Unbounded Continuous Function at 1 2 Q

Continuous Function

The continuous function describing deformation, elastic force, geometry and material relations includes the limiting elastic deformation, be it the limit of a material or a specific limit generated by the geometry of a structure.

From: Non-Linear Theory of Elasticity and Optimal Design , 2003

Stability

Derong Liu , in The Electrical Engineering Handbook, 2005

2.6.1 Definiteness of a Function

A continuous function v: Rⁿ → R [resp., v: B(h) → R] is said to be positive definite if:

1)

v (0) = 0, and

2)

v(x) > 0 for all x ≠ 0 [resp., 0 < ||x|| ≤ r for some r > 0].

A continuous function v is said to be negative definite if − v is a positive definite function.

Positive semidefinite: A continuous function v: Rⁿ → R [resp., v: B(h) → R] is said to be positive semidefinite if:

1)

v (0) = 0, and

2)

v(x) ≥ 0 for all x ∈ B(r) for some r > 0.

A continuous function v is said to be negative semidefinite if −v is a positive semidefinite function.

A continuous function v: Rⁿ → R is said to be radially unbounded if:

1)

v(0) = 0,

2)

v(x) > 0 for all x ≠ 0, and

3)

v(x) → ∞ as ||x|| → ∞.

Example 9

The function v: R ³ → R is given by:

$v\left(x\right)=x_1^2+x_2^2+x_3^2,$

which is positive definite and radially unbounded.

Example 10

The function v: R ³ → R is given by:

$v\left(x\right)=x_1^2+{\left(x_2-x_3\right)}^2,$

which is positive semidefinite. It is not positive definite because it is zero for all x ∈ R ³ such that x ₁ = 0 and x ₂ = x ₃.

Example 11

The function v: R ³ → R is given by:

$v\left(x\right)=x_1^2+x_2^2+x_3^2-{\left(x_1^2+x_2^2+x_3^2\right)}^3,$

which is positive definite in the interior of the ball given by $x_1^2+x_2^2+x_3^2<1$ . It is not radially unbounded since v(x) < 0 when $x_1^2+x_2^2+x_3^2>1$ .

Example 12

The function v: R ³ → R is given by:

$v\left(x\right)=\frac{x_1^4}{1+x_1^4}+x_2^4+x_3^4,$

which is positive definite but not radially unbounded.

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'BEST' APPROXIMATION

G.M. PHILLIPS , P.J. TAYLOR , in Theory and Applications of Numerical Analysis (Second Edition), 1996

Definition 5.8

A continuous function E is said to equioscillate on n points of [a, b] if there exist n points x _i with a ⩽ x ₁ < x ₂ < … < x _n ⩽ b, such that

$\left|E\left(x_i\right)\right|=\underset{a⩽x⩽b}{\max }\left|E\left(x\right)\right|,i=1,\cdots ,n,$

and

$E\left(x_i\right)=-E\left(x_{i+1}\right),i=1,\cdots ,n-1.$

Thus for the minimax approximation from P _n in Example 5.13 the error function equioscillates on n + 2 points and the same is true with n = 1 in Example 5.14. The equioscillation of the error on n + 2 points or more turns out to be the property which characterizes minimax approximation from P _n for any f ∈ C[a, b].

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EVALUATION OF THE INVERSE PROBLEM FOR NDT OF MAGNETISED STEEL CORD BELT SPLICES

A. HARRISON , S. GHYS , in Non-Destructive Testing 1989, 1989

BUILDING BLOCK MODELLING

A continuous function of the form in Equation (1) can be used effectively with an IBM AT PC computer to simulate the total composite induced voltage that results from the progressive offset of cord ends by the splice skive. The effect of fringing near the cord end has been approximated by a series of progressively reduced-length cords, each with a reduced external field. The actual field φ, modified along th cord for fringing (shown in Fig. 2(b)), is

(3) ${\phi }_{\text{a}\text{c}\text{t}}={\displaystyle \sum _{\text{i}=-15}^{50}{\text{e}}^{-\left|\text{i}\right|}{\phi }_0\left(\text{x}+\text{i}\Delta \text{x}\right)}$

where the increment along the x axis is Δx and the external field of the hypothetical coincident cords is made to reduce exponentially up to 50 iterations.

