3.5.5 Root Mean Square Deviation comparison of the dierent methods
The output of the RSC4BI-algorithm was compared not only to the convolved reference signal but to the two conventional methods as well. These results are presented in Fig. 4647 for theK= (20, 30)cases whileN = 20. IfN =Kwas considered, the precision of the calculations depended on the algorithm, and as Fig. 46 shows, the RMSD of the system of linear equations was lower compared to the inversion and RSC4BI. Due to the underdeterminated system, if K= 30, then the inversion could not nd the proper solution (Sec. 3.5.4), while the system of linear equations gave unchanged precision, and the RMSD of RSC4BI was higher but still at an acceptable level.
Thesis II: Optimisation of the RSC4BI-algorithm
I investigated the RSC4BI-algorithm presented by A. Kraker, which is used to evaluate the Wiener-Hopf equation in a sliding window manner. I proved that this new method has a lower computational complexity than the conventional inversion calculation; therefore, it is a preferable solution for a sliding Wiener lter. I derived and compared the two calculation methods for RSC4BI based on the dyadic product and based on the permutation. Furthermore, I decreased the computational complexity to its half by using the fact that the input matrix is always symmetric; therefore, it is satisfactory to evaluate only the upper triangle parts.
Subthesis II1: I specied, which calculation method dyadic or permutation is more preferable for updating the auto-correlation matrix and the cross-correlation vector during evaluation of Wiener-Hopf equation. The preferable algorithm depends on the lter lengthK and the sliding window sizeN. Subthesis II2: I proved that the computational complexity of the RSC4BI-algorithm is O K2, while the conventional inversion methods require at least O K2.37286
operations. I also implemented and veried the proposed algorithm; furthermore I compared its performance to the conventional methods using Matlab and embedded hardware.
Subthesis II3: I improved the RSC4BI-algorithm, and I showed that only the upper triangle parts are required for the calculations because the input matrix is symmetric. Furthermore, I presented an improved algorithm, where the reuse of the already calculated variables can reduce the number of the required calculations.
(Please refer to Section 3.3 and Section 3.4.)
The results presented in this thesis were published in: J3.
4 Complex-valued extension of RSC4BI-algorithm
In telecommunication, the eciency of the spectrum usage is one of the main aspect of the network designing. Considering that the number of the available channels are limited in a given frequency band, the physical properties of these channels require such conguration that can maximize the transmission data rate. Studying the spectrum of a real-valued signal, it can be seen that its negative side is the same as the positive side. This means that the spectral eciency is only 50%, and the half of the frequency band is needlessly occupied. To avoid this issue, two solutions can be applied: either the unnecessary half has to be cut o, or additional information should be given to the transmission to ll both halves with unique data. A further aspect of the eciency is, how close can the channels be located to each other, which is limited by the sidelobe compression, but this topic is outside of scope in this thesis.
To improve the spectral eciently, rstly the so called side band transmission was applied. In this case, only the useful half of the spectrum was transmitted, then the receiver reconstructed the whole spectrum before processing the received data. The other opportunity is to give additional information to the transmission: a second signal source, which is independent of the original one. Because of their independence, after combining these sources, they can be considered as real and imaginary parts of a complex-valued baseband signal. To transmit it, two carriers are required: a cosine and a sine wave, which are orthogonal to each other; therefore, the receiver can reconstruct the original signal and process the complex-valued data.
Having complex-valued data, the conventional processing methods should be extended for handling complex numbers as well. In this thesis, the Kraker's RSC4BI-algorithm is improved to evaluate the Wiener lter in sliding window manner if complex data is given. This extension makes possible to identify the transmission channel by sliding Wiener lter (SWF) for such solutions like OFDM.
The operations in complex-valued data processing are deduced to real-valued calculations, which leads to the two following aspects of the design: computational complexity and latency (or runtime). As it is presented in Appendix B, a complex addition is performed through two real-valued additions, while a complex multiplication may require four multipliers and two adders; furthermore, it may contain more stages than the real-valued multiplication. Because of these, it is investigated in this section that after extension of the algorithms, how do the computational complexity and latency increase by performing complex-valued signal processing.
This chapter is organized as follows. Section 4.1 gives a short overview about the complex-valued extension of adaptive ltering and the WH equation. Section 4.2 presents, how to modify Kraker's al-gorithms to process complex signals. In Sec. 4.3, I investigate the complexity of these complex-valued calculations. The simulation results for the proposed methods in terms of runtime and error are pre-sented in Sec. 4.4. Finally, the conclusions are drawn in the last section, and the new scientic results are summarized.
4.1 Principle of adaptive ltering for complex-valued signals
The principle of adaptive ltering, which is presented in Sec. 3.1 and in Fig. 23, can be applied for complex-valued signals as well. In this case,x[n]is composed as a sum of two independent signals
x[n] =xR[n] +jxI[n] (41)
wherexR[n]is the real (or in-phase) component whilexI[n]is the imaginary (or quarature) component, which is rotated by π/2. To lterx[n], the FIR lter can be used that is described in (31) and (32).
Having a complex-valued input, the outputy[n]may also be complex depending on the mapping of the lter:C→R,C→C, etc. Considering these dierent mappings, the impulse response of the lterh[n]
can contain complex-valued coecients as well.
To identify this lter, an adaptive algorithm should minimise the cost function C of the lter coe-cients. Based on (38), when complex signals are present,C can be expressed as
C(h[n]) = E
dH[n]d[n] +hH[n]R[n]h[n]−2hH[n]p[n] (42) wherehHis the Hermitian transpose ofh, thushH =hT where overbar denotes the complex conjugation.
Furthermore,
R[n] = E
XH[n]X[n] (43)
is the auto-correlation matrix of the complex-valued input signal while p[n] = E
XH[n]d[n] (44)
is the cross-correlation vector of the input and output signals, which are complex-valued. After evaluating this equation and calculating the derivative of C(h[n]) with respect to h[n], the optimal coecients hbopt[n]can be given in the same form as in (312).