Chapter 5 Conclusion and Future Work
5.1 Future Work
1. Since the ICI indicator can identify whether the subcarrier experiences good or bad channel, to utilize it in the transmitter side to enhance the overall system performance such as adaptive loading or allocation is a potential future research topic. Furthermore, our ICI indicator originally derived from the time linearly varying model that holds only when the normalized Doppler frequency is lower than 0.2. For even faster changing channels, the model that considers higher order terms is necessary. It is worth further investigation on the questions that how to modify the ICI indicator and extend its usability for this scenario.
2. For OFDMA, we set the un-used sub-channels nearing the boundaries to zero.
In PSA, we also have the flexibility to choose different ordering for cancella-tion. Further investigations are needed for these two issues, i.e., consider the co-channel interference from the un-used sub-channels and optimal ordering for ICI cancellation in PSA.
3. Nonlinear ICI cancellation approaches such as MAP or turbo ICI equalizers are still too complex even with the help of ICI indicator and thus further explorations are needed. MIMO ICI equalization is another important topic. In spatial multi-plexing, our proposed ICI indicator and equalizers can be easily extended. How-ever, when space-time or space-frequency codes are applied, it is not so trivial.
4. The inter-channel variations have been exploited to gain the diversity in this dissertation. The intra-channel variation may also be used to further improve
the diversity. An initial step on this has been provided in Appendix II. It is interesting to further investigate the diversity order.
5. In the current work, we did not consider the ICI caused by CFO which will result in performance degrades if CFO is left untreated or residual CFO exists; however, other ICI cancellation approaches focused on the Doppler spread induced ICI like us will also have performance degradation. It is worthwhile to study in the future the two kinds of ICIs jointly for practical applications.
Appendix I: Asymptotic Analysis on the Diversity Order of BICM-OFDM in
Doubly Selective Channels
With the system model described in Chapter 4, the stacked received signal y = [yT1, yT2, · · · , yTP]T can be given as
y = X(IP ⊗ FN ×L)h + z , Xheq+ z (I.1) where IP is a P × P identity matrix, ⊗ denotes the Kronecker product, h is an length-P L vector defined as
h = [˜h(1; 0), ..., ˜h(1; L − 1), ˜h(2; 0), ..., ˜h(2, L − 1), ..., ˜h(P, 0), ..., ˜h(P, L − 1)]T (I.2)
with ˜h(i; l) (1 ≤ i ≤ P , 0 ≤ l ≤ L − 1) representing the average channel impulse response in the i-th OFDM symbol, X is an P N × P N diagonal matrix given by
X = diag(X(0), ..., X(N − 1), X(N ), ..., X(2N − 1), ..., X(P N − 1)), (I.3) and z is an P N × 1 noise vector representing the residual ICI plus noise.
We derive the asymptotic diversity order of BICM-OFDM by first bounding the PEP. Let X be the coded transmit signal corresponding to codeword c and ˆX be the detected signal corresponding to codeword ˆc where c 6= ˆc. The assumption that the in-terleaver maps consecutive coded bits to different constellation points and transmitted onto different OFDM subcarriers is adopted [27]. Assume that z is complex Gaussian distributed with zero mean and variance N0. The maximum-likelihood decision rule is to choose the error signal ˆX if
1
(πN0)P N/2exp(−||y − ˆXheq||2
N0 ) ≥ 1
(πN0)P N/2exp(−||y − Xheq||2
N0 ). (I.4)
So, the conditional error probability P (c → ˆc|h) is equivalent to
By using the Q-function, the error probability in (I.5) can be written as
Q
Then, the pairwise error probability (PEP) can be derived by averaging over all channel realization
where SNR = 1/N0 since the average power of channel and transmitted signal are normalized to one and D = X − ˆX is the coded symbol difference matrix. According to the assumption made in the interleaver, there are at least dfree non-zero terms in D [27]. In the following we consider the worst case that there are only dfree different terms between X and ˆX and each of these terms has Euclidean distance d2min. Note that the typical parameters of BICM-OFDM are chosen as N ≥ L and N ≥ dfree.
A typical PEP analysis such as that in [27] assumes that the vector heq has inde-pendent elements and the rank analysis is focused on the term DHD. Here, however, the correlation of channels over several OFDM symbols needs to be considerd. We adopt the approach in [29] to extract the statistical independent components in heq which are considered as the source of diversity. After the extraction the usual PEP analysis can proceed.
Assume the Kronecker model [30,70,71] for the channel, i.e., the autocorrelation matrix of the channel can be decoupled as
R = E[hhH] = ΦT ⊗ ΦL, (I.9)
where ΦT and ΦL are the time and path gains autocorrelation matrices. The rank of the channel autocorrelation matrix is [75, Fact 7.4.20]:
rank(R) = rank(ΦT ⊗ ΦL) = rank(ΦT) × rank(ΦL) = rT × L (I.10)
where rT and L are the ranks of ΦT and ΦL, respectively. Note that rT×L is bounded by P × L. The correlation matrix of Heq is Req = E[heqhHeq] = (IP⊗ FN ×L)E[hhH](IP ⊗ FN ×L)H = (IP ⊗ FN ×L)R(IP ⊗ FN ×L)H. Since FN ×L has full rank L, the matrix (IP ⊗ FN ×L) also has full rank P L.
