Chebyshev I and Chebyshev II filters: what are the advantages and disadvantages? And what is the syntax of Chebyshev, explained with ASN Filterscript

What are Finite Impulse Respsonse (FIR) Filters? And how to design FIR Filters in ASN Filter Designer and which filters does ASN Filter Designer support?

Butterworth filters have a magnitude response that is maximally flat in the passband and monotonic overall. Good choice for eg DC and loadcells

The P-Z (pole-zero) editor, comprehensive and easy to use pole-zero editor. Together with useful options not commonly found in other filter design software.

DSP for engineers: the ASN Filter Designer is the ideal tool to analyze and filter the sensor data quickly. Create an algorithm within hours instead of days. When you are working with sensor data, you probably recognize these challenges:

- My sensor data signals are too weak to even make an analysis. So, strengthening of the signals is needed
- Where I would expect a flat line, the data looks like a mess because of interference and other containments. I need to clean the data first before analysis

Until now, you’ve probably spent days or even weeks working on your signal analysis and filtering? The development trajectory is **generally slow and very painful**.

In fact, just think about the number of hours that you could have saved if you had design tool that managed all of the algorithmic details for you. ASN Filter Designer is an **industry standard solution** used by thousands of professional developers worldwide working on IoT projects.

Our close collaboration with Arm and ST ensures that all designed filters are 100% compatible with all Arm Cortex-M processors, such as ST’s popular STM32 family.

## Challenges for engineers

- 90% of IoT smart sensors are based on Arm Cortex-M processor technology
- Sensor signal processing is difficult
- Sensors have trouble with all kinds of interference and undesirable components
- How do I design a filter that meets my requirements?
- How can I verify my designed filter on test data?
- Clean sensor data is required for better product performance
- Time consuming process to implement a filter on an embedded processor
- Time is money!

Designers hit a ‘brick wall’ with traditional tooling. Standard tooling requires an iterative, **trial and error approach** or expert knowledge. Using this approach, a considerable amount of valuable engineering time is wasted. ASN Filter Designer helps you with an interactive method of design, whereby the tool automatically enters the technical specifications based on the graphical user requirements.

## Fast DSP algorithm development

- Fully validated filter design: suitable for deployment in DSP, micro-controller, FPGA, ASIC or PC application.
- Automatic detailed design documentation: expediting peer review and lowing project risks by helping the designer create a paper trail.
- Simple handover: project file, documentation and test results provide a painless route for handover to colleagues or other teams.
- Easily accommodate other scenarios in the future: Design may be simply modified in the future to accommodate other requirements and scenarios, such as 60Hz powerline interference cancellation, instead of the European 50Hz.

## ASN Filter Designer: the fast and intuitive filter designer

The ASN Filter Designer is the ideal tool to analyze and filter the sensor data quickly. When needed, you can easily deploy your data for further analyze for tools such as Matlab and Python. As such it’s ideal for engineers who need and powerful signal analyser and need to create a data filter for their IoT application. Certainly, when you have to create data filtering once in a while. Compared to other tools, you can create an algorithm within hours instead of days.

**Easily deploy your algorithms to Matlab, Python, C++ and Arm**

A big timesaver of the ASN Filter Designer is that you can easily deploy your algorithms to Matlab, Python, C++ or directly on an Arm microcontroller with the automatic code generators.

## Instant pain relief

Just think about the number of hours that you could have saved if you had design tool that managed all of the algorithmic details for you.

ASN Filter Designer is an industry standard solution used by thousands of professional developers worldwide working on IoT projects. Our close collaboration with Arm and ST ensures that the all filters are 100% compatible with all Arm Cortex-M processors.

**How much pain relief can 125 Euro buy you? **

Because a lot of engineers need our ASN Filter Designer for a short time, a 125 Euro license for just 3 months is possible!

Just ask yourself: is 125 Euro a fair price to pay for instant pain relief and results? We think so. Besides, we have a license for 1 year and even a perpetual license. Download the demo to see for yourself or contact us for more information.

