How To Pick Size Of Moving Average Filter Window

Moving Average Filters

Steven W. Smith , in Digital Point Processing: A Practical Guide for Engineers and Scientists, 2003

Relatives of the Moving Average Filter

In a perfect world, filter designers would merely have to deal with time domain or frequency domain encoded data, simply never a mixture of the two in the aforementioned signal. Unfortunately, there are some applications where both domains are simultaneously important. For example, television signals fall into this nasty category. Video data is encoded in the fourth dimension domain, that is, the shape of the waveform corresponds to the patterns of brightness in the image. However, during transmission the video point is treated according to its frequency composition, such as its total bandwidth, how the carrier waves for sound and color are added, elimination & restoration of the DC component, etc. As some other case, electromagnetic interference is best understood in the frequency domain, even if

the indicate's information is encoded in the time domain. For instance, the temperature monitor in a scientific experiment might be contaminated with lx hertz from the power lines, 30 kHz from a switching power supply, or 1320 kHz from a local AM radio station. Relatives of the moving boilerplate filter have better frequency domain performance, and can exist useful in these mixed domain applications.

Multiple-pass moving average filters involve passing the input signal through a moving average filter 2 or more times. Figure 15-3a shows the overall filter kernel resulting from ane, two and four passes. Two passes are equivalent to using a triangular filter kernel (a rectangular filter kernel convolved with itself). After iv or more passes, the equivalent filter kernel looks like a Gaussian (recall the Key Limit Theorem). Every bit shown in (b), multiple passes produce an "s" shaped step response, as compared to the straight line of the single laissez passer. The frequency responses in (c) and (d) are given by Eq. 15-2 multiplied by itself for each laissez passer. That is, each time domain convolution results in a multiplication of the frequency spectra.

Figure xv-three. Characteristics of multiple-laissez passer moving average filters. Figure (a) shows the filter kernels resulting from passing a seven indicate moving average filter over the data once, twice and four times. Figure (b) shows the respective step responses, while (c) and (d) show the corresponding frequency responses.

Effigy xv-iv shows the frequency response of two other relatives of the moving average filter. When a pure Gaussian is used equally a filter kernel, the frequency response is too a Gaussian, every bit discussed in Chapter xi. The Gaussian is important because information technology is the impulse response of many natural and manmade systems. For example, a cursory pulse of light entering a long fiber optic transmission line volition exit as a Gaussian pulse, due to the different paths taken past the photons within the fiber. The Gaussian filter kernel is also used extensively in image processing considering it has unique properties that allow fast 2-dimensional convolutions (run into Chapter 24). The second frequency response in Fig. xv-iv corresponds to using a Blackman window as a filter kernel. (The term window has no pregnant here; it is just part of the accepted name of this curve). The exact shape of the Blackman window is given in Chapter 16 (Eq. sixteen-2, Fig. 16-two); still, information technology looks much like a Gaussian.

FIGURE 15-4. Frequency response of the Blackman window and Gaussian filter kernels. Both these filters provide better stopband attenuation than the moving average filter. This has no advantage in removing random dissonance from time domain encoded signals, but it can be useful in mixed domain problems. The disadvantage of these filters is that they must utilise convolution, a terribly slow algorithm.

How are these relatives of the moving boilerplate filter ameliorate than the moving average filter itself? Three ways: Showtime, and most important, these filters have better stopband attenuation than the moving average filter. Second, the filter kernels taper to a smaller amplitude near the ends. Remember that each point in the output signal is a weighted sum of a group of samples from the input. If the filter kernel tapers, samples in the input signal that are farther away are given less weight than those close by. Third, the step responses are polish curves, rather than the sharp directly line of the moving average. These last two are usually of express do good, although y'all might find applications where they are 18-carat advantages.

The moving average filter and its relatives are all about the aforementioned at reducing random noise while maintaining a sharp footstep response. The ambivalence lies in how the risetime of the step response is measured. If the risetime is measured from 0% to 100% of the step, the moving boilerplate filter is the best y'all can do, equally previously shown. In comparison, measuring the risetime from 10% to xc% makes the Blackman window better than the moving average filter. The point is, this is only theoretical squabbling; consider these filters equal in this parameter.

The biggest divergence in these filters is execution speed. Using a recursive algorithm (described next), the moving average filter will run like lightning in your reckoner. In fact, it is the fastest digital filter available. Multiple passes of the moving average will be correspondingly slower, but still very quick. In comparison, the Gaussian and Blackman filters are excruciatingly slow, because they must use convolution. Recall a cistron of x times the number of points in the filter kernel (based on multiplication being about 10 times slower than addition). For instance, expect a 100 indicate Gaussian to be m times slower than a moving boilerplate using recursion.

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Basic Linear Filtering with Application to Image Enhancement

Alan C. Bovik , Scott T. Acton , in The Essential Guide to Epitome Processing, 2009

10.three.one Moving Boilerplate Filter

The moving average filter can be described in several equivalent means. Beginning, using the notion of windowing introduced in Affiliate four, the moving average can exist defined as an algebraic operation performed on local image neighborhoods according to a geometric rule defined by the window. Given an epitome f to exist filtered and a window B that collects greyness level pixels co-ordinate to a geometric rule (defined past the window shape), then the moving boilerplate-filtered paradigm yard is given by

where the operation AVE computes the sample average of its. Thus, the local boilerplate is computed over each local neighborhood of the image, producing a powerful smoothing effect. The windows are usually selected to be symmetric, as with those used for binary morphological image filtering (Chapter iv).

Since the average is a linear operation, it is also true that

ten.23 $\mathrm{thousand}\left(\mathrm{due\; north}\right)=\text{AVE}\left[Bo\left(n\right)\right]+\text{AVE}\left[Bq\left(n\right)\right].$

Because the noise procedure q is assumed to exist nothing-mean in the sense of (10.20), then the last term in (10.23) volition tend to zero equally the filter window is increased. Thus, the moving average filter has the desirable result of reducing zero-hateful image racket toward zero. However, the filter also effects the original paradigm data. It is desirable that $\text{AVE}\left[Bo\left(\mathrm{northward}\right)\right]\approx o\left(n\right)$ at each northward, but this will not be the instance everywhere in the prototype if the filter window is too large. The moving average filter, which is lowpass, will blur the image, especially as the window span is increased. Balancing this tradeoff is often a hard task.

