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Digital Signal Processing 17 (2007) 530–540 www.elsevier.com/locate/dsp

Additional correction to the impulse invariance method for the design of IIR digital filters Sudarshan Rao Nelatury School of Engineering and Engineering Technology, Penn State University, Erie, 5101 Jordan Road, Erie, PA 16563-1701, USA Available online 14 November 2006

Abstract Almost four decades after the impulse invariance (i.i.) method has been in vogue, a correct way of its application has been emphasized independently, but at the same time, by Jackson and Mecklenbräuker. When there is a discontinuity at t = 0 in the analog impulse response, the correct way requires that one should take half of its value. In spite of this correction, due to undersampling there persists aliasing error in the frequency domain. In this paper the effect of aliasing is viewed as a 3 dB frequency warping and a pre-warping step prior to its application is proposed to reduce it, thereby making the technique appropriate from the viewpoint of frequency domain also. Simulation examples comparing this approach with the traditional method reveal significant improvement in the frequency response. Further, according to Jackson and Mecklenbräuker and also as written in many a standard textbook available today, i.i. technique cannot be used when the degrees of the numerator and denominator polynomials of the parent analog transfer function are equal. An example under this category is the Pre-emphasis circuit or a simple high pass filter. We shall show how the technique can still be used even for a Pre-emphasis circuit. Antoniou proposed (an intelligently) modified impulse invariance technique for certain types of filters by which, aliasing error is nearly eliminated. These cases are neither admitted by the traditional i.i. method nor by that of Jackson and Mecklenbräuker at all. Antoniou’s modified method results in a digital filter of order almost twice as great with an additional task of stabilization. The method in this paper retains the stability and order but gives satisfactory response. In the standard textbooks, the i.i. technique is explained with the help of a first-order low-pass filter and for higher orders, partial fraction expansion is used. For the situation of repeated poles, application of i.i. method calls for extreme care and special handling. This paper alternatively employs the use of state variables to make i.i. technique more elegant. © 2006 Elsevier Inc. All rights reserved. Keywords: Design of IIR filters; Impulse invariance technique; State variables

1. Introduction In the design of discrete systems based on their analog counterparts from their known transfer functions, we come across two well-known techniques—impulse invariance (i.i.) and bilinear transform (b.t.). Many standard textbooks deal with them [1,2]. The i.i. method is attributed to Kaiser [3] and hence i.i. method may be called k.i.i. method. In the b.t. technique the complex frequency variable s in the analog transfer function is replaced with a bilinear rational function of z. There is no aliasing error introduced in b.t and prominent features of the analog filter such as stability are retained. However, given an arbitrary analog rational transfer function or the impulse response, describing a E-mail address: [email protected]. 1051-2004/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.dsp.2006.10.005

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simulated plant for example, b.t. cannot be used; because, b.t. affects the phase response and results in frequency warping; restricting the technique to the cases, where the frequency response is supposed to be piecewise constant. Only the ideal filter specifications follow this pattern, so, the edge frequencies in the specifications are pre-warped before one proceeds with the design using b.t. In the traditional i.i. technique, frequencies are not pre-warped, but from the causal continuous time impulse response ha (t), for a sampling period T , we derive the discrete time response h[n] using h[n] = T ha (nT ).

(1)

The basic building block for application of the k.i.i. is the simple first-order analog low-pass filter with the transfer function  −1 α = α s1 − [−α] 1, α > 0. (2) Ha (s) = s+α The reason for introducing unities would be clear later. The corresponding causal and stable impulse response is ha (t) = αe[−α]t u(t),

(3)

where u(t) is the continuous time unit step function. The usual practice [1–3] is to sample T ha (t) leading to the impulse response f [n] given by f [n] = αT e−αT n u[n],

(4)

where u[n] is the discrete time unit step sequence and the digital filter frequency response is given by F (ej ωT ) =

αT . 1 − e−(α+j ω)T

(5)

