12.1 INTRODUCTION

The ever‐increasing demand for channel capacity and bandwidth has resulted in the widespread use of adaptive signal processing techniques that compensate for the inevitable signal interference and distortion that results from the competitive needs for capacity and bandwidth. The significant advantages in efficient spectrum utilization, gained by high‐order symbol modulation with spectral containment, have been facilitated by the use of adaptive processing algorithms that compensate for the interference and distortion under crowded channel conditions. The development of adaptive processing for improved communications was jump‐started when wireline telephone networks [1] were under pressure for more capacity and higher data rates, and was well understood and developed when wireless communications entered the marketplace.

In the following sections, the mathematical background and algorithms are developed for adaptive systems as they apply to waveform equalization of intersymbol interference (ISI), cancellation of interfering signals, and waveform identification. These three objectives are obtained using subtle alterations in the adaptive processing configurations. The adaptive processing generally uses finite impulse response¹ (FIR) filters with weights that are adaptively adjusted to minimize error between the sampled filter output and the received input signal with known or estimated data using the minimum mean‐square error [2] (MMSE) algorithm. With known data, a preamble or training sequence is available to aid in the adaptive acquisition or convergence processing. In these cases, the processing following the known data is based on the estimates of the received data using a decision‐directed adaptive processing algorithm. After the training interval, the decision error probability is sufficiently low to maintain the MMSE condition. In applications where a training sequence is not available, the adaptive algorithm is referred to as a blind or self‐recovering algorithm; however, using the training sequence significantly reduces the adaptation time.

In the next section, the orthogonality principle is introduced and applied in the derivation of Weiner’s optimum linear estimation filter [3–5]. In Section 12.2, the Wiener estimation filter is described using the orthogonality principle and the MMSE criteria. In Section 12.3, the optimum FIR filter is examined, with the tap weights adaptively estimated using the lease mean‐square (LMS) algorithm. The LMS algorithm results in lower implementation complexity than the MMSE algorithm. Section 12.4 considers various forms of equalizers and Section 12.5 discusses adaptive interference cancellation using the LMS algorithm. The preceding adaptive techniques are based on symbol‐by‐symbol processing and an entirely different approach, using the recursive least‐squares (RLS) algorithm, is discussed in Section 12.6. The RLS algorithm converges to the steady‐state condition in considerably less time than that of the MMSE or LMS algorithms. Case studies involving the application of ISI equalization and narrowband interference cancellation are given in Sections 12.7 and 12.8; this chapter concludes with Section 12.9 with a case study of the RLS‐adaptive equalizer.

12.1.1 The Orthogonality Principle

The orthogonality principle is the fundamental principle used in the evaluation of the MMSE between a set of parameters and their estimates. The principle applies in the analysis of linear or nonlinear estimation, involving constant or time‐varying, real or complex parameters. However, to simplify the following description, the linear mean‐square estimation (LMSE) of a single, constant, and real parameter is used. The orthogonality principle states that the MMSE results when the estimation error is orthogonal to the measurement x, that is, and, under this condition, the MMSE is computed as the expectation .

Consider that a real zero‐mean parameter y is observed as² x = y + n, where n represents zero‐mean additive noise and the estimate of y is formed using linear combination of the observations x_i, expressed as

(12.1)

The weights a_i form a filter that decreases the noise and, thereby, improves the estimate. In this description, the index i simply associates x with the sampled noisy value of the parameter y. Using (12.1), the mean‐square error (MSE) is expressed as

(12.2)

To establish the MMSE, the derivative of (12.2) with respect to a_i is formed and set to zero resulting in

(12.3)

Therefore, based on the last equality in (12.3), the requirement for the MMSE is

(12.4)

The first equality in (12.2) is expanded as

(12.5)

and, upon substituting (12.1) for the expectation , the MMSE error expression is evaluated as

(12.6)

The last equality in (12.6) follows from (12.4) and confirms the requirement that the MMSE occurs when the expectation of the product of the measurement error and data are orthogonal, that is, when .

12.2 OPTIMUM FILTERING—WIENER’S SOLUTION

Wiener’s approach to determining the optimum filter of the noisy estimate of an event (t) is based on the filter that provides the MMSE of (t). The solution is based on the application of the orthogonality principle outline in the introduction. Consider the noisy observation of the event (t) expressed as

(12.7)

with the filtered estimate of (t) expressed as

(12.8)

In the following analysis, the stochastic processes are stationary and characterized as complex baseband functions and the filter is linear and time‐invariant. It is desired to determine the optimum linear filter h_o(t) that results in the MMSE of the measurement error.³ Referring to Section 12.1.1, the MMSE occurs when the orthogonality principle applies and, under the above conditions, is expressed as

(12.9)

The solution to (12.9) is expresses in terms of the expectations as

(12.10)

Upon using the wws property and defining , (12.10) simplifies to

(12.11)

The rhs of (12.11) is expressed as the convolution integral by substituting with dλ = dζ resulting in

(12.12)

Therefore, the optimum filter is expressed as having frequency domain response

(12.13)

Evaluation of the MMSE corresponding to the optimum filter h_o(t) is left as an exercise. The LMSE algorithm can also be applied to the prediction [6] of from the data .

