Abstract The Complex Block Least Mean Square (LMS) technique is widely used inadaptive filtering applications because of its simplicity and efficiency from a theoret-ical and implementation standpoint. However, the limitations of the Complex BlockLMS technique are slow convergence and dependence on the proper choice of thestepsize or convergence factor. Moreover, its performance degrades significantly intime-varying environments. In this paper, a novel adaptive LMS technique namedthe Complex Block Conjugate LMS algorithm, CBC-LMS, is presented. Based onthe Conjugate Gradient Principle, the proposed technique searches orthogonal direc-tions to update the filter coefficients instead of the negative gradient directions usedin the Complex Block LMS algorithm. In addition, the CBC-LMS algorithm derivesoptimal stepsizes to adjust the adaptive system coefficients at each iteration. As aresult, the developed method overcomes the inherent limitations of the existing Com-plex Block LMS algorithm. The performance of the CBC-LMS technique is tested inwireless channel estimation and equalization applications, using both computer sim-ulations and laboratory experiments. Furthermore, the developed technique is com-pared to the Complex Block LMS method and a recently proposed method, which iscalled Complex Optimal Block Adaptive LMS (OBA-LMS). The experimental andsimulation results confirm that the proposed CBC-LMS technique achieves faster
Y. Liu (�) · R. Ranganathan · W.B. MikhaelSchool of Electrical Engineering and Computer Science, University of Central Florida, 4000 CentralFlorida Blvd, Orlando, FL, 32816, USAe-mail: [email protected]
In the area of modern wireless communications, complex signal processing has at-tracted enormous research efforts [2, 4, 7, 9, 11, 13, 15, 21]. This is mainly attributedto the fact that wireless communication systems frequently utilize the quadrature re-lationship between a pair of signals, which can be considered as “complex” quan-tities [11]. Several spectrally efficient and high-bit-rate modulation schemes arebased on the complex signal concepts, e.g., Quadrature Phase Shift Keying (QPSK),Quadrature Amplitude Modulation (QAM), and Orthogonal Frequency DivisionMultiplexing (OFDM). As a result, a variety of applications for complex digital fil-ters have been proposed and investigated, including one fast-growing field, complexadaptive filtering [2, 7, 15].
The Complex Block LMS algorithm has been frequently applied in complex adap-tive filtering applications due to its ease of implementation and its computational effi-ciency [4, 5, 9]. However, the main drawback of the Complex Block LMS method isthat the choice of the stepsize (learning rate, convergence factor) is made empirically,depending on the type of application and input signals. Moreover, a small stepsizeresults in slow convergence, and a large stepsize may cause unstable gradient as-cent or descent, leading to divergence. The recently proposed OBA-LMS algorithmovercomes this limitation by automatically deriving an optimal stepsize at each iter-ation [12].
Newton’s method [7] and the method of steepest descent [19] are two traditionalmethods of searching a performance surface. Recently, the Conjugate Gradient prin-ciple (CG) [3, 17] is widely used to solve unconstrained optimization problems suchas energy minimization and adaptive filtering [1, 6]. The main advantage of the CGmethod is that it converges faster than the steepest descent method by employing or-thogonal search directions to update the filter coefficients. On the other hand, it haslower computational complexity than Newton’s iteration approach, which involvesmatrix inversion. This unique combination of convergence speed and computationalcomplexity gives CG desirable properties for applications in numerous mathematicaloptimization problems.
In this paper, a novel algorithm of complex adaptive filtering called CBC-LMS isproposed. The CBC-LMS method utilizes the CG principle to adjust the filter coeffi-cients. As a result, the CBC-LMS technique converges faster than the Complex BlockLMS technique which employs the steepest descent method. Moreover, along eachconjugate direction, an optimal stepsize is generated at each iteration using the Taylorseries approximation [10]. The performance of the CBC-LMS technique is comparedto the Complex Block LMS method and the Complex OBA-LMS method. The re-sults from our computer simulations and lab experiments confirm that the CBC-LMS
Circuits Syst Signal Process (2012) 31:1067–1087 1069
Fig. 1 Signal model for complex channel estimation
technique demonstrates the highest convergence speed, while yielding low Normal-ized Error Energy (NEE) after convergence.
