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Convergence of Frequency-Domain Iterative MF-DFE for Single-Carrier Modulation Su Huang, Jun Wang, Jintao Wang, Senior Member, IEEE, Chao Zhang, Senior Member, IEEE, and Jian Song, Senior Member, IEEE
Abstract—An error transfer chart is introduced to trace the iterative process of the decision feedback equalizer for the singlecarrier modulation. In the investigated iterative MF-DFE structure, the forward filter is set to the matched filter to maximize the signal power, while the feedback filter is used to cancel the inter-symbol interference. MMSE detection is applied to minimize the decision error power. The trajectory of iterations is shown in the error transfer chart, in which MF-DFE maps a decision error power to an SINR and MMSE detection further converts the SINR to a new decision error power. The final state of iterations is shown to be the intersection of the MMSE curve and error power transfer curve corresponding to the MF-DFE algorithm. It is shown by simulation that the error transfer chart is a useful tool to analyze the convergence of non-linear iterative equalizers. Index Terms—Matched filter, minimum mean square error, decision feedback, single-carrier modulation, error transfer chart.
I. I NTRODUCTION
S
INGLE-CARRIER modulation was a hot topic a few years ago. Combined with non-linear frequency-domain equalization, it is possible to achieve a closer performance to orthogonal frequency division multiplexing (OFDM) [1]. The easier deployment of transmitter and the low peak-to-average-power ratio (PAPR) make single-carrier modulation a candidate for uplink communications. For example, LTE adopted the singlecarrier frequency division multiple access (SC-FDMA) as its uplink scheme [2]. Unlike the traditional time-domain (TD) decision feedback equalizer (DFE), the algorithms proposed in the recent years adopted the frequency-domain signal processing techniques. For example, hybrid DFE (HDFE) [3] uses a frequency-domain forward filter and time-domain feedback filter. A noise predictor was proposed by [4] to take advantage of the correlation of
Manuscript received February 1, 2015; revised June 26, 2015; accepted September 12, 2015. Date of publication September 17, 2015; date of current version November 13, 2015. This work was sponsored by Science Fund for Creative Research Groups of NSFC (61321061) and National Natural Science Foundation of China (Grant No. 61471219, 61471221). The associate editor coordinating the review of this paper and approving it for publication was T. Q. S. Quek. S. Huang, J. Wang, J. Wang, and C. Zhang are with the Tsinghua National Laboratory for Information Science and Technology (TNList), Department of Electronics Engineering, Tsinghua University, Beijing 100084, China (e-mail:
[email protected]). J. Song is with the Tsinghua National Laboratory for Information Science and Technology (TNList), Department of Electronics Engineering, Tsinghua University, Beijing 100084, China, and also with Shenzhen City Key Laboratory of Digital TV System, Shenzhen 518057, China (e-mail: jsong@tsinghua. edu.cn). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCOMM.2015.2479629
the time-domain noise after the equalization. [5] proposed a bi-directional feedback structure to improve the DFE performance. [6] proposed the iterative block DFE (IBDFE), which operates entirely in the frequency domain both for the forward and feedback filter. An insightful conclusion from it is that when there is no feedback, the optimum forward filter is the linear minimum mean square error (LMMSE) filter in the frequency domain, and when the feedback is correct, the optimum forward filter is the matched filter (MF) with the feedback to eliminate the multipath propagation introduced by the concatenation of the channel and the forward filter. The decision is also of relevance. Hard decision has long been used because it is easier to implement, while the error propagation is severe. Soft decision is also studied in the literature. The most common used soft information is the minimum mean square error (MMSE) estimate; it combines the discrete alphabet with the a posteriori probability. [6] proposed the soft detection IBDFE (SD-IBDFE) using the MMSE estimate and solved the optimization for the forward and feedback coefficients, both in the frequency domain, while [7] solved the coefficients in the time domain for soft detection model. Soft detection can also be performed using the bit-wise loglikelihood ratio (LLR) by taking advantage of the bit-symbol mapping rules [8], [9]. Soft-in-soft-out (SISO) structure is introduced in [10] to implement the turbo equalizer, where the information is passing between SISO equalizer and SISO convolutional decoder. [11] mainly focused on the multilevel modulation for turbo equalization. In summary, there are two ways to perform the soft decision, namely bit-wise or symbol-wise. The bit-wise SD usually calls for turbo equalization with an interleaver and a SISO decoder, while the symbol-wise SD does not break the symbol into bits and thus requires less complexity and latency. For the single-carrier modulation, we can treat the channel as a generalized convolutional encoder. However, as the modulation order and the channel length increase, the complexity to perform the maximum-likelihood sequence (MLS) detection is growing exponentially. So the detection, no matter soft or hard, is often symbol-by-symbol, which implies that the intersymbol interference (ISI) is usually detrimental. The equalizer should also balance between the interference and the noise. This explains why DFE has been proposed to cancel the ISI and minimize the mean square error. Novel technologies are proposed in IMT-2020 [12] by International Telecommunication Union (ITU), including a series of non-orthogonal waveforms [13]. Those waveforms introduce ISI for the sake of maximizing the throughput.
