Code Acquisition in Direct Sequence Spread Spectrum Communication Systems Using an Approximate Fast Fourier Transform

Transcription

1 26 IEEE Ninth International Symposium on Spread Spectrum Techniques and Applications Code Acquisition in Direct Sequence Spread Spectrum Communication Systems Using an Approximate Fast Fourier Transform Ivan Periša and Jürgen Lindner Department of Information Technology, University of Ulm Albert-Einstein-Allee 43, 898 Ulm, Germany Abstract This paper deals with code acquisition in communication systems with large frequency offsets. In such a situation, the receiver has to perform a 2-dimensional search in a time-frequency plane to detect a known signal. In previous publications it has been proposed to use a fast Fourier transform (FFT) to implement the search in frequency direction [], [2], [3]. We would here like to examine the effect of using approximate FFTs. For this puropose, we restrict the angles of the complex multiplications that are used in the FFT to four or eight states. This helps to reduce the total number of real multiplications. Further, there is no need to store trigonometric values. The drawback is that the approximations lead to a degradation at the FFT output. We will show, however, that in a two-stage acquisition scheme, where the FFT is only used to obtain an estimate of the frequency offset, approximate FFTs achieve almost the same performance as a conventional FFT. complexity of the FFT without sacrificing too much performance. The idea is that we do not really need to obtain a very good precision in our calculation. We only need to approximately maintain the absolute values of the individual FFT-elements. Therefore, we give an example for a very simple FFT-structure, which can be evaluated with a reduced number of multiplications or even without multiplications. This is accomplished by restricting the number of possible angles that are used in the multiplications in the FFT-operation to 4 or 8. In the case of 4 angles all the complex multiplications can be evaluated by additions, while for 8 angles only scaling factors are required. If the absolute value is not needed with high precision (e.g. if we just need a rough frequency offset estimate in a two-stage acquisition scheme), the scaling factors could be omitted in the case of 8 angles, further reducing complexity. I. INTRODUCTION Code acquisition is an important part of a direct sequence spread spectrum (DSSS) receiver, which has been discussed by many authors - see [4], [5] and the references therein. The problem becomes more difficult, when frequency offsets are present [4]. For small offsets, segmentation schemes using a subsequent non-coherent combining can be used [6]. While it is simple to implement, segmentation leads to a performance degradation and is not suitable for high frequency offsets (e.g. about % of the signal bandwidth). A fully differential scheme offers a large acquisition range and a very low complexity [7], [8], but its performance at low signal-to-noise ratios is significantly worse than a 2Dsearch in time and frequency. In [9] Reed suggested a two stage procedure, which implemented a filter-bank without any multiplications. Another possibility to perform the 2D search is by means of a fast Fourier transform (FFT) [], [2]. Especially, when a large frequency range has to be covered, it seems to be a good approach, because it can easily be adjusted to the acquisition range (see [3]) and the complexity is smaller than in a parallel filter bank. In different fields there has been research about ways to reduce the computational complexity of FFTs, by reducing memory references or simplifying the multiplication operations [], [], [2]. The aim of this work is to look at further ways to reduce the II. SYSTEM MODEL We assume that the receiver has already removed data modulation (by multiplying the received samples with the complex conjugate of the known PN-sequence). This greatly simplifies the system model, since now we receive a single-frequency complex tone, that is disturbed by additive white Gaussian noise (AWGN). Sampling the signal at rate T yields y(k) y(k) = E s e j(2πδfkt+φ) +n(k), k L. () Here E s is the energy of each transmit symbol, Δf is the frequency offset, k is the discrete time-index, Φ is the phase-offset, n(k) is a noise-sample,and L is the length of the received sequence. Δf and Φ are deterministic, unknown constants and the noise-samples are zero mean, complex, white Gaussian random variables with variance σ 2 n in real and imaginary part. Note, that we have assumed perfect timing, since the focus is on acquisition with frequency offsets. The same model is often used in frequency offset estimation schemes. This shows, that principally any frequency offset estimator could be used at this point. However, an FFT-based estimator is expected to have the best low-snr performance (which is important for acquisition). Another advantage is, that the FFT directly computes the values of the correlation at different frequency offsets. The absolute correlation values are generally used in acquisition schemes to decide, whether a preamble is present or not /6/$2. 26 IEEE 54

