FIELD OF THE INVENTION

The present invention relates to a Continuous Phase Modulation (CPM) receiver, and more specifically 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 de-interleaver 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. An exemplary technical background publication is US 2002034264 A1.

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 M-ary alphabet, memory L and CPM modulation index h = μ/q 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 q with at worst a small performance loss.

A scenario with CPM receiver 100 in the presence of multipath propagation is now considered. In a situation where the channel impulse response is perfectly known, 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 channel impulse response is n _c CPM symbols long, the number of states in the SISO CPM demodulator is . This represents an increase by a factor of at least M ⁿ c in comparison to the AWGN case. In this situation, the computational complexity of the first CPM receiver 100 then increases exponentially with delay spread. In addition, scenarios with a perfectly known channel impulse response almost never arise in practice. If the channel is unknown it typically needs to be estimated and the performance is limited by the quality of the channel estimate. To avoid significant performance impairments, advanced channel estimation schemes have been proposed for use in combination with the quasi-optimum CPM receiver described above. However, these channel estimation schemes may also dramatically exacerbate the receiver’s computational burden. Hence, for many real multipath propagation scenarios and reasonably sized Field Programmable Gate Arrays (FPGAs), many such solutions are too computationally intensive for practical use. Fess computationally intensive designs have therefore also been proposed, as described in US 2012069936 A1, but at the expense of a performance penalty with respect to the quasi-optimum CPM receiver described above.

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 de -interleaver 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, such as for instance in technical background documents 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.

In some instances the proposed channel equalizers have been relatively simple, leading to CPM receivers with manageable complexity 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 states. In other instances, the proposed equalizers were deemed too complex to be attractive. Frequency -domain equalization (FDE) is proposed in the above referred 2009 publication. Its major drawback is that it requires the addition of a cyclic prefix, which in the case of short dwells can drastically increase the pay load -to-overhead ratio.

There is thus 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.

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 estimate is obtained using a pseudo-inverse (PINV) method and subsequent channel estimates are obtained using a least mean squares (EMS) 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 de-interleaver, 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 filters f, g ₁ and g ₂ based on the channel estimates an equalizing filter f adapted to perform linear equalization of the multipath channel input signal,

CPM signal reconstructors ρ ₁ and ρ ₂ adapted to reconstruct a CPM signal sample j based on the soft information on CPM signal sample j, where for permitted values for a sample j with probabilities the reconstruction is either a soft reconstruction or a hard reconstruction a _i with where the lowest value of i' is selected if the maximum is attained for more than one i', the feedback filter g ₁ adapted to produce estimates for precursor inter - symbol interference (ISI) remaining in sample j of an output of the filter f based on the reconstructed samples (j + 1, j + 2, . . . ), the feedback filter g ₂ adapted to produce estimates for postcursor ISI remaining in sample j of an output of the filter f, based on the reconstructed samples (..., j - 2,j — 1), and an adder (312) adapted to subtract the ISI estimates received from the filters g ₁ and g ₂ from the output of filter f, such 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 reconstructors ρ ₁ and ρ ₂ at iteration n 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 n — 1 and extrinsic log-likelihood information from the SISO demodulator output at turbo iteration n — 1.

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 ρ ₁ at iteration n based on the a-priori log-likelihood information 317 from the first CB+S block of the SISO demodulator at turbo iteration n — 1 and the extrinsic log-likelihood information from the SISO demodulator output at turbo iteration n — 1, and the second log-likelihood adder adapted to provide soft information to the CPM signal reconstructor ρ ₂ at iteration n based on a-priori log-likelihood information from the second CB+S block of the equalizer at iteration n and the extrinsic log-likelihood information 315 from the SISO demodulator output 301 at turbo iteration (n — 1).

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 EMS 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 h _true , the convolution h of h _true and the impulse response of the low-pass filter 313 of Fig. 3, and channel estimates obtained respectively via the pseudo-inverse method and reduced-complexity soft EM channel estimator. Fig. 8 shows the magnitudes of example filters f from channel equalizer 307 for a synthetic example.

Fig. 9 compares for a synthetic example.

Fig. 10 compares for a synthetic example.

Fig. 11 shows the magnitudes of filters g ₁ and g ₂ after turbo iteration 10 in a synthetic example.

