NO347523B1 - Continuous phase modulation receiver - Google Patents

Description

CONTINUOUS PHASE MODULATION RECEIVER

FIELD OF THE INVENTION

The present invention relates to a Continuous Phase Modulation (CPM) receiver, in particular to a CPM receiver for use in a scenario with multipath propagation.

BACKGROUND AND PRIOR ART

Continuous Phase Modulation (CPM) techniques are well known in digital communications as they offer good spectral and power efficiency properties.

In state-of-the-art CPM communication systems, CPM is typically used together with convolutional error-correcting codes and decoded iteratively via the so-called turbo-principle, where soft information (involving probabilities that a transmitted bit be 0 or 1 rather than hard decisions on the values of those bits) is exchanged between two component Soft-Input Soft-Output (SISO) decoders / demodulators. Fig. 1 schematically illustrates a first prior art CPM receiver 100. The first CPM receiver 100 comprises a Soft-Input Soft-Output (SISO) demodulator 101, a deinterleaver 102, a Soft-Input Soft-Output (SISO) decoder 103, an interleaver 104, a signal input 105 and a signal output 106.

Multipath propagation, a phenomenon that results in radio signals propagating from a transmitting to a receiving antenna via two or more paths due e.g. to terrain, often creates challenges for wireless communication systems. For example, in Norwegian terrain, multipath delay spreads of ~50 µs in 30-1000 MHz range are common. The multipath arrivals combine at the receiving antenna in different ways, leading to a resultant signal which can vary widely in amplitude and phase. Waveform and receiver design must therefore take multipath propagation into account. Exemplary technical background publications include US2002034264A1.

In the absence of multipath propagation, i.e. with only one path from the transmitting to the receiving antenna, a scenario which is typically modelled via the so-called Additive White Gaussian Noise (AWGN) channel model, the computational complexity of the prior art CPM receiver 100 is manageable.

Specifically, for a CPM modulation with -ary alphabet, memory and CPM modulation index ℎ = / the number of states in the SISO CPM demodulator – which often bears the brunt of the receiver’s computational burden – is <>. This is if the CPM demodulator is based on Rimoldi’s tilted-phase trellis, in which case CPM demodulator 101 is optimum. If most of the total CPM signal energy is contained in the so-called principal components, which is the case for many relevant CPM schemes, the number of states in the SISO CPM demodulator can be reduced from to with at worst a small performance loss.

Now considering a scenario with CPM receiver 100 in the presence of multipath propagation. If most of the total CPM signal energy is contained in the principal components, a quasi-optimum CPM receiver can be obtained with a bank of filters (correlator bank) matched to the convolutions of the CPM principal components and the channel impulse response, followed by a SISO CPM demodulator 101 operating in a turbo loop. Such solutions have been examined in prior art. However, if the

complexity of the first CPM receiver 100 then increases exponentially with delay spread. Hence, for many real multipath propagation scenarios and reasonably sized Field Programmable Gate Arrays (FPGAs), such a solution is too computationally intensive for practical use. Less computationally intensive designs have therefore been proposed as exemplary described in technical background publication US 2012069936A1.

As an alternative, multipath CPM receivers with the structure from Fig. 2 can be used. Here, a channel equalizer 207 is provided in front of a second CPM receiver 200 similar to the first CPM receiver 100. The second CPM receiver 200 comprises a Soft-Input Soft-Output (SISO) demodulator 201, a deinterleaver 202, a Soft-Input Soft-Output (SISO) decoder 203, an interleaver 204, a signal input 205 and a signal output 206. The channel equalizer 207 addresses the complexity problem in the presence of multipath propagation. It compensates for the effects of multipath propagation and outputs an approximately multipath-free equalized signal 205. Since the second CPM receiver 200 experiences little or no multipath propagation, the second CPM receiver 200 may be implemented as a standard AWGN CPM receiver. The second CPM receiver 200 will in the following be referred to as an AWGN CPM receiver 200.

CPM receivers with the structure of Fig. 2 have been considered in prior art with simple channel equalizers 207. The complexity of such CPM receivers has been manageable, but this comes at the expense of a performance degradation in comparison to the first CPM receiver 100 where the SISO demodulator is based on a trellis with ates. Furthermore, when dealing with equalization, perfect channel knowledge (i.e. perfect information on arrival times, phases and amplitudes of all multipath components) at the receiver has often been assumed for simplicity.

Thus, there is a need in the art for a CPM receiver that keeps the receiver complexity sufficiently low in the presence of multipath propagation, whilst at the same time reducing the performance penalty.

