1 Introduction

A transceiver system based on Bit-Interleaved Coded Modulation with Iterative Decoding (BICM-ID) has been introduced for the first time by X. Li and J. A. Ritcey in [1, 2]. In their work they have shown that a BICM system with soft-decision feedback and iterative decoding significantly outperforms Trellis-Coded Modulation (TCM) which has been introduced much earlier by G. Ungerboeck in [3]. It also performs better compared to BICM without iterative decoding which was introduced by E. Zehavi in [4]. G. Caire et al. in [5] provided tools for evaluating the performance of BICM especially the metric of the Harmonic Mean of the minimum squared Euclidean Distance (HMMSED) which is widely used in this paper.

A BICM system is given by a serial concatenation of channel encoder, bit-interleaver, and modulator. At the receiver side, the arrangement of a demodulator, de-interleaver and channel decoder is used to reconstruct the originally transmitted signal. Compared to G. Ungerboeck’s TCM scheme, BICM improves in terms of the Bit Error Rate (BER) especially under Rayleigh fading channels. The asymptotic performance of BICM depends on the choice of channel coding, bit-interleaving, modulation scheme and modulation symbol labeling. For example, a Gray symbol labeling is often used in BICM in order to minimize the number of bit errors that appear within the information bits after demodulation. The bit-interleaver as another key element in BICM prevents long burst errors and correlations between neighbored bits in such a way that the decoder is able to improve decoding. The channel decoder further uses the redundant information introduced at the transmitter side for protection and reconstructs the net bits with high reliability. With the introduction of the well-known turbo principle of digital signal processing [6] it was possible to improve BICM by extending it to BICM-ID. In BICM-ID an additional feedback line is used to exchange information between channel decoder and demodulator. In [1], a hard-decision feedback version of BICM-ID has been introduced. Additional improvements can be made when extending BICM-ID with soft-decision [7]. Then, reliability information in terms of so-called extrinsic information or log-likelihood ratios (LLR or shortly L-values) is exchanged between channel decoder and demodulator. In a BICM-ID system, the BER performance improves with the number of iterations and converges until the bit error floor is reached. The bit error floor is the part of the BER curve that can be interpreted as the lowest achievable BER for the considered BICM-ID system and can be simulated or analyzed according to an Error Free Feedback (EFF) scheme. To investigate the EFF performance, a priori knowledge from the transmitter side is relayed as error free and reliable LLR values to the demodulator (perfect knowledge). This is an ideal assumption, because in real systems no information from the transmitter is available at receiver side. Another typical observation in performance investigation for BICM-ID is the waterfall region where the BER is reduced considerably in a small range of E S /N 0 . For a good performance of BICM-ID the channel coding, bit-interleaving, modulation scheme and modulation symbol labeling must be jointly optimized. But significant improvements in BICM-ID, e.g., reaching a very low bit error floor, can be realized when a symbol labeling with a high HMMSED [5] is used. When perfect a priori knowledge is available - this is considered for a sufficient number of iterations in BICM-ID - we only need to consider symbols where the labeling differs only in one bit position. An increased minimum Euclidean Distance between those pairs of neighbored labels helps the receiver to distinguish much better between them when noise is present. Such a symbol labeling is contrary to the optimization of a symbol labeling for BICM. In BICM neighbored symbols shall be neighbored labels, i.e., Gray coding scheme. However, the improved performance for good channel conditions comes at the cost of complexity at the receiver side.

In the literature, Hierarchical Modulation [8, 9] or Layered Modulation has been introduced to give a transceiver system the possibility to receive data under different conditions. Hierarchical Modulation allows the operator to send multiple data streams modulated to a single symbol stream. The different data streams are called base layer (BL) which is the 1st layer or enhancement layer (EL) for all others (2nd, 3rd and so on). Depending on the channel condition and computational capability of the receiver the modulated symbols can be demodulated in such a way that all or a subset of data streams are recovered. Therefore, the BL provides the most important information being transmitted in a robust way to all radio devices. In addition, each EL carries optional information used to provide further valuable information. This information can be used at the receiver to improve the communication in several ways, e.g., a higher data rate (quality), reliability, or range. The major challenge in the design of hierarchical systems is the impact of the different layers on each other. However, hierarchical modulation is very attractive because of its possibility to switch easily between different receiver configurations. Therefore, it is used in broadcast systems like Digital Video Broadcasting over Satellite (DVB-S2) [10] or over Terrestrial Antennas (DVB-T2) [11] to provide different information qualities (data rates).

In the literature, digital communication systems exploiting both the benefits of BICM-ID and the advantages of Hierarchical Modulation have not been extensively discussed so far. In [12, 13], first design solutions to a Hierarchically Modulated BICM-ID (HM-BICM-ID) have been introduced. In our previous work [14], we proposed a HM-BICM-ID system based on a hierarchical 8 × 8-PSK where each constellation point is composed by a group of eight labels with Hamming Distance d ham  = 2 (even and odd parity). The scheme has been found in a systematic way and reduces the performance degradation significantly. Now, we propose a novel procedure to give the system designer the possibility to develop further hierarchical modulation schemes with improved coded BER performance.

This paper is structured as follows: Chapter 2 introduces the two-layered HM-BICM-ID system. Furthermore, the effect of Inter-Layer Interference (ILI) as well as reasons for typical performance degradations in hierarchical systems are given in detail. In Chapter 3, a novel procedure is developed which moves the constellation points of a certain modulation scheme to a proper direction considering two main criteria, the HMMSED and the Bit Error Probability (BEP) of each layer. Then, the procedure is used in Chapter 4 to develop a novel hierarchical modulation scheme for a HM-BICM-ID. The HM-BICM-ID system is simulated and results for BER performance are presented and discussed in the context of so-called EXtrinsic Information Transfer (EXIT) charts. The paper concludes with Chapter 5.

2 Hierarchically Modulated BICM-ID

The block diagram of the simulation environment for an HM-BICM-ID system with different receivers is depicted in Fig. 1 as proposed in [14]. It shows a high capacity transmitter, an Additive White Gaussian Noise (AWGN) channel, and two different receiver types (configurations). Assuming that different radios of unequal capabilities can be available at the receiver side, two different configurations for a communication link can be considered for analysis. The high capacity radio is considered as the EL radio in this paper. Similarly, the low capability of legacy radio is equal to the BL radio.

Figure 1
figure 1

Block Diagram of a Hierarchical BICM-ID composed by BL and EL considering two independent configurations for the receiver.

