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Wireless CommunicationChapter:
Analog and
Digital Transmission
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We propose the use of a correlation window in the receiver which is equal to the full period of the spreading code. However, the spreading sequence used by the transmitter is longer than the period used by the correlator for detection. The extension consists of a cyclic prefix. In the event of Maximum Length (m) Linear Feedback Shift Register (LFSR) codes, the total code used to spread one user symbol then becomes the concatenation of one full period of the code plus the prefix. Cyclic prefixes are well established for Orthogonal Frequency Division Multiplexing (OFDM), and also used in equalizer training sequences to estimate the channel. Their use for Direct Sequence CDMA, in particular in conjuction with a rake receiver, has been explored to a much lesser extent.
2. A Simple Example
We assume that a logical "1" is represented by a user bit b0 = 1 and a logical "0" is represented by b0 = –1. In Direct Sequence Spread Spectrum, the user data symbols are multiplied with a fast code sequence. For instance the maximum length LSFR code {1,1,1,-1,1,-1,-1} has autocorrelation value 7 for zero time-offset and –1 for time-offsets 1, 2, .. or 6.
After transmission over a multipath channel, the received signal consists of multiple delayed copies of the transmit signal.
Figure 1: Direct Sequence Spread Spectrum, Three user bits spread by code sequence {1,1,1,-1,1,-1,-1}
The rake receiver for this signal consists of multiple correlators, each synchronized to one of the time offsets of the received signal. In the i-th finger of the rake, the incoming signal is multiplied with the code sequence synchronized to the signal arriving over path i. Let hi be the amplitude of the signal received over path i. If the autocorrelation is good, crosstalk into other fingers is weak, typically M times weaker that the wanted signal, with M the length of the sequence. Because of modulation, the crosstalk is much more severe in a practical situation. The signal Fi seen in the i-th finger consists of three terms, corresponding to contribution from the signal over all three paths. In this case,
Fo= – 7 h0 – h 1 – 3 h 2
F1= – 7 h1 – h 0 + h 2
F2= – 7 h2 – 3 h 0 – h 1
Note that the self-interference from the signal over path 2 is substantially (3 times) larger than would be case for unmodulated carriers, i.e., if b-1 = b0. Hence, ISI occurs.
Figure 2: Reception of three delayed waves, over Path 0, 1 and 2. The integration period used in finger 0 of the rake receiver is not well aligned with bit transitions in delayed paths. ISI occurs.
In the prefix example of an m-sequence {1, 1, 1, –1, 1, –1, –1}, the transmitter creates the spreading code {–1, –1, 1, 1, 1, –1, 1, –1, –1}. The receiver correlates during the same window, for every branch of the rake, but uses different, cyclically shifted sequences.
Figure 3: Delayed signals. A cyclic prefix allows the choice of a detection window that avoids ISI.
Now the ISI is reduced to levels determined by the periodic autocorrelation function, thus significantly smaller,.
Fo= – 7 h0 + h 1 + h 2
F1= – 7 h1 + h 0 + h 2
F2= – 7 h2 + h 0 + h 1
4. Performance Evaluation
Based on the analysis for partial correlations, for a single user channel, the decision variable recovered by the i-th finger (0 £ i £ LR) in the rake is
(8)
Here, the first term represents the "wanted bit" b0, the second and third term describe intersymbol (ISI, or self-) interference from bit b-1 and b+1.
For a system with a cyclic prefix longer than Lp, we can use the same expression, if we replace
and
.
Due to cyclic prefix, all ISI terms disappear. In summary, three effects associated with the use of a prefix influence the BER performance:
· Intersymbol interference. Depending on the value of the first partial cross-correlation function R<(k), paths with a different delay offset introduce interference components. It typically results in a floor in the symbol error rate. Above a certain signal-to-noise ratio, the BER does not significantly improve anymore if the signal-to-noise ratio is further increased. A cyclic prefix can entirely avoid this effect.
· Cyclic prefixes waste some transmit bandwidth and energy, because only M chips are considered by he detector. For a channel with delay spread Td one would need a prefix larger than Td, so effectively only a fraction MTc / (MTc + Td) of Eb can be used. To address this aspect, we define g I as Eb /N0 × MTc/(MTc + Td) × Gi as the effective energy received in the i-th finger.