Further examples of technical background documents include KOCA M., DELIÇ H. Doubly Iterative Equalization of Continuous-Phase Modulation, IEEE Transactions on Communications, December 2007, DOI: 10.1109/TCOMM.2007.908550, and KOCA M., DELIÇ H. Double Turbo Equalization of Continuous Phase Modulation with Frequency Domain Processing, IEEE Transactions on Communications, March 2009, DOI: 10.1109/TCOMM.2009.02.060674.

SUMMARY OF THE INVENTION

The present invention is set forth and characterized in the independent claims, while the dependent claims describe other characteristics of the invention.

In one aspect the invention relates to a Continuous Phase Modulation (CPM) receiver, comprising

a channel estimator adapted to perform channel estimation based on a multipath channel input signal, wherein a first channel estimat is obtained using a pseudo-inverse (PINV) method and subsequent channel estimates are obtained using a least mean squares (LMS) or expectation-maximization (EM) algorithm, where the LMS or the EM estimation is adapted to receive soft input information from a first log-likelihood adder,

a channel equalizer adapted to receive the multipath channel input signal, the channel estimates and soft information from a Soft-Input Soft-Output (SISO) CPM Demodulator and the first log-likelihood adder and output an equalized channel signal,

an Additive White Gaussian Noise (AWGN) CPM receiver adapted to demodulate and decode the equalized channel signal from the channel equalizer, the AWGN CPM receiver comprising the Soft-Input Soft-Output (SISO) demodulator, a deinterleaver, an interleaver, a Soft-Input Soft-Output (SISO) decoder, and an output adapted to output the demodulated and decoded channel signal from the SISO decoder,

the first log-likelihood adder adapted to compute soft information based on extrinsic and a-priori soft information from the SISO CPM demodulator, wherein, the equalizer comprising:

a linear equation solver adapted to determine filte based on the channel estimate

an equalizing filter adapted to perform linear equalization of the multipath channel input signal,

CPM signal reconstructors and adapted to reconstruct a CPM signal sample based on the soft information on CPM signal sample , where for permitted values

recons or a hard recons is selected

if the m

the feedback filte adapted to produce estimates for precursor intersymbol interference (ISI) remaining in sample of an output of the filter ased on the reconstructed samp , , ,

the feedback filter adapted to produce estimates for postcursor ISI remaining in sample of an output of the filte based on the reconstructed samples and

<an adder (312) adapted to subtract the ISI estimates received from the filters >rom the output of filtesuch as to output the equalized signal 305.

Computer simulations show that the new CPM receiver allows to reduce the performance penalty observed with simple equalizers whilst keeping computational complexity reasonable.

In one embodiment, the first log-likelihood adder may provide the soft information to the channel estimator and the CPM signal reconstructor at iteration based on a-priori log-likelihood information from a first correlation bank and scaling (CB+S) block, the CB+S block comprising a correlator bank (CB) followed by multiplicative scaling (S), in the SISO demodulator at turbo iteration and extrinsic log-likelihood information from the SISO demodulator output at turbo iteratio

In one embodiment, the equalizer may comprise a second log-likelihood adder and a second correlation bank and scaling block comprising a correlator bank (CB) followed by multiplicative scaling (S) provided at the adder output, wherein

the first log-likelihood adder provides soft information to the channel estimator and the CPM signal reconstructor at iteration based on the a-priori log-likelihood information 317 from the first CB+S block of the SISO demodulator at turbo iteratio and the extrinsic log-likelihood information from the SISO demodulator output at turbo iteration and

the second log-likelihood adder adapted to provide soft information to the CPM signal reconstructor at iteration based on a-priori log-likelihood information from the second CB+S block of the equalizer at iteration and the extrinsic log-likelihood information 315 from the SISO demodulator output 301 at turbo iteration

In one embodiment that may be combined with any of the above embodiments, the CPM receiver may further comprise a low-pass filter adapted to receive a digital input signal and output a low-pass filtered digital signal.

In one embodiment that may be combined with any of the above embodiments, the LMS or the EM estimation is either a soft or a hard estimation.

In one embodiment that may be combined with any of the above embodiments, the EM estimation is based either on a full or reduced-complexity EM algorithm.

BRIEF DESCRIPTION OF THE FIGURES

The following figures are appended to facilitate the understanding of the invention. Some of these figures are drawings showing embodiments of the invention. The figures are now described by way of example only:

Fig. 1 is a schematic illustration of a prior art CPM receiver.

Fig. 2 is a schematic illustration of another prior art CPM receiver.

Fig. 3 is a schematic illustration of an exemplary CPM receiver according to the present invention.

Fig. 4 is a schematic illustration of a prior art CPM transmitter.

Fig. 5 shows exemplary structures for data frames transmitted by a system using a CPM receiver of the present invention.

