# Discretization Methods:

There are a variety of approaches to solving continuous-time system equations using discrete-time methods. In general, a higher sampling rate will produce more accurate results. In this section, we overview several methods, including Finite Differences and the Bilinear Transform.

## Backward Finite Differences

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.

## Centered Finite Differences

• 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

• 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.

## Finite-Differences vs. the Bilinear Transform

• 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.

## Impulse Invariance

• 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 hc(t), the impulse invariance method defines a discrete-time approximation of the form:

h[n] = T hc(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.

## Boundary Elements and Finite Elements

• 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.