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1bit quantization and oversampling at the receiver: Sequencebased communication
 Lukas T. N. Landau^{1, 2}Email authorView ORCID ID profile,
 Meik Dörpinghaus^{2} and
 Gerhard P. Fettweis^{2}
https://doi.org/10.1186/s136380181071z
© The Author(s) 2018
 Received: 24 August 2017
 Accepted: 2 March 2018
 Published: 23 April 2018
Abstract
Receivers based on 1bit quantization and oversampling with respect to the transmit signal bandwidth enable a lower power consumption and a reduced circuit complexity compared to conventional amplitude quantization. In this work, the achievable rate for systems using such analogtodigital conversion with different modulation schemes is studied. The achievable rate and the spectral efficiency with respect to a given power containment bandwidth are considered. The proposed sequencebased communication approach outperforms the existing methods known from the literature on noisy channels with 1bit quantization and oversampling at the receiver. It is demonstrated that the utilization of 1bit quantization and oversampling can be superior in terms of the spectral efficiency in comparison to conventional amplitude quantization using a flash converter with the same number of comparator operations per time interval.
Keywords
 1bit quantization
 Oversampling
 ADC
 FasterthanNyquist signaling
 Achievable rate
 Markov capacity
1 Introduction
The achievable rate in case of Nyquist rate sampling is limited by the quantization resolution of the analogtodigital converter (ADC). In this regard, a flash converter consisting of N_{Comp} comparators limits the maximum achievable rate to \(\log _{2}(N_{\text {Comp}}+1)\) bits per Nyquist interval [1]. Differently, by time interleaving N_{Comp} comparator operations per Nyquist interval, \(2^{N_{\text {Comp}}}\phantom {\dot {i}\!}\) quantization regions exist, which enhances the limit of the achievable rate to N_{Comp} bits per Nyquist interval. In this regard, employing 1bit quantization and oversampling at the receiver is promising in terms of the achievable rate. Moreover, a 1bit ADC at the receiver is robust against amplitude uncertainties such that the automatic gain control can be simplified, and linearity requirements of the analog frontend are relaxed. Last but not least, a 1bit ADC requires only simple circuitry and does not need much headroom for amplitude processing, which makes it appropriate for low supply voltages and with this low energy consumption. All these motivate us to study the achievable rate of channels with 1bit output quantization and oversampling at the receiver.
A first study of the achievable rate with 1bit quantization and oversampling at the receiver has been carried out by Gilbert [2] showing a marginal benefit in terms of the achievable rate by oversampling. Subsequently, by using a Zakai bandlimited channel input processes, Shamai [3] has shown that oversampling can significantly increase the achievable rate. Both of these works consider a noiseless channel. For noisy channels, in [4] a benefit of oversampling has been proven in the low signaltonoise ratio (SNR) regime by studying the capacity per unit cost. Moreover, in [5] the achievable rate at high SNR has been studied by considering generalized mutual information, which did not confirm the high rates promised in [3].
Besides these papers on strictly bandlimited channels, also cases with less strict spectral constraints on the transmit signal have reported benefits from 1bit quantization and oversampling. For example, in [6, 7], where the channel is treated as memoryless, it has been observed that random processes such as additive noise and intersymbol interference can yield an increase of the achievable rate due to dithering. The same strategy, namely treating the channel as memoryless, has been applied for the utilization of fasterthanNyquist (FTN) signaling [8, 9] for channels with 1bit quantization and oversampling at the receiver [10]. An alternative strategy for communication with 1bit quantization and oversampling at the receiver is to transmit sequences which generate a unique output signal after 1bit quantization. In this regard, a waveform design supporting a unique detection of symbols with 16 quadrature amplitude modulation (16QAM) has been proposed in [11]. Without being exhaustive, the named papers show some benefit of oversampling when using 1bit channel output quantization. Nevertheless, none of these approaches provide achievable rates comparable to those which are presented in [3] for the noiseless channel.
