Why do we need linear phase filters?

Digital filters with linear phase have the advantage of delaying all frequency components by the same amount, i.e. they preserve the input signal’s phase relationships. This preservation of phase means that the filtered signal retains the shape of the original input signal. This characteristic is essential for audio applications as the signal shape is paramount for maintaining high fidelity in the filtered audio. Yet another application area that requires this, is ECG biomedical waveform analysis, as any artefacts introduced by the filter may be misinterpreted as heart anomalies.

The following plot shows the filtering performance of a Chebyshev type I lowpass IIR on ECG data – input waveform (shown in blue) shifted by 10 samples (\(\small \Delta=10\)) to approximately compensate for the filter’s group delay. Notice that the filtered signal (shown in red) has attenuated, broadened and added oscillations around the ECG peak, which is undesirable.

In order for a digital filter to have linear phase, its impulse response must have conjugate-even or conjugate-odd symmetry about its midpoint. This is readily seen for an FIR filter,

With the following constraint on its coefficients,

which leads to,

Analysing Eqn. 3, we see that roots (zeros) of \(\small H(z)\) must also be the zeros of  \(\small H^\ast (1/z^\ast)\). This means that the roots of \(\small H(z)\) must occur in conjugate reciprocal pairs, i.e.  if \(\small z_k\) is a zero of \(\small H(z)\), then \(\small H^\ast (1/z^\ast)\) must also be a zero.

Why IIR filters do not have linear phase

A digital filter is said to be bounded input, bounded output stable, or BIBO stable, if every bounded input gives rise to a bounded output. All IIR filters have either poles or both poles and zeros, and must be BIBO stable, i.e.

\(\displaystyle \sum_{k=0}^{\infty}\left|h(k)\right|<\infty \tag{4}\)

Where, \(\small h(k)\) is the filter’s impulse response. Analyzing Eqn. 4, it should be clear that the BIBO stability criterion will only be satisfied if the system’s poles lie inside the unit circle, since the system’s ROC (region of convergence) must include the unit circle. Consequently, it is sufficient to say that a bounded input signal will always produce a bounded output signal if all the poles lie inside the unit circle.

The zeros on the other hand, are not constrained by this requirement, and as a consequence may lie anywhere on z-plane, since they do not directly affect system stability. Therefore, a system stability analysis may be undertaken by firstly calculating the roots of the transfer function (i.e., roots of the numerator and denominator polynomials) and then plotting the corresponding poles and zeros upon the z-plane.

Applying the developed logic to the poles of an IIR filter, we now arrive at a very important conclusion on why IIR filters cannot have linear phase.

A BIBO stable filter must have its poles within the unit circle, and as such in order to get linear phase, an IIR would need conjugate reciprocal poles outside of the unit circle, making it BIBO unstable.

Based upon this statement, it would seem that it’s not possible to design an IIR to have linear phase. However, a discussed below, phase equalisation filters can be used to linearise the passband phase response.

Phase linearisation with all-pass filters

All-pass phase linearisation filters (equalisers) are a well-established method of altering a filter’s phase response while not affecting its magnitude response. A second order (Biquad) all-pass filter is defined as:

Where, \(\small f_c\) is the centre frequency, \(\small r\) is radius of the poles and \(\small f_s\) is the sampling frequency. Notice how the numerator and denominator coefficients are arranged as a mirror image pair of one another.  The mirror image property is what gives the all-pass filter its desirable property, namely allowing the designer to alter the phase response while keeping the magnitude response constant or flat over the complete frequency spectrum.

Cascading an APF (all-pass filter) equalisation cascade (comprised of multiple APFs) with an IIR filter, the basic idea is that we only need to linearise the phase response the passband region. The other regions, such as the transition band and stopband may be ignored, as any non-linearities in these regions are of little interest to the overall filtering result.

The challenge

The APF cascade sounds like an ideal compromise for this challenge, but in truth a significant amount of time and very careful fine-tuning of the APF positions is required in order to achieve an acceptable result. Each APF has two variables: \(\small f_c\) and \(\small r\) that need to be optimised, which complicates the solution. This is further complicated by the fact that the more APF stages that are added to the cascade, the higher the overall filter’s group delay (latency) becomes. This latter issue may become problematic for fast real-time closed loop control systems that rely on an IIR’s low latency property.

Nevertheless, despite these challenges, the APF equaliser is a good compromise for linearising an IIRs passband phase characteristics.