The moving average filter operation (x.22) is actually a linear convolution. In fact, the impulse response of the filter is defined equally having value 1/R over the span covered by the window when centered at the spatial origin (0, 0), and zero elsewhere, where R is the number of elements in the window.

For case, if the window is $\text{SQUARE}\left[{\left(\mathrm{ii}P+1\right)}^2\right]$ , which is the most common configuration (it is defined in Chapter 4), and then the boilerplate filter impulse response is given by

10.24 $h\left(\mathrm{one\; thousand},\mathrm{north}\right)=\{\begin{array}{ll}\mathrm{ane}/{\left(2P+1\right)}^2;\hfill & -P\le \mathrm{1000},\mathrm{northward}\le P\hfill \\ 0;\hfill & \text{else}\hfill \end{array}.$

The frequency response of the moving boilerplate filter (ten.24) is:

10.25 $H(U,V)=\frac{\sin \left[\left(2P+\mathrm{ane}\right)\pi U\right]}{\left(\mathrm{ii}P+1\right)\sin \left(\pi U\right)}·\frac{\sin \left[\left(\mathrm{two}P+1\right)\pi V\right]}{\left(2P+1\right)\sin \left(\pi \mathrm{Five}\right)}.$

The one-half-peak bandwidth is often used for prototype processing filters. The half-acme (or 3 dB) cutoff frequencies occur on the locus of points (U, V) where $\left|H\left(U,V\right)\right|$ falls to one/2. For the filter (10.25), this locus intersects the U-axis and V-centrality at the cutoffs $U_{\text{half-peak}},V_{\text{half-summit}}\approx 0.6/(2P+1)$ cycles/pixel.

As depicted in Fig. 10.2, the magnitude response $\left|H\left(U,5\right)\right|$ of the filter (10.25) exhibits considerable sidelobes. In fact, the number of sidelobes in the range $\left[0,\pi \right]$ is P. Every bit P is increased, the filter bandwidth naturally decreases (more high-frequency attenuation or smoothing), but the overall sidelobe energy does not. The sidelobes are in fact a pregnant drawback, since at that place is considerable noise leakage at high noise frequencies. These residual dissonance frequencies remain to degrade the image. Nevertheless, the moving average filter has been unremarkably used because of its full general effectiveness in the sense of (10.21) and because of its simplicity (ease of programming).

Figure 10.2. Plots of $|H(U,5)|$ given in (10.25) along $5=0$ , for $P=\mathrm{ane},2,3,4$ . Equally the filter span is increased, the bandwidth decreases. The number of sidelobes in the range [0, π] is P.

The moving average filter can be implemented either as a direct second convolution in the space domain, or using DFTs to compute the linear convolution (meet Chapter 5).

Since application of the moving average filter balances a tradeoff between racket smoothing and image smoothing, the filter span is ordinarily taken to be an intermediate value. For images of the most common sizes, east.g., $256\times 256$ or $512\times 512$ , typical (Square) average filter sizes range from $3\times \mathrm{iii}$ to $15\times 15$ . The upper stop provides significant (and probably excessive) smoothing, since 225 epitome samples are being averaged to produce each new paradigm value. Of course, if an epitome suffers from severe noise, then a larger window might be used. A large window might also be acceptable if information technology is known that the original image is very smooth everywhere.

Effigy 10.iii depicts the awarding of the moving boilerplate filter to an image that has had zip-mean white Gaussian noise added to it. In the electric current context, the distribution (Gaussian) of the noise is not relevant, although the meaning tin can exist found in Chapter 7. The original image is included for comparing. The paradigm was filtered with SQUARE-shaped moving boilerplate filters of window sizes $5\times \mathrm{v}$ and $9\times 9$ , producing images with significantly dissimilar appearances from each other as well as the noisy epitome. With the $\mathrm{v}\times 5$ filter, the noise is inadequately smoothed, however the image has been blurred noticeably. The issue of the $\mathrm{ix}\times 9$ moving boilerplate filter is much smoother, although the noise influence is nevertheless visible, with some higher racket frequency components managing to leak through the filter, resulting in a mottled appearance.

Effigy 10.3. Instance of application of moving average filter. (a) Original image "eggs"; (b) image with additive Gaussian white noise; moving average-filtered image using; (c) Foursquare(25) window $5\times 5$ ; and (d) SQUARE(81) window $\mathrm{nine}\times 9$ .

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Stage-locked loops and their pattern

Wenzhao Liu , Frede Blaabjerg , in Command of Power Electronic Converters and Systems, 2021

10.3.ii Moving average filter–based PLLs

Fig. 10.7 shows a conventional SRF-PLL with moving average filter (MAF), which is referred as MAF-PLL [ xv]. The MAF is a linear-phase filter which tin can be described as

Figure 10.7. Schematic diagram of the moving average filter–based phase-locked loop.

(10.11) $G_{MAF}\left(s\right)=\frac{1-e^{-T_{\omega \mathrm{due\; south}}}}{T_{\omega \mathrm{southward}}}$

Discover that including the MAF inside the SRF-PLL command loop can significantly improve its filtering capability, but slow downwards its dynamic response considerably [xv]. The reason is that the MAF in-loop will cause a stage delay. Information technology is on the condition that the window length of MAF is equal to the nominal period of the input signals. The selection for the window length is recommended to exist equal to the primal period of the grid voltage (T _ω   = T), when the filigree harmonic is unclear and DC beginning may exist presented in the PLL input [fifteen,16]. T _ω is the window length of MAF, more choices for the window length of the MAF such as T _ω   = T/2 and T _ω   = T/half-dozen are suitable for applications when there are possible odd-order harmonics in the input of PLLs [17]. The MAF will pass the DC component and completely block frequency components of integer multiples of 1/T _ω .

