Designs an FIR notch filter from a lowpass filter by computing the difference between the prototype lowpass filter and its amplitude complementary

### Posts

ASN Filter Designer’s new ANSI C SDK framework, provides developers with a comprehensive automatic C code generator for microcontrollers and embedded platforms. This allows developers to directly deploy their AIoT filtering application from within the tool to any **STM32, Arduino, ESP32, PIC32, Beagle Bone and other Arm, RISC-V, MIPS microcontrollers **for direct use.

**Arm’s CMSIS-DSP library vs. ASN’s C SDK Framework**

Thanks to our close collaboration with Arm’s architecture team, our new ultra-compact, highly optimised ANSI C based framework provides outstanding performance compared to other commercial DSP libraries, including Arm’s optimised CMSIS-DSP library.

As seen, using o1 complier optimisation, our framework is able to surpass Arm’s CMSIS-DSP library’s performance on an M4F and M7F. Although notice that performance of both libraries is worse on the Cortex-M3, as it doesn’t have an FPU. Despite the difference, both libraries perform equally well, but the ASN DSP library has the added advantage of extra functionality and being platform agnostic, making it ideal for variety of **biomedical (ECG, EMG, PPG), audio (sound effects, equalisers) , IoT (temperature, gas, pressure) and I4.0 (flow measurement, vibration analysis, CbM) applications**.

AIoT applications designed on the newer Cortex-M33F and Cortex-M55F cores can also take advantage of extra filtering blocks, double precision arithmetic support, providing a simple way of implementing high performance **AI on the Edge **applications within hours.

## Advantages for developers

- A developer can now develop, test and deploy a complete DSP filtering application within the ASN Filter Designer within a few hours. This is very different from a traditional R&D approach that assigns a team of developers for several days in order to achieve the same level of accuracy required for the application.

- Open source and agnostic code base: In order to allow developers to get the maximum performance for their applications, the ASN-DSP SDK is provided as open source and is written in ANSI C. This means that any embedded processor and any level of compiler optimisation can be used.

- Memory size required for the ASN-DSP SDK is relativity lower than other standard DSP libraries, which makes the ASN-DSP SDK extremely suitable for microcontrollers that have memory constrains.

- Using the ASN Filter Designer’s signal analyser tool, developers now can test the performance, accuracy and assess the frequency response of their designed filter and get optimised C code which they can directly use in their application.

- The SDK also supports some
**extra filtering functions**, such as: a median filter, a moving average filter, all-pass, single section IIR filters, a TKEO biomedical filter, and various non-linear functions, including RMS, Abs, Log and Sqrt. These functions form the filter cascade within the tool, and can be used to build signal processing applications, such as EMG and ECG biomedical applications.

- The ASN-DSP SDK supports both
**single and double precision floating point arithmetic**, providing excellent numerical accuracy and wide dynamic range. The library is unique in the sense that it supports double precision arithmetic, which although is not the most optimal for microcontrollers, allows for the implementation of high-fidelity filtering applications.

The ANSI C SDK framework is further extended by our new **C# .NET framework**, allowing .NET developers to build high performance desktop applications with signal processing capabilities.

## Find out more and try it yourself

Benchmarks on a variety of 32-bit embedded platforms, including a biomedical EMG filtering example, are covered in the following application note.

The both framework SDKs are available in ASNFD v5.0, which may be downloaded here.

Many Audio/acoustics engineers and researchers and audio hobbyists work with DSP (Digital Signal Processing). Some now and then, some on daily basis. Working on DSP for audio and speech, common challenges to create digital filters are: Noise cancellation, due to interference and Audio and Speech enhancement. And for whom DSP is not his daily job: filter design in general. In this blog, you’ll find out how the ASN Filter Designer may help for both experienced audio engineers and engineers where DSP is not their daily job to create digital filters for audio and speech.

