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BER Minimized OFDM Systems With Channel

Independent Precoders

Yuan-Pei Lin, Member, IEEE, and See-May Phoong, Member, IEEE

Abstract—We consider the minimization of uncoded bit error

rate (BER) for the orthogonal frequency division multiplexing (OFDM) system with an orthogonal precoder. We analyze the BER performance of precoded OFDM systems with zero forcing and minimum mean squared error (MMSE) receivers. In the case of MMSE receivers, we show that for quadrature phase shift keying (QPSK), there exists a class of optimal precoders that are channel independent. Examples of this class include the discrete Fourier transform (DFT) matrix and the Hadamard matrix. When the precoder is the DFT matrix, the resulting optimal transceiver becomes the single carrier system with cyclic prefix (SC-CP) system. We also show that the worst solution corresponds to the conventional OFDM system; the conventional OFDM system has the largest BER. In the case of zero forcing receivers, the design of optimal transceiver depends on signal-to-noise ratio (SNR). For higher SNR, solutions of optimal precoders are the same as those of MMSE receivers.

Index Terms—BER optimal multicarrier, OFDM, precoded

OFDM, single carrier.

I. INTRODUCTION

T

HE DISCRETE Fourier transform (DFT)-based trans-ceiver has found applications in a wide range of transmission channels, either wired [1]–[3] or wireless [4]–[8]. It is typically called discrete multitone (DMT) for wired digital subscriber loop (DSL) applications and orthogonal frequency division multiplexing (OFDM) for wireless local area network (LAN) and broadcasting applications, e.g., digital audio broadcasting [7] and digital video broadcasting [8]. The transmitter and receiver perform, respectively, -point indiscrete Fourier transform (IDFT) and DFT computation, where is the number of tones or number of subchannels. At the transmitter side, each block is padded with a cyclic prefix of length . The number is chosen to be no smaller than the order of the channel, which is usually assumed to be an FIR filter. The prefix is discarded at the receiver to remove interblock ISI. As a result, a finite impulse response (FIR) channel is converted into frequency-nonselective parallel

Manuscript received August 27, 2002; revised February 3, 2003. This work was supported in part by the National Science Council, Taiwan, R.O.C., under Grants NSC 90-2213-E-002-097 and 90-2213-E-009-108, the Ministry of Edu-cation, Taiwan, R.O.C., under Grant 89E-FA06-2-4, and the Lee and MTI Center for Networking Research. The associate editor coordinating the review of this paper and approving it for publication was Prof. Nicholas D. Sidiropoulos.

Y.-P. Lin is with the Department Electrical and Control Engineering, National Chiao Tung University, Hsinchu, Taiwan, R.O.C. (e-mail: ypl@cc.nctu.edu.tw). S.-M. Phoong is with the Department of Electrical Engineering and Graduate Institute of Communication Engineering, National Taiwan University, Taipei, Taiwan, R.O.C.

Digital Object Identifier 10.1109/TSP.2003.815391

subchannels. The subchannel gains are the -point DFT of the channel impulse response.

For wireless transmission, the channel profile is usually not available to the transmitter. The transmitter is typically channel independent, and there is no bit/power allocation. Having a channel-independent transmitter is also a very useful feature for broadcasting applications, where there are many receivers with different transmission paths. In OFDM systems, the channel-dependent part of the transceiver is a set of scalars at the receiver, and the transmitter is channel independent. In DSL applications, the channel does not vary rapidly. The transmitter has the channel profile, which allows bit and power allocation to be employed. Using bit allocation, the disparity among the subchannel noise variances is exploited in the DMT system for bit rate maximization. The DMT system has been shown to a be very efficient technique in terms of transmission rate for a given probability of error and transmission power.

In the context of transceiver designs for wireless applications, the single carrier system with cyclic prefix (SC-CP) system [9] is also a DFT-based transceiver with a channel independent transmitting matrix, i.e., the identity matrix. A cyclic prefix is also inserted like in the OFDM system. The receiver per-forms both DFT and IDFT operations. It is demonstrated that the SC-CP system has a very low peak to average power ratio (PAPR). Furthermore, numerical experiments demonstrate that it outperforms the OFDM system for a useful range of bit error rate (BER) [10]. We will see later that the SC-CP system can be viewed as the OFDM system with a DFT precoder. In [11], precoded vector OFDM systems are proposed for combating spectral nulls. When the channel has spectral nulls, the pro-posed system outperforms the conventional OFDM system. In the precoded vector OFDM scheme, more redundant samples are needed than in the conventional OFDM system. In [12], de-signs of linear precoding to maximize diversity gain are consid-ered.

