What Is The Order Of A Filter

The Roles of Filters in Power Systems and Unified Power Quality Conditioners

Mohammad A.South. Masoum , Ewald F. Fuchs , in Ability Quality in Power Systems and Electrical Machines (Second Edition), 2015

9.3.2.5 Composite Filter

Higher order filters are constructed by increasing the number of storage elements (e.yard., capacitors, inductors). However, their awarding in power systems is limited due to economic and reliability factors.

A tertiary-social club filter may exist constructed past adding a series capacitance C _f2 to the inductor bypass resistance (Fig. 9.10a) to limit I_bp and reduce the corresponding fundamental frequency losses or by including C _f2 in serial with L_f (Fig. 9.10b) and sizing it to grade a series resonant branch at the central frequency to reduce I_bp and to increase filter efficiency. If a second inductor L _f2 is added to the circuit of Fig. 9.10a, a fourth-order double ring-pass filter volition exist obtained, as shown in Fig. 9.10c.

Effigy 9.10. Higher order filters; (a) and (b) tertiary-order filters, (c) quaternary-order filter.

A common type of an nth-order composite filter for power quality improvement is shown in Fig. 9.11. Several band-pass filters are continued in parallel and individually tuned to selected harmonic frequencies to provide compensation over a broad frequency range. The terminal branch is a high-pass filter attenuating high-order harmonics, which unremarkably are a outcome of fast switching actions. Composite filters are only applied when fifty-fifty-order harmonics are modest since a parallel resonance will occur betwixt any ii adjacent band-pass filter branches and cause amplification of the distortion in that frequency range. For case, composite filter systems with shunt branches tuned at the fifth and 7th harmonics will have a resonant frequency at well-nigh the sixth harmonic. The impedance transfer function of the nth-order composite filter is

(ix-11) $\begin{array}{c}{H}_f\left(s\right)=Z_f\left(\mathrm{southward}\right)=\frac{V_f\left(\mathrm{due\; south}\right)}{I_f\left(s\right)}=\frac{\mathrm{one}}{\frac{1}{Z_f^{\left(\mathrm{v}\right)}}+\frac{1}{Z_f^{\left(7\right)}}+\frac{1}{Z_f^{\left(\mathrm{xi}\right)}}+....+\frac{\mathrm{i}}{Z_f^{\left(n\right)}}}\text{.}\end{array}$

It is time-consuming to derive the transfer functions of multiple-gild filters in terms of factorized expressions of poles and zeros [thirteen]. Therefore, numerical approaches are typically applied to plot the transfer part, and an iterative design procedure is used to optimize a filter configuration.

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Filters in Control Systems

George Ellis , in Command System Design Guide (Fourth Edition), 2012

9.two.1.4 Higher-Order Low-Pass Filters

High-order filters are used because they have the ability to coil off gain after the bandwidth at a sharper rate than depression-order filters. The attenuation of a filter above the bandwidth grows proportionally to the number of poles. When rapid attenuation is required, higher-lodge filters are ofttimes employed.

The southward-domain form of college-order filters is

(nine.iv) $T(s)=\frac{A_0}{s^M+A_{\mathrm{1000}-1}{s}^{\mathrm{Grand}-1}+\cdots +A_{1}s+A_0}$

where 1000 is the number of poles (the lodge) and A ₀ through A_{M −1} are the coefficients of the filter.

High-guild filters are often shown in the cascade course (that is, as a series of two-pole filters). Both analog and digital filters can exist built as a series of ii-pole filters (for a circuitous or real pole pair) and a single-pole filter for odd-ordered filters. This requires an alternative form of the transfer function as shown in Equation 9.4. Equation nine.5 shows the form for an even number of poles; for an odd number of poles, a single pole can be added in pour.