In addition, 1bit quantization—not necessarily with oversampling—received increased attention in the context of multipleinput multipleoutput (MIMO) systems, where the low SNR regime is discussed in [12, 13], and the high SNR case is investigated in [14]. It is shown that the power penalty for the 1bit quantization in the low SNR regime is less than 2 dB. For the high SNR regime, channel state information can be exploited at the transmitter for a channel inversion strategy for the construction of receive signals appropriate for 1bit quantization. Moreover, the sequence design approach described in [11] for the singleinput singleoutput channel has been recently extended for the massive multipleinput singleoutput scenario in [15] and for the massive MIMO scenario in [16].
Furthermore, 1bit quantization is considered in the context of phase quantization [17] and a related concept named overdemodulation [18], where the received signal is downconverted with more than two carrier phases, different to 90 degrees. The increased number of carrier phases provides additional information in cases where a coarse quantization at the receiver is considered. Another study is presented in [19], where multidimensional quantizer designs are investigated in the context of channels with memory. The proposed quantizers in [19] are optimized for channels with memory whose quantization regions incorporate multiple receive samples.

QAM and phaseshift keying (PSK) symbols at Nyquist rate

FasterthanNyquist signaling with quadrature phaseshift keying (QPSK) and QAM symbols
i.e., we either design transmit sequences corresponding to a conventional modulation or with an increased signaling rate. Moreover, we study specific signal design approaches, (1) reconstructible 4 amplitudeshift keying (4ASK) / 16QAM sequences for conventional signaling rate and (2) runlengthlimited (RLL) sequences for FTN signaling. We also propose a sequence optimization strategy, based on the approach in [22], which maximizes the achievable rate by optimizing the transition probabilities of a Markov source model. The present work goes clearly beyond the studies we have presented before on this subject. The main extensions are the consideration of PSK signaling, the consideration of the spectral efficiency with different outofband power thresholds, the extended description of the sequence optimization strategy including the explanation of the lower bound on the achievable rate and the overall performance comparison for a large number of transmit signaling schemes under the same conditions. Moreover, in the present work, we describe the constraints on the waveform for the reconstructable 16QAM sequences and discuss the zerocrossings in sequences composed of weighted cosine pulses.
In [23], we treat the channel with 1bit quantization and oversampling at the receiver and rootraisedcosine (RRC) transmit and receive filters with infinite memory. The study serves as a proof of concept for strictly bandlimited channels. The results in [23] in terms of the achievable rate are comparable to [3]. However, the utilization of RRC filters is impractical for many applications. In this regard, consider that the use of RRC filters implies an extensive memory of the channel when having 1bit quantization and oversampling at the receiver, which dramatically increases the computational complexity of the sequence demapping, e.g., by utilizing a trellis receiver. Differently to [23], in the present work, we consider transmit pulses with a shorter length in time domain such as the cosine pulse and the Gaussian pulse. These waveforms provide a good tradeoff between spectral efficiency and channel memory. We rely on the assumption that the residual outofband radiation can be tolerated for specific applications such as boardtoboard communication at subTerahertz carrier frequencies and intrachipstack communications, e.g., using throughsilicon vias. Our results show that the proposed methods outperform the existing methods in terms of the spectral efficiency. Furthermore, our results show that 1bit quantization with oversampling at the receiver can yield comparable and even superior spectral efficiency than conventional methods based on amplitude quantization when operating in the low quantization regime with the same number of comparator operations per time interval.
In the present work, we consider sequences with infinite length and optimal receivers which rely on the true or an auxiliary channel law. Alternative approaches based on fixedlength sequences and receive strategies with a lower complexity are presented in our prior work [24, 25].
The rest of the paper is organized as follows. Section 2 introduces the system model. In Section 3, we recall a method to lowerbound the achievable rate for channels with memory, which we will subsequently apply to evaluate the performance of the studied signaling schemes. Afterwards, in Section 4, we present an approach to generate reconstructible 4ASK/16QAM sequences. Moreover, the application of RLL sequences, which are used in combination with FTN signaling, is described in Section 5. In Section 6, we propose an optimization strategy for sequence design, which maximizes the given lower bound on the achievable rate. We discuss the numerical results in Section 7, and finally, a conclusion is given in Section 8.
