Patent Grant
8396168
The present invention generally relates to equalization, and more particularly relates to equalization based on serial localization with indecision.
MLSE (Maximum Likelihood Sequence Estimation) is a demodulation technique that also equalizes ISI (Inter-Symbol Interference) in a signal which is modulated in accordance with a particular constellation and transmitted over a channel. ISI causes the equalization complexity to increase as a power of the constellation size. Relatively large signal constellations such as 16-, 32- and 64-QAM (Quadrature Amplitude Modulation) have been adopted in EDGE (Enhanced Data Rates for GSM Evolution), HSPA (High Speed Packet Access), LTE (Long Term Evolution), and WiMax (Worldwide Interoperability for Microwave Access). In HSPA, multi-code transmission creates even larger effective constellations. Also, MIMO (Multiple-Input, Multiple-Output) schemes With two or more streams have been adopted in HSPA, LTE and WiMax. MIMO implementations also yield relatively large effective constellations. ISI causes equalization complexity to further increase when any of these techniques occur in combination, e.g. multi-code and MIMO.
In the ISI context, the ideal equalization scheme is MLSE, in the sense of maximizing the probability of correctly detecting the transmitted sequence of symbols, or sequences of symbols in the MIMO case. However, the complexity of MLSE increases substantially as a function of the size of the modulation constellation and/or because of the exponential effects of MIMO or multi-codes to the point where MLSE becomes impractical. Less complex solutions are available, such as DFSE (Decision-Feedback Sequence Estimation), DFE (Decision-Feedback Equalization), etc. Each of these solutions attempts to strike a balance between accuracy and complexity.
Another conventional equalization technique is MSA (Multi-Stage Arbitration). MSA involves sifting through a large set of candidates in multiple stages, where each stage rejects some candidates until a single candidate is left after the final stage. One specific example of MSA is generalized MLSE arbitration where the first stage is a linear equalizer and the second stage implements MLSE based on a sparse irregular trellis over a reduced state space.
Iterative Tree Search (ITS) has also been used for performing equalization in MIMO QAM environments. ITS exploits the triangular factorization of the channel. In addition, ITS uses the M-algorithm for reducing the search for the best candidate. ITS breaks down the search further, by dividing the QAM constellation in its four quadrants, and representing each quadrant by its centroid in intermediate computations. The selected quadrant itself is subdivided again into its 4 quadrants, and so on. This results in a quaternary tree search. Other conventional approaches give particular attention to the additional error introduced by the use of centroids instead of true symbols. The error is modeled as Gaussian noise whose variance is determined and incorporated in likelihood computations. However, a tight connection is typically made between the centroid representation and the bit mapping from bits to symbols. That is, if a so-called multi-level bit mapping is employed, then identifying a quadrant is equivalent to making a decision on a certain pair of bits. Such constraints place a restriction on bit mappings, restricting the design of subsets.
Serial localization with indecision (SLI) is another approach to equalization that approximates an MLSE equalizer with reduced complexity. In a receiver using SLI, equalization is performed in a series of stages for suppressing ISI. Each non-final stage attempts to further localize the search for a solution for the benefit of the next stage, based on input from the previous stage.
Viewed in isolation, a given SLI equalization stage can be quite indecisive, but makes progress and avoids an irreversible wrong decision. A given equalization stage localizes the solution by inputting a subset representative of the constellation and outputting a further reduced subset. Each stage makes a choice among candidate reduced subsets. Indecision arises from representing the modulation constellation with overlapping subsets. Indecision is beneficial in a multi-stage structure, because indecision discourages an irreversible bad decision in an early stage.
Referring now to the drawings,
The basic idea underlying SLI is to localize the search for the final sequence estimate in each preliminary stage 102 so that, in the final stage 104, only a subset of full symbol space is searched. The demodulators 110 in each stage are Maximum Likelihood Sequence Estimation (MLSE) demodulators with a reduced state trellis. As will be described in greater detail below, the demodulators 110 in the preliminary stages 102 of the multi-stage equalizer 100 use centroids rather than symbols in the transmit signal constellation for demodulation. The demodulator 110 in the final stage 104 of the multi-stage equalizer 100 uses a reduced state trellis based on a subset of the transmit constellation. The SLI equalization technique is described in U.S. patent application Ser. No. 12/572,692 filed Nov. 2, 2009 titled “Method for Equalization and Whitening of ISI using SLI” and U.S. patent application Ser. No. 12/549,143 filed Aug. 27, 2009 titled “Equalization using Serial Localization with Indecision.” Each of these pending applications is incorporated herein in its entirety by reference.