The above practice has been in vogue for almost four decades, but recently, a correction has been suggested independently and almost at the same time by Jackson and Mecklenbräuker [4,5]. The correction is made to an inherent error arising from sampling the analog impulse response ha (t) at t = 0 where there is a discontinuity. This error1 was also pointed out in [6]. It was based on the fact that for proper convergence at t = 0, we need to take half of f [0] and this results in a new impulse response  αT /2 for n = 0, (6) g[n] = αT e−αT n u[n] for n = 0 and in the new digital frequency response function given by   αT 1 + e−(α+j ω)T j ωT . G(e )= 2 1 − e−(α+j ω)T

(7)

Frequency response plots of (2), (5), and (7), for some selected values of αT are found in [5] and reveal that, whenever applicable, (7) is better than (5). 2. 3 dB frequency warping The i.i. method focuses only on preserving the impulse response in the time domain, but any deviation in the digital frequency response from its analog version due to a possible under-sampling is accepted as the aliasing error. This error is insignificant if the sampling rate is sufficiently high, but if not, the frequency domain specification, would not be fully met. After all, the frequency domain constraints are usually our target in filter design and a reasonable and the only descriptor of a first-order low-pass building block is the 3 dB frequency. The 3 dB frequencies measured in radians per sample in case of analog response of (2), traditional k.i.i. method of (5) and Jackson–Mecklenbräuker’s corrected i.i. method (j.m.i.i. method) of (7) respectively can be derived as follows: 1 Antoniou wrote [7] without this error, years before [4,5] appeared, but somehow the authors of [4,5] missed this reference. The author thanks one of the anonymous reviewers for pointing to the resources [7,8].

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Fig. 1. Showing the warping effect, dashed curve for the traditional k.i.i. method given by (9) or (11) and solid curve for the corrected j.m.i.i. method given by (10) or (12). Also seen is the y = x line.

Ω3HdB T = αT ,

   = cos 2 − cosh Ω3HdB T ,    H T ) − 3)  (cosh(Ω 3 dB ω3GdB T = π − cos−1 cosh(Ω3HdB T ) . (3 cosh(Ω3HdB T ) − 1)

ω3F dB T

−1

(8) (9) (10)

These equations reveal that either in k.i.i. or j.m.i.i. method, the 3 dB frequencies suffer some sort of warping as shown in Fig. 1. The deviation of the curves from the y = x line shows the degree of 3 dB warping, which is significant for the normalized frequencies greater than 0.5. Note that in k.i.i. method the whole curve is below the y = x line showing that 3 dB warping is inevitable, whereas, in j.m.i.i. method, the curve intersects it at a certain normalized frequency of about 2.2316 rad/sample. The frequencies below this value go up and those above are brought low. Suppose we are working with a first-order filter and the sampling rate is so adjusted that the normalized 3 dB frequency αT is equal to this particular value, say, α0 T = 2.2316 rad/sample (shown as P in Fig. 1), 3 dB warping would be absent, thereby making the analog to discrete approximation good both in time and frequency domains simultaneously. But in real life, we work with higher order filters and a chosen sampling rate, so the 3 dB warping is inevitable and needs to be reduced. The pre-warping equations corresponding to (9) and (10) can be readily written as    Ω3HdB T = cosh−1 2 − cos ω3F dB T , (11)    G  G   ω3 dB T ω3 dB T + 9 sin4 + cos(ω3GdB T ) . (12) Ω3HdB T = cosh−1 3 sin2 2 2 From Eqs. (11) and (12) for the extreme case corresponding to the digital frequency of π , the k.i.i. and j.m.i.i. analog images are respectively given by Ω3HdB T = cosh−1 (3) ≈ 1.7627, √   Ω3HdB T = cosh−1 3 + 8 ≈ 2.4485,

(13) (14)

which can be noticed from Fig. 1. Thus we find that, if we attempt to design a digital filter from an analog version whose 3 dB frequencies exceed the limits set by (13) or (14), they will not be preserved in the k.i.i. and j.m.i.i. methods respectively. Taking the possible aliasing error, the new expression for the frequency response of a j.m.i.i. digital filter after duly pre-warping, i.e., by replacing αT with α  T in (7) is now given by    α  T 1 + e−(α +j ω)T j ωT )= G(e , (15) 2 1 − e−(α  +j ω)T

S.R. Nelatury / Digital Signal Processing 17 (2007) 530–540

 

α T = cosh

−1

 2

3 sin

αT 2



 +

 9 sin

4

αT 2

533





+ cos(αT ) .

