0 0.005 0.01 0.015 0.02 0.025 0.03 Time (s)
-400 -200 0 200 400 600 800 1000 1200
RMSD (dB)
K = 10 K = 20 K = 30 K = 40
Figure 58: RMSD of RSC4BI if the imaginary part of the main diagonal not removed
lter coecients in general. At the same time, the system of the linear equations method calculated the corresponding results because it is an optimised algorithm without using matrix inversion (cf. Sec. 3.5.1).
4.4.6 Round-o error of the main diagonal in the inverse of the auto-correlation matix As it was shown in Sec. 4.2.5, the imaginary part of the main diagonal in R−1 had to be removed after updating the inverse of a matrix by RSC4BI to avoid the round-o error and its accumulation. In this simulation, the behavior of the algorithm was investigated if these imaginary parts were not set to zero. The results of these simulations are presented in Fig. 58 for various values ofK whileN = 40and the input signal was a Gaussian white noise withN2(0.1,1).
As it can be seen, when K was much lower than N, then the RMSD was also low, and it did not diverge, but the results could be distorted (seeK= 20in Fig. 58) if no correction was applied in the main diagonal ofR−1. By adding more taps to the lter (K= [30,40]), the algorithm became unstable, despite that its stability and correctness were demonstrated previously (Sec. 4.4.3), when the main diagonal was R−1 is compensated (cf. theK=N cases in Fig. 53 and in Fig. 58 theK= 40curve).
The hardware requirements of the new version are also analysed, and it is shown that the complexity is stillO K2, which is better than the conventional inversion methods. However, the numbers and the arithmetical units are complex-valued; therefore, they have to be deduced to addition and multiplication of real numbers. These implementations, the ways to perform complex-valued operations, are out of topic in this dissertation. Although∼3−5 times more adders and multipliers are required, the complexity is still squarely proportional to the number of lter coecients.
As the simulated results show, the complex-valued RSC4BI is a suitable solution to update the WF;
however, it suers from some limitations, which have to be taken into account and from round-o error, which has to be compensated during the inversion.
Thesis III: Extension of the RSC4BI-algorithm for complex-valued signals
I generalized the RSC4BI-algorithm presented by A. Kraker to generate the Wiener Filter in a sliding window manner if complex-valued signals are applied. As it is known, an auto-correlation matrix and its inverse are always Hermitian. Using this fact, I removed the imaginary parts of the elements in the main diagonal of the inverse of the auto-correlation matrix as the last step of the inversion. This step stabilizes the algorithm, and it protects that against the accumulation of round-o errors of the imaginary parts. Furthermore, I investigated the complexity of the extended algorithm. I showed that it is squarely proportional to the lter length, even though the addition and the multiplication of two complex numbers require additional computations.
Subthesis III1: I extended the algorithm of A. Kraker for complex-valued signals to evaluate the inverse of the auto-correlation matrix in a sliding window manner. I showed that the elements of the main diagonal in the inverse of the auto-correlation matrix are not purely real because of round-o errors of the algorithm. Therefore, these imaginary parts have to be removed to stabilize the algorithm.
Subthesis III2: I proved that the complexity of the algorithm is squarely proportional to the lter length if complex-valued signals are considered.
(Please refer to Section 4.2 and Section 4.3.)
5 Conclusion and future work
It is very advantageous in 3G or 4G systems to apply Golay sequences for dierent parameter es-timation purposes; furthermore, they can be a very promising candidates for 5G and even for 6G networks. During the design of network concepts, the following aspects can be taken into account if Golay sequences are used: easy generation, excellent auto- and cross-correlation properties, signal processing in time or in frequency domain, and opportunity to use of time-distributed algorithms. However, in this thesis, these sequences are concatenated after each other to get a preamble for a data packet, but as it was briey mentioned they can be sent simultaneously as well. This approach makes it possible to use these sequences for parameter estimation in MIMO systems or for moving object detection by radars.
Firstly, an overview presented about the history of the Golay sequences and about their dierent mathematical descriptions. After that, the channel estimation was detailed based on complementaray Golay sequences, where both the time domain approach and the frequency domain approach were deduced not only for real-valued but for complex-valued data as well. The channel impulse response estimation correlates the incoming preamble with the original one resulting such auto-correlation function, which includes the impulse response as well. This method suers from a notable disadvantage: during the compensation of the channel distortions, the channel impulse response has to be Fourier transformed neutralizing the advantage of the method: the low computational complexity. Therefore, in this section, new method was introduced to evaluate the channel estimation in frequency domain. This algorithm uses the Fourier transform unit, which is already given due to the compensation; thus, no further computational units are required. Furthermore, this transform can be evaluated in sliding window manner as well to further reduce the computational complexity by distributing the the computations over the samples. The presented methods were compared by simulations, and the resulted BER curves showed that there is no signicant dierence between them.