Full size image

The BL signal processing modules used by the low capability radio as well as the high capability radio are shown in Fig. 1 as blue/dark gray boxes. The situation at the receiver is equivalent to state-of-the-art signal processing in BICM-ID. Due to the higher computational power available by a high capability radio an HM-BICM-ID system can be developed and used at the transmitter and receiver side. The low capability radio will still be able to receive the information when a high capability radio has been used as transmitter. The EL signal processing blocks are shown in Fig. 1 as green/light gray boxes. Please note that the EL implies additional multiplexer and de-multiplexer blocks to separate the independent data streams from the BL and EL. To support reliable communication while being flexible in the choice between specific configurations, a fully hierarchical transmitter system including channel encoding, bit-interleaving and modulation is used to guarantee interoperability over the air. Therefore, HM-BICM-ID is a suitable solution to manage such communication scenarios. A fully hierarchical transceiver as depicted in Fig. 1 is based up on the idea behind incremental redundancy [15, 16]. However, it is not used in the context of ARQ (Automatic Repeat Requests), but in the context of forward error correction only.

The challenge in the context of the channel code is to find convolutional codes with maximum free distance d free to perform best in terms of the BER. Beside this, the convolutional code of the BL must be a subset of the EL to keep interoperability. Therefore, we investigated some low rate convolutional codes from [17–19] and listed them in Table 1.

Table 1 Some convolutional codes of coding rate R c and constraint length K from [17–19].
Full size table

However, the best matching result was found for the convolutional code with constraint length K = 3. All other convolutional codes with R c  = 1/2 and R c  = 1/4 cannot be combined to the best convolutional code of coding rate R c  = 1/6. The bold italic numbers indicate a common incidence between the combined convolutional code and the best convolutional code from literature. As a consequence, we decided to use the convolutional code with K = 3 for the hierarchical convolutional code, which was also used in [14] as well. For the BL the convolutional code with generator polynomial G = (5,7)8, d free  = 5, and R c  = 1/2 is used. The EL uses G = (5,7,5,7,7,7)8 with d free  = 16 and R c  = 1/6. Both are proposed by [18, 19] due to the maximum free distance and optimal distance spectra. Because the first code is a subset of the latter one, a separation of the encoded data stream into BL and EL information is possible. Furthermore, the convolutional code generating the additional information (EL information excluding BL information) is the best code of constraint length of K = 3 in literature with G = (5,7,7,7)8, R c  = 1/4 and d free  = 10.

After channel coding, the data stream is de-multiplexed to independent data streams. These data streams are each bit-interleaved and finally multiplexed to groups of bits. These groups of bits form bit patterns for the modulator symbol. The hierarchical modulator is designed in such a way that the BL modulator scheme is a subset of the EL modulator scheme.

A typical hierarchical modulation scheme with 2 bits for the BL and 4 extra bits for the EL is shown in Fig. 2 (a) and (b), respectively. Thus, groups of 6 bits are mapped to one single symbol by the hierarchical modulator. This keeps the overall rate of the hierarchical system constant. Such a HM-BICM-ID system guarantees the use of high capability radios at receiver side while keeping signal interoperability over the air with a low capability receiver. In the case where a high capability receiver is used, HM-BICM-ID helps to demodulate and decode the additional information carried by the EL. In our work HM-BICM-ID is designed to provide additional information to improve robustness and range. This is novel compared to broadcast systems where the throughput is increased under good channel conditions [10, 11].

Figure 2
figure 2

Hierarchical labeling of signal constellation points for the BL (a) and the EL (b).

Full size image

2.1 System Design

A big challenge in HM-BICM-ID is the design of the modulation scheme and channel encoder. Because of the hierarchical design two effects influence the system. The first effect affects the EL. The BL is often state-of-the-art and fixed. Therefore, especially in the case where a EL transmitter and BL receiver are used, interoperability must be kept in mind during the design process. This can be solved when both the BL modulator and BL decoder are subsets of the corresponding EL modulator and EL decoder. With this side constraint the design freedom of the EL for the channel encoding and modulation is restricted. For the modulator, the restriction is related to the decision bounds and limits the choice for the modulation labels and positions of the constellation points. For the channel encoder, the choice for a specific generator polynomial is restricted. However, the operator with the low capability radio is able to receive the BL information even when an EL transmitter was used. Unfortunately, in the context of BICM-ID the restrictions in the design of the modulation scheme leads to a reduced HMMSED.

A modulation scheme with a high HMMSED causes BICM-ID systems to improve the coded BER performance under good channel conditions. Therefore, we expect a degradation of the coded BER performance for HM-BICM-ID systems compared to classical equivalent BICM-ID.

The second effect in the design of the hierarchical system affects the BL itself. If the EL transmits information to the BL receiver, the additional EL information cannot be processed and will be ignored. Nevertheless, this additional EL information has an impact on the BL modulator and indicates extra interference beside AWGN. This is called Inter-Layer Interference (ILI) because the interference is coming from the mismatch between EL and BL modulation scheme. ILI can be described as:

$$ I L I=\frac{1}{M_{BL}}\cdot \frac{M_{BL}}{M_{EL}}\kern.2em {\displaystyle \sum_{k=1}^{M_{BL}}\kern.2em {\displaystyle \sum_{\overset{l=1}{\mu_k={\mu}_{l, BL}}}^{M_{EL}}{\parallel {\boldsymbol{x}}_k-{\boldsymbol{z}}_l\parallel}^2}} $$
(1)

The inner part sums over all EL symbols with same BL bit pattern. It is the mean square error (MSE) between one specific BL symbol and all related EL symbols with same BL bit pattern. Finally, the definition of ILI can be used to calculate a modified signal-to-ILI-plus-noise ratio SINR value:

$$ SINR=\frac{S}{ILI+ N} $$
(2)

The SINR value asymptotically converges to the signal-to-ILI ratio when no noise is present. Therefore, the drawback of ILI is the higher sensitivity to noise.

For example, the EL modulator symbol 4910 / MSB1100012 as depicted in Fig. 3 carries the same information within the first two bits of the equivalent BL bit pattern of the BL symbol 310 / MSB112. In the presence of AWGN, the EL symbol 4910 is now noisy received (solid line close to EL symbol 4310) and crosses the decision bound (ordinate) of the BL modulation scheme. The received symbol is finally demodulated to the wrong symbol 210.

Figure 3
figure 3

The EL symbol 4910 which carries the BL information of the BL symbol 310 is demodulated to the wrong BL symbol 210 in the presence of a noisy reception.

Full size image

3 Novel Procedure for Joint Multi-Layer Optimization of Hierarchical Modulation Schemes

Due to the fact that ILI causes performance degradations in the BL while design restrictions causes performance degradations in the EL, our idea is to reduce these two effects. Therefore, we propose a novel procedure that starts from a hierarchical modulation scheme with labels already optimized for BICM-ID. The procedure moves the constellation points towards the direction of interest in a serial iterative manner. The design process is done offline and results in a novel hierarchical modulation scheme. We propose to optimize two main criteria for each layer. On the one hand, we want to increase the HMMSED to improve the asymptotic BER performance of BICM-ID in each layer. On the other hand, we want to decrease the BEP to prevent grouping of symbols during HMMSED optimization. The HMMSED and BEP optimizations are described in Section 3.1 and Section 3.2. Afterwards the procedure itself is described in Section 3.3.

