EXPLOITING EQUIVALENCE BETWEEN CONTINUOUS AND DISCRETE TIME SYSTEMS Djordje BABIC Health Center "Dr Laza K. Lazarevic" Sabac, Popa Karana -2 -4 15000 Sabac, SERBIA AND MONTENEGRO Tel.: +381-64-153-6642;
[email protected] ABSTRACT This paper presents several novel structures for conversion between arbitrary sampling rates. The method allows arbitrary number of zeros at multiples of both input and output sampling rates, thus it has both good anti-imaging and good anti-aliasing properties. The novel structures are derived from continuous-time equivalents of CIC and modified comb filters. The implementation can be based on either the modified Farrow or transposed modified Farrow structure. By choosing one of these two alternatives appropriately, it is possible to shift most of the operations to lower sampling rate. Furthermore, it is possible to improve frequency domain performance by choosing continuoustime equivalent of the modified comb filter. In this way, we obtain very efficient filters for arbitrary sampling rate conversion, which are optimized directly in digital domain. 1.
INTRODUCTION
In many DSP applications it is required to change the sampling rate by a non-integer ratio. In this case, the required filter order becomes very large, and the overall number of coefficients in conventional polyphase implementation can be impractical [1]. In the case when the SRC factor is a ratio of two large relatively prime integers or irrational number it is beneficial to use polynomial-based filters [2]-[7]. Often in practice, the principle of oversampling is used together with sigma-delta AD conversion. For an oversampled signal, the ratio of sampling rate and desired signal bandwidth may be a very large number. Thus, the filter requirements are relaxed. The principle of oversampling makes it possible to use simple, power efficient structures as the first stage in the decimation chain [1]. A typical example is integer decimation by using cascaded integrator-comb (CIC) and modified comb filters as the first stage in a decimation chain [8]. However, CIC and modified comb filters cannot be directly used for SRC by an arbitrary non-integer factor. Several power efficient non-integer SRC structures have D. Babic was with the Institute of Communications Engineering, Tampere University of Technology and part of this work was done there. At the moment, the author is doing his civil service with the institution given above.
been proposed in the Literature. These structures are built as combination of CIC or modified comb and simple polynomial-based filters [2]-[9]. The common property of all these structures is the fact that the zeros of the transfer function are clustered at integer multiples of input sampling rate Fin, while the aliasing bands around the integer multiples of the output sample rate Fout are not attenuated very well. An effective solution to deal with this problem is the so-called transposed Farrow structure [3], [4]. However, the implementation complexity of the transposed Farrow structure as the first stage in the decimation chain may be costly. This paper presents several novel efficient noninteger sample rate conversion structures. The proposed structures are based on the continuous-time (CT) equivalent of the CIC filter and modified comb filters. Each of these novel structures can be realized by using two different alternatives. The choice of the structure depends on the overall SRC factor R, as well as on the signal and system wordlengths. The frequency response has an arbitrary number of zeros placed at integer multiples of both input and output sampling rates. The structures are power efficient because they have a small number of integer coefficient multipliers whose implementation is not costly 2.
CONTINUOUS-TIME EQUIVALENT OF CIC FILTER
The CIC filter has impulse response of finite length with all taps having value equal to unity [8]. The Nth order CIC decimation filter has the following transfer function: H CIC = H IN ( z ) H CN ( z ) =
(1 − z − R ) N . (1 − z −1 ) N
(1)
It consists of N cascaded digital integrator stages (see Fig. 1(a)) operating at high input data rate Fin, followed by N cascaded comb or differentiator stages (see Fig. 1(b)) operating at low sampling rate Fin / R (here R is an integer). +
z −1
z −1
−
(a) (b) Fig. 1.(a) Integrator with the transfer function H IN ( z ) = 1/(1 − z −1 ) N for N = 1. (b) Comb filter with the transfer function H CN ( z ) = (1 − z −1 ) N for N = 1.
ha(t) . The impulse response ha(t) can be represented as follows:
v(nl + N/2) cM(−N/2)
Fin
c1(−N/2)
c0(−N/2)
N −1 M
−1
Z
−1
cM(−N/2 + 1)
−1
c1(−N/2 + 1)
−1
Z
Z
c0(−N/2 + 1)
where basis functions fm(n, T, t) are given by
Z
c1(N/2 − 1)
v M (nl)
c0(N/2 − 1)
v1(nl)
v0(nl)
Fout u(l)
2 µ l −1
Fig. 2. Modified Farrow Structure.
The CT equivalent of CIC is an ideal integrator (zeroth order hold) [5]. The impulse response of the ideal integrator is rectangular pulse of length T. In our application, T is chosen to be equal to the input Tin or output Tout sampling interval of the discrete-time system. The impulse response of corresponding CIC filter is obtained by sampling the impulse response of the ideal integrator at T/R. The zero-phase frequency response of the ideal integrator is given by H i (ω ) =
sin(ωT ) . ωT
(2)
The cascade of N ideal integrators in CT domain corresponds to Nth order CIC filter in DT domain: ha (t ) = hsuuuuuuuuuuuuuuuuuuuuu ). i (t ) * hi (t ) *...* hi (tr
(3)
N
The impulse response of N cascaded ideal integrators is a piecewise polynomial, having N polynomial segments of order M=N−1. The resulting ha(t) is characterized by the following properties: (i) ha(t) can be nonzero for 0 ≤ t < NT and zero elsewhere, (ii) in each subinterval nT ≤ t < (n +1)T for n = 0, 1, …, N−1, ha(t) is expressible as a polynomial of degree M=N−1, and (iii) ha(t) is symmetric around t = NT/2, that is, ha(NT−t) = 2µk −1 v(k) Fin b0(k) −1
b1(kl) −1
Z
bM(k) −1
Z
Z
(4)
n =0 m=0
−1
Z
cM(N/2 − 1)
ha (t ) = ∑ ∑ d m (n) f m ( n, T , t )
−1
Z
I&D ov(l)
Fout
t − nT m ) ( f m (n, T , t ) = T 0
for nT ≤ t
