Terahertz Wireless Communications with Flexible Index Modulation Aided Pilot Design

Classical channel estimation methods usually employ a preamble sequence inserted at the beginning of each data block [20, 21], as illustrated in Fig. 2 (a), where $L_{\text{pre}}$ denotes the preamble length for each data block. Then the channel estimate of the $k$-th data block can be obtained with least-squared (LS) algorithm as [33]

where $\mathbf{P}_{k}$ is the $L_{\text{pre}}\times 2$ preamble matrix consisting of the pilot symbols and their conjugates, formulated as

and $\mathbf{y}_{\text{p},k}$ is defined as the corresponding received signal vector. Based on the block structure in Fig. 2 (a), the system spectral efficiency can be calculated as

It is readily seen from (10) that the classical fixed preamble results in throughput degradation.

To address this issue, a flexible pilot design strategy based on IM is proposed for the THz communication system, as illustrated in Fig. 2 (b) and Fig. 3. At the transmitter, the incoming $b_{k}$ bits for the $k$-th data block is firstly divided into $b_{\text{S},k}$ data symbol bits and $b_{\text{I},k}$ index bits. On one hand, the $b_{\text{S},k}$ binary bits are fed into a constellation mapper of an $M_{\text{s}}$-ary alphabet $\mathcal{M}_{\text{s}}=\{S(1),S(2),\cdots,S(M_{\text{s}})\}$, yielding the data symbol vector $\mathbf{s}_{k}=\left[s_{k}(1),s_{k}(2),\cdots,s_{k}(L_{\text{s}})\right]^{T}$. On the other hand, $L_{\text{p}}$ pilots $\mathbf{p}_{k}=\left[p_{k}(1),p_{k}(2),\cdots,p_{k}(L_{\text{p}})\right]^{T}$ are generated using an $M_{\text{p}}$-ary constellation alphabet $\mathcal{M}_{\text{p}}=\{S_{\text{p}}(1),S_{\text{p}}(2),\cdots,S_{\text{p}}(M_{\text{p}})\}$ for channel estimation in each data block. Note that $L=L_{\text{s}}+L_{\text{p}}$. Unlike classical preamble assignment, the proposed IM-aided pilots are no longer fixed, whose indices in the data block are determined by the $b_{\text{I},k}$ index bits with an index selector. More specifically, as presented in Fig. 2 (b), the $k$-th data block $\mathbf{x}_{k}$ for $k=1,2,\cdots,G$ is further split into $G_{\text{s}}$ subblocks with size of $l=L/G_{\text{s}}$, denoted as $\mathbf{x}_{k,g}=\left[x_{k,g}(1),x_{k,g}(2),\cdots,x_{k,g}(l)\right]^{T}$ for $g=1,2,\cdots,G_{\text{s}}$. In each subblock, aside from the $l_{\text{s}}$ data symbols, there are totally $l_{\text{p}}$ pilots drawn from $\mathbf{p}_{k}$, denoted as $\mathbf{p}_{k,g}=\left[p_{k,g}(1),p_{k,g}(2),\cdots,p_{k,g}(l_{\text{p}})\right]^{T}$, corresponding to the index set $\mathcal{I}_{k,g}=\{i_{k,g}(1),i_{k,g}(2),\cdots,i_{k,g}(l_{\text{p}})\}$. Here $\mathcal{I}_{k,g}$ is determined by the incoming bits according to the mapping table between the index bits and the pilot positions, satisfying $1\leq i_{k,g}(1)<i_{k,g}(2)<\cdots<i_{k,g}(l_{\text{p}})\leq l$, which can be exemplified by Table I with $l=4$ and $l_{\text{p}}=2$. By such arrangement, additional $\left\lfloor\log_{2}\binom{l}{l_{\text{p}}}\right\rfloor$ bits are conveyed by the pilot indices in each subblock. Hence, the system spectral efficiency applying the proposed flexible IM-aided pilots becomes

which indicates that the proposed pilot design is capable of compensating for the throughput loss induced by pilots, and even attaining higher spectral efficiency than transmission without using pilots, given that $\left\lfloor\log_{2}\binom{l}{l_{\text{p}}}\right\rfloor>l_{\text{p}}\log_{2}(M_{\text{s}})$.

