WO2012175636A1 - Two-dimensional iterative processing for dab receivers based on trellis-decomposition - Google Patents

Abstract

Aspects of the instant disclosure are directed towards communication apparatuses (100), and methods of using the apparatuses, having a front-end circuit (105) configured for communicating/receiving encoded symbols which are received over time and using adjacent OFDM sub-carriers. A logic circuit (110) is also included to characterize the encoded symbols using a trellis-based network of states and subtrellises. Further, first and second processing modules (115) are provided for determining a-posteriori subtrellis probabilities for the encoded symbols corresponding to the subtrellises and, therefrom, determining a-posteriori probabilities for the encoded symbols. An indicator circuit (120) is provided to be responsive to the first and second determinations for indicating a-posteriori symbol probabilities.

Description

TWO-DIMENSIONAL ITERATIVE PROCESSING FOR DAB RECEIVERS BASED

ON TRELLIS-DECOMPOSITION

This patent document claims benefit under 35 U.S.C. § 119 to U.S. Provisional

Patent Application Serial No. 61/499,840, entitled "Two-Dimensional Iterative Processing For Dab Receivers Based On Trellis-Decomposition" and filed on June 22, 2011; this patent document and the Appendices filed in the underlying provisional application, including the references cited therein, are fully incorporated herein by reference.

Commonly used classical DAB receivers can perform non-coherent two -symbol differential detection (2SDD) with soft decision Viterbi decoding. Non-coherent detection schemes like 2SDD are not optimal and can be improved by multi-symbol differential detection (MSDD), which is a maximum likelihood detection of a block of information symbols by observing a block of received symbols. For very large numbers of observations, the

performance of MSDD approaches the performance of ideal coherent detection of Differentially Encoded Quadratic Phase Shift Keying (DE-QPSK). Non-coherent MSDD can also be used if channel coding is applied in a non-iterative way.

If MSDD is combined with iterative (turbo) processing (parallel concatenated systems), serial concatenation needs to be improved to get an acceptable complexity. Results of serial concatenation of convolutional encoding followed by differential encoding with turbo-like decoding techniques (also referred to as Turbo-DPSK) have been encouraging. Turbo-DPSK was previously investigated for single-carrier transmissions on AWGN channels, as well as for time -varying channels. The main objective of these methods was to reduce the complexity of the inner decoder. The two main methods can be distinguished: first an explicit estimation of the channel phase and followed by coherent detection; or secondly by directly calculating the a- posteriori probabilities of the information symbols.

Various aspects of the instant disclosure are directed towards methods, apparatuses, and systems useful for communication. For instance, according to various embodiments, the instant disclosure is directed towards communication apparatuses that incorporate various circuitry, including a front-end circuit configured for communicating/receiving encoded symbols which are received over time and using adjacent Orthogonal frequency-division multiplexing (OFDM ) sub-carriers. A logic circuit is provided to characterize the encoded symbols using a trellis- based network of states and subtrellises. First and second processing modules determine a- posteriori subtrellis probabilities for the encoded symbols corresponding to the subtrellises and, therefrom, determining a-posteriori probabilities for the encoded symbols. An indicator circuit, responsive to the first and second determinations, is provided for indicating a-posteriori symbol probabilities.

The above discussion/summary is not intended to describe each embodiment or every implementation of the present disclosure. The figures and detailed description that follow also exemplify various embodiments.

Various example embodiments may be more completely understood in consideration of the following detailed description in connection with the accompanying drawings, in which:

FIG. 1 A shows an example block diagram of communication apparatuses, consistent with various example embodiments of the instant disclosure;

FIG. IB shows an example DAB convolution encoder, an interleaver, differential encoders, and a multi-carrier modulator, in accordance with aspects of the instant disclosure;

FIG. 2 shows an example of three services mapped onto consecutive Orthogonal frequency-division multiplexing (OFDM) symbols, in accordance with aspects of the instant disclosure;

FIG. 3 shows an example trellis representation of certain states and differentially encoded symbols in an incoherent case, in accordance with aspects of the instant disclosure;

FIG. 4 shows an example bit-error performance for LLRs for different trellis-lengths, in accordance with aspects of the instant disclosure;

FIG. 5 shows another example bit-error performance for LLRs and approximated LLRs for different trellis-lengths, in accordance with aspects of the instant disclosure;

FIG. 6 shows an example two-dimensional block of symbols out of an OFDM stream, in accordance with aspects of the instant disclosure;

FIG. 7 shows an example bit-error performance for LLRs for a decomposed multi-carrier trellis, in accordance with aspects of the instant disclosure;

FIG. 8 shows an example embodiment of a receiver, in accordance with aspects of the instant disclosure; FIG. 9 shows an example bit-error performance of a Peleg method for trellis length, in accordance with aspects of the instant disclosure;

FIG. 10 shows another example bit-error performance of a Peleg method for trellis length, in accordance with aspects of the instant disclosure;

FIG. 1 1 shows an example bit-error performance with a dominant sub-trellis approach, in accordance with aspects of the instant disclosure;

FIG. 12 shows an example bit-error performance for a multicarrier case, in accordance with aspects of the instant disclosure;

FIG. 13 shows another example bit-error performance with a dominant sub-trellis approach, in accordance with aspects of the instant disclosure;

FIG. 14 shows Bit-error Performance (BER) for LLRs, in accordance with aspects of the instant disclosure;

FIG. 15 shows an example BER-performance with perfect knowledge and estimates of a channel phase, in accordance with aspects of the instant disclosure;

FIG. 1 6 shows an example detail of the BER-performance with perfect know ledge and estimates of the channel phase in accordance with aspects of the instant disclosure;

FIG. 17 shows an example BER-performance with perfect knowledge and estimates of the channel gain for different values of M(N +1) in accordance with aspects of the instant disclosure;

FIG. 18 shows an example detail of BER-performance with perfect knowledge and estimates of the channel gain for di ferent values of M(N +1) in accordance with aspects of the instant disclosure;

FIG. 19 shows an example block-diagram of the practically realized improved DAB receiver with 2D block-based joint and iterative demodulation and decoding, consistent with various aspects of the instant disclosure;

FIG. 20 shows an example of real-time bench-test results of the iterative decoding gain for DAB-transmissions in Mode -I on the AWGN channel, consistent with v arious aspects of the instant disclosure; and

FIG. 21 shows example real-time bench-test results of the iterative decoding gain for DAB-transmissions in Mode -I on the TU-6 channel, consistent w ith various aspects of the instant disclosure. While the invention is amenable to various modifications and alternative forms, specifics thereof have been shown by way of example in the drawings and will be described in detail. It should be understood, however, that the intention is not to limit the invention to the particular embodiments described. On the contrary, the intention is to cover all modifications, equivalents, and alternatives falling within the scope of the invention including aspects defined in the claims. In addition, the term "example" as used throughout this application is only by way of illustration, and not limitation.

Aspects of the present invention are believed to be applicable to a variety of different types of devices, systems and arrangements involving communication. While the present invention is not necessarily so limited, various aspects of the invention may be appreciated through a discussion of examples using this context.

Various example embodiments are directed to communication apparatuses. Included in the communication apparatuses is a front-end circuit configured and arranged for

communicating/receiving encoded symbols which are received over time and using adjacent Orthogonal frequency-division multiplexing (OFDM) sub-carriers. In some embodiments, the OFDM subcarrier channels are modulated via DE-QPSK. Further, the encoded symbols are in 2D-data blocks which include subsequent OFDM symbols and adjacent OFDM subcarriers. A logic circuit is also in the communication apparatuses to characterize the encoded symbols using a trellis-based network of states and subtrellises. Further, first and second processing modules are provided for determining a-posteriori sub trellis probabilities for the encoded symbols corresponding to the subtrellises and, therefrom, determining a-posteriori probabilities for the encoded symbols. An indicator circuit, responsive to the first and second determinations, indicates a-posteriori symbol probabilities.

In certain specific embodiments, the front-end circuit is further configured and arranged for receiving OFDM subcarrier channels. Additionally, the logic circuit and the processing modules can also use a trellis decomposition method to estimate an unknown channel phase that relates to the subtrellises, in certain specific embodiments. In those embodiments, the logic circuit and the processing modules are further configured and arranged to search for a dominant one of the subtrellises. In other instances where ,the logic circuit and the processing modules are used a trellis decomposition method to estimate an unknown channel phase that relates to the subtrellises, the logic circuit and the processing modules can search for a dominant one of the subtrellises and, in response thereto, use forward/backward processing for demodulation.

Alternatively, the logic circuit and the processing modules, can search for a dominant one of the subtrellises and, in response thereto, use iterative processing on at least one of the subtrellises. Further, the logic circuit and the processing modules in other embodiments can also search for a dominant one of the subtrellises. The logic circuit and the processing modules can also be further configured and arranged to iteratively repeat a search for a dominant one of the subtrellises and choose a different dominant subtrellises for at least two of the iterations. In certain embodiments of communication apparatuses, the first and second processing modules operate on received data via logic that assumes channel-phase and gain are fixed for certain adjacent subcarriers and consecutive symbols. Additionally, in other embodiments, the first and second processing modules operate on received data via logic that assumes channel-phase or gain are fixed for certain adjacent subcarriers and consecutive symbols.

Aspects of the instant disclosure are also directed towards methods, which include communicating encoded symbols which are received over time and using adjacent OFDM sub- carriers. The encoded symbols are characterized using a trellis-based network of states and subtrellises. The method follows by determining a-posteriori subtrellis probabilities for the encoded symbols corresponding to the subtrellises, and from those determined a-posteriori subtrellis probabilities, determining a-posteriori probabilities for the encoded symbols. In response to determined a-posteriori subtrellis probabilities and a-posteriori probabilities for the encoded symbols, a-posteriori symbol probabilities are indicated. In certain embodiments, using adjacent OFDM sub-carriers also includes receiving ORDM subcarrier channels.