For simplicity, the present invention will be described in a system having one transmit and one receive antenna. The extension to multiple transmit and/or receive antennas is straightforward. For a symbol-spaced channel model with memory M, the received signal can be modeled by:
rk=hMsk-M+L+h0Sk+vk Eq. (1)
where (h0, . . . hM) is the channel filter, which is assumed to be constant over the duration of the received signal, and vk is the noise. The signal sk has a symbol constellation Q of size q. The noise vk has correlation Rv(1) and power Pk. Assuming that the noise vk is white, the noise correlation is given by:
Rv(1)=Pvδ(1) Eq. (2)
For simplicity, it is assumed that the transmitter uses uncoded modulation. The transmitter may use partial response signaling, which in effect, is filtering at the transmitter. The effect of filtering at the transmitter of the receiver may be incorporated into the channel model given in Eq. 1.
For channel estimation, a block of N>M consecutive pilot symbols are transmitted, surrounded by data symbols on both sides. The pilot symbols are denoted p0, L, pN-1. Channel estimate may be obtained using a approach referred to herein as the block maximum likelihood (ML) approach. More specifically, the first M received samples r0, L, rM-1 are discarded to avoid interference the previous block, and the remaining samples rM, L, rN-1 are used for channel estimation. In vector notation, the channel estimation model is given by:
r=Aph+v Eq. (3)
where r=(rM,L,rN-1)T, h=(h0,L,hM)T, v=(vM,L,vN-1)T, and Ap is a matrix of pilot symbols given by:
The channel estimate may be computed according to:
{circumflex over (h)}=(ApHRv−1Ap)−1ApHRv−1r Eq. (5)
where the matrix Rv has dimensions (N−M)×(N−M), and the (i,j)th element is given by:
The channel estimate can, for example, be used for demodulation as well as other purposes.
Alternatively, an MMSE channel estimator can be obtained by
ĥ=ChApH(ApChApH+Rv)−1r, Eq. (7)
where Ch is the covariance matrix of the channel. For example, Ch, can be computed by taking the average of the outer products of past channel estimate vectors.
The demodulator 110 in each stage 102, 104 of the multi-stage equalizer operates as an MLSE equalizer with a reduced state trellis. The symbol constellation Q of size q is decomposed into a plurality of overlapping subsets. Using 8-ASK as one non-limiting example, the symbol constellation is given by:
Q={−7,−5,−3,−1,+1,+3,+5,+7} Eq. (8)
For a 2-stage SLI equalizer, the symbol constellation may be decomposed into three overlapping subsets given by:
with centroids {−4,0,+4}, respectively. Note that the middle subset coincides with 4-ASK. Also, the first subset is shifted from the middle subset by the centroid −4. Similarly, the last subset is shifted from the middle subset by the centroid +4.
The input to the first stage of the SLI equalizer is the received sample stream with received symbols rk. The demodulator 110 in the first stage uses a modified symbol constellation Q′[1], which is equal to the centroid set {−4,0,+4} to construct the equalizer trellis. The demodulator 110 produces an intermediate sequence estimate with intermediate symbol decisions ŝ′k[1], which belong to Q′[1].
The symbols ŝ′k[1] are re-modulated to generate a reconstructed signal {circumflex over (r)}′k[1] given by:
{circumflex over (r)}′k[1]=hMŝ′k-M[1]+L+h0ŝ′k[1] Eq. (10)
The received signal is modified by subtracting {circumflex over (r)}′k[1] to generate a residual signal rk|1| given by:
rk[1]=rk−{circumflex over (r)}′k[1] Eq. (11)
The demodulator 110 in the second stage operates as an MLSE demodulator with the residual rk[1] as its input and uses a reduced state trellis based on the constellation Q′[2], which is set to equal the middle subset of the symbol constellation {−3,−1,+1,+3}. The demodulator 110 in the final stage 104 generates an intermediate sequence estimate with intermediate symbol decisions ŝ′k[2] selected from the subset Q′[2]. The final sequence estimate is derived by summing the intermediate symbol decisions output from each stage 102, 104. Thus, the symbol decisions in the final sequence estimate are given by:
ŝk=ŝ′k[1]+ŝ′k[2] Eq. (12)
Those skilled in the art will appreciate that the overlap between subsets allows the demodulator in a given stage of the equalizer 100 to recover from an erroneous symbol decision made in the preceding stage. The first and second stage equalizers comprises 3M states and 4M states, respectively, compared to 8M states for conventional MLSE demodulation.
