Windowed Overlapped frequency-domain Block Filtering Approach for Direct Sequence Signal Acquisition
aa r X i v : . [ ee ss . SP ] O c t Windowed Overlappedfrequency-domain Block FilteringApproach for Direct Sequence SignalAcquisition
Ebrahim Karami ∗ a , Harri Saarnisaari b a Department of Engineering and Applied Sciences, Memorial University, Canada, b Centre for Wireless Communications, University of Oulu, Finland
Abstract
This paper applies a windowed frequency-domain overlapped block filtering approach for the acquisition of directsequence signals. The windows, as a novel viewpoint, not only allow pulse shaping without a front end pulse shap-ing filter, but also improve the performance of the spectrum sensing unit which can e ffi ciently be implemented intothis frequency-domain receiver and may further be used for spectrum sensing in cognitive radios or narrowbandinterference cancellation in military radios. The proposed receiver is applicable for initial time synchronizationof di ff erent signals containing a preamble. These signals include single carrier, constant-envelope single-carrier,multi-carrier and even generalized-multi-carrier signals, which makes the proposed receiver structure a universalunit. Furthermore, the receiver can be used to perform filtering with long codes and compute the sliding corre-lation of an unknown periodic preamble. It can further be modified to handle large Doppler shifts. We will alsodemonstrate the computational complexity and analysis of the acquisition performance in Rayleigh and Ricianfading channels. KEYWORDS:
Synchronization, pseudonoise coded communication, matched filters.
1. Introduction
Initial synchronization, or acquisition of a di-rect sequence (DS) signal appears to be a quitecommon first step that a communication receiverhas to perform after switching the power on, be-cause many wireless standards either use a DSsignaling or their preamble, used for synchro-nization purposes, is a DS signal. These stan-dards include GSM, LTE, UMTS, GPS, GALILEO,WIMAX, Zigbee, and many others wireless stan-dards [1–4]. For example, LTE systems use twoDS signals, i.e., a 62-length Zado ff -Chu sequenceand an 31-length M-sequence, as primary andsecondary synchronization signals [5]. On the ∗ Ebrahim karami is responsible for all correspondance(email:[email protected]). other hand, the performance of channel estima-tion and equalization, and data detection, algo-rithms is significantly a ff ected by the accuracyof the initial synchronization [6–13]. One so-lution to improve the synchronization robust-ness is to use interference cancellation (IC) sig-nal processing [14–17]. Notch filters are well-known examples of these. Another applicationfor these IC units is spectrum sensing in cog-nitive radios. A notch filter may be a sepa-rate stand-alone unit in the front of a conven-tional receiver, but they may also be integratedinto a frequency-domain receiver, which reducesthe complexity, because the required transfor-mations may be shared. frequency-domain re-ceivers require less complexity and hence, havefound many applications. One particularly inter- Ebrahim Karami, et al.esting type of filtering is matched filtering, whichallows fast acquisition [18, 19]. In traditionalfrequency-domain filtering, where the filter isin a one piece, overlap-save (OLS) or overlap-add (OLA) methods have to be acquired to prop-erly handle the convolution process [20]. More-over, the frequency-domain receivers may be ofinterest in multipurpose or universal receivers,because they can be naturally used not onlyto receive multi-carrier signals such as orthogo-nal frequency division multiplexing (OFDM) andits variants as MC-CDMA [21] and generalized-multi-carrier (GMC) [22] signals but also to re-ceive single-carrier signals [23].Some systems employ long DS codes and con-sequently require long filters which are di ffi cultto implement [24, 25]. In such cases, the fil-tering has to be divided into blocks and the re-quired filtering process has to be performed us-ing a process known as a block or partitionedfiltering [26]. This technique is well-known inaudio signal processing [27, 28]. Even overlap-ping blocks may be used [28, 29]. Block filtersmay also be adapted to acquire larger Dopplershifts than sole filters, see [30–33] for time-domain approach. Block filtering is equal toDFT filter banks (multi-rate filters) and linearperiodic time varying (LPTV) filtering [26] butalso short time Fourier transform (STFT)-based-filtering [34]. The STFT adds windows, not usedin DFT filter banks or LPTV filters, to the overallpicture. The windows may be used to performthe pulse shape filtering, i.e., to match the filterfrequency response to that of the signal and toimprove the performance of notch filters by re-ducing the spectral leakage. However, althoughessential for proper performance of notch filters,windowing is known to cause signal-to-noise ra-tio (SNR) losses which are up to 3 dB for goodwindows. This loss may be reduced almost downto zero dB using overlapping segments, whichare