Fig. 12 compares the frame-error rate (FER) vs E _s /N ₀ 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 M-ary mapper 404, and a CPM modulator 405, and produces the baseband waveform 406

(1) where is a sequence of N symbols taken from the M-ary alphabet or the quantity

(2) with

(3) denotes the information-bearing phase, E _s is the symbol energy and T _s is the symbol period. In equations (2) and (3) above, g(t) is the CPM phase pulse, ω(t) is the CPM frequency pulse, L is the CPM memory, and h. = p/q is the CPM modulation index with p and q relatively prime integers. The values a_ ₁ , ••• ,α- _(L-1) are 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 (t) denotes the multipath channel, * is the convolution operator, and v(t) is AWGN with double-sided power spectral density N ₀ /2. The signal r(t) is 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); e.g. r _k = r(t = t _k )

The sampling frequency at the receiver is chosen so as to obtain p samples per CPM symbol period where p 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 η.

It is standard practice to apply a low-pass filter 313 to the received discrete-time signal r _k 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 M input symbols, qM ^L output symbols and qM ^L-1 states. The output of the CPE is presented to the MM, which then outputs one of qM ^L possible waveforms. The relation between the transmitted waveform s(t,α) and the MM output stil(t,α) is then

(6) or for the corresponding discrete-time signals (7)

Note that combining equations (1) and (6) gives

(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 f, g ₁ , g ₂

A channel estimate consisting of I 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 is η/T _s , i.e. the channel equalizer 307 works with η samples per CPM symbol and is therefore fractionally spaced for η > 1. In the following description η ≥ 2, however, η = 1 is within the scope of the invention.

With reference to Fig. 3, in the channel equalizer 307, the received low-passed filtered signal r _k sampled at frequency p/T _s is first processed by a linear equalization filter f. Thereafter, in a feedback-equalization section, the outputs of filters g ₁ and g ₂ are subtracted away to cancel out residual inter-symbol interference (ISI) in the output of f. The feedback equalizer cancels interference arising from M ₁ future symbols and M ₂ past symbols via filters g ₁ and g ₂ , where M ₁ and M ₂ are design parameters. Estimates for pM _t future samples s _k are provided by CPM signal reconstructor ρ ₁ to g ₁ , which then outputs an estimate for the precursor inter-symbol interference. Likewise, estimates s _k for ηM ₂ + μ past samples s _k , with μ = I — 1, are provided by CPM signal reconstructor ρ ₂ to g ₂ , which then outputs an estimate for the postcursor inter-symbol interference. The reconstructors are discussed further below.

For filter coefficients t and and vectors the equalizer output (305) that is provided to the SISO CPM Demodulator (301) is then

(9) where † denotes complex conjugate transposition. A reasonable choice for M = M ₁ + M ₂ , the length of filter f in units of T _s , could be a number of times - e .g. between four and ten - the length of the channel impulse response in units of T _s , though other choices are also possible.

To obtain f, g ₁ and g ₂ the idea is to minimize the expectation Introducing it can be shown that is minimized if x is the solution of

(10) which using the definitions of x and y _k can also be written

(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 case η = 1, i.e. they are symbol spaced instead of fractionally spaced. The elements of the sequence s _k can then be assumed to be mutually independent - and hence also uncorrelated - and for zero-mean signals _; for j ≠ 0. This is not the case here due to high CPM signal autocorrelations both within a symbol period for which are relevant to us whenever η > 1) and across CPM symbols for |j| > η ) which are a consequence of the CPM signal memory. Fig. 6 illustrates two plots of example CPM signal autocorrelation functions If samples Sy and s _k are highly correlated, attempting to cancel out interference due to Sj in via 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 Sj corresponding to lags T such that We therefore introduce a cut-off value and exclude samples in the range (k — c, ••• , k — 1, k + 1, ••• , k + c) from feedback equalization on where c ≥ 0 is the lowest integer such that . Interference cancellation is thus carried out for n _g1 = η M ₁ — c future samples and n _g2 = η M ₂ + μ — c past samples; and both the last c entries of g ₁ and the first c entries of g ₂ are set to zero.