In order to detect the indices of pilots for channel estimation and index bit recovery, the constellation alphabets employed for data symbols and pilots should be distinguishable, i.e., $\mathcal{M}_{\text{s}}\cap\mathcal{M}_{\text{p}}=\varnothing$. As an example, by considering the phase-shift-keying (PSK) format, one feasible constellation design for $M_{\text{s}}=M_{\text{p}}=4$ is illustrated in Fig. 4, where $\mathcal{M}_{\text{s}}=\left\{e^{j\frac{1}{4}\pi},e^{j\frac{3}{4}\pi},e^{j\frac{5}{4}\pi},e^{j\frac{7}{4}\pi}\right\}$ and $\mathcal{M}_{\text{p}}=\left\{\sqrt{\gamma},\sqrt{\gamma}j,-\sqrt{\gamma},-\sqrt{\gamma}j\right\}$. Note that $\gamma$ denotes the average power ratio between $\mathcal{M}_{\text{p}}$ and $\mathcal{M}_{\text{s}}$, and is set as $4$ in this example, which however is not guaranteed to be optimal. The impacts of $\gamma$ on the system BER performance are discussed as follows.

Firstly, the normalized minimum Euclidean distance between constellations in $\mathcal{M}_{\text{s}}$ and $\mathcal{M}_{\text{p}}$, which determines the performance of index pattern detection [24], can be expressed as

whose minimum value equals $2-\sqrt{2}$ as $\gamma=2$. Then it can be seen that, the value of $\gamma$ should be sufficiently away from $2$ to enlarge $d_{\min}$, thus improving the accuracy of the index pattern detection. Besides, with the increase of $\gamma$, the average transmit power of pilots becomes larger, which enhances the channel estimation accuracy. Note that for our proposed flexible IM-aided pilot design, channel estimation errors not only negatively affect demodulation of data symbols, but also contribute to poor index pattern detection, which leads to increased BER of index bits, and further causes additional errors of data symbol bits due to wrong detections of the index pattern. Hence, $\gamma$ needs to be sufficiently large for accurate channel estimation.

Furthermore, in the $k$-th data block, by assuming that its channel estimation error is tolerable to correctly detect the index pattern, the channel estimate using LS criterion can be expressed as

where $\mathbf{P}_{k}=\left[\mathbf{p}_{k},\text{conj}(\mathbf{p}_{k})\right]$, and $\tilde{\mathbf{w}}_{k}$ denotes the distortion-plus-noise term following $\mathcal{(}0,(\kappa^{2}P_{\text{r}}+\sigma^{2})\mathbf{I}_{L_{\text{p}}})$. Here $\mathbf{I}_{L_{\text{p}}}$ is the $L_{\text{p}}$-by-$L_{\text{p}}$ identity matrix. By assuming that each candidate of $\mathcal{M}_{\text{p}}$ is chosen as the pilot symbol equiprobably, the channel estimation error, defined as $\Delta\mathbf{h}=(\mathbf{P}^{H}\mathbf{P})^{-1}\mathbf{P}^{H}\tilde{\mathbf{w}}_{k}$, approximately follows CSCG distribution as $\mathcal{CN}\left(0,\frac{L_{\text{p}}\gamma+L_{\text{s}}}{LL_{\text{p}}\gamma}(\kappa^{2}P_{\text{r}}+\sigma^{2})\mathbf{I}_{2}\right)$, where the transmitted power is set as $1$ without loss of generality. Afterwards, the received signal for data symbol detection can be formulated as

where we have the equivalent noise term written as

following $\mathcal{CN}\left(0,(\frac{2}{L_{\text{p}}\gamma}+1)(\kappa^{2}P_{\text{r}}+\sigma^{2})\right)$. Then the received signal-to-noise ratio (SNR) can be approximately written as

It can be seen in (16) that, as $\gamma$ increases to infinity, the SNR value approaches zero, causing significant BER performance degradation. Hence, an intuitive guideline for the power allocation issue is that, $\gamma$ should be set to be sufficiently over $2$ to ensure accuracy of channel estimation and index pattern detection, but cannot be too large in case of undesirable SNR degradation. This finding can be well supported by Fig. 11, where the system BER performance versus $\gamma$ is simulated. Note that a sophisticated trade-off is required in this optimal constellation design (power allocation) of $\mathcal{M}_{\text{p}}$ and $\mathcal{M}_{\text{s}}$ for the PSK format as Fig. 4, which may be out of the scope of the main idea in this paper. Hence, the derivative work of the optimal power allocation issue is set aside as our future work for detailed investigation. Instead, numerical methods can be applied to obtain the value of $\gamma$ that optimizes the BER performance as illustrated in Fig. 11.