The APF equaliser

ASN Filter Designer provides designers with a very simple to use graphical all-phase equaliser interface for linearising the passband phase of IIR filters. As seen below, the interface is very intuitive, and allows designers to quickly place and fine-tune APF filters positions with the mouse. The tool automatically calculates \(\small f_c\) and \(\small r\), based on the marker position.

Right clicking on the frequency response chart or on an existing all-pass design marker displays an options menu, as shown on the left.

You may add up to 10 biquads (professional version only).

An IIR with linear passband phase

Designing an equaliser composed of three APF pairs, and cascading it with the Chebyshev filter of Figure 1, we obtain a filter waveform that has a much a sharper peak with less attenuation and oscillation than the original IIR – see below. However, this improvement comes at the expense of three extra Biquad filters (the APF cascade) and an increased group delay, which has now risen to 24 samples compared with the original 10 samples.

The frequency response of both the original IIR and the equalised IIR are shown below, where the group delay (shown in purple) is the average delay of the filter and is a simpler way of assessing linearity.

Notice that the group delay of the equalised IIR passband (shown on the right) is almost flat, confirming that the phase is indeed linear.

Automatic code generation to Arm processor cores via CMSIS-DSP

The ASN Filter Designer’s automatic code generation engine facilitates the export of a designed filter to Cortex-M Arm based processors via the CMSIS-DSP software framework. The tool’s built-in analytics and help functions assist the designer in successfully configuring the design for deployment.

Before generating the code, the IIR and equalisation filters (i.e. H1 and Heq filters) need to be firstly re-optimised (merged) to an H1 filter (main filter) structure for deployment. The options menu can be found under the P-Z tab in the main UI.

All floating point IIR filters designs should be based on Single Precision arithmetic and either a Direct Form I or Direct Form II Transposed filter structure, as this is supported by a hardware multiplier in the M4F, M7F, M33F and M55F cores. Although you may choose Double Precision, hardware support is only available in some M7F and M55F Helium devices. The Direct Form II Transposed structure is advocated for floating point implementation by virtue of its higher numerically accuracy.

Quantisation and filter structure settings can be found under the Q tab (as shown on the left). Setting Arithmetic to Single Precision and Structure to Direct Form II Transposed and clicking on the Apply button configures the IIR considered herein for the CMSIS-DSP software framework.

Select the Arm CMSIS-DSP framework from the selection box in the filter summary window:

The automatically generated C code based on the CMSIS-DSP framework for direct implementation on an Arm based Cortex-M processor is shown below:

The ASN Filter Designer’s automatic code generator generates all initialisation code, scaling and data structures needed to implement the linearised filter IIR filter via Arm’s CMSIS-DSP library.

Arm deployment wizard

Professional licence users may expedite the deployment by using the Arm deployment wizard. The built in AI will automatically determine the best settings for your design based on the quantisation settings chosen.

The built in AI automatically analyses your complete filter cascade and converts any H2 or Heq filters into an H1 for implementation.

What we have learnt

The roots of a linear phase digital filter must occur in conjugate reciprocal pairs. Although this no problem for an FIR filter, it becomes infeasible for an IIR filter, as poles would need to be both inside and outside of the unit circle, making the filter BIBO unstable.

The passband phase response of an IIR filter may be linearised by using an APF equalisation cascade. The ASN Filter Designer provides designers with everything they need via a very simple to use, graphical all-pass phase equaliser interface, in order to design a suitable APF cascade by just using the mouse!

The linearised IIR filter may be exported via the automatic code generator using Arm’s optimised CMSIS-DSP library functions for deployment on any Cortex-M microcontroller.

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Matlab is a well-established numerical computing language developed by the Mathworks that allows for the design of algorithms, matrix data manipulations and data analysis. The product offers a broad range of algorithms and support functions for signal processing applications, and as such is very popular amongst many scientists and engineers worldwide.

ASN Filter Designer automatic code generator for Matlab

The ASN Filter Designer greatly simplifies exporting a designed filter to Matlab via its automatic code generator. The code generator supports all aspects of the ASN Filter Designer, allowing for a complete design comprised of H1, H2 and H3 filters and math operators to be fully integrated with an algorithm in Matlab.

The Matlab code generator can be accessed via the filter summary options (as shown on the right). Selecting this option will automatically generate a Matlab .m file based on the current design.