(16)

3. Extension to higher order filters Consider a higher order analog transfer function Ha (s) given by Ha (s) =

b0 s N + b1 s N −1 + · · · + bN −1 s + bN . s N + a1 s N −1 + · · · + aN −1 s + aN

(17)

The denominator and numerator polynomials in (17) may be factored to indicate poles and zeros. Suppose (17) has an overall band-pass characteristic, getting an expression for the low frequency and high frequency 3 dB points is very formidable. Approximate expressions for low and high 3 dB frequencies in some special cases are available such as the following taken from [9]:

2 −2 2 , ≈ ΩPn ΩZn (18) Ω3low dB n high

Ω3 dB ≈ 

1 2 Ω Pn n

n

1 −2



,

(19)

1 2 Ω Zn n

where −ΩPn and −ΩZn are pole and zero values of the nth factor in the denominator and numerator in (17), respectively. Then trying to relate (18) and (19) with their digital counter part would be a formidable pursuit. However, any higher order linear system can be viewed as a first-order complex system using state vectors. In the time domain, the system described by (17) may be represented as [10,11]

q˙ = Aq + Bx , (20) y = Cq + Dx where q is the state vector, x(t) and y(t) are input and output of the system. A is the companion. Likewise B, C, and D have the usual connotation. Also, let us recall that using a similarity we might choose altogether different parameters. The transfer function Ha (s) and the corresponding impulse response ha (t) are obtained as Ha (s) = C(sI − A)−1 B + D,

(21)

ha (t) = Ce Bu(t) + Dδ(t).

(22)

At

In particular, if D = 0, at t = 0+ , we observe that ha (0+ ) = CB.

(23)

By inspection, (2) can be identified as a specialization of (21) and we are in a position to extend the idea of i.i. to (21) and (22). According to Jackson and Mecklenbräuker and also as written in almost every standard textbook available today, this technique cannot be used when the orders of the numerator and denominator polynomials of the parent analog transfer function are equal. That is, for the i.i. technique to be applicable, b0 has to be zero. The k.i.i way would have been to use   f [n] = T CeAT n Bu[n] , (24)   −1 F (z) = T zC zI − eAT B. (25) The values in (24) are sometimes called Markov parameters. The j.m.i.i. method admits situations when there are two poles more than there are finite zeros and attempts to take half of (23) for n = 0. Thus the discrete impulse response and the corresponding transfer function respectively are   (26) g[n] = T CeAT n Bu[n] − CBδ[n]/2 ,   1 G(z) = T zC(zI − eAT )−1 B − CB . (27) 2

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Now we shall extend this to a bit further and include the D term also and write the above as   g[n] = Dδ[n] + T CeAT n Bu[n] − CBδ[n]/2 ,   1 AT −1 G(z) = D + T zC(zI − e t) B − CB . 2

(28) (29)