3.1 Maximization of the HMMSED

To maximize the HMMSED of a HM-BICM-ID we first need to define it for non-hierarchical modulation schemes in general. From [5] the definition of the HMMSED d 2 h (μ) of an M-ary modulation scheme (M = 2 m) for labeling μ is given by a slightly rewritten equation by:

$$ {d}_h^2\left(\mu \right)={\left(\frac{1}{M\cdot m}{\displaystyle \sum_{k=1}^M{\displaystyle \sum_{l=1}^m{\displaystyle \sum_{\boldsymbol{z}\in {X}_{\overline{b}\left({\boldsymbol{x}}_k\right)}^l}\frac{1}{{\left\Vert {\boldsymbol{x}}_k-\boldsymbol{z}\right\Vert}^2}}}}\right)}^{-1} $$
(3)

The mapping μ describes the relation between bit patterns and symbols. The inner sum goes across the subset \( {X}_{\overline{b}\left({\boldsymbol{x}}_k\right)}^l \). The subset \( {X}_{\overline{b}\left({\boldsymbol{x}}_k\right)}^l \) describes all symbols z with an inversed bit \( \overline{b} \) at the lth bit position of the considered symbol x k with k ∈ [1.. M]. Considering non-hierarchical modulation schemes the subset \( {X}_{\overline{b}\left({\boldsymbol{x}}_k\right)}^l \) has only one single element \( \left|{X}_{\overline{b}\left({\boldsymbol{x}}_k\right)}^l\right|=1 \). The outer sums go across all symbols and bit positions of the defined modulation scheme. Therefore, the HMMSED considers all symbol pairs which are direct neighbors in terms of bit labels, e.g., the bit pattern of the two symbols differ only for one bit. In a modulation scheme with m bits per symbol and M = 2m different symbols the sum of (M ⋅ m)/2 pairs are considered two times. The HMMSED can be used to compare different M-ary modulation schemes. In Fig. 4 an exemplary 8-PSK modulation scheme with Gray labeling from [4] is shown.

Figure 4
figure 4

Set partitioning of an exemplary 8-PSK modulation scheme with Gray labeling [4]. The Euclidean Distance between every pair of two symbols that differs at the 1st (a), 2nd (b) or 3rd (c) bit position are shown.

Full size image

As it can be seen, there are in total twelve pairs of neighbors that differ only at one bit position. Four pairs exist for each bit position of the bit pattern. They are shown for the 1st, 2nd, and 3rd bit position in Fig. 4 (a), (b), and (c), respectively.

In a hierarchical modulation scheme, the calculation of the HMMSED differs because the receiver can choose between different configurations. The calculation of the HMMSED for a BL receiver with EL transmitter must be developed in this context. Taking this into account, Eq. (3) can be modified and rewritten in such a way that the HMMSED for a lower layer is given by:

$$ {d}_{h, layer}^2={\left(\frac{1}{M\cdot {m}_{l ayer}}\cdot \frac{2^{m_{l ayer}}}{M}{\displaystyle \sum_{k=1}^M{\displaystyle \sum_{l=1}^{m_{l ayer}}{\displaystyle \sum_{\boldsymbol{z}\in {X}_{\overline{b}\left({\boldsymbol{x}}_k\right)}^l}\frac{1}{{\left\Vert {\boldsymbol{x}}_k-\boldsymbol{z}\right\Vert}^2}}}}\right)}^{-1} $$
(4)

HMMSED is now investigated over all BL bits m layer  < m related to the number of bits defined by the M layer -ary modulation scheme of the BL (\( {M}_{layer}={2}^{m_{layer}} \)). The outer sum is still summing over all M = 2m symbols defined by the EL modulation scheme. Therefore, d 2 h,layer is computed over all EL symbols x k with k ∈ [1.. M] considering only the bit positions from the BL. For hierarchical modulation schemes the subset \( {X}_{\overline{b}\left({\boldsymbol{x}}_k\right)}^l \) has now \( \left|{X}_{\overline{b}\left({\boldsymbol{x}}_k\right)}^l\right|={2}^{m_{l ayer}}/ M \) elements. The subset includes all symbols which are direct neighbors within the bit positions l ∈ [1.. m layer ] but not considering the bit positions from the EL. For example, the EL symbol 3210 / MSB1000002 (m ≡ m EL  = 6 bits) is related to the BL symbol 210 / MSB102 (m BL  = 2). Considering the 2nd bit position l = 2 all EL symbols with binary labeling MSB11xxxx2 are neighbors because they are all related to the BL symbol 310 / MSB112 which is a direct neighbor of 210 / MSB102.

This is shown in Fig. 5 by the dashed box at the bottom left corner. The second group of EL symbols with inversed bit at bit position l = 1 (bit pattern equal to MSB00xxxx2) is related to the BL symbol 010 / MSB002. This is shown by the dashed box at the top left corner in Fig. 5. As a consequence, the number of neighbors for a hierarchical modulation scheme for a lower layer, i.e., the BL, increases compared to the definition of the HMMSED in non-hierarchical modulation schemes.

Figure 5
figure 5

Neighborhood relationship between EL symbol 3210 which is related to BL symbol 210 and those EL symbols in the dashed boxes which are related to the BL symbols 010 and 310.

Full size image

With the definition from Eq. (4), we propose the maximization of the HMMSED by moving the constellation points far away from all neighbors with high influence. As a consequence, the vector for the movement considering a specific layer can be identified as follows:

$$ {\overrightarrow{\boldsymbol{r}}}_{k, layer}=\left(\begin{array}{c}\hfill Re\left\{{r}_{k, layer}\right\}\hfill \\ {}\hfill Im\left\{{r}_{k, layer}\right\}\hfill \end{array}\right);\kern1.25em {r}_{k, layer}={\xi}_{l ayer}\cdot \frac{2^{m_{l ayer}}}{M\cdot {m}_{l ayer}}{\displaystyle \sum_{l=1}^{m_{l ayer}}}{\displaystyle \sum_{\boldsymbol{z}\in {X}_{\overline{b}\left({\boldsymbol{x}}_k\right)}^l}}\frac{e^{j\cdot \measuredangle \left({\boldsymbol{x}}_k-\boldsymbol{z}\right)}}{{\left\Vert {\boldsymbol{x}}_k-\boldsymbol{z}\right\Vert}^2} $$
(5)