also elementary for STFT-based-processing[35]. A STFT-based-correlator DS-receiver is pre-sented in [36]. In addition to data demodula-tion investigated in [36], it may be used for se-rial search acquisition, which is known to result aslower acquisition than the matched filtering ac-quisition investigated herein.This paper presents a frequency-domain, win-dowed, overlapped block filtering approach forDS signal acquisition. In addition to introduc-ing the filtering and the acquisition concept, itsother possible applications in radio communica-tions will be briefly discussed. These include i)addition of a particular notch filter method [37]into the receiver chain, ii) processing of di ff er-ent signals like conventional DS, constant en-velope DS, OFDM (WIMAX), MC-CDMA and GMC, iii) adaption the receiver to handle largeDoppler frequency uncertainties and iv) possi-ble changes when receiver is turned to the de-modulation phase after acquisition. Further-more, the paper includes analysis of computa-tional complexity of the receiver compared tothe conventional (non-block) matched filter im-plementation in the time or frequency-domainas well as analysis of acquisition probabilitiesin additive white Gaussian noise (AWGN) andRayleigh flat fading channels, of which the latterare novel results. The probabilities include con-ventional detection and false alarm probabilities,maximum-search-based-probabilities and max-imum search followed by threshold-detection-based-probabilities o ff ering a very comprehen-sive picture of receiver’s performance. Theseprobabilities may be used to set the detectionthreshold and to predict the receivers perfor-mance in practice. As a summary of this itcould be said that the paper introduces a flexi-ble baseband architecture that may be used withmost existing and future signals and which of-fers spectrum sensing or narrowband interfer-ence rejection capability with a low additionalcost. Therefore, the proposed receiver structureis a candidate receiver architecture for futuremulti-waveform platforms.The rest of the paper is organized as fol-lows. Section 2 introduces the filtering conceptwhereas applications and modifications are dis-cussed in section 3. The acquisition process isanalyzed in section 4 and simulation results con-firming the analysis are shown in section 5. Fi-nally, conclusions will be drawn in section 6.
2. Block Filtering
This section first discusses block-wise convo-lution to provide an insight how the block fil-tering works and then present its mathematicalfrequency-domain basis, the STFT-based time-varying filtering.
A simple example is probably the best way toexplain how the block filtering di ff ers from theconventional one. Let x , x , x , x be the sig-nal block to be filtered. In the conventional fil-tering, the signal is continuously fed into thefilter whose impulse response is h , h , h , h .As a consequence, the response sequence is x h , x h + x h , x h + x h + x h , x h + x h + x h + x h (desired phase in acquisition) , x h + x h + x h , x h + x h , x h . The block-wise con-volution should end up to the same result.In the block processing, the signal and the fil-ter are divided into blocks using equal divisions.indowed Overlapped frequency-domain Block Filtering Approach for Direct Sequence SignalAcquisition 3In the example, the division of the signal couldbe (the filter is divided correspondingly) " x x x x , where the block size M = 2 and totality is 2 × x x x x . This is divided into 2 × " x x x x x x x x . In the block processing, only one input matrix isprocessed at each time instant, called as a filter-ing cycle. Each block (column) of an input ma-trix is convolved with the corresponding block(column) of a filter and the results are addedtogether. Then, the next input matrix in thenext cycle is received and the operations are re-peated. Therefore, M responses are calculated inone time cycle. To obtain the whole response,the operation has to be repeated for all L possi-ble cycles. Since the length of block convolutionis 2 M −
1, the tails have to be added to the corre-sponding convolutions in the next cycle. This isclarified next. It is assumed that each block (col-umn) of the signal (matrix) passes a filter blockfrom down to top. The corresponding convolu-tion results are added together from each filter-ing cycle. The cycles are separated by bars andtails are below the dot lines. This results x h | x h + x h | x h x h + x h | x h + x h | x h + x h | + x h + x h | . . . | . . . | . . .x h | x h + x h | x h . The tails of the convolution have to be added tothe head of the convolution in the next cycle.More precisely, let c k = [ h h t k ] denote the con-volution result in cycle k , where h k is the head(first M samples) and t k the tail. In the nextcycle c k +1 = [ h k +1 + t k t k +1 ]. Therefore, the re-sponse of the block convolution becomes equiva-lent to the conventional convolution. As a sum-mary: the signal stream is block by block fedthrough the filter, the column-wise convolutionbetween the signal and