The above approach of including D (but not DT ), the direct transmission term or (the high frequency gain term), allows us what would be impossible by traditional k.i.i. or even j.m.i.i. methods to obtain the digital transfer functions from their analog counter parts of systems like the Pre-emphasis circuit discussed in Section 4. For implementing the digital filter in real life, from the transfer function in (29), getting the numerator and denominator coefficients for higher orders is somewhat tedious, but straightforward using a symbolic toolbox. At this point, we can still bring in the idea of pre-warping as follows. Equations (11) and (12) are derived for a first-order example. The higher order transfer function in (17) can be split into partial fractions. Carrying over this idea, we need to choose for the state-space model, the diagonal canonical form or more generally, the Jordan form (for the case of multiple roots). The 3 dB frequencies that we mentioned in the preceding section are then related to the eigenvalues of the companion in (20) or to the diagonal elements in the diagonal form. Suppose the original analog filter has ˜ be the companion in distinct roots. Let us adopt the diagonal form denoting the model parameters with a tilde. Let A ˜ the diagonal form. Let a˜ ii = α˜ ii + j βii , i = 1, 2, . . . , N . We suggest that each of α˜ ii T should be modified as α˜ ii T ˜ replaced for i = 1, 2, . . . , N using (16). Then the proposed digital transfer function would be as given in (29) with A ˜  . This method attempts to restore the 3 dB points for each first-order subunit, even though the overall 3 dB by A points are not preserved since they depend on the zero frequencies also as is evident in (18) and (19). In the process of pre-warping, as mentioned in Section 2, the individual 3 dB frequencies at the left and right sides of α0 T are either decreased or increased as a reasonable compromise attempting to meet the overall objective. To distinguish the proposed method outlined with the aid of state-variables from others mentioned in this paper, we shall refer to it as the s.i.i. method. In the next section we shall provide few simulation examples. 4. Simulation examples Example 1. Pre-emphasis and De-emphasis circuits are used in frequency modulation systems to protect the message signal from undesired contaminations that tend to predominate at certain portions of the spectrum. For instance, the Dolby system for tape recording dynamically adjusts the amount of Pre-emphasis and De-emphasis in inverse proportion to the high frequency signal content. Typical analog R–C circuits used for Pre-emphasis and De-emphasis are shown in Figs. 2a and 2b, respectively. The analog transfer functions for Pre-emphasis and De-emphasis circuits shown in Figs. 2a and 2b are respectively:   ατ s + α pe , (30) Ha (s) = ατ s + 1

(a)

(b) Fig. 2. Typical Pre-emphasis (a) and De-emphasis (b) circuits used in FM. For our simulation we have chosen R1 = 10 k , R = 5 k , α = 1/3, and C = 7.5 nF. The lower corner frequencies are nearly 2.12 kHz. For both circuits, the time constant is τ = 75 µs.

S.R. Nelatury / Digital Signal Processing 17 (2007) 530–540

Fig. 3. Gain in dB of the Pre-emphasis filter for various cases: solid—analog transfer function, circles—s.i.i. response with prewarping for T = 3 µs, crosses—s.i.i. with pre-warping for T = 30 µs, boxes—s.i.i. without pre-warping for T = 30 µs. Note that the deviation at low frequencies is about 2 dB for T = 30 µs without pre-warping.

 Hade (s) =

535

Fig. 4. Magnitude response of the Pre-emphasis circuit of Fig. 2a shown here as a warning to the unwary users of MATLAB’s IMPINVAR function.

 1 , τs + 1

(31)

where τ = R1 C and α = R/(R + R1 ). From the degrees of numerator and denominator polynomials of (30), it is clear that k.i.i. or j.m.i.i. would not allow us to get a digital transfer function. Here the value of D is unity. Using the s.i.i. method in (29), after pre-warping, we get   0.96z − 0.9221 pe (32) G (z) = z − 0.8867 for a sampling period T = 3 µs and   0.6z − 0.3573 pe G (z) = z − 0.2552 for a sampling period T = 30 µs. If we repeat the same without pre-warping before hand, we get   0.6z − 0.4217 pe G (z) = z − 0.3012

(33)