(ix.5) $T(s)=\left(\frac{C_1}{{\mathrm{due\; south}}^2+B_1\mathrm{south}+C_{\mathrm{i}}}\right)\left(\frac{C_{\mathrm{ii}}}{{\mathrm{south}}^{\mathrm{ii}}+B_2\mathrm{due\; south}+C_{\mathrm{ii}}}\right)\cdots \left(\frac{C_{\mathrm{Thou}/\mathrm{ii}}}{s^2+B_{M/2}\mathrm{southward}+C_{K/\mathrm{two}}}\right)$

The form of Equation ix.5 is preferred to the form of Equation 9.four because pocket-sized variations in the coefficients of Equation 9.4 can move the poles enough to affect the functioning of the filter substantially. ⁷² In analog filters, variation comes from value tolerance of passive components; in digital filters, variation comes from resolution limitation of microprocessor words. Variations as small as 1 office in 65,000 can affect the operation of big-club filters.

An culling to the cascaded grade is the parallel form, where Equation 9.5 is divided into a sum of second-guild filters, as shown in Equation 9.6. Once again, this is for an fifty-fifty number of poles; a real pole can be added to the sum to create an odd-social club filter.

(ix.6) $T(\mathrm{south})=\frac{D_1}{(s^{\mathrm{ii}}+B_{1}s+C_1)}+\cdots +\frac{D_{M/\mathrm{ii}}}{(s^{\mathrm{two}}+B_{\mathrm{Grand}/2}s+C_{\mathrm{One\; thousand}/\mathrm{two}})}$

Both the cascaded form and the parallel grade have numerical properties superior to the directly form of Equation ix.iv for college-order filters. For more on the subject of sensitivity of higher-order filters, see Refs 1, 16, 45, and 72.

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Analog Low-Laissez passer Filters

Marc T. Thompson Ph.D. , in Intuitive Analog Circuit Pattern (Second Edition), 2014

Instance fourteen.1: Determining Butterworth filter order

Let us decide the filter order N needed to achieve a Butterworth filter with a cutoff frequency f _c  =   1   MHz and 50-dB attenuation at 2.vii   MHz. We could estimate the filter order by using the magnitude response plots of Effigy 14.iv, but for the Butterworth, we can use the magnitude equation directly. First, notation that an attenuation of −50   dB is equivalent to a gain magnitude of iii.xvi   ×   x⁻³. We tin use this in the attenuation equation equally follows:

(xiv.3) $\mathrm{three.sixteen}\times {\mathrm{x}}^{-3}=\frac{\mathrm{i}}{\sqrt{\mathrm{i}+{\left(\frac{f}{f_{\text{c}}}\right)}^{2N}}}=\frac{\mathrm{i}}{\sqrt{1+{\left(\frac{\mathrm{two.seven}}{1}\right)}^{\mathrm{two}N}}}\to \mathrm{Northward}=5.8$

So, to meet the magnitude specification, we need an Due north  =   6 Butterworth filter. An implementation of this filter as a doubly terminated ladder is shown in Figure xiv.vi(a) . (We will prove later how to derive the component values.) Also, in Figure 14.6(a), we note the use of a gain of 2.

FIGURE fourteen.6. Ladder filter implementation for an N  =   6 Butterworth filter with a cutoff frequency f _c  =   5   MHz for Case 14.1. (a) Circuit, with a gain of 2 added to recoup for the DC attenuation of the source and termination resistor. (b) Magnitude response. (For color version of this effigy, the reader is referred to the online version of this book.)

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Compliance and Resonance

George Ellis , in Control System Blueprint Guide (Fourth Edition), 2012

16.3.4.2 Second-Order Filters

Two types of second-order filters are as well used to deal with resonance. Two-pole notch filters provide a great bargain of attenuation over a narrow band of frequencies, and they do so without injecting significant stage lag at lower frequencies. Notch filters provide a narrow band of attenuation, which is effective for the narrow gain fasten of tuned resonance. The interested reader tin can verify this by using the notch filter of Experiment 16C afterward copying in the mechanical parameters of the tuned resonant system in Experiment 16A; fix the notch frequency to the natural frequency of the motor and load, and enable the filter. Notch filters are not effective for reduced-inertia instability considering that problem requires attenuation over a large frequency range.