Additionally, the ODFM subcarrier channels can be modulated using Differentially Encoded Quadratic Phase Shift Keying (DE-QPSK). Methods of the instant disclosure can also include a step of using a trellis decomposition method to estimate an unknown channel phase that relates to the subtrellises. Further, certain embodiments include an additional step of iteratively repeating a search for a dominant one of the subtrellises, and choosing a different dominant subtrellises for at least two of the iterations.

Moreover, an additional step of searching for a dominant one of the subtrellises can be included in certain methods of the instant disclosure. In those instances, certain more specific embodiments include a step of using forward/backward processing for demodulation in response to searching for a dominant one of the subrellises.

The instant disclosure also includes methods that involve receiving encoded symbols which are received over time and using adjacent OFDM sub-carrier channels. The encoded signals are characterized using a trellis-based network of states and subtrellises, and therefrom, a-posteriori subtrellis probabilities for the encoded symbols corresponding to the subtrellises are determined. From the determined a-posteriori subtrellis probabilities, methods of the instnat disclosure will determine a-posteriori probabilities for the encoded symbols, and in response to determined a-posteriori subtrellis probabilities and a-posteriori probabilities for the encoded symbols, a search for a dominant one of the subtrellises is iteratively repeated, and a-posteriori symbol probabilities are indicated. In certain specific embodiments, iteratively repeating a search for a dominant one of the subtrellises further includes choosing a different dominant subtrellises for at least two of the iterations. Additionally, in other embodiments, certain methods include a step of using a trellis decomposition method to estimate an unknown channel phase that relates to the subtrellises.

Certain aspects of 2D phase-estimate methods can be extended from the results of the multi-carrier case. Additionally, the hard-decision approach can be improved by considering soft-decision. Other such approaches rely on pilot symbols, which are not present in DAB- transmission. The proposed combination of the iterative scheme of on 2D phase-estimates and a fast converging blind channel estimator based on higher order asymmetrical modulation schemes are not used within a DAB-transmission. To obtain a-posteriori probabilities of information symbols in a 2D setting, iterative decoding schemes were considered for multi- carrier modulation with the soft-output Viterbi algorithm (SOVA). The SOVA was used for differential detection as well as for decoding of the convolutional code. Additionally, the SOVA used in the coherent setting an estimate of the phase based on a block of three by three received symbols, which are adjacent in time and frequency direction. It was proposed to use only the current received symbol to obtain a symbol-metric for the SOVA inner-decoder, ignoring the differential encoding. In the incoherent case, a transition-metric was used for the SOVA inner-decoder based on the current and previous received symbol. These a-posteriori detection schemes do not result in the ideal performance. To reduce complexity, the phase of the desired signal can be discretized into several equispaced values, but do not allow for "side-step" transitions to track small channel phase variations. In an efficient manner, the a-posteriori probabilities of the information symbols can be calculated using the BCJR-algorithm in a 2D setting. In this contribution, 2D-blocks and trellis decomposition can be considered. The 2D-blocks consist of a number of subsequent OFDM symbols and a number of adjacent sub-carriers. Channel coherence -time is typically limited to a small number of OFDM symbols, additionally, DAB-transmissions use time- multiplexing of services, which limits the number of OFDM symbols in a codeword. Therefore, the trellis-decomposition methods allow for estimation of the unknown channel-phase efficiently because this phase is related to sub-trellises of which the a-posteriori probabilities can be determined. With these probabilities a dominant sub-trellis can be chosen, which results in a significant complexity reduction. In addition, trellis-decomposition also allows for processing of, in an efficient way, several sub-carriers simultaneously and provides some extra gain compared to processing one subcarrier at a time.

The instant disclosure focuses on two-dimensional (2D) processing (in both the frequency- and time-domain), although it is not limited to only two-dimensional processing. The instant disclosure is additionally directed toward methods based on iteratively demodulating and decoding blocks of received symbols in a DAB-transmission stream.

Aspects of the instant disclosure are directed towards communication apparatuses incorporating various circuitry, including a front-end circuit configured for

communicating/receiving encoded symbols which are received over time and using adjacent Orthogonal frequency-division multiplexing (OFDM) sub-carriers. A logic circuit is provided to characterize the encoded symbols using a trellis-based network of states and subtrellises. First and second processing modules determine a-posteriori subtrellis probabilities for the encoded symbols corresponding to the subtrellises and, therefrom, determining a-posteriori probabilities for the encoded symbols. An indicator circuit, responsive to the first and second determinations, is provided for indicating a-posteriori symbol probabilities.

FIG. 1 A shows an example block diagram of communication apparatuses, consistent with various embodiments of the instant disclosure. For instance, the communications apparatus 100 includes a front-end circuit 105 for communicating/receiving encoded symbols which are received over time and using adjacent OFDM sub-carriers. A logic circuit 110 is also provided with the communication apparatus 100 to characterize the encoded symbols using a trellis-based network of states and subtrellises. First and second processing modules 115 determine a-posteriori subtrellis probabilities for the encoded symbols corresponding to the subtrellises and, therefrom, determine a-posteriori probabilities for the encoded symbols. An indicator circuit 120, responsive to the first and second determinations by the first and second processing modules 115, indicates a-posteriori symbol probabilities. In some embodiments, the OFDM subcarrier channels, used by the front-end circuit 105, are modulated via DE-QPSK. Further, the encoded symbols are in 2D-data blocks which include subsequent OFDM symbols and adjacent OFDM subcarriers.

The logic circuit 1 10 and the processing modules 1 15 can also use a trellis

decomposition method to estimate an unknown channel phase that relates to the subtrellises, in certain specific embodiments. In those embodiments, the logic circuit 1 10 and the processing modules 1 15 are further configured and arranged to search for a dominant one of the subtrellises. In other instances, the logic circuit 1 10 and the processing modules 1 15 can search for a dominant one of the subtrellises and, in response thereto, use forward/backward processing for demodulation. In certain specific embodiments, the front-end circuit 105 is further configured and arranged for receiving OFDM subcarrier channels.

Various embodiments of the communication apparatus 100 have a logic circuit 1 10 and processing modules 115 that also search for a dominant one of the subtrellises and, in response thereto, use iterative processing on at least one of the subtrellises. Additionally, the logic circuit 110 and the processing modules 115 iteratively repeat a search for a dominant one of the subtrellises, and choose different dominant subtrellises for at least two of the iterations. In other embodiments, the logic circuit 1 10 and the processing modules 1 15 search for a dominant one of the subtrellises (not necessarily using iterative processing). In certain embodiments of communication apparatus 100, the first and second processing modules 1 15 operate on received data via logic that assumes channel-phase and gain are fixed for certain adjacent subcarriers and consecutive symbols. The first and second processing modules 115 can also, in various embodiments, operate on received data via logic that assumes channel-phase or gain are fixed for certain adjacent subcarriers and consecutive symbols.

The following discussion relates to experimental embodiments and a detailed mathematical derivation of the embodiments discussed above. Terrestrial digital broadcasting systems like DAB, DAB+ and T-DMB, all members of the "DAB-family", include a combination of convolutional coding (CC), interleaving, - - DE-

QPSK modulation followed by OFDM, as can be seen in FIG. IB. This is accomplished through the DAB convolutional encoder 150, an interleaver 155, differential encoders 160, and a multi-carrier modulator 165. Time multiplexing of the transmitted services allows the receiver to perform per service symbol processing, as is shown in FIG. 2, hence the receiver can decode a certain service without having to process the OFDM symbols that do not correspond to this service. As shown in FIG. 2, three services 200 are mapped onto consecutive OFDM symbols 210 Note that there is overlap between the services because differential modulation is used.

The convolutional code that is used within DAB has basic code-rate R_c = ¼ , constraint length K = 7 and generator polynomials go = 133, gi = 171, g₂ = 145, and g₃ = 133. Larger code -rates can be obtained via puncturing of the mother code. The time and frequency interleavers in DAB respectively perform bit-pair interleaving. As a result, the code -bits leaving the convolutional encoder can be permuted and partitioned over the sub-carriers of a number of subsequent OFDM-symbols (in subsequent frames). The bits for each sub-carrier can be grouped in pairs, and each of such pair is mapped onto a phase (difference) that therefore can assume four different values. The mapping that is used here is based on the Gray principle (labels that correspond to adjacent phase differences differ only in a single bit-position).

For each sub-carrier, - - DE-QPSK modulation can be applied. A sequence

b = (bi, b₂, .. . , bN) consisting of N symbols (phase differences) b_n for n = [1, 2, ... , N] carries the information that is to be transmitted via this sub-carrier. The symbols b_n, n = [1 , 2, ... , N] assume values in the (offset) alphabet B = { (^{ρπΙ2+πΙ4}),ρ = 0, 1 , 2, 3} . The transmitted sequence s = (so, si, ... , S_JV) of length N + 1 follows from b by applying differential phase modulation, i.e., s_n = b„s„-i, for n = 1, 2, ... , N, (1) where for the first symbol s₀ = 1.

OFDM in DAB can be realized using a B-point complex Inverse Fast Fourier Transform (IFFT), where B is 256, 512, 1024, or 2048. To compute the «-th time-domain OFDM-symbol s_n = (s_{l n}, s_2in, ... , s_{B n}), the following is determined: ₁ . (t-i)(m-i)

n = ^∑m=i ^{eJ π B} > for t = 1 , 2, ... , B (2) and s_m_„ is the «-th differentially encoded symbol corresponding to the m^th sub-carrier, or equivalently the m element in the «-th frequency-domain OFDM-symbol, as seen in FIG. 1. Note, that the Inverse Fast Fourier Transform (IFFT) is a computational efficient IDFT for values of B that are powers of 2. To prevent Inter-Symbol Interference (ISI) resulting from multi-path reception, a cyclic prefix of length _φ is added to the sequence s_n. This leads to the sequence s_n = s_B__{L +1 n}, ... s_{B n}, s_{l n}, s_{2 n},—§_{B n} that is finally transmitted.

