A Low-Complexity Lattice-Based Low-PAR Transmission Scheme for ...
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 5, MAY 2004
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A Low-Complexity Lattice-Based Low-PAR Transmission Scheme for DSL Channels Iain B. Collings, Senior Member, IEEE, and I. Vaughan L. Clarkson, Member, IEEE
Abstract—This paper presents a new low-complexity multicarrier modulation (MCM) technique based on lattices which achieves a peak-to-average power ratio (PAR) as low as three. The scheme can be viewed as a “drop in” replacement for the discrete multitone (DMT) modulation of an asymmetric digital subscriber line modem. We show that the lattice-MCM retains many of the attractive features of sinusoidal-MCM, and does so with lower implementation complexity, ( ), compared with DMT, which requires ( log ) operations. We also present techniques for narrowband interference rejection and power profiling. Simulation studies confirm that performance of the lattice-MCM is superior, even compared with recent techniques for PAR reduction in DMT. Index Terms—Digital subscriber line (DSL), lattice codes, multicarrier modulation (MCM), peak-to-average ratio (PAR).
I. INTRODUCTION
S
INUSOIDAL-BASED multicarrier modulation (MCM) systems have recently attracted a great deal of interest for a wide range of applications, including discrete multitone (DMT) for asymmetric digital subscriber lines (ADSL), orthogonal frequency-division multiplexing (OFDM) for wireless LAN and 3G cellular, and also multicarrier code-division muiltiple access (CDMA). Unfortunately, there is a well-known drawback in terms of high peak-to-average ratio (PAR). Typically, the amplifier dynamic range must be large enough to achieve a clip-to-average ratio (CAR) of at least 14.2 dB. Various extensions have been shown to reduce the PAR [1]–[9]. Mostly, these are based on detecting high peaks before transmission and then either rotating certain subcarriers, filtering the waveform, or injecting a number of artificial tones. The best results are in the order of 3–6-dB reduction in PAR, with the exception of [8]; however, that technique is limited to coding schemes “for which the number of carriers is no more than around 32.” Interestingly, none of the schemes take the characteristics of the channel into account. This paper takes a fundamentally different approach to achieving low-PAR transmission. The aim is to avoid clipping altogether. While it sits within the general MCM framework,
Paper approved by C. Tellambura, the Editor for Modulation and Signal Design of the IEEE Communications Society. Manuscript received April 16, 2003; revised September 29, 2003. This paper was presented in part at the IEEE GLOBECOM Conference, San Francisco, CA, December 2003. I. B. Collings is with the Telecommunications Laboratory, School of Electrical and Information Engineering, University of Sydney, Sydney NSW 2006, Australia (e-mail: [email protected]). I. V. L. Clarkson is with the School of Information Technology and Electrical Engineering, University of Queensland, Brisbane, Qsld. 4072, Australia (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2004.826261
the key difference is that with current high-rate samplers, it is not necessary to limit attention to sinusoidal carriers. The problem, therefore, is to choose the best set of time-domain vectors which are maximally spaced in the receive domain, but which are restricted in the transmit domain such that each instantaneous sample is within the linear region of the amplifier. In this paper, we present a low-complexity lattice-MCM approach for transmission through clip-limited DSL twisted-pair channels.1 As with standard ADSL, we assume knowledge of the channel and any noise correlation at the transmitter. Previously, we have demonstrated that a lattice approach can have significant advantages compared with sinusoidal MCM for the case of a cyclic prefix and block processing [10]. A key to the technique is that we have means to ensure that all lattice points fall within a multidimensional cube. In the context of digital transmission, this guarantees that clipping is completely avoided at the transmitter. We present new techniques to achieve low-complexity transmission without sacrificing performance or requiring a cyclic prefix. We also introduce power profiling, colored noise mitigation, and narrowband interference (NBI) rejection. The new approach is shown to significantly outperform sinusoidal MCM in clip-limited channels, even when they employ PAR-reduction schemes. We will demonstrate that the new lattice-MCM technique , where only requires implementation complexity of is the length of the transmit symbol.2 This is a distinct advantage compared with sinusoidal-MCM schemes, which require . We also demonstrate that our low-complexity lattice-MCM implementation can be decoded serially. Unlike sinusoidal MCM, it is not necessary at the receiver to await all the samples of a symbol before the first decoded outputs are produced. Of course, this is ideal for delay-limited systems. It turns out that the specifics of the lattice formulation can be viewed as a generalized form of Tomlinson–Harashima (TH) precoding [11]. A related generalization of TH precoding has recently been proposed for “vectored transmission” within the sinusoidal-MCM framework to mitigate multiuser interference (MUI) [12], [13]. While there are clear similarities between the two, the aims and outcomes are quite different; namely, in our case, PAR reduction, low-complexity modulation, and a complete replacement of the sinusoidal MCM with a lattice approach. Another interesting and somewhat aligned area is that of polarization limitations in digital magnetic tape recording. 1The approach can also be considered for slowly time-varying fixed wireless channels, e.g., IEEE 802.16. 2The O(:) notation indicates the dominating complexity term, as the system parameter (in this case, transmit-symbol size) goes to infinity.
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For example, [14] provides interesting results, although the approach is quite different. In Section II, we first present the signal model, covering both cyclic and silent prefix transmission, and present the generalized lattice-MCM technique in Section III. Section IV presents the low-complexity implementations. The relationship with TH precoding and vectored transmission is discussed in Section V, implementation issues for DSL applications are considered in Section VI, and simulation studies in Section VII.
