I N WIRELESS communication systems, information sig- nals are communicated through a wireless medium. These signals undergo various distortions before being received at the receiver [1]. To estimate these distortions, the transmitter transmits reference signals called pilots, which are known to the receiver. Typically, these pilot symbols are transmitted at higher energy than the data symbols. The receiver carries out channel estimation, that is, an estimate of the channel is obtained using the received pilot symbols. For a rapidly varying channel, these pilot symbols are transmitted frequently. However, the pilot symbols do not carry any information and therefore frequent pilot transmissions lead to spectral efficiency loss. Different techniques are needed to reduce the frequency of pilot transmissions. Many schemes have been proposed to achieve this [2], [3], [4], [5], [6]. In this letter we focus on the differential communication scheme.
Differential communication has been studied before in the context of space time block codes (STBC) [4], [5], [6]. In [4] the authors use an initial pilot transmission to estimate the channel. Next, assuming that the channel is constant for the next data transmission, the initial channel estimate is used to detect the STBC encoded data. The detected data is modeled as a pilot and used to estimate the channel again which enables more data detection. In [5], the authors use differential encoding of information symbols to enable detection at the receiver without the need for a channel estimate. Authors in [6] demonstrate that in addition to the advantages of pilot free communication, differential communication also provides rate advantages at higher layers above the physical layer. In the STBC context, differential communication is possible thanks to the channel becoming predictable in a multi-antenna system [6]. However, this is not readily applicable for single antenna system in a doubly selective channel. New modulation schemes are required for achieving predictability in such communication systems.
Recently, the authors in [7], [8] proposed a delay-Doppler (DD) domain modulation scheme called Zak-transform based orthogonal time frequency space (Zak-OTFS). One of the salient features of Zak-OTFS is that the channel when viewed in DD domain is predictable. This is because the channel changes as fast as the physics of the reflectors would allow, which in practice is slowly varying. With each Zak-OTFS frame typically spanning about few milliseconds, the channel remains almost stationary for a few frame transmission durations.
For channel estimation in Zak-OTFS many DD pilot frames have been proposed [8], [9], [10]. These include the point pilot (PP) frame [8], embedded pilot (EP) frame [9], and spread pilot (SP) frame [10]. The PP frame has a single non-zero value corresponding to the pilot in the whole frame. The EP frame has both pilot and data symbols separated by a guard region to prevent interference between the two. The SP frame has pilot symbols superimposed on the data symbols in way that makes the two mutually unbiased. The PP frame has the least spectral efficiency while SP has full spectral efficiency with EP frame in between. On the other hand, estimation complexity is the highest for SP frame followed by EP and PP frames.
In this letter, we propose differential communication scheme for doubly-dispersive channel using Zak-OTFS. Our contributions can be summarized as below.
• We leverage the predictability of the DD domain channel in Zak-OTFS to design a communication scheme that reduces the frequency of pilot transmissions. • We propose a differential communication scheme in doubly-dispersive channels using Zak-OTFS. Since all practical channels are doubly-dispersive to various degrees, the method proposed in this letter has practical implications. • We analytically show that the detected data symbols can be used as pilot symbols to estimate the DD channel. We use a time-varying DD channel to show that the estimated channel in a given instant can enable data detection in the subsequent time instant and this process can be repeated over multiple time instants, alleviating the need for frequent pilot transmissions. • The proposed scheme makes for a low-complexity receiver while achieving full spectral efficiency and numerical results demonstrate that the proposed differential communication scheme achieves better biterror performance at lower complexity via spread pilots [10], [11]. Notation: x denotes a complex scalar, x denotes a vector with nth entry
In Zak-OTFS, each information symbol is mounted on a pulsone which is a pulse train modulated by a tone [7], [8]. A Zak-OTFS frame consists of MN DD bins, where M is the number of delay bins and N is the number of Doppler bins. The frame bandwidth and time are B = M ν p and T = N τ p , respectively. ν p =foot_0 /τp is the Doppler period and τ p is the delay period [7]. For a data frame, MN information symbols drawn from a constellation alphabet (e.g., 4-QAM) is mounted on these MN DD bins by modulating each of the MN pulsones. The input-output relation of the Zak-OTFS system [8] can be expressed as (we avoid providing a repeat of the system model for brevity) 1 :
where y ∈ C MN ×1 is the vector of received symbols in the DD domain, H ∈ C MN ×MN is the end-to-end channel matrix, x ∈ C MN ×1 is the vector of transmitted symbols in the timedomain, and n ∈ C MN ×1 is the additive Gaussian noise. For detection of x from y, we need to compute an estimate of H. This is carried out by transmitting pilots.