Finite impulse response (FIR) filters are useful for a variety of sensor signal processing applications, including audio and biomedical signal processing. Although several practical implementations for the FIR exist, the **Direct Form Transposed** structure offers the best numerical accuracy for floating point implementation. However, when considering fixed point implementation on a micro-controller, the **Direct Form** structure is considered to be the best choice by virtue of its large accumulator that accommodates any intermediate overflows.

This application note specifically addresses FIR filter design and implementation on a Cortex-M based microcontroller with the ASN Filter Designer for both floating point and fixed point applications via the Arm CMSIS-DSP software framework. Details are also given (including an Arm reference software pack) regarding implementation of the FIR filter in Arm/Keil’s MDK industry standard Cortex-M micro-controller development kit.

## Introduction

ASN Filter Designer provides engineers with a powerful DSP experimentation platform, allowing for the design, experimentation and deployment of complex FIR 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 FIR filters with the ASN Filter Designer

ASN Filter Designer provides engineers with an easy to use, intuitive graphical design development platform for 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:

Fs: | 500Hz |

Passband frequency: | 0-25Hz |

Type: | Lowpass |

Method: | Parks-McClellan |

Stopband attenuation @ 125Hz: | ≥ 80 dB |

Passband ripple: | < 0.01dB |

Order: | Small as possible |

Graphically entering the specifications into the ASN Filter Designer, and fine tuning the design marker positions, the tool automatically designs the filter), automatically choosing the required filter order, and in essence – automatically producing the filter’s exact technical specification!

The frequency response of a filter meeting the specification is shown below:

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

**Parks–McClellan algorithm**

The Parks–McClellan algorithm is an iterative algorithm for finding the optimal Chebyshev FIR filter. The algorithm uses an indirect method for finding the optimal filter coefficients, that offers a degree of flexibility over other FIR design methods, in that each band may be individually customised in order to suit the designer’s requirements.

The primary FIR filter designer UI implements the Parks-McClellan algorithm, allowing for the design of the following filter types:

Filter Types | Description |

Lowpass | Designs a lowpass filter. |

Highpass | Designs a highpass filter. |

Bandpass | Designs a bandpass filter. |

Bandstop | Designs a bandstop filter. |

Multiband | Designs a multiband filter with an arbitrary frequency response. |

Hilbert transformer | Designs an all-pass filter with a -90 degree phase shift. |

Differentiator | Designs a filter with +20dB/decade slope and +90 degree phase shift. |

Double Differentiator | Designs a filter with +40dB/decade slope and a +90 degree phase shift. |

Integrator | Designs a filter with -20dB/decade slope and a -90 degree phase shift. |

Double Integrator | Designs a filter with -40dB/decade slope and a -90 degree phase shift. |

These ten filter types provide designers with a great deal of flexibility for a variety of IoT applications. Design requirements may be simply specified via the use of the design markers. In all cases, the tool will automatically calculate the required filter order to meet the designer’s specification.

The Parks-McClellan algorithm is an optimal Chebyshev FIR design method. However, the algorithm may not converge for some specifications. In such cases, increasing the distance between the design marker bands generally helps.

**Other FIR design methods**

Designers looking to experiment with other types of FIR design methods may use the ASN FilterScript live symbolic math scripting language. The scripting language supports over 65 scientific commands and provides designers with a familiar and powerful programming language, while at the same time allowing them to implement complex symbolic mathematical expressions. The following functions are supported:

Function | Description |

movaver | Moving average FIR filter design. |

firwin | FIR filter design based on the Window method. |

firarb | Designs an FIR Window based filter with an arbitrary magnitude response. |

firkaiser | Designs an FIR filter based on the Kaiser window method. |

firgauss | Designs an FIR Gaussian lowpass filter. |

savgolay | Design an FIR Savitzky-Golay lowpass smoothing filter. |

Please refer to the ASN FilterScript reference guide for more details.

All filters designed in ASN FilterScript are designed using double precision arithmetic in the H2 filter sandbox. An H2 filter must be transformed to an H1 (primary) filter for deployment.

