The Design and Detection of Signature Sequences in Time-Frequency Selective Channel

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1 The Design and Detection of Signature Sequences in Time-Frequency Selective Channel Jiann-Ching Guey Ericsson Research 81 Development Drive Research Triangle Park, North Carolina 2779, USA Abstract In a slot-synchronized wireless communication system, a set of signature sequences can be derived by circularly shifting a base sequence with good auto correlation properties. In a time-dispersive channel, the sequences with different circular time shifts can be uniquely identified as long as their minimum separation in time is greater than the channel s maximum delay spread. In this paper, we extend this approach further to the Doppler domain by introducing an additional circular frequency offset to the delay shifted sequences. When the base sequence is carefully designed through applying some delay-doppler radar signal design principles, the results of this two-dimensional generalization are a much larger set of sequences with better correlation properties than some existing designs. Moreover, the new design is also suitable in channels that are both time and frequency selective as long as the minimum Doppler separation of the sequences is greater than the channel s maximum Doppler spread. The detection of the sequences essentially consists of correlating the received signal with all valid hypotheses, and can be carried out efficiently if the base sequence is properly designed. Simulation results show that given a sequence length that is large enough, exact number of multiple devices can be accurately detected in very challenging conditions with a single threshold test. I. INTRODUCTION The design of signature sequences with small auto and cross correlation has been extensively studied in a wide range of applications including wireless communication and radar. Of particular interest in wireless communication is the need to design a large number of unique sequences for the purpose of synchronization and device identification. Examples include the Barker sequence [1], the m-sequence [2] and the Gold sequence [3] derived from it. More recently, the Zadoff- Chu [4] sequence has been considered in the 3GPP Long Term Evolution (LTE) [5]. For a system that is synchronized coarsely to a slot boundary, these sequences with good auto correlation property may be extended to a larger set by introducing different circular delays. As long as the minimum separation in time between any pair of sequences is greater than the channel s maximum delay spread, they can be detected and uniquely identified. Therefore, they are often also suitable for timedispersive (or frequency-selective) channel. However, many of these existing designs do not consider the sequences property in a frequency-dispersive (or time-selective) channel. The channel s frequency dispersion may arise from the Doppler shifts resulting from the device s movement or frequency offset due to inaccuracy of the device s oscillator. Since the time-frequency selectivity in a wireless communication channel is essentially the results of the delay-doppler shifts incurred to the signal by the scatterers in the propagation environment, the signal at the channel output exhibits an expansion in the delay-doppler domain. It is therefore natural to apply the radar signal design principles of separating the targets in the delay-doppler plane to the design of signature sequences. This can be achieved by artificially introducing delay- Doppler shifts to a well designed base signal. In this paper, we will demonstrate that indeed the borrowed techniques can help construct larger sequence sets with better properties in time-frequency selective channel than existing designs. This two-dimensional generalization of circular shifts in both time and frequency also leads to an optimal detector consisting of a two-dimensional correlator in both delay and Doppler domains followed by a simple threshold test. The remainder of the paper is organized as follows. Sec. II describes the background and system model of the problem. A few examples of existing design are also given for reference. The detection of signature sequence in time-frequency selective channel is then derived in Sec. III. The analysis leads to the introduction of the ambiguity function, which in turn motivates the basic design principle given in Sec. IV. Simulation results supporting our claims are presented in Sec. V, followed by the conclusion in Sec. VI. II. BACKGROUND AND SYSTEM MODEL The ability of a pair of signature sequences to be distinguished from each other is often measured by their cross correlation function defined as s [n]s 1[n], (1) where N is the length of the sequence. In a time-dispersive (frequency-selective) channel, a good signature sequence also needs to be able to distinguish itself from its multipath echos. This is measured by its auto correlation function defined as s [n]s [n τ], (2)