We use the following proposition [75, Proposition 2.6.2]: let A be a m × n matrix, B is n × k with rank n, and C is l × m with rank m, then rank(AB) = rank(A) and rank(CA) = rank(A). Therefore, the rank of Req is
rank(Req) = rank(R) = rT × L. (I.11)
The eigenvalue decomposition is used to extract the statistically independent com-ponents in heq. Consider Req = V ΣhVH where V is a P N × rTL matrix satisfying
VHV = IrTL and Σh = diag(σ12, · · · , σ2r containing independent and identically distributed (iid) zero-mean complex Gaussian random variables with unit variance. It can be shown that the PEP is not affected by this substitution, since heq and V Σ
1 2
h¯heq have identical distributions which is known as the isotropy property of the standard Gaussian random vector.
The term hHeqDHDheq in (I.8) now becomes ¯hHeq(Σ that the central part (Σ
1 2
h)HVHDHDV Σ
1 2
h is a Hermitian matrix and can be diag-onalized by an unitary matrix U , i.e., (Σ
1 matrix DHD has rank dfree, it follows immediately that the rank of (Σ
1
Moreover, define ˘h = UH¯heqand notice that ˘h is iid complex Gaussian distributed with zero mean and unit variance since U is unitary. Thus, (I.8) becomes
P (c → ˆc) = E where ˘hn is the n-th element in ˘h and its magnitude |˘hn| is Rayleigh distributed with the PDF
Then using the PDF in (I.13), the RHS in (I.14) can be derived as
E[exp(−SNR · d2min 4
rT V
X
n=1
λn| ˘Hn|2)]
= Z
· · · Z
2|˘h1| × · · · × 2|˘hrT V| · exp(−SNR · d2min 4
rT V
X
n=1
λn|˘hn|2)×
exp(−|˘h1|2) × · · · × exp(−|˘hrT V|2)d|˘h1| · · · d|˘hrT V|
= 1
QrT V
n=1[1 + (λnSNR · d2min/4)].
(I.15)
Finally, when SNR is large enough and the eigenvalue is non-zero, the term 1 in the denominator in (I.15) can be neglected. Thus, Equation (4.14) in Chapter 4 can be obtained.
Appendix II: Regarding the Diversity Order From Intra-Symbol Channel
Variations
We investigate the impact of ICI on the diversity order here. We have shown that available ICI-cancellation techniques are good enough for the system to realize the po-tential diversity order resulting from the inter-symbol channel variation. As regarding the benefit of exploiting the intra-symbol channel variation, it has been observed that the diversity order slightly increases if ICI is utilized at the receiver [9,11,12,18] but the high complexity remains a concern. However, more delicate effects of ICI on the diversity order still need further investigation. We chose not to distract ourselves in Chapter 4 with this venture. Here we show how one may further pursuit this direction of analysis by making additional assumptions. With these assumptions, our framework can be extended to evaluate the diversity gain provided by the intra-symbol channel variation.
Consider the transmission of a single OFDM symbol and assume that the time variation of CIR is linear within the OFDM symbol [13,15]. It is a reasonable assump-tion in most practical cases when the normalized Doppler frequency is smaller than 0.1, which corresponds to 500 km/h in WiMAX standard. The CIR can be written as
h(pNS− N + t; l) = h(pNS −N − 1
2 ; l) +t − N −12 NS
δ(pNS; l) (II.1)
where NS denotes the length of CP plus N . h(pNS− N + t; l) and h(pNS− N −12 ; l) represent the l-th path at the t-sample instant and the center point of the p-th OFDM symbol, respectively, and δ(pNS; l) is the difference of CIR between the (p − 1)-th and
the p-th OFDM symbol. Note that with the linear variation assumption the channel at the center point is also the averaged channel in one OFDM symbol.
We shall drop the OFDM symbol index for simplicity for now since we are con-sidering the intra-symbol channel variation. The CIR matrix can be decomposed as
Ht = Mt+ ξ∆t (II.2)
where [Mt]i,j = h(pNS−N −12 ; i − j) and [∆t]i,j = δ(pNS; i − j) for i, j = 0, 1, . . . , N − 1. Mt and ∆t both are circulant matrices. The diagonal matrix ξ is defined as ξ = N1
Sdiag(−N −12 , −N −12 + 1, . . . ,N −12 ). Note that h(pNS − N −12 ; l) and δ(pNS; l) are assumed to be zero when l ≤ 0 or l ≥ L − 1.
The CFR can be derived as
Hf = F HtFH
= F MtFH + F ξFHF ∆tFH
= Hf+ Ξ∆f
(II.3)
The frequency-domain received signal model is y = Hfx + z
To obtain the signal model used in Chapter 4 (cf. Equation (4.6)), (II.4) is rewrit-ten in matrix form:
The diversity order can be analyzed via the framework established in Chapter 4.2; in essence, the asymptotic diversity order is determined by the rank of Req. It can be inferred that the intra-symbol channel variation potentially provides higher diversity orders. However, the power of the difference channel δt is usually two-order smaller than that of the average channel ht [13]. The diversity gain due to δt, unlike the fre-quency diversity provided by ht, is not accessible when SNR is moderate. Moreover, the optimal ML receiver fully exploiting the signal energy from the ICI terms is re-quired while the corresponding super-trellis in the Viterbi algorithm usually demands unaffordable computational resource. Instead of complicating the analysis by studying the marginal gain provided by intra-symbol channel variation, we chose to focus on the more pronounced benefit provided by inter-symbol variation.
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