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In this paper, we consider a simple and plain structure to perform DFE. We firstly use a LMMSE equalizer to get the initial source estimate. For the succeeding iterative rounds, we let the forward filter be MF to maximize the signal-to-noise-plusinterference ratio (SINR), and use the feedback filter to suppress the interference. The detection criterion is symbol-wise MMSE. The algorithm is referred to as the iterative MF-DFE. This structure is used as an example to show the underlying decision error power transfer in the iterative DFE. By decomposing the equalizer output into signal, filtered noise and filtered error (which we also referred to as interference in the following context), we are able to establish the power transfer of the error term between the iterations of MF-DFE. If we define the SINR as the power ratio of signal to noise-plus-error, MF-DFE maps the decision error power of the previous round to the current SINR, while the MMSE detection maps the current SINR to the current decision error power. The trace of iterations is plotted in the error transfer chart, from which we can find the final state of iterations. Using the error transfer chart, we will be able to predict the iterations and measure the performance loss compared to the ideal feedback scenario, which also corresponds to the case of a channel without frequency selectivity. The analysis can be extended to the demodulation of non-orthogonal waveforms other than the single carrier modulation. The rest of the paper is organized as follows. The system model is described in Section II. The error power transfer model is formulated in Section III. Section IV introduces the error transfer chart to trace the MF-DFE. Section V shows verification of the error power transfer model through simulation. Section VI concludes the paper. Throughout the paper, we use an extra ∼ , e.g., x˜ [k], to denote the discrete Fourier transform (DFT) of the corresponding discrete-time sequence, e.g., x[n], i.e., P−1 nk x˜ [k] = x[n] exp −2πi P
where we drop the length P for brevity. Equivalently, we treat x[n], y[n], h[n], v[n] as periodic extensions of the original sequences with the period P. Therefore, the equations described in the time domain can be derived in the frequency domain via DFT. A general equalization model with feedback is established as follows
Pr(X = x|Y = y) is the conditional probability of X = x under Y = y. The superscript ∗ denotes the conjugate of a complex number.
The DFE can be equivalently implemented in the frequency domain, shown in (6).
yl [n] =
P−1
fl [m]y[n − m] +
m=0
P−1
bl [m]xl−1 [n − m].
(2)
m=1
In (2), yl [n] is the lth iterative output to decode x[n]. fl [n] and bl [n] are the forward and feedback filter, respectively. xl−1 [n] is the detection of x[n] in the previous round. The extrinsic information is used so that we let bl [0] = 0 and the second sum starts with m = 1. Let gl [n] (3) and dl−1 [n] (4) denote the convolution fl [n] ∗ h[n] and the error in the (l − 1)th iterative detection output, respectively, gl [n]
P−1
fl [m]h[n − m]
(3)
m=0
dl−1 [n] x[n] − xl−1 [n]
(4)
and we assume that bl [n] = gl [0]δ[n] − gl [n] as it corresponds to the coefficient of ISI term after the forward filter fl [n]. For fl [n], we can adjust the coefficients fl [n] for different purposes, like LMMSE or MF. Other optimized fl [n] and gl [n] were derived in [6] and [14]. Then (2) is written as yl [n] =
P−1
fl [m]y[n − m] −
m=0
P−1
gl [m]xl−1 [n − m].
(5)
m=1
n=0
II. S YSTEM M ODEL We consider a general discrete-time multi-path equation for the block transmit scheme of the single-carrier modulation (1) y[n] =
P−1
h[m]x[n − m] + v[n]
y˜ l [k] = f˜l [k]˜y[k] − (˜gl [k] − gl [0]) x˜ l−1 [k].
Plugging (1) into (5) yields (7). It shows the components in the iterative DFE output, consisting of the desired signal, the filtered noise and the filtered decision error that we also refer to as interference.