For ease of exposition let and . Omitting equations corresponding to the entries of g ₁ and g ₂ that have been set to zero, the system of equations (11) becomes

(12)

The remaining hurdles to obtain f, g ₁ and g ₂ are evaluating the expectations in equation (12) and solving the system of equations. This is not straightforward due to expectations involving estimates or and ill-conditioning of the system matrix. A number of different approximations can be made, many of which do not lead to satisfactory performance. A few such approximations are now described, the last of which is used in the channel equalizer 307 of this invention.

Let the matrix

(13) with η (M ₁ + M ₂ + 1) + μ columns and η (M ₁ + M ₂ + 1) rows. Furthermore, let R be a Toeplitz square matrix of dimension n(M1 + M ₂ +1) + μ of CPM autocorrelation values, i.e. entry (i,j) of R is This matrix is only a function of the CPM modulation parameters. Let R ₁ be the matrix of the first n _g1 columns of R and R ₂ be the matrix of the last n _g2 columns of R. In addition, let R ₁₁ be the matrix of the first n _g1 rows of R ₁ and R ₂₂ be the matrix of the last n _g2 rows of R ₂ . _{Let R[ηM1+1]} be the

The expectations appearing in equation (12) are evaluated by making the following approximations: all entries of , and are zero; and when computing the remaining expectations from equation (12), correlations with estimates or are replaced by correlations with actual sample values Specifically, in the computation of it is assumed that and and in that of and the complex conjugate transposed of these two matrices it is assumed that and This gives

(14) where is the variance of the discrete AWGN signal Vj and I 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 matrices R ₁₁ and R ₂₂ . Therefore the following additional approximations are introduced: R ₁₁ and R ₂₂ are replaced by identity matrices and of sizes respectively n _g1 X n _g1 and n _g2 X n _g2 , and the remaining entries of R ₁ and R ₂ are set to zero, leading to H R ₁ = H ₁ and H R ₂ = H ₂ , where H 1 contains the first n _g1 columns of H and H ₂ the last n _g2 columns of H. 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 H R H † and where the definition of R has not changed (entry (i,j) of R is and Is the (η ₁ 1)th column of R). It is possible to eliminate M + and from (15) to obtain f as

(16)

Once f is known, and can be trivially obtained via (15). Computer simulations show that calculating f, and in this fashion leads to good performance for low signal-to-noise ratio (SNR) values; however for high SNR values the matrix may be ill-conditioned due to the low value of . Therefore, a threshold is introduced and f 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 filter f is preferably obtained via expression (17). Thereafter, and , and therefore also feedback equalizer filters g ₁ and g ₂ , are obtained via (15).

CPM signal reconstructors ρ ₁ , ρ ₂

The task of CPM signal reconstructors ρ ₁ and ρ ₂ shown in Fig. 3 is to reconstruct CPM signal samples based on soft information. The reconstructed CPM signals are then respectively input to filters g ₁ and g ₂ as shown in Fig. 3.

Concretely, for a CPM modulation with parameters h = p/q , M-ary alphabet and memory L, there is a total of Q ₀ = qM ^L possible tilted-phase waveforms in the signal interval corresponding to each CPM symbol, the jth of which is represented by p samples denoted Let the transmitter send a sequence of N CPM symbols α ₀ , ••• , α _N _ ₁ , which are mapped to a sequence of tilted- phase waveforms by the CPM modulator. Note that corresponds to the restriction of ^S til(t, a) from equation (8) to the interval

Sampling of equation (1) with p samples per symbol leads to a sequence Consider a sample s _k of this sequence. Note that each index 0 ≤ k ≤ ηN — 1 is uniquely associated with a pair where 0 ≤ n ≤ N — 1 is the CPM symbol index and 0 ≤ m ≤ η — 1 is the sample index within each symbol.

Every waveform is associated with a set of Q ₀ probabilities

1 ≤ j ≤ Q ₀ discussed in more detail below. The signal reconstructors’ task is to provide reconstructions of the original sequence based on the probabilities and knowledge of the fixed waveforms . The signal reconstruction may be soft or hard. The soft reconstruction of sample s _k is obtained as whereas the hard reconstruction is obtained as

In the unlikely event that the maximum in (19) is attained for more than one value j the lowest j' 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 signalling interval means the continuous phase property of CPM is not enforced in or , and is typically also lost unless reconstruction is perfect.

