Skip to content Skip to navigation

OpenStax-CNX

You are here: Home » Content » Conversion of Analog to Digital Transfer Functions

Navigation

Lenses

What is a lens?

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

This content is ...

Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
  • NSF Partnership display tagshide tags

    This module is included inLens: NSF Partnership in Signal Processing
    By: Sidney BurrusAs a part of collection: "Digital Signal Processing and Digital Filter Design (Draft)"

    Click the "NSF Partnership" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

  • Featured Content

    This module is included inLens: Connexions Featured Content
    By: ConnexionsAs a part of collection: "Digital Signal Processing and Digital Filter Design (Draft)"

    Click the "Featured Content" link to see all content affiliated with them.

Also in these lenses

  • UniqU content

    This module is included inLens: UniqU's lens
    By: UniqU, LLCAs a part of collection: "Digital Signal Processing and Digital Filter Design (Draft)"

    Click the "UniqU content" link to see all content selected in this lens.

  • Lens for Engineering

    This module is included inLens: Lens for Engineering
    By: Sidney Burrus

    Click the "Lens for Engineering" link to see all content selected in this lens.

Recently Viewed

This feature requires Javascript to be enabled.

Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.
 

Conversion of Analog to Digital Transfer Functions

Module by: C. Sidney Burrus. E-mail the author

For mathematical convenience, the four classical IIR filter transfer functions were developed in terms of the Laplace transform rather than the z-transform. The prototype Laplace-transform transfer functions are descriptions of analog filters. In this section they are converted to z-transform transfer functions for implementation as IIR digital filters.

There have been several different methods of converting analog systems to digital described over the history of digital filters. Two have proven to be useful for most applications. The first is called the impulse-invariant method and results in a digital filter with an impulse response exactly equal to samples of the prototype analog filter. The second method uses a frequency mapping to convert the analog filter to a digital filter. It has the desirable property of preserving the optimality of the four classical approximations developed in the last section. This section will develop the theory and design formulas to implement both of these conversion approaches.

The Impulse-Invariant Method

Although the transfer functions in Continuous Frequency Definition of Error were designed with criteria in the frequency domain, the impulse-invariant method will convert them into digital transfer functions using a time-domain constraint [7], [5], [8]. The digital filter designed by the impulse-invariant method is required to have an impulse response that is exactly equal to equally spaced samples of the impulse response of the prototype analog filter. If the analog filter has a transfer function F(s)F(s) with an impulse response f(t)f(t), the impulse response of the digital filter h(n)h(n) is required to match the samples of f(t)f(t). For samples at TT second intervals, the impulse response is

h ( n ) = F ( T ) | t = T n = F ( T n ) h ( n ) = F ( T ) | t = T n = F ( T n )
(1)

The transfer function of the digital filter is the z-transform of the impulse response of the filter, which is given by

H ( z ) = n = 0 h ( n ) z - n H ( z ) = n = 0 h ( n ) z - n
(2)

The transfer function of the prototype analog filter is always a rational function written as

F ( s ) = B ( s ) A ( s ) F ( s ) = B ( s ) A ( s )
(3)

where B(s)B(s) is the numerator polynomial with roots that are the zeros of F(s)F(s), and A(s)A(s) is the denominator with roots that are the poles of F(s)F(s). If F(s)F(s) is expanded in terms of partial fractions, it can be written as

F ( s ) = i = 1 N K i s + s i F ( s ) = i = 1 N K i s + s i
(4)

The impulse response of this filter is the inverse-Laplace transform of Equation 4, which is

f ( t ) = i = 1 N K e s i t f ( t ) = i = 1 N K e s i t
(5)

Sampling this impulse response every TT seconds gives

f ( n T ) = i = 1 N K i e - s i n T = i = 1 N K i ( e - s i T ) n f ( n T ) = i = 1 N K i e - s i n T = i = 1 N K i ( e - s i T ) n
(6)

The basic requirement of Equation 1 gives

H ( z ) = n = 0 [ i = 1 N K i ( e - s I T ) n ] H ( z ) = n = 0 [ i = 1 N K i ( e - s I T ) n ]
(7)
H ( z ) = i = 1 N K i z z - e s I T H ( z ) = i = 1 N K i z z - e s I T
(8)

which is clearly a rational function of zz and is the transfer function of the digital filter, which has samples of the prototype analog filter as its impulse response.