This may be simply achieved via the P-Z options menu:

The** re-optimise** method automatically analyses and converts the H2 filter into an H1 filter.

**Floating point implementation**

When implementing a filter in floating point (i.e. using double or single precision arithmetic) the **Direct Form Transposed** structure is considered the most numerically accurate. This can be readily seen by analysing the difference equations below (used for implementation), as the undesirable effects of numerical swamping are minimised, since floating point addition is performed on numbers of similar magnitude.

\(\displaystyle \begin{eqnarray}y(n) & = &b_0x(n) &+& w_1(n-1) \\ w_1(n)&=&b_1x(n) &+& w_2(n-1) \\ w_2(n)&=&b_2x(n) &+& w_3(n-1) \\ \vdots\quad &=& \quad\vdots &+&\quad\vdots \\ w_q(n)&=&b_qx(n) \end{eqnarray}\)

\(\displaystyle \begin{eqnarray}y(n) & = &b_0x(n) &+& w_1(n-1) \\ w_1(n)&=&b_1x(n) &+& w_2(n-1) \\ w_2(n)&=&b_2x(n) &+& w_3(n-1) \\ \vdots\quad &=& \quad\vdots &+&\quad\vdots \\ w_q(n)&=&b_qx(n) \end{eqnarray}\)

The quantisation and filter structure settings used to implement the FIR can be found under the **Q** tab (as shown below).

Despite the **Direct Form****Transposed **structure being the most efficient for floating point implementation, the Arm CMSIS-DSP library **does not currently** support the **Direct Form** **Transposed **structure for FIR filters. Only the ** Direct Form** structure is supported.

Setting **Arithmetic** to** Single Precision **and** Structure** to **Direct Form **and clicking on the** Apply **button configures the FIR considered herein for the CMSIS-DSP software framework.

The optimised functions within the Arm CMSIS-DSP framework currently support **Single Precision** arithmetic only.

Support for **Double Precision** and the **Direct Form****Transposed **structure will be added in future releases.

## Fixed point implementation

When implementing a filter with fixed point arithmetic, the **Direct Form** structure is considered to be the best choice by virtue of its large accumulator that accommodates any intermediate overflows. The Direct Form structure and associated difference equation are shown below.

\(\displaystyle y(n) = b_0x(n) + b_1x(n-1) + b_2x(n-2) + …. +b_qx(n-q) \)

The CMSIS-DSP Framework supports Q7, Q15 and Q31 coefficient quantisation only. The options may be simply specified via the quantisation tab Q as shown below:

The tool’s inbuilt analytics (shown in the textbox) are intended to help the designer choose the most suitable quantisation settings.

As seen on the left, the tool has recommended a RFWL (recommended fraction length) of 15bits (Q15) for the coefficients, which is as required.

The **Direct form** structure is chosen over the **Direct Form Transposed **as a single (40-bit) accumulator can be used. The tool’s automatic code generator makes use of CMSIS-DSP’s 64-bit accumulators functions, so that the final C code deployed to a Cortex-M device will not overflow.

## Deploying Arm CMSIS-DSP compliant code

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

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

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

This code may be directly used in any Cortex-M based development project.

**Arm Keil’s MDK (uVision)**

As mentioned above, 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 uVision MDK (micro-controller development kit).

The following Arm software pack is available on Keil’s website for using this code directly with Keil uVision MDK.

Although the design of FIR filters with linear phase is an easy task. This is **certainly not true for IIR filters that usually have a highly non-linear phase response**, especially around the filter’s cut-off frequencies. This article discusses the characteristics needed for a digital filter to have linear phase, and how an IIR filter’s passband phase can be modified in order to achieve linear phase using all-pass equalisation filters.

**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 must be based on **Single Precision** arithmetic and either a **Direct Form I** or **Direct Form II Transposed** filter structure. 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.

## 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.

A digital filter is a mathematical algorithm that operates on a digital dataset (e.g. sensor data) in order extract information of interest and remove any unwanted information. Applications of this type of technology, include removing glitches from sensor data or even cleaning up noise on a measured signal for easier data analysis. But how do we choose the best type of digital filter for our application? And what are the differences between a IIR filter and a FIR filter?

Digital filters are divided into the following two categories:

- Infinite impulse response (IIR)
- Finite impulse response (FIR)

As the names suggest, **each type of filter is categorised by the length of its impulse response**. However, before beginning with a detailed mathematical analysis, it is prudent to appreciate the differences in performance and characteristics of each type of filter.

**Example**

In order to illustrate the differences between an IIR and FIR, the frequency response of a 14th order FIR (solid line), and a 4th order Chebyshev Type I IIR (dashed line) is shown below in Figure 1. Notice that although the magnitude spectra have a similar degree of attenuation, the phase spectrum of the IIR filter is non-linear in the passband (\(\small 0\rightarrow7.5Hz\)), and becomes very non-linear at the cut-off frequency, \(\small f_c=7.5Hz\). Also notice that the FIR requires a higher number of coefficients (15 vs the IIR’s 10) to match the attenuation characteristics of the IIR.

These are just some of the differences between the two types of filters. A detailed summary of the main advantages and disadvantages of each type of filter will now follow.

**IIR filters**

IIR (infinite impulse response) filters are generally chosen for applications where linear phase is not too important and memory is limited. They have been widely deployed in audio equalisation, biomedical sensor signal processing, IoT/IIoT smart sensors and high-speed telecommunication/RF applications.

**Advantages**

**Low implementation cost**: requires less coefficients and memory than FIR filters in order to satisfy a similar set of specifications, i.e., cut-off frequency and stopband attenuation.**Low latency**: suitable for real-time control and very high-speed RF applications by virtue of the low number of coefficients.**Analog equivalent**: May be used for mimicking the characteristics of analog filters using s-z plane mapping transforms.

**Disadvantages**

**Non-linear phase characteristics**: The phase charactersitics of an IIR filter are generally nonlinear, especially near the cut-off frequencies. All-pass equalisation filters can be used in order to improve the passband phase characteristics.**More detailed analysis:**Requires more scaling and numeric overflow analysis when implemented in fixed point. The**Direct form II**filter structure is especially sensitive to the effects of quantisation, and requires special care during the design phase.**Numerical stability**: Less numerically stable than their FIR (finite impulse response) counterparts, due to the feedback paths.

**FIR filters**

FIR (finite impulse response) filters are generally chosen for applications where linear phase is important and a decent amount of memory and computational performance are available. They have a widely deployed in audio and biomedical signal enhancement applications. Their all-zero structure (discussed below) ensures that they never become unstable for any type of input signal, which gives them a distinct advantage over the IIR.

**Advantages**

**Linear phase**: FIRs can be easily designed to have linear phase. This means that no phase distortion is introduced into the signal to be filtered, as all frequencies are shifted in time by the same amount – thus maintaining their relative harmonic relationships (i.e. constant group and phase delay). This is certainly not case with IIR filters, that have a non-linear phase characteristic.**Stability**: As FIRs do not use previous output values to compute their present output, i.e. they have no feedback, they can never become unstable for any type of input signal, which is gives them a distinct advantage over IIR filters.**Arbitrary frequency response**: The Parks-McClellan and ASN FilterScript’s firarb() function allow for the design of an FIR with an arbitrary magnitude response. This means that an FIR can be customised more easily than an IIR.**Fixed point performance**: the effects of quantisation are less severe than that of an IIR.

**Disadvantages**

**High computational and memory requirement:**FIRs usually require many more coefficients for achieving a sharp cut-off than their IIR counterparts. The consequence of this is that they require much more memory and significantly a higher amount of MAC (multiple and accumulate) operations. However, modern microcontroller architectures based on the Arm’s Cortex-M cores now include DSP hardware support via SIMD (signal instruction, multiple data) that expedite the filtering operation significantly.**Higher latency**: the higher number of coefficients, means that in general an FIR is less suitable than an IIR for fast high throughput applications. This becomes problematic for real-time closed-loop control applications, where an FIR filter may have too much group delay to achieve loop stability.**No analog equivalent**: using the Bilinear, matched z-transform (s-z mapping), an analog filter can be easily be transformed into an equivalent IIR filter. However, this is not possible for an FIR as it has no analog equivalent.