It is assumed that the channel is slowly varying with an impulse response shorter than the cyclic-prefix length. Moreover, it is assumed that the channel coherence-bandwidth and coherence-time span multiple OFDM-subcarriers and multiple OFDM-symbols. Therefore, the channel-phase and gain might be assumed to be fixed for a number of adjacent subcarriers and consecutive symbols. This is the assumption on which we base our investigations. The channel phase and gain are assumed constant (yet unknown to the receiver) over a 2D-block of symbols, as shown in FIG. 6. FIG. 6 shows an example 2D-block of symbols out of an OFDM stream. M aligned sequences of N + 1 subsequent symbols are of specific interest, where each such sequence corresponds to one of M adjacent sub-carriers.

The receiver, in the case of perfect synchronization, removes the (received version of the) cyclic prefix, and then applies a B-points complex FFT on the time-domain received sequence r_n = (f_{l n}, f_2jTl, ... , f_{B n}), which results in the B received symbols r_m,n = -^∑^B _t=1 f_tine-^]2n B , for m = l , 2,... , B. (3)

OFDM reception can be regarded as parallel matched- filtering corresponding to B complex orthogonal waveforms, one for each subcarrier. This results in a channel model, holding for a 2D-block of symbols, that is given by

for some subsequent values of n and m, where the channel gain |h| and phase φ are unknown to the receiver. The sequence s that is transmitted via a certain subcarrier is now observed by the receiver as sequence r =(r₀, ri,... , r_N). Note that compared to the previous example, the subscript m has been dropped. Since it is relatively easy to estimate the channel gain, it is assumed here that it is perfectly known to the receiver, and to ease the analysis, it can be taken to be 1. The received sequence now relates to the transmitted sequence s as follows:

r_n = e³'^ps_n + n_n , for n = 0, 1 , · · · , N, where we assume that n„ is circularly symmetric complex Gaussian with variance σ² per component. It is assumed that the random channel-phase φ is real-valued and uniform over [0, 2π). This channel phase is fixed over all N + 1 transmissions, and unknown to the receiver.

Accepting a small performance loss, it can be assumed that the channel phase is discrete, and uniform over 32 levels which are uniformly spaced over [0, 2π), hence

Pr{( > = ,τ/; Ki | = 1/32, for I = 0, 1 , 2,^■

(6)

The following discussion focuses on the situation of a uniformly chosen channel phase in a single sub-carrier. Subsequently, the setting, in which a uniformly chosen channel phase is moreover constant over a number of (adjacent) sub-carriers, can be investigated.

The - -DE-QPSK modulation, which is performed in each of the sub-carriers, is equivalent to DE-QPSK. To see this, we define for n = 1 , 2, ... , N:

and for n = 0, 1 , ... , N ft

=i

(8)

It now follows that ^tt« ^{€ A} = i^^/2 ^ P = - ½ 3) , x₀ = L _{^ and} It

Additionally, ^x« ^fc -^*¾ for all « = 0, 1 , ... , N, and that w„, just like n„, is circularly symmetric complex Gaussian with variance ⁽r¹ per component. Moreover since b is Gray-coded with respect to the interleaved code -bits, so is a = (ai, a₂, ... , a_N), and therefore, the focus will be on DE-QPSK.

The following discussion relates to the single-carrier case. For some single sub-carrier, discussion focuses on DE-QPSK modulation with incoherent reception. Based on trellis decoding techniques, the a-posteriori symbol probabilities are determined under the assumption that the (quantized) channel phase is uniform and unknown to the receiver. It is also assumed that the transmitted symbols are independent of each other and uniform.

The derivations focus on non-iterative detection. It should be noted that that if z„ = ^,:c_ne^j0j [_s defined for n = 0, 1 , ... , N, then, since Xo = 1 and φ is uniform over

i -f/16, 1 = 0. 1 ,^■ - · . 31 } , it follows that

P i

1) I = 1 /32, for / = 0, 1 , ... , 31 , (10)

and ^zn fe

= 0, 1 , - · · , 31 } . Moreover, for n = 1 , 2, ... , N, ti Zfi— i, where

The variables z„ for n = 0, 1 , ..., N can now be regarded as states in a trellis and the independent uniformly distributed (iud) symbols ai, a₂, ... , a_N correspond to transitions between states. The resulting graphical representation of this trellis can be found in Fig. 3. If the standard BCJR algorithm is used for computing the a-posteriori symbol probabilities in the trellis in Fig. 3, 32 x 4 multiplications in the forward pass are completed, 32 x 4 multiplications in the backward pass, and 4 x 32 x 2 multiplications and 4 normalizations in the combination pass, per trellis section, if the a priori probabilities are all equal. In total, this is 512 multiplications and 4 normalizations per trellis section. Multiplications and normalizations are focused upon since additions have a smaller complexity than multiplications and

normalizations. In the log-domain, multiplications and normalizations are replaced by additions, and additions are typically approximated by maximizations. This would more or less suggest to consider multiplications, normalizations, as well as additions, but for reasons of simplicity we neglect the additions here. An important observation is that the trellis can be seen to consist of eight sub-trellises To, Tj, ... , Tj, that are not connected to each other.

Sub-trellis T_s consists of states *«€ Z, = {e-'^lir-^{/ 16} J = s + p,p = 0, 1, 2, 3}, for s = 0, I , ... , 7. Fig. 3 shows the entire sub-trellis To, and the first section of sub-trellis T\ and of sub-trellis T₇. The trellis representation of the states z₀, zi, · · · , z_N and the differentially encoded symbols ai, a₂, · ..., a_N in the incoherent case is shown in FIG.3. An edge between two subsequent states indicates that a transition between these states is possible. Note that the trellis can be decomposed into eight unconnected sub-trellises.

Note that for the likelihood In t ^»»_* ) corresponding to some state z_n€ Z for n = 0, 1 , ... , N in the trellis T or in a sub-trellis we can write that

Computing the a-posteriori symbol probabilities * ^β*}Λ>< fi* ^{" * "} < ϊ , for all n = 1, 2, ... , N and all values a„ e A is demonstrated by the following. The resulting a-posteriori probability is a convex combination of the a-posteriori probabilities corresponding to the eight sub-trellises. Computing the a-posteriori probabilities for each sub-trellis is simple and can be done without performing the BCJR algorithm. The coefficients of the convex combination do not depend on the trellis section index n, and are quite easy to determine as will be shown here. In our forward pass, focus is on the sub-trellis T_s, for some s e {0, 1 , ... , 7} . For that sub-trellis all the o's in that sub-trellis first are calculated first. Starting from ^£|ot^*¾) = 1/32 for all ¾ e 2, the tt's can be calculated recursively from

for n = 1 , 2, ... , N and ^::« € 2_a. χ¾_ε notation z, a→ z' stands for all states z and symbols a that lead to next state z'. Lemma 1: If for n = 0, 1 , ... , N we define ^^<

^_¾€2, χ ¾) _then.

i T T ^ , ...

O-n (Zn ) ⁼ J_.1 "4')'

i=0 for « = 0, 1, ... , N and *n€ 2* .

This proof is based on induction. Clearly for n = 0 the result holds. Now assume that αη- η- ι ) = -if HLte for ²« I ^Z% then from (13), (15) is obtained

for all * ^€ ^* .

Also, in the backward pass, first only sub-trellis T_s for some s is focused upon. In this sub-trellis the Ts are computed. Taking ¾v(¾v) = (¾v) for *" ^fc *-·^« all other 5's can be computed from

where again « = 0, 1 , .. , N - 1 and ^€: . Based on definition ^kf,ini = ^ ¾ ÷ 2, J^{* n} t^¾ ) for all « = 0, 1, ... , N:

II

i=.n + \. (17) for « = 0, 1 , ... , N and all %n fc 2_S _

Again, this proof is based on induction. Note first that for n = Nthe result holds. Now assume that »(·-:■» 4-1.) = ": ' fl ,„ , 2 f_or

Then

for all ¾€ Z_s.

To determine the a-posteriori symbol probability for symbol value °n t A the joint probability and density are computed ^ _r ⁽*n 1 { ¾» - 1 ) 7^η- 1 ί ~ «'· - J ) Ai (

If consider the "middle" term in (19) is considered, then

AK_a{n-i)K

(20)

From this, it is concluded that

with

(22) Now observing that

(24) for ^s fe (Ο' ¹ » · ^{* *} » ⁷) and <½ fc ^the following can be written

vi' = ∑ Pr{sly} Pr{a„fy. *}.

=0,1 ,- (25) The right-hand side of this equation can be interpreted as a convex combination of a- posteriori symbol probabilities Prf^t½ |y. sj_{^ one} f_or each sub-trellis, where the weighting- coefficients are the a-posteriori sub-trellis probabilities ' ^{1 1} " An a-posteriori sub-trellis probability is the conditional probability that the discrete channel phase modulo 8 equals s for some s = 0, 1 , ... , 7 given y.

The demodulator that operates according to (25) has three tasks. First, the eight weighting coefficients (23) are to be computed. Then for each of the eight sub-trellises for all symbol values a„ e A and all n e {1 , 2, ... ,N} the a-posteriori symbol probabilities are computed. Finally the weighting (25) has to be done. Computing the weighting coefficient requires for each sub-trellis s e {0,1 , ... ,7} requires the computation of the factors K_s(«) for n = 0, 1 , . . . , N. These factors should then be multiplied and normalized to form Pr{s|y} . For these computations, 8 multiplications per trellis section are needed. Computing the a-posteriori symbol probabilities Pr{a_n|y, s} can be done efficiently by applying a technique to each sub- trellis developed in Colavolpe, G., "Classical coherent receivers for differentially encoded M- PSK are optimal," Communications Letters, IEEE, vol. I, no. 4, pp. 21 1 - 213, Apr. 2004, which is fully incorporated by reference. Each such a-posteriori symbol probability is based on only two received symbols y„-_\ andy_n as is shown in (24). This avoids the use of the BCJR method in full generality and leads to significant complexity reductions, i.e., only 8x4x4 = 128 multiplications and 8x4 = 32 normalizations are needed per trellis section. The weighting operation requires 8 x 4 = 32 multiplications, and therefore in total this approach leads 8 + 128 + 32 = 168 multiplications and 32 normalizations, which is considerably less than what we need for full BCJR.

