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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 them?

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.

Figure 1: FIR vs IIR: frequency response of a 14th order FIR (solid line), and a 4th order Chebyshev Type I IIR (dashed line)

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,

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

and an FIR categorised by its finite impulse response,

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

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:

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

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,

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

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.

Figure 4: detailed specification.

FIR definition

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

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

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:

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

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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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 and form a critical building block in algorithmic design.

Advantages

  • Low implementation footprint: 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 coefficient footprint.
  • May be used for mimicking the characteristics of analog filters using s-z plane mapping transforms.

Disadvantages

  • Non-linear phase characteristics.
  • Requires more scaling and numeric overflow analysis when implemented in fixed point.
  • Less numerically stable than their FIR (finite impulse response) counterparts, due to the feedback paths.

Definition

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 \(p\) and zeros \(q\), as defined by the linear constant coefficient difference equation given by:

\(\displaystyle
y(n)=\sum_{k=0}^{q}b(k)x(n-k)-\sum_{k=1}^{p}a(k)y(n-k)
\)

where, \(a(k)\) and \(b(k)\) 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,

\(\displaystyle
\sum_{k=0}^{q}b(k)x(n-k)-\sum_{k=1}^{p}a(k)y(n-k)\quad\stackrel{\displaystyle\mathcal{Z}}{\longleftrightarrow}\quad\frac{\sum\limits_{k=0}^q b(k)z^{-k}}{1+\sum\limits_{k=1}^p a(k)z^{-k}}
\)

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.

A discussion of IIR filter structures for both fixed point and floating point can be found here.

Classical IIR design methods

A discussion of the most commonly used or classical IIR design methods (Butterworth, Chebyshev and Elliptic) will now follow. For anybody looking for more general examples, please visit the ASN blog for the many articles on the subject.

ASN Filter Designer’s graphical designer supports the design of the following four IIR classical design methods:

  • Butterworth
  • Chebyshev Type I
  • Chebyshev Type II
  • Elliptic

The algorithm used for the computation first designs an analog filter (via an analog design prototype) with the desired filter specifications specified by the graphical design markers – i.e. pass/stopband ripple and cut-off frequencies. The resulting analog filter is then transformed via the Bilinear z-transform into its discrete equivalent for realisation.

Biquad implementations are advocated for numerical stability.

The Bessel prototype is not supported, as the Bilinear transform warps the linear phase characteristics. However, a Bessel filter design method is available in ASN FilterScript.

As discussed below, each method has its pros and cons, but in general the Elliptic method should be considered as the first choice as it meets the design specifications with the lowest order of any of the methods. However, this desirable property comes at the expense of ripple in both the passband and stopband, and very non-linear passband phase characteristics. Therefore, the Elliptic filter should only be used in applications where memory is limited and passband phase linearity is less important.

The Butterworth and Chebyshev Type II methods have flat passbands (no ripple), making them a good choice for DC and low frequency measurement applications, such as bridge sensors (e.g. loadcells). However, this desirable property comes at the expense of wider transition bands, resulting in low passband to stopband transition (slow roll-off). The Chebyshev Type I and Elliptic methods roll-off faster but have passband ripple and very non-linear passband phase characteristics.

Comparison of classical design methods

The frequency response charts shown below, show the differences between the various design prototype methods for a 5th order lowpass filter with the same specifications. As seen, the Butterworth response is the slowest to roll-off and the Elliptic the fastest.

Elliptic

Elliptic filters offer steeper roll-off characteristics than Butterworth or Chebyshev filters, but are equiripple in both the passband and the stopband. In general, Elliptic filters meet the design specifications with the lowest order of any of the methods discussed herein.

Filter characteristics

  • Fastest roll-off of all supported prototypes
  • Equiripple in both the passband and stopband
  • Lowest order filter of all supported prototypes
  • Non-linear passband phase characteristics
  • Good choice for real-time control and high-throughput (RF applications) applications

Butterworth

Butterworth filters have a magnitude response that is maximally flat  in the passband and monotonic overall, making them a good choice for DC and low frequency measurement applications, such as loadcells. However, this highly desirable ‘smoothness’ comes at the price of decreased roll-off steepness. As a consequence, the Butterworth method has the slowest roll-off characteristics of all the methods discussed herein.


Filter characteristics

  • Smooth monotonic response (no ripple)
  • Slowest roll-off for equivalent order
  • Highest order of all supported prototypes
  • More linear passband phase response than all other methods
  • Good choice for DC measurement and audio applications

Chebyshev Type I

Chebyshev Type I filters are equiripple in the passband and monotonic in the stopband. As such, Type I filters roll off faster than Chebyshev Type II and Butterworth filters, but at the expense of greater passband ripple.

Filter characteristics

  • Passband ripple
  • Maximally flat stopband
  • Faster roll-off than Butterworth and Chebyshev Type II
  • Good compromise between Elliptic and Butterworth

Chebyshev Type II

Chebyshev Type II filters are monotonic in the passband and equiripple in the stopband making them a good choice for bridge sensor applications. Although filters designed using the Type II method are slower to roll-off than those designed with the Chebyshev Type I method, the roll-off is faster than those designed with the Butterworth method.