This method has its requirements set in the time domain, but the frequency response is important. In most cases, the prototype analog filter is one of the classical types, which is optimal in the frequency domain. If the frequency response of the analog filter is denoted by F(jω)F(jω) and the frequency response of the digital filter designed by the impulse- invariant method is H(ω)H(ω), it can be shown in a development similar to that used for the sampling theorem

H ( ω ) = ( 1 / T ) k = - F ( j ( ω - 2 π k / T ) ) H ( ω ) = ( 1 / T ) k = - F ( j ( ω - 2 π k / T ) )
(9)

The frequency response of the digital filter is a periodically repeated version of the frequency response of the analog filter. This results in an overlapping of the analog response, thus not preserving optimality in the same sense the analog prototype was optimal. It is a similar phenomenon to the aliasing that occurs when sampling a continuous-time signal to obtain a digital signal in A-to-D conversion. If F(jω)F(jω) is an analog lowpass filter that goes to zero as ωω goes to infinity, the effects of the folding can be made small by high sampling rates (small T).

The impulse-invariant design method can be summarized in the following steps:

  1. Design a prototype analog filter with transfer function F(jω)F(jω).
  2. Make a partial fraction expansion of F(jω)F(jω) to obtain the NN values for KiKi and sisi.
  3. Form the digital transfer function H(z)H(z) from Equation 8 to give the desired design.

The characteristics of the designed filter are the following:

  • It has N poles, the same as the analog filter.
  • It is stable if the analog filter was stable. This is seen from the change of variables in the denominator of (6.70) which maps the left-half s-plane inside the unit circle in the z-plane.
  • The frequency response is a folded version of the analog filter, and the optimal properties of the analog filter are not preserved.
  • The cascade of two impulse-invariant designed filters are not impulse-invariant with the cascade of the two analog prototypes. In other words, the filter must be designed in one step.

This method is sometimes used to design digital filters, but because the relation of the analog and digital system is specified in the time domain, it is more useful in designing a digital simulation of an analog system. Unfortunately, the properties of this class of filters depend on the input. If a filter is designed so that its impulse response is the sampled impulse response of the analog filter, its step response will not be the sampled step response of the analog filter.

A step-invariant filter can be designed by first multiplying the analog filter transfer function F(s)F(s) by 1/s1/s, which is the Laplace transform of a step function. This product is then expanded in partial fraction just as F(s)F(s) was in Equation 4 and the same substitution made as in Equation 8 giving a z-transform. After the z-transform of a step is removed, the digital filter has the step-invariant property. This idea can be extended to other input functions, but the impulse-invariant version is the most common. Another modification to the impulse-invariant method is known as the matched z transform covered in [7], but it is less useful.

There can be a problem with the classical impulse-invariant method when the number of finite zeros is too large. This is addressed in [3], [4].

An example of a Butterworth lowpass filter used to design a digital filter by the impulse-invariant method can be shown. Note that the frequency response does not go to zero at the highest frequency of w=pw=p. It can be made as small as desired by increasing the sampling rate, but this is more expensive to implement. Because the frequency response of the prototype analog filter for an inverse-Chebyshev or elliptic-function filter does not necessarily go to zero as w goes to infinity, the effects of folding on the digital frequency response are poor. No amount of sampling rate increase will change this. The same problem exists for a highpass filter. This shows the care that must be exercised in using the impulse-invariant design method.