Furthermore, in order to ameliorate the dynamic of the MAF-PLL while maintaining better harmonics-filter functioning, several methods are proposed in the literature. In Ref. [18], a proportional integral derivative (PID) controller is used instead of the conventional PI controller as the LF of the MAF-PLL, which can provide an additional degree of freedom. Therefore, it enables the designer to effectively recoup for the phase filibuster caused by the MAF past arranging a pole-nil counterfoil in the design [ix].

In addition, a special lead compensator is added before the PI controller in the MAF-PLL [19], where the transfer function of the compensator is inverse of the MAF's transfer function; therefore, it will be able to reduce the phase delay in the MAF-PLL control loop. It should exist emphasized here that, the MAF-PLL with a window length equal to the input fundamental period can remove all of the harmonics upwardly to the aliasing frequency in addition to the fundamental-frequency disturbance components.

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Assay of continuous and detached time signals

Alvar M. Kabe , Brian H. Sako , in Structural Dynamics Fundamentals and Avant-garde Applications, 2020

five.four.three.i Linear divergence equations

We brainstorm our discussion with the post-obit simple example that develops a recursive implementation of the FIR moving boilerplate filter.

Instance v.iv-3 Consider the moving average filter,

(5.4-89) $y\left[\mathrm{north}\right]=\frac{\mathrm{ane}}{\mathrm{Chiliad}}\sum _{m=0}^{M-\mathrm{i}}\mathrm{ten}[\mathrm{northward}-\mathrm{chiliad}]$

From the previous discussion, we annotation that Eq. (5.4-89) is a FIR filter with impulse response given by

(v.iv-ninety) $h_G\left[n\right]=\{\begin{array}{cc}\frac{1}{M}& 0\le n<M\\ 0& \text{otherwise}\end{array}$

Furthermore, $h_M\left[\mathrm{due\; north}\right]$ defines a causal LTI system. Consider the post-obit difference equation,

(five.iv-91) $y\left[n\right]=y[n-1]+\frac{1}{\mathrm{Thou}}\left(x\left[n\right]-\mathrm{10}[\mathrm{north}-\mathrm{Thou}]\right)$

with the boosted requirement that causality holds. In other words, if $x\left[\mathrm{due\; north}\right]=0$ for $n<0$ , then $y\left[\mathrm{northward}\right]=0$ for $\mathrm{northward}<0$ . For a causal input, $\mathrm{10}\left[\mathrm{north}\right]$ , it is piece of cake to show (see Trouble 5.12),

(v.4-92) $y\left[n\right]=\frac{1}{M}\sum _{\mathrm{grand}=0}^{\mathrm{northward}}x\left[k\right]-\frac{1}{M}\sum _{m=0}^{n-M}x\left[g\right]=\frac{\mathrm{ane}}{\mathrm{Chiliad}}\sum _{m=n-M+1}^{n}x\left[m\right]=\frac{\mathrm{ane}}{\mathrm{One\; thousand}}\sum _{m=0}^{K-\mathrm{i}}x[n-m]$

Hence, (v.4-91) is equivalent to the FIR filter in (5.four-89) and, therefore, defines a causal LTI organization with a finite impulse response, despite having a recursive definition. Also, note that the filter defined recursively by (v.four-91) has a linear phase response since the moving average is a linear stage FIR filter. This example illustrates that not all recursive filters accept space impulse responses and nonlinear phase responses.

It is worthwhile to examination (5.4-91) if we remove the requirement of causality. To evaluate $y\left[0\right]$ , we demand to know $y[-\mathrm{i}]$ . Suppose that we have defined the initial condition, $y[-1]=c\ne 0$ . For the causal input, $\mathrm{10}\left[n\right]=\alpha \delta \left[\mathrm{northward}\right]$ , we have for $\mathrm{due\; north}\ge 0$ ,

(v.4-93) $y\left[\mathrm{due\; north}\right]=\{\begin{array}{cc}c+\frac{\alpha }{\mathrm{One\; thousand}}& 0\le n<M\\ c& n\ge M\end{array}$

For $n<0$ , we can solve for $y[n-\mathrm{one}]$ in (5.4-91) and obtain

(five.4-94) $y[\mathrm{northward}-\mathrm{i}]=y\left[n\right]+\frac{\mathrm{ane}}{\mathrm{Thou}}\left(x[n-K]-x\left[\mathrm{north}\right]\right)$

which results in

(5.four-95) $y\left[n\right]=c\mathrm{northward}<0$

If $\alpha =0$ , then (5.four-93) and (five.iv-95) imply that $y\left[n\right]=c\ne 0$ . This shows that the difference equation is not linear, since linear systems produce zero output for zero input.

Let u.s.a. further investigate the reason for the nonlinearity by examining the general difference Eq. (5.4-88), to a causal input signal, $\mathrm{ten}\left[\mathrm{north}\right]$ . Offset notation that a zero input produces the homogeneous solution, $y_H\left[n\right]$ , that satisfies

(five.4-96) $y_H\left[n\right]+a_{\mathrm{i}}{y}_H[n-\mathrm{one}]+\cdots +a_N{y}_H[\mathrm{north}-N]=0$

Consider a solution of the form, $y_H\left[n\right]={\rho }^{-n}$ , then substitution into (5.four-96) shows that $\rho$ is a root of the characteristic equation,

(5.4-97) $1+a_{\mathrm{ane}}\rho +a_{\mathrm{two}}{\rho }^2+\cdots +a_{\mathrm{Northward}}{\rho }^N=0$

Suppose that at that place are $N$ distinct roots, ${\rho }_1,{\rho }_2,\cdots ,{\rho }_{\mathrm{North}}$ of (five.iv-97). Then the general homogeneous solution to Eq. (5.4-96) is given past