Intuitively, the soft reconstruction (18) is simply a weighted average of the Q ₀ 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 signalling interval is selected. Depending on system parameters such as SNR, computer simulations show best performance is at times achieved with and at other times with

AWGN iterative CPM receiver 300

The equalizer output z _k , k = 0, ••• , ηN — 1 from equation (9), a sequence of η 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 de-interleaver 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 z _k is first converted to the tilted-phase representation z _til,k by element-wise multiplication with following equatio (7), in view of later processing by a SISO decoder based on a tilted-phase trellis; second, the sequence ztil,k is split into sections _n of η samples each corresponding to each CPM symbol; and third the a-priori probabilities are obtained via the expression denotes the mth of η] samples in . The sum over terms corresponds to the correlation between each signal section and the complex conjugates of the Q ₀ possible waveforms γ _j (this operation is represented by CB in the CB+S block), and the multiplication by is a scaling operation (represented by S in the CB+S block). If the sequence z _k were the output of an AWGN channel, the noise affecting z _k would be i.i.d. Gaussian, and 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 z _k is 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 E _s /N ₀ ratios, computer simulations for several different multipath channels show that the AWGN value N _o based on the spectral density of the noise process v(t) from equation (4) is too high a value for . As attempts to analytically obtain an improved value for q have not been successful, in this invention an empirical correction factor ξ based on computer simulations is introduced. In equation (20), let , where the factor ξ is a function of the E _s /N ₀ ratio in dB. For E _s /N ₀ ≤ —3 dB, ξ is set to 2.6; for E _s /N ₀ > 1 dB, ξ is set to 1.0; and for —3 dB < E _s /N ₀ < 1 dB the value of ξ is obtained via linear 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 probabilities discussed above, related to the output symbols of the tilted-phase trellis, the inner SISO decoder is also provided with a-priori probabilities P _α (α _n = a) on the tilted-phase trellis input symbols α _n , for all a from the M-ary CPM modulation input alphabet. The probabilities P _α (α _n = a) are 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 probabilities and

Following prior art, the serially concatenated SISO CPM demodulator and outer SISO decoder illustrated in Fig. 3 iteratively exchange extrinsic information on the symbols α _n . At the end of the iterative process, hard decisions on the transmitted information bits b _k are made based on the output of the outer SISO decoder. Computation of CPM signal reconstructor input probabilities

The probabilities provided to the CPM signal reconstructors ρ ₁ and ρ ₂ are computed by combining a-priori and extrinsic probabilities and - In typical prior art iterative AWGN CPM receivers the CPM SISO demodulator output is not used.

The input to reconstructor ρ ₁ is obtained by combining the probabilities 317 and 315, respectively produced by the CB+S and inner SISO decoder blocks in the SISO CPM demodulator 301. Since and ^are assumed to be independent in a turbo receiver,

(21) where is a normalization constant which ensures that Z In the logarithmic domain equation (21) becomes

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

Now examining in more detail the assembly of prior to the (n)th turbo iteration, where parentheses in the superscript are used to distinguish the turbo iteration number (n) from the symbol index n. The information received by log-likelihood adder A1 arises from the SISO demodulator outputs at the (n — l)th turbo iteration. The output of log-likelihood adder A1 is provided to reconstructor ρ ₁ 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 reconstructor ρ ₁ is set to zero.

The output of log-likelihood adder A1 can also be provided to signal reconstructor ρ ₂ , in which case switch B from Fig. 3 is set to position 1. However, the necessary information to compute becomes available as soon as the channel equalizer outputs have been computed. Therefore, this information may already be used for reconstruction of past signal samples, which takes place at reconstructor ρ ₂ and is provided to filter g ₂ for estimating postcursor 1ST Hence it is possible to combine with 315 when computing at ρ ₂ . In this case switch B shown in Fig. 3 is 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 filters f, g ₁ and g ₂ via equation (15), knowledge of the channel impulse response h _k , 0 ≤ k ≤ I — 1, assumed to consist of I taps, is required. In practice I is unknown and may be set to η·τ _max /T _S , where T _max is 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 estimate 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 obtain a known training signal of length N _tr CPM symbols is sent by the transmitter, which after oversampling leads to the sequence Sampling and low-pass filtering the corresponding signal r _tr (t) = (see equation (4)) at the receiver leads to the sequence