(1)
m=0
x[n] and y[n], n = 0, · · · , P − 1, are the transmit and receive time-domain symbols of a block, respectively. P is the block length. x[n] is an independently and identically distributed variable chosen from the constellation set Q = {s0 , · · · , s|Q|−1 } with equal probability. The power of x[n] is σx2 . h[n] is the time-invariant channel impulse response (CIR) in the equivalent discrete-time model. v[n] is the complex additive white Gaussian noise (AWGN) with power σv2 . With proper framing like inserting the unique word (UW), (1) can be regarded as a circular convolution. In the paper, all the convolutions are treated as a P-point circular convolution,
(6)
yl [n] = gl [0]x[n] + desired signal
P−1 m=0
fl [m]v[n − m]
filtered noise P−1
+
gl [m]dl−1[n − m] .
m=1
(7)
filtered decision error
The following discussion is based on (7), but in order to analyze the statistics of three components, we have to make a few assumptions: 1) Ex[n]v ∗ [m] = Edl−1 [n]v ∗ [m] = 0
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∗ [m] = 0 (n = m) Ex[n]dl−1 ∗ Ex[n]x [m] = σx2 δ[n − m] Ev[n]v ∗ [m] = σv2 δ[n − m] ∗ [m] = σ 2 δ[n − m] Edl−1 [n]dl−1 dl−1 P−1 6) gl [m]dl−1[n − m] is approximately complex normal
2) 3) 4) 5)
m=1
according to the central limit theorem. The use of σd2l−1 in assumption 5 is yet to be justified, as the decision error could have been time-selective; it is possible that E|dl−1 [n1 ]|2 = E|dl−1 [n2]|2 . However, later deduction will show that Proposition 1: If ∀ n1 , n2
round, to the current σd2l , based on the signal, noise and error decomposition (8) and the power of each components (9). Next we will show how the relationship is established. A. Decision and Decision Error Power There are various decision criteria to obtain xl [n] from yl [n], e.g., hard detection and MMSE detection. In this paper, we use the MMSE detection, as shown in (11). It computes the a posteriori probability of the source to forge into an estimate with the minimum mean square error. q Pr (x[n] = q|yl [n]) . (11) xl [n] = E (x[n]|yl [n]) =
E |dl−1 [n1 ]|2 = E |dl−1 [n2 ]|2 σd2l−1
q∈Q
Based on (7), the a posteriori probability of x[n] given yl [n] is derived in (12), where we assume equal priori of x[n].
we can have E |dl [n1 ]|2 = E |dl [n2]|2 σd2l . As initially there is no feedback, which means that x−1 [n] = 0, making E |d−1 [n]|2 = E |x[n]|2 = σx2 the initial condition satisfies. The rest is mathematical induction. III. S TATISTICS FOR S IGNAL , N OISE AND E RROR To simplify the notations, we let ⎧ ⎪ Signal : sl [n] gl [0]x[n] ⎪ ⎪ ⎪ P−1 ⎪ ⎨ fl [m]v[n − m] Noise : vl [n] m=0 ⎪ ⎪ P−1 ⎪ ⎪ ⎪Error : ul [n] gl [m]dl−1[n − m]. ⎩
Pr (x[n] = q|yl [n]) ∝ Pr (x[n] = q) fyl [n]|x[n] (yl [n]|q) |yl [n] − qgl[0]|2 ∝ exp − . (12) E |vl [n]|2 + E |ul [n]|2 The MMSE estimate of x[n] (11) is the sum of all the possible value of x weighted by the corresponding a posteriori probability, shown in (13). 2 l [0]| q exp − |yl [n]−qg 2 2 q∈Q
xl [n] = q∈Q
(8)
m=1
The cross-correlations between sl [n], vl [n] and ul [n] are all zero, while the power of each component is given by ⎧ 2 P−1 ⎪ ⎪ 2 1 ⎪ 2 2 2 ⎪ |s |Ave(˜ E g [n]| = σ )| σ g ˜ [k] l l l ⎪ x x P ⎪ ⎪ k=0 ⎪ ⎪ 2 ⎪ P−1 ⎪ ˜ ⎪ ⎨E |vl [n]|2 = σv2 Ave |f˜l |2 σv2 1 fl [k] P k=0 (9) ⎪ 2 ⎪ gl ) ⎪E |ul [n]| = σd2l−1 Var(˜ ⎪ ⎛ ⎪ 2 ⎞ ⎪ ⎪ P−1 P−1 ⎪ ⎪ ⎪ ⎪ |˜gl [k]|2 − P1 σd2l−1 ⎝ P1 g˜ l [k] ⎠ ⎪ ⎩ k=0 k=0 where g-˜g and f -f˜ are two DFT-IDFT pairs. Ave(·) and Var(·) are the mean and the variance of a certain sequence, respectively. We here define the overall error term el [n] and derive its power estimation in (10). el [n] vl [n] + ul [n] (10) E |el [n]|2 = E |vl [n]|2 + E |ul [n]|2 . It consists of two Gaussian terms, and thus is also complex Gaussian. The power of el [n] is the sum of the two components power due to the independence. From the power transfer perspective, MF-DFE and the decision step maps σd2l−1 , the decision error power of the previous
exp −
E|vl [n]| +E|ul [n]|
|yl [n]−qgl [0]|2 E|vl [n]|2 +E|ul [n]|2
.