the filter is performed,the convolution results are added column-wisetogether and the tails have to be added to the head of the next cycle. Since the convolutionin the time-domain might be equally well per-formed in the frequency-domain, in each cyclethe FFT of the signal (matrix) can be element-wise multiplied by the FFT of the filter (matrix)and then is inverse transformed to obtain thetime-domain convolution. After that, the convo-lution results are added together and OLA pro-cessing is performed. However, only one FFTper incoming signal block has to be calculated,because these transformations flow matrix-wisethrough the filter.By using a similar example, one can easilysee that in the overlapping segments case (like x , x ; x , x ; x , x ), the response of the block-wise convolution is not equal to the one of theconventional convolution. Instead, the origi-nal signal and its overlapped version have to beprocessed separately and the results have to beadded afterwards. The filter has to be overlappedcorrespondingly. All this is put into the STFT framework as fol-lows. Let x ( n ) , n = 0 , . . . , N − X lm = N − X n =0 x ( n ) w ( n − lR ) e j πmn/M , (1)where the analysis window w ( n ) has length M with non-zero values being in the interval n =0 , . . . , M −
1. It is obvious that the signal is di-vided into blocks of M samples and the blocksmay overlap depending on the parameter R ; if R = M there is no overlapping, but just consec-utive blocks. As a result of the analysis process(1), the signal is presented by a M × LM/R ar-ray of coe ffi cients X lm . For the simplicity, as-sume that N = LM and M/R = 1 , , , . . . . The case M = R is called the critical sampling case. Theselected restrictions yield to a simple implemen-tation through FFT, but are still quite flexible.More general case is studied in [38], but withoutconsidering signal acquisition.There are several alternatives to recover thesignal [34]. One particularly interesting form is x ( n ) = L − X l =0 g ( n − lR ) M − X m =0 X lm e j πmn/M , (2)where g ( n ) is the synthesis window of length M .Assuming that w ( n ) and g ( n ) satisfy some restric-tions [34], the signal x ( n ) can be perfectly recon-structed (synthesized) from its STFT coe ffi cients X lm . In other words, the STFT columns are firstinverse-Fourier-transformed (rightmost sum in(2)), then windowed and finally added together Ebrahim Karami, et al.in OLA fashion. Note that since the (I)FFT is alinear operator the order of addition and (I)FFTcan be changed. Thus, if the synthesis window isrectangular, the complexity may be reduced per-forming addition before the IFFT. This naturallyis a sensible operation, only if partial filtering re-sults are not required like in Doppler processingor in filtering of several symbols during a filter-ing cycle.Let H lm be the STFT of the filter. It can beshown [34] that the output of the filter is theinverse STFT of X lm H lm (element-wise product).Thus, the filtering includes multiplication of thesignal’s STFT by that of the filter, and inversetransformation of the product. In the paper’scase, the frequency response of the filter is zerooutside an interval. Therefore, the output is com-puted as multiplying finite portion of signal’sSTFT with the filter’s STFT. Furthermore, to han-dle the heads and tails properly, the FFT size hasto be 2 M . The overlapping e ff ect is taken into ac-count by stepping the input signal STFT streamin steps of size M/R , the number of overlappingsegments. The filtering process is illustrated inFig. 1. Obviously, if N = M = R the describedblock FFT filtering method reduces to the con-ventional FFT OLA filtering [27]. Herein, the complexity of generic time, con-ventional frequency-domain and block filteringare compared in terms of complex multiplica-tions (CM). In the time-domain each outputneeds N CM and there are N outputs such thattotal complexity is N CM. The conventionalfrequency-domain OLA processing needs FFTand IFFT of size 2 N and multiplication by fil-ters frequency response of size 2 N yielding totalcomplexity of 2 N (log N + 1) CM.The proposed block filtering (assuming rectan-gular windows) requires FFT of size 2 M , whichhas to be repeated LM/R times, multiplying byfilter of size 2 M × LM/R which has to be re-peated
LM/R times. The conventional formstill needs
LM/R
IFFTs of size 2 M , whereasthe simpler form has only L IFFTs. Therefore,the total complexity of the conventional form is MNR (cid:16) log M + LMR (cid:17)
CM and that of the simplerform MNR (cid:16) (1 + RM ) log M + LMR (cid:17)
CM. It can beobserved that the complexity of the block filter-ing becomes equal than that of the conventionalOLA filtering if N = M = R , as it should.The complexity comparison results to a con-clusion that the block filtering is more complexthan the conventional one, but without overlap-ping the complexity increase is marginal. In ad-dition, both frequency-domain versions are sim-pler than the generic time-domain implementa- tion. However, possible windowing increases thecomplexity.