2 for τ =,,. Note here that unless otherwise specified, all the indexing in this paper is modulo N. This results in the circular operations that can be achieved in practice by introducing cyclic prefix of appropriate length commonly seen in an OFDM system. Therefore, the most commonly used metrics for sequence design in a time-dispersive channel is simply the cross correlation function defined as φ s,s 1 [τ] s [n]s 1[n τ]. (3) In the case where s [n] =s 1 [n], the cross-correlation function becomes auto correlation function. A good sequence set should then have small cross correlation between any pair of sequences at all lags and small auto correlation at non zero lag for all individual sequences. In cases where the system is coarsely synchronized to the sequence length, the same sequence can be circularly shifted and assigned to more than one devices as long as the relative circular shifts are more than the channel s maximum delay spread. The common pilot code for CDMA2 is such an example. One example of sequence set with good auto and cross correlation function is the Zadoff-Chu [4] sequence defined as { s u [n k] =exp j2πu } (n k)(n k +1), (4) 2N where n =, 1,,N 1 and the sequence index u can be any integer between to N 1 that is a relative prime to N. The auto correlation function of any individual Zadoff- Chu sequence is zero except for the zero lag where it is N. Moreover, if N is prime, it can be shown that the cross correlation between any pair of Zadoff-Chu sequences with distinctive u is a constant N for all lags. For identification, a device may be assigned a unique sequence index u and a circular shift k, as proposed to the Long Term Evolution in 3GPP [5]. Another example is the set of N +2 Gold sequences [3] derived from a pair of preferred m-sequences with a maximum cross-correlation of 2N. The good correlation properties of the existing designs described above are valid only when there is no frequency uncertainty in the communication environment. In reality, the channel may be time-selective (or frequency dispersive) due to Doppler spread. There may also be frequency offset among the communication devices due to unsynchronized oscillators. These frequency uncertainties, together with the channel s time dispersion, are best described by the received signal at the channel output given by r[n] = τ max 1 τ= ν max 1 ν= h[τ,ν]s[n τ]e j 2πνn N + z[n], (5) where h[τ,ν] is the channel s delay-doppler response with maximum delay-doppler spread (τ max,ν max ) and z[n] is the Additive White Gaussian Noise (AWGN). Note that the frequency offset is incorporated into the Doppler spread of the channel. III. DETECTION OF SINGLE SEQUENCE IN TIME-FREQUENCY SELECTIVE CHANNEL Since the channel response h[τ,ν] is not known, it is a nuisance variable that needs to be removed when detecting the sequence. We can do this by applying the same method used in the Generalized Likelihood Ratio Test (GLRT) [6]. The first step is to estimate the nuisance variable h[τ,ν]. Assuming that the only channel information available is the maximum delay-doppler spread (τ max,ν max ), it has been shown [7] that the Maximum Likelihood (ML) channel estimator for a properly designed signal s[n] is given by ĥ ML [τ,ν]= 1 E s I[τ,ν] (6) where E s = s[n] 2 is the signal s energy and I[τ,ν]= r[n]s [n τ]e j2πνn, (7) is the two-dimensional delay-doppler image defined over τ<τ max and ν<ν max. The next step is to replace the true channel response with its estimate in the log-likelihood function: νmax 1 Λ= r[n] τ 2 max 1 ĥ ML [τ,ν]s[n τ]e j 2πνn N. ν= τ= (8) After some rearrangement and removing irrelevant terms, the log-likelihood function for the detector becomes Λ= ν max 1 ν= τ max 1 τ= I[τ,ν] 2. (9) In the case where there is only one sequence to be detected, this metric can then be compared against a threshold and the resulting detector is optimal in the generalized likelihood sense. If multiple sequences may be present, on the other hand, further normalization is needed before the thresholding, as will be described in the simulation section. A. Ambiguity Function By substituting the received signal given by Eq. (5) into the delay-doppler image given by Eq. (7), we have I[τ,ν] = where = τ = ν = r[n]s [n τ]e j2πνn N e j2π(ν ν )τ h[τ,ν ]χ s [τ τ,ν ν ] +χ z,s [τ,ν], (1) χ s [τ,ν]= s[n]s 2πνn j [n τ]e N (11)

3 is the (circular) ambiguity function [8] of the base signal s[n] and χ z,s [τ,ν]= z[n]s 2πνn j [n τ]e N (12) is the two-dimensional noise term. From Eq. (1) it can be observed that the delay-doppler image is the two-dimensional convolution of the channel s delay- Doppler response with the sequence s ambiguity function. Therefore, the measure of a sequence s ability to be uniquely identified in a time-frequency selective channel should be its ambiguity function. The one-dimension auto and cross correlation functions conventionally used for measuring signature sequence properties fail to reveal the sequence s characteristics in the presence of frequency uncertainty. An ideal signature sequence for time-frequency selective channel should then have a thumbtack-like ambiguity function. Otherwise, it may become impossible