In the following part of the thesis, the Wiener lter and the evaluation of the Wiener-Hopf equation were investigated. These methods are commonly used for lter design to estimate or identify the coe-cients of an unknown lter, but it can be suitable even for channel estimation. Assuming that the transfer channel and its paths can be modelled by a FIR lter, the channel impulse response can be estimated by Wiener-Hopf equation using the original, transmitted signal and the received one. The solution of this equation requires the inversion of the auto-correlation matrix, which is a crucial step of the procedure.
For case of continuous evaluation, Kraker suggested a new method: the RSC4BI. If an inverse is already given, then this algorithm updates it in sliding window manner reducing the computational complexity down to O K2, as this thesis deduced and proved that. For further simplication, dierent methods were introduced to update the cross-correlation vector and the auto-correlation matrix either through permutation or through dyadic product. The eciency of the new methods were proven by simulations to compare them to the conventional calculation procedures.
The RSC4BI-algorithm was originally given for real-valued signals. As it was mentioned several times in thesis, the current transmissions apply IQ modulations; therefore, the baseband signals which have to be processed are complex-valued. This requires an extension for Kraker's algorithm to handle these signals. This modication is presented not only for the RSC4BI but for the update of the cross-correlation vector and the auto-correlation matrix as well. Studying these methods, it was also proven that their complexity increased but still remained squarely proportional to the lter length. The functionality and the applicability of these new complex-valued extensions were veried by simulations.
The investigations performed in this thesis are not complete, one major aspect was not studied: the eects of the quantisation. Considering that the systems are almost always digital, the incoming signals have to be converted by ADCs; therefore, they contain only certain values depending on the resolution.
In the current available ADCs, the resolution is reciprocally proportional to the conversion time leading to that the high frequency signals can be digitised only by 8, 10 bits ADCs. Thus, it is going to be a very important study in the future, how high is the quantisation fault tolerance of the algorithm. If this tolerance is high enough, then the number of bits can be reduced until an optimised value. This decreased resolution allows to increase the conversion speed while lower resolutions lead to cheaper ADCs.
In the future, the implementation opportunities have to be investigated as well. It is quite straightfor-ward, how to implement the algorithms in Matlab, but it can be dicult to give a eective implementa-tion for a micro-controller or for a FPGA. First quesimplementa-tion in these cases is the form of complex numbers:
should we use the Cartesian form or would the polar form be better? How large are the round-o errors for them? Finally, after choosing the proper form for the implementation, there are many remaining questions: how could these algorithms be parallelized on an FPGA? How large clock frequency can be applied in a real-time environment? These questions related to the theses are still open...
A Denition of Golay sequences using polynomials
In this section theG(z)G z−1product is expanded to prove the equality in (213). LetGbe a Golay complementary sequence with length ofN, andG(z), a complex-valued polynomialΓgenerated as follows
Γ =G(z) =G[0]z0+G[1]z1+· · ·+G[N−1]zN−1 (A1) whileG z−1
has the similar structure Γ−=G z−1
=G[0]z0+G[1]z−1+· · ·+G[N−1]z−(N−1). (A2) Now the product of (A1) and (A2) can be expressed and expanded
ΓΓ−=
G[0]z0+G[1]z1+· · ·+G[N−1]z(N−1) G[0]z0+G[1]z−1+· · ·+G[N−1]z−(N−1)
=
=G[0]G[0]z0+G[0]G[1]z−1+G[0]G[2]z−2+· · ·+G[0]G[N−1]z−(N−1)+ +G[1]G[0]z1+G[1]G[1]z0+G[1]G[2]z−1+· · ·+G[1]G[N−1]z−(N−1)+1+ +G[2]G[0]z2+G[2]G[1]z1+G[2]G[2]z0+· · ·+G[2]G[N−1]z−(N−1)+2+ +· · ·+
+G[N−1]G[0]zN−1+G[N−1]G[1]zN−2+· · ·+G[N−1]G[N−1]z−(N−1)+(N+1). (A3) After grouping the terms depending on the exponent ofz
ΓΓ−=
N−1
X
i=0
(G[i])2+ z1+z−1
N−2
X
i=0
G[i]G[i+ 1] + z2+z−2
N−3
X
i=0
G[i]G[i+ 2] +
+· · ·+
zN−1+z−(N−1)
N−(N−1)
X
i=0
G[i]G[i+ (N−1)] =
=
N−1
X
n=0
zn+z−n
N−1−n
X
i=0
G[i]G[i+n] (A4)
Using (27), the inner sum of (A4) can be substituted to prove the equality in (213):
ΓΓ−=G(z)G z−1
=
N−1
X
n=0
RA[n] zn+z−n
. (A5)
B Addition and multiplication of complex numbers
B.1 Addition of two complex numbers
If two complex numbers are given in Cartesian form, then their sum can be calculated through adding the real and imaginary parts separately as follows
(m1+jn1) + (m2+jn2) = (m1+m2) +j(n1+n2). (B6)
m1
+ jn1
m2
+ jn2
+
+
m1+m2
j(n1+n2)
Figure B1: Addition of two complex numbers in Cartesian form