Multiband signal processing by using nonuniform sampling and ...

Multiband signal processing by using nonuniform sampling and
iterative updating of autocorrelation matrix
Modris Greitāns
Institute of Electronics and Computer Science, University of Latvia,
Latvia
E-mail: modris@edi.lv
Abstract
The approach to multiband signal processing is considered.
The method discussed in this paper is
based on nonuniform sampling, Minimum Variance
filter and iterative updating of autocorrelation matrix.
That allows to process a multiband signal even
if the number of known signal samples is less than
equivalent of Nyquist criterion for uniform sampling.
The proposed approach of multiband signal processing
is suitable for spectral analysis, estimation of
power spectral density and autocorrelation functions
as well as for signal reconstruction. The information
about the limits of the signal’s subband frequencies
gives the possibility to reconstruct the waveform of
each signal part separately. It means that the suggested
multiband signal processing method provides
also some capability of signal subband filtering. The
performance of the method is illustrated by the conjoint
GSM900 and GSM1800 signal processing example.
1 Introduction
Often the multiband signal processing is based on calculation
of special sampling series in accordance with
the signal spectral region location [1]. The possibility
of achieving the minimum sampling density (equivalent
of the Nyquist rate for uniform sampling) depends
on possibility to tessellate frequency space by
a group of translations. In general, the calculated
sampling instants of a multiband signal are spaced
nonuniformly therefore any changes of the signal subband
limits leads to the necessity to recalculate signal
sampling series.
This paper discusses the multiband signal processing
if arbitrary nonuniform sampling series is used. If
it has a quality of frequency aliasing suppression [2],
then there are no special requirements regarding the
exact values of the sampling moment. Only two general
conditions should be considered – the maximum
gap between two known samples and the sampling
density should be conformed to the equivalent signal
bandwidth [3, 4].
Traditionally, signal processing methods take into
account only one parameter of signal power spectral
density (PSD) function P (f), namely, the frequency
boundaries of the spectrum. If such limits do exist,
the signals are called band-limited signals. The
PSD function of a multiband signal consists of several
separate frequency regions. It can take different
appearance within these spectral subbands. One way
to characterize the form of PSD is to use the equivalent
bandwidth of the band-limited signal defined as
[3]
σ e =
∫ ∞
−∞ P (f)df
max(P (f)) . (1)
It is obvious that for real applications the value
of the equivalent signal bandwidth is usually smaller
than the actual signal bandwidth. Therefore, as it
is shown in [4], nonuniform sampling allows to process
signals employing less signal samples than it is

equired by the Nyquist criterion in conformity with
the cumulative signal bandwidth.
2 Processing method
The existence of several subbands in a multiband signal
spectrum determines the necessity for developing
of a special processing method taking into account
the information about the boundaries of the signal
frequency regions. The suggested multiband signal
processing approach is based on the Minimum Variance
method [3, 5]. The basic idea of this method
is to minimize the variance of the narrowband filter
output signal. The frequency response of this filter
adapts to the input signal spectral components on
each frequency of interest. The variance of the output
process is determined as:
ρ = a H Ra, (2)
where a is the vector of filter coefficients, while R is
the signal autocorrelation matrix. In addition, filter
coefficients should guarantee that on the frequency
f 0 the gain of the filter response will be one. This
condition could be described as:
e H (f 0 )a = 1, (3)
where e i (f 0 ) = exp(j2πf 0 t i ). On the other hand, the
expression (3) means that a sinusoid at frequency f 0
passes through the filter designed for this frequency
without distortion. It is shown in [6] that the coefficients
of the filter under condition (3) for the frequency
f 0 are determined as:
a(f 0 ) =
R −1 e(f 0 )
e H (f 0 )R −1 e(f 0 ) . (4)
Taking into account expression (3), the output s of
the designed filter
s(f 0 ) = xa(f 0 ) (5)
can be interpreted also as a complex spectral value
(analogous of Fourier transform value) of signal x on
the frequency f 0 [7]. Therefore the PSD value of the
signal on this frequency can be calculated as
p(f 0 ) = s(f 0 )s ∗ (f 0 ). (6)
For the multiband signal processing task, each signal
subband should be covered by the set of such filters.
The distance between filter frequencies can be
chosen equal to the frequency step of Discrete Fourier
transform (DFT) – ∆f = 1 Θ
, where Θ is the length
of the signal to be analyzed.
According to expression (4) the filter coefficients
depend on the signal autocorrelation matrix. Usually
the values of this matrix are not known a priori.
Therefore the estimates of autocorrelation matrix
values should be calculated from known signal
samples. The traditional approach for obtaining correlation
matrix is based on averaging of the mutual
products of signal samples. It is not applicable in
the nonuniform sampling case, because the time intervals
between sampling points are not distributed
regularly. Instead of that the cross-relation of signal’s
autocorrelation and power spectral density functions
R(τ) =
∫ ∞
−∞
P (f)e j2πfτ df (7)
is employed for R calculation [8]. Moreover, the
expression (7) allows to take into account also the
known values of signal spectral subbands, because
the integration should be done only in the defined
frequency regions. The most popular and simple way
to obtain P (f) estimate from signal samples in the
nonuniform sampling case is to use DFT.
In accordance with approach described above the
spectral analysis is performed in the fixed set of frequencies
f = [f 1 , f 2 , ...f M ]. The known signal values
x = [x 1 , x 2 , ...x N ] are sampled at defined time
instants t = [t 1 , t 2 , ...t N ]. Thereby the values of autocorrelation
function can be obtained as:
ˆR (0) =
ˆxE H 2
∣ N ∣ E, (8)
where E = exp(−j2πf m t n ) and ·(0) means that it
is zero order estimate of R. Signal autocorrelation
matrix values obtained by (8) are rather rough estimates
that lead to rough estimation of signal complex
spectral function values
Ŝ (0) =
E ˆR (0)−1 x T
diag(ER (0)−1 E H )
(9)

