```Filtered Noisy Sine Wave```

• Sine Wave = `Original Sine Wave`

In the next step, you will set the color, style, and marker of each channel.

10. In the Time Scope window, select View > Line Properties, and set the following:

LineStyleMarkerColor
Noisy Sine Wave - NoneBlack
Filtered Noisy Sine Wave - diamondRed
Original Sine Wave None *Blue
11. The Time Scope display should now appear as follows:

You can see that the lowpass filter filters out the high-frequency noise in the noisy sine wave. You have now used Discrete FIR Filter blocks to build a model that removes high frequency noise from a signal. For more information about designing and implementing a new filter, see Using Digital Filter Design Block.

### Specify Static Filters

You can specify a static filter using the Discrete FIR Filter (Simulink) or Biquad Filter block. To do so, set the Coefficient source parameter to ```Dialog parameters```.

For the Discrete FIR Filter, set the Coefficients parameter to a row vector of numerator coefficients. If you set Filter structure to `Lattice MA`, the Coefficients parameter represents reflection coefficients.

For the Biquad Filter, set the SOS matrix (Mx6) to an M-by-6 matrix, where M is the number of sections in the second-order section filter. Each row of the SOS matrix contains the numerator and denominator coefficients of the corresponding section in the filter. Set Scale values to a scalar or vector of M+1 scale values used between SOS stages.

#### Tuning the Filter Coefficient Values During Simulation

To change the static filter coefficients during simulation, double-click the block, type in the new filter coefficients, and click . You cannot change the filter order, so you cannot change the number of elements in the matrix of filter coefficients.

### Specify Time-Varying Filters

Time-varying filters are filters whose coefficients change with time. You can specify a time-varying filter that changes once per frame. You can filter multiple channels with each filter. However, you cannot apply different filters to each channel; all channels use the same filter.

To specify a time-varying filter using a Biquad Filter block or a Discrete FIR Filter block:

1. Set the Coefficient source parameter to `Input port(s)`, which enables extra block input ports for the time-varying filter coefficients.

• The Discrete FIR Filter block has a `Num` port for the numerator coefficients.

• The Biquad Filter block has `Num` and `Den` ports rather than a single port for the SOS matrix. Separate ports enable you to use different fraction lengths for numerator and denominator coefficients. The scale values port, `g`, is optional. You can disable the `g` port by setting Scale values mode to `Assume all are unity and optimize`.

2. Provide matrices of filter coefficients to the block input ports.

• For Discrete FIR Filter block, the number of filter taps, N, cannot vary over time. The input coefficients must be in a 1-by-N vector.

• For Biquad Filter block, the number of filter sections, N, cannot vary over time. The numerator coefficients input, `Num`, must be a 3-by-N matrix. The denominator input coefficients, `Den`, must be a 2-by-N matrix. The scale values input, `g`, must be a 1-by-(N+1) vector.

### Specify the SOS Matrix (Biquadratic Filter Coefficients)

Use the Biquad Filter block to specify a static biquadratic IIR filter (also known as a second-order section or SOS filter). Set the following parameters:

• Filter structure```Direct form I```, or `Direct form I transposed`, or `Direct form II`, or ```Direct form II transposed```

• SOS matrix (Mx6) M-by-6 SOS matrix

The SOS matrix is an M-by-6 matrix, where M is the number of sections in the second-order section filter. Each row of the SOS matrix contains the numerator and denominator coefficients (bik and aik) of the corresponding section in the filter.

• Scale values Scalar or vector of M+1 scale values to be used between SOS stages

If you enter a scalar, the value is used as the gain value before the first section of the second-order filter. The rest of the gain values are set to 1.

If you enter a vector of M+1 values, each value is used for a separate section of the filter. For example, the first element is the first gain value, the second element is the second gain value, and so on. You can use the `ss2sos` and `tf2sos` functions from Signal Processing Toolbox software to convert a state-space or transfer function description of your filter into the second-order section description used by this block.

`$\left[\begin{array}{cccccc}{b}_{01}& {b}_{11}& {b}_{21}& {a}_{01}& {a}_{11}& {a}_{21}\\ {b}_{02}& {b}_{12}& {b}_{22}& {a}_{02}& {a}_{12}& {a}_{22}\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ {b}_{0M}& {b}_{1M}& {b}_{2M}& {a}_{0M}& {a}_{1M}& {a}_{2M}\end{array}\right]$`

The block normalizes each row by a1i to ensure a value of 1 for the zero-delay denominator coefficients.