(13)
The mean square error (MSE) of the estimate is present in (14). One can find that if E|sl [n]|2 , E|vl [n]|2 and E|ul [n]|2 are irrelevant with n, so are E|dl [n]|2 and E|ul+1[n]|2 . Proposition 1 is thus proved. σd2l = E |dl [n]|2 = E |x[n] − xl [n]|2 = σx2 MMSE(SINRl ) 2 |s [n]| E l . (14) σx2 MMSE E |vl [n]|2 + E |ul [n]|2 MMSE(snr) is a non-linear function, which measures the mean square error of the MMSE estimate under √ Gaussian channels. In mathematical words, for Y = X + V/ snr where EX = 0, EX 2 = 1 and V ∼ CN (0, 1), MMSE(snr) = E |X − E(X|Y)|2 .
(15)
For more information about the MMSE estimator and MMSE(snr), please refer to Appendix I. The SINR of the lth iteration, which is the argument of MMSE(·) in (14), is shown in (16). Starting from here, we will drop the index n for E|sl |2 , E|vl |2 and E|ul |2 , as we have shown that the power estimates are irrelevant with n. SINRl
E|sl |2 σx2 |Ave(˜gl )|2 . = E|vl |2 + E|ul |2 σv2 Ave |f˜l |2 + σd2l−1 Var(˜gl ) (16)
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TABLE I DFE A LGORITHM AND E RROR T RANSFER
Combining (16) and (14) would generate the underlying error transfer in the iterative DFE for the single-carrier modulation. We summarize the equalizer and error transfer in Table I. Next we will show two equalizers, including the LMMSE and the MF-DF equalizer. B. LMMSE Equalizer Shown in [6], LMMSE achieves the minimum square error in linear equalizers under no feedback. (6) shows that the equalization can be equivalently applied in the frequency domain, facilitating the implementation of LMMSE. We only need to let ⎧ ˜∗ ⎪ f˜0 [k] = h [k] ⎪ 2 ⎪ ˜ +σ 2 ⎨ h[k] (17) ˜ 2 h[k] ⎪ ⎪ ⎪ g ˜ [k] = 0 ⎩ 2 ˜ +σ 2 h[k]
where σ 2 = σv2 /σx2 is the noise-to-signal ratio. As there is no feedback, we have σd2−1
=
σx2 .
(18)
The Appendix II would show that 1
SINR0 =
1
˜ |2 |h[k] 1 P ˜ |2 +σ 2 k=0 |h[k] P−1
−1
.
(19)
We can use (19) and (14) to obtain σd20 as the error power of the initial detection. C. Iterative MF-DFE MF-DFE is the optimum equalizer with ideal feedback, i.e., completely correct decision. For l ≥ 1, we let ⎧ ⎨f˜l [k] = h˜ ∗ [k] ∗ 2 ⇔ fl [n] = h [−n] (20) ˜ ⎩g˜ l [k] = h[k] gl [n] = h[n] ∗ h∗ [−n]
as much as possible. MF-DFE can be performed iteratively to increase the reliability of detection. Similarly, one can use (21) to estimate the SINR of the lth detection to further determine the the error power σd2l . 2 ˜2 σx2 Ave |h| . SINRl = ˜ 2 + σ 2 Var |h| ˜2 σv2 Ave |h| dl−1
(21)
˜ ˜ D. |h[k]| ≡ |h| ˜ A special case is the flat fading, i.e., |h[k]| is a constant for all k. It gets simple; both LMMSE and MF equalizers suggest constant |f˜ [k]| and g˜ [k] with g[n] = 0 for n = 0. It is worth noting that even g[0] is not zero, we do not use g[0] in the feedback, see Table I. Therefore, there is no feedback and no need for ˜2 feedback at all. The SINR is automatically maximized to |σh|2 . IV. E RROR T RANSFER C HART In this section, the error transfer chart of the iterative MF-DFE is introduced to trace the error power transfer between iterations. The final state of iterations can also be found in the chart, as will be shown to be the crossing point of two curves. A few qualitative properties of the error transfer chart will also be presented. A. Error Transfer Chart in Iterative MF-DFE Fig. 1 shows the steps to perform the iterative MF-DFE in the frequency domain. We first use LMMSE (17) to get the initial estimate, then we perform the MMSE detection (13) and MF-DFE (20), then the next round of the MMSE detection and MF-DFE and so on. For the error power transfer in the iterations, combining Table I and (21), we can established (22). ⎞ ⎛ 2 ˜2 Ave |h| σd2l ⎟ ⎜ = MMSE ⎝ σ2 ⎠ . (22) σx2 ˜ 2 + dl−1 ˜2 σ 2 Ave |h| Var |h| σ2 x
(20) shows that once there is decision feedback, we could use the matched filter to maximize the signal power, and the feedback to remove the interference of neighboring symbols
˜ and σ 2 , the iterations transfer the error power between Fix h[k] two curves (23) and (24). The x-axis is the SINR in iterations.