Notation: Bold symbols, e.g., y_{ k }, denote vectors, where k indicates the kth symbol, or more specifically, the samples which belong to the kth input symbol time interval. y_{ k } is a column vector with M entries, where M is the oversampling factor w.r.t. a transmit symbol. Sequences are indicated with x^{ n }=[x_{1},…,x_{ n }]^{ T }, and sequences of vectors are denoted as \(\boldsymbol {y}^{n}= \left [\boldsymbol {y}_{1}^{T},\ldots,\boldsymbol {y}_{n}^{T}\right ]^{T}\). A segment of a sequence is written as \({x}^{k}_{kL}=[ {x}_{kL}, \ldots, {x}_{k} ]^{T}\) and \(\boldsymbol {y}^{k}_{kL}=\left [ \boldsymbol {y}_{kL}^{T}, \ldots, \boldsymbol {y}_{k}^{T} \right ]^{T}\). Random quantities are denoted by upright letters, e.g., y_{ k } is random vector. A simplified notation for probabilities of random quantities is used with \(P\left (\boldsymbol {y}^{n} \vert x^{n} \right)= P\left ({\mathbf {{y}}}^{n} = \boldsymbol {y}^{n} \vert \mathrm {x}^{n} = x^{n} \right)\). Exceptions are explicitly declared.
2 System model
M_{Tx} larger than 1, e.g., M_{Tx}=2 or 3, corresponds to fasterthanNyquist signaling following the principle in [8, 9]. In this regard, a compression of channel input symbols in time is given, such that M_{Tx} channel input symbols are emitted in the unit time interval T_{s}. The compression of input symbols in time provides additional degrees of freedom which can be exploited for the waveform design. In order to avoid extensively complex trellisbased receivers, a transmit filter h(t) with short impulse response is favorable. In this context, different standard pulses (Gaussian pulse, cosine pulse, and rect pulse) will be examined in terms of the spectral efficiency for the considered channel.
whose short impulse response is favorable for a trellisbased sequence detection. The system impulse response is denoted as v(t)=(h∗g)(t).
The channel input symbols x_{ k } are taken from discrete modulation alphabets, specifically, a QPSK, QAM, or PSK symbol alphabet \(\mathcal {X}\) with the cardinality \(\left  \mathcal {X} \right \). While for QAM we use the standard constellation, for PSK constellations, the input symbols are given by \(\mathrm {x}_{k}= e^{j 2 \pi \frac {{\mathrm {m}_{k}} + \frac {1}{2} }{\left  \mathcal {X} \right } }\) with \(\mathrm {m}_{k} \in \left \{0, \dots, \left  \mathcal {X} \right 1 \right \}\).^{2} The channel including transmit and receive filtering and quantization is a discrete input discrete output channel with memory, for which it is known that the channel capacity can be asymptotically achieved by a stationary Markov source [21]. Thus, we consider a stationary Markov source model, such that each channel input symbol x_{ k } depends on L_{src} previous symbols \(P\left (x_{k} \vert x^{k1}\right) = P\left (x_{k} \vert x^{k1}_{kL_{\text {src}}} \right) = P\left (s_{k} \vert s_{k1} \right)\), where for the latter, we use the state variable \(\mathrm {s}_{k}=\mathrm {x}^{k}_{kL_{\text {src}}+1}\) to describe the current state of the source. To simplify the notation, we use the shorthand notation \({P}_{i,j}=P\left (\mathrm {s}_{k}=j \vert \mathrm {s}_{k1}=i \right)\). We denote the stationary distribution of the source states by \({\mu }_{i}=P\left (\mathrm {s}_{k}=i\right)\) for \(i=1, \ldots, \left  \mathcal {X} \right ^{L_{\text {src}}}\).
where i and j are positive integers accounting for the row and the column number, respectively.
