Sample Rate Conversion in Software Radio Terminals Tim Hentschel, Matthias Henker, Gerhard Fettweis Dresden University of Technology Mobile Communications Systems Chair D-01062 Dresden, Germany
[email protected] Abstract Feasibility of Software Radio mainly depends on the availability of a suitable hardware platform. A basic feature of this hardware platform is the ability to process signals of different mobile communications standards, which are usually based upon a diversity of master clock rates. Hence, the conversion between different sample rates is an important functionality of a Software Radio receiver. Preferably realized digitally, sample-rate conversion is conventionally regarded as interpolation. It will be shown that interpolation does not fulfill the anti-aliasing requirement which is the very criterion to be obeyed. Based upon this idea anti-aliasing filters for sample rate conversion are proposed. These filters are realized by a multi stage system with different stages for integer-factor and fractional sample rate conversion.
Introduction In the context of mobile communications systems the concept of software radio has become the subject of intensive study. Aiming for a universal receiver with virtually no restrictions regarding bandwidth, selectivity, frequency range and modulation schemes, the intended flexibility can only be achieved by digital signal processing while minimizing the number of analog components. All signal processing has to be realized on a fixed hardware platform, exclusively being controlled by software. This enables the adaptation of the terminal to present and future communications systems. A key component of a software radio receiver is the analog-to-digital converter (ADC) (Hentschel and Fettweis, 1999b). For reasons of simplification the AD-conversion is advantageously performed at a fixed clock rate as close to the antenna as possible. However, the different standards of operation require different symbol- or chiprates usually not having common integer divisors or ratios with respect to the digitization rate. Therefore the sample rate of the digital signal has to be converted to a standard-specific rate (Hentschel et al., 1998; Buracchini and Mastroforti, 1999). In order to cope with the wide-band nature of the signals comprising several channels, sophisticated solutions to sample-rate conversion have to be found. While sample rate conversion (SRC) by integer factors can be realized without difficulty it is much more complicated to find efficient software-controlled algorithms for fractional SRC. Because of a necessarily high intermediate sample rate most of them are only suitable for low input rates. On the other hand, polyphase structures avoiding high intermediate sample rates render the implementation of more than two rate change factors on a fixed hardware platform nearly impossible. A solution to the problem could be a periodically timevariant implementation allowing a parameterizable sample rate conversion by arbitrary rational factors without any clock rate higher than the input sample rate. Starting with some fundamental considerations on SRC the important difference between anti-aliasing and antiimaging is elaborated. Usually being regarded as the very solution to the problem of SRC, the disadvantages of pure interpolation are worked out in the course of the paper. Finally, investigations regarding the implementation of sample-rate converters and thereby derived solutions are presented.
Basics of Sample Rate Conversion (SRC) While sampling an analog time-continuous signal xa (t ) with a period T the digital signal x(k) is obtained. Sample rate conversion means converting this sequence x(k) to another sequence y(m) as if it were obtained from sampling xa (t ) with a period T . As a basic approach we use an analog interpretation: The time-continuous signal xa (t ) will be reconstructed from x(k) by digital-to-analog-conversion followed by low-pass filtering with ha (t ). Eventually, the reconstructed signal xa (t ) is re-sampled with a period T (Crochiere and Rabiner, 1983). Thus the filter ha (t ) has to fulfill two requirements:
to reconstruct the signal xa (t ), which corresponds to an interpolation of the signal x(k), and to perform band limitation, in order to avoid aliasing caused by the succeeding re-sampling.