While notch filters are effective for tuned resonance, the benefits are limited to a narrow frequency range. If the resonant frequency shifts even a small corporeality, the notch filter tin can become ineffective. Unfortunately, resonant frequencies frequently vary over time and from ane re-create of a machine to another. The load inertia may modify, equally was discussed in Department 12.4.four. The bound abiding can also change. For example, the compliance of a lead screw will vary when the load is located in dissimilar positions, although this effect is not as dramatic as it might at first seem, since much of the compliance of a ball spiral is in the ball-nut rather than the screw itself. Notch filters piece of work best when the machine structure and operation permit fiddling variation of the resonant frequency and when the notch can be individually tuned for each machine.

Two-pole low-pass filters are commonly used to deal with resonance. Ofttimes the damping ratio is set to between 0.5 and 0.vii. In this range, the two-pole low-pass filter can provide attenuation while injecting less phase lag than the unmarried-pole low-pass or lag filters. Bi-quad filters are also used to deal with resonance. The well-nigh mutual instance is where bi-quad is configured to benumb college frequencies, that is, where the poles (denominator frequencies) are smaller than the zeros (numerator frequencies). In this instance, the bi-quad operates similar a second-society lag filter, providing attenuation while injecting less phase lag than the low-pass filter.

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Active Filters

Rolf Schaumann , in The Electrical Engineering Handbook, 2005

The GIC Biquad

A very successful 2nd-order filter is based on the RLC band-pass circuit in Figure iv.vii(A) that realizes the following:

Figure 4.7. (A) RLC Epitome Band-Pass Filter (B) Inductor Simulation with a General Impedance Converter (C) The Concluding GIC Band-Pass Filter with GIC Enclosed

(4.17) $\frac{5_{\text{out}}}{V_{\text{in}}}=\frac{G}{\mathrm{Thousand}+sC+1/(sL)}=\frac{\mathrm{due\; south}\mathrm{Thousand}/C}{s^{\mathrm{two}}+sG/C+1/(LC)}.$

Since inductors have to be avoided, circuits that employ only capacitors, resistors, and gain were developed whose input impedance looks inductive. The concept is shown in Effigy 4.seven(B) where the box labeled GIC is used to convert the resistor R_L to the inductive impedance Z_L . The general impedance converter (GIC) generates an input impedance as Z_L = sTR_L , where T is a time constant. For the inductor Fifty in the RLC circuit, we then can substitute a GIC loaded by a resistor R_L = L/T.

I manner to develop a GIC is to construct a circuit that satisfies the following two-port equations (refer to Effigy iv.7(B)):

(4.eighteen) $\begin{array}{ll}{5}_1=V_2,\hfill & I_1=I_2/(\mathrm{southward}T).\hfill \end{array}$

In that case, nosotros take:

(4.19) $Z_L=\frac{V_1}{I_{\mathrm{one}}}=sT\frac{V_2}{I_{\mathrm{two}}}=sT{R}_{\mathrm{Fifty}},$

exactly as desired. A circuit that accomplishes this feat is shown in the dashed box in Figure 4.7(C). Routine assay yields V ₂ = 5 _one and I ₂ = s(C _ii R ₁ R ₃/R _iv)I ₁ = sTI ₁ to requite the inductive impedance:

(4.twenty) $Z_L(s)=\frac{{\mathrm{Five}}_1}{I_1}=s\left(C_2\frac{R_1{R}_3}{R_4}\right)R_{\mathrm{Fifty}}=sT{R}_L.$

Usually 1 chooses in the GIC ring-pass filter identical capacitors to save costs: C = C ₂ . Further, it tin exist shown that the best choice of resistor values (to brand the simulated inductor optimally insensitive to the finite gain-bandwidth product, ω_t, of the op-amps) is the following:

(4.21) $R_{\mathrm{one}}=R_3=R_4=R_L=\mathrm{ane}/({\omega }_0{C}_{\mathrm{ii}}),$

where ω₀ is a frequency that is critical for the application, such as the center frequency in our band-pass case of Effigy iv.seven or the passband corner in a low-pass filter. The faux inductor is then equal to $\mathrm{Fifty}=C_2{R}_1^2$ . Finally, the design is completed by choosing R = QR ₁ to determine the quality factor.