and PSD function values
ˆP (0) =
∣
∣Ŝ(0) ∣ ∣∣
2
. (10)
A special iterative updating algorithm, similar
to described in [4, 7], is used to improve the results
of processing. According to this algorithm the
(i+1) − th order estimate of signal autocorrelation
matrix is updated from the i−th order P (i) estimate
in the following way
m=M
∑
ˆR (i+1)
lk
=
m=1
ˆP (i)
m E ∗ mlE mk . (11)
Now, using matrix ˆR (i+1) , the estimates Ŝ(i+1) and
ˆP (i+1) can be calculated using expressions (9)-(10).
In effect, an iterative algorithm has been derived.
The iteration process (9)-(11) can be stopped when
the difference
∆ = ‖ ˆP (i+1) − ˆP (i) ‖
becomes small.
Although ˆR (i) is a positive definite symmetric matrix,
it becomes ill conditioned as the number of the
known samples increase [9]. In that case, the direct
inversion of ˆR (i) leads to processing errors. Therefore
the expression for obtaining PSD of signal could be
derived as
ˆP (i) =
dx T 2
∣diag(dE H ) ∣ , (12)
where matrix d is solution of following matrix equation:
ˆR (i) d = E. (13)
The equation (13) could be solved by iterative
methods [9, 10], for example, the conjugate gradient
method.
As it was mentioned above, the vector S can be
interpreted as complex spectral values, therefore the
inverse DFT could be used to obtain the values of
reconstructed signal [7]. If the multiband signal processing
task is to perform some subband filtering then
for inverse DFT input only certain parts of estimated
S values could be exploited.
3 Example
The conjoint GSM900 and GSM1800 signal processing
is chosen as an example of practical application of
proposed multiband signal processing approach. The
primary band of GSM900 includes two subbands of
25 MHz each, 890-915 MHz for uplink (Mobile to
Base) and 935-960 MHz for downlink (Base to Mobile).
The GSM1800 includes the two domains 1710-
1785 MHz and 1805-1880 MHz, i.e., twice 75 MHz:
tree times as much as the primary 900 MHz band
[11]. The central frequencies of the GSM channels are
spread evenly every 200kHz within these bands, starting
200 kHz away from the band borders. 124 different
frequency slots are therefore defined in 25 MHz
band, and 374 in 75 MHz band. The spectrum of
the GMSK modulation used in GSM is somewhat
wider than 200 kHz, resulting in some level of interference
between bursts on adjacent frequency slots.
Frequency planning must take the effect of adjacent
channel overlapping into account. Therefore, in practice,
not all of frequency slots are used in base station
and the equivalent signal bandwidth usually is
almost twice narrower than cumulative bandwidth.
That provides the possibility to gain certain benefits
from applying nonuniform sampling and using the
discussed signal processing approach.
The simulation example is considered, where altogether
40 frequency slots are used simultaneously.
The equivalent signal bandwidth in this case is approximately
few tens of MHz, while total signal bandwidth
to be processed is 200 MHz, because the positions
of currently used frequency slots are not known.
The equivalent of Nyquist rate for such a multiband
signal is 400 MSamples/s. Taken into account equivalent
bandwidth for discussed example signal, the
nonuniform sampling with 62.5 MSamples/s is exploited.
The nonuniform sampling series is obtained
from uniform sampling with frequency 4 GHz by
random selection 1
64−th of sampling instants. That
guarantees the necessary frequency aliasing suppression.
The performance of proposed processing approach
is compared with results obtained by processing
method based on Discrete Fourier transform. Figure
1 shows the PSD estimate calculated as a