3. Applications
Symbol or chip synchronization is convention-ally performed by correlation, but this results ina slow synchronization phase, see, e.g., [18]. Away to speed it up is to implement several cor-relators in parallel to simultaneously compute anumber of test variables (search cells) [39]. Ifthe signal, to be synchronized, consist of N sym-bols or chips, then the receiver usually has qN search cells in time-domain, where q is the over-sampling factor. Additionally, there might besearch cells in frequency as will be seen laterbut in this section only time uncertainty will beinvestigated. It is reminded that the receiverconventionally first includes a pulse shape fil-ter whose response in fed into the correlator asone sample per symbol or chip basis. In theoversampling case, the response has to be splitinto q steams and each stream is separately pro-cessed [40]. Alternatively, the pulse shape maybe taken into account in the correlation [41]. Inthis case, the receiver does not include a sepa-rate pulse shaping filter. This processing may becalled as waveform-based-correlation, whereasthe another processing may be called as training-symbol or chip-sequence-based-correlation. TheSTFT-based block-filtering may adopt both ways.In the former the analysis window, indeed, maybe matched to form a suitable response. Anotherway to speed it up is to calculate the test vari-ables through matched filtering either in time orfrequency-domain [39, 41, 42].In the serial search matched filter (MF) acqui-sition the outputs of the MF (computed in anypossible way) are compared to a threshold in a se-rial fashion. In the maximum search a period ofoutputs is calculated and the maximum is found.This maximum is then compared the threshold.In both cases, it is claimed that the signal ispresent and symbol or chip synchronization hasbeen acquired if the threshold is exceeded. Thiswill be considered more detailed later in section4. ff erent Modulations It is obvious that OFDM systems are a spe-cial case of the synthesized signal (2), i.e., thewindow is rectangular and L = 1. ConventionalWIMAX synchronization is performed in time-domain. In WIMAX the preamble DS code isput into even subcarriers whereas odd ones arezero. This makes the time-domain signal peri-odic with two periods of size N /
2, where N is theindowed Overlapped frequency-domain Block Filtering Approach for Direct Sequence SignalAcquisition 5 Input signal(cid:13)
M(cid:13) segmenting(cid:13)overlapping(cid:13)Window + FFT (2M)(cid:13)M/R, the step size of filter cycles(cid:13)
Example with L=2 and R=M/2(cid:13)
Form input matrix (2M x LM/R)(cid:13)multiply by filter(cid:13)(matrix (cid:13) x(cid:13) matrix)(cid:13)Input(cid:13)STFT(cid:13)stream(cid:13)
N(cid:13)
Columnwise IFFT + addition(cid:13)or(cid:13)columnwise addition + single IFFT(cid:13)tail head(cid:13) 1. cycle(cid:13)2. cycle(cid:13)Final output, 1 period(cid:13)Add tail to next head(cid:13)
N(cid:13)
Fig. 1: An illustration of the windowed and overlapped block filtering approach. number of subcarriers. The acquisition unit per-forms sliding correlation between two consecu-tive
N /
N /
N /
8. This is closely relatedto the overlap processing.The generalized multi-carrier (GMC) transmis-sion technique presented, e.g., in [23, 38, 45, 46],is a possible candidate for the future wirelesscommunication systems. This is due to its bet-ter time-frequency localization properties whichmay reduce intersymbol and intercarrier inter-ference, and remove need for the cyclic prefixneeded in conventional OFDM systems [22]. Therefereed papers consider di ff erent aspects of theGMC signal but not synchronization. In [23] itwas just mentioned that synchronization may beperformed on subcarrier basis. The GMC sig-nal may be explained as follows. The STFT co-e ffi cients X lm are the transmitted data symbols.In this case the signal (2) is called the GMC sig-nal, or, if considered during interval 0 , . . . , N − M = 1. However, to keep thereceiver universal one might to want to receivealso these using the frequency-domain process-ing instead of conventional time-domain pro- cessing. This is possible since filtering, essentialto all receivers, may be done either in the time