(five.4-98) $y_H\left[n\right]=c_{\mathrm{ane}}{\rho }_1^{-\mathrm{northward}}+c_{\mathrm{ii}}{\rho }_2^{-n}+\cdots +c_{\mathrm{Due\; north}}{\rho }_N^{-n}$

where $c_{\mathrm{thou}}$ are constants to be determined. If the characteristic polynomial does not have distinct roots, then boosted independent solutions must exist adamant from roots with multiplicities greater than ane. If a characteristic root, $\rho$ , has multiplicity $r>\mathrm{i}$ , and so $r-1$ boosted linearly independent solutions may be obtained past differentiating ${\rho }^{-\mathrm{north}}$ with respect to $\rho$ , which produces the solutions (Henrici, 1964),

(5.4-99) $n{\rho }^{-(n-1)},n(n-\mathrm{ane}){\rho }^{-(n-2)}\text{,}\cdots ,n(\mathrm{northward}-1)\cdots (n-k+2){\rho }^{-(n-\mathrm{1000}+\mathrm{ane})}$

A particular solution, $y_P\left[\mathrm{north}\right]$ , to Eq. (5.4-88) requires that nosotros specify the initial values for $\mathrm{due\; north}=-1,-2,\cdots ,-\mathrm{Northward}$ . For a causal input, $x\left[\mathrm{northward}\right]$ , permit us too consider a detail solution that is initially at remainder then that $y_P\left[n\right]=0,n<0$ . Then $y_P\left[\mathrm{northward}\right]$ tin be determined recursively past Eq. (5.4-88). Observe that for a nix input, $x\left[n\right]\equiv 0$ , we obtain the zero output, $y_P\left[n\right]\equiv 0$ , i.eastward., imposing causality avoids the nonlinearity issue we noted earlier. To bear witness that is sufficient and necessary, nosotros need to consider the full general solution of Eq. (five.4-88), which can exist expressed as

(5.4-100) $y\left[\mathrm{north}\right]=y_H\left[\mathrm{northward}\right]+y_P\left[n\right]$

Since $y_P\left[n\right]$ is causal, $y\left[n\right]$ satisfies the initial weather condition,

(5.iv-101) $y\left[n\right]=y_H\left[\mathrm{north}\right],\mathrm{northward}=-1,\cdots ,-\mathrm{Northward}$

Therefore, if we desire $y\left[\mathrm{north}\right]$ to satisfy a specific ready of initial conditions, $y\left[n\right]=y_n,n=-\mathrm{one},\cdots ,-\mathrm{Northward}$ , Eqs. (5.4-101) and (v.4-98) lead to the following $N\times N$ arrangement of linear equations (for singled-out roots),

(five.iv-102) $\underset{\mathbf{R}}{\underbrace{\left[\begin{array}{llll}{\rho }_{\mathrm{i}}^{-1}\hfill & {\rho }_2^{-\mathrm{ane}}\hfill & \cdots \hfill & {\rho }_N^{-1}\hfill \\ {\rho }_1^{-\mathrm{ii}}\hfill & {\rho }_{\mathrm{two}}^{-2}\hfill & \cdots \hfill & {\rho }_{\mathrm{North}}^{-2}\hfill \\ \hfill ⋮\hfill & ⋮\hfill & \ddots \hfill & ⋮\hfill \\ {\rho }_1^{-\mathrm{Northward}}\hfill & {\rho }_{\mathrm{two}}^{-\mathrm{Due\; north}}\hfill & \cdots \hfill & {\rho }_N^{-\mathrm{Northward}}\hfill \end{array}\right]}}\underset{\mathbf{c}}{\underbrace{\left\{\begin{array}{l}{c}_1\hfill \\ c_2\hfill \\ ⋮\hfill \\ c_{\mathrm{North}}\hfill \end{array}\right\}}}=\underset{\mathbf{b}}{\underbrace{\left\{\begin{array}{l}{y}_{-1}\hfill \\ y_{-2}\hfill \\ ⋮\hfill \\ y_{-\mathrm{North}}\hfill \end{array}\right\}}}$

If the characteristic roots are distinct, $\mathbf{R}$ is nonsingular and the coefficients of the homogeneous solution are given past $\mathbf{c}={\mathbf{R}}^{-1}\mathbf{b}$ . This as well holds in the example of roots having multiplicities past using the additional solutions in (5.4-99). Therefore, if any of the initial conditions are nonzero, and then $\mathbf{c}$ is besides nonzero and, therefore, $y_H\left[\mathrm{due\; north}\right]$ is nontrivial. Clearly then, $y_H\left[\mathrm{due\; north}\right]\equiv 0$ if and only if $y_n=0,n=-1,\cdots ,-N$ . Furthermore, since $y_P\left[n\right]$ is causal, we conclude that specifying zero initial conditions is necessary and sufficient for producing a causal solution with a causal input. In this case, observe that for a zero input, $y\left[n\right]\equiv 0$ . On the other manus, if nosotros impose nonzero initial conditions, then for a zero input, its output, $y\left[\mathrm{northward}\right]$ , is nonzero. This is the nonlinearity that we noted earlier. To summarize, in gild for the deviation Eq. (5.4-88) to stand for a linear system, nosotros must impose zero initial weather condition, which is equivalent to requiring that the arrangement be causal.