70
70
60
60
PSD (dB)
50
40
50
40
30
30
20
900 920 940 960
1720 1740 1760 1780 1800 1820 1840 1860 1880 20
frequency (MHz)
Figure 1: PSD estimate from DFT of uniformly sampled (4 GSamples/sec) GSM signal.
70
70
60
60
PSD (dB)
50
40
50
40
30
30
20
900 920 940 960
1720 1740 1760 1780 1800 1820 1840 1860 1880 20
frequency (MHz)
Figure 2: PSD estimate from DFT of nonuniformly sampled (62.5 MSamples/sec) GSM signal.
70
70
60
60
PSD (dB)
50
40
50
40
30
30
20
900 920 940 960
1720 1740 1760 1780 1800 1820 1840 1860 1880 20
frequency (MHz)
Figure 3: PSD estimate with iterative MV filter of nonuniformly sampled (62.5 MSamples/sec) GSM signal.

squared module of FFT result from uniformly sampled
(4 GHz) GSM signal. Eight sample series of
131072 samples each are averaged without any additional
windowing. The form of PSD estimate is
affected by sidelobes from used frequency slots. Figure
2 presents the PSD estimate obtained in a similar
way, but 1 64−th of previously processed samples are
left in random way and DFT instead of FFT is used.
It is obvious that only high-powered frequency slots
show up from noise floor originated by nonuniform
sampling. The described iterative multiband signal
processing approach eliminates both imperfections illustrated
before - sidelobes of used frequency slots
and noise floor of nonuniform sampling. It is demonstrated
in the Figure 3. The PSD estimate clearly
displays all used frequency slots frequency and relative
power.
4 Conclusions
The simulation results confirm the applicability of
discussed approach for multiband signal processing.
The traditional multiband signal processing methods
require specific calculation of sampling series in accordance
with limits of signal frequency bands. Instead,
the proposed method operates with arbitrary
nonuniform sampling, if it provides necessary frequency
aliasing suppression. Moreover, the number
of used signal samples could be significantly less then
equivalent of Nyquist rate, if the equivalent signal
bandwidth is less than cumulative signal bandwidth.
The serious disadvantage of method described in this
paper is its complexity in the terms of required mathematical
calculations due to iterative nature of obtaining
estimates of power spectral density and autocorrelation
functions.
References
[3] S. M. Marple Jr., Digital spectral analysis with
applications, Prentice-Hall, 1987.
[4] M. Greitans, “ Iterative Reconstruction of Lost
Samples Using Updating of Autocorrelation Matrix”,
in Proc. SampTA’97, Workshop on Sampling
Theory & Applications, Aveiro, Portugal,
pp. 155-160, Jun. 1997.
[5] S. M. Kay and S. L. Marple Jr., “Spectrum analysis
- a modern perspective”, Proc. of the IEEE,
vol. 69, no. 11, pp. 1525-1578, 1981.
[6] McDonough R. N., “Aplication of the
Maximum-Likelihood Method and the
Maximum-Entropy Method to Array Processing”,
Chapter 6 in Nonlinear Methods of
Spectral Analysis, 2nd ed., S. Haykin, ed.,
Springer-Verlag, New Yourk, 1983.
[7] V. Liepinsh, “An algorithm for estimation of discrete
Fourier transform from sparse data”, Automatic
control and computer sciences, vol. 29,
no. 3, pp. 27-41, 1996.
[8] J. S. Bendat, A. G. Piersol, Engineering applications
of correlation and spectral analysis, John
Wiley & Sons Inc., 1980.
[9] A. K. Jain and S. Ranganath, “Extrapolation algorithms
for discrete signals with application in
spectral analysis”, IEEE Trans. Acoust., Speech,
Signal Processing, vol. ASSP-29, no. 4, pp. 830-
845, 1981.
[10] G. H. Golub and C. F. van Loan, Matrix computations,
The Johns Hopkins University Press,
Baltimore, 1989.
[11] M. Mouly and M. B. Pautet, The GSM System
for Mobile Communications, Cell & Sys, 1992.
[1] J. R. Higgins, “Some gap sampling series for
multiband signals”, Signal Processing, vol. 12,
no. 3, pp. 313-319, 1987.
[2] I. Bilinskis and A. Mikelsons, Randomized Signal
Processing, Prentice-Hall, 1992.