As shown in Fig. 3, a turbo receiving algorithm is developed to jointly estimate the index pattern and the channel fading vector in an iterative manner, which can be clearly illustrated with mathematical induction as below.

Initialization: Before data transmission, a preamble $\mathbf{p}_{0}=\left[p_{0}(1),p_{0}(2),\cdots,p_{0}({L}^{\prime}_{\text{pre}})\right]^{T}$ of length ${L}^{\prime}_{\text{pre}}$, is utilized to obtain an initial channel estimate $\hat{\mathbf{h}}_{0}$ with LS algorithm. Then $\hat{\mathbf{h}}_{0}$ is employed as the prior knowledge of CSI to detect the index pattern of pilots for the 1-st data block.

Coarse index pattern detection: For the $k$-th data block ($1\leq k\leq N$), $\hat{\mathbf{h}}_{k-1}$ is employed as its prior knowledge of CSI. The received signals in the $g$-th data subblock can be formulated as

where

Here $\tilde{w}_{k,g}(i)$ follows CSCG distribution as $\mathcal{CN}(0,\kappa^{2}P_{\text{r}}+\sigma^{2})$. Based on (17), a symbol-wise log-likelihood ratio (LLR) detector can be applied to obtain the positions of pilots [24], where the logarithm ratio between a posteriori probabilities of the event that the symbol is modulated by $\mathcal{M}_{\text{p}}$ or $\mathcal{M}_{\text{s}}$ is calculated as

Since we have $\Pr\left(x_{k,g}(i)=S_{\text{p}}(m)\right)=\frac{l_{\text{p}}}{lM_{\text{p}}}$ and $\Pr\left(x_{k,g}(i)=S(n)\right)=\frac{l-l_{\text{p}}}{lM_{\text{s}}}$, (19) can be further derived as

where the likelihood function $\Pr\left(y_{k,g}(i)|x_{k,g}(i)\right)$ is formulated as

By substituting (21) into (20), the final expression of the LLR value can be derived as

where $\eta_{k,g}(i,\hat{\mathbf{h}}_{k-1})$ stands for the LLR value calculated using $\hat{\mathbf{h}}_{k-1}$ as the channel estimate, and $\hat{P}_{\text{r}}=\hat{\mathbf{h}}_{k-1}^{H}\hat{\mathbf{h}}_{k-1}P_{\text{t}}$ is the estimate of $P_{\text{r}}$. After all the LLR values of the $g$-th subblock are obtained using (22), the indices of pilots in each subblock can be coarsely determined by

where $\hat{i}_{k,g}^{(0)}(1)<\hat{i}_{k,g}^{(0)}(2)<\cdots<\hat{i}_{k,g}^{(0)}(l_{\text{p}})$, and the operator $l_{\text{p}}\text{-}\max$ finds the largest $l_{\text{p}}$ LLR values. Then the resultant $\hat{\mathcal{I}}^{(0)}_{k}=\{\hat{\mathcal{I}}_{k,1}^{(0)},\hat{\mathcal{I}}_{k,2}^{(0)},\cdots,\hat{\mathcal{I}}_{k,G_{\text{s}}}^{(0)}\}$ is defined as the coarse estimate of the pilot index pattern for the $k$-th data block.

Iterative joint index pattern detection and channel estimation: At the $n$-th iteration ($n\geq 1$), the positions of the flexible IM-aided pilots are firstly determined by $\hat{\mathcal{I}}^{(n-1)}_{k}$, and then they are employed to update the channel estimate, where the resultant CSI is further fed back for updating the pilot index pattern. More specifically, for the $g$-th subblock, the received symbols of the indices $\hat{\mathcal{I}}^{(n-1)}_{k}\setminus\hat{\mathcal{I}}^{(n-1)}_{k,g}$ are concatenated together, as illustrated in (III-C) (on the top of next page),

which is then utilized to estimate the channel fading vector

where $\tilde{\mathbf{P}}_{k,g}$ is an $(L_{\text{p}}-l_{\text{p}})\times 2$ pilot matrix defined as