, , is diagonal, and is upper triangular with ones along the diagonal. has full column rank, We make the usual assumption that which implies that the (diagonal) elements of , namely , are , then the lower right nonnegative. If has column rank diagonal elements of are set to zero. Now, consider transmit symbols of the following form:
II. SIGNAL MODEL
, , and where where is a vector of integers, related to , as we shall see later. Note that is upper triangular with ones along the diagonal, , and that columns of form an orthogonal . Moreover, columns of the (but not orthonormal) basis for are a basis for a lattice, and is a lattice prefiltering matrix point. We term the lattice whose basis matrix is the transmit which lattice and give it the symbol . It is the columns of shall be used in the lattice-based approach, as carriers for the data. Hence, we use the term lattice-MCM. . It remains to show how is mapped to such that Consider
be a time-domain Let the transmit symbol vector of transmit samples.3 The receive symbol is given by (1) where is the channel matrix, and is independent zero-mean is a vector function Gaussian noise with variance . Also, representing the transmitter nonlinearity, such that the th eleis clipped to or if or , ment of respectively, and otherwise, equals . samples is Assume that a guard period of length used between successive transmit symbols, where is the delay spread of the channel. In the generalized lattice approach presented in this paper, we allow the guard period either to contain a cyclic prefix (if the situation demands it), or zero transmit en1, ergy (thus saving power). In the latter case, is Toeplitz. For a cyclic prefix, is 1, and is and is circulant. In either case, has the following first column:
where
(4)
(5) Assume without loss of generality that the clipping level equals 1/2, so that the transmit cube has unit volume. to the Consider the following mapping from transmit symbol :
(2)
(6)
Assume that the data stream is divided into blocks of bits, not necessarily independent and identically distributed (i.i.d.), allowing for coded data. The aim is to design a set of transmit symbols which are widely spaced in the receive , where is called domain, but for which the transmit cube. In doing so, clipping would be completely avoided over the discrete-time channel while achieving a good BER. Furthermore, the mapping from data sequence to transmit symbol must be computationally efficient.4
is chosen such that . The elements of where can be found in an iterative way, since is upper triangular. It is possible to start by finding the value of which ensures , and then iterating backward to find the other values for , then we in turn. More specifically, if by subtracting the appropriate integer can force multiple of the th column vector of , namely . An explicit algorithm is given in [10]. It turns out that this can be viewed as a generalized form of TH precoding, applied to lattice basis vectors, as discussed later in Section V. , but it can also be Furthermore, not only do we have , and that shown that
III. LATTICE-BASED TRANSMISSION Each -bit data block is first divided into subsequences of length for . The subsequences are then mapped into the corresponding Gray-mapped binary subsequence, and then directly converted into a decimal integer, , where, of . We now show how this data vector course, (with elements ) is modulated in this lattice approach onto nonsinusoidal basis functions, or “carriers,” following the approach of [10]. Consider the following QR decomposition of :
(7) to Note that (7) is effectively adding integer multiples of (which is in the range , from above) to arrive at . The result is that the mapping in (7) can be inverted by division of each . simply taking the modulo, the function has no effect, We see now that since and in the receiver, we can compute the filtered measurement
(3) (8)
3The
basic assumptions on data formatting are identical to those for systems such as ADSL and fixed wireless IEEE 802.16. 4Note that sinusoidal-MCM schemes are computationally of ( log ) and fail to satisfy the clipping restriction.
ON
N
is a basis for the receive lattice, and where i.i.d. zero-mean Gaussian with variance . Since both
are and
COLLINGS AND CLARKSON: LOW-PAR TRANSMISSION SCHEME FOR DSL CHANNELS
are diagonal, the (unconstrained) maximum-likelihood (ML) estimate of is then simply [10] (9) is the th element of . The ML estimate of is where computed by calculating the positive remainder of the division , since from (7), we note that . of by Finally, the data estimates are obtained from by binary-toGray-code conversion. The resulting bit-error rate (BER) can be approximated by the lower bound derived by considering only the pairwise errors of nearest neighbor lattice points, and is given by (10)
The lattice carrier bit-loading values so as to minimize (10). First, let
can now be chosen
(11) In high-data-rate conditions, is sufficiently large so that each . In this case, we assign bits to basis vectors with the largest values of . To the remainder, we . It can be shown that the number of assigned assign bits is equal to , and that this allocation achieves the lowest , according to BER of all allocations for which each the low-noise approximation of (10), see [10]. In scenarios of either lower data-rate conditions, excess bandwidth, or channels with spectral nulls (e.g., bridged taps, or is relatively small, and we find that masked frequencies), there will be some ’s less than one. In this case, we propose an iterative bit-loading algorithm in order to select a subset of basis vectors, similar to the Chow algorithm in DMT [15]. The approach proceeds by excluding the subchannel with the lowest power, and recalculating the ’s. This is iterated until all the remaining ’s are greater than one. In other words, successively, with the lowest corresponding values of the columns of are allocated zero bits. The power is then spread amongst the remaining columns (carriers). While this is not necessarily guaranteed to provide an optimal allocation, simulations show that the performance is certainly more than acceptable.
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operations. Furthermore, the amount of require . We then show memory required for lattice MCM is also that these properties also hold for the cyclic-prefix case. A. Low-Complexity Modulation The arithmetic complexity of the coding process is dominated by the matrix calculation of from by (5) and from by (6). All other steps in the process (serial-to-parallel (S/P) conversion, Gray-to-binary encoding, etc.) are elementary logical operations. The QR decomposition is not included in the complexity calculation, since we are assuming fixed DSL channels, and as such, the QR operation only occurs once during the startup phase.5 We now make some key observations that will assist in showing that (5) and (6) can be combined into a single operation, and that even though they appear to be matrix calculations, the number of arithmetic operations required is, in fact, . only 1) Provided , has full column rank. 2) is banded with lower bandwidth and upper bandif . width 0. Hence, 3) The QR decomposition of is unique because has full column rank. Moreover, none of the diagonal elements of are zero [16]. is banded with lower bandwidth 0 and 4) The matrix upper bandwidth . , and therefore Proof: Observe that (12) for all , such that . For a given , the proof , (12) implies is by induction on , as follows. With . Assuming for all , then, from that also, since . (12), 5) Since is banded, and is diagonal, is banded also with the same bandwidths. 6) For the th row of , , we can write (13) where is the unit vector whose th element is one. Proof: Since , and is upper triangular with ones on the diagonal, then
IV. LOW-COMPLEXITY IMPLEMENTATION In this section, we will first show that in the zero-energy guard-period case (where silence is inserted between symbols rather than a cyclic prefix), the new lattice-MCM scheme arithmetic operations. This compares farequires only vorably with conventional sinusoidal-MCM techniques, which
5Even for extensions to slowly fading fixed wireless channels, the QR decomposition is only required at infrequent reinitialization phases.
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Fig. 1.
Modulator structure for the new coding scheme.