Using this building block, the shape of the induced voltage for any Lay–up can be obtained. Figure 3 shows two examples of Lay–ups and their computed magnetic signature.

Fig. 3. Simulated signatures for step 1 and step 2 splices.

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Plug and play control for islanded microgeneration systems

Boyu Qin , ... Shengwei Mei , in Renewable Energy Microgeneration Systems, 2021

6.2.2.1 Theoretical preliminaries

This chapter studies the following affine nonlinear system for its LISS/LIOS properties:

(6.1) $\begin{array}{c}\dot{x}=f\left(x,u\right)\\ y=h\left(x,u\right)\end{array}$

where x  ∈ R ⁿ , u  ∈ R ^m , f  : R ^{n  + m}   ⟶ R ⁿ , and h  : R ^{n  + m}   ⟶ R ^p . f and h satisfy two conditions: ① they are continuous and locally Lipschitz with respect to x, ② f(0,   0)   =   0, h(0,   0)   =   0.

Since the notion of ISS is defined based on comparison functions, this chapter gives the definitions of comparison functions as follows:

A continuous function γ _ISS   : R _{≥   0}  ⟶ R _{≥   0} is a $\mathcal{K}$ function when the conditions hold that γ _ISS (0)   =   0 and γ _ISS is strictly increasing. Furthermore, if γ _ISS (s)   ⟶   ∞ as s   ⟶   ∞, then γ _ISS is called ${\mathcal{K}}_{\infty }$ function. A bivariate function β _ISS (s, t)   : R _{≥   0}  × R _{≥   0}  ⟶ R _{≥   0} is a $\mathcal{KL}$ function when these conditions hold that for each fixed t  ≥   0, the function β _ISS (·, t) is a $\mathcal{K}$ function, and for each fixed s  ≥   0, β _ISS (s,   ·)is no-increasing for t, also β _ISS (s, t)   ⟶   0 as t  ⟶   ∞.

Definition 2.1

System (6.1) is ISS if functions ${\beta }_{\mathit{ISS}}\in \mathcal{KL}$ and ${\gamma }_{\mathit{ISS}}\in {\mathcal{K}}_{\infty }$ exist, such that for any x ₀  ∈ R ⁿ , u  ∈ R ^m , the following inequality holds

(6.2) $\left|x\left(t,x_0,u\right)\right|\le {\beta }_{\mathit{ISS}}\left(\left|x_0\right|,t\right)+{\gamma }_{\mathit{ISS}}\left({‖u‖}_{\infty }\right)\forall t\ge 0$

where |·| represents the Euclidean norm and ‖u‖_∞ denotes the supremum norm of u. It can be seen that β _ISS and γ _ISS represent the transient and asymptotic behavior of system (6.1), respectively.

In practice, most of the nonlinear systems are stable within local stability regions. Thus the estimated difficulty of the systems' ISS properties can be simplified by limiting the allowable input and initial conditions in local regions.

Definition 2.2

System (6.1) is locally input-to-state stable (LISS) if functions ${\beta }_{\mathit{LISS}}\in \mathcal{KL}$ and ${\gamma }_{\mathit{LISS}}\in {\mathcal{K}}_{\infty }$ exist, such that for any x ₀  ∈   Ω _LISS , u  ∈ U _LISS , the following inequality holds

(6.3) $\left|x\left(t,x_0,u\right)\right|\le {\beta }_{\mathit{LISS}}\left(\left|x_0\right|,t\right)+{\gamma }_{\mathit{LISS}}\left({‖u‖}_{\infty }\right)\forall t\ge 0$

where U _LISS is a local region of external inputs, which denoted by {u  ∈ R ^m   |   ‖u‖_∞  ≤ τ}, τ  >   0, Ω _LISS is a local region of initial states. U _LISS and Ω _LISS are called the LISS regions.

As discussed in Sontag and Wang (1999), we can get the concept of local input-to-output stability (LIOS) similarly.