This paper is organized as follows: Sect. 2 introduces the complex signal modelsfor channel estimation and equalization. Section 3 outlines the concept of ComplexConjugate Gradient Principle. The proposed CBC-LMS algorithm is formulated inSect. 4. Section 5 deals with several practical issues. In Sect. 6, the computer simu-lations of a wireless channel estimation model are carried out to evaluate the perfor-mance of the CBC-LMS algorithm in Sect. 6. Section 7 discusses the performance ofthe proposed algorithm in a laboratory experiment of complex channel equalization.Section 8 compares the computational complexity of the CBC-LMS, Complex OBA-LMS, and Complex Block LMS algorithms. Conclusions are presented in Sect. 9.
2 Complex Channel Estimation and Equalization
2.1 Mathematical Notation and Preliminaries
To avoid the ambiguity of the mathematical notation in the algorithm formulation,notational conventions are given in this subsection. Scalar variables appear in lowercase, vectors in bold lower case, and matrices in bold upper case. A vector x withN coefficients is denoted by xN , and the ith component of x is denoted by xi . Thereal and imaginary parts of a complex-valued quantity x, in turn, are denoted byRe{x} = xR and Im{x} = xI , respectively. Transposition, conjugate transposition, andscalar conjugation are indicated by superscripts (.)T , (.)H , and (.)∗, respectively. Theexpectation operator is E[.].
2.2 Channel Estimation Model
Figure 1 illustrates a block diagram for wireless channel estimation (identifica-tion) [7]. The unknown wireless channel can be modeled as a complex FIR filterF(z). Both the unknown system and the adaptive filter are driven by the same input
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Fig. 2 Signal model for complex channel equalization
s(k). In this scenario, the adaptive filter input x(k) is equal to s(k). In practical cases,noise n(k) is generally uncorrelated with the channel input, which can be modeledas an additive component of the channel output. To reduce the error signal e(k), theadaptive filter T (z) tries to emulate the channel’s transfer characteristics. After adap-tation, the unknown channel is “identified” in the sense that its transfer function isnow modeled by the adaptive filter. Adaptive system estimation can be used as such,to model an unknown channel whose input and output signals are available. If theadaptive model is an adaptive linear combiner whose weights are adjusted to mini-mize mean-square error, it can be shown that the LMS solution will not be affectedby the presence of the channel noise [20].
2.3 Channel Equalization Model
A block diagram of channel equalization (deconvolution, inverse filtering, or inversemodeling) is shown in Fig. 2 [7]. Similar to the channel estimation model in Sect. 2.2,the unknown wireless channel can be modeled as a complex FIR filter F(z). In thisapplication, the adaptive processor attempts to recover the transmitted signal s(k),which is assumed to have been altered by the unknown channel F(z) and containadditive noise n(k). n(k) is generally uncorrelated with the channel input. After con-vergence, the adaptive filter output y(k) is the best match to the channel input s(k),and the adaptive filter T (z) becomes the inverse of the unknown F(z) [20]. In thissense, the adaptive system equalizes the unknown channel.
2.4 Complex Signal Representation for Channel Estimation and Equalization
Assuming that the adaptive filter has N taps, the coefficients of T (z) at the kth itera-tion can be defined as
w(k) = [w1(k),w2(k), . . . ,wN(k)
]T (2.1)
xN(k) is defined as the input vector containing x(k) and the preceding N − 1 inputsof the adaptive filter, given by
xN(k) = [x(k), x(k − 1), . . . , x(k − N + 1)
]T (2.2)
Thus, the output of the adaptive filter y(k) at the kth iteration can be expressed as
y(k) = wT (k)xN(k) (2.3)
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The algorithms presented in this work are iterative. They are mathematical proceduresthat follow a succession of steps toward a better solution. Each step for adjusting theadaptive filter coefficients is referred to as “iteration.”
In Figs. 1 and 2, d(k) and e(k) denote the complex desired and complex errorsignals, respectively. In the block-based algorithm with the block size L, the errorvector, the desired signal vector, and the adaptive filter output vector are defined as
eL(k) = [e(k), e(k − 1), . . . , e(k − L + 1)
]T (2.4)
dL(k) = [d(k), d(k − 1), . . . , d(k − L + 1)
]T (2.5)
yL(k) = [y(k), y(k − 1), . . . , y(k − L + 1)
]T (2.6)
These three vectors have the relationship as follows:
eL(k) = dL(k) − yL(k) (2.7)
A block version of (2.3) can be expressed as
yL(k) = X(k)w(k) (2.8)
where
X(k) = [xN(k),xN(k − 1), . . . ,xN(k − L + 1)
]T (2.9)
Since the block size L is constant throughout the adaptation process, the subscript L
will be removed for simplicity of notation.