7) The calculation of (14) is 8) With
, since
is diagonal. , it follows from (6) and (13) that
), and these are banded matrices whose bandwidths are independent of . Fig. 1 shows an implementation of our new scheme, which . The shows how a single block is encoded for the case S/P conversion breaks up the incoming block of bits into parbits each. The pulse amplitude modulation allel channels of (PAM) encoders produce the elements of the vector as defined in (14).6 The modulo adders and feedback paths calculate the elements of according to (15). Observe that the maximum , in accordance with the number of inputs to any adder is banded structure of . B. Low-Complexity Demodulation
(15) This low-complexity algorithm for computing from can now be seen to be . First, is calculated in from (14). Second, the and are calculated iteratively from down to , using (15). Each is chosen as that unique in, i.e., , is set to the nearest teger that makes integer of the value that is computed on the right-hand side (RHS) of (15). Finally, note that the RHS of (15) requires a number of arithmetic operations that is not dependent on , but rather on , because of the banded nature of . Hence, the overall amount of arithmetic operations required to calculate for transmission is . In addition, it is important to point out that the memory required for encoding is also , since it is only necessary to store the nonzero coefficients of and (and therefore, also
The computational complexity of the receiver is dominated by evaluating (8). All other steps are easily shown to require at most arithmetic operations. Again, at first sight it would ; however, consider seem that the computation would be the following. 1) We can write , and so
2) The calculation of (16) requires only
operations, because
is banded.
6In fact, (14) is an asymmetric PAM, since the signal levels are not symmetric about zero, but this constant offset to the signal levels makes no difference to the analysis, so long as we are consistent in the decoding.
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Fig. 2. “Causal” demodulator structure for the new coding scheme.
3) With
Proof: Since
, we can write the th column
is
, we conclude that the entire decoding process is only . Once again, we point out that the memory required for de, since it is only necessary to store the coding is also nonzero coefficients of and . Fig. 2 shows the demodulator structure at the receiver. Following the analog–digital conversion (ADC), the samples are passed through a matched filter, then grouped according to their symbol and passed through a S/P converter, from which is output the elements of as defined in (16). The adders and where feedback paths compute the vector
is upper triangular
and we observe that 4) As a result, we have
and
(17) This low-complexity algorithm for computing the vector from clearly consists of iteratively evaluating the elements , starting at up to , according to (17). Note that evaluation of the RHS of (17) requires the summation of no more than elements, because is banded. Since the amount of computation required to compute each element of (from ), does not depend on , and because calculation of from
Therefore, passing through a modulo element before PAM detection recovers the bits encoded in . Note also that since the demodulation iteration in Fig. 2 starts , it is not necessary at the receiver to wait for an entire at symbol before the first decoded outputs are produced from it. samples of a symbol are received, beAfter the first comes available. The first decoded output can then be produced, and thereafter, a new output can be produced as each subsequent sample arrives from the ADC. In this sense, the structure can be called “causal.” If required, it is also possible instead to configure the modulation scheme to have a causal modulator, and then the demodulator becomes anti-causal (i.e., it requires the entire block of samples to be received prior to demodulation, as is the case for all existing sinusoidal-MCM schemes). To deduce the structure,
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of terms in the summation on the RHS is unaffected. However, require summing up to elements. the last values of Even so, the total computation required to calculate from is still . Therefore, the decoding complexity is still arithmetic operations. D. Complexity of Sinusoidal-MCM Techniques Fig. 3.
Structure of channel matrix with cyclic prefix.
Fig. 4. Structure of covariance matrix.
we simply reverse the order of the elements in the vectors so that
and
and and reconstruct to satisfy (1) and repeat the analysis as above. Computational and memory requirements are unchanged. C. Cyclic-Prefix Case We now show that the coding and decoding complexity is still arithmetic operations,when a cyclic prefix is applied to each symbol. This is because the matrix is no longer banded but “circulant banded.” This has implications for the structure of , in that it too is no longer banded. However, it still contains enough structure for fast coding and decoding to be performed. Note that now has the structure as shown in Fig. 3, where the circulant band has width , and the overall size of the matrix . As a result, it is possible that no longer has full is . However, its column rank is column rank, even when always greater than . If is rank deficient, we simply delete columns from the matrix until it has full rank. Equivalently, for each symbol, we set the last few samples to zero. if . Following the Now, same reasoning as followed in (12), we have that if . Therefore, (and also ) has the structure, as shown in Fig. 4. Let us now examine how this new structure for affects the amount of computation required for coding and decoding. The key to the low coding complexity is (15). We determined, for the case where there is no cyclic prefix, that each and could be determined in a fixed number of arithmetic operations, because in the sum on the RHS is the number of nonzero elements not a function of . This is still the case with the new structure for . Although there are now more nonzero elements of to be summed in calculating each and , the number of such elements is still independent of . Hence, coding complexity is arithmetic operations. still Changes to the structure of affect the decoding complexity only inasmuch as they affect the amount of computation re, the number quired to calculate each in (17). For
Standard sinusoidal-MCM transmission techniques are designed on the cornerstone of the fast Fourier transform (FFT). The scheme requires an inverse (I)FFT in the transmitter, and a corresponding FFT in the receiver. These operations cannot be avoided, and are well known to require computational com. Clearly, this is higher than for the latplexity of , as discussed tice-MCM approach, which requires only above. This is particularly significant when considering that the new lattice-MCM technique is totally free from clipping, with a corresponding superior performance, compared with sinusoidal MCM. It should also be noted also that the three leading PAR-reduction techniques for sinusoidal MCM achieve their performance improvements at the expense of higher computational complexity, since each scheme can be viewed as an add-on to the standard sinusoidal MCM. The selected mapping (SM) [1], [2] and partial transmit sequences (PTS) [3], [4] approaches calculate and check multiple transformed recalculations of the MCM symbol and pick the one with the lowest PAR. They also require that the receiver is informed whenever the transform is applied, via a reserved data bit in the DMT symbol. Tone reservation [5] is slightly different, since it selects unused (or even minimally used) subchannels and transmits energy at these frequencies in an attempt to lower the PAR of the overall transmit symbol. In doing so, there is no need to notify the receiver, which simply demodulates the data-carrying subchannels as usual. However, the computational cost of finding the best waveform to send down the unused channels can be extremely high. V. RELATIONSHIP WITH EXISTING PRECODING SCHEMES The signaling scheme presented in this paper has been derived from basic considerations of lattice theory. In this section, we highlight some relationships with vector coding and TH precoding. Vector coding was introduced in [17] as a linear technique for equalization of channels with intersymbol interference (ISI). As with our lattice scheme, it is a block coding MCM technique where the subchannels are derived from the QR decomposition of the channel matrix. The approach makes use of a partitioning of the data in into orthogonal subchannels, obtained by setting (18) This is clearly related to, but different from, (4). Unlike sinusoidal-MCM techniques, the vector coding subchannels are not orthogonal at the transmitter, only at the receiver, and decoding is straightforward, without the need for decision-feedback equalization, since
COLLINGS AND CLARKSON: LOW-PAR TRANSMISSION SCHEME FOR DSL CHANNELS
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Fig. 5. TH precoder structure.