The vector \( {\overrightarrow{\boldsymbol{r}}}_{k, layer} \) describes the vector to move the kth symbol of an M-ary EL modulation scheme in the direction where the Euclidean Distance increases. The definition is similar to Eq. (4) without the outer sum and introduces a complex normalized term \( {e}^{j\cdot \measuredangle \left({\boldsymbol{x}}_k-\boldsymbol{z}\right)} \) for the direction in ℂ. The complex result can be converted to describe a direction vector. The inverse of the Euclidean Distance is used to weight the direction vector. A lower distance between neighbors in the terms of the HMMSED causes an increased value and influence of the movement. The direction component in x k for the movement away from z is defined by the vector subtraction. The final direction vector is the superposition of the direction vectors of each neighbor. For hierarchical modulation schemes the number of neighbors for the BL is much higher. The scaling factor ξ layer is introduced to further control the convergence behavior of our procedure described in Section 3.3 and shall be a positive real value ξ layer  ∈ ℝ+. Please note, considering the EL (highest layer) the definition of the HMMSED from Eq. (4) falls back to the definition of Eq. (3) because the number of bits are identical m EL  = m. Beside this, the number of neighbors in the subset \( {X}_{\overline{b}\left({\boldsymbol{x}}_k\right)}^l \) reduces to 1.

3.2 Minimization of the BEP

As a consequence of the multi-labeling challenge, stated later in Section 3.4, we propose not only to use HMMSED as an optimization criterion, but also the BEP. The main idea is to prevent the collapse of several constellation points to groups and thus to achieve a sufficient coded BER for the first iteration in BICM-ID.

We propose the calculation of direction vectors for each modulation symbol in the I/Q-plane to reduce the BEP. For an AWGN channel the conditional Probability Density Functions (PDFs) are related to the two dimensional Gaussian distribution. The result of the direction vector depends highly on the position of the modulation symbols in the I/Q-plane, the modulation labels, the decision bounds, and the channel conditions (N 0/E S ). The main challenge to calculate the influence is the fact that the symbols are moving during the optimization process and therefore the decision bounds and PDFs are changing. Therefore, an analytical way of determining the BEP by mathematical integration becomes impractical. Therefore, we propose a numerical way of determining the BEP considering a division of the I/Q-plane into sufficiently small segments. Assuming AWGN, the joint PDF p(w, x k ) of w and the kth symbol x k is given by:

$$ p\left(\boldsymbol{w},{\boldsymbol{x}}_k\right)= Pr\left({\boldsymbol{x}}_k\right)\cdot p\left(\left.\boldsymbol{w}\right|{\boldsymbol{x}}_k\right)=\frac{1}{M}\cdot \frac{1}{2\pi {\sigma}^2}\cdot {e}^{-\frac{{\left\Vert \boldsymbol{w}-{\boldsymbol{x}}_k\right\Vert}^2}{2{\sigma}^2}} k\in \left[1.. M\right] $$
(6)

Because of AWGN, the equation depends on the variance σ 2 = N 0/E S . Pr(x k ) is the probability of occurrence for the modulated symbol x k . Considering that all modulation symbols are equally distributed, we can simplify Pr(x k ) = 1/M, where M is the total number of modulation symbols, i.e., M = 64 for a 64-QAM. With the definition, the side constraint ∬ W ∑ k p(w, x k ) = 1 is fulfilled. Now, we want to consider the Probability Pr(δ i,j , x k ) that the received noisy modulation symbol lies within the small segment with center δ i,j in the I/Q-plane and the modulation symbol x k has been transmitted. Therefore a two dimensional integration in w must be performed:

$$ Pr\left({\boldsymbol{\delta}}_{i, j},{\boldsymbol{x}}_k\right)={\displaystyle \underset{\delta_j-\frac{\Delta}{2}}{\overset{\delta_j+\frac{\Delta}{2}}{\int }}{\displaystyle \underset{\delta_i-\frac{\Delta}{2}}{\overset{\delta_i+\frac{\Delta}{2}}{\int }} p\left(\boldsymbol{w},{\boldsymbol{x}}_k\right)\ d{w}_I d{w}_Q}} $$
(7)

Due to the fact that i,j,k are all natural numbers the probability Pr(δ i,j , x k ) is discrete. Δ describes the width and length of a square with center δ i,j  = (δ i , δ j ) ∈ ℝ2 in the I/Q-plane. The value Δ shall be small enough to form a grid of sufficiently high resolution. The distance between neighbored segments is Δ in the direction of the ordinate / abscissa. For all segments δ i,j which are part of a border in the I/Q-plane, a modified bound of ±∞ for the integrals in Eq. (7) shall be used. Finally, to determine the BEP the labels of the investigated segment δ i,j and the transmitted modulation symbol x k needs to be taken into account:

$$ {P}_b\left({\boldsymbol{\delta}}_{i, j},{\boldsymbol{x}}_k\right)={P}_{b,\mu, {\boldsymbol{\delta}}_{i, j}, k}\left({\mu}_{{\boldsymbol{\delta}}_{i, j}},{\mu}_k\right)\cdot P r\left({\boldsymbol{\delta}}_{i, j},{\boldsymbol{x}}_k\right) $$
(8)

\( {P}_{b,\mu, {\boldsymbol{\delta}}_{i, j}, k}\left({\mu}_{{\boldsymbol{\delta}}_{i, j}},{\mu}_k\right) \) describes the BER between the label \( {\mu}_{{\boldsymbol{\delta}}_{i, j}} \) related to the segment δ i,j and the label μ k related to the modulation symbol of x k . Label \( {\mu}_{{\boldsymbol{\delta}}_{i, j}} \) is always equal to the labeling of the nearest symbol of the segment. Therefore, \( {\mu}_{{\boldsymbol{\delta}}_{i, j}} \) simplifies to μ l with mapping index l = arg min n ∈ [1.. M]‖δ i,j  − x n ‖2. Then, \( {P}_{b,\mu, {\boldsymbol{\delta}}_{i, j}, k}\left({\mu}_{{\boldsymbol{\delta}}_{i, j}},{\mu}_k\right) \) simplifies to P b,μ,l,k (μ l , μ k ) which is given by:

$$ {P}_{b,\mu, l, k}\left({\mu}_l,{\mu}_k\right)=\frac{1}{M}\cdot {d}_{hamming}\left({\mu}_l,{\mu}_k\right)=\frac{1}{M}\cdot {\displaystyle \sum_{n=1}^m{\mu}_{l, n}\oplus {\mu}_{k, n}} $$
(9)

d hamming (μ l , μ k ) is the Hamming Distance between the two labels μ l and μ k . It is simply the total number of different bits between the bit patterns related to μ l and μ k and can be expressed by the summation of XOR operations over all bit positions of the pattern. For hierarchical modulation schemes Eq. (6) can be modified to:

$$ p\left(\boldsymbol{w},{\boldsymbol{x}}_{k^{\prime }}\right)={\displaystyle \sum_k P r\left({\boldsymbol{x}}_k\right)\cdot p\left(\operatorname{}\boldsymbol{w}\Big|{\boldsymbol{x}}_k\right)\kern1em {k}^{\prime}\in \left[1..{M}_{layer}\right]} $$
(10)