3 Complex Conjugate Gradient Optimization
As mentioned before, CG derives a set of orthogonal search directions q(0), q(1), . . . ,
q(I − 1). In each search direction, CG will take exactly one step with an appropriatestepsize γ (i) to guarantee it will finish within I steps [18].
Figure 3 illustrates the CG method with a two-dimensional example. a(0) is thestarting point, and a∗ is the destination. Irrespective of the direction of the first stepq(0), the second step will always lead to the solution as long as these two steps areorthogonal and γ (i)’s are properly selected. It is clear that for an N -dimensionalspace, CG can achieve the optimal solution within N steps.
Notice that e(1) is orthogonal to q(0). In general, for each step, we choose a point
a(i + 1) = a(i) + γ (i)q(i) (3.1)
The error between the destination and the current point is
e(i + 1) = e(i) + γ (i)q(i) (3.2)
To find the value of γ (i), e(i + 1) should be orthogonal to q(i). In the real domain,the orthogonality of e(i + 1) to q(i) is defined as qT (i)e(i + 1) = 0, where T isthe transpose. In the complex domain, the conjugate transpose H replaces T . Theorthogonal relationship is then expressed as
qH (i)e(i + 1) = 0 (3.3)
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Fig. 3 Method of searching orthogonal directions
Substituting (3.2) into (3.3), the following is obtained:
qH (i)[e(i) + γ (i)q(i)
] = 0 (3.4)
γ (i) can be derived from (3.4) as follows:
γ (i) = − qH (i)e(i)
qH (i)q(i)(3.5)
Unfortunately, nothing is accomplished, because γ (i) cannot be computed withoutknowing e(i). If e(i) is available, the problem would already be solved.
The solution is to make the search directions A-orthogonal instead of orthogo-nal [18]. Two vectors are defined to be A-orthogonal, or conjugate, if
qH (i)Aq(j) = 0 (3.6)
A set of A-orthogonal search directions {q(i)} is generated using the conjugateGram–Schmidt process.
Suppose that there is a set of n linear independent vectors u(0), u(1), . . . , u(n−1).To derive q(i), all components that are not A-orthogonal to the previous q vectorsare subtracted from u. Therefore, the following formula is derived for i > 0:
q(i) = u(i) +i−1∑
j=0
βij q(j) (3.7)
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βij can be derived following the same procedure as in the real domain in [18]:
βij ={
rH (i)r(i)
rH (j)r(j), i = j + 1,
0, i > j + 1(3.8)
where r(i) is the residual, defined as the negative of the gradient estimate:
r(i) = −g(i) (3.9)
r(i) can be chosen as u(i) in (3.7) [18]. From (3.8) and (3.9) it is clear that it isno longer necessary to store old search vectors other than the previous one. As aresult, both the storage requirements and computational complexity per iteration aresignificantly reduced. Henceforth, the abbreviation β(i) is used instead of βi,i−1 asfollows:
β(i) = βi,i−1 = rH (i)r(i)
rH (i − 1)r(i − 1)(3.10)
Combining (3.7), (3.9), and (3.10), the method of complex conjugate gradient direc-tions can be summarized by the following equations:
In order to derive the CG directions, the gradient g(k) of the objective function isrequired. In this subsection, g(k) is derived based on the steepest descent method.