Assuming a rational ratio of T and T a direct digital approach can be found. This provides a means of replacing the ADC and analog filter ha (t ) by the digital filter h(n) whose coefficients are obtained from sampling ha (t ) with a period TL = TM . Usually the digital filter h(n) will be implemented directly (figure 1). In this case the sample rate of the input signal is to be increased to an intermediate sample rate which the filter is clocked at. This intermediate samplerate is TL = L fin . It has to be observed that the sample rate of the input signal is generally relatively high in the context of mobile communications. This is due to the large bandwidth being occupied by the signals. Therefore the intermediate sample-rate quickly reaches values which real systems cannot cope with. Thus, the direct implementation of sample-rate conversion is not feasible. Interpolation by L x(k) fin
L" up-sampler (zero-filling)
hI (n)
Decimation by M h(n) L fin
hD (n)
anti imaging filter anti aliasing filter
y(m)
M#
fout
L
= M fin
down-sampler
Figure 1: Direct digital approach for SRC with arbitrary rational factor and decimator with integer factors
L M
by cascading interpolator
Since the impulse response ha (t ) has to be sampled with a rate of L times the input sample rate, the digital impulse response h(n) can be divided into L distinct sets hm (k) = h(kL + m). This division is the basis for the time-varying implementation of the filter h(n). The signal, having its sample rate converted, is then j ∞ M k y(m) = ∑ x k hm (k) (1) m L k= ∞ T M with = (2) T ; T 2 R; M ; L 2 N T L The time-varying filter is clocked at a rate which equals the maximum of the input- and the output-rate. This rate is generally much lower that the intermediate rate of the direct implementation. It can be concluded that for high input sample rates the only sensible implementation of the filter h(n) is a time-varying filter hm (k).
Differences between Anti-Aliasing and Anti-Imaging Filters Sample rate conversion has so far been connected to synchronization tasks, where the conversion factor is always in the vicinity of 1 (Vesma and Saram¨aki, 1996). Therefore SRC has widely been considered as a process of interpolation, i.e. signal reconstruction. Classical interpolation is a well-known mathematical problem of calculating in-between values of tabulated functions. It can be applied to signal processing where the actual interpolation process is realized by filters. In this case the interpolation constraint is the reconstruction of the original signal by means of fitting polynomials to the given samples of the signal. With respect to this constraint the filters are polynomial filters. A main characteristic of polynomial interpolation filters is that they have zeros of the transfer function that tend to be clustered about integer multiples of fin = 1=T (Schafer and Rabiner, 1973). Such filters can only perform image rejection while the suppression of aliasing components is generally not possible. Still, in case the input signal is band-limited with Ωc π, image rejection might be sufficient due to the lack of aliasing components beside the image-components. Such a band-limitation of the signal can be achieved by suppressing all signal components for Ω > Ωc , e.g. quantization noise from the analog-to-digital conversion process or adjacent channel interferers. This band-limitation also suppresses potential aliasing components and thereby fulfills the second requirement being set upon the filter h(n). From a general point-of-view, this approach carries the filtering a bit too far. It is sufficient to keep the band-ofinterest (channel-of-interest among several channels which the signal comprises) free from aliasing. Moreover, signal reconstruction, i.e. interpolation, is not really necessary if only a small fraction of the total bandwidth is to be re-constructed. Therefore we have to tackle SRC with arbitrary rational factors ML with a different emphasis: For oversampled and non-band-limited signals is it much more important to suppress aliasing than imaging components.
This idea is very important since filters for fractional sample rate conversion are almost always realized as polynomial interpolation filters. These filters would yield very poor SNR characteristics in the band of interest, in case non-band-limited signals had to be processed. In figure 2 it is shown how the sample-rate of a nonband-limited signal is converted. The band-of-interest occupies only a fraction of the total available band-width. Potential aliasing components rather than imaging components are suppressed. This leads to a distortion-free signal in the band-of-interest after SRC. Outside this band aliasing is acceptable. An interpolation filter had suppressed the image instead of the aliasing-components. This would have led to aliasing-distortions after SRC. band of interest spectrum of input signal x(n)
neighboring channels, interferer, quantization noise 0 Ωc
H (Ω)
image
spectrum of real or virtual up-sampled and filtered signal
2π
+ 0 Ωc
L=2 M=3 2π
aliasing components
spectrum of down-sampled output signal y(m) 0 Ωc
2π
Figure 2: Typical spectra for SRC with arbitrary rational factor and down-sampling.
L M,
realized by up-sampling, filtering
In the following we show that it is possible to implement powerful anti-aliasing filters. With respect to low power-consumption efficient solutions are suggested. They are based on generalized polynomial filters and modified time-varying comb-filters.