Since the input and output detached-fourth dimension signals are causal, their z-transforms are given past

(five.4-103) $\widetilde{\mathrm{10}}\left[z\right]=\sum _{n=0}^{\infty }x\left[\mathrm{due\; north}\right]z^{-n}\text{and}\tilde{Y}\left[z\right]=\sum _{n=0}^{\infty }y\left[\mathrm{north}\right]z^{-n}$

where the ROC for each is given by the outside of a circle in the z-plane. The departure Eq. (5.four-88) tin at present be represented in terms of the z-transforms,

(5.4-104) $\tilde{Y}\left[z\right]=\frac{b_0+b_1{z}^{-1}+\cdots +b_M{z}^{-\mathrm{One\; thousand}}}{\mathrm{ane}+a_1{z}^{-1}+\cdots +a_{\mathrm{North}}{z}^{-\mathrm{Northward}}}\tilde{X}\left[z\right]=\tilde{H}\left[z\right]\widetilde{\mathrm{Ten}}\left[z\right]$

where $\tilde{H}\left[z\right]$ is the filter's transfer part. Since $\tilde{H}\left[z\right]$ represents a causal organization, it has a ROC that is equal to $\left\{\right|z|$ > R} for some radius, $R$ . If we farther require that the filter be stable, and so from our earlier remarks in Department 5.4.1, $R<1=0$ . This implies that the poles of $\tilde{H}\left[z\right]$ must lie within the interior of the unit of measurement circle.

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Digital Filters

James D. Broesch , in Digital Signal Processing, 2009

Instant Summary

In this chapter nosotros have worked with both Finite Impulse Response (FIR) filters and Infinite Impulse Response (IIR) filters.

The FIR filters are essentially sophisticated versions of the simple moving average filter. An FIR is designed by specifying the transfer part H(ω). The office H(ω) is then converted to a sequence using the IDFT. This sequence, h(n), then becomes the coefficients of the filter. The FIR is then realized by convolving the input with h(northward).

The FIR filter has a number of meaning advantages. It is unconditionally stable, easily designed, and easily implemented. Information technology is possible to design an FIR filter with a linear phase delay. The one major disadvantage of the FIR is that information technology can require a large number of computations to implement.

IIR filters are more circuitous and much more difficult to understand intuitively than FIR filters. We worked through a pattern of an IIR filter using the approach of placing poles and zeroes accordingly around the z-plane. From the pole/naught graph nosotros and then generated the z-transform in factored form and evaluated the partial fraction into a standard polynomial form. From in that location, we put the z-transform in the standard course of the definition and could find the coefficients of the IIR by simple inspection. The high potential operation of the IIR was noted, only we too pointed out the risks of using the IIR.

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Linear System Analysis

John Semmlow , in Signals and Systems for Bioengineers (Second Edition), 2012

8.3 Two-Dimensional Filtering—Images

FIR filters tin can be extended to two dimensions and applied to images. A wide range of useful paradigm filters tin exist designed using MATLAB's Image Processing Toolbox, but again a gustatory modality of paradigm filtering can be had using standard MATLAB.

Two-dimensional filters accept two-dimensional impulse responses. Ordinarily these filters consist of square matrices with an odd number of rows and columns. Implementation of a ii-dimensional filter tin can be accomplished using two-dimensional convolution. The equation for two-dimensional convolution is a straightforward extension of the one-dimensional discrete convolution equation (Equation vii.3):

(8.18) $y(m\text{,}n)={\displaystyle \sum }_{{\mathrm{chiliad}}_1=0}^{K_{\mathrm{two}}-1}{\displaystyle \sum }_{{\mathrm{grand}}_{\mathrm{two}}=0}^{K_1-\mathrm{one}}x({\mathrm{chiliad}}_{\mathrm{ane}}\text{,}{\mathrm{1000}}_2)b(\mathrm{thou}-k_1\text{,}n-k_2)$

where K _ane and Thou ₂ are the dimensions in pixels of the prototype, b(k ₁,k ₂) are the filter coefficients, x(g _one,thou ₂) is the original image, and y(k,north) is the filtered image. In two-dimensional convolution, the impulse part, now a matrix, moves across the image every bit illustrated in Figure 8.20. The filtered image is obtained from the center pixel (blackness dot in Figure 8.20) where the value of this pixel is the 2-dimensional summation of the product of the impulse response matrix (a three×iii matrix in Figure 8.20) and the original image. As the impulse response matrix slides horizontally and vertically across the original image, the new filtered image is constructed.

Effigy eight.20. Illustration of two-dimensional convolution. The impulse response, which is now a matrix, moves across the epitome one pixel at a time. After one row is complete then the impulse response matrix moves across the next row, i pixel down. At each position, another pixel in the output image is generated as the sum of the production of each value in the impulse response matrix times the corresponding original epitome pixel. The edges are usually handled by zero-padding.

Ii-dimension convolution is implemented in MATLAB using:

y=conv2(x,b,'options');   % Two-dimensional convolution

where y is the output paradigm, ten is the input image, b is the impulse response matrix, and the options are the same as for ane-dimensional convolution including the option same.

The next example applies a ii-dimensional moving average filter to a dissonance-filled image of red blood cells.

It would be squeamish to know the frequency characteristics of the moving average filter used in Example eight.eight. To determine the frequency characteristics of a two-dimensional filter we take the Fourier transform of the impulse response, but now we apply a ii-dimensional version of the Fourier transform. The two-dimensional discrete Fourier transform is divers equally:

Instance eight.8

Load the image of blood cells plant every bit variable cells in file blood_cell_noisy.mat. This image contains noise, blackness and white speckles known accordingly every bit salt and pepper noise. Filter the noise with a v×five moving boilerplate filter. Brandish the original and filtered images side by side.

Solution: Load the image and display using pcolor with the proper shading option as in previous examples. The matrix that generates a v×5 moving average is:

(viii.19) $b=\left|\begin{array}{ccccc}\mathrm{ane}/\text{25}& \mathrm{1}/\text{25}& \mathrm{i}/\text{25}& \mathrm{1}/\text{25}& \mathrm{i}/\text{25}\\ \mathrm{i}/\text{25}& \mathrm{1}/\text{25}& \mathrm{one}/\text{25}& \mathrm{ane}/\text{25}& \mathrm{1}/\text{25}\\ \mathrm{1}.\text{25}& \mathrm{ane}/\text{25}& \mathrm{1}/\text{25}& \mathrm{1}/\text{25}& \mathrm{1}/\text{25}\\ \mathrm{i}/\text{25}& \mathrm{one}/\text{25}& \mathrm{1}/\text{25}& \mathrm{1}/\text{25}& \mathrm{1}/\text{25}\\ \mathrm{1}/\text{25}& \mathrm{i}/\text{25}& \mathrm{ane}/\text{25}& \mathrm{1}/\text{25}& \mathrm{1}/\text{25}\end{array}\right|$

After generating this matrix, employ it to the image using conv2 with the aforementioned option and brandish.

