4 ROSENBROCK, H. H. : 'Connection between network theory and the theory of ... The solution of the problem of frequency transformation in digital filtering ...
3 ROSENBROCK, H. H.: 'Reduction of system matrixes', Electronics Letters 1967, 3, p. 368 4 ROSENBROCK, H. H. : 'Connection between network theory and the theory of linear dynamical systems', ibid., 1967, 3, pp. 296-297 5 BRYANT, p. R.: 'The order of complexity of electrical networks', Proc. IEE, 1959, 106C, pp. 174-188
further on the unit circle a rational function of z x. Therefore (3)
/(co) = - cot (kco)
(4)
— cot (A:co) = — j -
and
(c) Bandpass frequency variable /(co): Let us assume that one
band is required which is centred at the frequency co0. We have to satisfy the following conditions:
FREQUENCY TRANSFORMATIONS FOR DIGITAL FILTERS Transformations to convert lowpass systems to highpass, bandpass and band-elimination systems in the case of pulse transfer functions of digital filters are given in this paper. It is believed that these results, for z plane transformations, are the first ones to be published in this field.
The solution of the problem of frequency transformation in digital filtering schemes has not been available so far, mainly because the method of synthesis of digital filters was, until recently,1 an indirect one2 obtained through continuous filters. The need for frequency transformations on the z plane was thus not obvious and was apparently ignored. Recently, a move towards direct synthesis procedure for digital filters was made, 1 ' 3 ' 4 ' 5 and hence the need for frequency transformations on the zplane became more apparent. It is the aim of this letter to develop a theory of these transformations. Formulation of the problem: The problem with which we are concerned is as follows: Given the pulse transfer function (p.t.f.) .G(z-1) of a lowpass digital filter which has a certain type of characteristic, it is required to obtain a transformation g^z"1) for z-1 so that G{^(z~1)} becomes the p.t.f. of one ofthe following types of digital filters: highpass, bandpass or band-elimination. Furthermore, the type of characteristic should be preserved after transformation. This problem is analogous to the problem existing in the synthesis of continuous filters, where, in order to convert from a lowpass to any other type of filter, the complex frequency variable 5 is replaced by an appropriate reactance function; but the analysis in the case of digital filters is more complex. The requirements for the transformations on the frequency axis are similar to those of continuous filters with slight modifications. In continuous filters, the range of relevant frequencies extends over the whole of the frequency axis, i.e. — oo < co < + oo. The equivalent range in digitalfiltersis compressed between the limits ±CIJ2, the baseband of the filters. The following conditions concerning the digital frequency variables are applicable: (a) Lowpass frequency variable f(co): At co = 0,/(co) = 0 and at co = Q.J2, /(co) = oo. Bearing in mind the repetitive property of/(co) with respect to the sampling frequency Qs and that/(co) must be a rational function of z" 1 , one obtains tan (kco) (where k = 7RLQ.S), which satisfies these criteria. i.e.
/(co) = tan (kco)
(
1
)
(1)
1
On the unit circle C, \z~ \ = 1 and eqn. 1 becomes .1 -z"1 tan (A:co) = —j1+z-1 where
(2)
z = exp (J2kco)
co = 0
/(co) = -oo
co = co0
f(oj) = 0
co = Q,/2 /(co) = +oo Hence the required function is given by cos (2kco0) — cos (2kco)
(5)
sin(2*«) On the unit circle, eqn. 5 is written cos (2kco0) — cos (2kco) sin (2kco)
.z- 2 -2cos(2A:coo)z-1 + 1
(d) Band-elimination frequency variable f '(co): In this case we
assume again that one elimination band is required and that it is centred at co0. Hence the requirements become co = 0
/(co) = 0
co = co0
/(co) = oo
co = CIJ2 /(co) = 0 with the further constraint that
*M>0,for-% den
2
The required /(co) is therefore f( 1 = f( 1 = J v
'
sin (2kw) cos (2A:co) - cos (2A;co0)