The Bilinear Transformation

A second method for converting an analog prototype filter into a desired digital filter is the bilinear transformation. This method is entirely a frequency-domain method, and as a result, some of the optimal properties of the analog filter are preserved. As was the case with the impulse-invariant method, the time interval is not normalized to one, but is explicitly denoted by the sampling interval TT with units of seconds. The bilinear transformation is a change of variables (a mapping) that is linear in both the numerator and denominator [7], [5], [2], [6]. The usual form is

s = 2 T z - 1 z + 1 s = 2 T z - 1 z + 1
(10)

The zz-transform transfer function of the digital filter H(z)H(z) is obtained from the Laplace transform transfer function F(s)F(s) of the prototype filter by substituting for ss the bilinear form of Equation 10.

H ( z ) = F ( 2 ( z - 1 ) T ( z + 1 ) ) H ( z ) = F ( 2 ( z - 1 ) T ( z + 1 ) )
(11)

This operation can be reversed by solving Equation 10 for z and substituting this into H(z)H(z) to obtain F(s)F(s). This reverse operation is also bilinear of the form

z = 2 / T + s 2 / T - s z = 2 / T + s 2 / T - s
(12)

To consider the frequency response, the Laplace variable s is evaluated on the imaginary axis and the z-transform variable z is evaluated on the unit circle. This is achieved by

s = j u and z = e j ω T s = j u and z = e j ω T
(13)

which gives the relation of the analog frequency variable u to the digital frequency variable ωω from Equation 13 and Equation 10 to be

u = ( 2 / T ) tan ( ( ω T ) / 2 ) u = ( 2 / T ) tan ( ( ω T ) / 2 )
(14)

The bilinear transform maps the infinite imaginary axis in the-s plane onto the unit circle in the zz-plane. It maps the infinite interval of -<u<-<u< of the analog frequency axis on to the finite interval of -π/2<ω<π/2-π/2<ω<π/2 of the digital frequency axis. This is illustrated in Figure 1.

Figure 1: The Frequency Map of the Bilinear Transform
Figure one is a graph titled, bilinear transform from analog to digital. The horizontal axis is labeled frequency, ω, and ranges in value from 0 to 6 in increments of 1. The vertical axis is unlabeled, but it ranges in value from 0 to 5 in increments of 1. From (0, 4) to (1.1, 4), and then from (1.1,4) to (1.1, 3) are two blue line segments. From (0, 3) to (5, 3) is a horizontal blue line segment. Above this line segment is a box containing the phrase Analog Prototype. At the end of the line segment to the right is a small zig-zag. Below the long line segment is a box containing the phrase Digital Filter. There is an arrow drawn vertically downward from the Analog prototype box pointing to the digital filter box. From (0, 1) to (1, 1) and then from  (1, 1) to (1, 0) is a horizontal and vertical blue line segment. There is an arrow drawn pointing downward from (1.1, 3) to (1, 1). There is a blue horizontal segment drawn from (0, 0) to (4, 0), and another two segments drawn from (3, 0) to (3, 1) and (3, 1) to (4, 1). Pointing at (2, 0) is an arrow labeled π/T, and pointing at (4, 0) is an arrow labeled 2π/T. Two more arrows pointing downward reach across the figure from (6, 3) to (2.2, 1) and from (2.2, 1) to (2, 0).

There is no folding or aliasing of the prototype frequency response, but there is a compression of the frequency axis, which becomes extreme at high frequencies. This is shown in Figure 2 from the relation of Equation 14.

Figure 2: The Frequency Mapping of the Bilinear Transform
Figure two is a graph titled, Frequency Warping. The horizontal axis is labeled Digital frequency, ω, and ranges in value from 0 to 3 in increments of 0.5. The vertical axis is labeled Analog frequency, and ranges in value from 0 to 7 in increments of 1. There is one blue curve on the graph, beginning at (0, 0) and increasing slowly at first, and further across the page to the right, it begins increasing at an increasing rate, until it terminates with a nearly vertical slope at approximately (2.6, 7).