Equation (25) shows how the exact a-posteriori symbol probabilities can be determined. If the a-posteriori sub-trellis probabilities are such that one of the probabilities dominates the other ones then weighting (25) can be approximated by

Pr{only} ¾ Pr{a_n|y. *} , _with ^s ar maxPr{f|y}. (26)

Observe that this approach involves the computations of the a-posteriori symbol probabilities, as described in (24), only for the dominant sub-trellis s. This requires 4 x 4 = 16 multiplications and 4 normalizations only per trellis section. Together with the computation of the weighting coefficients 8+16 = 24 multiplications and 4 normalizations are necessary.

Therefore this reduces the number of multiplications with respect to full weighting by a factor of seven.

In simulations, the de-facto industry standard (R_c = ½) convolutional code is used with generator polynomials go = 133 and g = 171 , which is equal to the convolutional code with puncturing index PI = 8. The DAB, DAB+ and T-DMB bit-reversal time interleaver and block frequency interleaver is modeled by a bit-wise uniform block interleaver generated for each simulated code block of bits, hence, any permutation of the coded bits is a permissible interleaver and is selected with equal probability conform. The demodulator calculates, for each OFDM-subcarrier, the a-posteriori probability given by (25) for N +1 = 2, 4, 8 and 32. The demodulator is followed by a convolutional decoder which needs, as input, soft-decision information about the coded bits. Now, it follows from Gray mapping, i.e.,

1, 1», I 00 01 11 10 that the desired metrics related to transmission n, i.e. the log-likelihood ratios (LLRs) can be expressed as

with symbol metric wi o) = 1^'. ί r i_w = **Jy}) , ^28)

and where *¾ corresponds to bit bi and to bit b₂.

Fig. 4 shows the Bit-Error Rate (BER) performance with so-called ideal LLRs for a decompose r trellis-length N+l = 2, 4, 8, and 32. On the horizontal axis is the signal-to noise ratio 2σ*. The demodulator operates according to (25). Fig. 4 shows an example bit-error performance for LLRs computed as in (25), i.e., ideal LLRs, for different trellis-lengths, in accordance with various aspects of the instant disclosure.

The performance of this demodulator is compared with that of two well-known procedures described in the literature. Firstly, to "classical" DQPSK, i.e. two-symbol differential detection (2SDD). This leads to a-posteriori symbol probabilities as in (9), i.e., to

1

Pr{s„ji/_Tl, y„_i } x Jo [— + Vn- i I ) for a- e A ,_ _n.

' (29) where I₀( · ) is the zeroth order modified Bessel function of the first kind. Secondly, the results can be compared to coherently detected DE-QPSK. The received sequence is assumed to be perfectly de -rotated, i.e., >^{' =} 7^{e i '} . Then the a-poste iori symbol probabilities arc given by

Pr{f½ |y} oc T exp ( _j ¾ { - ΰ^ + Ife^ t) } , for a_fl€ A. (30)

Note that (30) is similar to (24) for s = 0. The simulation results, which are shown in Fig. 4, demonstrate that the BER

performance curves of 2SDD and trellis length N + 1 = 2 are practically identical as expected. Moreover, the coherent-detection curve and the curve for very large trellis-sizes (N→∞) are very close. The small performance loss is due to discretizing the channel-phase with 32 levels. Furthermore, Fig. 4 shows that: (a) larger values of N + 1 result in performance closer to the coherent-detection performance, and (b) for N + 1 = 32, ideally computed LLRs for a decomposed trellis perform quite close to coherent detection, i.e., the difference in signal-to- noise ratio (Eb No) is less than 0.15 dB at a BER of 10 ^{4 '}.

FIG. 5 shows an example bit-error performance for LLRs computed as in (25), i.e., ideal LLRs, and approximated LLRs computed as in (26), for different trellis -lengths of the dominant sub-trellis approach. Trellis-length N + 1 = 2, 8, and 32 are compared to the difference in performance between ideal LLRs based on the a-posteriori probabilities given by (25) and the approximated LLRs based on the dominant-sub-trellis a-posteriori probabilities specified in (26). It can be seen from Fig. 5 that: (a) for larger N+l , the difference between the exact and approximated LLRs becomes smaller, and (b) for N +1 = 32, the difference between the ideal LLRs and the approximated LLRs, is less than 0.1 dB.

These simulations demonstrate that for trellis length N + 1 = 32, the ideal LLRs and the approximated LLRs have a performance quite close to that of coherent detection. The difference in signal-to-noise ratio is less than 0.25 dB at a BER of 10^-4 for the dominant-trellis approach. Therefore, focusing on a BER of 10 ⁴ for obtaining an acceptable performance, with single sub- carrier transmission, a trellis -length of N + 1 > 32 is needed.

With a trellis-length of N + 1 > 32 received symbols, the channel coherence-time needs to be in the order of T_c ~ 32Γ₈, where T_s is the OFDM symbol time. This imposes quite a strong restriction on the time -varying behavior of the channel. In practice the channel may not be coherent so long, and therefore focusing on trellis-length N + 1 = 32 might not be realistic. This effect is discussed further below with reference to a typical urban channel. There is a second reason for arguing that large values of N are undesirable. DAB-system support, for complexity reduction, per service symbol processing. In such services, typically, at most N + 1 < 4 subsequent OFDM symbols are contained in a single convolutionally encoded word, see FIG. 2, and this does not match to processing more than four OFDM symbols in a demodulation trellis. After having concluded that cannot be N too large, it makes sense to investigate the possibility of using a number of (adjacent) sub-carriers to jointly determine the a-posteriori symbol probabilities for the corresponding DE-QPSK streams. Instead of using a single trellis with length N + 1 = 32, we could find out whether a similar performance can be obtained with a 2D-block of M = 8 trellises of length N + 1 = 4 corresponding to adjacent sub-carrier as shown in FIG. IB. This is discussed in further detail below.

As demonstrated above, the trellis-length N + 1 needs to be larger as 32. However, this may not always be true. In order to determine whether or not this is true, the question of jointly decoding a block (2D) of received symbols is investigated.

It is assumed that in each subcarrier m = 1 , 2, ... , M, a sequence a_m = (a_m, , a_mj2, ... , a_m,_N) is conveyed using differential encoding. For the components of the transmitted sequence x_m = (¾o, Xm,u■■■ , ΧΜ,Ν) the following can be written:

and x_m,o = 1 . The channel phase is assumed to be constant over the block of symbols, therefore jftn — c^{j i*'}*.i:_Fn.„ + ii n,« , (32) where φ e |ΐ^*~/ 1β, I = 0, 1· ^{* *} · .31 } and uniform just as before, and the noise variables w_m.„ are circularly complex Gaussian with variance σ² per component. The output sequence

corresponding to sub-carrier m is denoted by y_m = (y_m j, y_m , ..., y_m,N} . Just like in the single carrier case, the a-posteriori sub-trellis probabilities can be determined.

¹ '^{' 11 "}Jj '^; -\. ^" jl-L-^{' , ;} '^{i y} ' 1

> . I '' - ."^■ · t '. y ,, - · - . ^■ p|y (33)

Where Ρ } = 1/8 for s = 0, 1 , ... , 7 and

(34) where ^~™ ^ _{i i} 2, li^ ^d ^n ) '. Note that for the likelihood corresponding to some state z„ for n = 0, 1 , ... ,N in the trellis T or in a sub-trellis, the following can be written

Now the a-posteriori symbol probability for a_m,_n ε A can be written as

7

= * Pr{sfy j , y 2, · ^» · , y_M } I | n ._r,,_fI[yfjt , s} .

s=0 (36) where

and

,i im,n- 1 1, *-n~ - S ^Qt

Pr{a_m<n|y_TFl . s}

4J£^" _m,,{n - i)if,_riM,{nJ _^ (38) for 5 e {0, 1 , ... , 7} and _m>„ e _^4.

This suggests that the demodulator first determines the a-posteriori sub-trellis probabilities (weighting coefficients) using (37), for which the first 8 x M x (N + l) K- factors must computed. Using the weighting coefficients, the convex combination in (36) then leads to the a-posteriori symbol probabilities. Finding the a-posteriori symbol probabilities

s} again can be done using the Colavolpe method for each sub-carrier and for each sub-trellis, where again such an a-posteriori symbol probability is based on only the two received symbols y_m,_n-i and y_m,_n s is shown in (38). Again the BCJR method in full generality is not needed, and the number of required multiplications and normalizations per trellis section are the same as in the single carrier case.

Equation (36) shows how the exact a-posteriori symbol probabilities can be determined. Just like in the single-carrier case, if the a-posteriori sub-trellis probabilities are such that one of the probabilities dominates the other ones then weighting (36) can be approximated as follows:

Pr a_m> | i . |M } ¾ i «m.« |y . »} ,

with i = »i¾ inax Prf sjy^ ys . '^■■ . yif } .

Again this approach involves the computations of the a-posteriori symbol probabilities only for the dominant subtrellis s. The resulting number of multiplications and normalizations per trellis section is the same as for the single carrier case.

As discussed above, the single sub-carrier approach was simulated. As in that case, simulations corresponding to the multi-carrier method are discussed by investigating the coded

BER versus the signal-to-noise ratio ^ &,' ^J¾"o — a r . The BER performance for the ideal LLRs, based on a-posteriori probabilities computed as in (36) is shown in FIG. 7 with a fixed block- size of M(N + 1) = 16. This fixed block-size is realized by the parameter pair values (M, N + 1) = (1 , 16), (2, 8), (4, 4) and (8, 2). Fig. 7 shows an example bit-error performance with ideal LLRs for a decomposed multicarrier trellis for different values of M and Nbut with a fixed block-size of M(N + 1) = 16.

The detector operates according to (36). The performance of 2SDD and coherently detected DE-QPSK are shown as reference curves. From Fig. 7, it can be observed that, a 2D decomposition with a shortest possible trellis-length ofN + 1 = 2 and = 8 adjacent sub- carriers performs identical to the largest trellis length N+ 1 = 16 and M = 1 sub-carrier, i.e., the single-carrier case. Intermediate cases also have an identical performance. The results of the dominant sub-trellis approach for the multi-carrier case here are not specifically shown because these results are identical to the corresponding results for the single- carrier case shown in FIG. 5.