(34)

for a sampling period T = 30 µs. Actually, had j.m.i.i. been correctly formulated just as s.i.i., one would have arrived at (34). But the fact remains that j.m.i.i. misses the introduction of D. The magnitude response curves are shown in Fig. 3. One finds a significant improvement in the response with pre-warping even for T = 30 µs. It is widely known that in the signal processing toolbox of MATLAB, the k.i.i. method is implemented through the command IMPINVAR to get the digital transfer function coefficients from the analog version. As a warning to an unwary user, the response obtained by IMPINVAR command is plotted for T = 3 and 30 µs, in Fig. 4 along with the analog response. A very unsatisfactory response is obtained even for T = 3 µs. The pre-emphasis circuit is also viewed as a phase-lead compensator in control system applications. What would be worthwhile is to examine the phase response of this network. Figure 5 shows the phase response of the analog R–C circuit and that of the digital transfer functions obtained through the s.i.i. method as well as using MATLAB’s IMPINVAR command. Observe how unsatisfactory the latter is. For the De-emphasis circuit shown in Fig. 2b, we repeat the exercise. For this case, j.m.i.i. has the solution. Choosing a sampling period of T = 100 µs, we obtain the digital transfer function using the s.i.i. method with pre-warping as

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Fig. 5. Phase response of the analog transfer function of the Pre-emphasis circuit shown in Fig. 2a and that of the digital version obtained through the s.i.i. method and MATLAB’s IMPINVAR command.

 G (z) = de

0.6667z + 0.1460 z − 0.2190

Fig. 6. Gain in dB of the De-emphasis filter for various cases: solid—analog transfer function, circles—s.i.i. with Prewarping, boxes—j.m.i.i., and diamonds—k.i.i. or MATLAB. Ts = 100 µs.



and the same without it, that is, according to j.m.i.i. method, we obtain   0.6667z + 0.1757 Gde (z) = . z − 0.2636

(35)

(36)

The response curves are shown in Fig. 6, wherein we also included the k.i.i. method to reveal the difference. In this example, we notice that the pre-warping did not achieve a significant improvement, but appears to follow the analog response a bit more closely. Here we chose T = 100 µs just to underscore the differences more clearly. Example 2. Next let us consider a third-order analog system described by s Ha (s) = . (s + 4)(s + 6)(s + 8)

(37)

This example is not aimed so much as a filter design, but is chosen just to examine how well the s.i.i. method can emulate the analog response. The state space parameters (with top-row companion) for (37) are a11 = −18, a12 = −104, a13 = −192, a21 = a32 = 1, b1 = 1, c2 = 1, and all the remaining are zeros. After we diagonalize, we get a˜ 11 = −8, a˜ 22 = −6, a˜ 33 = −4, b˜1 = b˜2 = b˜3 = 1, c˜1 = −1, c˜2 = −1.5, c˜3 = −0.5, and the remaining elements are  = −6.5666, a ˜ 22 = −5.3316, a˜ 33 = −3.7842, zero. Pre-warping as mentioned in the preceding section, we get a˜ 11     b˜1 = b˜2 = b˜3 = 1, c˜1 = −1.9109, c˜2 = 2.7899, c˜3 = −0.879 using (11). Using (29), the rational digital transfer function that approximates (37) is given by G(z) =

0.009524131552z(z − 1.052574005) . (z − 0.1840294719)(z − 0.4526765648)(z − 0.6617400534)

(38)

Note that for this example from the initial value theorem, ha (0) = 0 and hence, as remarked in [4,5], correction is not required; for, the j.m.i.i. method degenerates to k.i.i. However, we hope to see slight improvement with pre-warping. For a sampling period T = π/15, the results are shown in Fig. 7. Note that the dc gain of the analog filter is infinite dB down. But the dc gain of the digital filters based on traditional i.i. and the proposed method is 7.9439 and 10.6492 dB down, respectively. Likewise, the gain of the analog filter corresponding to ωmax = 15 rad/s is 10.5202 dB down, but in case of the j.m.i.i. it is only 3.9415 dB down. In the s.i.i. method it is about 5.3130 dB down showing a reduction in aliasing. Just for comparison, the response obtained by using bilinear transformation (b.t.) is also depicted. We find that b.t. significantly alters the response due to warping effect. Warping is reduced by performing pre-warping in case of those filters with piecewise constant magnitude response, but in this counter example, b.t. cannot be directly used.

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537

Fig. 7. Frequency response of an analog filter in (37) and that of the digital counter parts resulting from the k.i.i., j.m.i.i., s.i.i., and b.t. methods.