Eqn. 7 on the unit circle is expressed in the form: sin (2kco) cos (2&co) — cos (2fcco0)
1-z-
2
*•
;
r-2 _2cos(2fcco0)z Transformations: We assume that the p.t.f. of the lowpass digital filters was synthetised using the lowpass frequency variable of eqn. 1. (i) Lowpass to highpass transformation Theorem 1: Given the p.t.f. ^(z - 1 ) of a lowpass digital filter, the p.t.f. of a highpass digital filter having the same type of characteristic as G(z-1) is obtained by replacing z" 1 by -z-
11
in
1
)
Proof: It is seen from eqn. 3 that, if instead of tan (kco) we use —cot (kco), the resulting digital filter becomes highpass. But this is not the complete answer because in the lowpass p.t.f. G(z~l) the complex variable z" 1 must be changed according to this replacement. Rewriting eqn. 2 and eqn. 4, we have tan(kco)= -
j
......(2)
k = TT/Q 5
Q.S = sampling angular frequency (b) Highpass frequency variable f(co): At co = 0, /(co) = — oo
and at co = Qs/2, /(co) = 0. The'function — cot (Axo) satisfies these conditions, and it is ELECTRONICS LETTERS
November 1967 Vol. 3 No. 11
- c o t (£co)= - j - 1
-
z
-
1
,
-
i
( 4 )
From eqn. 2 we have that i _, 1 - j tan (kco) l+jtCL
Corollary 2: The upper and lower frequencies of a bandpass filter are connected with the centre frequency of the filter by the relationship,
where COCH = cutoff frequency of highpass filter
cos (2kco0) =
Q.s = sampling angular frequency
where
COCL = cutoff frequency of lowpass filter Proof: Consider the moduli of eqn. 2 and eqn. 4 at their respective cutoff frequencies. At these points we have tan (ku>CL) = cot (kcoCH)
cos k(co2 cos k(co2 — cuj)
cu2 = the upper frequency
&>! = the lower frequency Proof: By constraining the frequency function (eqn. 5) to have the same values at the upper and lower frequencies, but of opposite sign, we have cos (2kco0) — cos (2koo2) _
hence k(coCL + coCH) = -(2n + 1) w = 0,1, 2 . . . Within the baseband (i.e. n = 0), we have
i.e.
cos (2kco0) — cos
sin (2kco2) sin {2kco{) sin (2&a>2)cos (2kcu{) +cos(2kio2) sin(2fca
cos(2&o>0) =
sin
+ sin
sin sin (2koo2) + sin (2kco{)'
But k = Ttins therefore ooCH = QJ2 — wCL
(12)
Using the simple trigonometric relationships
(ii) Lowpass to bandpass transformation Theorem 2: Given the p.t.f. G(z~l) of a lowpass digital filter, the p.t.f. of a bandpass digital filter having the same type of characteristic as G(z ~l ) and centred at the angular frequency to0 is given by
sin 29 = 2 sin0 cos 6 a • po = 2- sin . • —^+ JJ^ sm a +, sin - cos aa — -j8S
the result is obtained directly. Corollary 3: The upper and lower frequencies of a bandpass filter are connected with the cutoff frequency of the prototype low-pass filter by the equation
where a = cos (2kco0) co0 = centre frequency
where ooc = cutoff frequency of lowpass filter °°2 ~ upper frequency of resulting bandpass filter Wj = lower frequency of resulting bandpass filter
k = rr/Q s Qs = sampling angular frequency Proof: The required frequency function to replace tan (kco), so that the resultant p.t.f. is of bandpass form, is given in eqn. 5.
Proof: At the upper and lower frequencies we constrain the frequency function (eqn. 5) to have the values
Table 1 Lowpass
Transformation f(z Centre frequency co0 Relation to ooc 488
!
Low-high
Low-bandpass
_z-l(z-l _ a ) 1 -az-1
) 0
Low-band-elimination z~\z~x
-a)
1 - az~'
VJ2 6o 2 — co\
2
ELECTRONICS LETTERS
November 1967 Vol. 3 No. 11
cos (2A:aj0) — cos (2kco{) ^ - — — '-' = - tan (ka sin (2kwx) cos (2Arcu0) - cos
(13) (14)
sin (2ka>2)
preassigned frequencies. It is hoped that the solution of the problem of general frequency transformations, looked upon from the interpolation point of view, will be published in a forthcoming paper.
On substituting eqn. 12 into eqn. 13 (or eqn. 14) we have sin (2KCO2) cos (2A:ajj) + cos (2A:co2) sin (2kto{) _ sin (2ka)i) + sin (2kco2) ' sin
=
cos (2kco2) sin (2ka>{) — cos Qkoy^) sin (2kcoi) sin (2A:o>1){sin (2kaj{) + sin (2ka)2)} = — tan kojc cos (2&CO2) — cos (2Aro>,) __ — sin kwc
sin (2ka>2) + sin (2Ara>j)
cos ku>c
= — sin (2/:a>2) i.e.