Investigations for the non-iterative 2D-case show very close performance to the coherent detection of DE-QPSK even for small values of the trellis length N + 1 , by processing simultaneously several sub-carriers.

The following discussion investigates consider iterative decoding procedures. Iterative techniques could significantly increase the performance of the demodulation procedures of DE- QPSK streams. An iterative approach is specialized to DAB system, and solves a problem connected to the, in practice quite small, length of the trellises for each subcarrier, by turning to 2D-blocks for iterative demodulation.

Iterative decoding procedures for DAB-like systems are investigated, which are based on convolutional encoding, interleaving, and DE-QPSK modulation. If a DE-QPSK modulation is considered as the inner coding method and convolutional encoding as the outer code, then techniques developed for serially concatenated coding systems can be applied, as shown in FIG. 8. FIG. 8 shows a structure of an example receiver, consistent with aspects of the instant disclosure. The receiver includes a inner code soft-input-soft-output (SISO) 805 and an outer code soft-input-soft-output (SISO) 810, as well as a positive interleaver 815, a negative interleaves 820, and two adders 825. Iterating between the DPSK demodulator and

convolutional decoder for the incoherent has been investigated for a single carrier.

Additionally, applied iterative techniques have been applied to the incoherent case. However, both applications focused on channel estimation to be able to use coherent detection.

The following discussion focuses on Forward Backward procedures, where the assumption that the symbols a_n, n = 1 , 2, ... , N are uniform is dropped. Interleaving should still guarantee the independence of the symbols however. The entire trellis T is focused on, and tracking of small channel phase variations by adding "side-step" transitions is not used;

therefore, the trellis of the instant disclosure can be decomposed in eight unconnected sub- trellises. This decomposition can be utilized; however, first the un-decomposed trellis is considered. Again starting from ^ao(~o) = 1/32 for all 20€ 2 the a's can be recursively computed from o_t|; -I i «. 1 , 1 - i

(40)

for n = 1,2, ... ,N and z_n ε Z. Also, in the backward pass because the entire trellis T is considered, taking β_η(ζ_η) = γ_η (z_n)for z_n ε Z all other 's can be computed from

;ΓΓ. γ ! . _I .

(41) where again n = 0, I, ... ,N^~ I and ¾EZ.

To determine the a-posteriori symbol probability for symbol value a„ A, the joint probability and density are computed:

PrRiJp(y )

= 5 ^Q"~l ^'- " ■ i'- ;^< „ ! ^" .ι-ΐ(-η-ΐ)Λι(2η-ΐβη).

(42)

This expression also demonstrates how the resulting extrinsic information can be determined. It can be checked that multiplying by the factors Pr{ „} in (42) should be omitted. The extrinsic information is now further processed by the convolutional decoder. The results of the iterative procedure are discussed in further detail below.

Using the standard BCJR algorithm for computing the extrinsic symbol probabilities in the trellis in Fig.3, since a-priori symbol probabilities are non-uniform now, leads to 32 x 4 x 2 multiplications in the forward pass, 32 x 4 x 2 multiplications in the backward pass, and 32 x 4 x 2 multiplications and 4 normalizations in the combination pass for computing extrinsic information, per trellis section. In total, this is 768 multiplications and 4 normalizations per trellis section per iteration.

The following discussion focuses on whether the entire trellis can be decomposed for the case where the a-priori probabilities are non-uniform by decomposing (42) in such a way that (42) can be written for all a„E A. The question now is how to compute the a-posteriori sub-trellis probabilities Pr{s|y} for s = 0, 1 , ... , 7. It can be shown that

and therefore

Now for each sub-trellis the a-posteriori symbol probabilities can be determined using

Friary. "?} oc Pr{a_n, s}p(y|a„,

(46) and by omitting the factor Pr{a„, s\ in (46) the corresponding extrinsic information. Note that this approach requires a backward pass through the entire trellis T first to find the weighting probabilities Pr{s|y} , for s = 0, 1 , ... , 7. This requires 32 x (4 +1) = 160 multiplications per trellis section observing that in (41) γ_η (z_n) can be put in front of the summation sign. Then for all sub-trellises Ts, a forward pass is performed (requiring 8x4 4 2 = 256 multiplications per section), and the results are combined to obtain the extrinsic probabilities Pr{ „|y, s} for that sub-trellis (for which we need 8 x 4 x 4 x 2 = 256 multiplications and 8 x 4 = 32 normalizations per section). Finally these probabilities have to be weighted as in (43), which requires 8 x 4 = 32 multiplications. In total this results in 704 multiplications and 32 normalizations per iteration. It should be noted that decomposition of the trellis does not result in a significant complexity reduction with respect to previously developed approaches.

To achieve a complexity reduction, a method is investigated that is based on finding, at the start of a new iteration, the dominant sub-trellis first and then performing the forward- backward processing for demodulation only in this dominant sub-trellis. Finding the dominant sub-trellis for an iteration is performed based on the a-posteriori sub-trellis probabilities Pr{s|y} that are computed before starting this iteration. Now assuming that one of the a-posteriori sub- trellis probabilities dominates the other ones:

Pr{ ii_fl.[y | « Pr{B_n|v, l}, with I = arg niajc Pr{sjy } , (Λ Ί\

- a . (4 /)

This approach involves the computations of the α-posteriori symbol probabilities (and corresponding extrinsic information) as described in (40), (41), and (42) only for the dominant subtrellis s. Computing the a-posteriori sub-trellis probabilities for each iteration, and then focusing only on the forward pass and combination computations, results in a less complex method than following previously developed methods {e.g., the Peleg method as discussed in M. Peleg, S. Shamai, and S. Galan, "Iterative decoding for coded noncoherent MPSK

communications over phase-noisy AWGN channel," Communications, IEE Proceedings-, vol. 147, no. 2, pp. 87-95, Apr. 2000.). For the best sub-trellis , a forward pass is performed (4 x 4 2 = 32 multiplications per trellis-section), and the results are combined to obtain the a- posteriori (actually extrinsic) symbol probabilities Pr{ „|y, s\ for that sub-trellis (4 x 4 x 2 = 32 multiplications and 4 normalizations per section). In total, 224 multiplications and 4

normalizations per trellis section per iteration are needed.

A second approach involves choosing the dominant subtrellis only once, before starting with the iterations. Since before starting the iterations the a-priori probabilities Pr{ „} = 1/4, i.e., are all equal, the analysis in dominant sub-trellis approach, as discussed above, applies. The a-posteriori sub-trellis probabilities can be computed as in (23). The iterations only in the sub-trellis were chosen initially. This approach requires 84 multiplications and 4 normalizations per trellis section per iteration, and is therefore essentially less complex than previously developed techniques. In simulations, only this last technique will be used when addressing dominant sub-trellises.

The Peleg method was simulated, and the BER versus the signal-to-noise ratio was

i

determined: Eb/No = ai^3" . This BER performance is shown in Fig. 9 for trellis-lengths practically infinite, i.e., N→∞, and ideal LLRs based on the a-posteriori probability given by (42). Fig. 9 shows an example bit-error performance of the Peleg method for trellis length N→ co and up to L = 5 iterations. The BER performance is shown for L = 1 , 2, ... , 5 iterations, where L = 1 stands for no iterations. Note that since ideal LLRs and infinite trellis-lengths are utilized, the corresponding curves shown in Fig. 9 can be regarded as target curves for the iterative (single-carrier) case. In addition, also here, 2SDD and coherently detected DE-QPSK curves are shown for reference. Not in the figure are the curves corresponding to the approach based on decomposing the trellis and using weighting as in (43). As expected, the performance of this approach shows no differences with the Peleg approach in (42). From Fig. 9 it can be seen that for a BER = 10 ⁴, the improvement in required signal to noise ratio is ~ 4.1 dB after L = 5 iterations. FIG. 9 also shows that improvement decreases with the number of iterations and that the first iteration yields the largest improvement. Similar results were obtained in the Peleg method.

To see how the performance in the iterative case depends on the trellis length N, the Peleg approach was simulated forN= 2, 4, and 32, for L = 5 iterations. The results are in Fig. 10, which shows an example bit-error performance for the Peleg method for different values of N and L = 5 iterations. It can be seen that the "iterative coding gain" increases, as expected, with N and that, for N + 1 = 32, the performance is already quite close to that of N→∞.

For N+ 1 = 4 and 32, the difference in BER between the exact LLRs was compared based on the a-posteriori (extrinsic) probability given by (42) or (43), and the approximated LLRs based on the a-posteriori (extrinsic) probability given by (47). The results are shown in Fig. 11 , which shows bit-error performance with dominant sub-trellis for different values N and L = 5 iterations. From Fig. 11 , it was determined that for larger N+ 1 the difference in performance between the exact and approximated LLRs becomes smaller and that for N+l = 32 the difference between the ideal LLRs and the approximation versions, by selecting the dominant subtrellis before starting with the iteration process, is less than 0.3 dB.

Just as in the non-iterative multi-carrier case, processing is based on trellis

decomposition with focus on the computation of the a-posteriori sub-trellis probabilities:

Pi

Note that Pr{s} = 1/8 for s = 0, 1 , ... , 7 and therefore it follows from (44) that

(49) for each subcarrier m = 1 , 2, ... , M.

Now the a-posteriori symbol probability for a_m,n ε A can be written as in (36), i.e.,

Pt^'{ t½i,fi|yi , f 3 , - . - jM J

= ^{1 ' ;} >" · \ .· ' " · * ¥M i*r| e,„ ,„ \y_m , » f .

where Pr{ |y_m, s\ is computed as given by (46) for s e {0, 1 , ... , 7} and a_m,_n ε A. From these a-posteriori probabilities, the extrinsic information that is needed by the convolutional decoder can be computed. Computing extrinsic information is actually a bit easier since it involves fewer multiplications.