12.3 FINITE IMPULSE RESPONSE‐ADAPTIVE FILTER ESTIMATION

In this section, the Wiener filter is applied to the sampled signal where the index i represents the uniform sampling of at increments t = iT_s, where T_s is the reciprocal of the sampling frequency f_s. All of the discrete‐time functions in this section are considered to be complex baseband functions that are within the Nyquist bandwidth. Furthermore, the analysis involves discrete‐time stochastic processes that are ergodic and stationary in the wide‐sense, that is, the mean value is constant and the correlation and covariance functions depend only on the sample‐time increment T_s. The basic structure of the adaptive filter is shown in Figure 12.1. The filter input is represented by the current sampled complex data and N − 1 previous samples. If the sampled signal is the output of the demodulator matched filter then T_s is equal to the modulation symbol duration T, otherwise T_s may represent a suitable sub‐symbol sampling interval; the unit delay designation z⁻¹ corresponds to the normalized sample delay. For each set of N samples, the filter output is computed and compared to a known reference signal forming the error corresponding to the sample t = jT_s. The error is then applied to the weight update function that adjusts all of the weights resulting is a steady‐state condition that corresponds to the minimum error. The adaptive processing details are discussed in the remainder of this section.

The sampled complex inputs and the filter weights are defined, respectively, as the complex vectors

(12.14)

and

(12.15)

The reference, or desired response, of the adaptive filter is _j and the difference between _j and the estimated response⁴ is the instantaneous estimation error, expressed as

(12.16)

The optimum estimation filter corresponds to the MMSE that results when the filter has settled to the steady‐state condition corresponding to the optimum weight vector . Based on the orthogonality principle, the MMSE error occurs when the error is orthogonal to the data, that is, when

(12.17)

and, referring to (12.6), the corresponding MMSE is

(12.18)

Aside from the trivial condition corresponding to , the MSE criterion is based on the error defined in (12.16) and is expressed as the expectation

(12.19)

Upon expanding (12.19) and distributing the expectation, the MSE is evaluated as

(12.20)

where is the variance of the reference, is the N × 1 cross‐correlation vector between the reference and data, and R is an N × N correlation matrix of the data; and R are defined, respectively, as

(12.21)

and

(12.22)

The second equality in (12.20) is established using the following relationships:

(12.23)

(12.24)

(12.25)

and

(12.26)

The matrix R possesses unique properties based on the stationarity of the sampled stochastic processes and is referred to as a Hermitian matrix.⁵ The diagonal elements are real and the trace of R is . Indexing the elements of the row and column vectors and as x_i and x_j respectively, the elements . For k > 0 (k < 0), the matrix elements correspond to positive (negative) correlation lags that lie above (below) the matrix diagonal. Furthermore, the elements of the Hermitian matrix correspond to the condition , so the elements above and below the principal diagonal possess complex conjugate symmetry.

The MSE, defined by (12.20), characterizes a concave upward surface with a minimum value corresponding to the derivative equal to zero. The derivative is defined as the gradient vector and is evaluated as

(12.27)

The solution to the derivatives in (12.27), indicated by the third equality, is found in Section 1.12.5. Upon evaluation of (12.27) with the optimum filter weights are expressed as

(12.28)

From the evaluation of (12.20) under the optimum condition , the MMSE is expressed as

(12.29)

Based on the method of steepest‐descent [7], the filter tap weights are altered to reduce, or minimize, the slope of the N‐dimensional concave upward surface characterized by the weights. At each decision sample jT_s, the weight updated is based on the first‐order recursive filter response, expressed as

(12.30)

The second equality in (12.30) results from (12.27) and the step‐size parameter u is used to provide stability and control the rate of convergence. As the expectation of the gradient vector approaches zero, the filter approaches the optimum MSE estimate of the reference _j. The slope of the concave upward surface about the minimum point forms a discriminator S‐curve that ensures stable tracking about the optimum tap weights. The updating is identical for each tap weight and the required processing for the n‐th tap weight is determined for the corresponding elements at sample j of the column vectors, and , in (12.30); with the subscript n,j representing the n‐th row at sample j, the result is expressed as

(12.31)

The complex scalars and are evaluated as

(12.32)

and

(12.33)

where the Kronecker delta function arises from the evaluation of the gradient vector in (12.27) and the variance in (12.33) is simply . Substituting these results into (12.31), the weight updating for the n‐th tap weight and sample j is expressed as

(12.34)

The updating of the weight at sample j given the previous weight at sample is shown in Figure 12.2; the asterisk * denotes conjugation. The filter is typically initialized using  = 0 + j0; however, if feasible, the best guess of the steady‐state solution should be used.

The stability and convergence properties of the FIR‐adaptive filter have been analyzed by Widrow [8], Ungerboeck [9], Gitlin and Weinstein [10], and Haykin [5]. Their analysis focuses on the eigenvalues of the correlation matrix R and, based on the method of steepest‐descent, the adaptive is filter stable and converges to the steady‐state condition, regardless of initial conditions, if all of the eigenvalues satisfy the condition

(12.35)

Equation (12.35) leads to the conditions and, recognizing that the eigenvalues for the Hermitian matrix R are all positive real values, upon defining the maximum eigenvalue as λ_max, the condition on u is

(12.36)

Haykin [11] examines the transient behavior of the steepest‐descent algorithm for a two‐tap FIR filter with varying eigenvalue spread,⁶ fixed step‐sizes u, and varying step‐sizes u with fixed eigenvalue spread. Haykin’s salient observations are:

• The steepest‐descent algorithm converges most rapidly when the eigenvalues, λ₁ and λ₂ are equal or when the initial conditions are chosen properly.
• For a fixed step‐size u, as the eigenvalue spread increases the correlation matrix R becomes more ill conditioned.
• For small step‐sizes u, the transient behavior of the steepest‐descent algorithm becomes overdamped. For large step‐sized, approaching 2/λ_max, the transient behavior becomes underdamped and exhibits oscillations.