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Fig. 2. Error transfer chart for SNR = 20 and 16QAM.
function J2−1 ◦ J1 . The region to the right of P0 is not reachable for iterations. One can also find that the starting SINR does not affect the iterations so much. Therefore, we can also use f˜0 [k] = h˜ ∗ [k]
Fig. 1. Flow chart for iterative MF-DFE.
The y-axis is the MMSE, which is also the decision error power normalized by the signal power. y = J1 (x) = MMSE(x)
2 2
˜ Ave |h| 2 ⇔ 2 ˜ ˜ + yVar |h| σ 2 Ave |h| 2 2 ˜2 ˜ Ave |h| σ 2 Ave |h| y = J2 (x) = 2 − 2 . ˜ ˜ xVar |h| Var |h|
x = J2−1 (y) =
(23)
(24)
Now we use the following channel as an example. Consider the MMSE function for 16QAM and assume that σ 2 = σv2 /σx2 = 1/20 and the signal-to-noise ratio (SNR) is 13 dB. 1 0.5i h[n] = √ δ[n] + √ δ[n − 1]. 1.25 1.25
(25)
It is easily derived that ⎧ ⎨Ave |h| ˜ 2 = 1.0 ⎩Var |h| ˜ 2 = 0.32. Fig. 2 is referred to as the error transfer chart. It consists of two curves. One is the decision function, here as J1 (x), which shows the decision error power for an SINR. The other curve is the DFE function, here as J2−1 (y), which returns the SINR for the decision error power on a certain decision feedback algorithm. The iteration trajectory is a polygonal path that bounces between the two curves. It is worth noting that if the decision criteria or the forward and backward coefficients varies in iterations, the error transfer chart is turned into a chart that consists of two groups of curves. Decision is a segment that connects pointing from one of the DFE curves to one of decision curves, while DFE is a segment that connects the previous decision curve and another DFE curve. From Fig. 2, we can find that the iterations converge to a single point P0 , where the two curves (23), (24) intersect. The x-coordinate of P0 is also the fixed point of the composite
as the initial forward filter to replace LMMSE, even when there is no feedback. LMMSE actually provides a lower initial σd20 so that fewer rounds of iterations would be required. In Fig. 2, the vertical line that passes through the point 2 ˜ Ave |h| Pifb , 0 = (20, 0) 2 σ denotes the ideal feedback scenario, where y = 0 suggests that the MMSE of feedback is 0. As it is beyond P0 , it can never be reached via iterations. For the ideal feedback, the effect of multipath is completely eliminated; the multipath channel is converted into an AWGN channel under the same SNR. Therefore, the difference between P0 and ideal feedback indicates the performance loss. After the ideal feedback, the equalizer output is not ideal any more. It is because the feedback is used as extrinsic information to cancel the interference, while the correct symbol can not help in detecting itself. In conclusion, for the single-carrier modulation with decision feedback, ideal feedback is not possible; we can not convert a frequency-selective channel to a pure AWGN channel without changing the noise level. B. Qualitative Properties of Error Transfer Chart 1) Performance loss We assume P0 (sinr0 , mmse0 ) as the crossing point of J1 and J2 indicating the final state of iterations. (26)–(28) [15] show the SNR loss, the noise enhancement and the mutual information loss, respectively. 2 ˜ Ave |h| (26) SNRloss[dB] = 10 log10 2 σ sinr0 mmse0 (27) Noiseenh [dB] = 10 log10 ˜2 MMSE Ave |h| Iloss[nat] =
(
˜2 Ave |h| σ2
sinr0
)
σ2
MMSE(x) dx.
(28)
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Fig. 3. Performance loss: SNRloss , Noiseenh and Iloss . Fig. 5. Comparison between 16QAM and QPSK.
all reduced as well, as the new P0 is closer to the x-axis (Fig. 5). ˜ ˜ 4) |h[k]| ≡ |h| If the channel is flat fading, J2−1 is a constant. x = J2−1 (y) =
Fig. 4. Comparison between σ 2 = 0.05 and σ 2 = 0.5.