This suggests that, for each iteration, the demodulator first determines the a-posteriori sub-trellis probabilities using (48), for which first a backward pass in each of the M trellises corresponding to the sub-carriers is needed.

Using the weighting coefficients, the convex combination in (50) leads to the a- posteriori symbol probabilities. Finding the a-posteriori symbol probabilities Pr{ |y_m, s} should be done in the standard way, taking into account that the backward passes were already carried out.

Equation (50) shows how the exact a-posteriori symbol probabilities can be determined, in each iteration. Just like in the single-carrier case, if the a-posteriori sub-trellis probabilities are such that one of the probabilities dominates the other ones, then convex combination (50) can be approximated as follows:

' . . ,yi . 2, * " . y - ; ¾ l 'r | < - , y,„. .* ^' (_{5 1}) with

I = argm s Prfs|yi ,y_{2 !} - ·^■ , JM }. ^₅2)

Again this approach involves, in each iteration, the computations of the a-posteriori symbol probabilities only for the dominant sub-trellis s. Computing the dominant sub-trellis before the start of the iteration process obtains a significant complexity reduction since the analysis in dominant sub-trellis approach applies. Moreover, all iterations are done in the initially chosen sub-trellis. The methods described here are evaluated below. This type of behavior is also seen in the non-iterative multi-carrier case in that the performance is more or less determined by the size M(N+\) of the block. If the channel cannot be assumed to be constant for large values of N+1, the number of sub-carriers M can be increased (if the frequency selectivity allows). Note that keeping N+1 small also has advantages related to service symbol processing. Here the situation is slightly different than what is demonstrated in Fig. 12, which shows an example bit-error performance for the multi-carrier case for different values M, where N+ 1 = 4 and for L = 5 iterations. Increasing has a positive effect on the performance; however, since the trellis-length N+1 remains constant (and is quite small), the effect of iterating is limited. However, by increasing M from 1 to 8, an improvement of roughly 0.7 dB is achieved.

For Ν + 1 = 4 and 32, and for M = 8, the difference between the performance of exact LLRs based on the a-posteriori (extrinsic) probabilities given by (50) and the approximated LLRs based on the a-posteriori (extrinsic) probabilities given by (51) was compared, the results of which are shown in FIG. 13. Fig. 13 shows an example bit-error performance with dominant sub-trellis approach for different values N, M= 8 and L = 5 iterations. From Fig. 13 it can observed that, as expected, the larger Ν + 1 is, the smaller the difference between the exact and approximated LLRs becomes. For Ν + 1 = 4 the difference between the ideal LLRs and the approximation, by selecting the dominant subtrellis before starting to iterate, is roughly 0.3 dB.

The experimental discussions up to this point have utilized AWGN channels with unknown channel phase and fixed (unit) gain in the analysis and simulations. To investigate the performance in practice, a TU-6 (Typical Urban 6 taps) channel model, as demonstrated in "COST 207, Digital land mobile radio communications," Office for Official Publications of the European Communities, Final report, Luxembourg, 1989, which is fully incorporated by reference, is used. This model is commonly used to test DAB, DAB+ or T-DMB transmission. Two maximum Doppler frequencies are chosen, i.e., fd = 10 and 20 Hz, representing DAB transmission (in Band-Ill) movement-speeds between transmitter and receiver of ~ 45 and ~ 90 km/h, respectively.

The methods of the instant disclosure used DAB transmission in Mode-I, where the inverse subcarrier-spacing T_u = 1 ms and where the cyclic-prefix period _φ = 246 μ .

Now, with these settings, the fading-coefficients fd and T_u are 0.01 and 0.02 respectively

(a "slowly-varying" regime). Note that to prevent ISI, i.e., the impulse response length T_m must satisfy T_m < T_cp < (T 4), and therefore the coherence bandwidth B_c ~ (1/ T_m) > 4(1/ T_u), which is at least 4 OFDM-subcarriers. For Doppler frequency fa = 20 Hz,the coherence-time is

j _r _ _β m* _which is ~ 20 OFDM-symbols (including cyclic prefix).

The channel gain representative for a 2D-block, where it is assumed to be constant, is estimated:

The results of the simulations with the TU-6 model are shown in Fig. 14, which shows bit-error performance for the TU-6 COST-207 channel with the solid-lines for f T_u = 0.01 , the dashed-lines for f_dT_u = 0.02, N + 1 = (4, 18), M = (8, 1) and Z = 5 iterations. Iterative procedures with L = 5 iterations are considered. In these simulations, the dominant sub-trellis approach was used, where the dominant sub-trellis is chosen before starting the iterations.

The value for N + 1 = 4 might be seen as a representative frame-size for services broadcasted by the DAB-family in transmission Mode-I. In this mode, N+l = 18 is the maximum possible number of interleaved OFDM symbols. Note that N+l = 18 is close to the coherence-time of our TU-6 channel for a Doppler frequency of 20 Hz.

It can be concluded from Fig. 14 that for N + 1 = 18 and = 1 , reliable transmission is not possible for the TU-6 channel with movement speeds of ~ 45 km h and ~ 90 km h. For the 2D-decomposition approach however with N+l > 4 and M= 8 there is a considerable improvement of roughly 2.4 and 1.6 dB for 10, respectively, 20 Hz in required signal-to-noise ratio possible, compared to 2SDD.

Various aspects of the instant disclosure relate to decoding procedures for DAB, like systems focusing on trellis decoding and iterative techniques, with a special focus on obtaining an advantage from considering 2D-blocks and trellis decomposition. These 2D-blocks consist of the intersection of a number of subsequent OFDM symbols and a number of adjacent subcarriers. The channel coherence time is typically limited to a small number of OFDM symbols. Further, per service symbol processing is used, which limits the number of OFDM symbols in a codeword. Various aspects of the instant disclosure relate trellis decomposition methods that allow for estimation of the unknown channel-phase modulo π/2. This channel phase relates to sub- trellises of which we can determine the a-posteriori probabilities. Using these probabilities the contributions of all the sub-trellises can be weighted to compute the a-posteriori symbol probabilities. These probabilities can also be used to choose a dominant sub-trellis for providing us with these a-posteriori symbol probabilities. Working with dominant sub-trellises results in significant complexity reductions. A second important advantage of trellis- decomposition is that it allows for processing, in an efficient way, of several subcarriers simultaneously.

Additionally, investigation of non-iterative methods has shown the advantage of these methods, which are forward-backward procedures turned out to be extremely simple by the use of Colavolpe processing. The drawback of these non-iterative methods, however, is that their gain relative to the standard 2SDD technique is modest. Iterative procedures result in a significantly larger gain however. In this context we must emphasize that part of this gain comes from the fact that we can do 2D-processing.

Simulations for the non-iterative AWGN case show that (a) trellis-lengths of N + 1 > 32 are required and (b) that 2D dominant sub-trellis processing with M(N+ 1) = 32 outperforms 2SDD with 0.7 dB at a BER of 10 ⁴. For the iterative AWGN case with L = 5 iterations, simulations show that 2D dominant sub-trellis processing with M(N+ 1) = 32 where N + 1 = 32 and M= 1 outperforms 2SDD with 3.7 dB at a BER of 10 ⁴. However, simulations also reveal that with M(N + 1) = 32 where N + 1 = 4 and M = 8 the iterative coding gain is reduced to 2.5 dB, which is caused by the smaller value of N + 1.

Iterative simulations for a practical setting (i.e. the TU-6 model) show that (a) with trellis-length N + 1 = 18 and M = 1 (one sub-carrier), no reliable communication is possible but that (b) with a modest trellis-length N+l = 4 and M= 8 sub-carriers, the iterative coding advantage is maintained and that the gain is roughly 2.4 dB for 10 Hz Doppler frequency, and 1.6 dB for 20 Hz.

Based on the analysis above, a MAP channel phase estimator, with π/2 phase ambiguity can be derived. The MAP channel phase estimator determines the estimated channel phase Φ = ¾ - <≡ ' TT 16^, ί = 0. 1 _f - - - _ Q/AJ - 1 = 7}. For the derivation of the MAP channel phase estimator, it is assumed that the channel gain \h j is perfectly known at the receiver. Subsequent portions of the instant disclosure discuss a channel gain estimator to provide an estimate of the channel gain at the receiver. For the transmitted DE-QPSK symbols, x_m = (x_m,0,x_m,i,...,x_m,N ) corresponding to sub-carrier m and y_m = (y_m,₀ ,y_m,i,...,y_m,N ) for the received DE-QPSK symbols corresponding to sub-carrier m during the same fading period. Since, t he dominant sub-trellis was computed only before the start of the iteration process where iud symbols, i.e., Prja„_u, = ¼} , the trellis-decomposition analysis is used for the non-iterative multi-carrier case as discussed above. Hence, for the a-posteriori probability of the channel phase during the same fading period, assuming perfect knowledge of Pr can be rewritten as:

'"{ i. a A(,iA]} = tn qn;L,^.. _^|ft[)

X II K_mJn h\),

(54)

with

and with the likelihood

^,Α^ |Λ,ι =— «P{ Τ Γ - (55)

where »n — = x_m.f»e^i<?*.<¾ € ffe/lfi.i = 111.-- - ,7}, jf„ t Λ and

Pr{x_m,_n} =

Now, an estimate of the carrier phase might be obtained with the MAP decision rule: = arg max Fr s = e^Jf>* |y _:l , y₂, ... , J_M ,\^h\}

= arg

With this MAP decision rule, the dominant a-posteriori probability is selected amongst the different a-posteriori probabilities calculated by (54). Since, the eight sub-trellises are represented by * ⁼ " " ^* · " 1 , the dominant subtrellis are selected as discussed above.