orfrequency-domain. This paper has readily shownhow this is performed using frequency-domainblock filtering. It is worth noting that also con-stant envelope DS signals may be received usinga conventional matched filter [47], and thus theproposed frequency-domain block filter. In the ideal Doppler processing, the input sig-nal is transformed into di ff erent frequency o ff -set corresponding to the possible Doppler val-ues. In a simpler solution, the filter is divided(partitioned) into blocks and the outputs of theseblocks are then Fourier transformed as shown in[30] (and references therein). The reference usestime-domain processing, but as already shown,this partitioned matched filtering can be donealso in the frequency-domain. If the Doppleris chancing (due to accelerated motion) betweenthe blocks one may possible search over all pos-sible (but sensible) Doppler tracks in the result-ing time-frequency uncertainty grid. The acqui-sition probabilities concerning the Doppler pro-cessing are analyzed in [30] and not repeated inthis paper. Another way to increase Doppler re-sistance is to combine the partial responses ei-ther non-coherently or in a di ff erentially coher-ent way [48]. Some systems have a basic long code and di-rect implementation of a filter matched to it maybe infeasible. Block filtering is a possible solutionwith shorter elements that are feasible to imple-ment. Another case where block matched filtersmay be needed is when a long code is dividedinto subintervals and each subinterval contains Ebrahim Karami, et al.a symbol. In this case responses of the parti-tioned matched filters are variables used for sym-bol demodulation (naturally sampled at symbolsynchro position). This may be needed in a longcode system where data rate is adjusted usingcode partitioning, but for some reasons short DScodes are not willed to be used.A possible example where block filtering maybe applied is the UMTS system where the uplinkpreamble consist of several scrambled repeats ofa short code [49]. frequency-domain processing allows easyadaption of spectrum sensing algorithms sinceFFT is readily included into the processing chain.Spectrum sensing may be applied in cognitiveradios to found available spectrum holes [50].Another application is interference cancellation(IC) needed especially in military systems.In these cases the process is known as notchfiltering, but in both the cases the technique isbasically the same. The window, inherent to theproposed receiver, is helpful since it reducesspectral leakage. However, a drawback of thewindowing is the SNR loss, which may be 3dB. Luckily overlapping, also inherent to thereceiver, reduces this loss almost down to zerodB.An important aspect, to notify when doingspectrum sensing or IC, is that if the desiredunderlaying signal is not flat, or white, in thefrequency-domain also it may be detected (ifSNR is high enough) or, what is worse, can-celed. To avoid this unpleasant phenomena, thereceiver should be designed using one sample persymbol/chip processing, i.e., the receiver shouldhave a traditional pulse shaping filter at frontand parallel processing of over-sampled streams.In this case we may loose an advantage of win-dows, but the complexity remains (almost) thesame.
Once the acquisition is performed, the receiverturns its attention into tracking and data demod-ulation. In this turn the receiver may continuematched filtering if the signal has a DS compo-nent. The block filtering allows di ff erent codelengths; the short are needed at high data ratesand the long are used in low data rates or whenDS processing gain is needed for interference tol-erance. In addition, the time varying nature [34]of the filter allows de-spreading of scrambled sig-nals. However, in this case the filter’s or correla-tor’s frequency response has to be updated fre-quently. Alternatively, the receiver uses correla-tion in the DS component case, pure FFT in the OFDM case or frequency-domain pulse shape fil-tering in the single carrier case. In the latter thefilter may filter several symbols at one filteringcycle and the filter’s frequency response is justthe pulse shape. This pulse shaping goal mayalso be achieved using a suitable analysis win-dow.