3 Achievable rate
The considered channel in (3) has memory. A channel output y_{ k } depends on previous input symbols and previous channel outputs y^{k−1}, where the latter is induced by the correlation of the noise samples. Considering blockwise stationarity and ergodicity with respect to y_{ k }, the simulationbased methods in [26–29] can be applied for computing the achievable rate.
3.1 Lowerbounding by considering an auxiliary channel law
where we have used that y_{k−1} is independent of x_{ k }. Numerator and denominator in (14) can be computed directly when considering a specific system model.
3.2 Transition probabilities
Because the computation of the transition probabilities incorporates an integration over a multivariate circularly symmetric Gaussian distribution, it is favorable in terms of computational complexity to decompose them into statistically independent realvalued components. With \(\text {Re}\left \{{\mathbf {z}}_{k} \right \} = \acute {\mathbf {z}}_{k}\) and \(\text {Im}\left \{ {\mathbf {z}}_{k} \right \} = { \grave {\mathbf {z}}_{k}} \), a shorthand notation is used, which is also applied for the x_{ k } and n_{ k }.
with the mean vector \(\boldsymbol {\mu }_{x}\! =\! \boldsymbol {V}\!(N\!)\! \boldsymbol {U}\!(N\!) \acute {x}_{kLN}^{k}\) and the covariance matrix \(\boldsymbol {R}_{N\,+\,1}\,=\,\mathrm {E}\!\left \{ \boldsymbol {D}(N) \boldsymbol {G}(N) \acute {\mathbf {n}}_{k\!N\!\xi }^{k} \left (\acute {\mathbf {n}}_{kN\xi }^{k}\right)^{T}\right. \left.\boldsymbol {G}(N)^{T} \boldsymbol {D}(N)^{T}{\vphantom {\left (\acute {\mathbf {n}}_{kN\xi }^{k}\right)^{T}}} \right \}\), where G(N) is real valued.
where \( \acute {\mathbb {Y}}_{kN}^{k}=\left \{ \acute {\boldsymbol {{z}}}_{kN}^{k} \Big  Q \left \{ \acute {\boldsymbol {z}}_{kN}^{k} \right \} = \acute {\boldsymbol {y}}_{kN}^{k} \right \}\). QAM sequences are described by two independent ASK sequences. In case of a PSK input alphabet, the real and imaginary part of the received signal are independent when they are conditioned on the input, which allows to write the probability distribution as a product.
4 Reconstructible ASK sequences
Note that the principle can be applied for all waveforms which fulfill the constraints described in Appendix A, e.g., when h(t) is a cosine pulse with length 2T_{s}. For the illustrating example, we consider a 4ASK input alphabet, 3fold oversampling (M=3), and a signaling rate with M_{Tx}=1.
4.1 The reconstruction issue of sequences with i.i.d. symbols

x_{ k } and x_{k+1} can be directly reconstructed based on the current M+1 ADC output samples in the time interval kT_{s}≤t≤(k+1)T_{s} (“decision”)

x_{k+1} can be reconstructed based on the current M+1 ADC output samples in case x_{ k } is known at the receiver, or x_{ k } can be reconstructed in case x_{k+1} is known (“forward”)

Possible ambiguity with transitions in state D (“ambiguity1”)

Possible ambiguity with transitions in state C (“ambiguity2”).
4.2 A state machine representation for reconstructible ASK sequences
Source entropy rates of reconstructible sequences
Sequence property  L_{src}=1  L_{src}=2  L_{src}=3  L_{src}=4 

\({\lim }_{n \to \infty } \frac {1}{n} H(\mathrm {x}^{n})\) [bit/symbol]  1.585  1.7237  1.7583  1.7678 
5 Runlengthlimited sequences
An alternative approach to model transmit sequences which can be uniquely reconstructed at a receiver with a 1bit ADC is to use runlengthlimited (RLL) sequences [32] in combination with FTN signaling. RLL sequences are a natural choice because they convey the information in the distances of zerocrossings or runlengths. As the temporal positions of a change of the signal should be controlled on a more finegrained timegrid than T_{s}, we have to choose M_{Tx}>1 in (1), which corresponds to FTN signaling.