Since (54) is M(N + 1) times K_m^(n,[h|), i.e. , the sum of four exponential likelihoods as given by (55) it can be, to avoid exponential and logarithmic functions, of interest to select the dominant exponential in K_{a s}(n,\h\). To do so,

J #f « . I A I)

is rewritten with lm,n (Zm. n S,

and Vm.n is defined as

(59) to obtain

= 2 cueh + 2 cueh {Y_mM j

(60) with ' " ^{' "} '' ^' ·^"' · ¹¹ and ^m-^{n ~} Now, the a-postenon probability of the channel phase, given by (54), becomes:

From (61 ), it can be seen that the a-posteriori probability of the channel phase is proportional to the sum of two cosh-functions. To avoid underflow, calculations are often carried out in the logarithmic domain, and therefore instead of (60) the metric shown in (62) is applied: m₀(©) In (2 coeh [X_m._n) + 2 A IY_{m ,M})) , _{( 62 )}

The following discussion focuses on complexity reduct ion by introducing approximations for metric mo((p). Note, that these approximations are combined with selecting the dominant sub-trellis only before the start of the iterative process.

As an approximation for mr φ), the max-log-MAP approximation can be used:

Ηί_,Φ) * In ; ; .. . [ . ^{' ■ ■} : , = raHx(iA'_m.n |, |V„,. ) ,

(63) And the MAP decision rule, given by (56), can be rewritten as: φ = . J [J [2 cofih (_Y_m.n) + 2 co6li (r_m,„)]

= ar j)

(64) where, as (64) shows, by using this max-log-MAP approximation that calculation of exponentials and logarithms is avoided. This max-log-MAP approximation leads to a performance identical to that of MAP sequence detection for sequences of length M(N +1).

Note, that for estimating the channel phase, the MAP sequence decision rule can be written as: arg

iurg raa* I \ '^{, ,} 1.^'· ! ■ _,„//,„. ί^··- ^'" ·'*» }

(65)

then, for all φ, (67) can be obtained:

= max ( j _m.„,j, |V„,,ji |) _r

This is identical to the approximation given by (63). As a consequence, approximation τηι(φ) will result in the same phase estimates as MAP sequence detection applied to estimate the phase of the channel. Motivated by the improved results obtained with the piecewise-linear fitting of the logarithm of the hyperbolic cosine, the approximation of ηΐο(φ) can be introduced:

\ 2 / \ (68).

BER simulations have been performed on the AWGN-channel to investigate the proposed MAP channel phase estimator, based on choosing a dominant sub-trellis, and its proposed approximations to achieve complexity reduction and implementation. The simulations show the BER versus the signal-to-noise ratio — B , where Eb is the received signal energy of an information bit. The defacto industry standard was used: R, = ½, K =7, convolutional code with generator polynomials go = 133 and g i = 17 1 . Its output is randomly bit-interleaved and differential ly encoded after Gray mapping.

FIG. 15 shows simulation results if the channel phase φ is perfectly known, but also for the case where we estimate the channel phase with the MAP decision rule (56) with the a- posteriori probability mo(-) as given by (62), for the case in which we use the approximated a- posteriori probability mi(-), i.e. , MAP sequence estimation, see (63), and for the case in which the improved a-posteriori probability m₂(-) was used (68). As expected, and indeed shown by FIG. 15, the perfectly known channel phase and the ideal symbol-metrics together with ideal LLR-computation result in the best results. However, using the a-posteriori probability mo(-) or the approximation mi(-) or the improved approximation m₂(-) each achieve, in combination with the MAP-decision rule (56), similar results, as can be seen in the detail BER-curve of FIG. 16 around E_b/N₀ = 5 dB where the BER ~ 10 \

Moreover, the approximations in combination w ith the MAP-decision rule (56) are slightly worse than the best results obtained by perfect know ledge of the channel phase, as shown in FIG. 1 . Note that the combination of the "improved^" a-posteriori probability m₂(-) requires knowledge of the noise variance. Also, the max-log-MAP approximation, as given by (63), is an appropriate method, in terms of performance and complexity trade-offs, to perform channel phase estimation.

The following discussion introduces a channel gain estimator, since the receiv er w ill, in the practical case, not have perfect know ledge of the channel gain \h \ and, moreov er, practical channels are in general non-AWGN channels. For practical channels, i.e., non-AWGN

channels, the channel gain \h \ should also be made available at the receiver, as was shown by the simulations for the practical case demonstrated above, Therefore, a channel gain estimator, based on time-averaging, provides an estimate \fi\ of the channel gain jhj. Moreover, similar as with the channel phase and also verified for a practical case above, the channel gain is fixed (or nearly fixed) w ithin a 2D-block. Now, the received signal might be w ritten as:

2

w ith w_m._n as zero mean circular comple Gaussian noise w ith variance a per component. For obtaining an estimate of the channel gain, based on averaging, the fol lowing can be w ritten:

: ■ - , ^·' = K^■ , _n.»j „ } = E {

+ }

+ ¾P1« { ,„,-.^· i*w_m,_n } + [_UWt }

2JAP {xl,._ne-J+} ¾ |_tt , + l «m.»!² } (70) with Ε{·} is the statistical-average and where in the third equation that |x_m,_n| =1. Since the

2

noise w_m,_nis circular complex Gaussian with zero mean and variance σ per component, (70) be rewritten as:

I: !-¹- ! ·. ' ^" : ^'· ■ ! · K ! ^,|²1

= E{\h\²} +0-Q + E{\w_m - ' -- ]: \\ί· - +2er² _;

(71) where in the second equation E fit { _m,_n}} = ι,η}} = 0 because of Ν(0,σ ) for each component. Rewriting (71) yields the average of the squared channel gain

(72)

From (72), it can be deduced that the average channel gain is a s t at i s t i c a 1-a verage over, in principle, an infinite amount of received symbols, hence:

Kin -2σ^:

m "·». til

(73)

To avoid averaging over an infinity amount of received symbols, as given by (73), a complexity reduction, i.e., a practical realization, is introduced by limiting the average over a finite number of received symbols, as will be discussed in further detail below. From (73), it can also be seen that the channel gain estimator and, thus, the receiver need to have know ledge of the noise

2

variance 2σ . Simulations show the performance of the channel gain estimator w hen the noise variance is unknown at the receiver (for further reducing the complexity ).

As can be seen from (73), an infinite amount of receiv ed symbols is required to obtain the average channel gain. A practical realization of a channel gain estimator is of interest, therefore, a limit amount of M (N+1) received symbols is used for the channel gain estimator, i.e, a first

2

approximation of mo{\h\ ) by: ^Af( + ¹ ) _m-l.* .n-0..V

(74) which is similar to the single-carrier case. Note, maximizing avoids a negative value for the approximation of the squared value of the channel gain. In addition, to avoid that the receiver needs to have knowledge of the noise variance, such as based on a second approximation of m₀(\h\²) by:

BER simulations have been performed on the practical TU-6 channel to investigate the proposed channel gain estimator and its approximations. S imulations show the BER versus the signal-to-noise ratio ^* ^{« ™} ¾fr where j¾ is the received signal energy of an information bit. The de-facto industry standard R, = ½ is used: convolutional code with generator polynomials g = 133 and gi = 1 7 1 . Its output is randomly bit-interleaved and differential ly encoded after Gray mapping. For efficiency, 2SDD was simulated with ideal a-posteriori symbo l

2 2

probabi lities, (where XI a was replaced by the known ( perfectly or estimated) \h \lo ), ideal LL Rs, and Vitcrbi decoding.

FIG. 1 7 shows the performance of the channel gain estimator, for perfectly known gain, and for both the approximations. FIG. 1 8 shows that both approximations achieve simi lar

-4

results for different values of M (N +1) around Et/No =12 dB where the BER ~ 10 .

For (N +1 ) > 8, the approximations given by ( 74) and (75) are sl ightly worse than the best results obtained by perfect knowledge o f the channel gain, as shown in FIG. 1 8. Note that the

2

approximation, given by (74), requires knowledge of the noise variance σ . The second approx imat ion, given by (75), seems an appropriate method, in terms of performance and complexity trade-offs, for channel gain estimation.

FIG. 1 9 shows the relevant blocks of the practically realized improved DAB receiver, which uses 2D block-based joint and iterative demodulation and decoding based on trellis- decomposition as discussed in detail above. For clearness and consistency in notation, to discuss the practically realized improved DAB receiver based on DE-QPSK instead of (π/4)- DE-QPSK, for details see above on equivalence between DE-QPSK and (π 4)-DE-QPS .

Moreover, the time and frequency interleaving are modeled by a one bit-wise uniform block interieaver for each simulated code block of bits. Further, the incoming information bits {d_p} are encoded by a 1 4-rate convolution encoder with 64-states using the industry-standard polynomials go = 133s, gi = I 7 1 g2 = 145g and * = go = 133s. Higher code-rates are obtained via puncturing of the mother code, i.e., rate-compatible punctured convolutional codes ( CPC codes). To avoid burst errors, the coded bits {c,j are fed into the interieaver. The interleaved coded bits {¾^■} are divided into bit-pairs { .} . Each such bit-pair is converted, via a

Gray labeling map, into a QPSK-symbol a_m,_n of the finite alphabet A = | f '+'\ ρ ^~ 0. 1.2, 3} . Then, differential encoding for each OFDM-subcarrier between two consecutive OFDM- symbols is applied i.e. generating DE-QPSK symbol:

&m,n ⁼ ½,ti- l " &m,U' (76)

Each DE-QPSK symbol χ_Μι„ modulates an OFDM-subcarrier m. Moreover, OFDM is efficiently accomplished by a B- points complex Inverse Discrete Fourier Transform (IDFT).

th _

Now, by adding a cyclic prefix the n transmitted OFDM-symbol, ¾. is generated. Note that the ( Inverse) Fast Fourier Transform ((I)FFT) is an efficient implementation of the ( I )DIT for p .

lengths of 5 =2 .where p is an integer. For the AWGN-channei, thermal noise Wn is added to

¾,! ;, resulting in the «-th received OFDM-symbol y„ = x_n + ^' _B . (77)

The receiver, in the case of perfect synchronization, removes the (received version of the) cyclic prefix, and then applies a /i-points complex FI T on the time-domain received sequence J' ¹'* — tl?i,f*, l¾,». - - · . fi,« ) which results in the B received symbols

As stated previously, OFDM reception can be regarded as parallel matched-fi ltering corresponding to B complex orthogonal waveforms, one for each subcarrier and a transition pdf of the AWGN-channel can be given by

€ "⁰

ττΛ₀ (79) where x_m,n and y_m,_n are the «-th transmitted and received differentially encoded symbol corresponding to OFDM-subcarrier m, respectively, and w_m>„ is zero mean circular complex Gaussian noise with variance σ = (No/2). In addition, also here, it is assumed that the channel phase is constant over the block of symbols, therefore:

where φ ε {1π/16,1 = 0, 1 ,...,31 } and uniform just as before. In addition, the practically realized improved DAB receiver performs jointly decoding of a block (2D) of received symbols and decomposes the a-posteriori probabilities. Hence, what was received over several sub- carriers is focused upon. For the demodulation and decoding, the iterative decoding procedures are applied with SCCC. In the receiver, the trellis can be decomposed into a part that corresponds to the channel phase and a part that relates to differential encoding. Furthermore, the dominant sub-trellis approach is the principle that is applied to the receiver and results in acceptable performances, as will be shown by real-time tests.