2 see in the following section. This results in a serial search in time direction, while the search in frequency direction is carried out in parallel by the FFT. A. Acquisition Theory In general, the receiver searches for a known sequence at a set of time- and frequency-offsets. For each time instant this search is usually performed by evaluating a decision variable Z(f t ) at a set of different frequency offsets f t according to: L Z(f t )= y(k)e j(2πftkt ) 2. (2) This decision variable is then compared to a threshold θ, which is often chosen such, that the receiver yields a constant false alarm rate P FA - i.e. the case where we decide that the signal is present, although it is not [3]. This can be shown to be P FA = exp ( θ/(2lσ 2 n) ) (3) for AWGN-channels. Hence, θ is obtained to be θ = 2Lσ 2 n ln(p FA). (4) The noise power σ 2 is not known and it is therefore approximated by the received signal energy E y L E y = y(k) 2. (5) This yields the following threshold: θ = E y ln(p FA ). (6) Employing this threshold results in a false alarm rate, which is slightly smaller than the theoretical one. The reason for this is, that the signal power is estimated. The remaining task is to evaluate equation (2) for a set of frequencies in the search space. As mentioned before, we do this by means of an FFT, which is described next. B. The Fast Fourier Transform (FFT) The FFT has been described in many books and papers. We therefore do not want to give a detailed description of the algorithm. Instead we would like to depict the basic idea and explain how we have modified the FFT to reduce its complexity. Let us first define the complex number W : W = e j2π/l. (7) Then the n-th element of the Fourier transform of y(k) can be written as: L Y (n) = W nk y(k). (8) So this means that we basically multiply the received sequence with a complex sinusoidal of a certain frequency to obtain a value of the Fourier transform. The idea of the FFT is to exploit some of the symmetries in this multiplication procedure to reduce the complexity. There are different ways to do this, though we will just describe decimation in frequency (DIF) as an example. In DIF we split a DFT of length L in two DFTs of length L/2, where one DFT evaluates the even elements and the other evaluates the odd elements of the full length DFT: L 2 Y (2n)= W 2nk (y(k)+y( L + k)), (9) 2 L 2 Y (2n +)= W 2nk W k (y(k) y( L + k)). () 2 As we can see, the right hand side corresponds to DFTs of length L/2, with additional coefficients W k (often called twiddle factors) for the calculation of the odd elements. This approach can now be used recursively and thus leads to a calculation of several DFTs of short lengths instead of the full length DFT. Note, that there are many different approaches to simplify the DFT (e.g. higher radix, split radix algorithms etc.) and the above approach is not the optimum one. Nevertheless we chose it due to its simplicity. The suggested modification can be applied to any more sophisticated FFTalgorithm that is based on FFT coefficients W k. C. Partial Matched Filters The previously described FFT searches the whole frequency range, although in many applications the frequency offset is smaller than % of the signal bandwidth. This fact can be exploited to reduce the search range and thus reduce the computational complexity, by employing partial matched filters (PMFs) [2], [3]. In this approach we construct a new sequence r c (l) from the original sequence r(k): r c (l) = M r(k + lm). () Here M is the length of the PMF and specifies the factor by which the original sequence length is reduced. This also implies that the length of the subsequent FFT is reduced (which actually accounts for the complexity reduction). The drawback of using greater values of M is that we observe a performance degradation for higher frequency offsets even when they are within the search range. The reason for this degradation is, that in the summation in equation () we assume that no frequency offset is present. In reality we always have a degradation compared to the frequency corrected combining, which is given by: [ ] 2 sin(πmδft) Deg(Δf,M) =, (2) πmδf where Δf is the frequency offset. This means that for a given maximum frequency offset we have to limit M such 55