can be PAM detected in the traditional way, as is i.i.d. Gaussian. Notice that vector coding takes no care to limit the amplitudes of the transmit samples in (i.e., to limit the PAR). Instead, as in [17], the objective with vector coding is to minimize the transmit power while holding other factors constant. In contrast, the TH precoder [18], [19] is a technique suitable for transmission over amplitude-limited ISI channels. Like our lattice-MCM, it makes use of modulo arithmetic at the transmitter and receiver. However, it is not suitable for block coding. Rather, it is a “modulo inverse filter” which, for a finite impulse response (FIR) channel, has an infinite impulse response. Fig. 5 ) which is comshows the structure of a TH precoder (for patible with the signal and channel model used in this paper. Clearly, the modulator structure bears a resemblance to the lattice precoder, by virtue of the modulo addition and feedback arrangement. Indeed, our scheme could be classified as a generalization of the TH precoder for block transmission. Many generalizations of TH precoding have appeared since its invention. Indeed, vector coding was combined with a TH-style precoder in the earliest publication [20] of which the authors are aware, although the intention was not, as it is here, to seek to minimize PAR and computational complexity.
In this case the receive symbol can be written (21) and we finish by computing the filtered measurement (22) where is a basis for the receive lattice, and . Note that is a vector of i.i.d. zero-mean Gaussian random variables with variance , since it is an orthogonal transform of . B. Frequency-Domain Power Profiling
Colored noise, cross-talk, and indeed NBI, are commonly modeled as autoregressive filtered versions of white noise. For typical DSL noise environments, see [21, p. 475] and [15, p. 85]. The resulting generalization of (1) is
In a regulated environment, it is often necessary to apply power profiles to MCM systems. For sinusoidal MCM, it is a relatively straightforward process of allocating less power and bits to particular (frequency-domain) carriers and their neighbors, due to the fact that the sinusoidal basis functions are really only orthogonal at the center of the frequency band. We now show that even though the lattice-MCM carriers are not orthogonal in the frequency domain, it is still straightforward to enforce a power profile. Consider the power profile as a filter on the transmit symbol. One approach is to build an analog filter (ie. post digital-toanalog (D/A) conversion) which suppresses the required frequencies, and then design the lattice-MCM carriers based on the combined filter–channel combination. Of course, all-digital solutions are preferred. The proposed approach is to implement a digital profile filter immediately prior to D/A conversion. In this case
(19)
(23)
where , and where is the noise correlation matrix. In the case of a transmission system with a zero-energy guard matrix, and for systems period, is an employing a cyclic prefix, is an matrix. In either case, we can make the following QR decomposition:
where is the filter matrix with frequency components given by the desired power profile. The design process is simply to follow the procedure in Section III, but start by taking the QR decomposition of the modified channel matrix, . Since the profile filter will have the effect of rotating the transmit lattice, it will be necessary to scale back the (by a known factor) to ensure that the outtransmit symbol ermost points of the rotated lattice are in the transmit cube, i.e., . Of course, this is at the expense of reduced SNR, and therefore, increased BER relative to nonprofiled lattice MCM. The alternative is to operate without a scale-back factor, and suffer an increase in PAR. In either case, performance is still significantly superior to power-profiled sinusoidal-MCM techniques, as shown in Section VII.
VI. SYSTEM IMPLEMENTATION ISSUES A. Colored Noise and Cross-Talk
where is a square matrix, and . We propose a lattice-MCM design continuing as in Section III, but instead performing the channel QR decomposition on the noise-whitened channel matrix (20)
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BER versus SNR for clip level (CAR) 5 dB, with standard channel.
Fig. 7.
BER versus CAR for standard channel with E
=N
= 27 dB.
Note that a profile filter can also be used in extreme cases of high NBI levels, or when the NBI statistics cannot be measured accurately. The filter can be viewed as masking out the (narrowband) frequency range, and could be used in addition to the inverse noise correlation approach in Section VI-A. VII. SIMULATIONS AND EXPERIMENTAL RESULTS A. BER Performance To compare directly with ADSL in each of the simulations in , , and , and this section, we set simulated a 1-km copper twisted-pair AWG24 line terminated at each end by 110 . Fig. 6 shows the vastly superior performance of lattice MCM in the presence of clipping at CAR dB. In such conditions, sinusoidal-MCM PAR-reduction techniques have no effect, and the curves for SM, PTS, and tone reservation sit on top of the standard ADSL curve shown in the figure. We note also that the PAR for the lattice MCM was measured at 3.01, also in very good agreement with the theoretical asymptotic PAR of 3 (ie. 4.8 dB). Fig. 7 shows the effect of varying the clipping level. For the ADSL system, the bit allocation was done using the well-known Chow algorithm [15] based on the SNR of 27 dB, and filling the lower frequency subchannels first (in this case, all available subchannels are used). For the lattice scheme, the bit allocation is as described in Section III for high data rates (i.e., each in this case). The figure shows that for CARs below 11 dB, the lattice-MCM scheme has significantly superior performance, compared with ADSL, even when existing PAR-reduction techniques are used. Note that the lattice BER floor is just for this channel-SNR scenario, as also indicated by above Fig. 6. This is, of course, higher than the ADSL floor, due to the tighter packing of constellation points needed to avoid clipping is an acceptable level for in the lattice scheme. Even so, data applications. Simulations were also carried out for the colored noise case arising from AM radio interference. In particular, narrowband noise was injected between the normalized frequencies of 0.8
Fig. 8. Maximum bits possible versus noise power, for BER standard channel with transmitter clip level 0:5 V.
