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Pre- and Post-Processing for Optimal Noise Reduction in Cyclic Prefix

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Pre- and Post-Processing for Optimal Noise Reduction in Cyclic Prefix Pre- and Post-Processing for Optimal Noise Reduction in Cyclic Prefix Based Channel Equalizers Bojan Vrcelj and P. P. Vaidyanathan Dept. of Electrical Engineering 136-93 California Institute of Technology Pasadena, CA 91125-0001 Abstract— Cyclic prefix based equalizers are widely used for It is preceded (followed) by the optimal precoder (equalizer) high-speed data transmission over frequency selective channels. for the given input and noise statistics. These blocks are real- Their use in conjunction with DFT filterbanks is especially attrac- ized by constant matrix multiplication, so that the overall com- tive, given the low complexity of implementation. Some examples munications system remains of low complexity. include the DFT-based DMT systems. In this paper we consider In the following we first give a brief overview of the cyclic a general cyclic prefix based system for communication and show prefix system with DFT matrices used as the basic ISI can- that the equalization performance can be improved by simple pre- celer. Then, we introduce a way to deal with noise suppres- and post-processing aimed at reducing the noise at the receiver. This processing is done independently of the ISI cancellation per- sion separately and derive the optimal constrained pair pre- formed by the frequency domain equalizer.1 coder/equalizer for this purpose. The constraint is that in the absence of noise the overall system is still ISI-free. The per- formance of the proposed method is evaluated through com- I. INTRODUCTION puter simulations and a significant improvement with respect to There has been considerable interest in applying the eq– the original system without pre- and post-processing is demon- ualization techniques based on cyclic prefix to high speed data strated. transmission over frequency selective channels, such as twisted Throughout this paper we assume that the frequency selec- pair channels in telephone cables [1], [2]. Cyclic prefix systems tive channel is a known FIR system of order Ä, i.e. are invariably used in conjunction with the discrete Fourier Ä transform (DFT) matrices, and one example is the DFT-based X k ´Þ µ= c´k µÞ : discrete multitone modulation (DMT) system [3]. Although it C =¼ has been shown by Kalet [9] that the use of ideal transmit and k Ì receive filters in the multitone environment can lead to achiev- Ý A The notations A and represent the transpose and trans- ing signal to noise ratio (SNR) within 8-9 dB of the channel pose conjugate of a matrix A. capacity (depending on the probability of error), in practice the DFT filterbank is most commonly used because of its low com- II. CYCLIC PREFIX SYSTEM WITH DFT MATRICES plexity. The development of these systems lead, for example, In this section we give a review of a cyclic prefix based sys- to asymmetric digital subscriber loops (ADSL) and high-speed tem for channel equalization that employs DFT filterbanks. Be- digital subscriber loops (HDSL) [2], [3]. fore we demonstrate the complete system, we first introduce the The initial goal of cyclic prefix based methods for equaliza- concept of cyclic prefix. This is a way of inserting redundancy tion is to cancel the intersymbol interference (ISI) induced by into the input data stream and it becomes useful in the process the channel. This task is achieved by introducing some redun- of channel equalization as will be explained later. Consider dancy in the form of a cyclic prefix (see Sec. II). Depending on the symbol stream from Fig. 1(a). It is divided into blocks of the nulls of the channel in question, the noise can get severely Ä length Å . The last symbols from each block are copied and amplified by the equalizer alone. In this paper we consider a inserted at the beginning of that block (we assume hereafter general cyclic prefix based system with the aim of further re- that the redundancy Ä< Å). This is achieved by “squeezing” ducing the noise at the receiver. This is achieved by perform- the samples as explained in Fig. 1(b)-(c). Obviously, this op- ing the ISI cancellation and the noise suppression separately, eration of inserting the redundancy into the input data stream in different modules. The main module for ISI cancellation is results in the bandwidth expansion. In this case the bandwidth based on cyclic prefix with DFT filterbanks and is unchanged. = ´Å · ĵ=Å expansion ratio is given by « , which can be 1 Work supported in part by the ONR grant N00014-99-1-1002, USA. made sufficiently small by making Å large. The purpose of T R A N S M I T T E R R E C E I V E R block of M q(n ) s(n ) q(n ) r(n ) s(n ) -1 channel and Γ -1 W prefix system W e n IDFT DFT DFT domain (a) e(n ) equalizer Figure 3: Conventional cyclic prefix system with DFT matrices used for ISI cancellation. (b) n room for L samples copy copy copy x(n ) thus we have ¾ ¿ C [¼] ¼ ¼ ::: ¼ (c) n Å 6 7 ¼ C [½] ¼ ::: ¼ Å 6 7 = ; 6 7 e . (2) 4 Figure 1: (a) Input symbol stream, (b)-(c) explanation of . .. 