Haykin concludes that the transient behavior of the steepest‐descent algorithm is highly sensitive to the step‐size parameter and the eigenvalue spread. In the selection of the adaptive processing gain, a performance tradeoff exists between the variance of the tap weights and the responsiveness of adaptive filter to the channel dynamics. For example, decreasing the gain decreases the variance but increases the system performance sensitivity to the received signal dynamics and increasing the gain has the inverse affects. Gitlin and Weinstein have investigated the adaptive equalizer MSE as it relates to the tap weight precision, the number of taps, the channel characteristics, and the digital resolution. Gitlin and Weinstein conclude that the number of taps should be kept to a minimum, consistent with the steady‐state performance objectives and the weight precision should be ten to twelve bits. Insufficient processing precision will result in tap accumulation round‐off errors and dominate the system performance as the number of filter taps is increased. In general, the gain should be no greater than necessary to track the time‐varying fading as determined by the signal decorrelation time and the length or span of the equalizer should not exceed the multipath delay spread; these topics are also discussed in Chapters 18 and 20. Adaptive filtering in the frequency domain is discussed by Dentino, McCool, and Widrow [12] and by Bershad and Feintuch [13].

12.3.1 Least Mean‐Square Algorithm

The weight updating in the preceding section, using the Wiener FIR‐adaptive filter, is problematic, in that, the algorithm is dependent on the correlation matrix R. To circumvent the sensitivity of the weight update algorithm on the correlation matrix, Widrow and Hoff [14] developed the LMS algorithm using real data. The LMS algorithm was subsequently modified by Widrow, McCool, and Ball [15] using the complex data. The LMS algorithm is an efficient method of updating the FIR‐adaptive filter tap weights through the entire process of adaptation and estimation. The steepest‐decent algorithm is used, in that, the filter tap weights are updated using a single‐pole recursive filter that reduces the slope, or gradient vector , as given by the first equality in (12.30). However, in this case, the N‐dimensional instantaneous square‐error is defined as

(12.37)

and the gradient vector is evaluated as

(12.38)

The third equality in (12.38) is established using the rules for complex matrix differentiation discussed in Section 1.12.4. From (12.38), the filter tap weights are updated as

(12.39)

The LMS tap weight update algorithm is identical for each tap weight and is shown in terms of the n‐th complex weight in Figure 12.3. Although (12.39) is a convenient form in expressing the tap weight updating, an alternate expression is obtained by substituting (12.16) for and rearranging the terms as in (12.34) to obtain the algorithm for the n‐th tap weight update expressed as

(12.40)

The tap weight update algorithm processing for the LMS feedforward equalizer and canceler, depicted respectively in Figures 12.6 and 12.8, is shown in Figure 12.3a. Figure 12.3b shows the weight update processing for the peak‐distortion zero‐forcing equalizer (ZFE) depicted in Figure 12.5.

In remainder of this section, the stability and convergence properties of the FIR‐adaptive filter using the LMS algorithm are summarized based on the work by Haykin [5]. As before, the analysis focuses on the eigenvalues of the correlation matrix R and, based on the method of steepest‐descent, the adaptive filter is stable and converges to the steady‐state condition if the step‐size satisfies the condition

(12.41)

The sum of the eigenvalues represents the mean‐square value, or total power, of the filter taps. Since the sampled input represents a stationary process, the eigenvalues represent the zero lag correlation values of the Hermitian matrix R and the left‐hand‐side (lhs) of (12.41) follows from the positive real values .

The misadjustment parameter [16] M is defined as the ratio of the average excess MSE to the MMSE J_min, expressed in (12.29), that is,

(12.42)

The parameter M corresponds to the steady‐state condition evaluated as expressed in the second equality in (12.42) [16].

The average excess MSE is defined as

(12.43)

where J(∞) is steady‐state mean‐square convergence error of the LMS algorithm which exceeds J_min. In terms of the convergence samples j and the tap weight vector error ε(j) = w(j) − w_o, the LMS mean‐square convergence error is which leads to the second equality in (12.43).

Haykin’s salient performance observations for the LMS algorithm are [17]:

• The adaptive filter converges slowly for small values of u with a correspondingly low misadjustment. Conversely, for larger values of u the filter converges rapidly with a correspondingly large misadjustment.
• The convergence of the average MSE, as given by (12.41), also guarantees the convergence of the mean tap weight values given by (12.36) since .
• For increasing eigenvalue spreads, the convergence time of the excess MSE becomes slower and the average tap weight convergence is limited by the smallest eigenvalues.

The simplicity of the LMS algorithm stems from the fact that it is based on the instantaneous squared error and does not require the expectation associated with the evaluation of the correlation functions or inversion of the correlation matrix. Considerable attention has been given to the theoretical evaluation and measurement of the symbol‐error probability, with channel‐induced ISI, using the LMS algorithm [18–22].