To estimate the channel in a Zak-OTFS system, at the receiver, a DD domain cross-ambiguity function (which also happens to be the maximum-likelihood estimate of the channel [10]) is computed between y and x. The DD domain cross-ambiguity function is given by:
where Y[k , l ] = y[k + lM ] and X[k , l ] = x[k + lM ] are the matrix representations of the corresponding vectors, where k = 0, 1, . . . , M -1, l = 0, 1, . . . , N -1. The estimation of the channel matrix, H happens in two steps. First, A y,x gives the "model-free" [10] estimate, h eff , of the effective channel matrix h eff . Next, the estimate h eff ∈ C M ×N is used to construct the estimated channel matrix H (see [8, eq. ( 38)]). H is then used for subsequent data detection.
As mentioned earlier, in Zak-OTFS, information symbols are mounted on pulsone bases in the DD domain. The discrete DD domain point pulsone indexed by (k 0 , l 0 ) is:
where the term e j 2π N nl is to account for quasi-periodicity of the pulsone [10]. Information symbols modulate pulsone at each (k 0 , l 0 ) tuple where k 0 = 0, 1, . . . , M -1, l 0 = 0, 1, . . . , N -1. The cross-ambiguity between a pulsone indexed by (k 0 , l 0 ) and (k 1 , l 1 ) is (up to a phase) [12]:
where A X (p) ,X (p) [k , l ] is the self-ambiguity of the pulsone.
To generate the time-varying DD domain channel, we make use of the following equations. For ith path (i = 0, 1, . . . , P -1, for a P path channel) with delay τ i and Doppler spread ν i , the distance between the transmitter and receiver is:
where f c is the carrier frequency and c is the speed of light.
To generate the channel gains we assume a path loss model given by:
where α i is a function of power profile of the channel and θ ∼ U[-2π, 2π). For a given time t, the tuple (h i (t), τ i , ν i ) characterizes the DD channel.
In this section, we derive the conditions that enable differential communication in Zak-OTFS. The time-domain symbols mounted on pulsone bases is:
where
is the (k 0 , l 0 )th pulsone basis in the timedomain (obtained by the inverse discrete Zak transform [13] of (3)), X[k 0 , l 0 ] is the (k 0 , l 0 )th data symbol mounted on the corresponding pulsone bases. The input output relation in time-domain is given by [14]:
The maximum likelihood estimate of the channel is equivalent to computing the cross-ambiguity between the received and transmitted symbols [10,Appendix H]. Therefore, at the receiver, to estimate the channel, we compute the time-domain cross-ambiguity between the received time-domain symbols (in ( 8)) and the transmitted symbols (in ( 7)):
where step (a) follows because the pulsone samples and noise samples are uncorrelated. Substituting n = nk in (9):
[k , l ] (i.e., the timedomain ambiguity is same as the DD domain ambiguity), substituting (4) in B[k , l ], we have:
This implies that the cross-ambiguity between the transmitted data frame and the received frame is the twisted convolution between the effective channel h eff [k , l ] and the inner product between the transmitted data and its delay and Doppler shifted version. If the information symbols are chosen uniformly at random from a constellation, asymptotically we havefoot_2 :
where e d is the energy of information symbols. The crossambiguity between the received and transmitted data symbols is therefore approximately equal to the estimate of the effective channel up to a scale.
Remark 1: Notice that in the case of spread pilot ([10] or [11]),
We compare the complexity of the proposed differential communication approach against other approaches for Zak-OTFS in literature. The comparison is presented in Table I. The complexity associated with the proposed approach is O(M 2 N 2 ) which is incurred from the cross-ambiguity computation, while that for the spread pilot is twice this because of the extra step of pilot removal from the received frame. Using separate frame for sensing and communication incurs the same complexity as the proposed approach but has poor spectral efficiency since one frame is dedicated to the pilot transmission. Demonstrating differential communication scheme using data symbols as pilots. Perfect channel and channel estimated using pilots are added for reference. Noiseless system. Zak-OTFS system with M = 31, N = 37, νp = 30 kHz, root raised cosine (RRC) pulse shaping with roll-off parameter in delay (βτ ) and in Doppler (βν ) set as 0.6. For the DO frame, data symbols are drawn from a 4-QAM constellation. (a) Actual h eff (b) h eff estimated using SP [11] (c) h eff estimated using data. Fig. 2. Performance curves with proposed differential communication scheme, SP [10] with varying power distribution, separate sensing and communication, and perfect CSI for 4-QAM modulation. Zak-OTFS with M = 31, N = 37, νp = 30 kHz, RRC pulse shaping with parameter βτ = βν = 0.6.