Fig. 6 shows exemplary CPM autocorrelation functions.

Fig. 7 shows, for a synthetic example, the magnitudes of the true channel impulse response the convolution ulse response of the low-pass filter 313 of Fig. 3, and channel obtained respectively via the pseudo-inverse method and reduced-complexity soft EM channel estimator.

Fig. 8 shows the magnitudes of example filter from channel equalizer 307 for a synthetic example.

Fig. 9 compare a synthetic example.

Fig. 10 compa or a synthetic example.

Fig. 11 shows the magnitudes of filters after turbo iteration 10 in a synthetic example.

Fig. 12 compares the frame-error rate (FER) v performance of a receiver from this invention to that of a simple CPM receiver for a synthetic example.

DETAILED DESCRIPTION OF THE INVENTION

In the following, embodiments of the invention will be discussed in more detail with reference to the appended figures. It should be understood, however, that the figures are not intended to limit the invention to the subject-matter depicted in the figures.

The CPM transmitter, illustrated in Fig. 4, is taken from prior art and comprises the concatenation of a binary information source 401, a convolutional encoder 402, an interleaver 403, an y mapper 404, and a CPM modulator 405, and produces the baseband waveform 406

(1)

denotes the information-bearing phase the symbol energy and the symbol period. In equations (2) and (3) above is th is the CPM frequency pulse is the CPM memory, a odulation index with and relatively prime integers. Th re known at the transmitter and the receiver, and are present in summation (2) to ensure the CPM modulator 405 starts in a valid state.

The received multipath channel input signal can be written

(4) where ℎ? denotes the multipath channel, ∗ is the convolution operator and `? is AWGN with double-sided power spectral densit processed by an anti-aliasing filter to avoid noise folding and sampled at the receiver to obtain

(5)

where all discrete-time signals in (5) are obtained by sampling the corresponding continuous-time signals in (4)

The sampling frequency at the receiver is chosen so as to obtain d samples per CPM symbol period where d is the receiver oversampling factor. The approximation in equation (5) is a consequence of the infinite bandwidth of CPM, and would hold with equality if it were possible to sample at frequencies greater than or equal to the Nyquist frequency. As most of the CPM signal power is contained within a limited band this is a good approximation for sufficiently large

It is standard practice to apply a low-pass filter 313 to the received discrete-time signal from equation (5) to remove out-of-band noise. This will also filter out a small portion of the CPM signal as CPM has infinite bandwidth. For example, the low-pass filter can be setup to have a cut-off frequency equal to 99% of the bandwidth of the CPM signal. For ease of understanding, the effect of this filter is omitted from the equations in the description.

It is sometimes convenient to describe CPM signals with Rimoldi’s decomposition, where CPM is considered as a concatenation of a continuous-phase encoder (CPE) and a memoryless modulator (MM). In this decomposition, the CPE is based on a so-called tilted-phase trellis with input symbols, utput symbols and

<states. The output of the CPE is presented to the MM, which then outputs one of >

possible waveforms. The relation between the transmitted waveform >?, @

(6)

(7)

(8) Possible structures for the frames transmitted by the system are illustrated in Fig. 5, where T corresponds to known training symbols, FEC Term denotes termination symbols for the forward-error correction (FEC) convolutional encoder, and CPM Term denotes termination symbols for the CPM tilted-phase trellis. In a system which is meant for continuous transmission the frames may look as in Fig. 5a, with only one CPM frame per FEC frame. In case of a system meant for sporadic transmissions, which may additionally include frequency-hopping for e.g. protection against jamming, a FEC frame may consist of many shorter frequency dwells, each one of which may correspond to one CPM frame, such as illustrated in see Fig. 5b. Other exemplary variants are possible; such as using training midambles instead of preambles as illustrated in Fig. 5, or splitting the training preamble into a pre- and postamble.

Now, each component of the invention will be described in detail.

Channel equalizer 307 filters

A channel estima taps is assumed to be available. Procedures to obtain and update such estimates are described in the section describing the channel estimator.

The sampling frequency in the equalizer . the channel equalizer 307 works with samples per CPM symbol and is therefore fractionally spaced for

1. In the following descriptionhowever, d = 1 is within the scope of the invention.

With reference to Fig. 3, in the channel equalizer 307, the received low-passed filtered signa sampled at frequency first processed by a linear equalization filter Thereafter, in a feedback-equalization section, the outputs of filte are subtracted away to cancel out residual inter-symbol interference (ISI) in the output of he feedback equalizer cancels interference <uture symbols and past symbols via filters><where >

sign parameters. Estimates >̃a for d future samplese

provided by CPM signal reconstructor which then outputs an estimate for the precursor inter-symbol interference. Likewise, estimates past samples >a, with re provided by CPM signal reconstructo

which then outputs an estimate for the postcursor inter-symbol interference. The reconstructors are discussed further below.