Maximum entropy of dconstrained sources
Runlength constraint  d=1  d=2  d=3 

Max entropy rate [bit/symbol]  0.6942  0.5515  0.4650 
The d constraint implies redundancy within the channel input sequence. However, in combination with a higher signaling rate, the RLL sequences can yield a benefit in terms of spectral efficiency for the case of 1bit quantization at the receiver, which is different from the unquantized FTN [33]. This is due to the fact that the FTNcaused intersymbol interference cannot be corrected by the trellisbased receivers because of the loss of the additional amplitude information due to the 1bit ADC. In this regard, the sequences need to be well shaped, such that the intersymbol interference does not induce a flip of the sign of current symbols. In this regard, the RLL sequences can tolerate some intersymbol interference, e.g., of the considered channel, at a relatively low cost in redundancy. In addition, the RLL sequences yield a higher concentration of the signal power of the transmit symbol sequence at lower frequencies. Depending on the bandwidth criterion, this might further increase the spectral efficiency. For complex transmit symbol sequences, we consider independent RLL sequences for the real and the imaginary part.
6 Maximization of a lower bound on the achievable rate using an expectationbased BlahutArimoto algorithm
where λ_{max} is the largest real eigenvalue of \(\tilde {\boldsymbol {A}}_{\text {adj}}\) and b_{ i } and b_{ j } are entries of the corresponding eigenvector. The method is applied iteratively as \(\hat {T}_{i,j}\) itself is a function of P_{i,j}, where each iteration involves the generation of x^{ n } and y^{ n }.
Note that this optimization procedure does not take into account the power spectral density (PSD) of the resulting channel input signal. Moreover, the optimization has an influence on the average transmit power and, thus, on the SNR.
7 Numerical results
In this section, we numerically evaluate the achievable rate based on the lower bound in (10). The simulationbased computation of the RHS of (10), i.e., of the argument of the limit, is carried out based on a sequence of length n=10^{6} symbols. Whenever the proposed sequence optimizaton strategy is applied, 19 iterations of the loop in the algorithm described in Section 6 have been carried out. The power containment bandwidth and the SNR are postcomputed as the transmit signal bandwidth depends on the individual Markov source.
with the stationary input state distribution μ_{ i }. Hence, the corresponding PSD is given by the Fourier transform \( S_{\mathrm {x}}(f)= \frac {M_{\text {Tx}}}{T_{\mathrm {s}}} \sum _{k=\infty }^{\infty } c_{k} e^{j 2 \pi \frac {k T_{\mathrm {s}} }{ M_{\text {Tx}} }f} \), where the infinite sum can be approximated by considering a very large number of coefficients. Together with the transfer function H(f) of the transmit filter h(t), the PSD of the transmit signal is given by \(S(f)=S_{\mathrm {x}}(f) \left H(f)\right ^{2} \). In the following, we will refer to the twosided power containment bandwidth B_{90%} (or B_{95%}), which implies that a certain amount, e.g., 10% (or 5%), of the transmit power is emitted outside the nominal bandwidth^{8}.
Note that the transmit power depends on the Markov source modeling the input sequence x^{ n } and the transmit filter h(t). In the sequel, if not otherwise stated, we assume the 90% power containment bandwidth (B_{90%}).