The steepest descent method is employed in the Complex Block LMS tech-nique [12] where the weights are adjusted in the opposite gradient directions. In thisway, the weight update formula is given as follows:
w(k + 1) = w(k) − μ · g(k) (4.1)
where w(k) is defined in (2.1), μ is the fixed stepsize, and g(k) is the gradient vectorwhich is defined as
g(k) = ∂f (k)
∂w(k)(4.2)
where f (k) is the objective function to be minimized:
f (k) = E[e · e∗] ≈ 1
L· eH (k)e(k) (4.3)
Since the weight vector w(k) is complex, the real and imaginary components of w(k),namely wR(k) and wI (k) respectively, are updated by the following equations:
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wR(k + 1) = wR(k) − μ · ∂f (k)
∂wR(k)(4.4)
wI (k + 1) = wI (k) − μ · ∂f (k)
∂wI (k)(4.5)
The gradients can be derived as follows:
∂f (k)
∂wR(k)≈ 1
L· [−XT (k)e∗(k) − XH (k)e(k)
](4.6)
∂f (k)
∂wI (k)≈ 1
L· [−jXT (k)e∗(k) + jXH (k)e(k)
](4.7)
Substituting (4.6) and (4.7) into (4.4) and (4.5), respectively, and applying the relation
w(k + 1) = wR(k + 1) + jwI (k + 1) (4.8)
the weight update equation for the Complex Block LMS method is given as follows:
w(k + 1) ≈ w(k) + 2μ
L· XH (k)e(k) (4.9)
Combining (4.1) and (4.9), the gradient vector can be calculated as
g(k) ≈ − 2
L· XH (k)e(k) (4.10)
4.2 CG Directions and Optimal Stepsizes of CBC-LMS
In the proposed CBC-LMS technique, the weights are adjusted in the CG directionsinstead of the opposite gradient directions used in the steepest descent method. Inthis subsection, the complex conjugate gradient method described in Sect. 3 is em-ployed to develop the proposed CBC-LMS algorithm. Compared with the ComplexBlock LMS method, the CBC-LMS algorithm has two more steps: searching the CGdirections and deriving the optimal stepsizes.
In the proposed CBC-LMS technique, the complex weight update equation isgiven by
w(k + 1) = w(k) + α(k) · q(k) (4.11)
where q(k) is the vector version of the search direction defined in Sect. 3, expressedas follows:
q(0) = −g(0) (4.12)
q(k + 1) = −g(k + 1) + gH (k + 1)g(k + 1)
gH (k)g(k)· q(k) (4.13)
where g(k) is given in (4.10). Now the proper stepsize α(k) is desired to minimizethe square error of the next block, {eH (k + 1) · e(k + 1)} is minimized. Applyingthe complex Taylor series expansion, the error at the (k + 1)th iteration, e(k + 1), is
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expressed as follows:
e(k + 1) = e(k) +N∑
l=1
∂e(k)
∂wl(k)�wl(k)
+ 1
2!N∑
m=1
N∑
n=1
∂2e(k)
∂wm(k)∂wn(k)�wm(k)�wn(k) + · · · (4.14)
where wl(k) is the lth coefficient of the adaptive filter at the kth iteration, and
In (4.14), the derivatives higher than the first order can be removed due to the linearityof the error function. Thus, (4.14) becomes
e(k + 1) = e(k) +N∑
l=1
∂e(k)
∂wl(k)�wl(k) (4.16)
Combining (4.11), (4.15), and (4.16), we obtain
e(k + 1) = e(k) − α(k) · X(k) · q(k) (4.17)
The next step is to choose each stepsize α(k) such that the function E[e(k + 1) ·e∗(k + 1)] is minimized. Thus, the following condition must be satisfied:
In most wireless communication applications, there are multiple signal paths betweenthe transmitter and receiver. Different variations exist in different paths. As a result,there are significant fluctuations in the received signal’s amplitude, phase, and angleof arrival. This phenomenon, known as multipath fading [16], should be taken intoconsideration to describe channel behavior and predict the system performance.
In general, multipath fading can be categorized into two types: slow fading andfast fading. The terms slow and fast fading refer to the rate of the magnitude andphase change imposed by the channel on the signal. Slow fading arises when thecoherence time of the channel is large relative to the delay constraint of the channel.In this scenario, the amplitude and phase change imposed by the channel can beconsidered roughly constant (quasi-stationary) over the period of use. In other words,the parameters of the channel will remain approximately the same. Fast fading occurswhen the coherence time of the channel is small relative to the delay constraint of thechannel. In this scenario, the amplitude and phase change imposed by the channelvaries considerably over the period of use.
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5.2 Block Size Selection
In order to achieve a better estimation of the error signal, the block-based algorithmis employed in the CBC-LMS method. Theoretically, the block size L can be chosenin the range from 1 to a number much bigger than N . If L equals 1, the algorithmbecomes an online gradient-based technique, which updates the coefficients of theadaptive filter based on the current input sample only.
On the other hand, during a processing block, it is very important that the un-known channel is quasi-stationary, which means the parameters stay approximatelyconstant. Thus, a large L may violate the assumption of quasi-stationarity, resultingin divergence problems.