Example 3. In this simulation study, we shall design a sixth-order type II Chebyshev band-stop filter with stopband attenuation of 20 dB and the passband edge frequencies of ωp1 = 1 rad/s and ωp2 = 2 rad/s with a sampling interval Ts of π/8 s. As is well known, band-stop filters cannot be designed via k.i.i. or j.m.i.i. methods. But using the s.i.i. method, we can get the digital filter with a satisfactory response. The analog filter coefficients are taken using MATLAB command CHEBY2. The analog rational transfer function is Ha (s) =

s6 + 6.75s4 + 13.5s2 + 8.0000 (s6 + 2.3434s5 + 9.4958s4 + 11.8612s3 + 18.9917s2 + 9.3737s + 8)

.

(39)

Using the proposed method we obtain the digital transfer function as G(z) = 0.5371

(z2 − 1.5476z + 1.0517)(z2 − 1.6872z + 1.0466)(z2 − 1.8128z + 0.9805) . (z2 − 1.0965z + 0.7181)(z2 − 1.3874z + 0.6287)(z2 − 1.7643z + 0.8776)

(40)

In practice, finding the matrix inverse in (29) would pose a problem, but if we view it as the ratio of the adjoint to the determinant it would be easier. Also matrix exponential is made easy by writing the argument matrix as a sum of commutable nilpotent and diagonal matrix components. Most of these are built into many of the toolboxes available to researchers today. The frequency response curves for analog and the digital filter are shown in Fig. 8. As revealed in the inset we find that the s.i.i. method has a small error of about 1 dB in the right pass band, which is tolerable. The problem is repeated with eighteenth-order and the response curves are shown in Fig. 9. Once again we find a smaller error of about 0.8 dB in the pass band. One might increase the sampling rate to reduce it further. For some analog filters whose transfer functions are not strictly proper, with degrees of numerator and denominator polynomials being equal, the i.i. method cannot be used. For handling such cases Antoniou has suggested a modified impulse invariance technique [7,8]. He suggested matched z-transform technique also for some special cases, but, we are not dealing with that here. Antoniou’s modified i.i. method, (henceforth called a.m.i.i. method) attempts to approximate the numerator and denominator separately. Approximating the numerator and denominator by separate rational functions is indeed a mathematical trick but unfortunately this would double the order of the resulting digital filter. Further, the numerator might have zeros outside the unit circle, and they have to be mapped inside by taking their reciprocals. This stabilization process is not serious for the overall shape of the frequency response, but it would cause a slight vertical shift. Whenever, applicable, a.m.i.i. is considered to be the best i.i. approximation, even though the order of the resulting digital filter is almost twice higher than the analog parent. Needless to emphasize, this phenomenon of order doubling or loss of stability would not arise in the s.i.i. method; for, with (29) every stable analog transfer function would be converted into a stable digital transfer function. This is left as an exercise to the reader to verify mathematically. Next we shall reconsider Example 7.2 in [7] (or Example 11.2 in [8]) and compare the results.

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Fig. 8. A sixth-order type II Chebyshev bandstop filter responses, Analog (solid) and digital (circles) via the s.i.i. method. Sampling interval T = π/8 s.

Fig. 9. An eighteenth-order type II Chebyshev bandstop filter responses, Analog (solid) and digital (circles) using the s.i.i. method. Sampling interval T = π/8 s.

Example 4. Let us consider the Example 7.2 in [7] (or same as Example 11.2 in [8]). The analog transfer function is of a sixth-order elliptic low-pass filter√satisfying the specifications: passband √ ripple = 0.1 dB, minimum stopband loss = 43.46 dB, passband edge = 0.8 rad/s, and the stopband edge = 1/ 0.8 rad/s. The transfer function is (s 2 + 11.9934)(s 2 + 2.0001)(s 2 + 1.3024) . (s 2 + 0.9508s + 0.3582)(s 2 + 0.4423s + 0.6861)(s 2 + 0.1089s + 0.8633) Using the proposed s.i.i. approach, we derive the digital transfer function for ωs = 7.5 rad/s as Ha (s) = 0.006713