3 that the degradation is not too high. Regarding complexity, we would like to choose M to be as large as possible. So we have to find a compromise between complexity, estimation range, and performance at this point. D. Approximate FFT coefficients As we see in equation () we need to multiply a part of the newly constructed sequence with the complex FFT coefficients W k. These multiplications obviously account for a large part of the computational complexity of the algorithm. There are ways to reduce the number of multiplications (e.g. by exploiting symmetries, using higher radix algorithms), but still they are always present. The idea here is to restrict the coefficients W k to values for which the complex multiplications can be evaluated in asimpleway. At this point it should also be mentioned, that data modulation removal for 4- or 8PSK-modulation can be included in the first stage of the FFT, thus not increasing the operation count. ) Using 4 angles: In this case we restrict W k in equation () to ± and ±j. Obviously, any multiplication with these factors can be evaluated by a simple mapping and/or sign-change, thus leading to an FFT which could be evaluated without any multiplications at all. By simply rounding all FFT coefficients W k to these values we would obtain a very simple structure to perform this: ) For k < L 8, multiply with (i.e. perform no action) 2) For L 8 k< 3 8L, multiply with j 3) For 3 8 L k< L 2, multiply with This is repeated until we obtain sequences of length 4, for which the FFTs can be evaluated without degradation if we use the angles specified above. Considering this simple structure, we can directly evaluate the result in equation () without multiplications (i.e. we can directly substitute the subtraction by the corresponding addition/subtraction as needed). Thus, the whole FFT can be approximated by this fully multiplication-less structure. 2) Using 8 angles: Multiplications with complex numbers with angles that are multiples of π/4 can also be computed in a simple way. While a normal complex multiplication requires 4 real multiplications and 2 real additions, we could reduce this to 2 real multiplications (for scaling) and 2 real additions. Scaling is important, if we want to use the absolute FFT-values directly to decide whether a preamble is present or not. Otherwise we would obtain different probabilities of false alarm at different frequency offsets. If we just need a rough frequency offset estimate (which is the case in two-stage acquisition schemes), we could even omit the scaling and then only 2 real additions are necessary. In any case the structure would now be: Absolute value of approximate FFTs FFT bin Fig.. Amplitudes for different FFT-elements when using approximate FFTs with 4 (solid line) or 8 angles (dashed line). A conventional FFT would produce amplitude values of 64 for each element. ) For k< L 6, multiply with (i.e. perform no action) 2) For L 6 k< 3 6 L, multiply with ejπ/4 3) For 3 6 L k< 5 6L, multiply with j 4) For 5 6 L k< 7 6L, multiply with ej3π/4 5) For 7 6 L k< L 2, multiply with This is repeated until we obtain FFTs of length 8, which can anyway be evaluated without degradation if we use angles that are multiples of π/4. E. Degradation due to Approximations Obviously, we are using an approximation which leads to a signal degradation. This degradation does not occur for the even elements in a particular decimation step. To visualize this degradation for an FFT of length 64, we can have a look at figure. It shows the degradation for all evaluated frequencies (i.e. the degradation in signal amplitude if that particular complex sinusoidal was received) and for both approximations. Without degradation we would obtain amplitude-values of 64 and we can observe that as expected, the degradation for the approximation with 8 angles is smaller than for the approximation with 4 angles. Further there are also 4 (or 8) frequencies where no degradation occurs. This corresponds to complex sinusoidals which can be fully expressed with 4 (or 8) angles. Figure 2 shows the maximum degradation that occurs for different lengths of the FFT. We can see that with increasing length the degradation increases. This is expected, because a longer FFT means we have more decimation steps and thus more multiplications with FFT coefficients W k. Note, that we have plotted the result up to length L = 892, although in acquisition schemes lengths of L =64would be more realistic. In this region of practical interest, using 8 angles leads to a degradation of less then.5db, while using 4 angles leads to.5db degradation. 56