The adaptive filters are characterized as having a primary and reference input that determine the principal application of the adaptive processing. The primary input is associated with the received signal. For example, equalization uses the received signal as the primary input into the adaptive filter with the reference input derived from a locally generated replica of known data or data estimates derived from the received data. The reference data are often derived from a preamble or training sequence associated with the received data. With noise and interference cancellation, the reference input is a correlated replica of the interfering signal that is applied to the input to the adaptive filter. For sufficiently high interference‐to‐signal ratios (ISRs), the reference input may also be derived directly from the received signal. System or waveform identification is accomplished by applying the reference signal to the adaptive filter, possibly using multiple references that are suspected be correlated with the primary received signal.

12.4 INTERSYMBOL INTERFERENCE AND MULTIPATH EQUALIZATION

In this section, several commonly used equalizers are discussed with case studies of their performance presented in Sections 12.7–12.9.

12.4.1 Zero‐Forcing Equalizer

The ZFE is among the first equalizers to be analyzed and implemented [23]. The equalizer operates on the output of the channel filter that introduces ISI distortion on the transmitted data‐modulated symbols denoted by the baseband signal samples . The channel filter is shown in Figure 12.4 with the output expressed as the convolutional sum

(12.44)

The peak distortion, at the output of the channel, is defined in terms of the channel impulse response as

(12.45)

where is the maximum filter sample with the filter normalized such that  = 1. The peak distortion is a measure of the noise‐free dispersion about the data‐modulated symbol rest‐points resulting from the channel ISI. The dispersion between adjacent rest‐points characterizes the eye‐opening and when D_o < 1 the eye is said to be open [24].

The ZFE, shown in Figure 12.5, follows the channel filter and attempts to minimize the loss from the channel ISI. The equalizer output is evaluated as the convolutional sum

(12.46)

The peak distortion at the output of the equalizer is defined in terms of the impulse response of the channel cascaded with the equalizer and is expressed as

(12.47)

As indicated in Figures 12.4 and 12.5 the channel has L taps and the equalizer has 2N + 1 taps, so the extent or range of the non‐zero sampled impulse response is −N to N + L − 1 for a total of 2N + L samples. The cascaded impulse response in (12.47) is sampled at the time intervals jT_s and computed as

(12.48)

The peak distortion, given by (12.47), can be made arbitrarily small at the sampled points as the number of equalizer taps approaches infinity, and, although Lucky [25] has shown that D is a convex upward function of the equalizer coefficients, there is no assurance that the tap weights will converge for a finite number of taps. However, a solution does exist for a finite number of taps if the equalizer input peak distortion given by (12.45) satisfies the condition D_o < 1, but the distortion or decision error cannot be made arbitrarily small. The decision error at the output of the equalizer FIR filter is computed as

(12.49)

Using the data estimate results in a decision‐directed equalizer algorithm and using the known data _j, as with a preamble or training data sequence, results in a data‐directed equalizer algorithm. The relationship between the forward indexing of and the backward indexing of is expressed as . The weight update processing indicated in Figure 12.5 is based on the algorithm shown in Figure 12.3b.

12.4.2 Linear Feedforward Equalizer

The linear feedforward equalizer (LFFE) using the LMS tap weight update algorithm is shown in Figure 12.6. The LMS tap weight processing is indicated in Figure 12.3a. The complex tap weights are all initialized to zero and, as the sampled input data make its way through the FIR filter, the complex output of the summing bus is applied to the decision device to obtain the data decision estimate that is used to form the decision error . The data decision estimate is based on a complex limiter that computes the real and imaginary parts of based on the closest received waveform modulation rest states to the complex summation . For example, with binary phase shift keying (BPSK), if then otherwise . Under the input signal condition of wide‐sense stationarity,⁷ the mean value of the decision error will approach zero as the filter weights are adjusted during the adaptation processing. The decision estimate is subject to signal‐to‐noise ratio‐dependent decision errors; however, when a preamble is used, the known preamble data _j is used to reduce the learning time.

The performance of the LFFE is examined in the case study in Section 12.7. In the study, the recovery of the bit‐error performance loss resulting from ISI is examined for narrowband filtering of a BPSK‐modulated waveform.

12.4.3 Nonlinear Decision Feedback Equalizer

The decision feedback equalizer (DFE) [26, 27], shown in Figure 12.7, augments the LFFE symbol equalizer by forming the weighted linear combination of the previous symbols and reducing their contributions to the ISI. The DFE processing is equivalent to that involving linear prediction as embodied in forward linear prediction (FLP), backward linear prediction (BLP), and forward–backward linear prediction (FBLP) algorithms described by Haykin [28]. Belfiore and Park [29] and Proakis [30] also provide detailed analyses and compare the implementation and performance of the ZFE, LFFE, the DFE implementations. The DFE is used to correct multipath interference and signal distortion resulting from frequency selective fading [31] that is prevalent in wideband direct‐sequence spread‐spectrum (DSSS) applications. Lee and Hill [32] discuss the application of the DFE in terms of maximum‐likelihood sequence estimation (MLSE).