Overview on considered scenarios with 1bit quantization at the receiver
Modulation alphabet  Transmit pulse  Sequence design  M  M _{Tx}  N 

QPSK  Cosine  i.u.d.  1  1  0 
16QAM  Cosine  i.u.d.  2,3  1  1 
16QAM  Gaussian  i.u.d.  2,3  1  1 
16QAM  Rect  i.u.d.  2,3  1  1 
16QAM  Cosine  Optimized  2,3  1  1 
16QAM  Cosine  Reconstructible  3  1  1 
64QAM  Cosine  Optimized  2,3  1  0 
256QAM  Cosine  Optimized  2,3  1  0 
8PSK  Cosine  i.u.d.  2,3  1  0 
8PSK  Cosine  Optimized  2,3  1  0 
16PSK  Cosine  i.u.d.  2,3  1  0 
16PSK  Cosine  Optimized  2,3  1  0 
QPSK  Cosine  i.u.d.  1  2,3  1 
QPSK  Cosine  Optimized  1  2  1 
16QAM  Cosine  Optimized  1  2  0 
QPSK  Cosine  Optimized  1  3  0 
QPSK  Cosine  RLL, d=1  1  2  1 
QPSK  Cosine  RLL, d=2  1  3  1 
QPSK  Cosine  RLL, d=1  1  3  0 
To evaluate the burden for the use of 1bit quantization and oversampling, we compare our approach with the channel without output quantization and RRC filtering with a rolloff factor of 0.3. In terms of FTN signaling, we compare with a reference system without quantization and with a rolloff factor equal to 1 and with various compression factors τT, cf. the notation in [33]. Moreover, we compare our results on the spectral efficiency with the AWGN channel capacity, normalized with the power containment bandwidth, assuming a flat spectrum.
7.1 Transmit pulse
7.2 QAM
Based on the lower bound on the achievable rate in (10), Fig. 9 shows that the use of a higher order transmit symbol alphabet, namely 16QAM, is beneficial. While with 1bit quantization and without oversampling just 2 bits per channel use can be achieved (1 bit in the real and 1 bit in the imaginary component), with an increasing oversampling factor M the achievable rate increases. Moreover, it is illustrated that a sophisticated sequence design can further improve the achievable rate significantly compared to i.u.d. input symbols. In this regard, it is shown that the proposed method to model reconstructible sequences (Section 4), which is described for M=3, achieves an achievable rate fairly close to the optimized sequences (Section 6). With the approach based on reconstructible sequences, the achievable rate approaches the input entropy rate of 2·1.7678[bpcu], cf. Table 1, in the high SNR regime, where the factor 2 is due to the use of a complex modulation. The corresponding PSDs are shown in Fig. 10. Note that the sequence optimization depends on the SNR and that the illustrated spectra consider high SNR (30 dB). Figure 11 shows that the achievable rate can be further increased by utilizing even larger modulation alphabets, e.g., 64QAM or 256QAM. In this regard, note that the achievable rate for a 256QAM alphabet is larger than \(2 \log _{2} (M+1)\), for M=2 and M=3.^{11} This is remarkable, because it is higher than the upper limit for the noiseless channel without receive filter described in Appendix D. We explain this by the circumstance that with the receive filter the system impulse response is enlarged, such that new signal evolutions are enabled, leading to more zerocrossing patterns. This is in line with the data processing lemma because the subsequent quantization is a suboptimal processing step. Moreover, it is also remarkable, because \(2 \log _{2} (M+1)\) is the maximum achievable rate for flash ADC based sampling with M comparators. For 64QAM and 256QAM, the achievable rate is lowerbounded by the utilization of a simplifying auxiliary channel model with N=0. The sequence optimization only considers a peak power constraint and no bandwidth constraint. Because of this and the circumstance that our SNR definition involves the bandwidth, we expect that at low SNR the actual capacity is higher than that computed with our approach.
7.3 PSK
Unlike as for QAM, due to the constant modulus transmit symbols, the average transmit power is not strongly influenced by the applied sequence optimization strategy. However, as discussed for QAM modulation, the nominal bandwidth depends on the PSD of the transmit signal and, thus, on the applied Markov source which describes the transmit symbol sequences. Thus, the SNR in (35) depends on the chosen sequence design, which explains the slight horizontal shift of corresponding markers in Fig. 13.