6 Computer Simulations
To study the performance of the proposed CBC-LMS method, computer simulationshave been carried out to estimate a complex unknown FIR filter under the time-invariant and time-variant conditions. The performance measures are the number ofiterations for convergence and the Normalized Error Energy (NEE), representing con-vergence speed and accuracy, respectively. NEE is defined as the ratio of the estimatederror energy to the energy of the unknown transfer function, defined as
NEE =∫ ω=π
ω=0
∣∣F(ejω
) − T(ejω
)∣∣2dω
/∫ ω=π
ω=0
∣∣F(ejω
)∣∣2dω (6.1)
where F(ejω) and T (ejω) are the transfer functions of the unknown complex FIRfilter and the adaptive filter, respectively. The performance of the CBC-LMS algo-rithm is compared to the Complex Block LMS technique and the recently proposedComplex OBA-LMS technique.
6.1 Computer Simulations for Stationary Channel Estimation
In this subsection, the proposed CBC-LMS algorithm is applied to the estimation ofa stationary wireless channel modeled as a complex Finite Impulse Response (FIR)filter.
The length of the adaptive filter, N , is 10. The Complex Block LMS algorithm be-gins to diverge when the fixed stepsize μ is larger than 0.003. Hence, μ is chosen tobe 0.001 and 0.003 for the Complex Block LMS technique. The input x(k) used forchannel estimation is zero-mean complex white Gaussian signal. The performance ofthe algorithm is tested with different values of the channel order, the block size, andthe input Signal-to-Noise Ratio (SNR). In the simulation results shown in Figs. 4, 5,and 7, the unknown wireless channel is of 9th order with the following transfer func-tion:
Circuits Syst Signal Process (2012) 31:1067–1087 1077
(a)
(b)
Fig. 4 (a) Number of iterations for convergence vs. block size with input SNR = 30 dB. (b) Number ofsamples for convergence vs. block size with input SNR = 30 dB
+ (−0.0552 + j∗0.123)z−7 + (0.5530 + j∗0.5612)z−8
+ (−0.4720 − j∗0.1209)z−9
The complex coefficients of the unknown filter for the simulation results shown inFigs. 6 and 8 are generated randomly for each channel order.
Figures 4 to 6 illustrate the convergence speed of the CBC-LMS, Complex OBA-LMS, and Complex Block LMS algorithms, according to different simulation param-
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Fig. 5 Number of iterations for convergence vs. input SNR with L = 18
Fig. 6 Number of iterations for convergence vs. wireless channel order with L = 18, SNR = 30 dB
eters. Figure 4a illustrates that a small block size yields slow convergence and a largerblock size results in faster convergence for all these three techniques. In addition,choosing L less than N may result in numerical instability. In stationary channels,as L increases, better approximations of the expectation operators in (4.3) and (4.18)are computed. Consequently, the total number of iterations to achieve convergenceis decreased. This is confirmed in Fig. 4a. On the other hand, a larger block size re-quires more samples to fill the block before starting the adaptation process. Whenthe block size L is increased over a threshold value, the total number of signal sam-ples to achieve convergence, which is the sum of L and the number of iterations to
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Fig. 7 NEE vs. input SNR with block size L = 18
achieve convergence, is increased as shown in Fig. 4b. From our experimental work,choosing L close to 2N gives best result from the stability point of view and also interms of the number of samples needed to converge. For the following simulations,the block size L = 18, which is approximately equal to 2N , was found adequate. Thiswas confirmed to be true with a wide range of values of N in our experiments.
In Fig. 5, it is clear that the CBC-LMS and Complex OBA-LMS techniques al-ways converge within 20 iterations, irrespective of the input SNR. Figure 6 indicatesthat as long as the adaptive filter is overdetermined (i.e., the order of the adaptivefilter is higher than the order of the unknown channel), different orders of the un-known channel will not affect the convergence speed for all these three algorithms.Figures 4–6 indicate that the CBC-LMS algorithm has the fastest convergence speedamong these three adaptive methods, using different simulation parameters. From ourexperiments, it happens that the fixed stepsize algorithm can never converge as fastas the CBC-LMS method with any manually selected stepsize.