Figure 1 compares the estimate of effective channel matrix obtained from SP (Fig. 1(b)) and from data (Fig. 1(c)) using the proposed approach against the actual channel (Fig. 1(a)). A noiseless system is considered for demonstration purposes. The estimate obtained using data has undulations while the estimate from SP is free from the non-ideality. This corroborates the derivation from (12), where for a data frame, B[k , l ] is not a scaled delta function but for a SP frame it is (see Remark 1).
We provide the bit-error rate (BER) and normalized mean square error (NMSE) performance for the proposed differential communication scheme. For the simulations, we consider the Vehicular-A (VehA) channel [15] and generate channel realizations per Section II-D with f c = 3 GHz, and ν i = ν max cos(θ), ν max = 815 Hz, θ ∈ U [-π, π). The NMSE is computed as:
where h eff is the estimated effective channel matrix.
We consider an SP frame with pilot energy e p,dB = e d,dB -5 dB where e d,dB is the data energy in dB scale. The noise variance σ 2 = 1 and in the linear scale, e d,lin = 10 SNR d /10 , SNR d is the data SNR. For the data-only (DO) frame, we consider e do,lin = e p,lin + e d,lin to be the energy in the data symbols. Both the SP and DO frames, therefore, carry equal energy, for fair comparison. For all the DO simulations presented below, we transmit a pilot frame every 30 frame transmissions to curb error propagation.
Figure 2 shows the NMSE and BER performance of the Zak-OTFS system for various schemes using 4-QAM modulation. For the SP frame we consider various data and pilot distributions parameterized by α. Keeping the total energy in the frame to be e do,lin , we consider e d,lin = αe do,lin and e p,lin = (1α)e do,lin for various α values. The NMSE performance is seen to improve as α is decreased from 0.9 to 0.1. This is expected as the energy in the pilot increases leading to better estimates. SP frame with 5 turbo iterations [16] achieves the best NMSE as it involves cycling between channel estimation and data detection. The separate sensing and communication scheme involves transmitting separate pilot and data frames and the estimate from the pilot frame is used to detect data in the data frame. The NMSE of the proposed receiver with the DO frame floors at 0.2. Moving to the BER performance, separate sensing and communication achieves performance closest to that with perfect channel state information (CSI), followed by the SP frame with turbo iterations. The performance of the proposed receiver using the Fig. 3. Performance curves with proposed differential communication scheme, SP [10] with varying power distribution, separate sensing and communication, and perfect CSI for 16-QAM modulation. Zak-OTFS with M = 31, N = 37, νp = 30 kHz, RRC pulse shaping with parameter βτ = βν = 0.6.
DO frame is better than SP frames with various α values. It is interesting to note that the performance with DO frame is the lower bound for the performance with SP frame.
Complexity: Here we compare the complexity qualitatively for full spectral efficiency achieving methods only. Separate sensing and communication does not achieve full spectral efficiency. The complexity is the highest for SP frame with turbo iterations owing to repeated channel estimation and data detection. For the SP frame without turbo iterations, there is an additional step involved to remove the contribution of pilot symbols before data is detected. The DO frame incurs the least complexity because there is no additional interference cancellation step. The DO frame therefore achieves better performance than SP at lower complexity.
Figure 3 shows the NMSE and BER performance of the Zak-OTFS system with 16-QAM modulation. NMSE and BER trends are similar to that observed in 4-QAM. The SP frames with decreasing α values achieve better NMSE, while the DO frame achieves the best BER performance closest to the performance with perfect CSI, while being the least complex.
V. CONCLUSION In this letter, we proposed a novel differential communication scheme for Zak-OTFS systems. The proposed scheme leveraged the predictability of the DD channel in Zak-OTFS to reduce the frequency of pilot symbol transmissions. The channel estimate obtained at a time instant was used for data detection in the next time instant. This allowed the information symbols transmitted in a DO frame to be used as pilots thereby achieving full spectral efficiency. We analytically showed the cross-ambiguity between the received data frame and transmitted data frame provided a model-free estimate of the channel. Simulations results showed that the proposed detector using the DO frame achieved good NMSE and BER performance. SP and DO frames both achieve full spectral efficiency, but the proposed scheme with the DO frame achieved better BER performance at lower complexity than the SP frame.
In this letter we consider the system model in discrete baseband and quantization and synchronization errors are not considered. However evaluating the performance of the system under these practical non-idealities is an important direction of future research.
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The asymptotics are only in the frame size and not in the constellation size. The frame size must be chosen sufficiently large so as to enable the law of large numbers[12].