A further pocket-sized problem exists because the output voltage in the RLC circuit of Figure 4.seven(A) is taken at the capacitor node. In the active circuit, this node (5 _i in Figure 4.7(C)), is non an op-amp output and may non be loaded without disturbing the filter parameters. A solution is readily obtained. The voltage V _out in Figure 4.7(C) is obviously proportional to V₁ because 5 ₂ = V ₁: V _out/V _one = 1 + R _iv /R_L = 2. Therefore, we may take the filter output at 5 _out for a preset gain of 5 _out/V _one = 2. The band-pass transfer function realized by the circuit is and so

(4.22) $\frac{V_{\text{out}}}{{\mathrm{Five}}_{\text{in}}}=\frac{\mathrm{two}\mathrm{south}/(CR)}{s^{\mathrm{ii}}+s/(CR)+1/(C^{\mathrm{two}}{R}_1^{\mathrm{two}})}=\frac{2\mathrm{southward}{\omega }_0/Q}{s^2+\mathrm{southward}{\omega }_0/Q+1/{\omega }_0^{\mathrm{two}}}.$

Transfer functions other than a band-pass tin can also be obtained by generating additional inputs to the GIC band-pass kernel in Figure four.7(c).

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Active Filter Design

Rolf Schaumann , in Reference Information for Engineers (Ninth Edition), 2002

Two-Amplifier Filters

The literature contains several multiamplifier second-order filters whose main advantage over unmarried-amplifier circuits is that the sensitivities both to the passive components and to the op amps' finite gain-bandwidth product, ω_t, are reduced. Based on a sensitivity comparison, only a few of these circuits take been shown to result in applied designs. Among these, the biquad based on inductance simulation using an alternative version of the general impedance converter (GIC) of Fig. nine has emerged as i of the all-time biquads attributable to the reduced dependence of filter parameters on ω_t.

The circuit capable of realizing a general biquadratic transfer office (except depression-pass) is shown in Fig. 18. Simple assay yields, for ω_t → ∞,

Fig. 18. A GIC-based biquad.

(Eq. 63) $T\left(\mathrm{due\; south}\right)=\frac{{\mathrm{Five}}_2}{V_1}=\frac{\mathrm{Northward}\left(s\right)}{D\left(s\right)}=\frac{a_{\mathrm{ii}}{\mathrm{due\; south}}^{\mathrm{two}}+a_{1}s{\omega }_0+a_0{\omega }_0^2}{{\mathrm{due\; south}}^{\mathrm{two}}+\mathrm{southward}{\omega }_0(\mathrm{ane}\text{/}Q+R\text{/}{R}_c)+{\omega }_0^2}$

with

(Eq. 64a) ${\omega }_0=1\text{/}\left(RC\right)$

and

(Eq. 64b) $a_0=b$

(Eq. 64c) $a_1=\left[aH-b(H-1)\right]\left(\mathrm{ane}\text{/}Q+R\text{/}{R}_c\right)-aHR\text{/}{R}_c$

(Eq. 64d) $a_2=cH-b(H-1)$

The resistor R _c is a bounty resistor used merely for the case when H ≠ two (see Eq. 68). R _c = ∞ for H = 2.