Definition 2.3

System (6.1) is LIOS if functions ${\beta }_{\mathit{LOSS}}\in \mathcal{KL}$ and ${\gamma }_{\mathit{LOSS}}\in {\mathcal{K}}_{\infty }$ exist, such that for any x ₀  ∈   Ω _LOSS , u  ∈ U _LOSS ,the following inequality holds

(6.4) $\left|y\left(t,x_0,u\right)\right|\le {\beta }_{\mathit{LOSS}}\left(\left|x_0\right|,t\right)+{\gamma }_{\mathit{LOSS}}\left({‖u‖}_{\infty }\right)\forall t\ge 0$

where U _LOSS and Ω _LOSS are called the LOSS region, and they can be denoted by W  ={[x ₀, u]| x ₀  ∈   Ω _LOSS , u  ∈ U _LOSS }.

From Sontag (2008), it can be proved that the properties of ISS are equivalent to the existence of system (6.1)'s ISS Lyapunov function. The ISS Lyapunov function is defined as follows.

Definition 2.4

Dashkovskiy and Rüffer (2010). A function V  : D _LISS   ⟶ R _{≥   0} is system (6.1)'s LISS-Lyapunov function if $\mathcal{K}$ functions φ ₁, φ ₂, φ ₃, and χ exist, such that for any x  ∈ D _LISS , u  ∈ U _LISS , the following inequalities hold

(6.5a) ${\phi }_1\left(\left|x\right|\right)\le V\left(x\right)\le {\phi }_2\left(\left|x\right|\right)$

(6.5b) $V\left(x\right)\ge \chi \left(\left|u\right|\right)\Rightarrow \nabla V\left(x\right)·f\left(x,u\right)\le -{\phi }_3\left(\left|x\right|\right)$

where D _LISS   ⊆ R ⁿ , U _LISS   ⊂ R ^m , D _LISS , and U _LISS are local regions containing x  =   0 and u  =   0.

Remark 2.1

Fig. 6.2 presents the geometric meaning of LISS. It can be seen from this figure that if the x ₀ and ‖u‖_∞ are within the given LISS region, the system's state trajectory will converge into a sphere with the radius of γ _LISS (‖u‖_∞) at last. The function β _LISS represents the transient process of the nonlinear system states.

Fig. 6.2. The geometric meaning of LISS.

Theorem 2.1

(Qin et al., 2016a,b). System (6.1) is LISS when it has a LISS-Lyapunov function in the region D _LISS   ⊆ R ⁿ and U _LISS   ⊂ R ^m , where U _LISS is a LISS region of u. Furthermore, for a given scalar c, if there exists a level set Ω   =   {x  ∈ R ^N   |   V(x)   ≤ c}, where Ω   ⊂ D _LISS , then Ω is a local region of initial states for which the LISS conditions hold for system (6.1).

Since input signal u has the property of time-varying, checking the previous LISS conditions is difficult. This chapter gives the following theorems to estimate the local regions of initial states and external inputs and the asymptotic gain for system (6.1).

Corollary 2.1

(Qin et al., 2016a,b). A smooth function V  : D _LISS   ⟶ R _{≥   0} is an LISS-Lyapunov function when it satisfies the following conditions:

(1)

V(x)   >   0 and V(0)   =   0 for all the x  ∈ D _LISS \{0}.

(2)

For all the x  ∈ D _LISS and u  ∈ U _LISS , a $\mathcal{K}$ function χ exists such that

(6.6) $V\left(x\right)\ge \chi \left({‖u‖}_{\infty }\right)⟹\nabla V·f+{‖u‖}_{\infty }·\left|\nabla V·g\right|<0$

where D _LISS   ⊆ R ⁿ and U _LISS   ⊂ R ^m . D _LISS and U _LISS are local regions containing the equilibrium point of system (6.1) with zero input.

Remark 2.2

Assume a 0-AS Lyapunov function for system (6.1) is V. Then ∇  V  · f  +‖u‖_∞  ·   |∇  V  · g  | is monotonically increasing with respect to ‖u‖_∞. If for a given 0-AS Lyapunov function V and ‖u‖_∞  = v, the condition (6.6) in Corollary 2.1 is satisfied, then a $\mathcal{K}$ function $\tilde{\chi }$ can always be found that for all the x  ∈ D _LISS and ‖u‖_∞  ≤ ν, the following condition holds

$V\left(x\right)\ge \tilde{\chi }\left({‖u‖}_{\infty }\right)⟹\nabla V·f+{‖u‖}_{\infty }·\left|\nabla V·g\right|$

Then, we can take the set U  =   {u(t)|   ‖u‖_∞  ≤ ν} to represent the local region of external inputs, where ν  >   0, and ‖u‖_∞ can be selected as a constant because u is bounded by ‖u‖_∞.