Real-time and bit-true version tests were performed on the improved DAB receiver. The tests were performed for the AWGN-channcl, and the practical TU-6 channel. The following discussion demonstrates performance improvement tests of the proposed 2D block-based iterative processing with trellis decomposition for a different number of iterations, block-sizes and Doppler-frequencies, i.e., vehicular speeds at a given carrier frequency ( DAB transmission channel ).

For these bench tests, a DAB Ensemble has been created with different services that contain the all-zero sequence, w hich can be used to determine the BER. The DAB ensemble is applied to the DAB-PC tool to determine the performance. The sample frequency of the generated input fi le tor DAB-PC is 8.192 MSamples/s at a low- IF frequency of 2.048 MHz. The duration of the real-time bench-test file is ~ 600 seconds to avoid "wrap-around" of the

measurement data of 144 seconds for the AWGN tests and 432 seconds for the TU-6 tests, per S N R -value. The variance of the noise is based on the mean received signal level calculated over a frame duration in ode- 1 of 96 ms and a predefined SNR-vaiue. BER-measurements are performed for different S N R -values and each BER-measurement is an average based on 144 or 432 seconds of real-time bench-test stimuli. The iterative coding gain in dB yields the performance improvement i.e. the decrease in required signai-to-noise ratio with respect to 2SDD.

The AWGN channel bench- test measurements are obtained with the real-time test- stimuli. As stated before, for each S N R -value, the BER is mea.su red for a duration of 144

-3

seconds for the AWGN-channel, which corresponds to 144/96 1 0 = 1500 DAB transmission frames in Mode -I. With a bit-rate of, for example 128 kbit's, there will be on average 144 x 128

^{3 ~4} -4

- 10 x 10 ~ 1843 errors if the average BER - 10 , which is our point of interest.

As can be seen from FIG. 20, the bench-test results show that an iterative coding-gain of ~ 2.6 dB for L =4 iterations with M (N +\)= 80 is achieved.

The TU-6 bench-test measurements are obtained with the real-time bench-test stimuli. A TU-6 channel with a certain Doppler frequency contaminates this file. As stated before, for each SNR-value the BER is measured for a duration of 432 seconds for the TU-6 channel,

-3

which corresponds to 432/96 10 = 4500 DAB transmission frames in Mode-I. The bench-tests are performed for different Doppler frequencies/_d and FIG. 21 shows the iterative coding gain vs the movement speed in [km/h] between transmitter and receiver. The speed in [km/h] between transmitter and receiver, shown by FIG. 21 , is determined for a DAB-carrier frequency of f_c = 240 MHz in Band-Ill, i.e.,

where ^{c = ,J} " ms is the speed of light, thus, dividing the speed by 4.5 gives the according Doppler frequencies. As can be seen from FIG. 21 , the real-time bench test results show that an iterative coding-gain of ~ 2.4 dB with 1 = 4 iterations and M (N +1) = 80 for a speed of 45 km/h i.e., f_d = 10 Hz is possible. For 90 km/h, i.e., fi = 20 Hz, an iterative coding gain of- 1 .4dB with L = 3 iterations, is achieved for M (N + l) = 64. Note, channel gain \h \ is computed with (75). Various modules may be implemented to carry out one or more of the operations and activities described herein and/or shown in the figures. In these contexts, a "module" is a circuit that carries out one or more of these or related operations/activities (e.g., first and second processing modules). For example, in certain of the above-discussed embodiments, one or more modules are discrete logic circuits or programmable logic circuits configured and arranged for implementing these operations/activities, as in the circuit modules shown in FIG. 1 A.

In certain embodiments, the programmable circuit is one or more computer circuits programmed to execute a set (or sets) of instructions (and/or configuration data). The instructions (and/or configuration data) can be in the form of firmware or software stored in and accessible from a memory (circuit). As an example, first and second modules include a combination of a CPU hardware -based circuit and a set of instructions in the form of firmware, where the first module includes a first CPU hardware circuit with one set of instructions and the second module includes a second CPU hardware circuit with another set of instructions.

Certain embodiments are directed to a computer program product (e.g., nonvolatile memory device), which includes a machine or computer-readable medium having stored thereon instructions which may be executed by a computer (or other electronic device) to perform these operations/activities.

Based upon the above discussion and illustrations, those skilled in the art will readily recognize that various modifications and changes may be made to the present invention without strictly following the exemplary embodiments and applications illustrated and described herein. Such modifications do not depart from the true spirit and scope of the present invention, including that set forth in the following claims.

Claims

What is Claimed is:

1. A communication apparatus (100) comprising:

a front-end circuit (105) configured and arranged for communicating encoded symbols which are received over time and using adjacent OFDM sub-carriers;

a logic circuit (1 10) configured and arranged to characterize the encoded symbols using a trellis-based network of states and subtrellises;

first and second processing modules (115) configured and arranged for determining a- posteriori subtrellis probabilities for the encoded symbols corresponding to the subtrellises and, therefrom, determining a-posteriori probabilities for the encoded symbols; and

an indicator circuit (120) configured and arranged to be responsive to first and second determinations for indicating a-posteriori symbol probabilities.

2. The apparatus of claim 1, wherein the front-end circuit (105) is further configured and arranged for receiving OFDM subcarrier channels.

3. The apparatus of claim 2, wherein the OFDM subcarrier channels are modulated via DE- QPSK.

4. The apparatus of claim 1 , wherein the encoded symbols are in 2D-data blocks which include subsequent OFDM symbols and adjacent OFDM subcarriers.

5. The apparatus of claim 1 , wherein the logic circuit (1 10) and the processing

modules (1 15) are configured and arranged to use a trellis decomposition method to estimate an unknown channel phase that relates to the subtrellises.

6. The apparatus of claim 5, wherein the logic circuit (1 10) and the processing

modules (1 15) are further configured and arranged to search for a dominant one of the subtrellises.

7. The apparatus of claim 1 , wherein the logic circuit (1 10) and the processing

modules (1 15) are further configured and arranged to search for a dominant one of the subtrellises.

8. The apparatus of claim 1 , wherein the logic circuit (1 10) and the processing

modules (1 15) are further configured and arranged to iterative ly repeat a search for a dominant one of the subtrellises and choose a different dominant subtrellises for at least two of the iterations.

9. The apparatus of claim 5, wherein the logic circuit (1 10) and the processing

modules (1 15) are further configured and arranged to search for a dominant one of the subtrellises and, in response thereto, use forward/backward processing for demodulation.

10. The apparatus of claim 5, wherein the logic circuit (1 10) and the processing

modules (115) are further configured and arranged to search for a dominant one of the subtrellises and, in response thereto, use iterative processing on at least one of the subtrellises.

1 1. The apparatus of claim 1 , wherein the first and second processing modules (115) operate on received data via logic that assumes channel-phase and gain are fixed for certain adjacent subcarriers and consecutive symbols.

12. The apparatus of claim 1, wherein the first and second processing modules (115) operate on received data via logic that assumes channel-phase or gain are fixed for certain adjacent subcarriers and consecutive symbols.

13. A method comprising:

communicating encoded symbols which are received over time and using adjacent OFDM sub-carriers;

characterizing the encoded symbols using a trellis-based network of states and subtrellises; determining a-posteriori subtrellis probabilities for the encoded symbols corresponding to the subtrellises;

from the determined a-posteriori subtrellis probabilities, determining a-posteriori probabilities for the encoded symbols; and

in response to determined a-posteriori subtrellis probabilities and a-posteriori probabilities for the encoded symbols, indicating a-posteriori symbol probabilities.

14. The method of claim 13, wherein communicating encoded symbols which are received over time and using adjacent OFDM sub-carriers further includes receiving OFDM subcarrier channels.

15. The method of claim 14, further including a step of modulating the ODFM subcarrier channels via DE-QPSK.

16. The method of claim 13, further including a step of using a trellis decomposition method to estimate an unknown channel phase that relates to the subtrellises.

17. The method of claim 13, further including a step of searching for a dominant one of the subtrellises.

18. The method of claim 17, further including a step of using forward/backward processing for demodulation in response to searching for a dominant one of the subrellises.

19. The method of claim 13, further including a step of iteratively repeating a search for a dominant one of the subtrellises, and choosing a different dominant subtrellises for at least two of the iterations.

20. A method comprising:

receiving encoded symbols which are received over time and using adjacent OFDM sub- carrier channels; characterizing the encoded symbols using a trellis-based network of states and subtrellises;

determining a-posteriori subtrellis probabilities for the encoded symbols corresponding to the subtrellises;

from the determined a-posteriori subtrellis probabilities, determining a-posteriori probabilities for the encoded symbols; and

in response to determined a-posteriori subtrellis probabilities and a-posteriori probabilities for the encoded symbols, iteratively repeating a search for a dominant one of the subtrellises, and indicating a-posteriori symbol probabilities.

21. The method of claim 20, wherein the step of iteratively repeating a search for a dominant one of the subtrellises further includes choosing a different dominant subtrellises for at least two of the iterations.

22. The method of claim 20, further including a step of using a trellis decomposition method to estimate an unknown channel phase that relates to the subtrellises.