Different types of transfer functions are realized by an advisable selection of the parameters a, b, and c that are defined in Fig. 18. Table 2 gives details. Of grade, a, b, and c must be less than or equal to i, and H > 1. The circuit of Fig. eighteen cannot realize a good depression-pass function because the finite value of ω_t degrades the performance. Thus, a depression-pass filter is built with a slightly modified circuit as shown in Fig. nineteen; it realizes, with ω₀ = 1/(CR),

TABLE 2. Blueprint Data FOR THE Excursion IN FIG. 18

Filter Type	Northward 2(s)	a	b	c	Gain at ω =		Comments
Depression-Pass *	a 0	b/2	b	b/2	b	0	H = 2, b = a 0
Bandpass	a ane due south	i	0	0	H	ω0	Set H = a 1 Q/ω0, R c from Eq. 68
High-Laissez passer	a 2 s 2	0	0	one	H	∞	Prepare H = a two, R c from Eq. 68
All-Laissez passer	south 2 − sω 0/Q + ω0 two	0	1	1	ane	All	Prepare H = 2, R c = ∞
Notch	a 2(s ii + ω z ii) ${\omega }_z^2=\frac{b{\omega }_0^{\mathrm{two}}}{\mathrm{ii}c-b}$	b/2	b	b	b b b	0 ∞ 0, ∞	Ready H = 2, R c = ∞ Low-pass notch: b > c High-laissez passer notch: b < c Notch: b = c

*

Non realizable well with this circuit (run across Fig. nineteen).

Fig. 19. GIC depression-laissez passer filter.

(Eq. 65) $T\left(s\right)=V_2\text{/}{5}_1=\mathrm{ii}{\omega }_0^{\mathrm{two}}\text{/}\left(s^{\mathrm{two}}+\mathrm{due\; south}{\omega }_0\text{/}Q+{\omega }_0^2\right)$

A few observations are in guild nigh the performance of GIC-based filters: As was the case with the GIC in Fig. 9, the circuit is optimally insensitive to ω_t if all GIC-internal resistors are equal. In Fig. 18, this clearly requires H = two, which according to Table ii implies that the midband gain of the bandpass filter and the loftier-frequency gain of the high-pass filter are stock-still at H = two. However, in many applications, especially in high-social club filters based on second-order building blocks (meet next section), the gain must exist adjustable to maximize dynamic range.

If the required gain is less than 2, the lead-in element (QR for the bandpass filter, C for the loftier-pass filter), tin can simply exist split into a voltage divider of an appropriate ratio. This stride does not upset the desirable value H = 2. Yet, H > 2 tin be achieved simply by setting the ii resistors (H − ane)R in Fig. 18 to the appropriate value (or using an additional amplifier to raise the point). The effect of H > 2 can be shown to consequence in a very significant enhancement of Q in addition to a subtract of ω₀ with increasing values of ω₀/ω_t. The ω₀ error

(Eq. 66) $\Delta {\omega }_0\text{/}{\omega }_0=\left({\omega }_{\text{a}}-{\omega }_0\right)\text{/}{\omega }_0\approx -0.5\left[H^2\text{/}(H-\mathrm{i})\right]{\omega }_0\text{/}{\omega }_{\text{t}}$

(where ω_a is the really realized pole frequency) has its minimum at −2ω₀/ω_t for H = two. It can exist corrected just past predistortion. The Q deviation, expressed as

(Eq. 67) $Q_{\text{a}}=Q\text{/}\left\{1+\left[1-2Q(2-H)\text{/}H\right]\Delta {\omega }_0\text{/}{\omega }_0+2(3-\mathrm{iv}\text{/}H)Q{(\Delta {\omega }_0\text{/}{\omega }_0)}^{\mathrm{ii}}\right\}$

(where Q _a is the actually realized pole quality factor) also is minimized for H = ii and tin can exist shown to exist reduced approximately to nix, i.e., Q _a ≈ Q, by connecting a compensation resistor

