As it can be observed, the single and the continuous transmissions had the same properties and dierences (e.g. noise suppression with larger averaging) as in Sec. 2.5.2; therefore, the same conclusions can be drawn as in the case of real-valued transfer channel. Studying the impact of the length of Golay sequences, it can be observed that the longer length did not inuence signicantly the BER curves.
However, it has to be noted that in this case, the channel contained only 6 taps that could be estimated already by sequences length N = 32. In contrast, there are many applications, where more taps are required to give an adequate description about the transfer channel: for example a 60 GHz wireless system can be properly modelled by 40-50 taps [51, 52]; thus, longer Golay sequences are required (cf.
limitations for length of estimation in Sec. 2.3.1).
3 Investigation and optimisation of the RSC4BI-algorithm
In control engineering, a common challenge is to build a mathematical model of an unidentied dynamic system without decomposition. In order to identify the system, predened probe signals are applied to the input of the system to analyse the only measurable parameter, the output signal. In the case of signal processing applications, the observed systems are often lters, which have to be reproduced or compensated; therefore, it is necessary to identify (or at least estimate) their impulse responses or transfer functions. Based on these observations, the desired control operations or compensation algorithms can be designed for the given dynamic system.
In this section, it is assumed that the system is a Finite Impulse Response (FIR) lter. This FIR lter is modelled as a Wiener Filter (WF) [53], which is able to track a system by linear time-invariant ltering.
To perform the estimation of the lter coecients, two statistical functions have to be calculated in matrix form: the auto-correlation of the input signal and the cross-correlation of the input and output signals.
Using these matrices, the Wiener-Hopf (WH) equation can be formulated, and its solution provides the optimum for the lter coecients in a Least Square (LS) sense [54, 55, 56]. These LS-lters and their modied versions are widely used in signal processing to identify a given dynamic system [57, 58, 59, 60, 61, 62].
During the solution of the WH equation the auto-correlation matrix has to be inverted. Several algorithms can be applied to solve the inversion of matrices, depending on the properties of the matrix [63, 64]. The computational load of the inversion becomes critical if a low complexity hardware with limited performance is applied. Furthermore, this operation is also critical if real-time adaptive ltering has to be performed by the WF: during this adaptive method the lter coecients adapt over time to track the internal changes of the dynamic system.
The challenges, that require this adaptive and real-time ltering, are very diverse, such as compensa-tion of current transformers [65], noise cancellacompensa-tion in microphones [66], or to calculate charge density on a dielectric surface [67]. Studying these examples, it can be stated that the matrix inversion is a crucial step of these procedures due itsO n3
complexity. The applicability of this conguration is limited be-cause of the real-time environment, as the applied matrices have to be small enough to full the real-time conditions. Considering these circumstances, such algorithms have to be applied that can evaluate the WH equation in real-time, and they have low computational complexity. A common method for perform-ing matrix inversion in a recursive manner is the split Levinson-algorithm [68, 69]. Although the poor stability limits the usage of the method [70], but its behaviour can be predicted and compensated by computing the condition number [71].
Kraker presented a novel method for the ecient solution of the WH equation in a sliding window, where the direct calculation of the inverse of the auto-correlation matrix is avoided by recursive compu-tation [72]. Furthermore, not only the inverse but the auto-correlation matrix and the cross-correlation vector are also recursively evaluated using the results of the previous calculations. However, some ele-ments of his presented method had been already published in similar form such as the update of the auto-correlation matrix [73], [74] or the recursive inversion performed in two steps [75], but Kraker syn-thesized them into the following algorithm: Recursion with Splitting the Correlation matrix into 4 Blocks for Inversion (RSC4BI) for evaluating the WH equation.
This chapter is organized as follows. Section 3.1 gives a short overview of adaptive ltering and the WH equation. The applied notations for the investigated signals are introduced as well. Section 3.2 describes
the Kraker's recursive solution of the WH equation in a sliding window, i.e., the RSC4BI-algorithm is presented. In Sec. 3.3 and Sec. 3.4, I investigate the complexity of these recursive calculations, and they are compared to the conventional WH solution; furthermore, I present an optimised variant for the RSC4BI-algorithm. The simulation results for the proposed methods in terms of runtime and error are given in Sec. 3.5. Finally, the conclusions are drawn in the last section, and the new scientic results are summarized.
3.1 Principle of adaptive ltering
Adaptive
x [n]
ltery [n]
+
−+
ε [n]
d [n]
Adaptive algorithm
h [n]
Figure 23: Block diagram of an adaptive lter
The signal ow for adaptive ltering can be seen in Fig. 23 [56]. The idea is to tune the coecients of the adaptive lterh[n]so that the error between the output signaly[n]and the desired signald[n]is minimised. The coecients of the adaptive lter is determined through an adaptive algorithm based on the input signalx[n]and the error signalε[n].
The adaptive lter is modelled as a linear time-invariant (LTI) lter having a nite impulse response withK coecients. As a result, the output signal of the adaptive lter can be expressed through convo-lution using the coecients and the input signal as
y[n] =x[n]∗h[n] =
K−1
X
k=0
x[n−k]h[k]. (31)
The previous equation can also be expressed as matrix multiplication in an observation window containing N samples
y[n] =X[n]h[n] (32)
where vectorsy[n]andh[n]can be expressed using the signalsy[n]andh[n], respectively as yT[n] =
y[n] y[n+ 1] . . . y[n+N−1]
, (33)
hT[n] =
h[0] h[1] . . . h[K−1]
, (34)
and the data matrixX[n]can be expressed usingx[n]as
X[n] =
x[n] x[n−1] . . . x[n−K+ 1]
x[n+ 1] x[n] . . . x[n−K+ 1]
... ... ... ...
x[n+N−1] x[n+N−2] . . . x[n+N−K]
. (35)
The error signal vector ε[n] is the expected value of the dierence of the desired signal vectord[n]
and the output signal vectory[n],
ε[n] = E{d[n]−y[n]} (36)
whered[n] can be expressed as dT[n] =
d[n] d[n+ 1] . . . d[n+N−1]
. (37)
The adaptive algorithm should minimise the cost functionC as a function of the lter coecients, con-sidering (32) and (36). It can be expressed as
C(h[n]) =E{d[n]−X[n]h[n]}TE{d[n]−X[n]h[n]}=
=En
d[n]Td[n]o
+hT[n]R[n]h[n]−2hT[n]p[n] (38) where
R[n] = E
XT[n]X[n] (39)
is the auto-correlation matrix of the input signal while p[n] = E
XT[n]d[n] (310)
is the cross-correlation vector of the input and the desired signals.
The optimal solution for the lter coecients known as the WH equation can be given in LS sense through the derivative ofC(h[n])with respect toh[n],
∂C(h[n])
∂h[n] =−2p[n] + 2R[n]h[n] =0. (311)
Evaluating this equation, the optimal coecientsbhopt[n]are given in form [55] as
hbopt[n] =R−1[n]p[n]. (312)