Near zero frequency, the relation of uu and ωω is essentially linear. The compression increases as the digital frequency w nears π/2π/2. This nonlinear compression is called frequency warping. The conversion of F(s)F(s) to H(z)H(z) with the bilinear transformation does not change the values of the frequency response, but it changes the frequencies where the values occur.

In the design of a digital filter, the effects of the frequency warping must be taken into account. The prototype filter frequency scale must be prewarped so that after the bilinear transform, the critical frequencies are in the correct places. This prewarping or scaling of the prototype frequency scale is done by replacing s with Ks. Because the bilinear transform is also a change of variables, both can be performed in one step if that is desirable.

If the critical frequency for the prototype filter is uouo and the desired critical frequency for the digital filter is ωoωo, the two frequency responses are related by

F ( j u 0 ) = H ( ω 0 ) = F * F ( j u 0 ) = H ( ω 0 ) = F *
(15)

The prewarping scaling is given by

u 0 = 2 T tan ( ω 0 T 2 ) u 0 = 2 T tan ( ω 0 T 2 )
(16)

Combining the prewarping scale and the bilinear transformation give

u 0 = 2 K T tan ( ω 0 T 2 ) u 0 = 2 K T tan ( ω 0 T 2 )
(17)

Solving for KK and combining with Equation 10 give

s = u 0 tan ( ω 0 T / 2 ) z - 1 z + 1 s = u 0 tan ( ω 0 T / 2 ) z - 1 z + 1
(18)

All of the optimal filters developed in Continuous Frequency Definition of Error and most other prototype filters are designed with a normalized critical frequency of u0=1u0=1. Recall that ω0ω0 is in radians per second. Most specifications are given in terms of frequency ff in Hertz (cycles per second) which is related to ωω or uu by

ω = 2 π f ω = 2 π f
(19)

Care must be taken with the elliptic-function filter where there are two critical frequencies that determine the transition region. Both frequencies must be prewarped.

The characteristics of the bilinear transform are the following:

  • The order of the digital filter is the same as the prototype filter.
  • The left-half s-plane is mapped into the unit circle on the z-plane. This means stability is preserved.
  • Optimal approximations to piecewise constant prototype filters, such as the four cases in Continuous Frequency Definition of Error, transform into optimal digital filters.
  • The cascade of sections designed by the bilinear transform is the same as obtained by transforming the total system.

The bilinear transform is probably the most used method of converting a prototype Laplace transform transfer function into a digital transfer function. It is the one used in most popular filter design programs [1], because of characteristic 3 above that states optimality is preserved. The maximally flat prototype is transformed into a maximally flat digital filter. This property only holds for approximations to piecewise constant ideal frequency responses, because the frequency warping does not change the shape of a constant. If the prototype is an optimal approximation to a differentiator or to a linear-phase characteristic, the bilinear transform will destroy the optimality. Those approximations have to be made directly in the digital frequency domain.

Example 1: The Bilinear Transformation

To illustrate the bilinear transformation, the third-order Butterworth lowpass filter designed in the Example is converted into a digital filter. The prototype filter transfer function is

F ( s ) = 1 ( s + 1 ) ( s 2 + s + 1 ) F ( s ) = 1 ( s + 1 ) ( s 2 + s + 1 )
(20)

The prototype analog filter has a passband edge at u0=1u0=1. A data rate of 1000 samples per second corresponding to T=0.001T=0.001 seconds is assumed. If the desired digital passband edge is f0=200f0=200 Hz, then ω0=(2π)(200)ω0=(2π)(200) radians per second, and the total prewarped bilinear transformation from Equation 18 is

s = 1 . 376382 z - 1 z + 1 s = 1 . 376382 z - 1 z + 1
(21)

The digital transfer function in Equation 20 becomes

H ( z ) = 0 . 09853116 ( z + 1 ) 3 ( z - 0 . 158384 ) ( z 2 - 0 . 418856 z + 0 . 355447 ) H ( z ) = 0 . 09853116 ( z + 1 ) 3 ( z - 0 . 158384 ) ( z 2 - 0 . 418856 z + 0 . 355447 )
(22)

Note the locations of the poles and zeros in the z-plane. Zeros at infinity in the s-plane always map into the z = -1 point. The example illustrate a third-order elliptic-function filter designed using the bilinear transform.