% Ex 8_8 Awarding of moving average filter to blood prison cell paradigm with

% "table salt and pepper" noise.

%

load blood_cell_noisy;   % Load claret cell image

subplot(1,2,i);

pcolor(cells);   % Brandish epitome

shading interp;   % Same as previous examples

colormap(bone);

centrality('square');   % Adjust paradigm shape

%

b=[ones(v,5)/25;   % Define moving average impulse response matrix

y=conv2(cells,b,'same');   % Filter epitome using convolution

subplot(ane,2,2);

pcolor(y);   % Brandish image equally above

shading interp;

colormap(bone);

centrality('square');

Results: The original and filtered images are shown in Figure 8.21. The noise in the original paradigm is substantially reduced past the moving average filter.

(8.twenty) $F(p\text{,}q)={\displaystyle \sum }_{m=0}^{\mathrm{One\; thousand}-1}{\displaystyle \sum }_{\mathrm{due\; north}=0}^{N-1}b(\mathrm{thousand}\text{,}n){\mathrm{east}}^{-j(2pm/\mathrm{1000})}{\mathrm{eastward}}^{-j(\mathrm{two}qn/N)}p=0\text{,}\mathrm{i}\text{,}\dots \text{,}M-\mathrm{one};q=0\text{,}1\text{,}\dots \text{,}N-\mathrm{ane}$

where the impulse response part, b(thou,n), is an One thousand×N matrix. Of course the function can be whatever matrix, so this equation can also exist used to obtain the detached Fourier transform of an image. The 2-dimensional Fourier transform is implemented in MATLAB using:

X=fft2(ten,nrows,ncols);   % Two-dimensional Fourier transform

where x is the original matrix, X is the output, and nrows and ncols are optional arguments that are used to pad out the number of rows and columns in the input matrix. The output matrix is now in spatial frequency, cycles/altitude. When displaying the 2-dimensional Fourier transform, it is common to shift the zero frequency position to the center of the display and show the spectrum on either side. This improves visualization of the spectrum and is accomplished with routine fftshift:

Xshift=fftshift(10);   % Shift goose egg to center and reflect

where X is the original Fourier transform and Xshift is the shifted version. The spectrum can then be plotted using whatever of MATLAB's three-dimensional plotting routines such as mesh or surf. The next instance provides an example of the use of these routines to determine the spatial frequency characteristics of the moving boilerplate filter.

Example 8.9

Determine and plot the 2-dimensional spectrum of the 5×five moving boilerplate filter used in Case 8.8.

Solution: Generate the impulse response as in the last example, take the absolute value of the ii-dimensional Fourier transform, shift, and then display using mesh.

%EX eight.9 Spectrum of a v×5 moving boilerplate filter

% Construct moving average filter

b=ones(5,five)/25;   % Filter impulse response matrix

B=abs(fft2(b,124,124));   % Make up one's mind 2D magnitude Fourier transform

B=fftshift(B);   % Shift before plotting

mesh(B);   % Plot using mesh

Results: The plot generated past this program is shown in Figure 8.22. The spectrum is one of a ii-dimensional low-pass filter that reduces higher spatial frequencies every bit in both dimensions. The salt-and-pepper racket found in the paradigm of blood cells has high spatial frequencies since it involves rapid changes in intensity values within one or two pixels. Since this depression-pass filter significantly attenuates these frequencies, it is effective at reducing this type of racket. It is non possible to determine the spatial frequency calibration of the horizontal axes without knowing image size, the equivalent of signal catamenia in 2D data. Assuming the blood cell image is printed as 5   cm on a side, and then the fundamental spatial frequency is f _i =1/D=1/five=0.2   cycles/cm and the two axes should be multiplied by this scale factor.

Figure 8.22. Spatial frequency characteristic of a 5×5 moving average filter. This filter has the general features of a 2-dimensional low-pass filter reducing high spatial frequencies. If applied to an image v   cm on a side, then the horizontal axes should exist multiplied by 0.2   cycles/cm.

Other more complicated filters tin can exist designed, just most require support from software packages such as the MATLAB Epitome Processing Toolbox. An exception is the filter that does spatial differentiation. Utilize of the two-indicate central difference algorithm with a skip factor of 1 leads to an impulse response of b=[1 0 −1]. A two-dimensional equivalent might look like:

(eight.21) $b=\left|\begin{array}{ccc}1& 0& -1\\ \mathrm{ii}& 0& -2\\ \mathrm{one}& 0& -\mathrm{one}\end{array}\right|$

The corner values are half the master horizontal value, because they are on the edges of the filter array and should logically be less. This filter is known as the Sobel operator and provides an estimate of the spatial derivative in the horizontal direction. The filter matrix can be transposed to operate in the vertical management. The next example demonstrates the application of the Sobel operator.

Example 8.x

Load the MR image of the encephalon in variable brain of file Brain.mat. Apply the Sobel filter to this image to generate a new prototype that enhances changes that go from nighttime to light (going left to right). Flip the Sobel operator left to right, then use it once again to the original image to generate another paradigm that enhances modify going in the opposite management. Finally generate a quaternary epitome that is white if either of the Sobel filtered images is in a higher place a threshold, and black otherwise. Adjust the threshold to highlight the vertical boundaries in the original image.

Solution: Load the image, generate the Sobel impulse response and apply every bit is, and flipped, to the original image using conv2. Use a double for-loop to check the value of each pixel in the 2 images and set the pixel in the new image to one if either pixel is above the threshold and below if otherwise. Adapt the threshold empirically to highlight the vertical boundaries.