Multistage Sample Rate Conversion As we have seen in the previous sections SRC requires filtering. It has been shown that the emphasis lies on suppressing potential aliasing components. Furthermore it is well-known that the filter constraints are the stronger the smaller the oversampling ratio (OSR) of the signal-of-interest is, i.e. the larger the ratio between sample rate and bandwidth of the signal-of-interest is. If this ratio is large, there is a wide region where aliasing is acceptable. This could simplify the filter. In figure 2 this property is visualized by the relaxed slope of the transfer-function magnitude H (Ω). Thus it seems to be sensible to perform sample rate conversion in a cascaded manner with several stages, keeping the effort low at high sample rates and increasing it as the sample rate is reduced. Since the ADC will certainly oversample the signal-of-interest the overall SRC is effectively a reduction of sample rate. The cascade could therefore comprise several steps of integer factor SRC and one step that realizes the remaining fractional SRC by a factor between 1 and 0.5. The ordering of the different stages in such a cascade is still an open question. Integer factor sample rate decimation is a well investigated problem with a wide variety of solutions (Hogenauer, 1981; Crochiere and Rabiner, 1983; Hentschel and Fettweis, 1999a). However, SRC by rational factors certainly requires more effort than integer decimation. Therefore it seems to be most sensible to place it first in the cascade (see figure 3), enabling the highest degree of relaxation of filter constraints due to the high oversampling ratio. Only relatively narrow-band aliasing-components have to be suppressed. This point of view is completely different from conventional approaches, where first integer sample-rate conversion (decimation) is performed and finally the sample-rate of the signal is converted to the target rate. For reasons of efficiency multi-rate filters can be used to great advantage for integer factor SRC. These filters perform channel-selection, band-limitation, and decimation simultaneously. Thereby low oversampling ratios
SRC with rational factor
z
decimation by integer factors }|
{
stage 1
stage 2
stage n
SRC
SRC
SRC
x(n) fin
fin fout overall down-sampling
y(m) fout
requirements, effort, and word-length sample-rate (clock-rate) operations per time (effort clock-rate)
Figure 3: Effort vs. sample rate in cascaded sample rate conversion system result for the signal-of-interest. Hence the requirements on filters for fractional SRC being applied after integer factor SRC were very strong. This would result in complex and costly implementations. If simple interpolation were the issue, multi-rate filters would not be sensibly applicable, since the required band-limitation with Ωc π could not be realized. The novel approach being presented here is merely based upon the anti-aliasing constraint. This enables the application of both, multi-rate filters for band-limitation (and decimation) and relatively simple filters for fractional SRC. The anti-aliasing filters presented in the following have a complexity comparable to interpolation filters. In conjunction with multi-rate filters for integer factor SRC and band-limitation they offer a means for fractional SRC with minimum effort.
Proposed Filter Structures for SRC With anti-aliasing being the most important constraint, the filters for fractional SRC can be designed. Supposing that digitization will certainly be done with a comfortable oversampling ratio OSR 1, and SRC is performed in the first stage (see figure 3), the filters can have a very small pass-band, a set of equally spaced stop-bands attenuating the aliasing components, and a considerable number of “don’t-care-bands”. Possible solutions for such filters are
time-varying filters on a time-invariant or time-varying Farrow structure (Farrow, 1988; Ramstad, 1998; Lundheim and Ramstad, 1999), or time-varying comb-filters (Henker et al., 1999).