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A Low-Complexity Lattice-Based Low-PAR Transmission Scheme for DSL Channels Iain B. Collings, Senior Member, IEEE, and I. Vaughan L. Clarkson, Member, IEEE
Abstract—This paper presents a new low-complexity multicarrier modulation (MCM) technique based on lattices which achieves a peak-to-average power ratio (PAR) as low as three. The scheme can be viewed as a “drop in” replacement for the discrete multitone (DMT) modulation of an asymmetric digital subscriber line modem. We show that the lattice-MCM retains many of the attractive features of sinusoidal-MCM, and does so with lower implementation complexity, ( ), compared with DMT, which requires ( log ) operations. We also present techniques for narrowband interference rejection and power profiling. Simulation studies confirm that performance of the lattice-MCM is superior, even compared with recent techniques for PAR reduction in DMT. Index Terms—Digital subscriber line (DSL), lattice codes, multicarrier modulation (MCM), peak-to-average ratio (PAR).
I. INTRODUCTION
S
INUSOIDAL-BASED multicarrier modulation (MCM) systems have recently attracted a great deal of interest for a wide range of applications, including discrete multitone (DMT) for asymmetric digital subscriber lines (ADSL), orthogonal frequency-division multiplexing (OFDM) for wireless LAN and 3G cellular, and also multicarrier code-division muiltiple access (CDMA). Unfortunately, there is a well-known drawback in terms of high peak-to-average ratio (PAR). Typically, the amplifier dynamic range must be large enough to achieve a clip-to-average ratio (CAR) of at least 14.2 dB. Various extensions have been shown to reduce the PAR [1]–[9]. Mostly, these are based on detecting high peaks before transmission and then either rotating certain subcarriers, filtering the waveform, or injecting a number of artificial tones. The best results are in the order of 3–6-dB reduction in PAR, with the exception of [8]; however, that technique is limited to coding schemes “for which the number of carriers is no more than around 32.” Interestingly, none of the schemes take the characteristics of the channel into account. This paper takes a fundamentally different approach to achieving low-PAR transmission. The aim is to avoid clipping altogether. While it sits within the general MCM framework,
Paper approved by C. Tellambura, the Editor for Modulation and Signal Design of the IEEE Communications Society. Manuscript received April 16, 2003; revised September 29, 2003. This paper was presented in part at the IEEE GLOBECOM Conference, San Francisco, CA, December 2003. I. B. Collings is with the Telecommunications Laboratory, School of Electrical and Information Engineering, University of Sydney, Sydney NSW 2006, Australia (e-mail: [email protected]). I. V. L. Clarkson is with the School of Information Technology and Electrical Engineering, University of Queensland, Brisbane, Qsld. 4072, Australia (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2004.826261
the key difference is that with current high-rate samplers, it is not necessary to limit attention to sinusoidal carriers. The problem, therefore, is to choose the best set of time-domain vectors which are maximally spaced in the receive domain, but which are restricted in the transmit domain such that each instantaneous sample is within the linear region of the amplifier. In this paper, we present a low-complexity lattice-MCM approach for transmission through clip-limited DSL twisted-pair channels.1 As with standard ADSL, we assume knowledge of the channel and any noise correlation at the transmitter. Previously, we have demonstrated that a lattice approach can have significant advantages compared with sinusoidal MCM for the case of a cyclic prefix and block processing [10]. A key to the technique is that we have means to ensure that all lattice points fall within a multidimensional cube. In the context of digital transmission, this guarantees that clipping is completely avoided at the transmitter. We present new techniques to achieve low-complexity transmission without sacrificing performance or requiring a cyclic prefix. We also introduce power profiling, colored noise mitigation, and narrowband interference (NBI) rejection. The new approach is shown to significantly outperform sinusoidal MCM in clip-limited channels, even when they employ PAR-reduction schemes. We will demonstrate that the new lattice-MCM technique , where only requires implementation complexity of is the length of the transmit symbol.2 This is a distinct advantage compared with sinusoidal-MCM schemes, which require . We also demonstrate that our low-complexity lattice-MCM implementation can be decoded serially. Unlike sinusoidal MCM, it is not necessary at the receiver to await all the samples of a symbol before the first decoded outputs are produced. Of course, this is ideal for delay-limited systems. It turns out that the specifics of the lattice formulation can be viewed as a generalized form of Tomlinson–Harashima (TH) precoding [11]. A related generalization of TH precoding has recently been proposed for “vectored transmission” within the sinusoidal-MCM framework to mitigate multiuser interference (MUI) [12], [13]. While there are clear similarities between the two, the aims and outcomes are quite different; namely, in our case, PAR reduction, low-complexity modulation, and a complete replacement of the sinusoidal MCM with a lattice approach. Another interesting and somewhat aligned area is that of polarization limitations in digital magnetic tape recording. 1The approach can also be considered for slowly time-varying fixed wireless channels, e.g., IEEE 802.16. 2The O(:) notation indicates the dominating complexity term, as the system parameter (in this case, transmit-symbol size) goes to infinity.
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For example, [14] provides interesting results, although the approach is quite different. In Section II, we first present the signal model, covering both cyclic and silent prefix transmission, and present the generalized lattice-MCM technique in Section III. Section IV presents the low-complexity implementations. The relationship with TH precoding and vectored transmission is discussed in Section V, implementation issues for DSL applications are considered in Section VI, and simulation studies in Section VII.