% Example 8.10 Employ the Sobel operator to the image of bone

%

load Brain;   % Load prototype

%

b=[1 0 −1; two 0 −ii; i 0 −1];   % Define Sobel filter

brain1=conv2(brain,b,'aforementioned');   % Apply Sobel filter

brain2=conv2(encephalon,fliplr(b),'same');   % Apply Sobel filter flipped

........brandish brain, brain1, and brain2 images as in previous examples........

%

[M,N]=size(brain);   % Find size of prototype

thresh=.iv;   % Threshold. Determined % empirically.

brain3=zeros(Thousand,N);

for k=1:M

  for north=one:Northward

    if brain1(m,n) &gt; thresh || brain2(thou,northward) &gt; thresh % Exam each pixel in each image

      brain3(chiliad,north)=1;   % Set up new epitome pixel to white

    else

      brain3(thou,due north)=0;   % else set to blackness

    end

  stop

end

........display prototype brain4 and title........

Results: The images produced in this case are shown in Figure 8.23. The Sobel filtered images are largely grayness since, except for the edges of the brain, the prototype intensities do not change abruptly. Withal, slightly lighter areas can exist seen Trouble 11 through 14 where more precipitous changes occur. If these 2 images are thresholded at the appropriate level, an image delineating the vertical boundaries is seen. This is a rough case of edge detection which is oftentimes an important pace in separating out regions of interest in biomedical images. Identifying regions or tissue of interest in a biomedical image is known equally prototype segmentation. Other examples of image filtering are given in the Trouble 11 t.

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Digital Signal Processing

Kevin M. Lynch , ... Matthew Fifty. Elwin , in Embedded Computing in C with the PIC32 Microcontroller, 2016

22.3.1 Moving Boilerplate Filter

Suppose nosotros have a sensor indicate x(n) that has been corrupted by loftier-frequency dissonance (Figure 22.9). We would like to find the low-frequency signal underneath the racket.

Effigy 22.9. The original noisy signal with samples x(northward) given by the circles, the point z(n) resulting from filtering with a three-indicate MAF (P = 2), and the signal z(n) from a nine-point MAF (P = 8). The bespeak gets smoother and more than delayed as the number of samples in the MAF increases.

The simplest filter to try is a moving boilerplate filter (MAF). A moving average filter calculates the output z(n) as a running average of the input signals x(n),

(22.six) $\begin{array}{l}\hfill z(n)=\frac{1}{P+1}\sum _{j=0}^P\mathrm{ten}(\mathrm{northward}-j),\end{array}$

i.eastward., the FIR filter coefficients are b ₀ = b _i = ⋯ = b _P = ane/(P + 1). The output z(north) is a smoothed and delayed version of x(n). The more samples P + 1 we average over, the smoother and more than delayed the output. The delay occurs because the output z(north) is a function of just the current and previous inputs x(n − j),0 ≤ j ≤ P (come across Figure 22.ix).

To find the frequency response of a 3rd-order, four-sample MAF, we exam it on some sinusoidal inputs at different frequencies (Effigy 22.x). We observe that the phase ϕ of the (reconstructed) output sinusoid relative to the input sinusoid, and the ratio G of the amplitude of their amplitudes, depend on the frequency. For the 4 test frequencies in Figure 22.10, nosotros get the following table:

Figure 22.10. Testing the frequency response of a four-sample MAF with unlike input frequencies.

Frequency	Gain G	Phase ϕ
0.25f North	0.65	− 67.five°
0.5f N	0	NA
0.67f Northward	0.25	0°
f N	0	NA

Testing the response at many different frequencies, we tin can plot the frequency response in Effigy 22.11. Ii things to note about the gain plot:

Figure 22.eleven. The frequency response of a iv-sample MAF. Exam frequencies are shown as dotted lines.

•

Gains are normally plotted on a log calibration. This allows representation of a much wider range of gains.

•

Gains are frequently expressed in decibels, which are related to gains by the post-obit relationship:

$M_{\text{dB}}=\mathrm{twenty}{log}_{10}\mathrm{Thousand}.$

And then Yard = ane corresponds to 0 dB, G = 0.1 corresponds to − 20 dB, and G = 0.01 corresponds to − xl dB.

Examining Figure 22.eleven shows that depression frequencies are passed with a gain of G = 1 and no phase shift. The proceeds drops monotonically as the frequency increases, until information technology reaches K = 0 ( $-\infty$ dB) at input frequencies f = 0.5f _N . (The plot truncates the dip to $-\infty$ .) The gain then begins to rise again, before falling once again to G = 0 at f = f _{Due north} . The MAF behaves somewhat similar a low-laissez passer filter, but non a very good one; high frequency signals tin get through with gains of 0.25 or more. Still, it works reasonably well equally a signal smoother for input frequencies below 0.5f _Northward .

Given a set of filter coefficients b = [b0 b1 b2 …], we tin plot the frequency response in MATLAB using

Causal vs. acausal filters

A filter is chosen causal if its output is the issue of merely electric current and by inputs, i.due east., past inputs "cause" the current output. Causal filters are the just option for existent-time implementation. If we are post-processing data, even so, we tin choose an acausal version of the filter, where the outputs at fourth dimension n are a part of by too equally futurity inputs. Such acausal filters can eliminate the delay associated with just using past inputs to calculate the current value. For example, a five-sample MAF which calculates the average of the past two inputs, the current input, and the next ii inputs is acausal.

Zero padding

When a filter is first initialized, there are no past inputs. In this example we can assume the nonexistent by inputs were all aught. The output transient caused by this assumption will end at the (P + one)thursday input.

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AI and retinal paradigm analysis at Baidu

Yehui Yang , ... Le Van La , in Computational Retinal Image Analysis, 2019

2.1.1 Focus and clarity assessment

The input fundus paradigm is outset converted to greyscale. A 3   ×   three and 5   × 5 moving average filter is then applied to generate two filtered images. This step is important as a low-laissez passer filter will impact a "focused" retinal image more than a "blurred" ane.