The pseudo-inverse estimate is then the least-squares solution to the system or with the definitions for and r _tr following from (23). The least-squares solution is Computer simulations show that ill-conditioning of the matrix S for CPM signals often leads to solutions with undesirable properties. This problem is remedied with Tikhonov regularization, leading to where λ > 0 is a regularization parameter determined via standard prior art methods and I _l is an I X I identity 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 the turbo iterations have started, the output of log-likelihood adder A1, after the (n)th turbo iteration, ( ) 316 may be used to refine yielding estimates is a refinement of where is used as a starting 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 or via 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 when η = 1 and when η > 1 - and requires no further discussion for the skilled person to perform the invention. Using or 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 updating via the EM algorithm, which include novel and inventive steps, are now described.

Let the transmitter send a waveform consisting of N CPM symbols with index 0 ≤ n ≤ N — 1, which are mapped to a sequence of tilted-phase waveforms by the CPM modulator. After mapping to s(t, α) via equation (6), sampling of equation (1) with η samples per symbol leads to the length b = ηN sequence s = (s ₀ , s ₁ , .. , s _{b- 1} ) (27)

Bearing in mind equation (5), the received sequence sampled at η samples per symbol becomes after convolution with the l-tap channel and low-pass filtering r = (r ₀ , r ₁ , .. , r _b+l _ ₁ ). (28)

Let the so-called complete data w discussed in prior art literature on the EM algorithm be defined as w = (Θ, s), (29) where 0 is a random matrix with entry (j, k) equal to

(30)

Here, follows from a decomposition of the noise component v _j affecting r _j into I independent noise components such that

(31) must satisfy the normalization constraint The choice is made here.

Following standard practice, to obtain the EM estimate the auxiliary function

(32) is first computed, and thereafter is obtained by solving

(33) by requiring that After some algebra this leads to

(34) where denotes the set of all valid vectors denotes the

(k + 1)th component of and

(35) It is assumed without loss of generality that the transmitter sends a CPM signal with unit magnitude, hence . If instead it is still legitimate for the receiver to assume that and that it is instead the channel impulse response that has been scaled. The denominator of (34) is thus equal to By inserting (35) into (34), the numerator of (34) can be shown to be equal to and

The set of valid vectors appearing in equation (38) is obtained by appropriately rotating the vectors of the set of allowed tilted-phase vectors Any vector is obtained by sampling a sequence of tilted waveforms where is one of the allowed tilted- phase waveforms in a CPM symbol interval. If in the tilted-phase domain the relationship between the (k + 1)th element is s of is

It is possible to evaluate and N ₂ without any particular difficulties by using the information consisting of the Q ₀ probabilities associated with each one of the N transmitted CPM symbols.

Referring to equation (38), let the quantities be such that The probabilities in equation (40) can be decomposed into a product and each term evaluated using the probabilities provided by log-likelihood adder A1. This leads to what is referred to as the full EM algorithm.

To reduce the computational burden in the computation of the A _j,k s, an alternative is to limit the summation over in equation (39) to the sequences leading to the most likely sequences for each possible combination of index j and c _k , assuming The most likely sequence for each combination of j and c _k can be obtained from the inner SISO decoder. This is done as follows: first fix j and and then recursively obtain c _m for m < k CPM symbol by 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, c _m for m > k is obtained in the same fashion from the inner SISO decoder backward recursion. This leads to one unique sequence for each combination of j and and yields the approximation where the notation has been used to denote the values of c. m ≠ k in the most likely sequence for each j and c _k (obtained as described above), and the probabilities are obtained from the output of log-likelihood adder A1. This implementation leads 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 provided by log-likelihood adder A1 after the (n)th turbo iteration are directly used by the EM algorithm. In the hard variant, is computed for each n, after which is obtained as

The probabilities are then used in the EM algorithm. In the unlikely event that the maximum in (42) is attained for more than one i' the lowest index i' for which the maximum is attained is arbitrarily selected.

Linear equations solver 311

As discussed above, channel estimates are 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 (n)th turbo iteration, filter f must first be obtained by solving the system of equations

Af = b, (44) with the system matrix and the right-hand side (see equation (17)) based on the latest estimate Computing f - and thereafter also filters g ₁ and g ₂ which is easy once f is known - is the task of the linear equation 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, considering the structure of the receiver 308 presented in the invention is now described.