Frequency Transformations

For the design of highpass, bandpass, and band reject filters, a particularly powerful combination consists of using the frequency transformations described in Section elsewhere together with the bilinear transformation. When using this combination, some care must be taken in scaling the specifications properly. This is illustrated by considering the steps in the design of a bandpass filter:

  1. First, the lower and upper digital bandedge frequencies are specified as ω1ω1 and or ω1ω1, ω2ω2, ω3ω3, and ω4ω4 if an elliptic-function approximation is used.
  2. These frequencies are prewarped using Equation 16 to give theband edges of the prototype bandpass analog filter.
  3. These frequencies are converted into a single band- edge ωpωp or ωsωs for the Butterworth and Chebyshev and into ωpωp and ωsωs for the elliptic-function approximation of the prototype lowpass filter by using Equation 2 from Frequency Transformations and Equation 4 from Frequency Transformations.
  4. The lowpass filter is designed for this ωpωp and/or ωsωs by using one of the four approximations in the sections in Continuous Frequency Definition of Error or some other method.
  5. This lowpass analog filter is converted into a bandpass analog filter with the frequency transformation Equation 6 from Frequency Transformations.
  6. The bandpass analog filter is then transformed into the desired bandpass digital filter using the bilinear transformation Equation 10.

This is the procedure used in the design Program 8 in the appendix.

When designing a bandpass elliptic-function filter, four frequencies must be specified: the lower stopband edge, the lower passband edge, the upper passband edge, and the upper stopband edge. All four must be prewarped to the equivalent analog values. A problem occurs when the two transition bands of the bandpass filter are converted into the single transition band of the lowpass prototype filter. In general they will be inconsistant; therefore, the narrower of the two transition bands should be used to specify the lowpass filter. The same problem occurs in designing a bandreject elliptic-function filter. Program 8 in the appendix should be studied to understand how this is carried out.

An alternative to the process of converting a lowpass analog into a bandpass analog filter which is then converted into a digital filter, is to first convert the prototype lowpass analog filter into a lowpass digital filter and then make the conversion into a bandpass filter. If the prototype digital filter transfer function is Hp(z)Hp(z) and the frequency transformation is f(z)f(z), the desired transformed digital filter is described by

H ( z ) = H p ( f ( z ) ) H ( z ) = H p ( f ( z ) )
(23)

Since the frequency response of both H(z)H(z) and Hp(z)Hp(z) is obtained by evaluating them on the unit circle in the-zz plane, f(z)f(z) should map the unit circle onto the unit circle (|z|=1=>|f(z)|=1)(|z|=1=>|f(z)|=1). Both H(z)H(z) and Hp(z)Hp(z) should be stable; therefore, f(z)f(z) should map the interior of the unit circle into the interior of the unit circle (|z|<1=>|f(z)|<1)(|z|<1=>|f(z)|<1). If f(z)f(z) were viewed as a filter, it would be an “all-pass" filter with a unity magnitude frequency response of the form

f ( z ) = p ( z ) z n p ( 1 / z ) = a n z n + a n - 1 z n - 1 + + a 0 a 0 z n + a 1 z n - 1 + + a n f ( z ) = p ( z ) z n p ( 1 / z ) = a n z n + a n - 1 z n - 1 + + a 0 a 0 z n + a 1 z n - 1 + + a n
(24)

The prototype digital lowpass filter is usually designed with bandedges at ±π/2±π/2. Determining the frequency transformation then becomes the problem of solving the n+1n+1 equations

f ( e j ω i ) = e ± j π / 2 = ( - 1 ) i j f ( e j ω i ) = e ± j π / 2 = ( - 1 ) i j
(25)

for the unknown akak where i=0,1,2,ni=0,1,2,n and the ωiωi are the bandedges of the desired transformed frequency response put in ascending order. The resulting simultaneous equations have a special structure that allow a recursive solution. Details of this approach can be found in [6].