Figures 7 and 8 illustrate the convergence accuracies of the CBC-LMS, Com-plex OBA-LMS, and Complex Block LMS techniques. Figure 7 indicates that for allthe three adaptive techniques, the error energy after convergence is reduced to ap-proximately the level of the additive noise. Figure 8 shows that the performance ofthese three methods degrades significantly when the number of taps of adaptive filter(N = 10) is smaller than the number of taps of the unknown channel. From Figs. 7and 8, it can be inferred that after convergence, the CBC-LMS method is not as ac-curate as the Complex Block LMS method with the best stepsize. There is tradeoffbetween accuracy and the convergence speed.
6.2 Computer Simulations for a Dynamic Communication Channel
In this subsection, the performance of the CBC-LMS algorithm is compared to theComplex OBA-LMS and Complex Block LMS methods in the estimation of a lin-
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Fig. 8 NEE vs. channel order with block size L = 18
early time-varying fading channel. The fading channel is modeled as a 19th-ordercomplex FIR filter with the linear time-varying transfer function as follows:
where k is the symbol index, and Δ is the time-varying factor, which is equal to0.001. Accordingly, the adaptive filter length N is chosen to be 20. The input s(k) forthe unknown channel is a QPSK signal. The fixed stepsizes in our simulation for theComplex Block LMS algorithm are 0.003 and 0.005.
The NEE vs. the iteration index for the simulations with the block sizes 18 and36 are plotted in Figs. 9 and 10, respectively. Figure 9 indicates that the CBC-LMSmethod is superior to both the Complex OBA-LMS and Complex Block LMS tech-niques in terms of the convergence speed. Figure 10 shows that in a time-variant envi-ronment, a relatively large processing block size leads to divergence for the ComplexBlock LMS method when the fixed stepsize is increased beyond a threshold value.
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Fig. 9 NEE vs. iteration with block size L = 18 for a linearly time-varying channel
Fig. 10 NEE vs. iteration with block size L = 36 for a linearly time-varying channel
7 Laboratory Data for Equalization Model
In this section, laboratory experiments have been carried out to examine the perfor-mance of the proposed CBC-LMS algorithm in an equalization model. The real-worldsignal generation and processing diagram is illustrated in Fig. 11. Firstly, an Agilent
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Fig. 11 The diagram of the real-world laboratory experiment
Fig. 12 Input signal spectrum and bandpass filter responses
ESG Vector Signal Generator generates the source signal s(k), which is a QPSK sig-nal, 50% roll-off raised-cosine pulse-shaping, centered at 70 MHz. The symbol rateis 5 MHz, yielding roughly a 5 × (1 + 0.5) = 7.5 MHz signal bandwidth. The sig-nal then passes through a 70-MHz bandpass filter channel. The amplitude responseand group delay of the bandpass filter is analyzed with an Agilent E5071C NetworkAnalyzer and is shown in Fig. 12, along with the input signal spectrum. As can beseen from Fig. 12, the bandpass filter causes significant distortion of the input signal.This is also clearly illustrated in Fig. 15, which shows the unequalized constellationof the filtered signal. Subsequently, the signal is received and digitized by a ZTEC In-struments ZT8441 IF Digitizer. The ZT8441 samples and digitally downconverts theinput bandpass signal, creating the baseband I/Q components which are then loadedinto a personal computer. Finally, Matlab is used to process and recover the symbols.This real-world experimentation is carried out in a low-noise environment, which can
Circuits Syst Signal Process (2012) 31:1067–1087 1083
Fig. 13 Error (dB) vs. iterations for the CBC-LMS
Fig. 14 Error (dB) vs. iterations for the Complex Block LMS
be seen in the constellation plot of the equalized signal in Fig. 16. 300 symbols arecollected to test the performance of both the CBC-LMS and Complex Block LMSalgorithms. The order of the adaptive filter is set to 15, and the block size is 18. Ex-periments show that the best fixed stepsize for the Complex Block LMS method toachieve the same accuracy as the CBC-LMS method is 0.05. The performance accu-racy is measured by the error signal e(k).
In Figs. 13 and 14, the error is shown as a function of the iteration index. The twomethods are applied to yield comparable residual error. The proposed CBC-LMSmethod converges after 20 iterations, while the Complex Block LMS technique con-verges after 65 iterations. This faster convergence of the proposed technique wasconfirmed in extensive simulations, some of which are given in Sect. 6. The inputand output signal constellations of the adaptive filter employing the CBC-LMS tech-
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Fig. 15 QPSK constellation forthe output of the unknownchannel
Fig. 16 QPSK constellation forthe equalized signal
nique are plotted in Figs. 15 and 16. From the illustrations, it can be easily inferredthat the CBC-LMS algorithm effectively removes a considerable amount of channelinduced distortion.