Theorem 2.2

If there exists a smooth function V that satisfies Corollary 2.1, the state trajectory of system (6.1) will converge into a region when t  →   ∞. Let P represent this convergence region. Then

(6.7) $\mathrm{P}=\left\{x\in D|\left|\nabla V·f\right|-{‖u‖}_{\infty }·\left|\nabla V·g\right|\le 0\right\}$

Similar to the proof presented in (Qin et al., 2016a,b), the proof details are not given in this chapter for the sake of brevity.

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A three-step Taylor-Galerkin finite element method for orographic rainfall

K. KASHIYAMA , ... T. YAMADA , in Computational Wind Engineering 1, 1993

3.1 Three-Step Taylor-Galerkin Method

Denoting a continuous function at time t=t as F(t), the value at time t=t+Δt can be expressed using Taylor series expansion;

(12) $\text{F}\left(\text{t}+\text{Δ}\text{t}\right)=\text{F}\left(\text{t}\right)+\text{Δ}\text{t}\frac{\partial \text{F}\left(\text{t}\right)}{\partial \text{t}}+\frac{\text{Δ}{\text{t}}^2}{2}\frac{{\partial }^2\text{F}\left(\text{t}\right)}{\partial {\text{t}}^2}+\frac{\text{Δ}{\text{t}}^3}{6}\frac{{\partial }^3\text{F}\left(\text{t}\right)}{\partial {\text{t}}^3}+\text{O}\left(\text{Δ}{\text{t}}^4\right)$

where Δt is the time increment. Donea⁴ applied the equation (12) directly to the discretization in time and showed the efficiency of the method. In the method, a higher order term of the time derivative is replaced by the spatial derivative. On the other hand, by approximate equation (12) up to third-order accuracy, the following three-step scheme can be derived as:

(13) $\begin{array}{c}\text{F(t+}\frac{\text{Δt}}{3})=\text{F(t})+\frac{\text{Δ}\text{t}}{3}\frac{\partial \text{F(t)}}{\partial \text{t}}\\ \text{F(t+}\frac{\text{Δt}}{2})=\text{F(t)+}\frac{\text{Δt}}{2}\frac{\partial \text{F(t+}\frac{\text{Δ}\text{t}}{3}\text{)}}{\partial \text{t}}\\ \text{F(t+}\text{Δt})=\text{F(t})+\text{Δ}\text{t}\frac{\partial \text{F(t+}\frac{\text{Δ}\text{t}}{2}\text{)}}{\partial \text{t}}\end{array}$

The equations (13) are equivalent to the equation (12) and the method is referred to as the three-step Taylor-Galerkin method⁵. Applying the standard Galerkin finite element method, the finite element equation can be obtained.

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Sequences and series

Mary Attenborough , in Mathematics for Electrical Engineering and Computing, 2003

Example 12.5

A triangular wave of period 2 is given by the function

$\begin{array}{ll}f(t)=t& 0\le t<1\\ f(t)=2-t& 1\le t<2.\end{array}$

Draw a graph of the function and give a sequence of values for t ≥ 0 at a sampling interval of 0.1.

Solution To draw the continuous function, use the definition y = t between t = 0 and 1 and draw the function y = 2 − t in the region where t lies between 1 and 2. The function is of period 2 so that section of the graph is repeated between t = 2 and 4, t = 4 and 6, etc.

The sequence of values found by using a sampling interval of 0.1 is given by substituting t = Tn = 0.1n into the function definition, giving

$\begin{array}{ll}a(n)=0.1n& 0\le 0.1n<1(\text{for}n\text{between}0\text{\hspace{0.17em}and}10)\\ a(n)=2-0.1n& 1\le 0.1n<2(\text{for}n\text{between}10\text{\hspace{0.17em}and}20).\end{array}$

The sequence then repeats periodically.