This is an extremely general approach that allows multiple passbands of arbitrary width. If elliptic-function approximations are used, only one of the transition bandwidths can be independently specified. If more than one passband or rejectband is desired, f(z) will be higher order than second order and, therefore, the transformed transfer function H(f(z))H(f(z)) will have to be factored using a root finder.

To illustrate the results of using transform methods to design filters, three examples are given which are designed with Program 8 from the appendix.

Example 2: Design of an Chebyshev Highpass Filter

The specifications are given for a highpass Chebyshev frequency response with a passband edge at fp=0.3fp=0.3 Hertz with a sampling rate of one sample per second. The order is set at N=5N=5 and the passband ripple at 0.91515 dB. The transfer function is

H ( z ) = ( z - 1 ) ( z 2 - 2 z + 1 ) ( z 2 - 2 z + 1 ) ( z + 0 . 64334 ) ( z 2 + 0 . 97495 z + 0 . 55567 ) ( z 2 + 0 . 57327 z + 0 . 83827 ) H ( z ) = ( z - 1 ) ( z 2 - 2 z + 1 ) ( z 2 - 2 z + 1 ) ( z + 0 . 64334 ) ( z 2 + 0 . 97495 z + 0 . 55567 ) ( z 2 + 0 . 57327 z + 0 . 83827 )
(26)

The frequency response plot is given in Figure 3.

Figure 3: Fifth Order Digital Chebyshev Highpass Filter
Figure three is titled chebyshev highpass filter. Its horizontal axis is labeled Normalized frequency, and ranges in value from 0 to 1 in increments of 0.2. Its vertical axis is labeled Magnitude Response and ranges in value from 0 to 1 in increments of 0.2. There are two blue curves on the graph. The first is a simple horizontal line extending from (0, 0) to (1, 0). The second stretches out from this horizontal axis at approximately (0.3, 0) and begins increasing at an increasing rate all the way up to nearly the top of the page at approximately (0.6, 1). At this point the curve peaks and continues downward sharply, then back up to another peak more shallowly that occurs at approximately (0.75, 1), then back down and up in an even more shallow fashion to its termination point at (1, 1).

Example 3: Design of an Elliptic-Function Bandpass Filter

This filter requires a bandpass frequency response with an elliptic-function approximation. The maximum passpand ripple is one dB, the minimum stopband attenuation is 30 dB, the lower stopband edge f1=0.19f1=0.19, the lower passband edge f2=0.2f2=0.2, the upper passband edge f3=0.3f3=0.3, and the upper stopband edge f4=0.31f4=0.31 Hertz with a sampling rate of one sample per second. The design program calculated a required prototype order of N=6N=6 and, therefore, a total order of 10. The frequency response plot is shown in Figure 4.

Figure 4: Tenth Order Digital Elliptic Bandpass Filter
Figure four is titled tenth-order elliptic function filter. The horizontal axis is labeled normalized frequency and ranges in value from 0 to 1 in increments of 0.2. The vertical axis is labeled magnitude response and ranges in value from 0 to 1 in increments of 0.2. There are two blue curves on this graph. The first is a simple horizontal line from (0, 0) across the figure to (1, 0). The second is a more complex, but symmetric, curve that begins from the left with a very shallow positive slope. It decreases back down to the horizontal axis and wavers at (0.4, 0), where it then increases sharply to (0.4, 1). At this point, the graph peaks and follows a sinusoidal chape of an amplitude of approximately 0.05 with an increasing wavelength. There are two troughs and a wide peak in the middle of the graph. At this peak, the graph moves to the right in a completely symmetric fashion, mirroring the aforementioned description.