(z + 7.0605)(z + 0.1479)(z2 − 0.6632z + 1.0044)(z2 − 1.1454z + 1.0001) . (z2 − 1.2680z + 0.4417)(z2 − 1.3025z + 0.6890)(z2 − 1.3624z + 0.9128) Re-designing the filter for a sampling frequency of 30 rad/s, we get G(z) = 0.003099

(41)

(42)

(z2 − 1.4713z + 0.9992)(z2 − 1.9126z + 1.0001)(z2 − 1.9431z + 1.0000) . (43) (z2 − 1.8006z + 0.8152)(z2 − 1.8824z + 0.9111)(z2 − 1.9401z + 0.9774) For a sampling frequency of 7.5 rad/s, we show the Loss function, in stead of the gain in dB using both a.m.i.i. of order 10 and the s.i.i. techniques. The results are shown in Fig. 10. We find that even with the same order the digital response does meet the specifications. Far into the stop band, we find that the a.m.i.i. response closely follows the analog trace. The design is repeated with four times the sampling rate and the response is presented in Fig. 11, wherein we find that the s.i.i. method almost meets the analog response and in a way, slightly over-meets in the stop band by almost a dB. This shows that the s.i.i. technique also can perform well at a higher sampling rate if its lower order is thus made up for. In Fig. 10 an inset is laid showing a pass-band error of 1 dB, but the same is reduced to about 0.05 dB in Fig. 11. G(z) = 0.005686

Example 5. All the three i.i. methods viz., k.i.i., j.m.i.i., and a.m.i.i. tacitly assume that the poles of the underlying analog transfer function are distinct. The special case of repeated poles is dealt in [12] by Cavicchi. Based on Heaviside’s expansion of partial fractions, Cavicchi gave an algorithm that implements k.i.i. technique. Cavicchi’s paper appeared before [4,5]. Therefore, his method is still open for j.m.i.i. correction and the pre-warping mentioned here. The s.i.i. method described in (29) is more elegant. To examine how this compares, we shall repeat the example worked out in [12]. Take the analog transfer function, 2914.4(s + 2) . (s 4 + 14s 3 + 428.3s 2 + 3010.4s + 5828.8) Factoring the denominator we get Ha (s) =

(44)

S.R. Nelatury / Digital Signal Processing 17 (2007) 530–540

Fig. 10. Comparing the proposed method with Antoniou’s a.m.i.i. method for the Example 7.2 in [7] (or Example 11.2 in [8]) with ωs = 7.5 rad/s. Note that the order of the a.m.i.i. digital transfer function is 10 while in the s.i.i. method it is 6 itself like that of the parent analog transfer function.

Fig. 12. Analog and the s.i.i. digital magnitude responses for the case of repeated poles in Example 5 [12].

Ha (s) =

539

Fig. 11. Showing the same example appearing in Fig. 10, but with ωs = 30 rad/s and order = 6.

Fig. 13. Phase response for Example 5. Note that MATLAB’s IMPINVAR command furnishes unsatisfactory phase.

2914.4(s + 2) . (s + 4)2 (s 2 + 6s + 364.3)

(45)

Note that this transfer function has a double pole at s = −4. Though the synopsis on MATLAB’s IMPINVAR says that “the repeated pole case works, but is limited by the ability of the function ROOTS to factor such polynomials,” and although, the currently available version is updated subsequent to [12], actual simulations reveal that it does not offer full satisfaction. Using the sampling interval T = 0.03 s, symbolic computation of (29) readily produces the sought-for transfer function, G(z) =

(z4

z(0.033962z2 − 0.001843z − 0.028386) . − 3.3172z3 + 4.3595z2 − 2.6957z + 0.657)

(46)

The j.m.i.i. method does not admit this case. The magnitude and the phase responses are sketched in Figs. 12 and 13, respectively. We find the results satisfying.