## For whom? **For those **with some and experienced DSP knowledge alike

If you are a audio/acoustics engineer or researcher or hobbyist: ASN Filter Designer is tailored for whom DSP is not his daily job and with some knowledge DSP. For those who need filter design and have to create some signal analysis. But also for those whom DSP is his daily job and want to save time and money. Most engineers who are working with DSP on a daily basis, are usually working with extensive but also expensive tools. Or want to do the maths themselves. Also for them ASN Filter Designer is useful:

- Intuitive and easy to use.
- Save days of time spending calculating on your own for the price of 2-3 hours of work.
- Few lesser costs then extensive tooling with features you don’t use anyway.
- Automatic code generation: export for further analysis to Matlab, etc, or to Cortex-M Arm based processors via the Arm CMSIS-DSP software framework.

## How DSP for Speech and audio benefit from ASN Filter Designer:

- Experiment with a variety of equalisation, noise cancellation and sound effect audio filtering algorithms.
- Perform data analysis in the frequency domain and via specialised methods, including Cepstral analysis on the streaming data.
- Import your own
*wav*audio files (mono or stereo up to 48kHz) for streaming, and modify the filter characteristics in real-time while listening to the filtered audio stream.

## Some features for creating digital filters for Audio and Speech:

**The sampling frequency may be specified up to 4 decimal places **

which is useful for designing filters based on fractional sampling frequencies, such as multiples of the 44.1kHz audio standard. Common examples include audio interpolation filters: 44.1kHz × 128 = 5.6448MHz and 44.1kHz × 256 = 11.2896MHz.

**Filter orders of up to 499 **

may be constructed, where this is limited to 200 for streaming audio applications. As with the IIR filters, a FIR’s zeros may be modified by the P-Z editor (Method dropdown list changes to User defined), including the ability of adding poles and converting it into an IIR filter.

**Audio and user data playback streaming**

The signal analyzer allows designers to test their design on audio, real (user) data or synthetic data via the built-in signal generator. Default data playback is implemented as streaming data, providing a simple way of assessing the filter’s dynamic performance, which is especially useful for fixed point implementations. Both frequency domain and time domain charts are fully supported, allowing for design verification via transfer function estimation using the cross and power spectral density functions. As with all other charts, the signal analyzer chart fully supports advanced zooming and panning, as well as comprehensive chart data file export options.

**Load .wav for playback**

The signal generator allows you to load .wav audio files for playback via the Audio File method. Both mono and stereo formats are fully supported for 8.000, 11.025, 16.000, 22.05, 44.1 and 48kHz. sampling rates. There is no restriction as to the length of the .wav file.

**You may add extra signals to input audio stream **

Use the signal generator to add sinewaves and white noise to the data stream.

**Intuitive data analysis with the mouse**

Move the mouse over the chart will automatically produce data markers and data analytics (shown at the bottom right side of the GUI). The signal analyzer is directly coupled to the filter designer GUI. This means that you may modify the filter characteristics, and see the effects in real-time in the signal analyzer. This functionality is very useful when designing audio filters, as the new filter settings can be heard immediately on the streaming audio feed.

## Digital filters commonly used in audio and speech

The ASN Filter designer includes digital filters commonly used in audio such as:

- All-pass filter for improving phase response
- Bell filter/Peaking filter to perform magnitude equalization
- Linear Phase Filters for prevention of phase
- The APF equalizer for linearising the passband phase
- A-curve equalization, for modelling perceived sound
- Comb filter for powerline (50/60Hz) harmonic noise cancellation
- 3-band equalizer
- And many more…

## Read review here:

Top marks from Jacob Beningo

## UI e**xperience 2020 pack**

**Na het downloaden van de ASN Filter Designer willen de meeste mensen gewoon met de tool spelen, om een gevoel te krijgen of het voor hen werkt. Maar hoe ga je aan de slag met de ASN Filter Designer? Op basis van goede gebruikersfeedback wordt ASNFD v4.4 nu geleverd met het UI experience 2020-pakket. Dit pakket bevat gedetailleerde coachingstips, een verbeterde gebruikerservaring en stap-voor-stap instructies om je op weg te helpen met je ontwerp.**

**Een snel overzicht van de ASN Filter Designer v4.4 vindt je hieronder, een geweldige tool voorDSP IIR/FIR digitaal filterdesign!**

De ASN Filter Designer heeft een snelle, intuïtieve gebruikersinterface. Ontwerp interactief, valideer en implementeer jouw uw digitale filter binnen enkele minuten in plaats van uren. Het kan echter moeilijk zijn om een eerste begin te maken met DSP-filterontwerp, vooral wanneer je geen diepgaande kennis hebt van digitale signaalverwerking. De meeste mensen willen gewoon experimenteren met een tool om een gevoel te krijgen of het voor hen werkt (ja, natuurlijk zijn er tutorials en video’s). Maar waar begin je dan?

**Start onmiddellijk met experimenteren met het filterontwerp**

Daarom hebben we het UI Experience 2020-pakket ontwikkeld. Op basis van de feedback van de gebruikers hebben we gedetailleerde tooltips en animaties van de belangrijkste functionaliteiten gemaakt. Binnen enkele minuten krijgt u een kickstart in functionaliteiten zoals chart zoom, panning en design markers.

## Coachingstips, verbeterde gebruikerservaring, stap-voor-stap instructies

Gebaseerd op de feedback van de gebruiker bevat het UI Experience 2020-pakket:

- Uitgebreide coachingtips
- Gedetailleerde uitleg over ontwerpmethoden en soorten filters
- Verbeterde gebruikerservaring:
- cursors
- animaties
- visuele effecten

- Links naar gedetailleerde werkende oplossingen, tutorials en stap-voor-stap instructies

De feedback van de gebruikersgemeenschap is zeer positief! Door het verstrekken van gedetailleerde tooltips en animaties van de belangrijkste functionaliteit, is de eerste stap van het ontwerpen van een filter met jouw gewenste specificaties aanzienlijk vereenvoudigd.

Begin dus meteen met de ASN Filterontwerper en bespaar tot 75% op jouw ontwikkelingskosten!

DSP voor ingenieurs: de ASN Filter Designer is de ideale tool om de sensordata snel te analyseren en te filteren. Maak een algoritme binnen enkele uren in plaats van dagen. Wanneer u met sensorgegevens werkt, herkent u deze uitdagingen waarschijnlijk:

- Mijn sensordatasignalen zijn te zwak om zelfs maar een analyse te maken. Daarom heb ik versterking van de signalen nodig
- Waar ik een vlakke lijn zou verwachten, zien de gegevens eruit als een puinhoop door interferentie en andere vervuiling. Ik moet de gegevens eerst opschonen voordat ik ze analyseer.

Waarschijnlijk heb je tot nu toe dagen of zelfs weken gewerkt aan signaalanalyse en filtering. Het ontwikkelingstraject is over het algemeen langzaam en zeer pijnlijk. Denk maar eens aan het aantal uren dat je had kunnen besparen als je een ontwerptool had gehad die alle algoritmische details voor jou beheerde. ASN Filter Designer is een standaardoplossing voor de industrie die wordt gebruikt door duizenden professionele ontwikkelaars die wereldwijd aan iot-projecten werken.

Onze nauwe samenwerking met Arm en ST zorgt ervoor dat alle ontworpen filters 100% compatibel zijn met alle Arm Cortex-M processoren, zoals de populaire STM32-familie van ST.

## Uitdagingen voor ingenieurs

- 90% van IoT smart sensors zijn gebaseerd op Arm Cortex-M processor technologie
- Sensor signal processing is moeilijk
- Sensoren hebben moeite met interferentie en allerlei ongewenste componenten
- Hoe ontwerp ik een filter dat voldoet aan mijn requirements?
- Hoe kan ik mijn ontworpen filter controleren op testdata?
- Voor betere product performance is schone sensor data nodig
- Tijdrovend proces om een filter op een embedded processor te implementeren
- Tijd is geld!