Design of more general block transceivers, which are optimal in the sense of minimum transmission power or minimum total noise power, has been of great interest. In [13], general block transceivers, which are not constrained to be DFT matrices, are investigated. For the class of zero-padding transceivers, an op-timal solution that minimizes the total output noise variance is given in [13]. The optimal receiver and zero-padding transmitter can be given in terms of an appropriately defined channel matrix and the autocorrelation matrix of the channel noise. Information rate optimized DMT systems are considered in [14] and [15]. In [16], intersymbol interference (ISI)-free block transceivers are considered. Under an optimal bit allocation, optimal transmit-ters and receivers that minimize transmission power for a given 1053-587X/03$17.00 © 2003 IEEE

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Fig. 1. Block diagram of the OFDM system over a channelP (z) with additive noise (n).

bit rate and probability of error are derived. In all these systems [13]–[16], it is assumed that the transmitter has the channel pro-file.

Recently, based on the zero-forcing solution given in [13], Ding et al.1 consider a class of optimal precoders in [17], in

which a precoder refers to the transmitting matrix. It is assumed in both [13] and [17] that the transmitter has the channel profile. The optimal transmitter that minimizes the output noise vari-ance consists of a unitary matrix, a diagonal power loading ma-trix, and a second unitary matrix that is arbitrary [13]. The first unitary matrix exposes the eigenmodes of the channel, whereas power loading exploits the eigenmodes to reduce the total noise variance. The second unitary matrix is optimized in [17] to min-imize BER. For high SNR, the DFT matrix has been found to be optimal for BPSK modulation. The resulting transmitter is channel dependent.

In this paper, we will consider the minimization of BER for OFDM systems with orthogonal precoders. The underlying system is, in fact, the class of cyclic prefixed block transceivers with orthogonal transmitters. We will address the design of optimal precoders with the assumption that there is no bit and power allocation. Notice that the objective is BER, rather than mean squared error (MSE). In the conventional single-band transmission system, BER is directly tied to mean squared error. For multisubchannel systems like OFDM and SC-CP systems, this is no longer true. In the absence of bit and power allocation, transceivers with the same total noise variance can have different BER performances. This is because different transceiver designs distribute the noise among the subchannels differently. We will consider the design of optimal precoders for zero forcing and for MMSE receivers. In the MMSE case, we show that when the modulation symbols are QPSK, optimal precoders are not unique. In this case, there exists a whole class of channel-independent optimal precoders. Examples of pre-coders in this class include the DFT matrix and the Hadamard matrix. It turns out that when the precoder is chosen as the DFT matrix, the resulting transceiver becomes the SC-CP system [9]. On the other hand, we also show that the identity matrix is the worst precoder, and the conventional OFDM system has the largest BER. In the case of zero forcing receiver, solutions of optimal precoders are SNR dependent. For higher SNR, there also exists a class of channel-independent optimal precoders. The optimal solutions are the same as those of the MMSE receivers. We will derive the results for QPSK modulations. Generalizations to phase amplitude modulation (PAM), phase shift keying (PSK), and quadrature amplitude modulation (QAM), based on approximated BER from symbol error rate formulae, can be obtained with slight modifications. Some

1The authors would like to thank the anonymous reviewers for bringing this

reference to our attention.

preliminary results on the zero-forcing case can be found in [23] and [24].

The sections are organized as follows. In Section II, we present the schematic of an OFDM system with an orthogonal precoder. We will state results of the conventional OFDM and the SC-CP systems that will be useful for later discussion. In Section III, we consider zero forcing receivers and derive the optimal precoder for QPSK modulation. Extensions to modulation schemes other than QPSK are given in Section IV. The performance of a precoded OFDM system with an MMSE receiver is analyzed in Section V. Numerical examples of BER performances are given in Section VI. A conclusion is given in Section VII.