, , is diagonal, and is upper triangular with ones along the diagonal. has full column rank, We make the usual assumption that which implies that the (diagonal) elements of , namely , are , then the lower right nonnegative. If has column rank diagonal elements of are set to zero. Now, consider transmit symbols of the following form:
II. SIGNAL MODEL
, , and where where is a vector of integers, related to , as we shall see later. Note that is upper triangular with ones along the diagonal, , and that columns of form an orthogonal . Moreover, columns of the (but not orthonormal) basis for are a basis for a lattice, and is a lattice prefiltering matrix point. We term the lattice whose basis matrix is the transmit which lattice and give it the symbol . It is the columns of shall be used in the lattice-based approach, as carriers for the data. Hence, we use the term lattice-MCM. . It remains to show how is mapped to such that Consider
be a time-domain Let the transmit symbol vector of transmit samples.3 The receive symbol is given by (1) where is the channel matrix, and is independent zero-mean is a vector function Gaussian noise with variance . Also, representing the transmitter nonlinearity, such that the th eleis clipped to or if or , ment of respectively, and otherwise, equals . samples is Assume that a guard period of length used between successive transmit symbols, where is the delay spread of the channel. In the generalized lattice approach presented in this paper, we allow the guard period either to contain a cyclic prefix (if the situation demands it), or zero transmit en1, ergy (thus saving power). In the latter case, is Toeplitz. For a cyclic prefix, is 1, and is and is circulant. In either case, has the following first column:
where
(4)
(5) Assume without loss of generality that the clipping level equals 1/2, so that the transmit cube has unit volume. to the Consider the following mapping from transmit symbol :
(2)
(6)
Assume that the data stream is divided into blocks of bits, not necessarily independent and identically distributed (i.i.d.), allowing for coded data. The aim is to design a set of transmit symbols which are widely spaced in the receive , where is called domain, but for which the transmit cube. In doing so, clipping would be completely avoided over the discrete-time channel while achieving a good BER. Furthermore, the mapping from data sequence to transmit symbol must be computationally efficient.4
is chosen such that . The elements of where can be found in an iterative way, since is upper triangular. It is possible to start by finding the value of which ensures , and then iterating backward to find the other values for , then we in turn. More specifically, if by subtracting the appropriate integer can force multiple of the th column vector of , namely . An explicit algorithm is given in [10]. It turns out that this can be viewed as a generalized form of TH precoding, applied to lattice basis vectors, as discussed later in Section V. , but it can also be Furthermore, not only do we have , and that shown that
III. LATTICE-BASED TRANSMISSION Each -bit data block is first divided into subsequences of length for . The subsequences are then mapped into the corresponding Gray-mapped binary subsequence, and then directly converted into a decimal integer, , where, of . We now show how this data vector course, (with elements ) is modulated in this lattice approach onto nonsinusoidal basis functions, or “carriers,” following the approach of [10]. Consider the following QR decomposition of :
(7) to Note that (7) is effectively adding integer multiples of (which is in the range , from above) to arrive at . The result is that the mapping in (7) can be inverted by division of each . simply taking the modulo, the function has no effect, We see now that since and in the receiver, we can compute the filtered measurement
(3) (8)
3The
basic assumptions on data formatting are identical to those for systems such as ADSL and fixed wireless IEEE 802.16. 4Note that sinusoidal-MCM schemes are computationally of ( log ) and fail to satisfy the clipping restriction.
ON
N
is a basis for the receive lattice, and where i.i.d. zero-mean Gaussian with variance . Since both
are and
COLLINGS AND CLARKSON: LOW-PAR TRANSMISSION SCHEME FOR DSL CHANNELS
are diagonal, the (unconstrained) maximum-likelihood (ML) estimate of is then simply [10] (9) is the th element of . The ML estimate of is where computed by calculating the positive remainder of the division , since from (7), we note that . of by Finally, the data estimates are obtained from by binary-toGray-code conversion. The resulting bit-error rate (BER) can be approximated by the lower bound derived by considering only the pairwise errors of nearest neighbor lattice points, and is given by (10)
The lattice carrier bit-loading values so as to minimize (10). First, let
can now be chosen
(11) In high-data-rate conditions, is sufficiently large so that each . In this case, we assign bits to basis vectors with the largest values of . To the remainder, we . It can be shown that the number of assigned assign bits is equal to , and that this allocation achieves the lowest , according to BER of all allocations for which each the low-noise approximation of (10), see [10]. In scenarios of either lower data-rate conditions, excess bandwidth, or channels with spectral nulls (e.g., bridged taps, or is relatively small, and we find that masked frequencies), there will be some ’s less than one. In this case, we propose an iterative bit-loading algorithm in order to select a subset of basis vectors, similar to the Chow algorithm in DMT [15]. The approach proceeds by excluding the subchannel with the lowest power, and recalculating the ’s. This is iterated until all the remaining ’s are greater than one. In other words, successively, with the lowest corresponding values of the columns of are allocated zero bits. The power is then spread amongst the remaining columns (carriers). While this is not necessarily guaranteed to provide an optimal allocation, simulations show that the performance is certainly more than acceptable.
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operations. Furthermore, the amount of require . We then show memory required for lattice MCM is also that these properties also hold for the cyclic-prefix case. A. Low-Complexity Modulation The arithmetic complexity of the coding process is dominated by the matrix calculation of from by (5) and from by (6). All other steps in the process (serial-to-parallel (S/P) conversion, Gray-to-binary encoding, etc.) are elementary logical operations. The QR decomposition is not included in the complexity calculation, since we are assuming fixed DSL channels, and as such, the QR operation only occurs once during the startup phase.5 We now make some key observations that will assist in showing that (5) and (6) can be combined into a single operation, and that even though they appear to be matrix calculations, the number of arithmetic operations required is, in fact, . only 1) Provided , has full column rank. 2) is banded with lower bandwidth and upper bandif . width 0. Hence, 3) The QR decomposition of is unique because has full column rank. Moreover, none of the diagonal elements of are zero [16]. is banded with lower bandwidth 0 and 4) The matrix upper bandwidth . , and therefore Proof: Observe that (12) for all , such that . For a given , the proof , (12) implies is by induction on , as follows. With . Assuming for all , then, from that also, since . (12), 5) Since is banded, and is diagonal, is banded also with the same bandwidths. 6) For the th row of , , we can write (13) where is the unit vector whose th element is one. Proof: Since , and is upper triangular with ones on the diagonal, then
IV. LOW-COMPLEXITY IMPLEMENTATION In this section, we will first show that in the zero-energy guard-period case (where silence is inserted between symbols rather than a cyclic prefix), the new lattice-MCM scheme arithmetic operations. This compares farequires only vorably with conventional sinusoidal-MCM techniques, which
5Even for extensions to slowly fading fixed wireless channels, the QR decomposition is only required at infrequent reinitialization phases.
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Fig. 1.
Modulator structure for the new coding scheme.