<, >

)

that is provided to the SISO CPM Demodulator (301) is then

(9)

<where † denotes complex conjugate transposition. A reasonable choice >h l h f fil i i f could be a number of time

between four and ten – the length of the channel impulse response in units o

though other choices are also possible.

(10)

(11)

Prior art feedback equalizers typically consider linear modulations such as binary phase shift keying (BPSK) or quadrature amplitude modulation (QAM) and deal with the cas e. they are symbol spaced instead of fractionally spaced. The elements of the sequence >a can then be assumed to be mutually independent – and hence also uncorrelated – and for zero-mean signal his is not the case here due to high CPM signal autocorrelations both within a symbol pe re relevant to us whenev nd across CP which are a consequence of the CPM signal memory. Fig. 6 illustrates two plots of example CPM signal autocorrelation function samples >S and >a are highly correlated, attempting to cancel out interference due to > a feedback equalization will do more harm than good, as this actually removes a contribution that is close to the desired signal. In an ideal case would be either zero or one, and <feedback equalization would only be performed for samples >S corresponding to lags>

equation (12) and solving the system of equations. This is not straightforward due to expectations involving estimates conditioning of the system matrix. A number of different approximations can be made, many of which do not lead to satisfactory performance ew such approximations are now described, the last of which is used in the channel equalizer 307 of this invention.

Let the matrix

(13)

The expectations appearing in equation (12) are evaluated by making the following approximations: all entries of a a < >are zero; and when computing the remaining expectations from equation (12), correlations with estimates re replaced by correlations with actual sample values >a.

(14)

where is the variance of the discrete AWGN signal S` and ° is an identity matrix of size Whilst the system matrix and right-hand side of equation (14) can be easily assembled, computer simulations show that performance of CPM receivers based on such a channel equalizer is not satisfactory among others due to the ill-conditioning of matrice

following additional approximations are introduce

id i i ° d ° f sizes respectively

<e set to zero, lead >

m columns of ¤ a

The system of equations (14) then becomes

(15)

Note that while the effect of CPM signal autocorrelation has been neglected in some <entries of (14), it is still present in equation (15) via the products ¤ « ¤† and ¤>

O

s

performance for low signal-to-noise ratio (SNR) values; however for high SNR values the matrix<† † † >be ill-conditioned due to the low value of ® < ̄>. Therefore a threshold ® ̄<>

, eµ is introduced and is obtained as

(17)

which amounts to Tikhonov regularization for high SNR values, and leaves (16) unchanged for low SNR values.

In this invention, the linear equalization fi referably obtained via <expression (17). Thereafter><and therefore also feedback equalizer filters >

CPM signal reconstructors ,

The task of CPM signal reconstruct own in Fig. 3 is to reconstruct <CPM signal samples based on soft e reconstructed CPM signals >are then respectively input to filtshown in Fig. 3.

1U;.

<Sampling of equation (1) with d samples per symbol leads to a sequence>

In the unlikely event that the maximum in (19) is attained for more than one value

the lowest ′ is arbitrarily selected. Note that the constant magnitude property of CPM is not preserved in and is typically lost unless reconstruction is perfect. Moreover, independently estimating the signals in each CPM symbol signaling <interval means the continuous phase property of CPM is not enforced in>

>̂a,ÒÓ4Ô, and is typically also lost unless reconstruction is perfect.

Intuitively, the soft reconstruction (18) is simply a weighted average of the

possible waveforms in each signaling interval, the weights being given by the probabilities of each waveform. In the hard reconstruction (19) the most likely waveform in each signaling interval is selected. Depending on system parameters

such as SNR, computer simulations show best performance is at times achieved with >̂a,Î ¡3 and at other times with

AWGN iterative CPM receiver 300

The equalizer outpu − 1 from equation (9), a sequence of d samples per CPM symbol, is provided to a prior art AWGN iterative CPM receiver 300. As shown in Fig. 3, this receiver consists of a soft-input soft-output (SISO) CPM demodulator 301, an interleaver 304 and deinterleaver 302, an outer SISO decoder 303, and an output 306 of the CPM receiver 300. Now the SISO CPM demodulator 301 is described in more detail.