5 ¼ ¼ ::: C [Å ½] how cyclic prefix is inserted. ¼ Å È Ä j ¾kÒ=Å C [Ò]= c´k µe with Å . As we noted before, the k =¼ L L ISI cancellation is achieved by inverting C in (1). Therefore, q(n ) C(z ) discard ½ after moving the matrix Ï to the transmitter side, the com- channel M prefix plete DFT-based cyclic prefix system is shown in Fig. 3. The unblocking noise blocking M r(n ) M block labeled “channel and prefix system” is given in Fig. 2. ½ Matrix e is better known as the frequency domain equal- Figure 2: Channel with the system for cyclic prefix. izer and contains on its diagonal the reciprocals of the DFT ´Òµ coefficients of the channel. The input vector sequence × in Fig. 3 is obtained by blocking the symbol stream into blocks of e´Òµ length Å . The noise vector is obtained by blocking the · Ä physical noise process into blocks of length Å and then inserting the cyclic prefix is to allow the receiver to remove ISI discarding Ä samples out of each block (see Fig. 2). using some low-complexity operations. The main merit of the cyclic prefix system with DFT ma- As the first step in understanding how this is achieved, con- trices lies in its simplicity. From Fig. 3 we see that the trans- sider Fig. 2, ignoring the noise for the time being. The box la- mitter only needs to perform the inverse DFT operation fol- beled “blocking” is a serial to parallel converter, similarly “un- lowed by the cyclic prefix insertion. On the other hand, the blocking” performs the reverse operation and the block “prefix” receiver removes the cyclic prefix, performs the DFT operation inserts the cyclic prefix as described in Fig. 1. Notice that the and frequency domain equalization, which amounts simply to Å ´Òµ Ö´Òµ blocked sequences Õ and are related through a mul- multiplying each channel by a constant. Moreover, if is cho- ¢ Å tiplication by the Å right circulant [5] channel matrix sen as a power of two, both IDFT and DFT operations can be C Ä ·½ C. The first column of consists of the coefficients implemented using the fast radix-2 algorithm. The noise vec- e´Òµ ´k µ Å Ä ½ of the channel impulse response c followed by tor process at the receiver passes through the DFT ma- ½ 2 Ï Ï c´¼µ 6= ¼ zeros. Since we can safely assume , it follows that trix and the frequency domain equalizer e . Since C is nonsingular and the effect of the channel can be neutral- is a scaled unitary matrix, the main contribution to amplify- ized by inverting it. Next, note that any circulant matrix can be ing the noise power at the receiver comes from the multipliers ½ ½=C [Ò] C ´Þ µ e diagonalized by a DFT matrix [5]. Thus we can write [5] Å embedded in . Obviously, if the channel has zeros near the unit circle, these multipliers can get large, ½ consequently boosting the noise at the receiver and degrading C = Ï Ï ; e (1) the performance. In the next section we propose a simple mod- ification of the basic equalization structure from Fig. 3 that will Å ¢ Å where Ï is the DFT matrix and the diagonal matrix improve the performance significantly. C e has the eigenvalues of on its diagonal. Those eigenvalues, c´k µ in turn, are nothing but the Å -point DFT coefficients of , III. MODIFIED SYSTEM FOR NOISE SUPPRESSION Consider the system shown in Fig. 4(a)-(b). It consists of 2 ´Þ µ Otherwise we can shorten the impulse response of C . the same basic cyclic prefix transceiver (the middle portion) s(n ) x(n ) r(n ) -1 -1 channel and T W prefix system Our objective as mentioned before is to minimize the out- IDFT (a) put noise power (subject to the power constraint). This can be written as r(n ) x(n ) s(n ) -1 ½ ½ Ý Ý Γ Ý ÑiÒ fÌ ÏÊ Ï ´ Ì g; W e T µ e e Tr ee that is DFT DFT domain (b) Ì 4 ½ ½ ( ) Ý Ý e n equalizer Ý ÑiÒ fÈ ÈÉg; É = ÏÊ Ï ´ µ e e Tr ee (6) È Figure 4: Cyclic prefix system with separated ISI cancellation Notice that neither the objective (6) nor the constraint (5) de- and noise suppression. pend on the choice of the unitary matrix Í. This leads us to = conclude that without loss of generality we can choose Í Á Ì = È Å ,or .
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