RECURSIVE DIGITAL FILTERS WITH DELTA MODULATION V. A. Pogribnoi and B. I. Yavorskii
Izvestiya VUZ. Radioe1ektronika, Vol. 28, No. 8, pp. 53-58, 1985 UDC 621.372.542:376.56
Methods are presented for calculating the coefficients and modeling the operation of recursive digital filters with linear delta modulation.
Delta modulation (DM) with a uniform quantization step and a constant discretization frequency is used in digital filtration by virtue of its simplicity of realization and noise immunity [1]. Procedures for the calculation and design of recursive digital filters (RDFs) using DM are known for the case in which the readings are represented in linear DM format, while the coefficients are presented in a pulse-code-modulation (PCM) format [2,3].
Procedures have not been developed for calculation, computer modeling, and design of RDFs in which all quantities are presented in DM format (RDFDMs). This limits the applications of DM in physical RDP designs. In this paper we present methods derived with interacting PCM and linear DM formats [4,5] for the calculation and modeling of RDFDMs and one practical realization.
Notation: Td is the DM discretization period; e( x) is the input-signal quantizin step in the DM format;
x ,i yi are the approximations of the input and output signals x(t) and y(t) respectively in the PCM format;
) h
e( and e(g) are the steps of filter-coefficient quantization in DM format and h ,r gr are the approximations of the latter in PCM format. For a harmonic signal x(t), working from its maximum amplitude Um and frequency fU, the delta-modulator discretization frequency at which the modulator is no overloaded [1]
) x U ( 1 m
d 2 U f /e
T (1)
We introduce initial expression obtained on the basis of [6]:
1 for 0
;
;
;0 ,
;0 ,
1 ) ( 1
) ( ) ( ) (
e x e x k
x x
x x
x e x
k k
i x i k
x k k k
k k x
k k x x
k
(2)
The equality Lk (e(kx)(x))/2(x) applies for a single-digit of the delta-modulator output binary code Lk . We write for the values of the output readings:
).
y ( y
y ,
y y ) y (
; y y
y ,
y y y
n 1
n n 1
n n n
n 1 n n 1
n n n
whence
whence
The quantization steps er(h), e(rg) are related to h ,r gr by expressions analogous to (2). Generally, e() 1 but it is often assumed for simplification that e() 1 and the 0 code of Lk is changed to –1 in performing the subsequent arithmetic operations. Then cross-multiplication degenerates into modulo 2 addition with negation the Lk code is used in shifting and the DF hardware in DM format has an advantage over the PCM format. This advantage is a higher
discretization frequency Td1 T1 2fU. We present the algorithm of RDFDM operation on the basis of [7]:
1 ,
1 1 1
0
h y g n
x y
d
d M
m
m m n M
m m m n
n (3)
It was shown in [4] that for the DM format
1
1
) 1 (
0 ) ( )
( ( )
) (
d
d M
m
mg m k M
m
mh xm k
k e e y e
y (4)
For RDFs, the result of filtration in the time domain with (3), (4) in interaction of the formats of the mixed algorithm corresponding to the Stieltjes algorithm DM and PCM
n
i M
m
M m
m m i m
x m i n
d d
g y h
e y
1 1 0
1 1 )
( )
(
) )
( (
1 1
) ( 1
0
) ( ) ( 1
1
d d
M m
mg m k M
m
mh xm k i
k n i
n e e y e
y
(5)
(6)
i.e., additional summations are needed in the RDF to obtain the result in PCM format. Then, according to (5), (6), the transfer functions of the RDF are respectively
) 1
/(
) (
1 1 1
0
d Md
m
m m M
m
m
mz g z
h z
H
) 1
/(
) (
1 1
) ( 1
0 ) ( )
(
d d
M m
m mg M
m
m mh
d z e z e z
H
(7)
In these formulas, quantizing steps of equal absolute magnitude are usually put equal to 1, which is necessary when multiplication in the PCM format is replaced by modulo 2 addition with negation in the DM format. It follows from (7) that all complex-conjugate 1 poles of the RDF transfer function will be on the unit circle in the z-plane, which corresponds to work on the stability boundary. To ensure stable operation, it is necessary that 1
1 1
)
(
d M
m
m mg z
e which can be
brought about, for example, by choosing e(g) 1. However, the choice e(g) 1 with the other steps equal to 1 is an obstacle to modulo 2 summation with negation, since the latter is done only with the code values L(i) 0,1 into which the quantization steps of equal absolute magnitude are converted. This difficulty can be overcome as follows:
1 )
( ( ) ( ) ( )
y g x h ,
, 0
0 ,
)(
)(
)(
)(
)(
)(
)(
k k g m
k k g m g k m M g
k M g g
g g M M
,
where M(m) (0,1) is the scale factor in the PCM format. As a result, operations on the coded values