The first are digital filters having implemented the time-continuous envelope of the impulse response rather than having stored the samples of the (time-discrete) impulse response. At run-time the required coefficients are determined. Such filters have been widely employed for interpolation purposes. However, it is also possible to implement filters performing anti-aliasing on the Farrow-structure (for certain ratios ML only). Another approach is the application of comb-filters. They have been used for both, anti-imaging and antialiasing filtering (Hogenauer, 1981). The transfer-function of an Nth order CIC-filter (cascaded-integeratorcomb) is !N
R 1
H (z) =
∑z
k=0
k
=
1 1
z z
R 1
N =
|
1
z
1
{z
integrator
1 } |
N
z
{z
R N }
(3)
differentiator
where R is the up-sampling- or down-sampling factor, respectively the application. Equation (3) suggests the separate implementation of the differentiator- and the integrator section. Being periodic in R the differentiatorsection can always be implemented at the lower sample rate. If these filters are applied according to figure 1 the integrator section is to be clocked at the high intermediate sample rate. In order to avoid this it can be realized in a time-varying manner at a lower clock-rate. Exploiting the fact that the input signal to the integrator section is
x (k)
x(k) fin
D |
L"
D {z
y (k + ρL )
z1 (kL + 1 + ρ) zN (kL + 1 + ρ)
I
}
|
M#
I
L fin {z
DK
}
DK
|
{z
N =NI +ND
NI
y(m) = y(k + ρL ) fout
}
L
= M fin
ND
NI + ND
z
1 stages
}|
{
L5;1 L5;2 L4;1 +
L4;2
x(k) fin L fin
L3;1
NI 1
z
}|
D DDS
L5;3
{
D
z1
ρ(m)=L 1 ϕ(m) ρN
z2
Iv
L2;1
ρN
1
+
L3;2
z3
Iv
ρN
2
+
L4;3
3
fout
M
+
+
+
y (m) fout
DK
y(m)
DK
|
{z
}
fout
L
= M fin
ND
Figure 4: Time-varying CIC-filter (bottom) for SRC with arbitrary rational factors (with I = Iz 1 , D = 1 z 1 ) and equivalent cascade (top)
1 1 z
1
, Iv =
zero-padded due to the up-sampling process, the states of the integrators can be calculated at any time-instance without the need of calculating all intermediate states. In order to consider each input-sample (before up-sampling) the states of the integrator-section can be calculated at the time-instances defined by the input sample-rate: 0
1
0
z1 (k + 1) L1 1 B z2 (k + 1) C B L2 1 B C B C = B .. B .. @ A @ . .
L1 2 L2 2 .. .
;
;
;
zN (k + 1)
y (m) = y k +
;
LN 1 LN 2 ;
ρ(m) = ρN L
with
1 (m)
8 (N > < (N
Li
;
j=
> :
;
ρN
..
.
2 (m)
j)! (L 1+i j)! i)! (i j)! (L 1)!
1 0
10
L1 N L2 N C C .. C . A ; ;
1
z1 (k) + x (k) B C z2 (k) B C C B .. @ A . zN (k)
LN N ;
2N
(4)
ρ 1 (m )
0
1
B @
C A
z1 (k) + x (k) B C z2 (k) C B B C . .. 1 B C
for
i> j
for for
i= j i< j
zN 1 (k) zN (k)
(5)
(6)
i
ρi (m) = ∏ ρ(m) + h h=1
(7)
The coefficients Li j have to be calculated only once just before setting up the sample rate converter. Meanwhile the coefficients ρi (m) are time-varying and have to be calculated for every output sample y(m) at run-time. Equations (4,5) suggest a structure for the SRC which can be seen in figure 4. The word-length growth of the complete time-varying CIC-filter can be calculated as wlout d(NI 1) ld(L) + ND ld(M ) + ld((ND + NI 1)!)e + wlin (8) ;
where wlin and wlout are the word-lengths of the input- and the output-signal, respectively. NI and ND are the order of the interpolation- and the decimation-filter, respectively. Finally it should be noted that for NI = ND = 1 the whole structure can be realized multiplier-free if the analog-to-digital converter preceding the SRC is a 1-bit Sigma-Delta-ADC.
Conclusion Sample rate conversion is a vital functionality of software radio terminals. By analyzing the requirements and the area of application we have found that conventional interpolation is not well suited for realizing SRC in mobile communications terminals due to their lack of aliasing-component attenuation. Therefore we have proposed two filter structures which perform anti-aliasing while keeping the effort sufficiently low. Special emphasis has been given to the time-varying implementation of CIC-filters.
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