7.4 FasterthanNyquist signaling
In the following, we evaluate the achievable rate with FTN signaling, i.e., M_{Tx}>1, on the one hand for RLL sequences as discussed in Section 5 and on the other hand also for transmit sequences with i.u.d. symbols and for optimized sequences (Section 6) with QPSK and 16QAM input alphabets. Here, we choose an equal signaling and sampling rate, i.e., M=1.
Regarding the auxiliary channel law utilized for lowerbounding the achievable rate, the maximum can be practically approached by considering N=ξ=M_{Tx}. However, we have considered memories of N=1 or N=0, not necessarily N=M_{Tx}, to limit the computational complexity.
which corresponds to a larger receive bandwidth. In the figures, we refer to this exception by the notation wideband Rx. In this case, the achievable rate converges to the source entropy rate. However, due to the larger bandwidth of the receive filter, more noise is captured such that the achievable rate saturates at higher SNR.
Moreover, it can be observed that the optimized sequences (Section 6) yield a slightly larger achievable rate than RLL sequences. Compared to the RLL sequences, the sequence optimization strategy has more degrees of freedom for the construction of zerocrossings. Surprisingly, M_{Tx}=3 yields an even larger achievable rate in the high SNR as compared to M_{Tx}=2, which is counter intuitive. On one hand, increasing the signaling rate implies a relative expansion of the system impulse response w.r.t. \(\frac {T_{\mathrm {s}}}{M_{\text {Tx}}}\) which in our case strongly attenuates fast signal transitions. This is why at low SNR, M_{Tx}=2 holds a benefit in the achievable rate w.r.t. to a channel use in comparison to M_{Tx}=3. At high SNR, utilization of M_{Tx}=3 can effectively exploit more bandwidth for communication. This is possible due to the fact that the considered transmit pulse is not strictly bandlimited. Finally, the expansion of the system impulse response provides more degrees of freedom which is in general favorable for the construction of zerocrossings.
In addition, a 16QAM alphabet has been considered for sequence optimization (Section 6) with M_{Tx}=2. Due to the additional degrees of freedom, this approach shows a much better performance in terms of achievable rate compared to the other waveforms with M_{Tx}=2.
We have compared our results with RRCmatched filteringbased FTN signaling without quantization. The compression in time is such that the transmit pulses have a distance of τT·T_{ x }, where T_{ x } would be the conventional transmit symbol duration without FTN. We have computed a lower bound on the achievable rate by using a truncationbased auxiliary channel law where we have used for τT=0.5, 0.4, and 0.3 a truncated system impulse response of length (L+1)=3, 5, and 6, respectively.
Moreover, by the comparison of Figs. 17 and 12, we make the important observation that the communication based on the FTN signaling scheme requires a significantly lower SNR. This can be explained by the fact that the transmit filter h(t) in (36) is not strictly bandlimited. In this regard, the spectral copies at a signaling rate of \(\frac {1}{T_{s}}\) when M_{Tx}=1 implicitly restrict the sequence design which cannot be compensated by a large input alphabet. The faster signaling rate offers more degrees of freedom for the sequence design at higher frequencies. However, in a scenario with strict bandlimitation [23], e.g., by considering Nyquist pulses, this effect vanishes.
7.5 Relation of the spectral efficiency and the oversampling factor in the high SNR limit
8 Conclusions
We have studied the achievable rate for an additive Gaussian noise channel with 1bit output quantization and oversampling at the receiver, which is promising in terms of a simplification of circuitry and a reduction of the energy consumption at the receiver. As the transmit signal is not strictly bandlimited, we have considered power containment bandwidth criteria with 90% and alternatively 95% power containment. The transmit sequences are constructed based on various QAM and PSK input symbol alphabets and various signaling rates. Concrete sequence designs, namely reconstructible 4ASK (and with this 16QAM) sequences and runlengthlimited sequences for fasterthanNyquist signaling rates, are proposed. Furthermore, a sequence optimization strategy is studied which approaches the Markov capacity in the high SNR regime. The performance is evaluated in terms of the achievable rate and the spectral efficiency. We have observed that the proposed approaches outperform the existing methods on communication with 1bit quantization and oversampling at the receiver. For a number of methods, it has been shown that 1bit quantization and oversampling at the receiver yields a comparable or even superior spectral efficiency than conventional amplitude quantization using a flash converter with the same number of comparator operations per time interval.