The joint PDF \( p\left(\boldsymbol{w},{\boldsymbol{x}}_{k^{\prime }}\right) \) is the summation of all PDFs related to the two dimensional Gaussian distribution where the related BL symbols have equal BL bit patterns, i.e., the right left quadrant of Fig. 2 (b) becomes the group with bit pattern MSB10xxxx2. The other Eqs. (7), (8), and (9) can be used with slightly modified variables for k = k’, M = M layer , and l = l’.

A colored projection of the mapping μ δi,j is shown for the BL and EL in Fig. 6 (a) and (b), respectively. The values in the color bar are arranged from 1 to M BL  = 4 for the BL and 1 to M = 64 for the EL according to the index of the hierarchical modulation schemes from Fig. 2 (a) and (b). The colors of the four quadrants of the BL in Fig. 6 (a) are close to the “mean color” of those quadrants in Fig. 6 (b) of the EL which is typical for hierarchical modulation schemes.

Figure 6
figure 6

Colored projection of the mapping μ δi,j of a hierarchical 64-QAM modulation scheme from Fig. 2 for the BL (a) and the EL (b).

Full size image

The superposed PDFs for all symbols of a 64-QAM are depicted in Fig. 7 (a). The corresponding BEP P b (δ i,j , x k ) of all symbols x k over the I/Q-plane is shown in Fig. 7 (b). With the given BEP P b (δ i,j , x k ) for a small segment in the constellation diagram the direction of influence for x k can be expressed as follows:

$$ {\overrightarrow{\boldsymbol{s}}}_{k, layer}=\left(\begin{array}{c}\hfill Re\left\{{s}_{k, layer}\right\}\hfill \\ {}\hfill Im\left\{{s}_{k, layer}\right\}\hfill \end{array}\right);\kern1em {s}_{k, layer}={\beta}_{layer}{\displaystyle \sum_{{\boldsymbol{\delta}}_{i, j}\in {\mathrm{\mathbb{R}}}^2}{P}_{b, layer}\left({\boldsymbol{\delta}}_{i, j},{\boldsymbol{x}}_k\right)\cdot {e}^{j\cdot \measuredangle \left({\boldsymbol{x}}_k-{\boldsymbol{\delta}}_{i, j}\right)}} $$
(11)

\( {\overrightarrow{\boldsymbol{s}}}_{k, layer} \) is a direction vector to move the constellation point x k in the direction where the BEP is minimized. It can be computed by the complex value s k,layer which describes the direction vector in the complex domain ℂ. Complex number s k,layer is a result of the superposition of the movements expressed by each small segment. For all segments a direction based on the vector subtraction of x k  − δ i,j can be determined where the direction is defined by the normalized complex term \( {e}^{j\cdot \measuredangle \left({\boldsymbol{x}}_k-{\boldsymbol{\delta}}_{i, j}\right)} \). The weight for the direction is directly related to the BEP of each segment. Therefore, a low E s / N 0 and a short distance to a neighbor will result in a higher weight of the segment. Due to the nature of the Gaussian distribution the segments with highest impact are arranged at the borders of the decision bounds. Consequently, the direction of movement is very often similar to the task of increasing the distance between close constellation points. The convergence behavior can be controlled by the positive real value β layer  ∈ ℝ+.

Figure 7
figure 7

a Colored projection of the superposed PDFs of all symbols of 64-QAM. b Colored projection of the BEP for all symbols of a given two-layered 64-QAM as depicted in Fig. 2 (b) for E s /N 0  = 15 dB.

Full size image

3.3 Novel Procedure for Optimizing Hierarchical Modulation Schemes

In the previous sections, the maximization of the HMMSED and the minimization of the BEP have been discussed in detail. Eq. (5) and (11) are used to compute direction vectors for moving the constellation points towards a maximized HMMSED and minimized BEP, respectively. However, to prevent an uncontrolled growth of the energy per symbol E S during the movement of constellation points normalization is done after each optimization step during the procedure. Thus, the energy per symbol is kept constant. A description for a comprehensive optimization of a hierarchical modulation scheme is given by procedure 1.

Procedure 1: Comprehensive optimization

1: Initialize constellation points and labels

2: normalize

3: calculate d 2 h,layer & P b,layer for each layer

4: set optional stop criteria for lower bound (LB) and upper bound (UB), e.g. d 2 h,layer,LB ; d 2 h,layer,UB ; P b,layer,LB ; P b,layer,UB

5: set flag as true

6: set maximum number of procedure iterations

7: loop until stop criteria fulfilled or flag equals false or number of procedure iterations exceeded

8: set for each layer: d 2 h,layer,old as d 2 h,layer

9: set for each layer: P b,layer,old as P b,layer

10: optimize d 2 h,layer with Procedure 2 for each layer

11: optimize P b,layer with Procedure 3 for each layer

12: calculate d 2 h,layer & P b,layer for each layer

13: if d 2 h,layer  < d 2 h,layer,old or P b,layer  > P b,layer,old then

14: set flag as false

15: end if

16: end loop

First, the initial modulation scheme, i.e., constellation points and labels, is normalized and the HMMSED and the BEP of each layer are calculated. Additionally, optional stopping criteria for the procedure are defined, e.g., an upper and/or lower bound for the HMMSED and/or BEP are considered. For example, a very high HMMSED for the EL might be a good choice to optimize BICM-ID performance for the EL. The HMMSED and BEP are calculated in every procedure iteration step and compared with the old values before. This is important because the optimization of the HMMSED degrades the BEP and vice versa. A comparison between the modulation scheme of iteration n + 1 and the one of iteration n is done to recognize an improvement during the iteration process. If one parameter degrades, the procedure stops immediately and outputs the final modulation scheme. Inside the loop of procedure 1 the procedure 2 and 3 are used to maximize the HMMSED and to minimize the BEP, respectively.