8 Computational Complexity
In this subsection, the computational complexities of the CBC-LMS, Complex OBA-LMS and Complex Block LMS techniques have been studied. The numbers of realMultiplications Per Iteration (MPI) for these three methods are summarized in Ta-ble 1. To properly assess the computational complexity of the three algorithms, thehardware requirements are compared. Digital multiplication can be implemented by asequence of shifting and adding operations, while digital division requires the addedstep of quotient digit selection or estimation [14]. The divisions per iteration re-quired for the Complex Block LMS, Complex OBA-LMS, and CBC-LMS techniquesare 1, 4, and 2, respectively. These are insignificant compared with the number of
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Table 1 Computational complexities
MPI Complex Block LMS Complex OBA-LMS CBC-LMS
Error computation 4LN 4LN 4LN
Gradient computation 4LN 4LN 4LN
Direction search – – 4N
Stepsize generation – 8(LN + 2L) 4(LN + 2L)
Weight update 2N 4N 2N
Total 8LN + 2N 16LN + 4N + 16L 12LN + 6N + 8L
multiplications per iteration. Consequently, divisions do not influence the computa-tional requirements significantly.
In Table 1, it is clearly shown that the algorithms with optimal stepsizes, the CBC-LMS and Complex OBA-LMS algorithms, require more computations at each iter-ation than the Complex Block LMS method, which has a fixed stepsize. However,if the number of operations needed per iteration is within the capability of the digi-tal processor, optimal stepsize algorithms will converge much faster in real time byrequiring fewer samples. Recent advances in digital signal processing hardware aremaking high-performance algorithms desirable [8], even at the expense of increasedcomputation.
Table 1 clearly indicates that the calculation of the optimal stepsize at each iter-ation results in more computations for the CBC-LMS algorithm (Direction Searchis the prestep of Stepsize Generation for CBC-LMS), and the Complex OBA-LMSmethod. On the other hand, these two algorithms converge in fewer iterations thanthe Complex Block LMS method. Thus, it is not sufficient to compare the computa-tional complexity by only employing the criterion of MADPI. Instead, the overall realMultiplications required for convergence is adopted as a measure of computationalcomplexity, which is defined as follows:
Multiplications = MPI × Nc (8.1)
where Nc denotes the number of iterations for convergence. In this regard, the pro-posed CBC-LMS algorithm is compared to the other two techniques. Comparing theCBC-LMS and Complex OBA-LMS method, it can be inferred that the CBC-LMSmethod has lower MADPI and Nc, resulting in a lower MADS than the ComplexOBA-LMS method. With regards to the CBC-LMS and Complex Block LMS meth-ods, it can be seen that the MPI of the CBC-LMS algorithm is approximately 1.5times the MADPI of the Complex Block LMS algorithm. However, the experimentresults in Sects. 6 and 7 show that the Nc for the Complex Block LMS method ismuch bigger than 1.5 times the Nc of the CBC-LMS method. As a result, the CBC-LMS algorithm requires less overall computations. Therefore, the CBC-LMS algo-rithm has the least overall computational complexity among the three adaptive tech-niques.
1086 Circuits Syst Signal Process (2012) 31:1067–1087
9 Conclusion
In this paper, a complex adaptive digital filtering technique, called CBC-LMS, ispresented. It applies the conjugate gradient optimization technique to minimize thequadratic mean square error function. The presented results clearly indicate that inorder to guarantee convergence and achieve acceptable convergence speed, the fixedstepsize for the Complex Block LMS method has to be found by trial and error. Thisis not an easy task since the stepsize depends on the source signals, the environmen-tal parameters, and the simulation conditions. In contrast, the presented CBC-LMSalgorithm uses the Taylor series approximation to derive an optimal stepsize at eachiteration. In addition, the CBC-LMS method converges faster by utilizing CG princi-ple than the steepest descent approach used in the traditional Complex Block LMS al-gorithm. The simulation and laboratory results confirm that the CBC-LMS algorithmovercomes the inherent drawback of the fixed stepsize and demonstrates excellentconvergence speed, while maintaining computational efficiency.
Acknowledgements The authors would like to thank Astronics DME Corporation and ZTEC Instru-ments for technical support in performing the laboratory measurements reported in this paper.
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