7) The calculation of (14) is 8) With
, since
is diagonal. , it follows from (6) and (13) that
), and these are banded matrices whose bandwidths are independent of . Fig. 1 shows an implementation of our new scheme, which . The shows how a single block is encoded for the case S/P conversion breaks up the incoming block of bits into parbits each. The pulse amplitude modulation allel channels of (PAM) encoders produce the elements of the vector as defined in (14).6 The modulo adders and feedback paths calculate the elements of according to (15). Observe that the maximum , in accordance with the number of inputs to any adder is banded structure of . B. Low-Complexity Demodulation
(15) This low-complexity algorithm for computing from can now be seen to be . First, is calculated in from (14). Second, the and are calculated iteratively from down to , using (15). Each is chosen as that unique in, i.e., , is set to the nearest teger that makes integer of the value that is computed on the right-hand side (RHS) of (15). Finally, note that the RHS of (15) requires a number of arithmetic operations that is not dependent on , but rather on , because of the banded nature of . Hence, the overall amount of arithmetic operations required to calculate for transmission is . In addition, it is important to point out that the memory required for encoding is also , since it is only necessary to store the nonzero coefficients of and (and therefore, also
The computational complexity of the receiver is dominated by evaluating (8). All other steps are easily shown to require at most arithmetic operations. Again, at first sight it would ; however, consider seem that the computation would be the following. 1) We can write , and so
2) The calculation of (16) requires only
operations, because
is banded.
6In fact, (14) is an asymmetric PAM, since the signal levels are not symmetric about zero, but this constant offset to the signal levels makes no difference to the analysis, so long as we are consistent in the decoding.
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Fig. 2. “Causal” demodulator structure for the new coding scheme.
3) With
Proof: Since
, we can write the th column
is
, we conclude that the entire decoding process is only . Once again, we point out that the memory required for de, since it is only necessary to store the coding is also nonzero coefficients of and . Fig. 2 shows the demodulator structure at the receiver. Following the analog–digital conversion (ADC), the samples are passed through a matched filter, then grouped according to their symbol and passed through a S/P converter, from which is output the elements of as defined in (16). The adders and where feedback paths compute the vector
is upper triangular
and we observe that 4) As a result, we have
and
(17) This low-complexity algorithm for computing the vector from clearly consists of iteratively evaluating the elements , starting at up to , according to (17). Note that evaluation of the RHS of (17) requires the summation of no more than elements, because is banded. Since the amount of computation required to compute each element of (from ), does not depend on , and because calculation of from
Therefore, passing through a modulo element before PAM detection recovers the bits encoded in . Note also that since the demodulation iteration in Fig. 2 starts , it is not necessary at the receiver to wait for an entire at symbol before the first decoded outputs are produced from it. samples of a symbol are received, beAfter the first comes available. The first decoded output can then be produced, and thereafter, a new output can be produced as each subsequent sample arrives from the ADC. In this sense, the structure can be called “causal.” If required, it is also possible instead to configure the modulation scheme to have a causal modulator, and then the demodulator becomes anti-causal (i.e., it requires the entire block of samples to be received prior to demodulation, as is the case for all existing sinusoidal-MCM schemes). To deduce the structure,
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of terms in the summation on the RHS is unaffected. However, require summing up to elements. the last values of Even so, the total computation required to calculate from is still . Therefore, the decoding complexity is still arithmetic operations. D. Complexity of Sinusoidal-MCM Techniques Fig. 3.
Structure of channel matrix with cyclic prefix.
Fig. 4. Structure of covariance matrix.
we simply reverse the order of the elements in the vectors so that
and
and and reconstruct to satisfy (1) and repeat the analysis as above. Computational and memory requirements are unchanged. C. Cyclic-Prefix Case We now show that the coding and decoding complexity is still arithmetic operations,when a cyclic prefix is applied to each symbol. This is because the matrix is no longer banded but “circulant banded.” This has implications for the structure of , in that it too is no longer banded. However, it still contains enough structure for fast coding and decoding to be performed. Note that now has the structure as shown in Fig. 3, where the circulant band has width , and the overall size of the matrix . As a result, it is possible that no longer has full is . However, its column rank is column rank, even when always greater than . If is rank deficient, we simply delete columns from the matrix until it has full rank. Equivalently, for each symbol, we set the last few samples to zero. if . Following the Now, same reasoning as followed in (12), we have that if . Therefore, (and also ) has the structure, as shown in Fig. 4. Let us now examine how this new structure for affects the amount of computation required for coding and decoding. The key to the low coding complexity is (15). We determined, for the case where there is no cyclic prefix, that each and could be determined in a fixed number of arithmetic operations, because in the sum on the RHS is the number of nonzero elements not a function of . This is still the case with the new structure for . Although there are now more nonzero elements of to be summed in calculating each and , the number of such elements is still independent of . Hence, coding complexity is arithmetic operations. still Changes to the structure of affect the decoding complexity only inasmuch as they affect the amount of computation re, the number quired to calculate each in (17). For
Standard sinusoidal-MCM transmission techniques are designed on the cornerstone of the fast Fourier transform (FFT). The scheme requires an inverse (I)FFT in the transmitter, and a corresponding FFT in the receiver. These operations cannot be avoided, and are well known to require computational com. Clearly, this is higher than for the latplexity of , as discussed tice-MCM approach, which requires only above. This is particularly significant when considering that the new lattice-MCM technique is totally free from clipping, with a corresponding superior performance, compared with sinusoidal MCM. It should also be noted also that the three leading PAR-reduction techniques for sinusoidal MCM achieve their performance improvements at the expense of higher computational complexity, since each scheme can be viewed as an add-on to the standard sinusoidal MCM. The selected mapping (SM) [1], [2] and partial transmit sequences (PTS) [3], [4] approaches calculate and check multiple transformed recalculations of the MCM symbol and pick the one with the lowest PAR. They also require that the receiver is informed whenever the transform is applied, via a reserved data bit in the DMT symbol. Tone reservation [5] is slightly different, since it selects unused (or even minimally used) subchannels and transmits energy at these frequencies in an attempt to lower the PAR of the overall transmit symbol. In doing so, there is no need to notify the receiver, which simply demodulates the data-carrying subchannels as usual. However, the computational cost of finding the best waveform to send down the unused channels can be extremely high. V. RELATIONSHIP WITH EXISTING PRECODING SCHEMES The signaling scheme presented in this paper has been derived from basic considerations of lattice theory. In this section, we highlight some relationships with vector coding and TH precoding. Vector coding was introduced in [17] as a linear technique for equalization of channels with intersymbol interference (ISI). As with our lattice scheme, it is a block coding MCM technique where the subchannels are derived from the QR decomposition of the channel matrix. The approach makes use of a partitioning of the data in into orthogonal subchannels, obtained by setting (18) This is clearly related to, but different from, (4). Unlike sinusoidal-MCM techniques, the vector coding subchannels are not orthogonal at the transmitter, only at the receiver, and decoding is straightforward, without the need for decision-feedback equalization, since
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Fig. 5. TH precoder structure.