Proakis shows that, for moderate to severe multipath profiles, there is about a 2 dB loss in E_b/N_o performance between using known data decisions, as would occur during the reception of a learning or training data sequence, and using the locally detected data estimates. When using the LFFE, a severe multipath profile results in bit‐error probabilities that asymptotically approach with increasing signal‐to‐noise ratio; however, with the DFE equalizer the bit‐error probabilities continue to decrease. These results are demonstrated using symbol‐spaced equalizer (SSE) taps with an optimally sampled matched filter that is matched to the waveform and channel. Typically the channel is unknown so using the fractionally‐spaced equalizer (FSE) taps, as described in the following section, is necessary. The performance of the DFE is sensitive to the detected data because each decision error is shifted through the entire feedback path of the FIR filter so the selection of the step‐size gain is a particularly important consideration.

12.5 INTERFERENCE AND NOISE CANCELLATION

The adaptive filter, used as an interference signal canceler, has many applications as discussed by Widrow et al. [42] in their comprehensive paper on the subject. The adaptive canceler requires two inputs, usually identified as the primary signal and the reference signal. The primary input to the system is the desired signal corrupted by an additive interfering signal and the reference input is a correlated version of the interfering signal, ideally derived from the source of the interference. The reference input may be generated internally as in the case of decision‐directed applications or obtained locally in the case of nearby interfering signal and noise sources. The applications are nearly endless; however, include the broad subject of interference cancellation involving: antenna sidelobes, adaptive antenna arrays, electric motors, power lines, speech, and electrocardiography.

Figure 12.8 shows the general configuration of the LMS‐adaptive canceler with the reference input passing through the FIR filter and subtracted from the primary signal forming the estimation error that is minimized by the recursive weight update processing. The canceler performance is evaluated in the case study of Section 12.8 where the primary input is a BPSK‐modulated signal and a continuous wave (CW) interfering signal. In this case, the canceler reduces the interfering signal and the performance measure is the bit‐error probability based on the optimally sampled output of the matched filter.

12.6 RECURSIVE LEAST SQUARE (RLS) EQUALIZER

The adaptive RLS equalizer [43] algorithm processes data block‐by‐block to update the filter weights and results in faster convergence than the previously discussed equalizers. The implementation of the RLS‐adaptive equalizer is shown in Figure 12.9 and the following description of the processing follows the work of Haykin [44]. This configuration uses a known training sequence which is stored in a separate array containing the M most current values. The values of can be generated locally at the receiver without the need for storage. During reception of the message data, the detected data estimates are used resulting in a decision‐directed equalizer.

Since the RLS algorithm processes a block of data, it is convenient to characterize the filter tap values and weights in term of the vectors and respectively. The number of FIR filter taps (N) is selected to span the duration of the expected ISI of the received signal. The value of the M determines the relative location of the reference signal and is typically about N/2; however, if the duration of the pre‐ and post‐symbol ISI is not equal, then M must be selected proportionately to encompass all of the ISI samples. The computations also require computing a gain vector and the N × N matrix C = Φ⁻¹; where Φ is the N × N correlation matrix defined as

(12.50)

This definition of the correlation matrix is different from the correlation matrix R defined by (12.22), in that, it includes the exponential weighting factor where λ < 1 is a positive constant close to unity. The application of the scalar parameter λ forms a recursive‐adaptive infinite impulse response (IIR) filter that increasingly diminishes the influence of the older data blocks with decreasing values of λ. From a filtering point‐of‐view, λ controls the bandwidth of the recursive tap adjustments, influences the convergence time of the weight processing, and the responsiveness to the channel dynamics. Conventional inversion of the correlation matrix is processing intense; however, to obtain the recursive solution for the tap weight updating, the matrix C is computed based on the matrix inversion lemma discussed by Haykin. These quantities are initialized as indicated in Table 12.1. The vector and matrix C_j−1 provide auxiliary data storage for computing the recursively updated values and, upon completion, the contents of the updated arrays are transferred to these arrays in preparation for the next iteration.

Parameter	Initial Value	Comments
λ	0.99	Recursive scalar coefficient (typically: 0.999–0.95)
	(0,0)	Complex weight vector
	(0,0):	Elements of complex matrix Cj−1 (small positive constant δ = 0.005)
	(δ−1,0):

After initialization, the adaptive processing computations are described as follows. Upon reception of each received signal sample, the FIR filter output is computed as

(12.51)

The complex output of (12.51) is used for data detection and updating the adaptive filter weights for the next iteration. In computing the new weight values, the estimation error is determined as

(12.52)

The complex error is input to the weight processing function where the following operations are performed. Starting with the inverse correlation matrix, the gain vector is computed as

(12.53)

where the vector ṽ_j and scalar u are computed, respectively, as and . Using (12.52) and (12.53), the FIR filter taps are updated as

(12.54)

The final step involves updating the matrix inverse for the next iteration and the computation for this step is expressed as

(12.55)

In preparation for the next iteration and . Following these computations, the processing is repeated with the next received block of samples until receipt of the communication message is completed. The message must be prefixed with a training data sequence and possibly periodically repeated training sequences throughout the entire message. The inclusion and frequency of the periodic training sequences depends on the channel and system dynamics. As mentioned above, the training sequence reference bit must be replaced with the M‐th tap value of the stored data upon declaring acquisition. With periodic training sequences, the equalizer must switch back‐and‐forth between the training sequence and the message data. A case study demonstrating the performance of the RLS equalizer is given in Section 12.9.