By replacing $\hat{\mathbf{h}}_{k-1}$ with $\hat{\mathbf{h}}^{(n)}_{k,g}$ in (22), the LLR values of the $g$-th subblock are re-calculated, denoted as $\eta_{k,g}(i,\hat{\mathbf{h}}^{(n)}_{k,g})$ for $i=1,2,\cdots,l$. Then (23) is applied to these latest LLR values, where the pilot indices of the $g$-th subblock are updated as $\hat{\mathcal{I}}^{(n)}_{k,g}=\{\hat{i}_{k,g}^{(n)}(1),\hat{i}_{k,g}^{(n)}(2),\cdots,\hat{i}_{k,g}^{(n)}(l_{\text{p}})\}$, satisfying $\hat{i}_{k,g}^{(n)}(1)<\hat{i}_{k,g}^{(n)}(2)<\cdots<\hat{i}_{k,g}^{(n)}(l_{\text{p}})$. By repeating the operations above on each of the $G_{\text{s}}$ subblocks, the output index pattern of the $k$-th data block in the $n$-th iteration can be obtained, expressed as $\hat{\mathcal{I}}_{k}^{(n)}=\{\hat{\mathcal{I}}^{(n)}_{k,1},\hat{\mathcal{I}}^{(n)}_{k,2},\cdots,\hat{\mathcal{I}}^{(n)}_{k,G_{\text{s}}}\}$.

With the iterations going on, when $\hat{\mathcal{I}}_{k}^{(n)}$ equals $\hat{\mathcal{I}}_{k}^{(n-1)}$, it is implied that the detection of the pilot index pattern has converged. Therefore, the stopping criterion for the iterations is set as $\hat{\mathcal{I}}_{k}^{(n)}=\hat{\mathcal{I}}_{k}^{(n-1)}$. By defining $\hat{\mathcal{I}}_{k}$ as the final output of the proposed turbo receiving algortihm for the $k$-th data block, the fine channel estimate, denoted as $\hat{\mathbf{h}}_{k}$, can be calculated using LS criterion as (25) with all the detected flexible pilots corresponding to $\hat{\mathcal{I}}_{k}$. Afterwards, $\hat{\mathcal{I}}_{k}$ and $\hat{\mathbf{h}}_{k}$ for $k=1,2,\cdots,G$ are fed into the subsequent stages for signal demodulation. Besides, $\hat{\mathbf{h}}_{k}$ is also employed as the prior knowledge of CSI for coarse index pattern detection of the $(k+1)$-th data block. The aforementioned turbo receiving algorithm can be summarized as Algorithm 1, where the initialization phase is omitted for brevity.

Remark 1: As illustrated in (III-C)-(26), $\hat{\mathcal{I}}^{(n-1)}_{k}\setminus\hat{\mathcal{I}}^{(n-1)}_{k,g}$ is employed as the extrinsic information in the $n$-th iteration to update the index pattern of the $g$-th subblock, instead of $\hat{\mathcal{I}}^{(n-1)}_{k}$. Such arrangement is to eliminate the positive feedback that hinders the convergence of the proposed receiving algorithm, inspired by the principles of iterative decoding for turbo codes.

Remark 2: For the $k$-th and $(k+1)$-th data blocks, when the physical channel coefficients and I/Q imbalance parameters remain approximately unchanged, it is reasonable to employ $\hat{\mathbf{h}}_{k}$ as the prior CSI knowledge of the $(k+1)$-th data block for coarse index pattern detection, since there only exists a relatively small phase noise difference between $\mathbf{h}_{k}$ and $\mathbf{h}_{k+1}$. On the other hand, for fast-fading channel, where the physical channel coefficient also varies across adjacent data blocks, $\hat{\mathbf{h}}_{k}$ is considered to be outdated for the $(k+1)$-th block. Under such “worst” case, it is validated that the proposed turbo receiving algorithm is still capable of achieving superior performance of index pattern detection and channel estimation. Aside from the relevant simulation results, an intuitive geometrical analysis is provided below to support the validity of the proposed receiving algorithm under relatively fast-fading channel.

In this paper, flat-fading channel is mainly considered for THz communication systems with high-gain directional antennas [31]. However, the multi-path effects in the THz band, which are explicitly described in [34], cannot be fully eliminated by highly directional transmission under certain cases [35]. For instance, there may exist possible reflectors close to the LoS path, yielding additional reflected paths; Besides, combined reflection and transmission of THz signals on a multi-layer objects like window glass also contribute to multiple available paths. Furthermore, when the communication bandwidth is sufficiently high, frequency-selective channel fading should be considered even for LoS scenario without multi-path propagation [36]. Hence, in this subsection, the proposed flexible IM-aided pilot design is extended to the frequency-selective fading channel, where some guidelines about the generalized design will be provided.