(Eq. 68) $R_c\approx R{(\mathrm{ane}+\mathrm{yard})}^2\text{/}\left[\mathrm{grand}(H-2-\mathrm{chiliad})\right]$

in shunt with the GIC input, every bit indicated in Fig. xviii. In Eq. 68,

(Eq. 68a) $m=\left[H\text{/}(H-\mathrm{i})\right]{\omega }_0\text{/}{\omega }_1$

Equation 68 is valid for pole frequencies satisfying (approximately)

(Eq. 69) ${\omega }_0\le 0.3{\omega }_{\text{t}}\text{/}H$

For larger values of ω₀, Eq. 68 becomes increasingly inaccurate, just Q bounty can still be achieved by functional tuning of R _c. A note of caution must exist made: The value of R _c is seen to depend on ω_t, which is process-, temperature-, and bias-dependent, is very inaccurate, and changes from op amp to op amp. Compensating this parameter with a passive resistor can, at best, be only approximate.

The function of R _c is only to eliminate ω_t-caused Q enhancement for H > two; otherwise, information technology has no effect on the realized transfer part apart from the slight dependence of a _ane on R _c shown past Eq. 64c. Since past Eq. 69a yard is small, as is seen from Eq. 68, R _c is eliminated, i.e., R _c → ∞ for H = 2.

In the all-laissez passer and notch circuits, zero pregnant is gained by choosing H ≠ two, and no usable compensation exists for H ≠ two in the depression-laissez passer filter of Fig. nineteen. Thus, all GIC-internal resistors are set equal in these circuits.

Finally, it should be mentioned that Q is quite sensitive to capacitor losses. Labeling the capacitor loss resistor R _L and using Eqs. 5 and 64a, we obtain Q _c = ω₀ CR = R _Fifty/R. The actual quality cistron, Q _a, can be shown to exist

(Eq. 70) $Q_{\text{a}}=Q\text{/}\left(1+2Q\text{/}{Q}_c\right)$

where past Q we labeled the platonic quality cistron, contained of such furnishings every bit finite ω_t and H ≠ ii. Thus, loftier-quality capacitors should be used to build these filters. Further details nigh GIC-filter functioning can be found in References 2, 6, and seven, on which nigh of this word is based.

As a design example, presume that the bandpass function

(Eq. 71) $T\left(s\right)=0.8s_n\text{/}\left(s_{\text{n}}^{\mathrm{ii}}+0.1{\mathrm{due\; south}}_{\text{north}}+1\right)$

with southward _{due north} = s/ω₀ and ω₀ = 2π × 6 kHz has to be realized. Assume farther that op amps with ω_t = 2π × 900 kHz are available. From Eq. 71, pole frequency, quality gene, and midband gain are, respectively, f ₀ = 6 kHz, Q = 10, and H = viii. From Tabular array ii for a bandpass filter: a = 1 and b = c = 0. From Eq. 69 it follows that f ₀ = half-dozen kHz < 33.7 kHz so that Eq. 68 applies. Thus, R _c = 22.24R will recoup for ω_t-caused Q enhancement that by Eq. 67 would be 79%; i.e., without R _c, Q _a = i.79Q = 17.9. Remember though that this compensation depends on the specific value of ω_t assumed in the blueprint. Further, from Eq. 66, Δf ₀ ≈ −183 Hz, a −three.0% error. To eliminate this nominal deviation, the filter is designed with predistortion to realize f ₀ = 6.189 kHz and so that a −3% error gives f _a = 6 kHz. Thus, from Eq. 64a with C = 10 nF, R = 2572 Ω; therefore, QR = 25.seven kΩ and (H − i)R = xviii.0 kΩ. The circuit is shown in Fig. 20.

Fig. 20. GIC filter realizing Eq. 71.