One key observation is that among the proposed methods, the spectral efficiency is maximized by FTN signaling. This suggests that for the channel input signal, the resolution in time is preferable in comparison to the resolution in amplitude. However, it is known for the unquantized case that FTN exploits the excess bandwidth [33], such that it can be expected that the advantage of FTN vanishes for more strict spectral constraints, cf. [23]. In summary, the results show that the use of receivers with oversampled 1bit quantization is promising. The proposed ideas are a first step to a more complete understanding of the achievable rate and of an optimal transmit sequence design for such channels. Aspects like the robustness of these signaling schemes towards jitter and timing synchronization errors remain for further study. It is shown that the presented methods based on 1bit quantization and oversampling at the receiver require only 2−3 dB more transmit energy (at 5−10 dB SNR and 90% power containment bandwidth) in comparison to a conventional communication system design with Nyquist sampling and high resolution in amplitude.
9 Appendix A
9.1 The system impulse response for reconstructible sequences
which describe the combinations of the input symbols. Note that some of the constraints are redundant. Moreover, symmetries have been exploited. Besides the illustrated triangular waveform with \(\left [ v_{0},\ldots,v_{4} \right ]=\left [1, 0.666, 0.333, 0, 0 \right ]\), the waveform with the transmit pulse given in (36) jointly with the assumptions on the receive filter in Section 2 corresponding to the coefficients \(\left [ v_{0},\ldots,v_{4} \right ]=\left [0.9449, 0.759, 0.387, 0.1037, 0.0042 \right ]\) fulfills these constraints.
10 Appendix B
10.1 Reconstructable 4ASK sequences with finite memory
11 Appendix C
11.1 A lower bound based on the auxiliary channel law (reverse)
where D(·∥·) is the KullbackLeibler divergence [35] which is always nonnegative [36, Th. 8.6.1].
12 Appendix D
12.1 Upperbounding the capacity of the noiseless channel without receive filter
which describes a raised or lowered cosine function in the interval with the running time variable τ. Its frequency is such that x(t) has at max one zerocrossing per time interval kT_{s}≤τ<(k+1)T_{s}. Now, we consider that this signal is quantized with 1bit and sampling rate \(\frac {M}{T_{\mathrm {s}}}\). The fact that there is at most one zerocrossing in the time interval T_{s} implies that the maximum output entropy and with this also the capacity are upperbounded by \(2\log _{2}(M+1)\), where the factor 2 accounts for the complex equivalent.
A matched filter would also depend on the sequence design, i.e., on the statistical dependencies of the individual x_{ k }.
Thus, the input symbols are not placed on the real and imaginary axes which are the thresholds of the applied 1bit quantizer.
The system impulse response v(t) is normalized implicitly, because it is considered that h(t) has unit energy normalization.
The considered integrateanddump receiver is an exceptional case, where the noise correlation can be perfectly described on the sampling grid (D=1), although there is no bandlimitation.
For the computation, symmetries in the input sequences can be exploited to reduce the number of integrations.
The case of QAM sequences follows by using the concept for the real as well as for the imaginary axis.
In case of asymmetric spectra, it is considered that the power of the outofband radiation is equally splitted into the frequency range towards \(f=\infty \) and the frequency range towards \(f= \infty \).
Notes
Declarations
Acknowledgements
We thank the referees for very useful comments.
Funding
This work has been supported in part by the German Research Foundation (DFG) within the Collaborative Research Center SFB 912 “Highly Adaptive EnergyEfficient Computing.”
Availability of data and materials
Not applicable.
Authors’ contributions
All authors contributed to the conception and design of the study. LL drafted the manuscript and did the simulation work. All authors contributed to the interpretation of the results. MD reviewed and edited the manuscript and helped with the revisions. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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Authors’ Affiliations
References
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