The structure of the SLI equalizer 100 can be easily extended to 3 or more stages 102, 104. Each preliminary stage 102 uses a centroid constellation Q′[i] whose symbols represent a subset of the transmit symbol constellation to construct the equalizer trellis and localizes the search for the succeeding stages. The final stage 104 uses a reduced state trellis wherein each trellis stage is constructed using a subset of the transmit signal constellation Q. The input to the first stage 102 is the original received signal. The input to each subsequent stage i is the residual signal rk[i-1] from the preceding stage i−1 given by (11). Each stage i generates an intermediate sequence estimate comprising intermediate symbol decisions ŝ′k[i]. The re-modulated signal at each preliminary stage 102 is given by:
{circumflex over (r)}′k[i]=hMŝ′k-M[i]+L+h0ŝ′k[i] Eq. (13)
The re-modulated signal {circumflex over (r)}′k[i] in each preliminary stage 102 is subtracted from the signal rk[i-1] output from the preceding stage to generate the residual signal rk[i] for input to stage i+1. The final sequence estimate is the sequence derived by summing the symbols in the intermediate sequence estimates one by one. Thus, the final estimate for a symbol is given by:
ŝk=ŝ′k[1]+L+ŝ′k[N] Eq. (14)
Consider the 2 stage structure again. By using the centroid constellation Q′[1], the demodulator 110 in the preliminary stage 102 is effectively modeling the received signal rk as:
rk=(hMŝk-M[1]+L+h0ŝ′k[1])+xk Eq. (15)
The noise term xk absorbs the contribution of sk[2]. That is,
The term wk is due to the use of centroids instead of actual constellation points. It is an interference signal that affects the performance of the demodulator 110 adversely. In the 2-stage SLI equalizer 110, the demodulator 110 in the final stage 104 will explicitly estimate ŝ′k[2] but that doesn't help the demodulator 110 in the preliminary stage 102.
Fortunately, wk is a correlated signal, due to the filtering by the channel response hk, as can be seen in Eq. (16). The correlation of wk is given by
where by definition, hk=0 outside the interval 0≦k≦M. The power P2 is given by
The power spectrum of wk is the Fourier transform of Rw(1). It can be written as
Sw(f)=P2|H(f)|2 Eq. (19)
where H(f) is the Fourier transform of hk.
The total noise xk=wk+vk has correlation
Rx(1)=Rw(1)+Rv(1) Eq. (20)
The power spectrum of xk can be written as
Sx(f)=P2|H(f)|2+Sv(f) Eq. (21)
If vk is white, then
Sx(f)=P2|H(f)|2+Pv Eq. (22)
A multi-stage-equalizer 100 can be constructed based on nested subsets. For example, a three-stage equalizer 100 may be used for demodulating 8-ASK. The first stage constructs a reduced state equalizer trellis using the centroid constellation Q′[1]={−4,0,+4}, which may be used in the first stage of the equalizer. The second stage constructs a reduced state trellis using the constellation given by:
Q′[2]={−2,0,+2}. Eq. (23)
The final stage constructs a reduced state trellis using the BPSK subsets of transit signal constellation given by:
Q′[3]={−1,+1} Eq. (24)
The invention is generally applicable to any wireless systems where pilot symbols are seen as a block in the time domain, i.e., an entire block is dedicated to pilot.
where of Δf=15 kHz, Ts is the inverse of sampling frequency for obtaining the receive samples r. Assuming that r is obtained after discarding the cyclic prefix, which is assumed to be greater than the channel memory. Using pi to formulate matrix Ap as defined in (4), the ML channel estimator given in (5) applies.