&a>c — sin (2kto\) sin kwc
cos (2ka> 2 ) c o s (A:a>c) + sin (2kco2) sin (A:a>c) ){) cos (A:a>c) — sin (2ku>{) sin A:coc
= cos therefore
- ^
cos
i.e.
3rd October 1967
G. CONSTANTENIDES
Department of Electrical & Electronic Engineering City University London EC1, England References
cos (2koj2) cos (ka)c) — cos (2ka){) cos kcoc sin
A.
— t a n ka)c
= cos 2k
: (a>2 - ^- e ) =
1 HOLTZ, H., and LEONDES, C. T. : 'The synthesis of recursive digital
filters', / . Comput. Syst., 1966, 13, pp. 262-280 2 KAISER, J. F.: 'Design methods for sampled data filters', Proceedings of the first Allerton conference on circuit and system theory, Monticello, Illinois, 1963, pp. 221-236 3 RADER, c. M., and GOLD, B. : 'Digital filter design techniques in the frequency domain', Proc. Inst. Elect. Electronics Engrs., 1967, 55, pp. 149-171 4 CONSTANTINIDES, A. G.: 'Synthesis of Chebyshev digital filters', Electronics Letters, 1967, 3, pp. 124-126 5 CONSTANTINIDES, A. G.: 'Elliptic digital filters', ibid., 1967, 3, pp. 255256 6 BROOME, P.: 'A frequency transformation for numerical filters', Proc. Inst. Elect. Electronics Engrs., 1966, 54, pp. 326-327 7 CARATHEODORY, c.: 'Theory of functions', Vol. 1 (Chelsea Publishing Co., NY, 1954)
Within the baseband (n = 0), we have n = 0 , 1 , 2 , . . Wc
=
O) 2
(15)
— 60!
Hence the corollary (iii) Lowpass to band-elimination transformation: Theorem 3: Given the p.t.f. G{z~x) of a low-pass digital filter, the p.t.f. of a band-elimination digital filter, having the same type of characteristic as G{z~x) and centred at the angular frequency CU0, is given by
z-'jz1 -
1
- a)
TRANSIENT RESPONSE OF A CLASS OF NONLINEAR SYSTEMS An algorithm is given to be used in conjunction with the parameter-plane method and the describing-function method for rapid calculation of transient oscillations in the design of a class of nonlinear systems.
ocz~l
where a = cos (2A:co0) Proof: The proof of this theorem is similar to the proof of theorem 2. It follows the same lines using eqns. 2, 7 and 8. Two corollaries follow from this theorem. Their proofs are similar to corollaries 2 and 3, respectively. Corollary 4: The upper and lower frequencies of a bandelimination digital filter are connected with the centre frequency, by the relationship cos
(co2 —
Corollary 5: The upper and lower frequencies of a bandelimination digital filter are connected with the cutoff frequency of the prototype lowpass filter by the equation:
Summary of the results: The results are summarised in Table 1, where G^z"1) is the p.t.f. of a lowpass digital filter and g(z~x) is the required transformation for z"1 of Giz ~ l ) The following general theorem summarises all the given transformations: Theorem 4: Frequency transformations on the zplane, which convert lowpass digital filters into high-pass and bandpass (band-elimination) digital filters, are essentially functions of the form
n
- —1
(16)
v=i
which are known as unit functions (see eqn. 7). The transformations are obtained by constraining an appropriate function of the form of expr. 16 to have specific values at ELECTRONICS LETTERS
November 1967 Vol. 3 No. 11
This letter gives a straightforward algorithm for plotting the zero-input response of a class of nonlinear systems with arbitrary initial conditions. The classes of nonlinear systems considered are those for which the stability of self-excited oscillations are determined by the nonlinear differential equation
C(s)x + B(s)F(x) =
0
C
D
(
1
)
where s = d\dt, C(s) and B(s) are polynomials in s with the degree of C(s) being higher than the degree of B(s), and the function F(x) represents the nonlinearity. It will be assumed that the relative stability of nonlinear control systems described by eqn. 1 may be studied in the parameter plane using the approach of Krylov and Bogoliubov and the describing-function method. l~3 On the basis of this approach, the nonlinear function F(x) is 'linearised' as
F(x) = Nl(4)x
(2)
where ma i 1 r t h\(A) = — \F{A s i n ) s i n d
TTAJO
. . . . ( 3 )
The linearised differential equation corresponding to eqn. 1 is thus {C(s) + B{s)Nl{A)}x = 0
(4)
The advantage of using the parameter-plane approach in conjunction with the describing-function method is that insight is given as to the effects of the various parameters on transient behaviour in the preliminary stages of design.3 Also, 489
