Communications over phase-noise channels: A tutorial review

Abstract

In this tutorial paper, we first review the main information-theoretic results on channels affected by a timevarying phase noise. The main maximum a posteriori symbol detection algorithms to be employed in such a challenging scenario are then described considering linear modulations and advanced coding schemes based on iterative detection and decoding. The role of pilot symbols will be also discussed. , as well as for powerful channel codes to be decoded iteratively, such as turbo codes or lowdensity parity-check (LDPC) codes (see [3][4][5][6][7][8][9][10][11][12] and references therein). In this latter case, which is of interest in this paper, the algorithms for turbo or iterative detection/synchronization and decoding can be classified according to the way detection/synchronization is obtained.A first family of algorithms can be applied to turbo codes and serially concatenated convolutional codes (CCs) but not to LDPC codes (unless an LDPC code is used as outer code in a serial concatenation). These algorithms modify the component decoders so that they can also compute an implicit (e.g., see [3,4,8,20]) or explicit (e.g., see [7-9]) phase estimate. We have, in this case, joint detection/synchronization and decoding. These algorithms usually require to work on an expanded trellis and the adoption of techniques for complexity reduction [21,22] becomes mandatory. A different approach, able to effectively reduce the computational complexity especially when the component encoders are rotationally invariant (RI), is adopted in [7]. As an example, in the simplest case of a differentially encoded M -ary phase-shift keying (M -PSK) modulation, it is shown in [7] that the exact maximum a posteriori (MAP) symbol detector, under the assumption that the phase noise can be modeled as a Wiener process, can be implemented by a forward-backward estimator of the phase probability density function (PDF), followed by a symbol-by-symbol completion to produce the a posteriori probabilities of the information symbols. To practically implement the forward-backward carrier phase estimator, a couple of schemes with different complexity have been proposed in [7] along with the extension to general RI encoders. The resulting algorithms do not require the use of known (pilot) symbols to trigger the iterative detection/decoding process and are, up to now, the most robust in the literature. This latter approach can be also extended to CPMs [9].A second family is composed of algorithms that, on the contrary, leave the decoder unmodified and complement it with a separate detector/synchronizer whose aim is to estimate and compensate for the carrier phase and frequency uncertainties prior to decoding (e.g., see [5,6,10,11]). Hence, they can be used for LDPC codes also. This separate detector/synchronizer operates in soft-decisiondirected mode in the sense that it employs the soft information provided by the decoder to refine the estimates at each iteration, using, to a larger extent, symbols with highest reliability. In other words, a soft-decis...

Bootstrapping Iterative Demodulation and Decoding Without Pilot Symbols

“…Prominent examples of channels with memory are the intersymbol interference channel, the phase noise channel, and the fading channel. Among channels with memory, the phase noise channel is considered here, a recent tutorial on the subject being [1].…”

Capacity Bounds for MIMO Microwave Backhaul Links Affected by Phase Noise

“…We elaborate further on the accuracy of the model (1) in Section II-B. We assume that the phase-noise samples {θ k } form a Wiener process [7], i.e.,…”

“…Unfortunately, the capacity of the phase-noise channel is not known in closed form even for simple channel models, although capacity bounds and asymptotic results in the limiting regime of high SNR have been reported in the literature. Lapidoth characterized the capacity of the general class of stationary phase-noise channels (the widely used Wiener model [7] belongs to this class) in the high-SNR regime [8]. Specifically, he showed that whenever the phase-noise process has finite differential-entropy rate, the high-SNR capacity is equal to half the capacity of an AWGN channel with the same SNR, plus a correction term that accounts for the memory in the phase-noise process.…”

“…Capacity lower bounds obtained by numerically computing the information rates achievable with various families of finite-cardinality independent and identically distributed (i.i.d.) input processes (e.g., QAM, PSK, and APSK constellations) have been reported e.g., in [14], [7], [15]. The numerical evaluation of these bounds is often based on the algorithm for the computation of the information rates for finite-state channels proposed in [16] (see [17] for different approaches based, e.g., on particle filtering).…”

On discrete-time modeling of the filtered and symbol-rate sampled continuous-time signal affected by Wiener phase noise

“…Phase noise is due to both laser oscillators used for up-and down-conversion [4], and to crossphase modulation that arises in wavelength-division-multiplexing systems [5]. 5 A recent tutorial on transmission over phase noise channels is [6].…”