Fig. 3 shows the performance loss in the Error Transfer Chart. One is the frequency selectivity, which is inherent in the channel. The ideal feedback is equivalent to an AWGN channel free from frequency selectivity with the same SNR, while in reality, no matter single-carrier modulation and OFDM, frequency selectivity is present and is to degrades the performance. The other factor is the algorithm itself. MF as the forward filter is not the optimum under non-ideal feedback, contributing to another share of loss. 2) Effect of noise power If we increase the noise power, i.e., σ 2 , the corresponding curve J2 is moved downward. 2 ˜2 ˜2 Ave |h| σ 2 Ave |h| − . J2 (x) = ˜2 ˜2 xVar |h| Var |h| In Fig. 4, P0 moves up along J1 . As the slope of J2 at the point P0 is growing larger as P0 climbs up, we claim that SNRloss is reduced. 3) Effect of constellation If we change the constellation from 16QAM to QPSK, J1 is shifted to a new MMSE curve. The mean square error is greatly reduced. SNRloss, Noiseenh and Iloss are
˜ |h| . σ2
The figure of y = J2 (x) is actually a vertical line, corresponding to the ideal feedback. The performance losses defined in (26)–(28) are all 0, which matches the discussion in Section III-D. 5) Effect of turbo equalization with coding Channel coding actually adds a constraint on the transmit bit-sequence, and therefore, there is also a constraint on the symbol sequence. Equivalently, implementing turbo equalization with coding pulls down the MMSE J1 curve, similar to shrinking the constellation size. Normally in this case EXIT chart [16] is used to analyze the iterative behavior, and it is beyond the scope of this paper.
V. S IMULATION V ERIFICATION Simulations are performed to verify the error power transfer model. As previously formulated, we use the LMMSE equalizer to get the initial estimate, and the MF-DFE for the following iterative equalizations. Fig. 6 shows the distribution of the overall error for QPSK modulation under SNR = 3 dB. The normalized overall error is expressed as P−1
el [n] = gl [0]
m=0
fl [m]v[n − m] +
P−1 m=1
gl [0]
gl [m]dl−1[n − m] .
From the figure, it can be shown that the simulated error distribution is close to the theoretical one. Therefore, the assumptions of Gaussian distribution and independence are justified. The SINR for 16QAM modulation at the output of MF-DFE is presented in Fig. 7. We specially focus on the SNR = 13 dB scenario corresponding to σ 2 = 0.05, which we used as an
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Fig. 8. SER of QPSK and 16QAM. Fig. 6. The distribution of the normalized overall error for QPSK and SNR = 3 dB.
Fig. 9. MMSE estimator for QPSK and 16QAM.
Fig. 7. SINR of 16QAM modulation at equalizer output.
example in the previous context. The channel model in (25) is used. From Fig. 7, one can find that there is a significant gain of the SINR for the 1st and 2nd iterations of MF-DFE. The final state of iterations predicted by Fig. 2 is sinr0 ∼ 16.7, while the SINR at the output of the 2nd and 3rd iterations is proximately 101.22 ∼ 16.6. The MF-DFE is also compared with the BCJR algorithm to show the performance gap to the optimal equalization algorithm. The channel is treated as a rate-1.0 convolutional encoder. As the constraint length is short, we can perform the BCJR algorithm with tolerable complexity. Fig. 8 shows the symbol error rate under different SNR for QPSK and 16QAM. It can seen that the gap between the iterative MF-DFE of this paper and the BCJR algorithm is less than 1 dB. At higher SER as 10−2 , the difference gets even smaller. VI. C ONCLUSION In this paper, we have investigated the iterative MF-DFE and its convergence behavior. The error transfer chart is used to trace the iterations, of which the final state is the crossing point of the MMSE curve and error power transfer curve cor-
responding to the MF-DFE algorithm. Simulations have show the validity of the error transfer chart, making it a useful tool to analyze the non-linear equalizer for single-carrier modulation. Future works would involve the applications of other decision feedback algorithms. The model is to be optimized to incorporate an generalized equalizer scheme. A PPENDIX I As previously defined, for a Gaussian channel √ Y = X + V/ snr the MMSE estimate of source X is given by E(X|Y). The MSE of the estimate is given by E |X − E(X|Y)|2 . Fig. 9 shows the I/O relationship of the MMSE estimator and the curve of MMSE(snr) for QPSK and 16QAM, respectively. The constellations are normalized, and the SNR in the figure is the linear SNR. The MMSE estimate under certain of SNR is presented in the upper figures. The subscript I in YI represents the real part and can be replaced with Q for the imaginary part. One can also notice the floors on the curve √ for E(XI |YI ) under high SNRs. For QPSK, the floors are ±1/ 2 and for 16QAM, the floors
HUANG et al.: CONVERGENCE OF FREQUENCY-DOMAIN ITERATIVE MF-DFE FOR SINGLE-CARRIER MODULATION
√ √ are {±1/ 10, ±3/ 10}, corresponding to the discrete value taken by the real and imaginary part of a normalized X. MMSE(snr) curve is presented in the lower figures. It is a monotonically decreasing function, and the MMSE-mutual information relationship [15] holds for complex source (29). snr MMSE(x)dx. (29) I(X; Y)[nat] = 0
A PPENDIX II For the LMMSE equalizer without feedback, one can find that E|u0 |2 = σx2 Var(˜g0 ). As E|s0 |2 = σx2 |Ave(˜g0 )|2 , we can have
E|s0 |2 + E|u0 |2 = σx2 Ave |˜g0 |2 .
