Multiband signal processing by using nonuniform sampling and ...

Multiband signal processing by using nonuniform sampling and iterative updating of autocorrelation matrix Modris Greit¯ans Institute of Electronics and Computer Science, University of Latvia, Latvia E-mail: [email protected]

Abstract

sampling instants of a multiband signal are spaced nonuniformly therefore any changes of the signal subThe approach to multiband signal processing is con- band limits leads to the necessity to recalculate signal sidered. The method discussed in this paper is sampling series. based on nonuniform sampling, Minimum Variance This paper discusses the multiband signal processfilter and iterative updating of autocorrelation ma- ing if arbitrary nonuniform sampling series is used. If trix. That allows to process a multiband signal even it has a quality of frequency aliasing suppression [2], if the number of known signal samples is less than then there are no special requirements regarding the equivalent of Nyquist criterion for uniform sampling. exact values of the sampling moment. Only two genThe proposed approach of multiband signal process- eral conditions should be considered – the maximum ing is suitable for spectral analysis, estimation of gap between two known samples and the sampling power spectral density and autocorrelation functions density should be conformed to the equivalent signal as well as for signal reconstruction. The information bandwidth [3, 4]. about the limits of the signal’s subband frequencies Traditionally, signal processing methods take into gives the possibility to reconstruct the waveform of account only one parameter of signal power spectral each signal part separately. It means that the sug- density (PSD) function P (f ), namely, the frequency gested multiband signal processing method provides boundaries of the spectrum. If such limits do exalso some capability of signal subband filtering. The ist, the signals are called band-limited signals. The performance of the method is illustrated by the con- PSD function of a multiband signal consists of sevjoint GSM900 and GSM1800 signal processing exam- eral separate frequency regions. It can take different ple. 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 1 Introduction [3] R∞ P (f )df σe = −∞ . (1) Often the multiband signal processing is based on calmax(P (f )) culation of special sampling series in accordance with the signal spectral region location [1]. The possibility It is obvious that for real applications the value of achieving the minimum sampling density (equiv- of the equivalent signal bandwidth is usually smaller alent of the Nyquist rate for uniform sampling) de- than the actual signal bandwidth. Therefore, as it pends on possibility to tessellate frequency space by is shown in [4], nonuniform sampling allows to proa group of translations. In general, the calculated cess signals employing less signal samples than it is

required by the Nyquist criterion in conformity with the cumulative signal bandwidth.

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 1 2 Processing method , where Θ is the length transform (DFT) – ∆f = Θ of the signal to be analyzed. The existence of several subbands in a multiband sigAccording to expression (4) the filter coefficients nal spectrum determines the necessity for developing depend on the signal autocorrelation matrix. Usuof a special processing method taking into account ally the values of this matrix are not known a prithe information about the boundaries of the signal ori. Therefore the estimates of autocorrelation mafrequency regions. The suggested multiband signal trix values should be calculated from known signal processing approach is based on the Minimum Varisamples. The traditional approach for obtaining corance method [3, 5]. The basic idea of this method relation matrix is based on averaging of the mutual is to minimize the variance of the narrowband filter products of signal samples. It is not applicable in output signal. The frequency response of this filter the nonuniform sampling case, because the time inadapts to the input signal spectral components on tervals between sampling points are not distributed each frequency of interest. The variance of the outregularly. Instead of that the cross-relation of signal’s put process is determined as: autocorrelation and power spectral density functions Z ∞ ρ = aH Ra, (2) R(τ ) = P (f )ej2πf τ df (7) 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 is employed for R calculation [8]. Moreover, the f0 the gain of the filter response will be one. This expression (7) allows to take into account also the known values of signal spectral subbands, because condition could be described as: the integration should be done only in the defined eH (f0 )a = 1, (3) frequency regions. The most popular and simple way to obtain P (f ) estimate from signal samples in the where ei (f0 ) = exp(j2πf0 ti ). On the other hand, the nonuniform sampling case is to use DFT. expression (3) means that a sinusoid at frequency f0 In accordance with approach described above the passes through the filter designed for this frequency spectral analysis is performed in the fixed set of frewithout distortion. It is shown in [6] that the coefquencies f = [f1 , f2 , ...fM ]. The known signal valficients of the filter under condition (3) for the freues x = [x1 , x2 , ...xN ] are sampled at defined time quency f0 are determined as: instants t = [t1 , t2 , ...tN ]. Thereby the values of autocorrelation function can be obtained as: R−1 e(f0 ) a(f0 ) = H . (4) H 2 e (f0 )R−1 e(f0 ) x ˆ E (0) ˆ E, (8) R = Taking into account expression (3), the output s of N the designed filter where E = exp(−j2πfm tn ) and ·(0) means that it s(f0 ) = xa(f0 ) (5) is zero order estimate of R. Signal autocorrelation can be interpreted also as a complex spectral value matrix values obtained by (8) are rather rough esti(analogous of Fourier transform value) of signal x on mates that lead to rough estimation of signal complex the frequency f0 [7]. Therefore the PSD value of the spectral function values signal on this frequency can be calculated as ˆ (0)−1 xT ER (0) ˆ S = (9) p(f0 ) = s(f0 )s∗ (f0 ). (6) diag(ER(0)−1 EH )

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and PSD function values ˆ (0) P

ˆ (0) 2 = S .

(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 ˆ (i+1) = R lk

m=M X

(i) ∗ Pˆm Eml Emk .

(11)

m=1

ˆ (i+1) , the estimates S ˆ (i+1) and Now, using matrix R (i+1) ˆ P 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 ˆ (i+1) − P ˆ (i) k ∆ = kP becomes small. ˆ (i) is a positive definite symmetric maAlthough R trix, it becomes ill conditioned as the number of the known samples increase [9]. In that case, the direct ˆ (i) leads to processing errors. Therefore inversion of R the expression for obtaining PSD of signal could be derived as 2 dxT ˆ (i) = P diag(dEH ) ,

(12)

where matrix d is solution of following matrix equation: ˆ (i) d = E. R (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.

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 17101785 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 1 random selection 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

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Figure 1: PSD estimate from DFT of uniformly sampled (4 GSamples/sec) GSM signal. 70

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Figure 2: PSD estimate from DFT of nonuniformly sampled (62.5 MSamples/sec) GSM signal. 70

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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 1 −th of previously processed samples are way, but 64 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.

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Conclusions

[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.

The simulation results confirm the applicability of discussed approach for multiband signal processing. The traditional multiband signal processing methods [8] J. S. Bendat, A. G. Piersol, Engineering applirequire specific calculation of sampling series in accations of correlation and spectral analysis, John cordance with limits of signal frequency bands. InWiley & Sons Inc., 1980. stead, the proposed method operates with arbitrary nonuniform sampling, if it provides necessary fre[9] A. K. Jain and S. Ranganath, “Extrapolation alquency aliasing suppression. Moreover, the number gorithms for discrete signals with application in of used signal samples could be significantly less then spectral analysis”, IEEE Trans. Acoust., Speech, equivalent of Nyquist rate, if the equivalent signal Signal Processing, vol. ASSP-29, no. 4, pp. 830bandwidth is less than cumulative signal bandwidth. 845, 1981. The serious disadvantage of method described in this paper is its complexity in the terms of required math- [10] G. H. Golub and C. F. van Loan, Matrix comematical calculations due to iterative nature of obputations, The Johns Hopkins University Press, taining estimates of power spectral density and autoBaltimore, 1989. correlation functions. [11] M. Mouly and M. B. Pautet, The GSM System for Mobile Communications, Cell & Sys, 1992.

References [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.