Version 5 of the tool has a completely revamped filter summary UI, and now includes built in AI to analyse the filter cascade for any potential problems.  The project wizard bundles all of the necessary SDK framework files needed to run the designed filter cascade without the need for any other dependencies or 3^rd party plugins.

Framework files and examples

In order to use the generated code in Matlab without the need for Signal Processing Toolbox, the following three framework files are provided in the ASN Filter Designer’s \Matlab directory:

ASNFDMatlabFilterData.m
ASNFDMatlabImport.m
ASNFDFilter.m

These framework files do not have any special Matlab toolbox dependences, and the example script ASNFDMatlabDemo.m demonstrates the simplicity with which the framework can be integrated into your application for your designed filter. Several example filters generated via the automatic code generator are given within ASNFDMatlabDemo.m in order to get you going!

An example of the summary of all of generated files (including the framework files) is shown below.

These files can be used directly in your Matlab/Octave project.

Comparing the results to Matlab’s Signal Processing Toolbox

It’s sometimes informative to compare the results of the ASN Filter Designer’s DSP library functions to that of Matlab’s Signal Processing Toolbox.

Designing an IIR Chebyshev Type I filter with the following specifications:

Graphically entering the specifications into the ASN Filter Designer, and fine tuning the design marker positions, the tool automatically designs the filter as a Biquad cascade. Notice that the tool automatically finds the required filter order, and in essence – automatically produces the filter’s exact technical specification!

The frequency response of a 5th order IIR Chebyshev Type I lowpass filter meeting the specifications is shown below:

The resulting filter coefficients are:

Designing the same filter in Matlab using Signal Processing Toolbox:

Fs=500;
Rp=0.1;
Rs=80;
F=2*[25,125]/Fs;

[N,Wn]=cheb1ord(F(1),F(2),Rp,Rs)
[z, p, k] = cheby1(N,Rp,Wn,'low'); % design lowpass

[sos,g]=zp2sos(z,p,k,'up')  % generate SOS form

Running the script, we get the following, where each row of sos is a biquad arranged as: b0 b1 b2 a0 a1 a2

Analysing both sets of numerator and denominator coefficients, we get exactly the same result! But what about the gain? Matlab outputs a net gain, g = 3.0096e-05 but the ASN Filter Designer optimally assigns a gain to each biquad. Thus, combining the biquad section gains, i.e. 0.078643, 0.013823 and 0.027685 results in a net gain of 3.0096e-05, which is exactly the same net gain as Matlab!

Conclusion: the ASN Filter Designer’s DSP IIR library functions completely match Matlab’s Signal Processing Toolbox results!!

The complete automatically generated code is shown below, where it can be seen that the biquad gains have been pre-multiplied with the feedforward coefficients.

Using the generated code with Signal Processing Toolbox

If you have Signal Processing Toolbox installed, then you may directly use the generated coefficients given in SOS with the sosfilt() command, e.g.

Clear all;

ASNFD_SOS=[ 0.07864301814, 0.07864301814, 0.00000000000, 1.00000000000,-0.84271396371, 0.00000000000;...
 0.01382289248, 0.02764578495, 0.01382289248, 1.00000000000,-1.70536517618, 0.76065674608;...
 0.02768538360, 0.05537076720, 0.02768538360, 1.00000000000,-1.79181447713, 0.90255601154;...
];

y=sosfilt(ASNFD_SOS, x); %  x is your input data

plot(x,y); % plot results

As seen, it is as simple as copying and pasting the filter coefficients from the ASN Filter Designer’s filter summary into a Matlab script.

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Details are also given (including a reference example project) regarding implementation of the IIR filter in Arm/Keil’s MDK industry standard Cortex-M microcontroller development kit.

Introduction

ASN Filter Designer provides engineers with a powerful DSP experimentation platform, allowing for the design, experimentation and deployment of complex IIR and FIR (finite impulse response) digital filter designs for a variety of sensor measurement applications. The tool’s advanced functionality, includes a graphical based real-time filter designer, multiple filter blocks, various mathematical I/O blocks, live symbolic math scripting and real-time signal analysis (via a built-in signal analyser). These advantages coupled with automatic documentation and code generation functionality allow engineers to design and validate a digital filter within minutes rather than hours.