Example 4: Design of an Inverse-Chebyshev Bandreject Filter

The specifications are given for a bandreject Inverse- Chebyshev frequency response with bandedges at fs=0.1fs=0.1 and 0.20.2 Hertz with a sampling rate of one sample per second. The order is set at N=11N=11 and the minimum stopband attenuation at 30 dB. The frequency response plot is given in Figure 5.

Figure 5: Twenty second Order Digital Inverse Chebyshev Band Reject Filter
Figure five is a graph titled Inverse chebyshev band reject filter. The horizontal axis is labeled Normalized frequency, and ranges in value from 0 to 1 in increments of 0.2. The vertical axis is labeled Magnitude response and ranges in value from 0 to 1 in increments of 1. There are two blue curves on the graph. The first is a simple horizontal line from (0, 0) to (1, 0). The second begins at (0, 1) and extends to the right horizontally until just before (0.2, 1) where it sharply moves downward to (0.2, 0). At this point, the curve contains 10 peaks of a sinusoidal wave-shape of an extremely small amplitude. The wavelength of this section of the curve starts out small, then by the fifth  peak is its longest, then by the tenth peak is back to the wavelength of the first trough. This sinusoidal segment occurs horizontally from 0.2 to 0.4. After (0.4, 0), the curve returns sharply to the top of the figure at (0.4, 1), where it then continues horizontally to the right edge of the figure and terminates at (1, 1).

Summary

This section has described the two most popular and useful methods for transforming a prototype analog filter into a digital filter. The analog frequency variable is used because a literature on analog filter design exists, but more importantly, many approximation theories are more straightforward in terms of the Laplace-transform variable than the z-transform variable. The impulse-invariant method is particularly valuable when time- domain characteristics are important. The bilinear-transform method is the most common when frequency-domain performance is the main interest. Use of the BLT warps the frequency scale and, therefore, the digital band edges must be prewarped to obtain the necessary band edges for the analog filter design. Formulas that transform the analog prototype filters into the desired digital filters and for prewarping specifications were derived.

The use of frequency transformations to convert lowpass filters into highpass, bandpass, and bandreject filters was discussed as a particularly useful combination with the bilinear transformation. These are implemented in Program 8 and design examples from this program were shown.

There are cases where no analytic results are possible or where the desired frequency response is not piecewise constant and transformation methods are not appropriate. Direct methods for these cases are developed in other sections.

References

  1. DSP Committee, (Ed.). (1979). Programs for Digital Signal Processing. New York: IEEE Press.
  2. Gold, B. and Rader, C. M. (1969). Digital Processing of Signals. New York: McGraw-Hill.
  3. Jackson, L. B. (2000, October). A Correction to Impulse Invariance. IEEE Signal Processing Letters, 7(10), 273–275.
  4. Mecklenbrauker, W. F. G. (2000, August). Remarks on and Correction to the Impulse Invariant Method for the Design of IIR Digital Filters. Signal Processing, 80, 1687–1690.
  5. Oppenheim, A. V. and Schafer, R. W. (1999). Discrete-Time Signal Processing. (Second). [Earlier editions in 1975 and 1989]. Englewood Cliffs, NJ: Prentice-Hall.
  6. Parks, T. W. and Burrus, C. S. (1987). Digital Filter Design. New York: John Wiley & Sons.
  7. Rabiner, L. R. and Gold, B. (1975). Theory and Application of Digital Signal Processing. Englewood Cliffs, NJ: Prentice-Hall.
  8. Taylor, F. J. (1983). Digital Filter Design Handbook. New York: Marcel Dekker, Inc.

Content actions

Download module as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Add module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks