1.1 Introduction
1.1.1 Applications of ADF
The basic block diagram of an ADF is shown in Fig. 1.1. It processes the input signal x(n) and produces the output signal y(n) which is subsequently used to compute the error signal e(n) by subtracting from the desired signal d(n), as per e(n) =d(n)−y(n). The error signal is used by the adaptation algorithm to update the coefficient vector w(n) based on some error minimizing criteria.
In particular, the adaptation process aims to minimize some metric of the error signal in order to make the output of ADF as close as to the desired signal in a statistical sense. The ADF as shown in Fig. 1.1 can be perfectly fit in several practical applications such as system identification, channel equalization, noise cancellation etc [5, 6]. Each application requires the knowledge of underlying basic adaptive filtering framework. Precisely, the distinctive feature of each application is the primary factor on the basis of which the input signal and the desired signal of the ADF are chosen. Once these signals are determined, any known properties of them can be used to understand the expected behavior of the ADF when attempting to minimize the error function. In this thesis, DSP applications such as system identification, channel equalization and noise cancellation are considered to validate the proposed low-complexity architectures of ADF. Therefore, it is important to discuss them from ADFs point-of-view.
x(n) y(n)
d(n)
w(n)
e(n)
FIR Filter
Algorithm Adaptation ADF
Fig. 1.1: Block diagram of an ADF.
1.1.1.1 System Identification
The typical set up of the system identification application is depicted in Fig. 1.2, wherewu(n) is the impulse response of the unknown system,η(n) is the noise associated at the output of unknown system, w(n) is the coefficient vector of ADF,yu(n) is the output of the unknown system andx(n) is the input to both the systems. Observably, the desired signal in this case is given byd(n) =yu(n) +η(n). Here
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wu(n)
Adaptive
x(n) y(n)
Filter Unknown
w(n)
d(n)
e(n) System
η(n) yu(n)
Fig. 1.2: Block diagram of system identification configuration of an ADF.
the ADF estimates the signal yu(n) at its output. If the output of ADF y(n) approaches to yu(n), then the ADF is able to model a portion of the unknown system. The primary aim is to identify the best possible model of the unknown system for some set of system parameters. As the filter reaches to convergence, the desired responsed(n) is expressed as
d(n) =xT(n)wopt(n) +η(n) (1.1) where wopt(n) is the optimum coefficient vector of the unknown system at time instant n. While identifying an unknown system, it is important for an ADF to have fast convergence and low steady- state error in order to provide close approximate at its output. The ideal adaptation procedure adjust w(n) such that wopt(n) = w(n) when n approaches to infinity. Some real-world applications of the system identification scheme include modeling of multi-path communication channels, control systems, seismic exploration etc [8, 9].
1.1.1.2 Channel Equalization
Most of the channels in communication system are band-limited in nature due to which distortions in the amplitude and phase of the transmitted signal are bound to occur. This may even lead to over- lapping of symbols with nearby symbols over several signaling periods of time. The phenomenon of overlapping of successive symbols is commonly known as inter-symbol interference (ISI). If the trans- mission data rate is high, more and more symbols get overlapped, thereby inhibiting the transmission of data at higher rates. Channel equalization is therefore of prime importance for reliable detection of symbols. ADF is widely used in channel equalization to counter the ISI to decipher the received information. This can be explained by considering a system model of adaptive channel equalization
T 2T
0 −α 0 αT T+α
t(n)
η(n) CHANNEL
x(n) y(n)
r(n)
Delay
d(n)
e(n) ADF
Fig. 1.3: Block diagram of channel equalization configuration of an ADF.
shown in Fig. 1.3 whereh(n) is the channel impulse response and η(n) is the additive white noise in the channel. Note that the channel impulse response h(n) contains causal, anti-causal component or both which results in pre-cursor ISI, post-cursor ISI component or both in the received symbol x(n) respectively. However, in the presence of noise, the actual received symbol becomesx(n) =r(n)+η(n).
The ISI at the receiver can be eliminated by using an ADF whose transfer function is reciprocal of the channel transfer function. The direct application of ADF in channel equalization leads to noise ampli- fication, especially when the channel contains the spectral (or deep) nulls. This can be well understood by calculating the error between the transmitted signal and the equalized signal e(n) = t(n)−y(n).
Assuming that the z-transforms of e(n),t(n) and y(n) are respectivelyE(z),T(z) and Y(z), we get
E(z) =T(z)−Y(z) (1.2)
Substituting Y(z) =X(z)W(z),X(z) =R(z) +N(z) and R(z) =T(z)H(z) in (1.2), we have
E(z) =T(z)(1−H(z)W(z))−N(z)W(z) (1.3) Now, the power spectrumSe(z) of the error signal can be calculated as
Se(z) =St(z)|1−H(z)W(z)|2+Sη(z)|W(z)|2 (1.4) whereSt(z) and Sη(z) are the power spectra of the transmitted data symbols and the additive noise.
It is clear from (1.4) that the ADF eliminates the ISI component from the power spectrum, if and only ifW(z) = 1/H(z). However, it leads to noise amplification at spectral nulls (|H(z)|= 0) or deep nulls (|H(z)| ≈0) due to the second term in (1.4), where |•| denotes the magnitude operator. Moreover, such a configuration only eliminates the pre-cursor component of ISI. This has been addressed by a more complex system which is commonly known as adaptive decision feedback equalizer (ADFE) [10].
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1.1.1.2.1 ADFE: It consists of two ADFs in series with one in feedforward (FF) path and other in feedback (FB) path with a decision device. It utilizes the regenerative effect of a decision device in the feedback loop to overcome the noise amplification. Hence, it can perform well even when channel contains spectral and/or deep nulls. The typical configuration of such filter is shown in Fig. 1.4. The purpose of FF-ADF is to remove the pre-cursor ISI component from the received data, while FB-ADF acts on the residual data to remove the post-cursor ISI component by subtracting from the output of FF-ADF, according to
z(n) =yf(n)−yb(n) (1.5)
where z(n) is the output of ADFE, yf(n) =xT(n)wf(n), yb(n) = dT(n)wb(n), x(n) = [x(n), x(n− 1), ..., x(n−Nf+1)]T,wf(n) = [w0f(n), w1f(n), ..., wNf
f−1(n)]T,d(n) = [δ(n−1), δ(n−2), ..., δ(n−Nb)]T, wb(n) = [wb0(n), wb1(n), ..., wNb
b−1(n)]T and,Nf andNb are the orders of FF and FB filters respectively.
D yb(n) δ(n)
e(n)
yf(n)
FB-ADF
x(n) FF-ADF z(n)
Fig. 1.4: Block diagram of conventional ADFE.
The most recent tentative decision δ(n) is obtained by quantizing the output of ADFE, according to
δ(n) =Q[z(n)] (1.6)
where Q[•] denotes the quantizer operation. An error signale(n) is computed, as per
e(n) =δ(n)−z(n) (1.7)
The coefficients of FF and FB filters are usually updated based on LMS criterion, according to
wf(n+ 1) =wf(n) +µfe(n)x(n) (1.8)
wb(n+ 1) =wb(n) +µbe(n)d(n) (1.9) where µf and µb are the step-sizes of FF and FB filters respectively. It is clear that every iteration involves 2Nf+ 2Nb+ 2 multiplications (Nf+Nb for filtering,Nf+Nb+ 2 for coefficient adaptation), 2Nf + 2Nb additions (Nf +Nb−2 for filtering,Nf +Nb for coefficient adaptation, 1 for filter output, 1 for error computation) and 1 quantizer (or decision device).
1.1.1.3 Noise Cancellation
The aim of noise cancellation is to eliminate the background noise by recreating its replica. In this, ADF plays a key role in estimating the noise over the time in order to cancel with the actual signal.
A typical configuration of adaptive noise cancellation (ANC) of an ADF is shown in Fig. 1.5. The desired primary input signal comprises of the original loudspeaker ˆd(n) and a component of interfered noise signal η(n). The ADF is used to estimate the noise signal from the desired signal. Subtracting the estimated noise signal by ADF from the desired signal ideally leads to the actual signal. An error signale(n), is therefore calculated as
e(n) = ˆd(n) +η(n)−y(n) (1.10) This error signale(n) is used to update the filter coefficients until the mean square error is minimized.
x(n) y(n)
ADF
e(n) = ˆd(n) +η(n)−y(n) Primary Input
Reference Signal
Noiseη(n)
d(n)ˆ d(n) = ˆd(n) +η(n)
Fig. 1.5: Block diagram of adaptive noise cancellation configuration of an ADF [6].