The Arm CMSIS-DSP (Cortex Microcontroller Software Interface Standard) software framework is a rich collection of over sixty DSP functions (including various mathematical functions, such as sine and cosine; IIR/FIR filtering functions, complex math functions, and data types) developed by Arm that have been optimised for their range of Cortex-M processor cores.

The framework makes extensive use of highly optimised SIMD (single instruction, multiple data) instructions, that perform multiple identical operations in a single cycle instruction. The SIMD instructions (if supported by the core) coupled together with other optimisations allow engineers to produce highly optimised signal processing applications for Cortex-M based micro-controllers quickly and simply.

ASN Filter Designer fully supports the CMSIS-DSP software framework, by automatically producing optimised C code based on the framework’s DSP functions via its code generation engine.

Designing IIR filters with the ASN Filter Designer

ASN Filter Designer provides engineers with an easy to use, intuitive graphical design development platform for both IIR and FIR digital filter design. The tool’s real-time design paradigm makes use of graphical design markers, allowing designers to simply draw and modify their magnitude frequency response requirements in real-time while allowing the tool automatically fill in the exact specifications for them.

Consider the design of the following technical specification:

Graphically entering the specifications into the ASN Filter Designer, and fine tuning the design marker positions, the tool automatically designs the filter as a Biquad cascade (this terminology will be discussed in the following sections), automatically choosing the required filter order, and in essence – automatically producing the filter’s exact technical specification!

The frequency response of a 5th order IIR Elliptic Lowpass filter meeting the specifications is shown below:

This 5th order Lowpass filter will form the basis of the discussion presented herein.

Biquad IIR filters

The IIR filter implementation discussed herein is said to be biquad, since it has two poles and two zeros as illustrated below in Figure 1. The biquad implementation is particularly useful for fixed point implementations, as the effects of quantization and numerical stability are minimised. However, the overall success of any biquad implementation is dependent upon the available number precision, which must be sufficient enough in order to ensure that the quantised poles are always inside the unit circle.

Figure 1: Direct Form I (biquad) IIR filter realization and transfer function.

Analysing Figure 1, it can be seen that the biquad structure is actually comprised of two feedback paths (scaled by \(a_1\) and \(a_2\)), three feed forward paths (scaled by \(b_0, b_1\) and \(b_2\)) and a section gain, \(K\). Thus, the filtering operation of Figure 1 can be summarised by the following simple recursive equation:

\(\displaystyle y(n)=K\times\Big[b_0 x(n) + b_1 x(n-1) + b_2 x(n-2)\Big] – a_1 y(n-1)-a_2 y(n-2)\)

Analysing the equation, notice that the biquad implementation only requires four additions (requiring only one accumulator) and five multiplications, which can be easily accommodated on any Cortex-M microcontroller. The section gain, \(K\) may also be pre-multiplied with the forward path coefficients before implementation.

A collection of Biquad filters is referred to as a Biquad Cascade, as illustrated below.

The ASN Filter Designer can design and implement a cascade of up to 50 biquads (Professional edition only).

Floating point implementation

When implementing a filter in floating point (i.e. using double or single precision arithmetic) Direct Form II structures are considered to be a better choice than the Direct Form I structure. The Direct Form II Transposed structure is considered the most numerically accurate for floating point implementation, as the undesirable effects of numerical swamping are minimised as seen by analysing the difference equations.

Figure 2 – Direct Form II Transposed strucutre, transfer function and difference equations

The filter summary (shown in Figure 3) provides the designer with a detailed overview of the designed filter, including a detailed summary of the technical specifications and the filter coefficients, which presents a quick and simple route to documenting your design.

The ASN Filter Designer supports the design and implementation of both single section and Biquad (default setting) IIR filters. However, as the CMSIS-DSP framework does not directly support single section IIR filters, this feature will not be covered in this application note.

The CMSIS-DSP software framework implementation requires sign inversion (i.e. flipping the sign) of the feedback coefficients. In order to accommodate this, the tool’s automatic code generation engine automatically flips the sign of the feedback coefficients as required. In this case, the set of difference equations become,

\(y(n)=b_0 x(n)+w_1 (n-1)\)
\(w_1 (n)= b_1 x(n)+a_1 y(n)+w_2 (n-1)\)
\(w_2 (n)= b_2 x(n)+a_2 y(n)\)

Figure 3: ASN filter designer: filter summary.