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[10] J. Wu and Y. R. Zheng, “Low complexity soft-input soft-output block decision feedback equalization,” IEEE J. Sel. Areas Commun., vol. 26, no. 2, pp. 281–289, Feb. 2008. [11] H. Lou and C. Xiao, “Soft-decision feedback turbo equalization for multilevel modulations,” IEEE Trans. Signal Process., vol. 59, no. 1, pp. 186– 195, Jan. 2011. [12] International Telecommunication Union, “IMT vision—Framework and overall objectives of the future development of IMT for 2020 and beyond,” Geneva, Switzerland, Rev. 1/Doc. 5D/TEMP/625-E, Jun. 2015. [13] 5th Generation Non-Orthogonal Waveforms for Asynchronous Signalling, “5G waveform candidate selection,” presented at the Mobile World Congress, Barcelona, Spain, Mar. 2015, Paper D3.1. [14] S. Huang, J. Wang, and J. Song, “A receiver diversity scheme for singlecarrier systems with unique word,” IEEE Trans. Broadcast., vol. 58, no. 2, pp. 305–309, Jun. 2012. [15] D. Guo, S. Shamai, and S. Verdu, “Mutual information and minimum mean-square error in Gaussian channels,” IEEE Trans. Inf. Theory, vol. 51, no. 4, pp. 1261–1282, Apr. 2005. [16] S. Brink, “Convergence behavior of iteratively decoded parallel concatenated codes,” IEEE Trans. Commun., vol. 49, no. 10, pp. 1727–1737, Oct. 2001.
(30)
Plugging (30) and (17) into the SINR expression (16) with a little transformation yields E|s0 |2 = E|v0 |2 + E|u0 |2 =
1 E|v0 |2 +E|u0 |2 +E|s0 |2 E|s0 |2
−1
1
|f˜0 |2 +σx2 Ave(|˜g0 |2 )
σv2 Ave
σx2 |Ave(˜g0 )|2
=
1
Ave σ 2 |f˜0 |2 +|˜g0 |2 |Ave(˜g0 )|2
= −1
−1 1 1 Ave(˜g0 )
−1
Su Huang was born in Jiangsu, China. He received the bachelor’s and doctoral degrees in information and communication engineering from the Department of Electronic Engineering in Tsinghua University, Beijing, China, in 2010 and 2015, respectively. He was a Research Assistant in the DTV Technology R&D Center of Tsinghua University, where his major research area included broadband communication and its hardware implementation. He is currently working at Spectrum Co., Ltd. His major researching area includes 5G standardization and the design and demodulation of candidate waveforms for the next generation mobile communications.