While SLI provides a low complexity approximation to MLSE, consecutive reconstruction of signals with noisy channel estimates may degrade SLI performance. Therefore, in an alternate embodiment of the present invention, shown in
The preliminary stage 102 shown in
In vector notation, the channel estimation model using effective pilot symbols is given by:
r1=A{circumflex over (p)}
where r1, and xi have size (N′−M), and A{circumflex over (p)}
ĥ[i]=(ApHRv−1Ap+A{circumflex over (p)}
where the matrix Rx
The extension from one block of effective pilots to multiple separate blocks of effective pilots follows naturally. One reason to use multiple blocks is to be able to use data from different spots in the received signal. Another reason is to keep the block size reasonably small, which keeps the complexity of the matrix inverse Rx
In another exemplary embodiment, the average of the initial channel estimates ĥ and the contribution from the effective pilot block is used to generate the refined channel estimates ĥ[i]. That is the refined channel ĥ[i] estimates is given by:
ĥ[i]=ĥ+(A{circumflex over (p)}
Again, this simplified method extends naturally to the case of multiple blocks of effective pilots.
Note that Rx
For MMSE channel estimation, a similar approach can be applied. Using the effective pilot block to refine the channel estimate obtained from the pilot symbol block,
ĥ[i]=ĥ+ChA{circumflex over (p)}
The channel estimate obtained from the previous stage can be used to calculate the channel covariance given by:
Ch≈ĥ[i−1](ĥ[i−1])H Eq. (30)
We may choose to refine the channel estimates at selected stages in a multi-stage SLIC. One approach is to refine the channel estimates in early stages only, in a pre-determined manner. The idea is that once the estimates are good enough, there is no reason to refine them further. Another approach is to consider the quality of the channel estimates, and decide accordingly whether to refine them. For instance, if the noise correlation matrix Rv is relatively small, then the channel estimate ĥ is considered reliable, and there is no need to refine it. To decide if the correlation matrix Rv is small, we can consider its diagonal elements, or its eigenvalues, or use any classical matrix norm.
Another approach is to include the data from the effective pilots only if it is considered reliable. For instance, we can compare the noise correlation matrix Rx for the effective pilots with the noise correlation matrix measured from the original pilot symbols. If Rx is relatively large (in the norm or eigenvalue sense etc as discussed above), then we may skip the refinement.
Since the effective pilots are demodulator decisions, their reliability is in question. Fortunately, because of the use of centroids in SLI, the Euclidean distance between centroids is larger than the Euclidean distance between true symbols, so they tend to be more reliable in relative terms. Nevertheless, it may be useful to reduce the contribution of the effective pilots to the refined channel estimates. One approach is to scale down the terms corresponding to the effective pilots. That is, we can choose a parameter α, say between 0 and 1, that reflects the reliability of the effective pilots. Then we can modify the refined estimate as follows
ĥ[i]=(ApHRv−1Ap+αA{circumflex over (p)}
Similarly, we can choose a parameter β, and modify the simplified refined estimate as follows
ĥ[i]=ĥ+β(A{circumflex over (p)}
The parameters α and β may be chosen based on the expected reliability of the effective pilots, which is reflected in the noise covariance Rx.
The present invention offers an enhancement to the SLI receiver, by improving channel estimation, which is critical since channel estimates a repeatedly used in signal reconstruction and subtraction.
In some embodiments, a whitening filter may be used to transform the noise term xk into a white noise. If G(f) denotes the frequency domain representation of the whitening filter, then whitening may be performed by choosing a filter G(f) such that
The filter G(f) is not unique. Thus, there is flexibility in designing G(f) to have certain attractive features, such as minimum phase.
In one exemplary embodiment, shown in
The symbols ŝ′k[1] out of the MLSE are re-modulated using the original filter hk according to Eq. 11. The modified received signal is produced according to Eq. 12. The second stage is unchanged.
The present invention offers an enhancement to the SLI receiver, by improving channel estimation, which significantly improves performance because the channel estimates are repeatedly used in signal reconstruction and subtraction.
The present invention may, of course, be carried out in other specific ways than those herein set forth without departing from the scope and essential characteristics of the invention. The present embodiments are, therefore, to be considered in all respects as illustrative and not restrictive, and all changes coming within the meaning and equivalency range of the appended claims are intended to be embraced therein.
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