can be PAM detected in the traditional way, as is i.i.d. Gaussian. Notice that vector coding takes no care to limit the amplitudes of the transmit samples in (i.e., to limit the PAR). Instead, as in [17], the objective with vector coding is to minimize the transmit power while holding other factors constant. In contrast, the TH precoder [18], [19] is a technique suitable for transmission over amplitude-limited ISI channels. Like our lattice-MCM, it makes use of modulo arithmetic at the transmitter and receiver. However, it is not suitable for block coding. Rather, it is a “modulo inverse filter” which, for a finite impulse response (FIR) channel, has an infinite impulse response. Fig. 5 ) which is comshows the structure of a TH precoder (for patible with the signal and channel model used in this paper. Clearly, the modulator structure bears a resemblance to the lattice precoder, by virtue of the modulo addition and feedback arrangement. Indeed, our scheme could be classified as a generalization of the TH precoder for block transmission. Many generalizations of TH precoding have appeared since its invention. Indeed, vector coding was combined with a TH-style precoder in the earliest publication [20] of which the authors are aware, although the intention was not, as it is here, to seek to minimize PAR and computational complexity.
In this case the receive symbol can be written (21) and we finish by computing the filtered measurement (22) where is a basis for the receive lattice, and . Note that is a vector of i.i.d. zero-mean Gaussian random variables with variance , since it is an orthogonal transform of . B. Frequency-Domain Power Profiling
Colored noise, cross-talk, and indeed NBI, are commonly modeled as autoregressive filtered versions of white noise. For typical DSL noise environments, see [21, p. 475] and [15, p. 85]. The resulting generalization of (1) is
In a regulated environment, it is often necessary to apply power profiles to MCM systems. For sinusoidal MCM, it is a relatively straightforward process of allocating less power and bits to particular (frequency-domain) carriers and their neighbors, due to the fact that the sinusoidal basis functions are really only orthogonal at the center of the frequency band. We now show that even though the lattice-MCM carriers are not orthogonal in the frequency domain, it is still straightforward to enforce a power profile. Consider the power profile as a filter on the transmit symbol. One approach is to build an analog filter (ie. post digital-toanalog (D/A) conversion) which suppresses the required frequencies, and then design the lattice-MCM carriers based on the combined filter–channel combination. Of course, all-digital solutions are preferred. The proposed approach is to implement a digital profile filter immediately prior to D/A conversion. In this case
(19)
(23)
where , and where is the noise correlation matrix. In the case of a transmission system with a zero-energy guard matrix, and for systems period, is an employing a cyclic prefix, is an matrix. In either case, we can make the following QR decomposition:
where is the filter matrix with frequency components given by the desired power profile. The design process is simply to follow the procedure in Section III, but start by taking the QR decomposition of the modified channel matrix, . Since the profile filter will have the effect of rotating the transmit lattice, it will be necessary to scale back the (by a known factor) to ensure that the outtransmit symbol ermost points of the rotated lattice are in the transmit cube, i.e., . Of course, this is at the expense of reduced SNR, and therefore, increased BER relative to nonprofiled lattice MCM. The alternative is to operate without a scale-back factor, and suffer an increase in PAR. In either case, performance is still significantly superior to power-profiled sinusoidal-MCM techniques, as shown in Section VII.
VI. SYSTEM IMPLEMENTATION ISSUES A. Colored Noise and Cross-Talk
where is a square matrix, and . We propose a lattice-MCM design continuing as in Section III, but instead performing the channel QR decomposition on the noise-whitened channel matrix (20)
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BER versus SNR for clip level (CAR) 5 dB, with standard channel.
Fig. 7.
BER versus CAR for standard channel with E
=N
= 27 dB.
Note that a profile filter can also be used in extreme cases of high NBI levels, or when the NBI statistics cannot be measured accurately. The filter can be viewed as masking out the (narrowband) frequency range, and could be used in addition to the inverse noise correlation approach in Section VI-A. VII. SIMULATIONS AND EXPERIMENTAL RESULTS A. BER Performance To compare directly with ADSL in each of the simulations in , , and , and this section, we set simulated a 1-km copper twisted-pair AWG24 line terminated at each end by 110 . Fig. 6 shows the vastly superior performance of lattice MCM in the presence of clipping at CAR dB. In such conditions, sinusoidal-MCM PAR-reduction techniques have no effect, and the curves for SM, PTS, and tone reservation sit on top of the standard ADSL curve shown in the figure. We note also that the PAR for the lattice MCM was measured at 3.01, also in very good agreement with the theoretical asymptotic PAR of 3 (ie. 4.8 dB). Fig. 7 shows the effect of varying the clipping level. For the ADSL system, the bit allocation was done using the well-known Chow algorithm [15] based on the SNR of 27 dB, and filling the lower frequency subchannels first (in this case, all available subchannels are used). For the lattice scheme, the bit allocation is as described in Section III for high data rates (i.e., each in this case). The figure shows that for CARs below 11 dB, the lattice-MCM scheme has significantly superior performance, compared with ADSL, even when existing PAR-reduction techniques are used. Note that the lattice BER floor is just for this channel-SNR scenario, as also indicated by above Fig. 6. This is, of course, higher than the ADSL floor, due to the tighter packing of constellation points needed to avoid clipping is an acceptable level for in the lattice scheme. Even so, data applications. Simulations were also carried out for the colored noise case arising from AM radio interference. In particular, narrowband noise was injected between the normalized frequencies of 0.8
Fig. 8. Maximum bits possible versus noise power, for BER standard channel with transmitter clip level 0:5 V.
6