12.7 CASE STUDY: LMS LINEAR FEEDFORWARD EQUALIZATION

In this case study, the LMS algorithm is used with the LFFE to reduce the ISI loss of a coherently detected BPSK‐modulated waveform. The performance with ISI is examined for three 0.4 dB ripple Chebyshev filters. Two of the filters have 6‐poles with respective normalized bandwidths BT = 1.5, 2.0; the phase response of these filters corresponds to the intrinsic Chebyshev filter phase function. The third filter has 4‐poles, a normalized bandwidth of BT = 3.0, and includes quadratic phase equalization with 6.5° at the band edges. The filtered received signal spectrums are shown in Figure 12.10 for the three filters.

The bit‐error performance results, shown in Figure 12.11, are based on Monte Carlo (MC) simulations using 1 M bits at each signal‐to‐noise ratio <8 dB, otherwise 10 M bits are used. The dotted curve is the theoretical or antipodal performance of BPSK and the circled data points verify the fidelity of the simulations program and correspond to the AWGN channel without the equalizer. The filters are symmetrical baseband filters and the received signal frequency error is zero. Ideal demodulator phase and bit‐time tracking is used with an integrate‐and‐dump (I&D) matched filter so the performance degradation is solely based on the channel and equalizer. The solid and dashed cures in each figure correspond to the channel filter without and with the equalizer. The equalizer uses a nine tap bit‐spaced FIR filter with a tap weight gain step‐size u = 0.002. The learning time, before demodulation bit‐error counting, is 400 bits. The simulation results demonstrate the performance improvement for the worst case filter shown in Figure 12.10a; however, for the less severe channel filters the equalizer has little effect. Reducing the equalizer step‐size during the data detection will improve the equalizer bit‐error performance while maintaining an acceptable learning time, although the channel dynamics will influence the misadjustment parameter restricting the lower limit of the step size.

12.8 CASE STUDY: NARROWBAND INTERFERENCE CANCELLATION

In this case study, the cancellation of narrowband interference is examined using CW interfering signals characterized as

(12.56)

where V_i = k_vV_s, and V_s is the peak carrier voltage of the received waveform and T is the modulation symbol duration. The simulation program provides an arbitrary number of narrowband interfering tones; however, the following results apply for a single tone centered on the desired signal spectrum with discrete phases relative to the phase of the received signal. The total power (P_tot) of the interfering tones, the modulated signal, and the receiver noise are adjusted to the automatic gain controlled power of P_agc according to the relationships , , and where ρ is defined as and the total power is computed as

(12.57)

The noise power is established from the specified signal‐to‐noise ratio γ_b = E_b/N_o according to the relationship

(12.58)

where γ_s is the signal‐to‐noise ratio in the bandwidth of the transmitted symbol so γ_s = kr_cγ_b with k bits/symbol and a forward error correction (FEC) code rate of r_c. The performance simulations use uncoded BPSK so k = r_c = 1. Using these normalized conditions results in a common received power level that provides for establishing a common adaptive filter gain, or step size u, independent of the number of interferers or the ISR; it also simplifies the specification of the analog‐to‐digital converter (ADC) range and the maximum amplitude level when the effect of finite amplitude quantization is being examined.

The model for evaluating the performance with the signal and interference is depicted in Figure 12.12. Under ideal conditions, the interfering signal is input as the canceler reference; however, the model provides for scenarios where the reference also contains the signal. To accommodate these situations, the parameter defines the signal level that is added to the canceler reference input. In the following simulated performance results, the parameter is specified as the signal attenuation or loss and is alternately used to denote the signal attenuation in decibels.

Based on Figure 12.12, the interference into the canceler can be characterized, in terms of the ISR, as

(12.59)

Equation (12.59) is plotted in Figure 12.13 as a function of I_in/I for various values of the parameter k_s. The departure from the condition I_in = I results from the influence of the additive signal power to the interference with decreasing I/S. From Figure 12.12 it is seen that as k_s increases in value from 1 (0 dB), the interference power P_Iin → P_I which is a more authentic representation of the interference.

As defined in (12.57), the total receiver power, including the desired signal, interference, and channel noise, is adjusted to the AGC power of 0.5 W and the resulting signal carrier power is computed using as

(12.60)

where R_b is the user bit rate. In the performance simulation program R_b = 1 kbps so the modulated carrier power is 30 dB above the energy‐per‐bit. For a single interfering CW tone, the power (P_cw) is given by

(12.61)

As shown in Figure 12.14, the canceler gain adjustment, or step‐size Δ = 2u, is the key parameter that determines the canceler convergence characteristics given a specified FIR filter length. As seen from Figure 12.14, the canceler convergence time can be decreased by increasing Δ; however, the bit‐error performance of the demodulator is improved by decreasing Δ following an acceptable convergence time. Consequently, a lower adjustment step‐size is typically used when convergence is detected based on the statistical properties of the canceler error. In the following bit‐error performance simulations, the bit‐error counting is delayed until the canceler has reached, or nearly reached, the steady‐state condition.

