- Backward Finite Differences
- Centered Finite Differences
- The Bilinear Transform
- Finite-Differences vs. the Bilinear Transform
- Impulse Invariance
- Boundary Elements and Finite Elements

The Finite Difference technique replaces derivative expressions in differential equations with discrete-time difference approximations.

- The backward finite difference
(FD) approximation is given by:

where*T*is the sampling period. - In the frequency-domain, the FD approximation is defined by the mapping:

where*s*is the Laplace Transform frequency variable and*z*is the*z*-Transform frequency variable. - The inverse finite difference substitution is given by:

- The FD approximation maps analog dc (
*s*=0) to digital dc (*z*=1). - By noting that the FD approximation maps an infinite analog frequency () to
*z*=0, it should be clear that non-zero poles and zeros are warped in potentially undesireable ways. - The FD approximation does not alias because the conformal mapping
*s*= 1 -*z*^{-1}is one to one. - By applying the FD twice, the second derivative is found as:

- The Matlab example, msd_fd.m, demonstrates the use of the finite difference approach to simulate the motion of the mass-spring-damper system.

- Assuming that one sample of ``look-ahead'' is available, another finite difference scheme can be defined as:

and

- These equations represent zero phase filters, producing no delay at any frequency.
- Note, however, that the second derivative approximation is not formed by applying the first derivative approximation to itself. Thus, the
*s*- to*z*-plane mapping is different for the two approximations. Nor is it straight-forward to evaluate the*s*- to*z*-plane mapping in either case. - For the first derivative approximation, we can find:

- For frequencies above
(or frequencies above ), the resulting values of
*z*are outside the unit circle, and thus unstable. - Thus, while the centered finite difference approximation has desirable qualities, it is limited by frequency range considerations.

- The bilinear transform is an algebraic transformation between the continuous-time and discrete-time frequency variables
*s*and*z*, respectively. It is therefore appropriate only when a closed-form filter representation in*s*exists. - The bilinear transform
is defined by the substitution:

where*T*is the sampling period. - The bilinear transform maps the entire continuous-time frequency space,
onto the discrete-time frequency space
Continuous-time dc () maps to discrete-time dc () and infinite continuous-time frequency (
) to the Nyquist frequency (). Thus, a nonlinear warping of the frequency axes occurs.
- Since the -axis in the
*s*plane is mapped exactly once around the unit circle of the*z*plane, no aliasing occurs. - The constant
*c*is typically given by*c*= 2/*T*. This parameter provides one degree of freedom for the mapping of a particular finite continuous-time frequency to a particular location on the*z*-plane unit circle. The warping of frequencies is given for the discrete-time frequency in terms of*c*and as

- If one wishes to map a specific ``analog'' frequency () to the equivalent ``digital'' frequency (), the constant
*c*is found as

- Because of the nonlinear compression of the frequency axis, use of the bilinear transformation should be limited to filters with a single transition band (lowpass and highpass) or resonance (narrow bandpass and bandstop).
- The Matlab example, msd_fdbt.m, compares the results of using both the finite difference and bilinear transform approaches to simulate the motion of a mass-spring-damper system.

- The FD approximation is given by the mapping:

- The Bilinear transform is given by the mapping:

- The
*s*- to*z*-plane mapping for both the finite difference and bilinear transform is shown in Fig. 2.4. Note that the bilinear transform maps the axis exactly onto the unit circle. - Both methods preserve order, stability, and are free from aliasing. Both methods provide an ideal frequency mapping at zero frequency but compressively warp higher frequencies.
- Only the finitie difference method introduces artificial damping at higher frequencies. Because of this, it is even possible that unstable
*s*-plane poles could be mapped to stable*z*-plane poles.

- In some instances, we may have a continuous-time impulse response characterization of a system that we wish to model. The Impulse Invariance method offers a simple means for discretizing this representation in the form of a finite impulse response (FIR) filter. This method, however, is susceptible to aliasing.
- If given a continuous-time impulse response
*h*_{c}(*t*), the impulse invariance method defines a discrete-time approximation of the form:

*h*[*n*] =*T h*_{c}(*n T*),

where*T*is the sampling period. - The frequency response of the discrete-time system obtained in this way is related to the frequency response of the continuous-time system by:

- If for (the system is bandlimited to the discrete-time Nyquist frequency), this method provides a good discrete-time system approximation. If this is not the case, however, aliasing will occur.

- The Boundary Element Method (BEM) is a numerical technique for solving linear partial differential equations that have been formulated as integral equations.
- The BEM technique involves ``meshing'' the surface of an object.
- Because these equations are only solved over the surface of an object, BEM is typically much more efficient than techniques that involve meshing the entire volume of an object.
- The Finite Element Method (FEM) is another numerical technique for finding numerical approximations to solutions of partial differential and integral equations.
- With FEM, the entire volume of an object is ``meshed''.
- Many commercial software systems are available for evaluating the vibratory behavior of systems using BEM and FEM techniques.

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