This gives the sequence:

$0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1,0.9,0.8,0.7,0.6,0.5,0.4,0.3,0.2,0.1,0,0.1,0.2,...$

The continuous function is plotted in Figure 12.4(a) and the digital function in Figure 12.4(b).

Figure 12.4. (a) A triangular wave of period 2 given by f(t) = t, 0 &lt; t ≤ 1, f(t) = 2 − t, 1 &lt; t &lt; 2. (b) The function sampled at a sampling interval of 0.1.

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Mathematical Background

Dimitrios G. Pavlou PhD , in Essentials of the Finite Element Method, 2015

2.4.1 Definition of the Variation of a Function

Let us assume a continuous function u(x) (e.g., the axial displacement of a bar due to axial loads). For simplicity, we have chosen a polynomial function

(2.95) $u\left(x\right)=a_0+a_{1}x+a_2{x}^2+\cdots +a_n{x}^n$

For infinitesimal changes of the above coefficients, a ₀, a ₁, a ₂,   …, a _n, namely, δa ₀, δa ₁, δa ₂,   …, δa _n , a closely related function is written as follows:

(2.96) $\overline{u}\left(x\right)=\left(a_0+\delta a_0\right)+\left(a_1+\delta a_1\right)x+\left(a_2+\delta a_2\right)x^2+\cdots +\left(a_n+\delta a_n\right)x^n$

The variation δu(x) of u(x) is defined as the following difference

(2.97) $\delta u\left(x\right)=\overline{u}\left(x\right)-u\left(x\right)=\delta a_0+\delta a_{1}x+\delta a_2{x}^2+\cdots +\delta a_n{x}^n$

Very often, apart from the variation of a function we need to calculate the variation of a functional (i.e., function of a function) U(u(x)):

(2.98) $\delta U=U\left(\overline{u}\left(x\right)\right)-U\left(u\left(x\right)\right)$

The calculation of the variation of a functional can be simplified by using the following properties of variations.

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Inverse Kinematics of Dextrous Manipulators

David DeMers , Kenneth Kreutz-Delgado , in Neural Systems for Robotics, 1997

Definition 4.1

[51] A continuous function g: W _i → Θ such that ∀x ∈ W _i f [g (x)] = x is called an inverse kinematic function for f on $W_i\subset W$ . An invertible workspace is any subset $W_i\subset W$ for which there exists an inverse function.

An inverse kinematic function is just a right inverse of f. We call the maximally sized singularity-free subsets of W the w-sheets, following the terminology of [7]; the w-sheets, henceforth denoted by W _i, are the connected regions of w bounded by Jacobian surfaces. The w-sheets are invertible in the sense of Definition 4.1.

An inverse function specifies one of the joint configurations which solves Equation (4.1) for each specified end-effector position x. An obvious approach to developing an inverse kinematic function is to fix n – m of the n joint variables, resulting in a nonredundant invertible mapping on the w-sheets. Varying the remaining m variables results in inverse solution sets which form n – m dimensional manifolds in the configuration space. This inverse kinematic mapping is a local diffeomorphism, thus the solution sets are topologically equivalent to the topological structure of the n – m variables. If the joints are rotational, in which case each joint variable can be represented as an angle (equivalently, a point on the circle S ¹), then this structure is the n-torus, Tⁿ = S ¹ × … × S ¹ [7], which is known as the fiber above the point x.

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RAZUMIKHIN TYPE THEOREM FOR DIFFERENTIAL EQUATIONS WITH INFINITE DELAY*

Junji Kato , in Dynamical Systems, 1977

Theorem A

Suppose that there exists a continuous function V(t,ϕ) defined on (-∞,∞) × C such that

$\text{a}\left(\left|\text{ϕ}\left(0\right)\right|\right)\leqq \text{V}\left(\text{t},\text{ϕ}\right)\leqq \text{b}\left(\Vert \text{ϕ}\Vert \right)$

for continuous positive-definite functions a(r) and b(r) and that for a continuous function c(t,r) ≧ 0