The canceler convergence is derived in Section 12.8.1, and Figure 12.14 is based on the generic normalized expression

(12.62)

where N_s is the number of canceler taps, Δ is the canceler weight incremental adjustment, and j is the incremental time index. The bit‐error performance of the demodulator with the CW interfering tone is shown in Figure 12.15 without the canceler. The ISR is indicated in decibels and the results clearly demonstrate the impact on the performance when the canceler is not used. To achieve a performance loss less than a few tenths of a decibel requires ISR ≤ −30 dB, which is virtually no interference.

With the exception of the performance in Figure 12.15 for ISR ≥ 0 dB, the MC simulations involve 1 M symbols or trials for signal‐to‐noise ratios <8 dB otherwise 10 M symbols for each signal‐to‐noise ratio are used. The range of signal‐to‐noise ratios is 0–14 dB in 1 dB steps. Furthermore, the phase of the interfering tone is indexed over 360° using ten deterministic phases given by for i = 1, …, 10. Therefore, the bit‐error performance involving 10 M MC trails results by averaging the bit errors over ten 10 M MC trials with each trial corresponding to a different signal phase. These simulation conditions apply to the simulated bit‐error results in the remainder of this case study.

Figure 12.16 shows the bit‐error performance under the steady‐state conditions following the acquisition or learning time using a canceler step‐size of Δ = 5 × 10⁻⁵. The signal attenuation parameter is k_s = ∞ dB, that is, the reference signal into the canceler is the CW interference tone without any additive signal. Upon examining the expanded plot in Figure 12.16b, it is seen that the performance loss is negligible for ISR ≤ 0 dB and increases to 0.4 dB at P_be = 10⁻⁶ when the ISR is increased to 30 dB; the corresponding loss is 0.23 dB at P_be = 10⁻⁵. Consequently, for a step‐size of Δ = 5 × 10⁻⁵ the losses were reduced to that shown in Figure 12.16 for ISR ≤ 30 dB. These losses can be further reduced by decreasing the step‐size.⁸

In Figure 12.17, a series of bit‐error performance curves show the impact of the additive signal level to the canceler reference input as depicted in Figure 12.12. For a given ISR, the bit‐error performance approaches that in Figure 12.16 as k_s approaches infinity. The objective is to characterize the minimum value of k_s (k_smin) to achieve a loss in E_b/N_o ≤ 0.2 dB relative to that shown in Figure 12.16. Examination of the results in Figure 12.17, supplemented with the additional performance for ISR = 30 and 50 dB, indicates that k_s ≥ 20 dB results in an E_b/N_o performance that is consistent with the above objective. These performance results are a direct consequence of the user bit‐rate of R_b = 1 kbps corresponding to the signal carrier power (P_s) being 30 dB‐Hz above the energy‐per‐bit as expressed in (12.60).

The interpretation of these results is shown in Figure 12.18 with the signal carrier power S = P_s = 30 dBW. Referring to Figure 12.12 and using the canceler reference input is where () is the interference input power resulting from the signal. Therefore, the signal‐related interference power is k_s(dB) below the signal power and, from Figure 12.17, k_s ≥ 20 dB so k_smin = 20 dB corresponding to . Consequently, cannot exceed the maximum margin of Δ_max = 10 dB/s relative to E_b, otherwise, the presence of the signal in the canceler reference degrades the bit‐error probability. Noting that R_b and Δ have units of 1/s, these results can be generalized for BPSK modulation by the requirement k_smin = R_b/Δ_max = R_b/10.

12.8.1 Theoretical Canceler Convergence Evaluation

This case study is concluded by examining the theoretical tap weight convergence based on the characteristic equation of the canceler. The analysis is based on the minimum shift keying (MSK) symbol matched filer expressed as

(12.63)

where ω_m = 2πf_m with f_m = 1/2T where T is symbol duration. The analysis considers the input as a CW interfering signal given by

(12.64)

The optimally sampled MSK filter outputs, , are input to the canceler as and the canceler output is expressed as

(12.65)

where N is the number to FIR filter tap weights and is the delay of the interfering tone into the canceler. The optimum weight vector w_o corresponds to the weights that minimize the MSE . The weight error is defined as the vector

(12.66)

Applying the method of steepest‐decent, discussed in Section 12.3, the transient response of the weights is formulated in terms of the weight error vector as

(12.67)

where v_o is the initial weight error vector, Δ = 2u is the canceler gain step‐size, I is an N × N identity matrix, and R is the N × N correlation matrix, given by

(12.68)

The matrix R is Hermitian, so there exists N linearly independent characteristic vectors,⁹ and, upon using the diagonal transformation where Q is a matrix composed of the N independent characteristic vectors and Λ is a diagonal matrix of characteristic values. Using these results, (12.67) is expressed as

(12.69)

The solution to (12.69) is based on the trace tr[Λ], defined as the summation of the diagonal elements of Λ, that is,

(12.70)

so the normalized solution to (12.69) is

(12.71)

At this point the analysis is simplified by assuming real signals corresponding to ω_ε and ϕ equal to zero, in which case, s(t) = V_p, independent of time, and the optimally sampled matched filter output is given by

(12.72)

In this example, the corresponding characteristic values are identical and evaluated as

(12.73)

and (12.71) becomes

(12.74)

Equation (12.74) is plotted in Figure 12.19 with V_p = 0.125 V. The simulated performance, based on (12.72), is indicated by the circled data points.