4. Acquisition Analysis
One usually requires detectors insensitive tosignal level variations called as constant falsealarm rate (CFAR) detectors. These CFAR detec-tors may also be derived using generalized like-lihood ratio detectors [51]. A CFAR detector ispresented in [52, 53]. Let ~y k = a k ~s + ~n k denote the k th received signal including N samples, where a k is a channel amplitude, ~s a preamble signalsuch that k ~s k = 1 ( k k denotes the Euclidean norm)and ~n k a complex white Gaussian noise with vari-ance σ . In addition, let r ( n ) be an output of theMF (a test variable). If the signal is not present a k = 0. The detector is | r ( n ) | > γ k ~s k k ~y k k , (3)where γ is a parameter depending on the desiredfalse alarm rate. The average signal power in theright hand side makes the detector a CFAR de-tector. It basically is an estimator of the thermalnoise level, but it also makes the detector insen-sitive to interference. Note that if an IC unit isused, the average signal power has to be mea-sured after the IC unit, i.e., after mitigation. Thisis so because mitigation may remove the inter-ference that otherwise could deny detection. Inother words, the mean signal power would be toohigh.Another concern is that the e ff ect of windowon the threshold since it a ff ects the signal power.This is more important, if input signal is win-dowed but the reference (filter) is not. Let ~w a and ~w r denote window vectors used for signal and fil-ter analysis. Then, one has to use the power dif-ference of windows as a normalizing factor. Fur-thermore, overlapping means that the computedresponse is replicated M/R times. As a conse-quence, the signal power should be modified as k ~s k ≡ MR k ~w a kk ~w r k k ~s k . (4)It can be shown [52, 53] that the false alarm prob-ability P FA , i.e., the probability that the thresholdis exceeded even though the signal is not present,can be approximated as P FA = e − γN , (5)from which γ can easily be obtained as γ = N ln( P FA ), where . Another useful probability isindowed Overlapped frequency-domain Block Filtering Approach for Direct Sequence SignalAcquisition 7the probability that the maximum exceeds thethreshold. It can be shown to be [52, 53] P FA,M = 1 − (1 − P FA ) N . (6)The probability of detection P D , i.e., the prob-ability that the test cell exceeds the thresholdwhen the actual synchro position is investigated,can be approximated [52, 53] as P D = Q (cid:16) p µ, q γ ( N + µ ) (cid:17) , (7)where µ = | a k | /σ is the signal-to-noise ratio(SNR) of the preamble signal and Q m ( a, b ) is thegeneralized Marcum Q-function [20].Another useful probability is the probability P m that the maximum occurs at the actual syn-chro position. The approximation in [52, 53] isnot too accurate. Therefore, another attempt thatwill result a closer approximation is provided.Briefly explained, the analysis tool in [52, 53]considers the distribution of r ( n ) as it is and as-sumes that k ~y k k converges to its average. Thissimplifies analysis since only one random vari-able has to be considered, but the method stillhas its roots on probability and statistics [54].Now, at the synchro position r ( n ) is a complexGaussian variable with mean a k and variance σ . Thus, | r ( n ) | has a non-central chi-squaredistribution. Assuming insignificant sidelobeson the autocorrelation function of the preamblesignal, the non-synchro positions are zero meanGaussian variables with variance σ . Now, theprobability of interest is P m = P ( | r synchro ( n ) | > | r non-synchro ( i ) | , ∀ i ). A direct application of theanalysis principle yields to result in [52, 53].However, this probability is equivalent the prob-ability that the decision variable at the synchroposition is larger than one of the largest non-synchro position. It is well-known that 98 % ofGaussian variables are within 2.33 standard de-viations from the mean. Thus, the novel approx-imation is P m = Q (cid:16) p µ, q . (cid:17) . (8)For very long (large N ) preamble signals theconfidence probability may be higher, e.g., 99.5%, since it is natural that then, on average,the largest test variable at non-synchro positionsmay be larger than with short signals. See [55]for another solution to this problem.Still another probability of interest is the prob-ability that the maximum exceeds the thresholdindependent of the fact is it the synchro positionor not. This is [53] P D,M = 1 − (1 − P FA ) N − (1 − P D ) . (9) Finally, the probability that the maximum is atthe synchro position and it exceeds the thresholdis P M = P m P D,M .The above results are derived in additive whiteGaussian noise (AWGN) case. In fading chan-nels the situation is di ff erent. In Rayleigh fad-ing channels, at the synchro position variable r ( n )follows a zero mean complex Gaussian distribu-tion with variance σ s + σ , where µ = E {| a k | } /σ = σ s /σ is the average SNR, i.e., P ( | r ( n ) | ) ≡ P ( y ) = 1 − e − y/ ( σ s + σ ) . (10)If the analysis