to unambiguously determine its presence and time-frequency offset, as demonstrated by the following example. Fig. 1 shows the ambiguity function of a length N =29 Zadoff-Chu sequence with u =6. It is clear that for ν = (no frequency uncertainty), the correlation property is ideal. However, there are two peaks at (τ =24,ν =1)and (τ = 5,ν = 28). This implies that the sequence is identical to itself shifted in time and frequency by the corresponding amounts. Therefore, if there is a frequency uncertainty of ±1/N, it is impossible to determine if the peaks detected around τ =24 and τ =5correspond to the self image of a sequence with zero time-frequency shift or another device assigned a circular shift of τ =5or τ =24. IV. BASIC DESIGN PRINCIPLES To avoid the ambiguity in time-frequency channel illustrated by the Zadoff-Chu sequence example, the basic design approach is to select a base sequence s[n] with good (thumbtacklike) ambiguity function. The existence of such sequences have been documented in the radar/sonar literature. From this base sequence, a set of sequences can be derived by introducing a circular delay-doppler 1 shift to the base sequence as follows: s l,m [n] =s[n lτ d ]e j 2πmν d n N, (14) where (τ d, ν d ) is the minimum delay-doppler separation between any pair of derived sequences and (l, m) is the unique identification index associated with the derived sequence. As long as τ d >τ max and ν d >ν max, the signals from multiple devices passing through the time-frequency selective channel are completely separable and uniquely identified in the delay-doppler domain. The total number of available sequences is given by N 2 /(τ d ν d ), a very large set in most typical wireless communication applications. Fig. 2 shows an example of sequence assignment in a multi-cell environment. Each terminal in a cell is assigned a sequence with index on a regularly spaced grid. The neighboring cells may use different grids that are substantially separated from each other. The channel expands a signal s time-frequency footprint, but never to the extent that it overlaps with that of other signals in the same cell or in a neighboring cell. τ=5,ν=28 3 N=29,u=6 τ=24,ν=1 (τ,ν) Fig. 2. Sequence assignment in multi-cell environment Fig. 1. Ambiguity function of a length-29 Zadoff-Chu sequence The Zadoff-Chu sequences with different parameter u, however, have excellent cross correlation property even in timeselective channel, since the cross ambiguity function defined as χ su1,s u2 [τ,ν]= is a constant N for any τ and ν. s u1 [n]s 2πνn j u 2 [n]e N, (13) A. Detection Metric for Multiple Sequences Eq. (9) gives the log-likelihood function of a sequence with zero delay-doppler shift. The likelihood function for any sequence generated by Eq. (14) can be similarly derived as γ[l, m] = lτ d +τ max 1 τ=lτ d mν d +ν max 1 ν=mν d I[τ,ν] 2, (15) 1 Note that the circular Doppler shift does not alter the spectral characteristic of the signal nor does it actually introduce a frequency offset since it is applied to the discrete samples of the sequence, not to the analog transmit signal.

4 for all hypotheses of [l, m]. After some normalization to be described in Sec. V, these likelihood metrics can then be compared with a threshold to determine the presence of multiple sequences. Two examples of good base signals will be given in the following sub-section. B. Examples A BPSK (±1) modulated m-sequence s[n] has the special property that the product s[n]s [n τ] is another m-sequence for any nonzero integer τ [2]. Furthermore, the DFT of an m-sequence is given by { 2πnk j s[n]e N 1,k = =. (16) N +1,k Therefore, the ambiguity function of an m-sequence, as shown in Fig. 3 for N = 255, has a mainlobe to sidelobe ratio of χ s [, ] [τ,ν] = N (17) (N +1) throughout the entire delay-doppler plane except for the two axes along zero delay and zero Doppler. In other words, the cross-correlation between any pair of sequences in the set of N 2 distinct sequences derived from a length-n m-sequence is at most N/ N +1. This is clearly a better alternative to the N +2 Gold sequences derived from a pair of preferred m- sequences with a maximum cross-correlation of approximately N/ 2N. are notably suppressed as shown in Fig. 4 for Q =16.The ideal Costas sequence has a certain advantage when it comes to complexity of the detection. However, this is out of the scope of this paper and will be addressed in a future publication. [τ,ν] Fig length 16x17 Costas sequence Ambiguity function of a length Costas sequence From Eq. (1) it can be seen that the time-frequency selective channel expands the ambiguity mainlobe in the delay- Doppler domain. Fig. 5 shows an example where two m- sequences of different delay-doppler shifts are present in the system. Visually, as long as the delay-doppler image footprints of the multiple sequences do not overlap with each other, they can be uniquely distinguished and identified length 255 m sequence delay Doppler image 25 x target [τ,ν] 15 1 I[τ,ν] target Fig. 3. Ambiguity function of a length-255 m-sequence Our second