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Discrete-time Signals and Systems

Luis Chaparro , in Signals and Systems Using MATLAB (Second Edition), 2015

ix.3.four.1 Linear Filtering

To illustrate the way a linear filter works, consider getting rid of a random disturbance η[north], which we model as Gaussian noise (this is ane of the possible noise signals MATLAB provides) that has been added to a sinusoid 10[north]   =   cos (πn/16). Let y[due north]   = 10[northward]   + η[n]. Nosotros will use an averaging filter having an input/output equation

$z[n]=\frac{\mathrm{one}}{M}{\displaystyle \sum _{\mathrm{yard}=0}^{\mathrm{1000}-1}}y[\mathrm{north}-k]$

This M-order filter averages K past input values {y[n  − g], k  =   0,   …   ,Chiliad  −   1} and assigns this average to the output z[n]. The effect is to smooth out the input signal by attenuating the high-frequency components of the signal due to the noise. The larger the value of One thousand the ameliorate the results, but at the expense of more complexity and a larger delay in the output indicate (this is due to the linear phase frequency response of the filter, as we will encounter later).

We use 3rd- and xv ^th -order filters, implemented by our office averager given below. The de-noising is washed past means of the following script:

  %%

  % Linear filtering

  %%

  N   =   200;n   =   0:Due north–1;

  x   =   cos(pi*northward/sixteen);        % input indicate

  dissonance   =   0.two*randn(1,N);  % noise

  y   =   x   +   racket;            % noisy bespeak

  z   =   averager(3,y);        % averaging linear filter with Chiliad   =   iii

  z1   =   averager(15,y);      % averaging linear filter with Yard   =   fifteen

Our function averager defines the coefficients of the averaging filter and then uses the MATLAB function filter to compute the filter response. The inputs of filter are the vector b  =   (1/K)[one   ⋯   1], or the coefficients continued of the numerator, the coefficient of the denominator (1), and x a vector with entries the signal samples we wish to filter. The results of filtering using these 2 filters are shown in Figure ix.12. As expected the performance of the filter with 1000  =   xv is a lot meliorate, but a delay of eight samples (or the integer larger than 1000/2) is shown in the filter output.

Figure 9.12. Averaging filtering with filters of order Thousand  =   3 (acme figure), and of order G  =   fifteen upshot (lesser effigy) used to get rid of Gaussian noise added to a sinusoid x[n]   = cos(πn/16). Dark blue line corresponds to the noisy betoken, while the light blue line is for the filtered signal. The filtered betoken is very much like the noisy signal (see top figure) when Grand  =   3 is the order of the filter, while the filtered signal looks like the sinusoid, but shifted, when M  =   15.

office y   =   averager(M,x)

% Moving boilerplate of signal x

%M: order of averager

%ten: input signal

%

b=(1/Chiliad)*ones(i,M);

y   =   filter(b,ane,x);

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Analog Electric Filters

Lawrence P. Huelsman , in Encyclopedia of Concrete Scientific discipline and Applied science (3rd Edition), 2003

V.D Direct Realizations

In add-on to the cascade realizations of second-lodge filters described in the preceding sections, higher-club active filters can also be realized in a general grade called a direct realization. Such a realization uses a passive, lossless-ladder filter equally a prototype and replaces the passive reactive elements of inductance and capacitance by unmarried-time-constant analog active circuits. Effectively, elements with impedance Z ₅₀ (s)   = sL and admittance Y _C (s)   = _southward C are replaced past integrators with voltage transfer functions 5 ₂/Five _one  = Due north(s)   =   1/southward RC, where R and C are the elements of the integrator. Such a filter configuration is frequently called a leapfrog filter because of the multiple negative feedback paths for signal flow. In effect, the leapfrog filter replaces the voltage and electric current variables of the passive filter configuration with a new set of voltage (simply) variables in a fashion similar to that used in old-time analog computers. Since the replacement of variables retains the depression sensitivity properties of the original passive filter, Active-RC filters based on the leapfrog blueprint have excellent sensitivity properties. An example of a 3rd-order leapfrog filter is shown in Fig. 18. The single-resistance, terminated-LC ladder filter which realizes a third-order Butterworth approximation is shown in part (a). The resulting leapfrog configuration is shown in role (b).