Procedure 2: Maximize d 2 h,layer of a specific layer

1: Initialize normalized modulation scheme

2: for k = 1 to number of constellation points do

3: compute direction vector \( {\overrightarrow{\boldsymbol{r}}}_{k, layer} \) using Eq. (5)

4: end for

5: for k = 1 to number of constellation points do

6: move constellation point to direction of \( {\overrightarrow{\boldsymbol{r}}}_{k, layer} \)

7: end for

8: normalize

Procedure 2 initializes the normalized modulation scheme which has been committed by procedure 1. Then the direction vector \( {\overrightarrow{\boldsymbol{r}}}_{k, layer} \) for the specific layer is computed. The computation is performed for each symbol of the M layer -ary modulation scheme. Finally, the constellation points are moved towards the computed direction and are normalized afterwards. The concept of procedure 2 can easily be adapted to the BEP in a similar way and is described by procedure 3. The direction vector for the BEP is identified as \( {\overrightarrow{\boldsymbol{s}}}_{k, layer} \).

Procedure 3: Maximize P b,layer of a specific layer

1: Initialize normalized modulation scheme

2: for k = 1 to number of constellation points do

3: compute direction vector \( {\overrightarrow{\boldsymbol{s}}}_{k, layer} \) using Eq. (11)

4: end for

5: for k = 1 to number of constellation points do

6: move constellation point to direction of \( {\overrightarrow{\boldsymbol{s}}}_{k, layer} \)

7: end for

8: normalize

3.4 Convergence Behavior during Optimization of the HMMSED with Procedure 2

In Section 3.3 we described the procedures for optimizing the HMMSED and BEP for each layer. A high value of HMMSED guarantees a convergence to an asymptotic low coded BER for BICM-ID / HM-BICM-ID systems. This is mainly because the extrinsic information improves with the number of iterations during the exchange between decoder and demodulator at receiver side. Therefore, the demodulator can distinguish much better between two neighbored symbols when only one bit of the bit pattern is unreliable.

However, the optimization of the HMMSED with procedure 2 is not sufficient because modulation schemes collapse into a multi-labelled Binary Phase-Shift Keying (ML-BPSK). ML-BPSK has the drawback that in the first receiver iteration of BICM-ID no extrinsic information can be generated without having reliable a priori information at the input. In the first iteration reliable a priori knowledge is not available at all. Therefore, ML-BPSK is not performing well in context of BER.

In Fig. 8 the principle of the optimization of HMMSED within the procedure 2 presented in Section 3.3 is shown in detail. The circles represent the constellation points after the 1st iteration of the procedure 2. The lines/arrows indicate the computed overall vector for the movement of a specific constellation point before normalization. During the development of procedure 2, it has been observed for 16-QAM-Ray [20, 21] (the modulation scheme with the best HMMSED value in literature) that the constellation points collapse to ML-BPSK. ML-BPSK has two groups of eight constellation points where each symbol of one group has only even or odd parity bit patterns. This is somehow obvious because BPSK has the highest possible HMMSED of d 2 h,BPSK  = 4. Therefore, procedure 2 may converge to the best possible constellation scheme when considering only HMMSED.

Figure 8
figure 8

a, b 16-QAM within 1st iteration of procedure 2. c is the result after movement.

Full size image

However, this causes a new challenge. Without having any a priori knowledge at the first receiver iteration step in BICM-ID, the demodulator, based on ML-BPSK, can only distinguish between the even and odd parity group but not between labels of the same group. This is because the labels within one group have the same constellation points. Thus, the demodulator cannot produce any valuable extrinsic information and the BER performance will degrade to a BER of 0.5. Furthermore, we analyzed the behavior of the HMMSED under the influence of the number of iterations and the scaling factor ξ. The results are shown in Table 2.

Table 2 Influence of the scaling factor ξ on the value of HMMSED for different number of iterations of procedure 1.
Full size table

For this, the procedure has been started with an initial 16-QAM-Ray, as given in [20, 21], and without any restrictions, i.e., no partitioning into different layers and no further stopping criteria. 16-QAM-Ray labeling has been chosen because of its HMMSED of d 2 h,16 − QAM − Ray  = 2.719 which is the highest value compared to any other 16-QAM labeling. Within Table 2, a general convergence behavior towards an ML-BPSK scheme can be observed for all parametrizations. But with a higher value for ξ a reduced number of iterations is needed to reach the ML-BPSK and therefore a faster convergence behavior is observed. But this might also result in the instability of the procedure when used with a hierarchically modulated scheme. Thus, the movement must be controlled by a carefully chosen ξ to ensure that no constellation point will cross a decision bound of another layer as this would violate the premise of one layer being a subset of another layer.

4 Performance Analysis of HM-BICM-ID With Optimized Modulation Scheme

In the previous sections we introduced our proposed procedure. Two parameters, the HMMSED and the BEP, have been used as optimization criteria for the procedure. Now, we want to use the procedure to develop a novel hierarchical modulation scheme with optimized performance in terms of coded BER for both the BL and EL of a two-layered modulation scheme.

4.1 An Optimized Two-Layered Modulation Scheme Based on a Hierarchical 64-QAM Modulation Scheme

In [14], HM-BICM-ID has been introduced for several configurations. For both cases, three-layered and two-layered hierarchical modulation schemes have been investigated and it has been shown that a reduced number of layers give the designer more freedom to optimize the modulation scheme. Therefore, we propose a HM-BICM-ID system with a BL and one EL to keep the freedom in the design of the hierarchical modulation as high as possible. Further, we propose the BL modulation scheme to be a QPSK as shown in Fig. 2 (a) with d 2 h,QPSK  = 2.6667.

The initial EL is a hierarchical 64-QAM which can be constructed by superposition of QPSK mapping (first 2 bits fixed) and a 16-QAM-Ray labeling from [20, 21] for each quadrant. The mapping is identical to Fig. 2 (b). The parameter set for the procedure has been chosen to β EL  = 25.0, ξ EL  = 0.01, β BL  = 0.2, and ξ BL  = 0.01. The energy per symbol to noise power spectral density ratio E S / N 0 has been set to 8 dB. The maximum number of iterations for the procedure was 10. There were no further restrictions, e.g., no optional stopping criteria. Executing procedure 1 results in a constellation diagram as depicted in Fig. 9.

Figure 9
figure 9

64-QAM-BF modulation scheme.