4 It should also be noted that, while simple, the proposed Degradation [db] III. SIMULATION RESULTS In this section we want to compare the simulation results in AWGN environments. In all simulations we set a false alarm probability P FA = 9 and use a sequence length of L = 256, which is then shortened through PMFs. The original sequence length is kept constant so that we are able to focus on the degradation arising from the approximate FFTs of different lengths. In figure 3 the degradation Approximation with 8 angles Approximation with 4 angles log2(fft_len) Fig. 2. Maximum degradation for different FFT-lengths when using approximate FFTs with 4 or 8 angles algorithms are not optimum with respect to degradation. A DFT which is restricted to the same number of angles, would suffer a much smaller degradation for long sequences. On the other hand, a DFT would also have a much higher computational complexity. If longer Fourier transforms are required, one would have to think about a different approximate FFT. F. Complexity Reduction There are a lot of different ways to implement the FFT and to reduce its complexity compared to the simple radix- 2 approach described in section II-B. One area of optimization is the storage or generation of FFT coefficients []. Besides, higher radix and split radix approaches can reduce the number of operations somewhat but lead to more complicated structures (especially for higher radices, which is a reason why they are sometimes not used). Further, there are ways to simplify the complex multiplications so that only 3 real multiplications instead of 4 real multiplications are required []. The complexity reduction for each of these optimization approaches depends on the length of the FFT and the hardware that is used. It is not the goal of this work to optimize the FFT for a certain architecture or FFT-length. We just note that when the total number of operations is considered the approximate FFTs are significantly less complex than any of the previously discussed approaches, while maintaining a very simple structure. When we use the 4-angles approximation no multiplications are required at all and when we use 8 angles only /8-th of the original multiplications are required (which is for scaling purposes - if no scaling is performed we would have an even lower complexity in this case). This compares to a complexity reduction of 2-3 % for the previously mentioned approaches [4]. Detection Probability P D Conventional FFT Approx. FFT, 8 phases, length 28 Approx FFT, 4 phases, length 32 Approx. FFT, 4 phases, length Fig. 3. Detection probability, Sequence Length L = 256, False Alarm Probability P FA = 9, ΔfT for approximate FFTs chosen to coincide with worst case. Comparison of the effect of different lengths in approximate FFTs. due to approximate FFTs is shown. Note, that in the figure we have always chosen the worst case when using the approximate FFTs (obviously in the best case we would not have any degradation at all). For an FFT-length of 28, we can observe half a db degradation for the approximate FFT with 8 angles and more than 2dB degradation for the approximate FFT with 4 angles. When the FFT length is reduced to 32, the degradation in the latter case is roughly.5db. This corresponds to what we observed in the previous sections. Note, that degradations observed in figure 2 cannot be translated into the same loss for probability of detection. When we observe the probability of detection at a higher SNR-value (so that the degradation is compensated) the corresponding threshold and noise level are different. The loss appears to be very much acceptable for the case of an 8-angles FFT and somewhat higher than one would like for the 4-angles FFT. The reason is that in the acquisition scheme that we discussed so far, the decision variables to decide whether a preamble is present or not, is formed from the maximum FFT-element directly. The situation is different in a two-stage scheme, where the FFT is just used to obtain a rough frequency offset estimate. The motivation for a two-stage scheme is that when we use PMFs, the detection performance degrades for higher frequency offsets. This is clear, since the PMFs are only optimum for zero-offset. Nevertheless, they are very important in order to reduce the overall system complexity. Figure 3 shows this degradation for a frequency offset ft =27/256 and a conventional FFT. While the degradation for PMF- 57