The block labelled CB+S (correlation bank and scaling) in the SISO CPM demodulator 301 performs the following standard operations: the sequence is first converted to the tilted-phase representation 3ef,a by element-wise

multiplication with llowing equation (7), in view of later <processing by a SISO decoder based on a tilted-phase trellis; second, the sequence > 3ef,a is split into sec amples each corresponding to each CPM symbol; and third th ilities re obtained via the expression

(20)

<Ð denotes th><amples i><he sum over terms>

corresponds to the correlation between each signal sectionnd

the complex conjugates of the ossible waveforms this operation is represented by CB in the CB+S block), and the multiplication b is a scaling operation (represented by S in the CB+S block). If the sequence he output of an AWGN channel, the noise affecting would be i.i.d. Gaussian, and ® ̄<>,6Ø in equation (20) would be chosen equal to the variance of this noise process. However, due to the presence of the channel equalizer 307, the noise process affecting

coloured, leading to suboptimal performance of the AWGN iterative CPM receiver. If it were possible to introduce an interleaver between the channel equalizer and the inner SISO decoder – with a corresponding block at the transmitter – the noise samples affecting neighbouring CPM symbols could be assumed to be approximately independent. Such an approach is usually followed in related prior art on turbo-equalization with linear modulations. However, such an interleaver would destroy the continuous-phase property of CPM and is thus not suitable for the present invention.

Experiments involving various strategies for setting in equation (20) have been carried out. For low atios, computer simulations for several different multipath channels show that the AWGN value ased on the spectral density of the noise process rom equation (4) is too high a value fo attempts to

interpolation using the entries from the following table:

Reference is now made to the inner SISO decoder block in the SISO CPM demodulator 301. In addition to the a-priori probabilitie discussed above, related to the output symbols of the tilted-phase trellis, the inner SISO decoder is also provided with a-priori pro on the tilted-phase trellis input sym from the on input alphabet. The probabilitie re obtained from the output of the outer SISO decoder block 303. The inner SISO decoder, which operates with the logarithms of these probabilities for enhanced numerical stability, then uses the structure of the <tilted-phase trellis to output the extrinsic probabilitie >

Following prior art, the serially concatenated SISO CPM demodulator and outer SISO decoder illustrated in Fig. 3 iteratively exchange extrinsic information on the symbols At the end of the iterative process, hard decisions on the transmitted information bits Þa are made based on the output of the outer SISO decoder.

Computation of CPM signal reconstructor input probabilities

<mbining the probabilities >

spectively produced by the CB+S <and inner SISO decoder blocks in the SISO CPM demodulator 301. Since>

which is the operation carried out in log-likelihood adders A1 and A2 shown in Fig. 3.

Now examining in more detail the assembly prior to the

turbo iteration, where parentheses in the superscript are used to distinguish the

É S

Note therefore that at the beginning of the iterative process (before the first turbo iteration), no information is available at log-likelihood adder A1, and the output of reconstructo is set to zero.

<The output of log-likelihood adder A1 can also be provided to signal reconstructor >n which c o position 1. However, the necessary information to mes available as soon as the channel

equalizer outp ave been computed. Therefore, this information may already be used for reconstruction of past signal samples, which takes place at reconstructor and is provided to filte r estimating postcursor ISI. Hence it is poss

315 when computinÉ S

in position 2. This requires a second CB+S block 314 in the channel equalizer 307, which processes one symbol at a time rather than one frame at a time. In all other respects the second CB+S block 314 is identical to the first CB+S block of the SISO CPM demodulator 301. This also requires a second log-likelihood adder A2 in the channel equalizer 307. The second log-likelihood adder A2 is identical to the first log-likelihood adder A1.

Channel estimator 310

In order to compute the channel equalizer f 15), <knowledge of the channel impulse response onsist of >is required. In practice { is unknown here s the maximum delay spread the system is designed to support. The role of the channel estimator 310 is to provide estimates for the channel impulse response.

<\>

The first channel estim is obtained via the well-known pseudo-inverse method, implemented in the block labelled PINV in Fig. 3. The superscript 1 indicates this is the first such estimate, necessary for the computation of the channel equalizer output used in the first turbo iteration. To

with the definitions f om (23). The least-squares solution is

(25)

Computer simulations show that ill-conditioning of the matrix nals often leads to solution ith undesirable properties. This problem is remedied with Tikhonov regularization, leading to

(26)

egularization parameter determined via standard prior art methods c dentity matrix. This is the procedure that is used in the invention.

Since the training sequences are known in advance at both the transmitter and the receiver, the matrices corresponding to the different training sequences can be precomputed and stored in memory to avoid expensive on-the-fly matrix inversions.

Once th fter the t ding estimate

point. In the invention, the computation of <>is performed via the least mean square (LMS) or expectation-maximization (EM) algorithms. This is the function of the block labelled LMS or EM shown in Fig. 3. As discussed further below, both algorithms can be used either with soft or hard information.