Full size image

Due to the look of the scatter plot we propose to call the constellation 64-QAM-ButterFly (64-QAM-BF). The normalized symbols of all labels are given by the following vector S i where i is the index of the mapping as depicted in Fig. 9:

$$ \begin{array}{l}{S}_i=\Big(-1.1176+0.6701\mathrm{i},-0.7242+0.1625\mathrm{i},-0.7566+0.1831\mathrm{i},-0.6058+0.8386\mathrm{i},-0.7322+0.2010\mathrm{i},-0.6066+0.8749\mathrm{i},\hfill \\ {}-0.5489+1.0047\mathrm{i},-1.0410+0.2303\mathrm{i},-0.6073+0.2088\mathrm{i},-0.7861+0.8201\mathrm{i},-0.7261+0.9005\mathrm{i},-0.9419+0.1735\mathrm{i},\hfill \\ {}-0.8910+0.8805\mathrm{i},-0.8464+0.1646\mathrm{i},-0.8521+0.1918\mathrm{i},-0.5292+0.7662\mathrm{i},+0.5303+0.7667\mathrm{i},+0.9420+0.1734\mathrm{i},\hfill \\ {}+0.8463+0.1646\mathrm{i},+0.7863+0.8201\mathrm{i},+0.8521+0.1917\mathrm{i},+0.7262+0.9004\mathrm{i},+0.8909+0.8806\mathrm{i},+0.6079+0.2086\mathrm{i},\hfill \\ {}+1.0413+0.2301\mathrm{i},+0.6063+0.8386\mathrm{i},+0.6072+0.8750\mathrm{i},+0.7243+0.1625\mathrm{i},+0.5500+1.0045\mathrm{i},+0.7567+0.1830\mathrm{i},\hfill \\ {}+0.7323+0.2009\mathrm{i},+1.1176+0.6703\mathrm{i},+0.6084-0.2106\mathrm{i},+0.8916-0.8815\mathrm{i},+0.7264-0.9021\mathrm{i},+0.8510-0.1921\mathrm{i},\hfill \\ {}+0.7864-0.8226\mathrm{i},+0.8475-0.1664\mathrm{i},+0.9412-0.1754\mathrm{i},+0.5301-0.7678\mathrm{i},+1.1191-0.6707\mathrm{i},+0.7330-0.2020\mathrm{i},\hfill \\ {}+0.7569-0.1840\mathrm{i},+0.5499-1.0050\mathrm{i},+0.7246-0.1648\mathrm{i},+0.6066-0.8763\mathrm{i},+0.6060-0.8405\mathrm{i},+1.0410-0.2324\mathrm{i},\hfill \\ {}-1.0408-0.2326\mathrm{i},-0.5489-1.0052\mathrm{i},-0.6061-0.8763\mathrm{i},-0.7329-0.2021\mathrm{i},-0.6054-0.8405\mathrm{i},-0.7569-0.1841\mathrm{i},\hfill \\ {}-0.7245-0.1648\mathrm{i},-1.1191-0.6706\mathrm{i},-0.5290-0.7672\mathrm{i},-0.8509-0.1922\mathrm{i},-0.8475-0.1665\mathrm{i},-0.8917-0.8813\mathrm{i},\hfill \\ {}-0.9413-0.1755\mathrm{i},-0.7263-0.9021\mathrm{i},-0.7863-0.8226\mathrm{i},-0.6078-0.2108\mathrm{i}\Big)\hfill \end{array} $$

4.2 Performance Analysis for 64-QAM-BF

In a first step, we calculated the HMMSED for those modulation schemes depicted in Table 3. The upper schemes are calculated for single-layered modulation schemes. The values are given from literature. We observed that hierarchical schemes used for BICM-ID systems have poor coded BER performance due to a reduced HMMSED. Therefore, in [12] performance investigation have been proposed. The best coded BER performance has been reached by the 8 × 8-PSK modulation scheme because of the increased HMMSED. The reason for this is a relaxed design of the constellation points which is no more fixed to the 64-QAM scheme. Compared to the latter one, the novel 64-QAM-BF modulation scheme has a slightly reduced HMMSED. But, considering the HMMSED of the BL in a HM-BICM-ID system we find the HMMSED of the BL to be increased dramatically to d 2 h,BL  = 2.847. Therefore, we expect the 64-QAM-BF to outperform all other two and three-layered hierarchical modulation schemes from [14].

Table 3 HMMSED d h 2 for several modulation schemes.
Full size table

In a second step, we performed a coded BER simulation for a two-layered HM-BICM-ID system with 64-QAM-BF. The coded BER curves and the corresponding EFF curves have been simulated for both the BL and EL configuration. The setup for the simulation environment has been chosen equivalent to [14] and is depicted in detail in Table 4.

Table 4 Setup of simulation parameters for HM-BICM-ID.
Full size table

In Fig. 10 the coded BER performance according to the energy per symbol to noise power spectral density ratio E S /N 0 is depicted for three reference systems. The non-iterative BICM system with QPSK and Gray labeling is depicted by the black solid curve. The second system is the HM-BICM-ID with relaxed design constraints from [14]. The BER performance of the BL and EL are depicted by the blue solid (square marker) and green solid (diamond marker) curves. The BER performance of the BL and EL of our proposed HM-BICM-ID with 64-QAM-BF is depicted by the violet solid curve (star marker) and yellow ocher solid curve (triangle marker). Each dashed colored curve describes the EFF of the associated HM-BICM-ID system. The EFF describes the lowest possible coded BER performance that can be achieved with a particular iterative decoding system.

Figure 10
figure 10

Coded BER performance of BICM with QPSK Gray labeling, HM-BICM-ID with relaxed design constraint from [14], and HM-BICM-ID with 64-QAM-BF.

Full size image

Our novel approach of HM-BICM-ID with 64-QAM-BF outperforms the system from [14]. While the coded BER performance of the EL is kept nearly constant in the EFF zone the coded BER performance for the BL is improved significantly. The gain between the BL of the HM-BICM-ID with 64-QAM-BF and HM-BICM-ID from [14] is approximately 2 dB for a coded BER of 10−6. Furthermore, the coded BER performance of the novel approach is close to the BICM reference system with Gray labeling considering the EFF zone at the coded BER of 10−6. Beside this, the EL configuration compared to the BL configuration has still a significant gain of approximately 2.7 dB considering a coded BER of 10−6. These results are still valid for BER < 10−6. Of course, the better BER performance in the BL comes with the cost of a shifted waterfall region in both the BL and EL. For the BL, the novel HM-BICM-ID improves for E S /N 0  > 4 dB and BER < 10−3. Other regions of the coded BER and E S /N 0 are not of interest. Therefore, only in the EL the novel approach performs slightly worse because of the shifted waterfall region. For higher E S /N 0  > 4 dB the novel approach performs similar to [14]. Finally, we can conclude that the procedure presented in Section 3.3 gives the designer more freedom to improve specific layers without degrading the performance of another layer in the same way. Thus, the coded BER of HM-BICM-ID is improved by an optimized modulation scheme, i.e., 64-QAM-BF. Further, we can deduce that performance degradation is mainly caused by ILI and design restrictions coming from the side constraints of hierarchical systems. Finally, the novel HM-BICM-ID system outperforms the BICM reference system (coded BER curve marked by a black solid line) in the EL for values of E S /N 0  > 2 dB and in the BL between 4 dB < E S /N 0  < 7 dB. Only in the relevant range E S /N 0  > 7 dB a small loss of the coded BER performance in the BL can be observed although the EFF curve is below the BER curve of the BICM reference system.