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S.R. Nelatury / Digital Signal Processing 17 (2007) 530–540

5. Conclusions The impulse invariance technique, known for a long time, along with a recently proposed correction is revisited. In most standard textbooks, this technique is introduced initially for a first-order filter. For a higher order filter, the transfer function is subjected to partial fraction expansion and the technique is applied on each term. This paper has given an equivalent but a more elegant treatment of this technique with the aid of state variables. Addition of the direct transmission term and the pre-warping step are considered as added correction to the k.i.i. or j.m.i.i. methods. Examples like Pre-emphasis circuits and bandstop filters are worked out. Even the case of repeated poles is addressed. The aliasing error occurring in the frequency domain due to inadequate sampling is viewed as a warping effect on the 3 dB frequency. To account for this error, a pre-warping expression is proposed, which is valid for the first-order filters. Since higher order filter transfer functions can be considered as a combination of several first-order building blocks, to extend the same for higher order filters, we used a state variable model with a diagonal companion. In principle, the technique attempts to restore the 3 dB frequencies of individual blocks, resulting in an improved overall frequency response. For some classes of analog transfer functions, allowing higher order for the digital transfer function, the a.m.i.i. method is excellent. The proposed s.i.i. method with nearly half the order is satisfactory even though, the aliasing error is not equally reduced; however, if the sampling frequency is raised, the s.i.i. method can give equally or slightly better design. References [1] A.V. Oppenheim, R.W. Schafer, Discrete-Time Signal Processing, Prentice Hall, Englewood Cliffs, NJ, 1989, pp. 407–410. [2] J.G. Proakis, D.G. Manolakis, Digital Signal Processing: Principles, Algorithms, and Applications, third ed., Prentice Hall, Upper Saddle River, NJ, 1996, pp. 671–676. [3] J.F. Kaiser, Digital filters, in: F.F. Kuo, J.F. Kaiser (Eds.), System Analysis by Digital Computer, Wiley, New York, 1966, p. 245 (chap. 7). [4] L.B. Jackson, A correction to impulse invariance, IEEE Signal Process. Lett. 7 (10) (2000) 273–275. [5] W.F.G. Mecklenbräuker, Remarks on and correction to the impulse invariant method for the design of IIR digital filters, Signal Process. 80 (2000) 1687–1690. [6] R.A. Gabel, R.A. Roberts, Signals and Linear Systems, Wiley, New York, 1987, pp. 408–422. [7] A. Antoniou, Digital Filters—Analysis, Design and Applications, second ed., McGraw–Hill, New York, 1993. [8] A. Antoniou, Digital Signal Processing—Signals, Systems, and Filters, McGraw–Hill, New York, 2006. [9] R.C. Jaeger, T.N. Blalock, Microelectronic Circuit Design, second ed., McGraw–Hill, New York, 2004. [10] L.A. Zadeh, C.A. Desoer, Linear System Theory: The State Space Approach, Series in System Science, McGraw–Hill, New York, 1963. [11] K. Ogata, Modern Control Engineering, fourth ed., Prentice Hall, Englewood Cliffs, NJ, 2002. [12] T.J. Cavicchi, Impulse invariance and multiple-order poles, IEEE Trans. Signal Process. 44 (9) (1996) 2344–2347.

Sudarshan Rao Nelatury received the B.Tech. degree in 1981 from JNT University, Hyderabad, India, the M.E. and Ph.D. degrees in 1985 and 1996, respectively, both from Osmania University, Hyderabad, India. From 1983 to 1999 he taught at the Department of Electronics and Communication Engineering, Osmania University. During 1999–2003 he was a Visiting Faculty Member at the Electrical and Computer Engineering Department, Villanova University. Presently, he is with the Department of Electrical, Computer and Software Engineering, at the School of Engineering and Engineering Technology, Penn State, Erie, PA. His research interests include electromagnetics and digital signal processing. Dr. Nelatury is a Life Member of the IETE, Delhi, India, Life Member of the ISTE, Calcutta, India, Member of the ASEE, Member of the ITE, and a Senior Member of the IEEE.