Ontwerpers verzanden vaak met traditionele tooling. Deze vereist meestal een iteratieve, trial and error aanpak of deskundige kennis. Met deze aanpak gaat kostbare tijd verloren. ASN Filter Designer helpt u met een interactieve ontwerpmethode. Hierbij voert de tool automatisch de technische specificaties in op basis van eisen die de gebruiker grafisch heeft ingevoerd.

## Snelle ontwikkeling van het DSP-algoritme

- Volledig gevalideerd filterontwerp: geschikt voor toepassing in DSP, Arm microcontroller, FPGA, ASIC of PC-toepassing
- Automatische gedetailleerde ontwerpdocumentatie: de Filter Designer helpt je met documenatie, waardoor je de peer review kunt versnellen en projectrisico’s verlaagt
- Eenvoudige overdracht: projectdossier, documentatie en testresultaten bieden een gemakkelijk manier voor overdracht aan collega’s of andere teams
- Gemakkelijk in te passen in nieuwe scenario’s: het ontwerp kan eenvoudig worden aangepast aan andere eisen en scenario’s, zoals 60Hz interferentieonderdrukking op de voedingslijn, in plaats van de Europese 50Hz.

## ASN Filter Designer: de snelle en intuitieve filter designer

De ASN Filter Designer is het ideale hulpmiddel om sensorgegevens snel te analyseren en filteren. Indien nodig kun je jouw gegevens eenvoudig naar tools als Matlab en Python exporteren voor verdere analyse. Daarom is het ideaal voor ingenieurs die een krachtige tool voor signaalanalyse nodig hebben en een datafilter voor hun IOT-toepassing moeten maken. Zeker als je af en toe een datafilter moet maken. Vergeleken met andere tools creeer je een algoritme binnen enkele uren in plaats van dagen.

**Exporteer jouw algoritmes naar Matlab, Python of een Arm microcontroller**

Je kunt veel tijd besparen doordat je met ASN Filter Designer algoritmes eenvoudig kunt implementeren in Matlab, Python of direct op een Arm-microcontroller omdat de Filter Designer automatisch code generateert.

## Onmiddelijke verlichting

Denk eens aan het aantal uren dat je had kunnen besparen als je een ontwerptool had gehad die alle algoritmische details voor je beheerde.

ASN Filter Designer is een standaardoplossing in de sector die wordt gebruikt door duizenden professionele ontwikkelaars die wereldwijd aan ivd-projecten werken. Onze nauwe samenwerking met Arm en ST **zorgt ervoor dat alle filters 100% compatibel zijn met alle Arm Cortex-M processoren**.

**Hoeveel pijnverzachting kun je voor 145 Euro kopen?**

Omdat veel technici onze ASN Filterontwerper voor korte tijd nodig hebben, is een licentie van 145 euro voor slechts 3 maanden mogelijk!

Vraag jezelf maar af: is 145 Euro een eerlijke prijs om te betalen voor onmiddellijke pijnverlichting en resultaat? Wij denken van wel. Bovendien hebben we een licentie voor 1 jaar en zelfs een eeuwigdurende licentie. Download de demo om het zelf te zien of neem contact met ons op voor meer informatie

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?

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.

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 an IIR filter and an 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 a linear phase FIR is less suitable than an IIR for fast high throughput applications. This becomes problematic for real-time closed-loop control applications, where a linear phase FIR filter may have too much group delay to achieve loop stability.**Minimum phase filters**: A solution to ovecome the inherent N/2 latency (group delay) in a linear filter is to use a so-called**minimum phase filter,**whereby any zeros outside of the unit circle are moved to their conjugate reciprocal locations inside the unit circle. The result of the**latency will be reduced from N/2 to something much smaller**(although non-constant), making it suitable for real-time control applications.

For applications where phase is less important, this may sound ideal, but the difficulty arises in the numerical accuracy of the root-finding algorithm when dealing with large polynomials. Therefore, orders of**50 or 60 should be considered a maximum**when using this approach. Although other methods do exist (e.g. the Complex Cepstrum), transforming higher-order linear phase FIRs to their minimum phase cousins remains a challenging task.**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.

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