4.3 EXIT Chart Analysis for HM-BICM-ID With 64-QAM-BF

In his work [22], ten Brink invented the so-called EXtrinsic Information Transfer (EXIT) charts. EXIT charts are very helpful to predict the behavior of iterative receiver systems, e.g., turbo coders or BICM-ID. While the EXIT charts of an EL modulation scheme can be investigated by state-of-the-art simulation tools known from literature, an investigation of the BL modulation scheme must be analyzed with slightly different tools. Therefore, we propose a novel EXIT chart simulation system to support BL analysis under EL influence as depicted in Fig. 11.

Figure 11
figure 11

Block diagram of an EXIT chart simulation, for the BL of a hierarchical receiver.

Full size image

As it can be seen, in Fig. 11 the bit stream of the binary random source is divided into two bit streams separating the BL and EL information for a two-layered hierarchical modulation scheme. From the two streams the symbols are modulated by the EL and then superposed by AWGN. The noisy symbols are then fed to the BL demodulator. To investigate the EXIT characteristic T BL of the BL modulation scheme the simulation is done over the range of the mutual information I a between 0 and 1. But only the BL-related source bits after de-multiplexing are used for further EXIT chart analysis. Therefore, those BL bits are then used to generate the a priori L-Values by the LLR generator. After BL demodulation the extrinsic information given by L-values is compared with the BL source information and the extrinsic mutual information I e is computed. Both I a and I e are used to describe the EXIT characteristic (or transfer function) T with the relation I e  = Τ(I a ). We simulated the EXIT characteristics of the BL and EL for our proposed 64-QAM-BF.

The EXIT charts are depicted in Fig. 12 for the BL (b) and the EL (a). The blue solid curves with squared markers represent the EXIT characteristics for the convolutional codes for the BL and EL. The green solid curves with diamond, asterisk, and triangle markers are the reference characteristics for typical 64-QAM modulation schemes known from literature. Especially the green curve with asterisk markers describes the characteristic of a hierarchical 8 × 8-PSK proposed in our previous work in [14]. The green curves with diamond markers are QPSK / 64-QAM modulation schemes with Gray labeling. They are often used in context of BICM and characterized by a typical flat characteristic. The main drawback of Gray labeling is that demodulation cannot improve even when very good a priori knowledge is present. The red solid curve with triangle markers is our novel and proposed 64-QAM-BF. The black solid line is the average trajectory for the investigated HM-BICM-ID. The trajectory iterates not exactly through the two characteristics of the channel decoder and the 64-QAM-BF demodulator. This has several reasons related to character of the hierarchical communication system and the introduced ILI. For example, the area below the BL demodulator characteristic in the EXIT chart in Fig. 12 (b) (red curve with triangle markers) is much smaller than the area under the reference curve of the QPSK Gray demodulator. This effect can be explained by the mismatch of the transmitter and receiver modulation scheme which causes ILI. As the demodulator expected only symbols affected by AWGN the resulting L-values does not match exactly and therefore a sub-optimal demodulator is used. A comparison of the area under the EXIT characteristics of QPSK Gray (BL) and 64-QAM-BF (EL) gives a hint to the influence of ILI. The values for the areas are given in Table 5.

Figure 12
figure 12

EXIT charts of the EL (a) and BL (b) for the hierarchical 64-QAM-BF.

Full size image
Table 5 d h 2 and area under EXIT characteristic (E S /N 0  = 8 dB) for some modulation schemes.
Full size table

In this context the novel 64-QAM-BF outperforms the 8 × 8-PSK. This observation matches with those results given by the coded BER performance from Section 4.2. Furthermore, it is obvious that HM-BICM-ID with 64-QAM-BF performs slightly better than the QPSK with Gray labeling. This can be explained, because both characteristics meet at the right border of the EXIT chart. Therefore, the asymptotic coded BER performance must be close together.

The results given in the EXIT chart of Fig. 12 (a) for the EL includes several effects to be discussed. The area under the EXIT characteristics of the 8 × 8-PSK and 64-QAM-BF is smaller compared to 64-QAM SSP (with best d h 2) or 64-QAM Gray. The reason is the reduced Euclidean Distance between the symbols in the I/Q plane. Furthermore, the 64-QAM-BF characteristic (red curve with triangle marker) ends up with a lower mutual information I e compared to 8 × 8-PSK (under perfect a priori knowledge; see right border in EXIT chart). This can be explained when comparing the HMMSED of the EL of both modulation schemes in Table 5. Nevertheless, the asymptotic coded BER performances of the corresponding hierarchical communication systems are the same. This can be explained, by the behavior of the trajectory which shows overshooting effects in the EXIT charts. On the one hand, it is due to the hierarchical interleaver arrangement which does not fully remove the correlation between bit positions in the modulation scheme and the bit positions related to specific elements of the generator polynomial of the channel code. On the other hand, BL and EL exchanges information between each other during the iteration process. Therefore, the EL benefits from the improvement of the BL optimization and consequently the coded BER performance must be understood in that way.

5 Conclusions

In our article we proposed the concept of HM-BICM-ID. To further improve hierarchical modulation schemes, we developed a novel procedure to move constellation points of a certain modulation scheme to a direction where critical parameters for a specific layer, i.e., the HMMSED and the BEP are optimized. For a given parametrization, e.g., the number of procedural iterations and the convergence behavior of each optimization step, we found a novel modulation scheme termed 64-QAM-BF. To demonstrate the coded BER performance of the novel modulation scheme we designed a HM-BICM-ID with 64-QAM-BF and compared it to those already known from literature. It has been observed that the novel procedure provides more design freedom to improve coded BER performance. Finally, it has been shown that the novel HM-BICM-ID outperforms a non-iterative BICM reference system for a wide range of E S /N 0 in both the BL and EL. The HM-BICM-ID system with the novel 64-QAM-BF scheme performs better compared to [14] when considering the BL and performs nearly the same when considering the EL. The results have been discussed in detail by additional EXIT chart analysis. In our future research work, we plan to modify the procedure to improve the convergence behavior and the balance between the different optimization criteria. This shall help to further fine tune modulation schemes. HM-BICM-ID uses BICM-ID in each layer. In future, we also plan to analyze asymmetric forward error correction and alternative receivers using powerful codes, e.g., turbo codes or LDPC codes. Thus, an unequal FEC for each layer shall help to balance between the different layers and to give additional degree of freedom to the designer. Another aspect of research in the future could be the analysis of the peak-to-average power ratio for the different hierarchical modulation schemes.