tool in [52, 53] is adopted, it fol-lows that P D = e − γN/ ( µ +1) , (11)whereas the paper’s approach yields P m = (cid:18) e − (2 . / ( µ +1) (cid:19) . (12)In Rician fading channels, the decision vari-able of interest follows a complex Gaussian dis-tribution with mean a k and variance σ s + σ . Letthe ratio of the power of the constant and randomelement be | a k | /σ s = κ and let µ = | a k | /σ be theSNR of the constant element. Then, σ s + σ = σ ( µ/κ + 1). As a consequence, P D = Q (cid:18) s µ µκ + 1 , s γ ( N + µ ) µκ + 1 (cid:19) , (13)which reduces to that (7) in the AWGN channel(as it should) if the random element is weak sincewhen σ s = 0, then κ = ∞ . Correspondingly, P m = Q (cid:18) s µ µκ + 1 , s . µκ + 1 (cid:19) . (14) This section provides more exact analysis of P m in AWGN, Rayleigh, and Rician channels. Obviously, in an AGWN channel P m can be cal-culated as P AW GN m = Q (cid:16) p µ, α N − ) (cid:17) , (15)where α N is defined as, α N = E { max | r ( n ) | , n = 0 , . . . , N − } E {| r (0) | } , (16)where E { . } is the expectation operator and r (0)is the decision variable at the actual delay. Thevalue of α N closely follows a logarithmic func-tion of N . Ebrahim Karami, et al.In the random channels the integral (15) has tobe averaged over channel variations, i.e, integral P m = Z ∞ Q (cid:16) p µ, α N − (cid:17) P ( µ ) dµ (17)has to be solved, where P ( µ ) is the distribution ofthe SNR in a channel. In a Rayleigh channel P ( µ ) = µ ¯ µ exp( − µ ¯ µ ) (18)where ¯ µ is the average SNR. To solve the aboveintegral, the generalized Marcum Q-function isreplaced by its integral form. After some manip-ulations we will have P Rayl m = ( K ¯ µK ¯ µ + 1 ) − K exp( − Kα N − K ¯ µ + 1 ) , (19)where K is the number of PN sequences used forsynchronization. In a Rician channel P ( µ ) = µ ˜ µ exp( − µ + µ ˜ µ ) I (cid:16) √ α N − µ ˜ µ (cid:17) , (20)where ˜ µ is the average of the variable part of theSNR, µ is fixed part of the SNR such that ¯ µ = µ + ˜ µ , and I ( . ) is the zero order modified Besselfunction. To solve the needed integral, we have toreplace the generalized Marcum Q-function withits equivalent integral form whereas the Besselfunction is replaced by its Taylor series expan-sion. This series of integrals results P Rice m = ∞ X n =0 K − ( K ˜ µ ) n ( K ˜ µ + 1) n +1 F ( n + 1 , , µ µ ( K ˜ µ + 1) ) · e n + K ( Kα N − ) , (21)where F ( ., ., . ) is the hyper geometric function and e n + K ( Kα N − ) is the incomplete exponential func-tion defined as e n + K ( Kα N − ) = n + K − X m =0 ( Kα N − ) m . (22)Solution of (21) converges slowly. Convergencecan be speed up by manipulating (21) into form P Rice m =1 − ∞ X n =0 K − ( K ˜ µ ) n ( K ˜ µ + 1) n +1 F ( n + 1 , , µ µ ( K ˜ µ + 1) ) · (exp( Kα N − ) − e n + K ( Kα N − ) . (23) p r obab ili t y P D theorP m theorP D simul.P m simul. Fig. 2: Simulation and analysis results for a flat Rayleigh fad-ing channel.
5. Numerical Results
In this section the proposed windowedfrequency-domain acquisition technique issimulated and then compared to the analyticalresults which provide bounds on the acquisitionperformance. As a reference, it is reminded thatconventional non-windowed, non-overlappedapproached achieve the theoretical bounds inthe AWGN channel. Herein, it is trusted toa “fact” that if the analysis holds in Rayleighfading channels it holds also in AWGN channels.Therefore, only Rayleigh fading channels areused in simulations. In all the simulations a 64chips preamble sequence is used. It was a 63chip Gold code extended by one. The signal issampled one sample per chip. Simulation resultsare averaged over 1000 independent trials. SNRis expressed per preamble sequence. The desiredfalse alarm rate was quite high 10 − .Figure 2 shows the results in a flat Rayleighfading channel when M = R = 32, i.e., overlap-ping is not used, and the window is rectangular.The results show that the simulated and theoret-ical results coincide, i.e., the approximative anal-ysis is a proper one.Figure 3 shows interesting results concerninga frequency selective Rayleigh fading channel,which has two equal power multipath compo-nents with one chip separation. SNR is definedper path. In practice, the receiver does not knowis the detected signal sample from the first orsecond path. Therefore, also probability that ei-ther the first or second path exceed the thresh-old ( P D ), and probability that either the first orsecond path provides the maximum ( P m ) are re-ported. It can be concluded from the results thatdiversity in the multipath channels is very ben-eficial for the synchronization. Of course, thisbenefit is lost if the second path is weak andsituation becomes close to that in a single pathindowed Overlapped frequency-domain Block Filtering Approach for Direct Sequence SignalAcquisition 9 p r obab ili t y SNR [dB] P D theorP m theorP D simul.P m simul.P D2 simul.P m2 simul. Fig. 3: Simulation results for a frequency selective Rayleighfading