In case the LMS algorithm is used for updating the channel estimates, the output <316 of log-likelihood adder A1 is first used to obtain CPM signal reconstructions >

equations (18) or (19). These reconstructions may be directly

provided to prior art LMS algorithms developed for updating channel estimates in the presence of linear modulations such as BPSK. The extension of these algorithms to the CPM case is relatively straightforward – both w

and requires no further discussion for the skilled person to perform the invention. UsÒÓ4Ô in combination with the LMS algorithm leads to what here is respectively referred to as the soft LMS and hard LMS algorithms. Channel estimation via the soft or hard LMS algorithms in combination with the remaining components of the invention is deemed to be within the scope of the invention.

Examples of use of the EM algorithm for updating channel estimates based on soft or hard information can be found in prior art for linear modulations; however, adapting the EM algorithm to the CPM case is not straightforward. The procedures used in the invention for updatin a the EM algorithm, which include novel and inventive steps, are now described.

sequence

<>. (27)

Bearing in mind equation (5), the received sequence sampled a samples per symbol becomes after convolution with the {-tap channel and low-pass filtering

. (28)

Let the so-called complet prior art literature on the EM algorithm be defined as

(29) (30) into {

(31) <⋯ >

nction (32)

(33) to

34)

<a denotes the>

(35) It is assumed with sends a CPM signal with unit magnitude, he t is still legitimate for the receiver to assume channel impulse response that has been scaled. The denominator of (34) is thus equal t

<inserting (35) into (34), the numerator of (34) can be shown>

(36) (37) and

(38) SV= óô m ô m

∈ ö ÐV=

The set of valid vecto appearing in equation (38) is obtained by appropriately <rotating the vectors of the set of allowed tilted-phase vectors ><, Any vector>

<n >

s

Referring to equation (38), let the quantities

c

recursively obtain y CPM symbol from the inner SISO decoder forward recursion, by only allowing the most probable trellis transition, defined to have probability one for the purpose of the current approximation, at each step of the forward recursion. Similarly obtained in the same fashion from the inner SISO decoder backward recursion. This leads to one unique sequenc for each combination of a yields the approximation

<(41)>

where the notatio h b d d te the valu

most likely sequ ¢a (obtained as described above), and the p re obtained from the output of log-likelihood ad s to what is referred to as the reduced-complexity EM algorithm.

Both the reduced complexity and full EM algorithms may be used with soft and hard <information In the soft variant of the EM algorithm, the probabilities >ovided by log-likelihood adder A1 after the th turbo iteration are directly used by the EM algorithm. In the hard variant,

(42)

tained as

<(43)>

algorithm. In the

unlikely event that the maximum in (42) is attained for more than one &′ the lowest index &′ for which the maximum is attained is arbitrarily selected.

Linear equations solver 311

As discussed above, channel estimate re provided by the channel estimator 310 to the channel equalizer 307. In order to obtain the equalizer output 305 that is provided to the SISO CPM Demodulator 301 for the th turbo iteration, filter must first be obtained by solving the system of equations

(44)

is the task of the linear equations solver in the channel equalizer (block labelled LES in Fig. 3).

This is both computationally intensive and difficult to achieve in e.g. FPGA-based implementations. Hence a new method, taking into account the structure of the receiver 308 presented in the invention is now described.

First, it is observed that the system matr

and were excluded. In prior art literature on equalization for linear modulations, the Levinson-Durbin recursion is typically portrayed as the method of choice for solving Toeplitz systems. This allows savings in comparison to methods for general ch as Gaussian elimination. However, in the present case is not Toeplitz, though close to being so, and hence the Levinson-Durbin recursion cannot be used. In addition, even if the latter problem could be overcome, implementation of the Levinson-Durbin recursion would present a significant obstacle if e.g. an FPGA-implementation is desired.

An iterative solver is therefore utilized in this invention, which as will be discussed leads to a number of advantages. However, the use of other linear solvers such as Gaussian elimination is still deemed to lie within the scope of the invention.

Iterative solvers for Toeplitz systems have received significant attention as the latter arise in a wide number of applications, also outside the field of wireless communications. For Hermitian positive definite e gradient (CG) algorithm is guaranteed to converge. For Hermiti the more complex minimum residual (MINRES) method can be used.

To achieve rapid convergence of the iterative solver (e.g. CG or MINRES), a good preconditioner is essential. For Toeplitz systems, excellent convergence rates can be achieved with Strang’s or T. Chan’s circulant preconditioners. If desired, the preconditioning matrix can be forced to be positive definite by taking the absolute value of its eigenvalues.