Automatic code generation to Arm processor cores via CMSIS-DSP

The ASN Filter Designer’s automatic code generation engine facilitates the export of a designed filter to Cortex-M Arm based processors via the CMSIS-DSP software framework. The tool’s built-in analytics and help functions assist the designer in successfully configuring the design for deployment.

All floating point IIR filters designs should be based on Single Precision arithmetic and either a Direct Form I or Direct Form II Transposed filter structure, as this is supported by a hardware multiplier in the M4F, M7F, M33F and M55F cores. Although you may choose Double Precision, hardware support is only available in some M7F and M55F Helium devices. As discussed in the previous section, the Direct Form II Transposed structure is advocated for floating point implementation by virtue of its higher numerically accuracy.

Quantisation and filter structure settings can be found under the Q tab (as shown on the left). Setting Arithmetic to Single Precision and Structure to Direct Form II Transposed and clicking on the Apply button configures the IIR considered herein for the CMSIS-DSP software framework.

Select the Arm CMSIS-DSP framework from the selection box in the filter summary window:

The automatically generated C code based on the CMSIS-DSP framework for direct implementation on an Arm based Cortex-M processor is shown below:

As seen, the automatic code generator generates all initialisation code, scaling and data structures needed to implement the IIR via the CMSIS-DSP library. This code may be directly used in any Cortex-M based development project – a complete Keil MDK example is available on Arm/Keil’s website. Notice that the tool’s code generator produces code for the Cortex-M4 core as default, please refer to the table below for the #define definition required for all supported cores.

Automatic code generation of complex coefficient IIR filters is currently not supported (see below for more information).

Arm deployment wizard

Professional licence users may expedite the deployment by using the Arm deployment wizard. The built in AI will automatically determine the best settings for your design based on the quantisation settings chosen.

The built in AI automatically analyses your complete filter cascade and converts any H2 or Heq filters into an H1 for implementation. A complex coefficient filter will be automatically converted to real filter for implementation.

Implementing the filter in Arm Keil’s MDK

As mentioned in the previous section, the code generated by the Arm CMSIS-DSP code generator may be directly used in any Cortex-M based development project tooling, such as Arm Keil’s industry standard μVision MDK (microcontroller development kit).

A complete μVision example IIR biquad filter project can be downloaded from Keil’s website, and as seen below is as simple as copying and pasting the code and making minor adjustments to the code.

The example project makes use of μVision’s powerful simulation capabilities, allowing for the evaluation of the IIR filter on M0, M3, M4 and M7 cores respectively. As an added bonus, μVision’s logic analyser may also be used, allowing for comparisons between the ASN Filter Designer’s signal analyser and the reality on a Cortex-M core.

Fixed point implementation

As aforementioned, the Direct Form I filter structure is the best choice for fixed point implementation. However, before implementing the difference equation on a fixed point processor, several important data scaling considerations must be taken into account. As the CMSIS-DSP framework only supports Q15 and Q31 data types for IIR filters, the following discussion relates to an implementation on a 16-bit word architecture, i.e. Q15.

Quantisation

In order to correctly represent the coefficients and input/output numbers, the system word length (16-bit for the purposes of this application note) is first split up into its number of integers and fractional components. The general format is given by:

Q Num of Integers.Fraction length

If we assume that all of data values lie within a maximum/minimum range of \(\pm 1\), we can use Q0.15 format to represent all of the numbers respectively. Notice that Q0.15 (or simply Q15) format represents a maximum of \(\displaystyle 1-2^{-15}=0.9999=0x7FFF\) and a minimum of \(-1=0x8000\) (two’s complement format).

The ASN Filter Designer may be configured for Fixed Point Q15 arithmetic by setting the Word length and Fractional length specifications in the Q Tab (see the configuration section for the details). However, one obvious problem that manifests itself for Biquads is the number range of the coefficients. As poles can be placed anywhere inside the unit circle, the resulting polynomial needed for implementation will often be in the range \(\pm 2\), which would require Q14 arithmetic. In order to overcome this issue, all numerator and denominator coefficients are scaled via a biquad Post Scaling Factor as discussed below.

Post Scaling Factor

In order to ensure that coefficients fit within the Word length and Fractional length specifications, all IIR filters include a Post Scaling Factor, which scales the numerator and denominator coefficients accordingly. As a consequence of this scaling, the Post Scaling Factor must be included within the filter structure in order to ensure correct operation.

The Post scaling concept is illustrated below for a Direct Form I biquad implementation.

Figure 4: Direct Form I structure with post scaling.