channel. Theoretical values are for the flat fadingchannel. channel. It can be seen that multipath propa-gation causes SNR losses to P D . This is due tonon-zero autocorrelation sidelobes, which are in-versely proportional to the preamble length. An-other observation is that P m becomes close to half.This is easily understood since half of the timethe second path is stronger than the first path ifthe paths have an equal power. It appears, al-though not shown in the figure for clarity rea-sons, that a good explanation of for P i , where i is either D or m , is P i = 1 − Y k (cid:16) − P i (SNR k ) (cid:17) , (24)where the probabilities are expressed as a func-tion of SNR and SNR k is the SNR of the k thpath. This result follows from a though chainthat probability that either the first or second (or k th) path exceeds the threshold is equal to prob-ability that they all are below the threshold.The last set of simulations concerns e ff ects ofwindowing and overlapping. The used analysiswindow is the Kaiser window with the parame-ter 8, which has very low tail values. The win-dow for the reference is rectangular. The over-lapping is either non, 50 % or 75 %, i.e., R =64, 32 or 16 while M = N = 64. The chan-nel is a flat fading Rayleigh channel. The simu-lated false alarm rates with the original thresholdsetting are 0.01 (the desired value as it should),0.16, 0.54, respectively, without the window and4 . × − , 3 . × − , 0.05 with the window(640000 samples). This shows that thresholdtuning is needed if a desired false alarm rateis needed with windows and overlapping. Thetrend seems to be that a non-rectangular windowdecreases the false alarm rate, whereas overlap-ping increases it. As a consequence, simulationswith the original threshold setting would not befair with respect the false alarm rate. There- p r obab ili t y theoryR=64R=32R=16R=64 & windR=32 & windR=16 & wind Fig. 4: Probability of detection P D simulation results for a flatRayleigh fading channel when a window and overlapping areused. fore, a proper threshold (multiplier of the orig-inal) was determined by simulations for over-lapped and windowed cases. The results withequal false alarm rates are shown in Fig. 4. Theresults show that overlapping does not a ff ect theperformance significantly, but windowing does.Overlapping and windowing is even worse (by 2dB) than just windowing the conventional non-blocked MF (M=N=R=64). The windowing losswith the conventional MF is 2–3 dB with thiswindow.The last result is contrary to the expecta-tion that overlapping reduces windowing losses.Therefore, the last simulations use window alsofor the reference to see if that a ff ects the sit-uation. Fig. 5 demonstrates that adding ofthe reference window reduces the performance(decreases sensitivity), but now the overlappingdoes not further decrease it. The total loss com-pared to the theory is 5 dB. The results indicatethat if the same sensitivity is required, then win-dowed cases have to have a higher false alarmrate. Maybe the mentioned expectation resultsfrom the fact that overlapping increases sensitiv-ity if the threshold is kept constant. Therefore,paper’s results might not be in contradiction toearly ones.
6. Conclusions
The paper has provided insight into the win-dowed, overlapped, frequency-domain block fil-tering approach by explaining it and then show-ing (some of) its possible applications in radiocommunications. It was shown that this filteringapproach may be used as a universal basebandreceiver in communication systems, i.e., a singlebaseband architecture was shown to be able to re-ceive all kind of signals. This is especially helpfulin multipurpose platforms, which can (hereafter)0 Ebrahim Karami, et al. p r obab ili t y theoryR=64 & windR=64 & both windR=132 & both windR=16 & both wind Fig. 5: Probability of detection P D simulation results for a flatRayleigh fading channel when analysis and reference win-dows and overlapping are used. be based on single architecture simplifying thedesign. Further investigations will be needed tosee if this would reduce also other aspects in thereceivers such as power consumption or siliconarea.In particular, the proposed approach was ap-plied to signal acquisition with some novel anal-ysis of acquisition probabilities in fading chan-nels. This application and provided analysis andsimulation results verify usefulness of the ar-chitecture for a wide range of the channel con-ditions. In addition, simulations showed thatwindowing reduces sensitivity if a desired falsealarm rate is the receiver design goal. Therefore,one has to use windows with a care, e.g., in envi-ronments where they are really needed.One future research topic with the proposedfilter is that could a proper synthesis window beused to reduce the sensitivity losses the windowsproduce. Such a finding would improve useful-ness of the filter. A way to find an answer mightbe the dual window. 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