5 Detection Probability P D PMF length= PMF length=2 PMF length= Fig. 4. Detection probability, Sequence Length L = 256, False Alarm Probability P FA = 9, ΔfT = 27/256. Effect of using partial matched filters (PMF) of different lengths for large frequency offsets in a conventional acquisition scheme. Detection Probability P D Two Stage, Conventional FFT. Two Stage, Approx. FFT 8 phases Two Stage, Approx. FFT 4 phases Conventional, one stage only Fig. 5. Detection probability, Sequence Length L = 256, False Alarm Probability P FA = 9, ΔfT =27/256. Comparison of a two-stage scheme to a conventional acquisition scheme. length 2 is small, it is about 3dB for PMF-length 4. From this point of view it would appear that PMF-length 4 is not a good choice. The results are much better when we look at a two-stage scheme, however. In figure 5, we see that when using PMF-length 4, the two-stage scheme outperforms the conventional one. Overall the degradation compared to acquisition without PMF is just half a db (with a significantly reduced complexity). The interesting point is that now a conventional and an 8-angles FFT have almost the same performance. Even the 4-angles FFT is very close to the other cases. This shows that for rough frequency offset estimation the approximations are much more robust than for the conventional case, where we directly form decision variables from the FFT-results. IV. CONCLUSION We have proposed to use approximate FFTs with reduced complexity for acquisition. The approximations restrict the FFT coefficients (twiddle factors) to a few angles with good properties regarding the computational complexity. We have seen that when 8 angles are used, we get an acquisition performance close to the conventional FFTs, while for 4 angles the degradation is higher. In two-stage acquisition schemes, which can reduce overall system complexity, the degradation when using approximate FFTs is even smaller. We obtain almost the same performance as in a conventional FFT, while the complexity is reduced. REFERENCES [] R. A. Stirling-Gallacher, A. Hulbert, and G. Povey, A fast acquisition technique for a direct sequence spread spectrum signal in the presence of a large doppler shift, Proc. ISSSTA, pp. 56 6, 996. [2] C. L. Spillard, S. M. Spangenberg, and G. J. Povey, A serialparallel fft correlator for pn code acquisition from leo satellites, Proc. ISSSTA, pp , 998. [3] P. M. Grant, S. Spangenberg, I. Scott, S. McLaughlin, G. Povey, and D. Cruickshank, Doppler estimation for fast acquisition in spread spectrum communication systems, Proc. ISSSTA, pp. 6, 998. [4] A. J. Viterbi, CDMA: Principles of Spread Spectrum Communication. Addison-Wesley, 995. [5] S. Glisic and B. Vucetic, Spread Spectrum CDMA Systems for Wireless Communications. Artech House Inc., 997. [6] E. Sourour and G. E. Bottomley, Effect of frequency offset on ds-ss acquisition in slowly fading channels, Proc. WCNC, vol. 2, pp , September 999. [7] C.-D. Chung, Differentially coherent detection technique for direct-sequence code acquisition in a rayleigh fading mobile channel, IEEE Trans. Commun., vol. 43, pp. 6 26, February/March/April 995. [8] M. H. Zarrabizadeh and E. S. Sousa, A differentially coherent pn code acquisition receiver for cdma-systems, IEEE Trans. Commun., vol. 45, pp , November 997. [9] M. C. Reed, A low complexity ds/cdma acquisition technique for large frequency offsets, Proc. Wireless, Calgary, july 2. [] Y. Tang, L. Qian, Y. Wang, and Y. Savaria, A new memory reference reduction method for fft implementation on dsp, Proc. ISCAS 3, vol. 4, pp. IV 496 IV 499, May 23. [] J. Lee, J. Lee, M. Sunwoo, S. Moh, and S. Oh, A dsp architecture for high-speed fft in ofdm systems, ETRI Journal, vol. 24, pp , October 22. [2] S. Y. Park, N. I. Cho, S. U. Lee, K. Kim, and J. Oh, Design of 2k/4k/8k-point fft processor based on cordic algorithm in ofdm receiver, Proc. PACRIM, vol. 2, pp , August 2. [3] K. Choi, K. Cheun, and K. Lee, Adaptive pn code acquisition using instantaneous power-scaled detection threshold under rayleigh fading and gaussian pulsed jamming, IEEE Trans. Commun., vol. 5, pp , August 22. [4] V. K. Madisetti and e. Douglas B. Williams, The Digital Signal Processing Handbook. CRC Press, Inc.,