Pre-multiplying the numerator coefficients with the section gain, \(K\), each coefficient can now be scaled by \(G\), i.e. \(\displaystyle b_0=\frac{b_0}{G}, b_1=\frac{b_1}{G}, a_1=\frac{a_1}{G}, a_2=\frac{a_2}{G}\) and etc. This now results in the following difference equation:

\(\displaystyle y(n)=G \times\Big [b_0 x(n) + b_1 x(n-1) + b_2 x(n-2) – a_1 y(n-1)-a_2 y(n-2)\Big]\)

All IIR structures implemented within the tool include the Post Scaling Factor concept. This scaling is mandatory for implementation via the Arm CMSIS-DSP framework – see the configuration section for more details.

Understanding the filter summary

In order to fully understand the information presented in the ASN Filter Designer filter summary, the following example illustrates the filter coefficients obtained with Double Precision arithmetic and with Fixed Point Q15 quantisation.

Applying Fixed Point Q15 arithmetic (note the effects of quantisation on the coefficient values):

Configuring the ASN Filter Designer for Fixed Point arithmetic

In order to implement an IIR fixed point filter via the CMSIS-DSP framework, all designs must be based on Fixed Point arithmetic (either Q15 or Q31) and the Direct Form I filter structure.

Quantisation and filter structure settings can be found under the Q tab (as shown on the left): Setting Arithmetic to Fixed Point and Structure to Direct Form I and clicking on the Apply button configures the IIR considered herein for the CMSIS-DSP software framework.

The Post Scaling Factor is actually implemented in the CMSIS-DSP software framework as \( \log_2 G\) (i.e. a shift left scaling operation as depicted in Figure 4).

Built in analytics: the tool will automatically analyse the cascade’s filter coefficients and choose an appropriate scaling factor. As seen above, as the largest minimum value is -1.63143, thus, a Post Scaling Factor of 2 is required in order to ‘fit’ all of the coefficients into Q15 arithmetic.

Comparing spectra obtained by different arithmetic rules

In order to improve clarity and overall computation speed, the ASN Filter Designer only displays spectra (i.e. magnitude, phase etc.) based on the current arithmetic rules. This is somewhat different to other tools that display multi-spectra obtained by (for example) Fixed Point and Double Precision arithmetic. For any users wishing to compare spectra you may simply switch between arithmetic settings by changing the Arithmetic method. The designer will then automatically re-compute the filter coefficients using the selected arithmetic rules and the current technical specification. The chart will then be updated using the current zoom settings.

Automatic code generation to the Arm CMSIS-DSP framework

As with floating point arithmetic, select the Arm CMSIS-DSP framework from the selection box in the filter summary window:

The automatically generated C code based on the CMSIS-DSP framework for direct implementation on an Arm based Cortex-M processor is shown below:

As with the floating point filter, the automatic code generator generates all initialisation code, scaling and data structures needed to implement the IIR via the CMSIS-DSP library. This code may be directly used in any Cortex-M based development project – a complete Keil MDK example is available on Arm/Keil’s website. Notice that the tool’s code generator produces code for the Cortex-M4 core as default, please refer to the table below for the #define definition required for all supported cores.

The main test loop code (not shown) centres around the arm_biquad_cascade_df2T_f32() function, which performs the filtering operation on a block of input data.

Complex coefficient IIR filters are currently not supported.

Validating the design with the signal analyser

A design may be validated with the signal analyser, where both time and frequency domain plots are supported. A comprehensive signal generator is fully integrated into the signal analyser allowing designers to test their filters with a variety of input signals, such as sine waves, white noise or even external test data.

For Fixed Point implementations, the tool allows designers to specify the Overflow arithmetic rules as: Saturate or Wrap. Also, the Accumulator Word Length may be set between 16-40 bits allowing designers to quickly find the optimum settings to suit their application.

Extra resources

1. Digital signal processing: principles, algorithms and applications, J.Proakis and D.Manoloakis
2. Digital signal processing: a practical approach, E.Ifeachor and B.Jervis.
3. Digital filters and signal processing, L.Jackson.
4. Step by step video tutorial of designing an IIR and deploying it to Keil MDK uVision.
5. Implementing Biquad IIR filters with the ASN Filter Designer and the Arm CMSIS-DSP software framework (ASN-AN025)
6. Keil MDK uVision example IIR filter project

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All-pass filters

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