**Mathematical definitions**

As discussed in the introduction, the name IIR and FIR originate from the mathematical definitions of each type of filter, i.e. an IIR filter is categorised by its theoretically infinite impulse response,

y(n)=\sum_{k=0}^{\infty}h(k)x(n-k)

\)

and an FIR categorised by its finite impulse response,

y(n)=\sum_{k=0}^{N-1}h(k)x(n-k)

\)

We will now analyse the mathematical properties of each type of filter in turn.

**IIR definition**

As seen above, an IIR filter is categorised by its theoretically infinite impulse response,

\(\displaystyle y(n)=\sum_{k=0}^{\infty}h(k)x(n-k) \)

Practically speaking, it is not possible to compute the output of an IIR using this equation. Therefore, the equation may be re-written in terms of a finite number of poles \(\small p\) and zeros \(\small q\), as defined by the linear constant coefficient difference equation given by:

y(n)=\sum_{k=0}^{q}b_k x(n-k)-\sum_{k=1}^{p}a_ky(n-k)

\)

where, \(\small a_k\) and \(\small b_k\) are the filter’s denominator and numerator polynomial coefficients, who’s roots are equal to the filter’s **poles** and **zeros** respectively. Thus, a relationship between the difference equation and the z-transform (transfer function) may therefore be defined by using the z-transform delay property such that,

\sum_{k=0}^{q}b_kx(n-k)-\sum_{k=1}^{p}a_ky(n-k)\quad\stackrel{\displaystyle\mathcal{Z}}{\longleftrightarrow}\quad\frac{\sum\limits_{k=0}^q b_kz^{-k}}{1+\sum\limits_{k=1}^p a_kz^{-k}}

\)

As seen, the **transfer function** is a frequency domain representation of the filter. Notice also that the **poles** act on the **output** **data**, and the **zeros** on the **input** **data**. Since the poles act on the output data, and affect stability, it is essential that their radii **remain inside the unit circle **(i.e. <1) for BIBO (bounded input, bounded output) stability. The radii of the zeros are less critical, as they do not affect filter stability. This is the primary reason why all-zero FIR (finite impulse response) filters are always stable.

**BIBO stability**

A linear time invariant (LTI) system (such as 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, as

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

Where, \(\small h(k)\) is the LTI system’s impulse response. Analyzing this equation, 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.

An interesting situation arises if any poles lie on the unit circle, since the system is said to be **marginally stable**, as it is neither stable or unstable. Although marginally stable systems are not BIBO stable, they have been exploited by digital oscillator designers, since their impulse response provides a simple method of generating sine waves, which have proved to be invaluable in the field of telecommunications.

**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 2. 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 2: Direct Form I (biquad) IIR filter realization and transfer function.*

Analysing Figure 2, it can be seen that the biquad structure is actually comprised of two feedback paths (scaled by \(\small a_1\) and \(\small a_2\)), three feed forward paths (scaled by \(\small b_0, b_1\) and \(\small b_2\)) and a section gain, \(\small K\). 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, \(\small K\) 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 3 – Direct Form II Transposed strucutre, transfer function and difference equations*

The filter summary (shown in Figure 4) 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.

**FIR definition**

Returning the IIR’s linear constant coefficient difference equation, i.e.

y(n)=\sum_{k=0}^{q}b_kx(n-k)-\sum_{k=1}^{p}a_ky(n-k)

\)

Notice that when we set the \(\small a_k\) coefficients (i.e. the feedback) to zero, the definition reduces to our original the FIR filter definition, meaning that the FIR computation is just based on past and present inputs values, namely:

y(n)=\sum_{k=0}^{q}b_kx(n-k)

\)

**Implementation**

Although several practical implementations for FIRs exist, the direct form** **structure and its transposed cousin are perhaps the most commonly used, and as such, all designed filter coefficients are intended for implementation in a Direct form structure.