Figure xviii. Leapfrog filter.

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Analog Filters With Arbitrarily Adjustable Frequency Response

Jaroslav Koton , ... Jan Dvorak , in Partial Guild Systems, 2018

5 Conclusion

Traditionally, in the case of designing the integer-gild filters, the type of approximation or the quality factor Q, pole frequency f ₀, and pass-band gain Thousand ₀ are defined and can be controlled.

The fractional-order filters introduce a new variable – the partial parameter α that defines the (n + α) fractional society of the circuit, where north is the integer. Although a passive chemical element that would feature partial behavior, that is, the ± 90 ⋅ α phase shift between the cross-voltage and through-electric current, is non readily available, suitable approximation techniques were described and are used to design analog fractional-order filters. Assuming the α to be selected from and controlled in the range 0 < α < 1, filters generally featuring any noninteger order can exist designed using Butterworth-, Chebyshev-, or changed Chebyshev-like approximation and implemented using suitable types of active elements.

Every bit the just-mentioned approximation types (discussed in Sections 3.1, 3.2, and 3.3) are limited to a specific value of the quality factor, the technique of capricious quality factor partial-society transfer part has likewise been described (in Section 3.4). Such a technique by and large enables the design of a fractional-gild filter featuring whatsoever value of pole frequency, any value of order, and fifty-fifty any value of the quality gene. Based on that, using active elements that are electronically controllable, it is possible to design systems with arbitrarily adjustable frequency response. As an example of such electronic adaptability, the circuit solutions providing fractional-guild low- and high-pass frequency responses have been presented and analyzed to show the concept of arbitrarily adjustable partial-gild filters.

Following the theory provided in this chapter information technology is possible to design own-circuit solutions of frequency filters with capricious and controllable frequency response that fully fit the designer's requirements and after use such excursion in more complex systems.

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Active Filter Pattern Techniques

Bruce Carter , Ron Mancini , in Op Amps for Anybody (Fifth Edition), 2018

sixteen.2.4 Quality Factor Q

The quality factor Q is an equivalent design parameter to the filter gild north . Instead of designing an nth order Tschebyscheff depression-pass, the trouble tin be expressed as designing a Tschebyscheff depression-pass filter with a certain Q.

For band-pass filters, Q is defined as the ratio of the mid-frequency, f_grand, to the bandwidth at the two −three   dB points:

$\text{Q}=\frac{{\text{f}}_{\text{m}}}{\left({\text{f}}_2-{\text{f}}_{\mathrm{one}}\right)}$

For low-pass and loftier-pass filters, Q represents the pole quality and is defined as:

$\text{Q}=\frac{\sqrt{{\text{b}}_{\text{i}}}}{{\text{a}}_{\text{i}}}$

High Qs can exist graphically presented as the altitude between the 0   dB line and the peak bespeak of the filter's gain response. An example is given in Fig. sixteen.ten, which shows a 10th-order Tschebyscheff low-pass filter and its v partial filters with their individual Qs.

Figure 16.10. Graphical presentation of quality gene Q on a tenth-order Tschebyscheff low-pass filter.

The gain response of the 5th filter stage peaks at 31   dB, which is the logarithmic value of Q₅:

${\text{Q}}_{\mathrm{v}}\left[\text{dB}\right]=\mathrm{xx}\log {\text{Q}}_5$

Solving for the numerical value of Q₅ yields

${\text{Q}}_5=10\frac{31}{20}=35.48$

which is within 1% of the theoretical value of Q   =   35.85 given in Department 16.9, Table 16.eleven, terminal row.

The graphical approximation is good for Q   >   three. For lower Qs, the graphical values differ from the theoretical value significantly. Even so, only higher Qs are of concern, since the higher the Q is, the more a filter inclines to instability.

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What Is The Order Of A Filter,

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