(2) $\dot{\text{v}}(\text{t},{\text{x}}_{\text{t}})\leqq -\text{c}(\text{t},\text{v}(\text{t},{\text{x}}_{\text{t}}))$

along any solution x(t) of (1). Then, the zero solution of (1) is uniformly asymptotically stable, if for any r > 0

${\displaystyle {\int }_{\text{t}}^{\text{t}+\text{T}}\text{c}\left(\text{s},\text{r}\right)\text{ds}\to \text{∞}\text{as}\text{T}\to \text{∞}\text{uniformly}\text{in}\text{t}\geqq 0.}$

Since the solutions become more restrictive as the time elapses, the following theorem is expected to be more effectively (refer [3] and [4]).

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Subband Coding and Wavelet Transform

Anke Meyer-Baese , Volker Schmid , in Pattern Recognition and Signal Analysis in Medical Imaging (Second Edition), 2014

3.3.2 The Continuous Wavelet Transform

The CWT transforms a continuous function into a highly redundant function of two continuous variables, translation and scale. The resulting transformation is important for time-frequency analysis and is easy to interpret.

The CWT is defined as the mapping of the function $f(t)$ on the timescale space by

(3.46) $W_f(a\text{,}b)={\int }_{-\infty }^{\infty }{\psi }_{\mathit{ab}}(t)f(t)\mathit{dt}=〈{\psi }_{\mathit{ab}}(t)\text{,}f(t)〉$

The CWT is invertible if and only if the resolution of identity holds:

(3.47) $f(t)={\displaystyle \underset{\text{summation}}{\underbrace{\frac{1}{C_{\psi }}{\int }_{-\infty }^{\infty }{\int }_0^{\infty }\frac{\mathit{dadb}}{a^2}}}}{\displaystyle \underset{\text{Wavelet coefficients}}{\underbrace{W_f(a\text{,}b)}}}{\displaystyle \underset{\mathrm{Wavelet}}{\underbrace{{\psi }_{\mathit{ab}}(t)}}}$

where

(3.48) $C_{\psi }={\int }_o^{\infty }\frac{\mid \mathrm{\Psi }(\omega ){\mid }^2}{\omega }d\omega$

assuming that a real-valued $\psi (t)$ fulfills the admissibility condition. If $C_{\psi }<\infty$ , then the wavelet is called admissible. Then we get for the DC gain

(3.49) $\mathrm{\Psi }(0)={\int }_{-\infty }^{\infty }\psi (t)\mathit{dt}=0$

We immediately see that $\psi (t)$ corresponds to the impulse response of a bandpass filter and has a decay rate of $\mid t{\mid }^{1-∊}$ . It is important to note that based on the admissibility condition, it can be shown that the CWT is complete if $W_f(a\text{,}b)$ is known for all $a\text{,}b$ .

The Mexican-hat wavelet

(3.50) $\psi (t)=\left(\frac{2}{\sqrt{3}}{\pi }^{-\frac{1}{4}}\right)(1-t^2)e^{-\frac{t^2}{2}}$

is visualized in Fig. 3.16. It has a distinctive symmetric shape, and it has an average value of zero and dies out rapidly as $\mid t\mid \to \infty$ . There is no scaling function associated with the Mexican-hat wavelet.

Figure 3.16. Mexican-hat wavelet.

Figure 3.17 illustrates the multiscale coefficients describing a spiculated mass. Figure 3.17a shows the scanline through a mammographic image with a mass (8   mm) while Fig. 3.17b visualizes the multiscale coefficients at various levels.

Figure 3.17. Continuous wavelet transform: (a) scan line, (b) multiscale coefficients.

Images courtesy of Dr. A. Laine, Columbia University.

The short-time Fourier transform finds a decomposition of a signal into a set of equal-bandwidth functions across the frequency spectrum. The WT provides a decomposition of a signal based on a set of bandpass functions that are placed over the entire spectrum. The WT can be seen as a signal decomposition based on a set of constant-Q bandpasses. In other words, we have an octave decomposition, logarithmic decomposition, or constant-Q decomposition on the frequency scale. The bandwidth of each of the filters in the bank is the same in a logarithmic scale, or equivalently, the ratio of the filters' bandwidth to the respective central frequency is constant.

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