0,1
, ( ) ( )
) ( )
(kxm Lmh Lkym Lmg
L are corresponded to the products ek(x)mem(h) and (y)kme(mg), can be performed in the usual manner. With the steps and scale factor chosen in this way, expression (4) assumes the form
1 , 0 ) ( ,
) ( )
( ( )
1 1
) ( )
( 1
0
) ( )
(
e e M y e y r
y rx r
M m
mg m m k
M m
mh x
m k k
d
d (8)
We note that multiplication of the number M(m) by the corresponding sum is accomplished in position code, since the result of the modulo 2 summations with negation goes to reversible counters whose output signals represent the corresponding sums in the right-hand side of (8) in position code. Stable operation of the RDF is ensured at the cost of a certain imprecision of the AFC, that arises with this procedure, since the poles 1 lie within the unit circle:
) 1
/(
) (
1 1
) ( ) ( 1
0 ) ( )
(
d d
M m
m mg m
M m
m mh
Md z e z M e z
H
In light of the above, the value of M(m) should be chosen to ensure stability of fee RDF and precision of the AFC. In practice, the value of M(m) depends on the ratio TTd1, T (2fU )1, where fU is the upper-limit frequency of the signal spectrum and the
(a) (b)
Fig. 1
maximum value of yk, and the multiplication by M(m) reduce to shifting of the position code of the corresponding number in the direction of lower-order digits.
A concrete RDFDM system usually consists of cascaded resonator links, which represent direct or canonical second-order forms of recursive and nonrecursive parts that contain shift registers, various adders, and reversible counters [1-3].
The following procedure is proposed for determination of RDFDM coefficients (on the basis of calculation of the RDF with PCM) [8].
The amplitude values u(max)y Hru(max)x , where u(max)x is the maximum amplitude of the harmonic input signal of resonant frequency fr and H r is the absolute value of the digital-resonator transfer function in PCM format for fr are used to find the discretization frequency Td1 according to (1); in this case (max) ( ) (min)x
y
m u ,e u
u
where u(min)x is the lowest input-signal amplitude. If Td1 is too large, it can be reduced to acceptable amplitude by appropriate selection of the ratio um /()u(max)y /u(min)x . Another way to lower it is to use an amplifier with automatic gain control.
The sequences P[I] and Q[I] of coefficients obtained according to [8] for the recursive and nonrecursive parts of the RDF resonators (links), presented in PCM format, are quantized by DM (Fig. la), where AC is the old value of the approximation in DM, ACC is the new value of the approximation in DM, DEL is the quantization step in DM, VYX is the output code of the delta modulator, ND is the number of filled cells of register GW (or HW), N is the number of coefficients in the original sequence, I is the present serial number of the initial-sequence coefficient, DN is the number of the present DM cycle, and DL is the number of DM cycles on the PCM period.
The calculation is checked with the aid of a simulation model of the RDFDM. (Fig. 1b, JX is the number of filled cells of register SX, AS1 is the contents of the adder in the nonrecursive part of the RDFDM, RC is the state of the reversible counter, AS is the old value of the approximation in the input DM, J is the number of filled cells of register SY, AC1 is the contents of the adder in the recursive part of the RDFDM, AC is the old value of the approximation in the recursive part of the RDFDM, ASS is the new value of the approximation in the input DM, HW and GW are the registers for the DM coefficient sequences, BXD is the new reading of the input signal in PCM format and the remaining notation is the same as in Fig. 1a). There the pulse characteristics or AFCs of the original RDF in PCM format and the RDFDM simulation model are compared by any known method (for example, with the rms. error as a criterion). If the result is
unsatisfactory, it is necessary to raise the frequency Td1 by lowering ().
In the RDFDM simulation model, the readings of the input signal xn (BXD) are floating-point decimal numbers. In delta modulation of these readings, the preceding approximated quantized reading (AC) and the value of the quantum step 1 (DEL) also floating-point decimal numbers, are added or subtracted. The delta-modulator output code
Lk (VYX) assumes the value 1 or 0. The product of two DM sequences is the sum of the terms Lj Lk .