First, it is observed that the system matrix A would be Toeplitz if the terms 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 A such as Gaussian elimination. However, in the present case A 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 several 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 A the conjugate gradient (CG) algorithm is guaranteed to converge. For Hermitian indefinite A 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, A is 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 products A • x. As documented in prior art, for Toeplitz A 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 A itself is not Toeplitz, the matrices H, R, H ₁ and H ₂ are all Toeplitz, hence a matrix-vector product A • x can be obtained from a sequence of Toeplitz matrix - vector 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 that Hence for (n) > 1 a good initial guess for the solution vector f for turbo iteration (n) is the value of f from turbo-iteration (n — 1). Computer simulations show that for (n) > 1 it is sufficient to carry out one iteration of the preconditioned CG solver to update f, thereby allowing for significant computational savings. For the purposes of the invention a small residual error in f 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 (n) only one CG update to f is carried out. The advantages of this approach over the much- discussed Levinson-Durbin recursion in the context of prior art equalization methods are (1) applicability to quasi-Toeplitz A; (2) 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 η is 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 ^h true consists of equal-magnitude, equal-phase arrivals at taps 1, 11 and 20, i.e. arrivals in the first, third and fifth CPM symbols since η = 4. Its magnitude is plotted in Fig. 7. Let h - the magnitude of which is also plotted in Fig. 7 - denote the result of the convolution of h _true and the impulse response of the low-pass filter from Fig. 3 (EPF block).

The channel estimator is set up to produce channel estimates with a length of 20 taps. The magnitudes of the pseudo-inverse estimate (before the beginning of the turbo iterations) and estimate after 10 soft EM channel updates following 10 turbo iterations are shown in Fig. 7 .

The corresponding filters f for the channel equalizer, obtained by solving system (44) with H, Hi and H ₂ obtained either from (see equation (13) and the explanations thereafter for details) are shown in Fig. 8. The length of the filters f was set to 101 taps. An ideal equalizing filter would satisfy with φ (k) a discrete time Dirac pulse.

The smaller the difference between (h * f*) and the better the cancellation of interference- symbol interference via feedback filters g ₁ and g ₂ can be expected to perform. The quantities are compared to respectively in Figs. 9 and 10. As the figures show, is closer to than which illustrates the benefit of updating the channel estimate after each turbo iteration. The magnitudes of filters g ₁ and g ₂ after turbo iteration 10 are shown in Fig. 11. Comparing Figs. 10 and 11, it can be seen that g ₁ and g ₂ attempt to cancel out the portions of (h * f*) respectively before and after main lobe in order for the combined effect of h,f g ₁ and g ₂ to be closer to that of a dirac pulse. In this example filter g ₁ is 52 taps long and is aligned with the first 52 taps of (h * f), while filter g ₂ is 67 taps long and is aligned with the last 67 taps of (h * f), which is 120 taps long. To avoid canceling out portions of the main lobe of (h * f), which as previously discussed would do more harm than good due to high CPM signal autocorrelation values, a cut-off value Rcut-off was introduced which in this example ensures the last c = 4 entries of g ₁ and the first c = 4 entries of g ₂ vanish.

The frame-error rate (FER) vs E _s /N ₀ (ratio 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 response h _true and frame parameters discussed above. A gap of approximately 4 dB is observed for FER = 10 ^-2 .

A possible approach for a simple CPM receiver would be to use a linear equalizing filter f without any feedback-equalization (i.e. no filters g ₁ or g ₂ ). In such a receiver, the estimation of h may be completely bypassed, and estimates for f 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 both h (or an estimate and f are required.

Computer simulations for the simple CPM receiver described above have not been carried out, but instead for a receiver where is obtained via the pseudo-inverse method, updates are obtained via the soft LMS algorithm, g ₁ and g ₂ are identically zero, and f is obtained as the minimum mean-square error (MMSE) solution in the absence of feedback filters g ₁ and g ₂ . The behaviour of this receiver is expected to be similar to that of the simple CPM receiver described above. Indeed, both are based on a linear equalizing filter f, the coefficients of which attempt to minimize (refer 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.