example is an ideal Costas sequence [8], [9] of length N = Q(Q + 1) given by the following expression: s[n] = Q l= p[n lq]e j 2πν l (n lq) Q, (18) where p[n] is a unit-amplitude rectangular pulse function with support between and Q 1 and ν l is a frequency hopping sequence resulting from permuting Q consecutive integers in a particular manner as described in [8]. An ideal Costas sequence has the unique characteristics of having a constant mainlobe to sidelobe ratio of Q on the grids formed by (τ,ν) =(lq, mq), for l Q and m Q 1. Although the values between the grids are not constant, they Fig. 5. delay-doppler image of two length-255 m-sequences in dispersive channel V. SIMULATION RESULTS Table I gives a list of the key parameters used in the simulation. A Zadoff-Chu sequence, a Costas sequence and an m-sequence of similar length are evaluated in different channel conditions for comparison. For the Costas sequence and m-sequence, the sequence assignment follows the design principle described in Sec. IV and the minimum delay- Doppler separation between two derived sequences, (τ d,ν d ), are designed to yield the same number of available sequences for both cases. For the Zadoff-Chu sequence, each device is assigned a unique circular delay k and slope u. The separation

5 TABLE I SIMULATION PARAMETERS 1 Doppler spread ± 25 Hz, frequency offset ± 1 Hz Parameter Value sequence Costas m-sequence Zadoff-Chu N τ d 69 ν d 5 5 u =4, 8, 12, 16 # of sequences 124 bandwidth 5MHz τ max ν max 1 cyclic prefix (chips) pulse shaping RRC β =.35 SNR (db) 4 ρ (db) 6 # of active devices 1, 2, 3, 4, 5 (equally likely) in delay is similar to that in the Costas sequence and m- sequence whereas the slope u can be 4, 8, 12 or 16. The received power of each signal is distributed uniformly within a dynamic range of ρ =6dB. In each test, any number between 1 and 5 (equally likely) users may be transmitting simultaneously. The detector determines which and how many devices are present based on a single normalized threshold given by γ[l,m ]= γ[l,m ] (19) l,m γ[l, m]. A detection error occurs whenever the detected set does not match exactly the active devices. The performance of these sequences are tested in a 5 Mhz channel with a power delay profile that decays linearly for 9 db in log scale in 48 taps. Fig. 6 shows the detection error rate as a function of the threshold in a channel with little frequency ambiguity. The Doppler spectrum based on Jakes model has a maximum Doppler spread of 5 Hz (±25 Hz) and the frequency offset is uniformly distributed between ±1 Hz. In other words, the channel is only frequency selective (due to the multipaths), but not time-selective. The Costas sequence has the worst performance due to its ambiguity sidelobe variation shown in Fig. 4. Fig. 6 shows the results in a channel with maximum Doppler spread of 4 Hz (±2 Hz) and frequency offset uniformly distributed between ±1 Hz. It is clear that the Zadoff-Chu sequence performs much worse than the other two designs as predicted by the ambiguity analysis. VI. CONCLUSIONS By introducing circular delay-doppler shifts to a well designed base sequence, we are able to derive a large set of sequences that are unique and easily separatable in the delay-doppler domain. A sequence set constructed this way has better ambiguity properties than existing designs especially in time-frequency selective channel and can be efficiently detected by a delay-doppler correlator. Results from simulation with parameters similar to those in LTE show that even a simple thresholding detector can detect a two-symbol long sequence close to 99% of the time in a wide range of operating conditions. Average Error Rate Average Error Rate Zadoff Chu Costas m sequence Threshold λ Fig. 6. Performance in frequency selective channel Doppler spread ± 2 Hz, frequency offset ± 1 Hz Zadoff Chu Costas m sequence Threshold λ Fig. 7. Performance in time-frequency selective channel REFERENCES [1] R. H. Barker, Group Synchronization of Binary Digital Systems, Communication Theory. W. Jackson, Ed. New York: Academic, 1953, pp [2] R. J. McEliece, Finite Fields for Computer Scientists and Engineers, Kluwer Academic Publishers, [3] R. Gold, Optimal Binary Sequences for Spread Spectrum Multiplexing, IEEE Trans. on Information Theory, vo1. IT-13, pp , October [4] D. C. Chu, Poluphase codes with good periodic correlation properties, IEEE Trans. on Information Theory, vo1. 18, pp , July [5] Third Generation Partnership Project, TSG RAN, E-ULTRA Physical Channels and Modulation, 3GPP TS , v8.1., Nov. 27. [6] H. L. Van Trees, Detection, Estimation, and Modulation Theory, part I, Wiley-Interscience, reprint edition, September 27, 21. [7] J. Guey, D. Hui and A. Hafeez, Classical Channel Estimation for OFDM Based on Delay-Doppler Response, Proc. IEEE PIMRC 27, Athens, Greece. [8] J. Guey, Synchronization Signal Design for OFDM Based On Time- Frequency Hopping Patterns, Proc. IEEE ICC 27, Glasgow, Scotland. [9] S. W. Golomb and H. Taylor, Constructions and Properties of Costas Arrays, Proc. IEEE, vol. 72, no. 9, pp , Sept