and thus (19) is obtained. R EFERENCES [1] N. Benvenuto, R. Dinis, D. Falconer, and S. Tomasin, “Single carrier modulation with non-linear frequency domain equalization: An idea whose time has come—Again,” Proc. IEEE, vol. 98, no. 1, pp. 69–96, Jan. 2010. [2] Third-Generation Partnership Project, “Technical specification group radio access network; Evolved Universal Terrestrial Radio Access (E-UTRA); Physical channels and modulation (Release 12),” SophiaAntipolis, France, 3GPP TS36.211 V12.4.0, Dec. 2014. [3] N. Benvenuto and S. Tomasin, “On the comparison between OFDM and single carrier modulation with a DFE using a frequency domain feedforward filter,” IEEE Trans. Commun., vol. 50, no. 6, pp. 947–955, Jun. 2002. [4] Y. Zhu and K. B. Letaief, “Single carrier frequency domain equalization with time domain noise prediction for wideband wireless communications,” IEEE Trans. Wireless Commun., vol. 5, no. 12, pp. 3548–3557, Dec. 2006. [5] J. K. Nelson, A. C. Singer, U. Madhow, and C. S. McGahey, “BAD: Bidirectional arbitrated decision-feedback equalization,” IEEE Trans. Commun., vol. 53, no. 2, pp. 214–218, Feb. 2005. [6] N. Benvenuto and S. Tomasin, “Iterative design and detection of a DFE in the frequency domain,” IEEE Trans. Commun., vol. 53, no. 11, pp. 1867–1875, Nov. 2005. [7] R. R. Lopes and J. R. Barry, “The soft-feedback equalizer for turbo equalization of highly dispersive channels,” IEEE Trans. Commun., vol. 54, no. 5, pp. 783–788, May 2006. [8] R. Dinis, P. Montezuma, N. Souto, and J. Silva, “Iterative frequencydomain equalization for general constellations,” in Proc. 33rd IEEE Sarnoff Symp., Princeton, NJ, USA, Apr. 2010, pp. 1–5. [9] J. Silva, R. Dinis, N. Souto, and P. Montezuma, “Single-carrier frequency domain equalisation with hierarchical constellations: An efficient transmission technique for broadcast and multicast systems,” IET Commun., vol. 6, no. 13, pp. 2065–2073, Sep. 2012.
Jun Wang was born in Henan, China, on October 5, 1975. He received the B.Eng. and Ph.D. degrees from the Department of Electronic Engineering in Tsinghua University, Beijing, China, in 1999 and 2003, respectively. He has been an Assistant Professor and member of DTV Technology R&D Center of Tsinghua University since 2000. His main research interests focus on broadband wireless transmission techniques, especially synchronization and channel estimation. He is actively involved in the Chinese national standard on the digital terrestrial television broadcasting technical activities, and has been selected by the Standardization Administration of China as the Standard Committee Member for drafting.
Jintao Wang (SM’12) received the B.Eng. and Ph.D. degrees in electrical engineering both from Tsinghua University, Beijing, China, in 2001 and 2006, respectively. From 2006 to 2009, he was an Assistant Professor in the Department of Electronic Engineering at Tsinghua University. Since 2009, he has been an Associate Professor and Ph.D. Supervisor. He is the Standard Committee Member for the Chinese national digital terrestrial television broadcasting standard. His current research interests include space-time coding, MIMO, and OFDM systems. He has published more than 100 journal and conference papers and holds more than 40 national invention patents.
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Chao Zhang (SM’15) received the B.Eng. and Ph.D. degrees in 2001 and 2008, respectively, from the School of Electronic and Information Engineering, Beijing University of Aeronautics and Astronautics, Beijing, China. He is now a Research Assistant in the Department of Electronic Engineering of Tsinghua University Beijing, China. His primary research interests focus on broadband wireless communication and digital television broadcasting, especially synchronization, channel estimation, and single carrier frequency domain equalization.
Jian Song (SM’10) received the B.Eng. and Ph.D. degrees from Electronic Engineering Department, Tsinghua University, Beijing, China, in 1990 and 1995, respectively, and worked for the same university upon his graduation. He then conducted postdoctoral research work with the Chinese University of Hong Kong and University of Waterloo, Canada, in 1996 and 1997, respectively. He joined the industry in 1998 and has been with the Advanced Development Group of Hughes Network Systems in the United States for seven years before joining the faculty team in Tsinghua as a Full Professor in 2005. He is now the Director of the DTV Technology R&D Center, Tsinghua University and the Director of Key Laboratory of Digital TV System, both for Shenzhen City and Guangdong Province in China. He is very active in serving IEEE community and has been a Senior Member of IEEE since 2010 and IET Fellow since 2012. He now serves as the Associate Editor of IEEE T RANSACTIONS ON B ROADCASTING as well as the AdCom member of the IEEE Broadcasting Technology Society. He founded the IEEE BTS Beijing chapter in 2007 and has been the Chairman since. He has served as a technical committee member, panelist for many conferences, and also given invited talks. He has successfully organized several IEEE conferences such as IEEE BMSB 2014, IEEE Healthcom 2012, and IEEE ISPLC as the General or TPC Chair. Dr. Song has published more than 240 peer-reviewed journal and conference papers, holds two U.S. and more than 50 Chinese patents with several pending. He is also the co-author of three books in the area of DTV and co-translator of one book in the area of PLC. His current research interest includes digital broadcasting, wireless communications, powerline communications (PLC), and visible line communications (VLC).
