The paper describes certain general transformations for digital filters in the frequency domain. ... certain distortion errors were introduced into the amplitude.
PROCEEDINGS THE INSTITUTION OF ELECTRICAL ENGINEERS-,
volume 117
Electronics
Spectral transformations for digital filters A. G. Constantinides, B.Sc.(Eng.), Ph.D. Indexing term: Digital filters
Abstract The paper describes certain general transformations for digital filters in the frequency domain. The term digitalfilteris used to denote a processing unit operating on a sampled waveform, so that the input, output and intermediate signals are only defined at discrete intervals of time; the signals may be either p.a.m. or p.c.m. The transformations discussed operate on a lowpass-digital-filter prototype to give either another lowpass or a highpass, bandpass or band-elimination characteristic. The transformations are carried out by mapping the lowpass complex variable z" 1 [where z~x = exp (—j'toT) and T is the time interval between samples] . by functions of the form
known as unit functions.
List of principal symbols a, b — real constant coefficients / = frequency, Hz Fs-= sampling frequency, Hz G() = pulse-transfer function of a digital filter g() = unit function; spectral transformation T = sampling period (T = 1/FS) z = unit advance, z = exp (JcoT) z~l = unit delay, z = exp (—jcoT) , as is the case for the lowpasslowpass transformation in R, L, C and M filters where s is replaced by ks (k > 0 and constant). Mathematically, it is required to map a unit circle into PROC. IEE, Vol. 117, No. 8, AUGUST 1970
itself in a 1:1 correspondence, with the points A and D invariant. Let the mapping be denoted by so that, for one complete rotation of z~x around the unit circle, g(z~l) also takes one complete rotation. Furthermore, = 1
(3)
The transformation given by eqn. 4 has no effect on the amplitude characteristic except for the change in the position of the frequencies on the unit circle. Thus, for negative a (i.e. (DC > /3), the arc BDC of Fig. 1 is compressed on the circle, and, since A is invariant, the arc BAC is stretched. For positive a (i.e. coc < j8) the opposite effect occurs. Fig. 2a shows the amplitude characteristic of a lowpass digital filter whose cutoff frequency is 2-5kHz for a sampling frequency of 10kHz. If tojlTr of eqn. 5 is chosen to be 1kHz, as a result of transformation (eqn. 4) the amplitude characteristic is as shown in Fig. 2b.
10
= 05
2
3
frequency, kHz
10
10% 1-0
05
I-0-5
2
3
frequency. kHz b
2
3
frequency. kHz e
10
Fig. 2 S 05
Amplitude characteristics from spectral transformations a 4th-order lowpass Chebyshev characteristic fc = 2-5kHz, F, = 10kHz b Lowpass characteristic,/ bandpass and band-elimination pulse-transfer functions. Let G(z~l) be the pulse-transfer function of a lowpass digital filter. Then H(z~x) =G{z~1 exp (-Ja>0T)} + G{z~l exp (+jto0T)} is a translated version of G(z~l). Several comments have been produced by Kaiser in Reference 6. Basically, the above formula represents two rotations on the z" 1 plane, one in the clockwise and the other in the counter-clockwise direction. The sum of the two rotations of functions of the complex variable z~l is a real rational function z~l if the rotations are at equal angles. Because the pulse-transfer function resulting from the above formula is the sum of two individual pulse-transfer functions rotated to the angles ±co0T, it will exhibit what Broome3 and Kaiser6 term a 'distortion error' which results from the tails of the image of G(z~l) centred at — OJ0. It is suggested in these References that a good lowpass-filter design will reduce the distortion error to a very small level. No distortion will occur when the lowpass pulse-transfer function is rotated by ±180° or multiples thereof, because, when o)0T= ±180°, exp(ja)0T)
= -1 1589
and hence H(z~l) = 2GC-Z-1) which is recognised as the restricted case of the general lowpass-highpass transformation of eqn. 7. If COQT = 0, the pulse-transfer function remains unaltered except for a multiplier of value 2. This is not, however, a lowpass-lowpass transformation, because no change in cutoff frequency occurs. Thus the Broome translation formula is only satisfactory under very special conditions where the distortion errors are small and when co$T = 180°, so that the special case of the lowpass-highpass transformation is obtained, and, in this restricted sense, the translation formula is exact. The general transformations given in this paper, however, are therefore superior to the above translation formula, sWe they not only introduce no distortion errors but they are also unrestricted in their application, be it for wideband or narrowband digital filters.