12.9 CASE STUDY: RECURSIVE LEAST SQUARES PROCESSING

The RLS processing is examined using MC computer simulations of the bit‐error performance with MSK modulation using 10 M bits for each signal‐to‐noise ratio. The application involves the acquisition of the received signal when the signal phase and/or amplitude are unknown over the respective ranges |φ| ≤ 180° and . The signal phase increment is Δφ = 18° with φ_i = −180 + Δφ(i − 1) : i = 1, …, 11 and the amplitude increment is with A_i = −A_m + ΔA(i − 1) : i = 1, …, 10. For example, when both phase and amplitude are unknown, the simulation indexes over 100 K bits for each parameter result in 11 × 10 × 100 K = 11 M bits for each signal‐to‐noise ratio. At each signal‐to‐noise ratio, the acquisition processing is re‐initiated for each parameter and the detected bit‐error counting starts following completion of the training sequence. The training data sequence is generated using the same random process as used for generating the message data, so no attention is given to finding a good training sequence. The functions of the simulation processing are shown in Figure 12.20.

Upon completion of the training data sequence, the acquisition switch is changed to use either the fixed tap weights or the decision‐directed RLS processing during receipt of the message data.

The channel is characterized by a cosine‐weighted impulse response expressed as

(12.75)

where T_s is the sampling interval, N_p = odd integer is the number channel impulses, and K_s determines the impulse spread. The characteristics of the four channels considered, ranging in severity from the worst‐case to best‐case, are summarized in Table 12.2.

The details of RLS processing are described in Section 12.6 and the data detection includes the MSK matched filter detection and ideal tracking following the training sequence in use.

12.9.1 Performance with Fixed Weights Following Training

The bit‐error performance results in this section characterize the acquisition and subsequent bit‐error evaluation with combinations of known and unknown received signal phase and amplitude as described above. For these results the RLS algorithm uses δ = 0.1 for initialization of the matrix C and λ = 0.999; this value of λ results a long memory of past events resulting in a narrow recursive bandwidth. Figure 12.21 shows the performance using 4 samples/bit and a block length of 2‐bits or 8 samples. The training data ranges in length from 4 to 64 blocks with various combinations of known and unknown parameters. Figure 12.21a corresponds to the ideal conditions with zero signal phase and a fixed carrier amplitude using channel No.1. These known received signal phase and amplitude conditions are ideal so the evaluations in Figure 12.21a characterize the performance of the RLS processing as a channel equalizer. The circled data points represent the ideal simulation conditions corresponding to antipodal detection performance and the dashed curve represents the performance without equalization. The solid curve demonstrates the performance improvement when the RLS equalizer is used. The results in Figure 12.21b–d represent the performance improvements with increasing training blocks under the indicated received signal parameter conditions. In these cases, the ideal channel is used, that is without filtering, so the RLS processing estimates the unknown received signal phase and/or amplitude parameters. A training sequence of 64 blocks results in near optimum performance.

Figure 12.22a–c shows the performance results with unknown received signal phase and amplitude when using channels No. 1 and 3 for the respective training sequence lengths of 12, 20, and 64 blocks. These results demonstrate the ability of the RLS‐adaptive processing to perform equalization, parameter estimation and correction, and acquire a noisy received signal with sufficient fidelity to provide near optimal bit‐error performance. However, as mentioned above, these results are based on λ = 0.999 which severely restricts the dynamic behavior of the input signal.

ACRONYMS

BLP

Backward linear prediction

BPSK

Binary phase shift keying

CW

Continuous wave

DFE

Decision feedback equalizer

DSSS

Direct‐sequence spread‐spectrum

FBLP

Forward–backward linear prediction

FIR

Finite impulse response (filter)

FLP

Forward linear prediction

FSE

Fractionally‐spaced equalizer

I&D

Integrate‐and‐dump (filter)

IIR

Infinite impulse response (filter)

ISI

Intersymbol interference

ISR

Interference‐to‐signal ratio

LFFE

Linear feedforward equalizer

LMS

Least mean‐square (algorithm)

LMSE

Linear mean‐square estimation (algorithm, equalizer)

MC

Monte Carlo (simulation)

MLSE

Maximum‐likelihood sequence estimation

MMSE

Minimum mean‐square error (algorithm)

MSE

Mean‐square error (algorithm)

MSK

Minimum shift keying

RLS

Recursive least‐squares (equalizer)

SSE

Symbol‐spaced equalizer

TDL

Tapped delay line

ZFE

Zero‐forcing equalizer

PROBLEMS

1. Evaluate the MMSE for the optimum filter h_o(t) developed in Section 12.2 using the relationship expressed in (12.6). Express the result in terms of the respective spectrums S_ỹỹ(ω), , and . Note that the Fourier transform of is .
2. Consider that the optimally sampled matched filter output of an isolated symbol through a bandlimited channel is characterized as
   
   where c_j is the matched filter sample with c₀ ≫ c_j : j ≠ 0 and d_n = ±1 is the isolated n‐th source data symbol of duration T seconds, such that, E[d_nd_m] = δ_nm where δ_nm is the Kronecker delta function.

   For the estimator shown in the following figure, determine the open‐loop estimates E_k[•] : k = 1, 2, 3 considering contiguous source data symbols corresponding to the matched filter samples

   

   Redraw the figure to include an ideal integrator following the appropriate estimators and a transversal filter with adjustable taps that will adaptively eliminate the ISI terms c_j : j ≠ 0.

   

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