Several assumptions are made under the frequency-selective fading channel: a) The length of channel impulse response (CIR) is assumed to be known, denoted as $L_{h}$; b) The CIR keeps approximately constant within each data block of length $L$, since the symbol period is extremely short due to the ultra-high communication bandwidth. Under these assumptions, the generalized data block structure with the proposed flexible IM-aided pilots is modified as illustrated in Fig. 5. On one hand, for each data block of length $L$ with classical fixed preamble in Fig. 5 (a), a pilot sequence is added at the beginning of the block, consisting of the cyclic prefix (CP) part of length $L_{c}\geq L_{h}$ and pilot symbol part with length $L_{p}$. By considering the existence of I/Q imbalance at the transmitter, $L_{p}$ should be no less than $2L_{h}$ for channel estimation [20]. Besides, CP of length $L_{c}$ is also applied to the data symbol part. On the other hand, unlike its classical counterpart, the pilot sequence is no longer fixed in each of the data block with the proposed flexible IM-aided pilots, as shown in Fig. 5 (b). By such operations, additional information bits can be conveyed by the positions of pilots to compensate for the spectral efficiency loss caused by channel estimation, which have totally $(L-2L_{c}-L_{p}+1)$ possible choices for each data block. Then each data block could convey additional $\left\lfloor\log_{2}(L-2L_{c}-L_{p}+1)\right\rfloor$ index bits, yielding the system spectral efficiency calculated as

where $M_{\text{s}}$ denotes the modulation order of data symbols. At the receiver, for the $k$-th data block, the index pattern can be coarsely estimated using the channel estimate of the $(k-1)$-th data block as the prior knowledge of CSI. Note that a fixed pilot sequence is required to provide initial channel estimate for the $1$-st data block, as stated in the Initialization phase in Section III-C. Before coarse index pattern detection, equalization techniques such as frequency-domain equalization (FDE) or decision-feedback equalization (DFE) based on zero-forcing (ZF) or minimum mean squared error (MMSE) criterions are employed to eliminate the inter-symbol interference (ISI) caused by the frequency selectivity of THz channels [36]. Afterwards, a correlation-based method could be utilized to detect the index of pilot sequences. More specifically, sliding correlation is performed on the equalized signal vector, denoted as $\hat{\mathbf{x}}_{k}=\left[\hat{x}(1),\hat{x}(2),\cdots,\hat{x}(L-L_{c})\right]$, using the pilot sequence (including CP) ${\mathbf{p}}^{\prime}_{k}=\left[{p}^{\prime}_{k}(1),{p}^{\prime}_{k}(2),\cdots,{p}^{\prime}_{k}(L_{c}+L_{p})\right]$, which yields cross-correlation values $R_{k}[n]$ for $n=1,2,\cdots,L-2L_{c}-L_{p}+1$ as

Then the start index of the pilot sequence in the $k$-th data block can be obtained as

Finally, the index bits can be recovered from $\hat{n}_{k}$ using a look-up table between different index patterns and binary bits, and the demodulation of data symbols is straightforward. The aforementioned extension of the proposed flexible IM-aided pilot design to the THz frequency-selective fading channel will be discussed into more detail in our near-future research.

Under relatively fast-fading channel, there exists apparent difference between the physical channel coefficients of the $k$-th and $(k+1)$-th data blocks for $k=1,2,\cdots,G-1$, aside from the variations of phase noise. Then $\hat{\mathbf{h}}_{k}$, which is served as the prior CSI knowledge for the $(k+1)$-th data block, is likely to be outdated. To validate the feasibility of the proposed turbo receiving algorithm, an intuitive geometrical demonstration is provided to figure out the error probability of the LLR-based index pattern detection. For tractable analysis, several assumptions are made as below: (1) $v_{\text{t}}$ is approximated as zero; (2) $\mathcal{M}_{\text{s}}=\left\{e^{j\frac{1}{4}\pi},e^{j\frac{3}{4}\pi},e^{j\frac{5}{4}\pi},e^{j\frac{7}{4}\pi}\right\}$ and $\mathcal{M}_{\text{p}}=\left\{\sqrt{\gamma},\sqrt{\gamma}j,-\sqrt{\gamma},-\sqrt{\gamma}j\right\}$ are employed for modulation, where $\gamma$ is empirically set to be larger than $2$; (3) There is only phase difference in the physical channel coefficients between adjacent blocks, whilst the amplitude remains approximately unchanged⁵⁵5Due to the ultra-wide bandwidth employed in the THz communication system, the change of transmission distance is marginal across hundreds of symbol periods, where the path gain is approximately constant.. Then the received signal model in (7) can be simplified as (the indices are omitted for brevity)