Filter characteristics

  • Maximally flat passband
  • Faster roll-off than Butterworth
  • Slower roll-off than Chebyshev Type I
  • Good choice for DC measurement applications

 

 

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All-pass filters

All-pass filters provide a simple way of altering/improving the phase response of an IIR without affecting its magnitude response. As such, they are commonly referred to as phase equalisers and have found particular use in digital audio applications.

In its simplest form, an All-pass  filter can be constructed from a first order transfer function, i.e.,

\( A(z)=\Large{\frac{r+z^{-1}}{1+r z^{-1}}}  \, \,  \normalsize{; r<1} \)

Analysing \(\small A(z)\), notice that the pole and zero lie on the real z-plane axis and that the pole at radius \(\small r\) has a zero at radius \(\small 1/r\), such that the poles and zeros are reciprocals of another. This property is key to the all-pass filter concept, as we will now see by expanding the concept further to a second order all-pass filter:

\( A(z)=\Large\frac{r^2-2rcos \left( \frac{2\pi f_c}{fs}\right) z^{-1}+z^{-2}}{1-2rcos \left( \frac{2\pi f_c}{fs}\right)z^{-1}+r^2 z^{-2}} \)

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.

Frequency response of all-pass filter:
Notice the constant magnitude spectrum (shown in blue).

Implementation

An All-pass filter may be implemented in ASN FilterScript as follows:

ClearH1;  // clear primary filter from cascade

interface radius = {0,2,0.01,0.5};   // radius value
interface fc = {0,fs/2,1,fs/10};     // frequency value

Main()
Num = {radius^2,-2*radius*cos(Twopi*fc/fs),1};
Den = reverse(Num); // mirror image of Num
Gain = 1;

For a detailed disucssion on IIR filter phase equalisation, and the ASN Filter designer’s APF (all-pass filter) design tool, please refer to the following article.

 

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Comb filters have found use as powerline (50/60Hz) harmonic cancellation filters in audio applications, and form the basis of so called CIC (cascaded integrator–comb) filters used for anti-aliasing in decimation (sample rate reduction), and anti-imaging in interpolation (sample rate increase) applications.

The frequency response of a comb filter consists of a series of regularly-spaced troughs, giving the appearance of a comb. As seen in the plot below, the spacing of each trough appears at either odd or even harmonics of the desired fundamental frequency.

Frequency response of a typical FIR comb filter (odd harmonics cancellation):
\(f_s=500Hz\),  \(f_c=25Hz\), \(L=10\) and \(\alpha=1\)

An FIR comb filter can be described by the following transfer function:

\(H(z)=1+\alpha z^{-L}\)
\(\Rightarrow Y(z)=X(z)\left[1+\alpha z^{-L}\right]\)

Clearly, the comb filter is simply a weighted delayed replica of itself, specifiied by \(L\). Taking inverse z-transforms, we obtain the difference equation needed for implementation,

\(y(n)=x(n)+\alpha x(n-L)\)

where, \(\alpha\) is used to set the Q (bandwidth) of the notch and may be either positive or negative depending on what type of frequency response is required. In order to elaborate on this, negative values of \(\alpha\) have their first trough at DC and their second trough at the fundamental frequency. Clearly this type of comb filter can be used to remove any DC components from a measured waveform if so required. All subsequent troughs appear at even harmonics up to and including the Nyquist frequency.

Positive values of \(\alpha\) on the other hand, only have troughs at the fundamental and odd harmonic frequencies, and as such cannot be used to remove any DC components.

Application to powerline interference cancellation

The affectivity of the comb filter is dependent on the sampling frequency, \(f_s\), as \(L\) is limited to integer values only. Also, a relationship between \(f_s\), as \(L\) and will be dependent on the sign of \(\alpha\). Thus, for the purposes of the mains cancellation application considered in this discussion, only positive values of will be considered, as we need only cancel odd harmonics.

A simple relationship for determining  \(L\) can be summarized for positive values of \(\alpha\) as follows:

\(L=ceil\left( \large{\frac{f_s}{2f_c}}\right)\)

where, \(f_c\) is the desired centre point of the fundamental notch frequency. Based on this expression, we can re-calculate the sampling frequency, such that \(f_c\) is a true multiple of \(f_s\)

\(f_{snew}=2f_c L\)

Example

For the example considered herein, i.e. \(f_s=500Hz\) and \(f_c=25Hz\), we obtain \(L=10\). However, if \(f_c=60Hz\), we would need \(L=5\), and a new sampling rate of \(600Hz\) respectively, although it’s interesting to note that \(f_s=480Hz\) for \(L=4\) would also suffice.

Implementation

An FIR comb filter may be implemented in ASN FilterScript as follows:

ClearH1;  // clear primary filter from cascade
interface L = {4,20,1,5}; // delay
interface alpha = {-1,1,0.01,1};

Main()
Num = {1,zeros(L-1),alpha}; // numerator coefficients
Den = {1};  
Gain = 1/sum(abs(Num));

 

 

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