In the present case s neither positive definite nor Toeplitz. It is however close to being both. Computer simulations show that for typical cases of interest, laudable convergence properties can be achieved by simply using the CG algorithm in combination with Strang’s circulant preconditioner. Breakdowns were occasionally observed with T. Chan’s preconditioner. Further, whilst the MINRES method together with a modified positive-definite version of Strang’s preconditioner guarantees convergence, this adds complexity to each iteration, and hence setting in motion the heavier machinery does not seem worthwhile.

Most of the computational burden in an iterative solver lies in the evaluation of matrix-vector product As documented in prior art, for Toeplit this can be efficiently done via a combination of two Fast Fourier Transforms (FFTs) and one inverse Fast Fourier Transform (IFFT). This is not only computationally efficient but can also be easily implemented in FPGAs by making use of extensively researched and optimized FFT implementations for FPGAs In the present case, whilst self is not Toeplitz, the matric are all Toeplitz, hence a matrix-vector produc an be obtained from a sequence of Toeplitz matrixvector products.

A good initial guess can help reduce the number of iterations required for <convergence of iterative solvers. The norms of the updates to the channel estimates > carried out by the LMS or EM algorithms in the channel estimator will typically be small in the sense tha

good initial guess for the solution

from turbo-iteration − 1. Com

sufficient to carry out one iteratio

thereby allowing for significant computational savings. For the purposes of the invention a small residual error in should not lead to any trouble and hence the iterative process can be stopped early.

According to an embodiment of the invention an iterative CG solver is used in combination with Strang’s preconditioner, and starting from turbo iteration only one CG update to s carried out. The advantages of this approach over the muchdiscussed Levinson-Durbin recursion in the context of prior art equalization methods are (1) applicability to quasi-Toeplitz ease of implementation in e.g. FPGAs; (3) computational savings via re-use of previous solution vectors starting at the second turbo iteration; and (4) possibility to stop the CG iterations before convergence to achieve computational savings.

The above procedure for obtaining a solution to system of equations (44), or a reasonably accurate approximation to such a solution, at each iteration is an important aspect of this invention. Indeed, the procedure can easily be implemented in FPGAs and the computational complexity is reasonable. On the other hand, general-purpose solvers such as Gaussian elimination not only are difficult to implement in FPGAs but would also lead to large computational complexity increases.

Example

Now some of the topics discussed above are illustrated with a synthetic example based on the reduced-complexity soft EM channel estimator and 10 turbo iterations. The receiver oversampling factor set to 4. The frames consist of 4 dwells (see Fig. 5b), each one of which is 92 CPM symbols long, the first 14 symbols of each <dwell being reserved to training. For each dwell, the true channel impulse response >

consists of equal-magnitude, equal-phase arrivals at taps 1, 11 and 20, i.e.

arrivals in the first, third and fifth CPM symbols sinc magnitude is plotted in Fig. 7. Le e magnitude of which is also plotted in Fig. 7 – denote the result of the convolution of d the impulse response of the low-pass filter from Fig. 3 (LPF block).

The channel estimator is set up to produce channel e with a length of 20 taps. The magnitudes of the pseudo-inverse estim ore the beginning of the turbo iterations) and estimate r 10 soft EM channel updates following 10 turbo iterations are shown in Fig. 7 (labelled |ℎB|.

The corresponding filt or the channel equalizer, obtained by solving system (44) w tained either fro

explanations thereafter for details) are show

was set to 101 taps. An ideal equalizing fil

wit a discrete-time dirac pulse.

The smaller the differenc

of interference-symbol in

to perform. The quantitie

respectively in Figs. 9 and

tha hich illu

each turbo iteration.

The magnitudes of filteer turbo iteration 10 are shown in Fig. 11. Comparing Figs 10 and 11 it can be seen that and attempt to cancel out the portions

combined

example

while filt

is 120 tap

as previo

autocorre

example

The frame-error rate (FE atio between signal energy and noise-power spectral density) performance of the claimed CPM receiver, with reduced

complexity soft EM channel estimation, soft signal reconstruction and switch B from Fig. 3 in position 2, is compared to that of a simple CPM receiver in Fig. 12, for the three-tap impulse respons nd frame parameters discussed above. A gap of approximately 4 dB is observed for FER = 10 <>.

A possible approach for a simple CPM receiver would be to use a linear equalizing fil without any feedback-equalization (i.e. no filt In such a receiver, the estimation of may be completely bypassed, and estimates for

directly updated via e.g. the soft LMS algorithm after every turbo iteration. This has the advantage of avoiding difficulties related to the solution of system (44) but precludes ISI cancellation via feedback equalization, as for the latter knowledge of bot ( ) are required.