(a) (b) Fig. 2
(c)
Then the result of subsequent summations AC1 will be presented in integer form, and there may be cases in which AC1 yn,n0,. Therefore the result AC1 is scaled (it is the output adder in such a way that its value is larger by a factor /, where is product of the actual values of () and is the product of the values used for
) (
. In this case, () is determined from (1), =1, 2. Then the stability-ensuring scale coefficient in the recursive part equals 2 (in Fig. I b, M(m)corresponds to DEL2).
To preserve the absolute value of the transfer coefficient, multiplication by analogous coefficient is also performed in the nonrecursive part of the RDFDM. In a specific practical realization that multiplication is performed by shifting the code in the reversing counter.
To illustrate the procedure, we present written results from calculation of a recursive resonator with DM, the algorithm of which is based on the expression
1 k
) g (
k 2 2 n
1 i 1
k ) i g (
k 1 1 n
1 i
1 k
) x i( ) h (
k 2 2 n
1 i 1
k
) x (i ) h (
k 0 n
1 i 2 n 2 1 n 1 2 n 2 n 0 n
yi
e y
e
e e e
e yˆ
gˆ yˆ gˆ xˆ hˆ xˆ hˆ yˆ
2 1
0 1;hˆ 1;gˆ
hˆ 1.5588458; g2 0,81, which corresponds to Tfr 1/12 (Fig. 2 a and b). The registers HW and GW of this RDFDM contain the individual difference-equation coefficients, represented in the DM format in accordance with rule (2). The number of approximating readings is the same for each coefficient at 64, which
gives = DEL = 2/64. For coefficients whose absolute values are smaller than the maximum, these registers contain the corresponding number of symbols of the zero delta code of the form ...010101. For example, for the coefficients P[I], I = 1, 2, of the recursive part of the RDF (in this case P[1] = 1.5588458, P[2] = -0.81), the corresponding RDFDM coefficients GS(ND), ND = 1,128 will be: 1...1 for ND = 1...50; 01...01 for ND = 51...64; 0...0 for ND = 65...90; and 10...10 for ND = 91...128. The reversing counters PC perform step-by-step modulo 2 summation of the products with negation in position code. Their respective outputs are
d
d M
1 m
m ) i
g (m l
1 i M
1 m
) x ( m ) i h (m l
1 i
) y ( e e
e and ,
p
Md , p is the number of coefficients. Each of these sums is accumulated in adder AS1 or AC1, respectively, scaled with the multiplier M(m)and then fed to the output adder. Figure 2a shows the AFCs of an PCM RDF and an RDFDM. The minor disagreement between the AFCs is the result of slope overload and coefficient drift due to drain noise at the that was used. Equipment and speed gains are obtained when the RDFDM is used as compared to the
nonrecursive-type DFDM, since the register length Md for storage of the coefficients and the DM signal sequences is determined in the latter case by the number of PCM readings of the pulse characteristic that is necessary to produce a DP with small truncation effects (Gibbs oscillations); for the case in the example, this is at least three periods, i.e., 36 weighting coefficients (as against only four coefficients for the RDFDM).
REFERENCES
1. R. Steele, Delta Modulation Systems, Pentech Press, 1974.
2. N. Kouvaras, "Operation on delta-modulated signals and their application on the realization of digital filters," REE, vol. 48, no. 9, PP. 431-438, 1978.
3. N. Kouvaras, "Some novel elements for delta-modulated signal processing," REE, vol. 51, no. 5, pp. 241-249, 1981.
4. V. A. Pogribnoi, "Digital spectral analysis using delta modulation," Dokl. AN SSSR, Ser. A, no. 6, pp. 74-79, 1982.
5. V. A. Pogribnoi, "Delta modulation in apparatus spectral analyses," Radiotekhnika i Elektronika, vol. 27, no. 7, pp.
1352-1361, 1982.
6. L. F. Rocha, B. Cernuschi-Frias, and C. Orda, "Correction and correlation using delay modulators," IEEE Proc., vol.
68, no. 8, pp. 90-92, 1980.
7. L.-Rabiner and B. Gold, Theory and Application of Digital Signal Processing, Prentice-Hall, 1975.
8. B. I. Yavorskii and Z. I. Dombrovskii, "Calculation of Chebyshev-type digital band filters," Radiotekhnika, vol. 36, no.
10, pp.. 79-81, 1981.
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