To quantify the difference between the filtered prototype and the original epitome, Sobel operators are applied to the aforementioned images (i.e., the greyscale version of the original image, the three   ×   3 filtered image, and the v   ×   five filtered image). The Sobel filter is a gradient operator that captures focus data [2]. For each prototype, four Sobel operators (every bit shown in Fig. v) are used and the resulting 4 slope maps are added to produce a single map. The mean of the resulting feature map describes the focus data of the respective image. At this point, three numbers (i.due east., the sum of the Sobel feature maps of nonfiltered, three   ×   3 filtered, and five   ×   5 filtered) are generated for each fundus image. The deviation betwixt these three numbers is used to ascertain whether the image is in focus.

Fig. 5. Sobel operators used for gradient map. The arrows indicate the direction of gradient calculated by the respective operators.

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CGM filtering and denoising techniques

Andrea Facchinetti PhD , ... Claudio Cobelli PhD , in Glucose Monitoring Devices, 2020

Possible approaches to CGM denoising

Moving-average (MA) filtering is a first candidate arroyo to bargain with CGM denoising. MA filters are commonly used in denoising in many applications, including processing in commercial CGM devices [21]. Briefly, having fixed the lodge, thou, the output of the filter relative to the northwardth sample is given by a weighted sum of the concluding g measured samples

(10.2) $\hat{u}\left(n\right)=\frac{w_{\mathrm{ane}}y\left(\mathrm{due\; north}\right)+w_{2}y\left(n-1\right)+\cdots +w_{k}y\left(n-k+1\right)}{{\sum }_{i=1}^{\mathrm{grand}}{\mathrm{westward}}_i}$

where y(n) represents glucose of the due northth sample. The parameters of the filter are the order k and the weights w ₁, …, w _chiliad . The higher is one thousand, the longer is the "memory" of the past information. Increasing k usually produces a more meaning racket reduction and, at the same fourth dimension, a larger indicate baloney, for case, û(due north) is significantly delayed, thus being unable to rails fast changes of the true u(northward). Having fixed the social club yard, the weights due west _one, …, due west _k tin can be chosen in several ways. The most common strategy is an MA with exponential weights, where w _i   = μ ⁱ , with μ (a real between 0 and i) interim as a "forgetting factor" (the higher μ, the higher the memory of past data). The major weakness of MA is that, once weights have been chosen, it treats all the time series in the same style, irrespectively of possible differences of their bespeak-to-racket ratio (SNR) due to sensor and individual variability (meet Fig. 10.i). As a consequence, a filter with fixed parameters is at chance of beingness suboptimal in denoising CGM data.

A different CGM denoising procedure, proposed by Hunt et al. [vii], was based on an integral-based plumbing equipment and filtering method. Fifty-fifty if the procedure can exist used in existent time during clinical trials, its major limitation is, in fact, that some of its components (e.g., the concentration of plasma insulin) cannot be identified if only CGM data are available. This hinders the possibility of using the method in daily-life conditions.

Finally, another proposed denoising process, in our stance, more suited to CGM applications, resorts to Kalman filter (KF). Pioneering applications of KF to process CGM data were presented by Knobbe et al. [23], with the aim of reconstructing blood glucose concentration past employing a model of claret-to-interstitium glucose kinetics and blood glucose concentration references, by Palerm et al. [24,25], with the aim of predicting the glucose profile and detecting hypoglycemia, and by Kuure-Kinsey et al. [three], with the purpose of improving CGM calibration.

In the next department, we will provide an application of KF to the GCM denoising trouble originally proposed by Facchinetti et al. [viii,nine].

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Digital Filtering in the Fourth dimension Domain

Jonathan Thousand. Blackledget , in Digital Signal Processing (Second Edition), 2006

The Moving Average Filter

The moving average filter computes the average value of a prepare of samples within a predetermined window.

Example For a 3 × 1 window:

$\begin{array}{cc}\hfill ⋮\hfill & \hfill \hfill \\ \hfill f_i\hfill & \hfill \hfill \\ \hfill f_{i+\mathrm{one}}\hfill & \hfill {\mathrm{southward}}_{i+1}=\left(f_i+f_{i+1}+f_{i+2}\right)/3\hfill \\ \hfill f_{i+2}\hfill & \hfill s_{i+\mathrm{two}}=\left(f_{i+1}+f_{i+2}+f_{i+3}\right)/3\hfill \\ \hfill f_{i+\mathrm{iii}}\hfill & \hfill {\mathrm{south}}_{i+3}=\left(f_{i+\mathrm{two}}+f_{i+\mathrm{three}}+f_{i+4}\right)/3\hfill \\ \hfill f_{i+\mathrm{iv}}\hfill & \hfill \hfill \\ \hfill ⋮\hfill & \hfill \hfill \end{array}$

As the window moves over the data, the average of the samples 'seen' within the window is computed, hence the term 'moving average filter'. In mathematical terms, we can express this type of processing in the grade

$s_i=\frac{1}{\mathrm{Thou}}{\displaystyle \sum _{j\in \mathrm{\omega }\left(i\right)}{f}_j}$

where ω(i) is the window located at i over which the average of the data samples is computed and M is the total number of samples in ω. Notation that the moving average filter is simply an FIR of the form

$s_i={\displaystyle \sum _{i=-N}^N{p}_{j-i}{f}_i}$

and so for a three×i kernel

$\mathrm{p}=\frac{\mathrm{i}}{3}\left(1,1,1\right)$

and for a v×1 kernel

$\mathrm{p}=\frac{1}{5}\left(1,\mathrm{one},1,\mathrm{one},1\right)\text{.}$

This filter tin be used to shine a signal, a feature which tin can be taken to include the reduction of noise. Notation that this filter is in effect, the convolution of an input with a tophat function; the spectral response is therefore a sine function.

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How To Pick Size Of Moving Average Filter Window,

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