9
Conclusions
We have given in this paper the complete set of transformations on the z~l plane necessary to transform a given lowpass-digital-filter pulse-transfer function into a pulsetransfer function having the same type of amplitude characteristic and belonging to one of the following classes: (i) lowpass (ii) highpass (iii) bandpass (iv) band-elimination. Furthermore, along with the transformations, the necessary design formulas were given which relate the critical frequencies of the required digital filter to the cutoff frequency of the lowpass-digital-filter prototype. The transformations are quite general, and, their restricted forms that have been published elsewhere1*2 are easily obtainable as indicated. Proofs of the most important theorems are given in Appendix 12. The spectral transformations possess the following important and useful features: (a) They are allpass functions, and hence, because of their inherent form, the multipliers in the digital implementation can be reduced by a factor of 2 by multiplexing. (b) Since the application of the transformations involves a substitution of z~l in a lowpass-digital-filter transfer function by a unit function, it follows that the structure of the lowpass digital filter remains unaltered (i.e. the adders and multipliers are the same), and only an extra elementary transfer function, representing the transformation, is inserted in place of z~K (c) As a consequence of property (b) above, certain forms of the transformations (e.g. eqns. 13 and 20) that have one variable parameter exhibit simple characteristics where the centre frequency can be varied over the Nyquist range but the bandwidth is kept'constant. 10
Acknowledgments The author wishes to thank the authorities of Northampton College of Advanced Technology, London (The City University, London) for financial support, and the Senior Director of Development, UK Post Office Telecommunications Headquarters, for facilities provided. 11
References
1 CONSTANTINIDES, A. c.: 'Frequency transformations for digital filters', Electron. Lett., 1967, 3, pp. 487-489 2 CONSTANTINIDES, A. c : 'Frequency transformations for digital filters', ibid., 1968, 4, pp. 115-116 3 BROOME, P. : 'A frequency transformation for numerical filters', Proc. Inst. Elect. Electron. Engrs., 1966, 54, pp. 326-327 4 JURY, E. i.: 'Theory and application of the z-transform method' (Wiley, 1964) 5 CRYSTAL, T. H. : 'The design and applications of digital filters with complex coefficients', IEEE Trans., 1968, AU-16, pp. 330-335 6 KAISER, J. F.: 'Digital filters' in KUO, P. F., and KAISER, J. F. (Eds.): 'System analysis by digital computer' (Wiley, 1966), pp. 218-253 7 CONSTANTINIDES, A. G.: 'Synthesis of recursive digital filters from prescribed amplitude characteristics'. Ph.D. thesis, University of London,1968 1590
12
Appendix
Proof of theorem 2: Rewriting eqn. 9 in a different form, we have y2z
+ yjz i + l
From Table 2 we see that #(1) = — 1. Hence, by using this constraint on the above form of eqn. 9, we obtain eJQ = — 1. Furthermore, since ^{exp (—Jco0T)} = —1, we have (y2 + 1) + 2yt exp (-Jco0T) + (y2 + 1) exp (-j2co0T) = 0 and hence (1 + y2) cos co0T + yx = 0 . Let cos COQT = a, so that u
(
_
=
y 2 )z ~
8
2
y2
and also let y2 = y. Hence we can write the above equation in the form
and, since a = cos coT, |a| < 1, so that the function . . . i / z~x — a \ . l z [. ) is a unit function. \1 — az~W Furthermore \y\ < 1, since this quantity represents the product of two zeros of the unit function which are inside the ^-, J unit circle (^ * j = 1. Hence the quantity — (is a unit function. \1 + yz El
=
and E2 = , -
-
yz so that g(z !) = EX(E2). Now let us form i.e.
+ g(z ')
1 + yE2 — y — E2
1 + y 1 +E2 i -r gK*
)
1 — y 1 — E2
_ 1 + y z~2 - 2az~l + 1 ~ 1- y 1 -z-2 When o> = cj[ in the bandpass characteristic, the function g(z~l) corresponds to exp (j^T), and, when co = co2, g(z~1)= exp (-/j8r). Hence 6r 2 and
y + 1 coscuiT — a y — 1 sin co\T
BT y + 1 cos co2T - a —^—=— tan ;— = 2 y - 1 sin co2T
Therefore a = cos _ , so that Let y^ +' i == —k, y - 1
and
y =
k - 1 k +1
and the transformation becomes ,-2
*
-
_
.-2
Jfc + 1 *+ 1 . Hence the theorem is proved. The proof for theorem 3 is similar to the above. PROC. IEE, Vol. 117, No. 8, AUGUST 1970