The Direct form structure and associated difference equation are shown below. The **Direct Form** is advocated for fixed point implementation by virtue of the single accumulator concept.

\(\displaystyle y(n) = b_0x(n) + b_1x(n-1) + b_2x(n-2) + …. +b_qx(n-q) \)

The recommended (default) structure within the ASN Filter Designer is the **Direct Form Transposed** structure, as this offers superior numerical accuracy when using floating point arithmetic. This can be readily seen by analysing the difference equations below (used for implementation), as the undesirable effects of numerical swamping are minimised, since floating point addition is performed on numbers of similar magnitude.

\(\displaystyle \begin{eqnarray}y(n) & = &b_0x(n) &+& w_1(n-1) \\ w_1(n)&=&b_1x(n) &+& w_2(n-1) \\ w_2(n)&=&b_2x(n) &+& w_3(n-1) \\ \vdots\quad &=& \quad\vdots &+&\quad\vdots \\ w_q(n)&=&b_qx(n) \end{eqnarray}\)

## What have we learned?

Digital filters are divided into the following two categories:

- Infinite impulse response (IIR)
- Finite impulse response (FIR)

IIR (infinite impulse response) filters are generally chosen for applications where linear phase is not too important and memory is limited. They have been widely deployed in audio equalisation, biomedical sensor signal processing, IoT/IIoT smart sensors and high-speed telecommunication/RF applications.

FIR (finite impulse response) filters are generally chosen for applications where linear phase is important and a decent amount of memory and computational performance are available. They have a widely deployed in audio and biomedical signal enhancement applications.

ASN Filter Designer provides engineers with everything they need to design, experiment and deploy complex IIR and FIR digital filters for a variety of sensor measurement applications. These advantages coupled with automatic documentation and code generation functionality allow engineers to design and validate an IIR/FIR digital filter within minutes rather than hours.

Advanced Solutions Nederland is happy to announce that the ASNFD filtering Arm MDK5 software pack now avalailable for download! The filtering pack provides MDK users with an easy way of ASN’s IP.

Keil MDK is the most comprehensive software development solution for Arm-based microcontrollers. For MDK, additional software components and support for microcontroller devices is provided by software packs. Download here

## UI e**xperience 2020 pack**

**After downloading the ASN Filter Designer, most people just want to play with the tool, in order to get a feeling of whether it works for them. ** **But how do you get started with the ASN Filter Designer? ** **Based on some great user feedback, ASNFD v4.4 now comes with the UI experience 2020 pack. This pack includes detailed coaching tips, an enhanced user experience and step-by-step instructions to get you started with your design. **

**A quick overview of the ASN Filter Designer v4.4 is given below, and we’re sure that you’ll agree that it’s an awesome tool for DSP IIR/FIR digital filter design! **

The ASN Filter Designer has a fast, intuitive user interface. Interactively design, validate and deploy your digital filter within minutes rather than hours. However, getting started with DSP Filter Design can be difficult, especially when you don’t have deep knowledge of digital signal processing. Most people just want to experiment with a tool to get a feeling whether it works for them (sure, there are tutorials and videos). But where do you start?

**Start experimenting with filter design immediately**

That’s why we have developed the **UI Experience 2020 pack. **Based on user feedback, we’ve created detailed tooltips and animations of key functionality. Within minutes, you’ll get a kick-start into functionalities such as chart zooming, panning and design markers.

## Coaching tips, enhanced user experience, step-by-step instructions

Based on user feedback, the *UI Experience 2020 pack* includes:

- Extensive coaching tips
- Detailed explanations of design methods and types of filters
- Enhanced user experience:
- cursors
- animations
- visual effects

- Short cuts to detailed worked solutions, tutorials and step-by-step instructions

Feedback from the user community has been very positive indeed! By providing detailed tooltips and animations of key functionality, the initial hurdle of designing a filter with your desired specifications has now been significantly simplified.

So start with the ASN Filter Designer right away, and cut your development costs by up to 75%!

### Advanced Solutions Nederland B.V.

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