Computer simulations for the simple CPM receiver described above have not been carried out, bu

method, updat

identically zer

<the minimum mean-square error (MMSE) solution in the absence of feedback filters >

. The behavior of this receiver is expected to be similar to that of the PM receiver described above. Indeed, both are based on a linear equalizing

he coefficients of which attempt to minimiz er to

equation (9) and the discussion on the equalizer filters in that section).

In the preceding description, various aspects of the CPM receiver according to the invention have been described with reference to the illustrative embodiment. For purposes of explanation, specific numbers, systems and configurations were set forth in order to provide a thorough understanding of the system and its workings. However, this description is not intended to be construed in a limiting sense.

Various modifications and variations of the illustrative embodiment, as well as other embodiments of the system, which are apparent to persons skilled in the art to which the disclosed subject matter pertains, are deemed to lie within the scope of the claims.

Claims (6)

1. Continuous Phase Modulation (CPM) receiver (308), comprising

a channel estimator (310) adapted to perform channel estimation based on a multipath channel input signal (309), wherein a first channel estima

<obtained using a pseudo-inverse (PINV) method and subsequent channel estimates >re obtained using a least mean squares (LMS) or expectation-maximization (EM) algorithm, where the LMS or the EM estimation is adapted to receive soft input information (316) from a first log-likelihood adder (A1),

a channel equalizer (307) adapted to receive the multipath channel input signal (309), the channel estimates nd soft information (315, 316) from a Soft-Input Soft-Output (SISO) CPM Demodulator (301) and the first log-likelihood adder (A1) and output an equalized channel signal (305),

an Additive White Gaussian Noise (AWGN) CPM receiver (300) adapted to demodulate and decode the equalized channel signal (305) from the channel equalizer (307), the AWGN CPM receiver (300) comprising the Soft-Input Soft-Output (SISO) demodulator (301), a deinterleaver (302), an interleaver (304), a Soft-Input Soft-Output (SISO) decoder (303), and an output (306) adapted to output the demodulated and decoded channel signal from the SISO decoder (303),

the first log-likelihood adder (A1) adapted to compute soft information (316) based on extrinsic (315) and a-priori (317) soft information from the SISO CPM demodulator (301);

wherein, the equalizer (307) comprising:

a linear equation solver (311) adapted to determine filters ased on the channel estimat

an equalizing fi pted to perform linear equalization of the multipath channel input signal (309),

CPM signal reconstructors dapted to reconstruct a CPM signal sample based on the soft information on CPM signal sample , where for permitted

a hard

selected,% ,

if the maximum is attained for more than on

the feedback fil adapted to produce estimates for precursor intersymbol interference (ISI) remaining in sample of an output of the filte based on the reconstructed sampl

the feedback filter adapted to produce estimates for postcursor ISI remaining in sample of an output of the filte sed on the reconstructed samples

<an adder (312) adapted to subtract the ISI estimates received from the filters >uch as to output the equalized channel signal (305).

2. CPM-receiver (308) according to claim 1, wherein the first log-likelihood adder (A1) provides the soft information (316) to the channel estimator (310) and the CPM signal reconstructor and at iteration ased on a-priori loglikelihood information (317) from a first correlation bank and scaling (CB+S) block, the CB+S comprising a correlator bank (CB) followed by multiplicative scaling (S), in the SISO demodulator (301) at turbo iteration and extrinsic log-likelihood information (315) from the SISO demodulator (301) output at turbo iteration

3. CPM-receiver (308) according to claim 1, wherein the equalizer (307) comprises a second log-likelihood adder (A2) and a second correlation bank and scaling block (314) comprising a correlator bank (CB) followed by multiplicative scaling (S) provided at the adder (312) output, wherein

the first log-likelihood adder (A1) provides soft information (316) to the channel estimator (310) and the CPM signal reconstructor at iteration based on the a-priori log-likelihood information (317) from the first CB+S block of the SISO demodulator (301) at turbo iteratio and the extrinsic log-likelihood information (315) from the SISO demodulator (301) output at turbo iteration

1, and

the second log-likelihood adder (A2) adapted to provide soft information to the CPM signal reconstructo iteratio based on a-priori log-likelihood information (318) from the second CB+S block (314) of the equalizer (307) at iteration and the extrinsic log-likelihood information (315) from the SISO demodulator (301) output at turbo iteration

4. CPM receiver (308) of any of the preceding claims, further comprising a low-pass filter (313) adapted to receive a digital input signal and output a low-pass filtered digital signal (309).

5. CPM receiver (308) of any of the preceding claims, wherein the LMS or the EM estimation is either a soft or a hard estimation.

6. CPM receiver (308) of any of the preceding claims, wherein the EM estimation is based either on a full or reduced-complexity EM algorithm.