· Correlated fading of the signal amplitudes in the various diversity branches of the rake. Its effect on performance is determined by the value of the second partial correlation function R>n,n(k), but its influence presumably is small as long as the partial cross-correlation functions do not become too poor. With a cyclic prefic this can be guaranteed more easily even if the spreading code is short, since it leads to a crosstalk described by Rn,n(k) = -1 as opposed to larger values (|R>n,n(k)| ³ 1 and |R<n,n(k)| ³ 1) for a system without a prefix. The best performance is achieved with uncorrelated fading. Correlation reduces performance.
5. Numerical Results
The foregoing development of a new model for partial correlations now allows us to adapt and use commonly known expressions for the BER for a rake receiver.
We consider an M = 15 LFSR sequence, for which R(0) = 15, R(¹ 0) = -1. We plot the average bit error rate, averaged of the ensemble of all channels and over all codes. The averaging over all codes comes in when we consider only the variance of the partial correlation functions. The method followed here extends the analysis of a rake in the presence of AWGN as for instance treated in [1]. ISI is modelled as Gaussian noise, because according to the Rayleigh channel model, the in-phase component from other (interfering) paths is Gaussian, and its amplitude is independent of the amplitude of the signal amplitude in the wanted path. Such assumption would be less realistic for a specific fixed (non-fading) channel, in which the ISI is binary and the eye-pattern exhibits a fixed reduction in opening width. The variance of the ISI in finger i is
(9)
We define h I as
This variable can be interpreted as the normalized multipath self interference. The effective local-mean signal-to-noise ratio in each finger is
(10)
If the rake receiver can optimally weight signals, such that the SNR at the output is the sum of the SNRs in all fingers, the BER for CDMA with rake receiver is
(11)
where the factors p j are [1]
This assumes accurate estimation of the signal power and interference power levels in all fingers. Also, Eq. (11) inherently assumes that weight factors are chosen to account for different noise and ISI levels in the fingers. If the rake just weighs the signal in each branch proportionally to the total amplitude per finger (as it would do if no crosstalk or ISI were present), the BER becomes somewhat larger.
Figure 6: Average BER versus local mean Eb/N0 in a two-path Rayleigh fading channel. Single user spread-spectrum with (¾ ) and without (- × - ) cyclic prefix. Single-user narrowband transmission (
· ×× ). N = 15.
Figure 6 plots the average BER versus the signal-to-noise ratio for a Rayleigh channel with delay profile Ehh*T = diag(1, 0.5, 0.25). For a narrowband signal, the BER is ½[1-Ö (g t / 1+g t)], with g t the local-mean signal-to-noise ratio. It equals the sum of the expected powers in the entire delay profile. Because of the diversity achieved by the rake, the spread-spectrum system has a steeper, thus better, slope of the curve for BER versus g t. However, if no cyclic prefix is applied, the curve levels off for a BER a little better than 10-4 for M = 15.
6. Applications in Systems
The cyclic prefix described here can be used in systems for spread spectrum communications. Examples are proprietary systems that operate in ISM bands, such as the 2.4 GHz band. The application is useful for channels with a delay spread that is on the order of a few chip times.
An other application is in synchronous multi-user systems. The synchronous downlink from a cellular base station would be a prime example of a potential application. Here one can think of new standards for universal personal communications, which extend the current cellular telephone systems, or indoor multimedia computing networks, such as those studied in the Berkeley Infopad project. Currently such systems are mostly inspired by the IS-95 US cellular CDMA standard, and use a Walsh-Hadamard spreading code for perfect user orthogonality, superimposed on a LFSR m-sequence. Let M be the spread factor and m the number of active users. In terms of resistance to self interference, the IS-95 solution is less optimal than pure m-sequences. In fact, in such systems the MUI is zero in a non-dispersive channel, but the self interference grows proportionally with m/Ö M [6]. Systems with a cyclic prefix have non-zero MUI (it is proportional with m/M) but the self interference is proportional to 1/M, thus it grows slower with m than in IS-95.
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7. Conclusions
We proposed and analyzed cyclic prefixes for DS. We made a comparison with systems that do not apply a cyclic prefix. This evaluation involved a statistical interpretation of the behavior of partial correlation functions. An accurate model has been proposed to model the average effect of partial correlation functions.
Numerical results for the BER performance in a Rayleigh channel with an exponential delay profile confirm the advantages of the use of a cyclic prefix. A comparison with IS-95 type of solutions has been initiated, but requires further study.