A. Notations and Preliminaries

1) Boldfaced lower case letters represent vectors, and bold-faced upper case letters are reserved for matrices. The no-tation denotes transpose-conjugate of .

2) The function denotes the expected value of a random variable .

3) The notation is used to represent the identity matrix.

4) The notation is used to represent the unitary DFT matrix given by

for

II. OFDM TRANSCEIVERSWITHORTHOGONALPRECODERS The block diagram of the OFDM system is as shown in Fig. 1. The modulation symbols to be transmitted are first blocked into by 1 vectors, where is the number of subchannels. Each input vector of modulation symbols is passed through an by IDFT matrix, followed by the parallel to serial (P/S) oper-ation and the insertion of redundant samples. The length of re-dundant samples is chosen to be no less than the order of the channel so that inter-block interference can be removed. Usually the redundancy is in the form of a cyclic prefix. At the receiving end the cyclic prefix is discarded. The samples are again blocked into by 1 vectors for -point DFT

compu-tation. The scalar multipliers , for ,

are the only channel dependent part of the transceiver design,

where are the -point DFT of the channel

impulse response . In this case, ISI is canceled completely, and the receiver is a zero-forcing receiver.

In this paper, we consider the class of block transceivers with an orthogonal transmitter, followed by cyclic prefix in-sertion. This class of system can be viewed as an OFDM system with a unitary precoding matrix , as shown in Fig. 2, where

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Fig. 2. OFDM system with a precoderT.

Fig. 3. Illustration of noise path at a zero-forcing receiver.

is unitary with . The resulting block transceiver has a unitary transmitting matrix . To have a zero-forcing receiver, is cascaded to the end of the receiver. The transmit-ting matrix and receiving matrix are as shown in Fig. 2. By considering the optimal solution of the precoder , we are ad-dressing the problem of designing optimal cyclic-prefixed block transceivers with orthogonal transmitters.

Bit Error Rate: We assume that the channel noise is complex circular AWGN with variance . The modulation scheme is QPSK, and modulation symbols

with symbol energy . Let the receiver output vector be as indicated in Fig. 2; then, the output error vector is . The vector comes entirely from the channel noise as the receiver is zero forcing. The noise vector can be ana-lyzed by considering the receiver block diagram in Fig. 3. The vector consists of a block of size of the noise process . The elements of are uncorrelated Gaussian random vari-ables with variance . The elements of continue to be uncorrelated Gaussian random variables with variance , due to the unitary property of . Therefore, the th element of the noise vector has variance given by . The output noise is related to by

where denotes the th element of . As are un-correlated, the th subchannel noise variance

. That is

for (1)

The real and imaginary parts of have equal variance. Let , which is the SNR of the th subchannel; then

where (2)

As is unitary, we have . Using this

fact, we can write the average mean square error (MSE) as

(3)

The average MSE is independent of . All zero-forcing OFDM transceivers with a unitary precoder have the same MSE given in (3).

For QPSK modulation, the BER of the th subchannel is [19]

where The average BER is

Although the MSE is the same regardless of , the choice of affects how the same amount of noise is distributed among the subchannels. We look at two important cases of and the respective BER analysis.

• OFDM System: The unitary precoder is . We have (4) For the th subchannel, the SNR is

(5) where is the SNR . The BER of the OFDM system becomes

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• SC-CP System: When the unitary precoder is the DFT matrix , the transmitting matrix . The unitary matrix appended to the receiver is . The resulting system shown in Fig. 4 becomes the SC-CP system [9]. The SC-CP system can be viewed as a precoded OFDM system with precoder . All the elements in the DFT matrix have the same magnitude, which is equal to . Using this fact and (1), we see that the noise vari-ances in all the subchannels are the same, and they are

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Fig. 4. Block diagram of the SC-CP system over a channelP (z) with additive noise (n).

equal to the average MSE . As a result, all the subchannels have the same SNR - or

-The BER of the SC-CP system can be written as

- - (7)

The BER performance of the precoded system is determined by subchannel SNRs. The unitary property of allows us to establish upper and lower bounds on the subchannel SNRs.

Lemma 1: For any unitary precoder , the th subchannel SNR is bounded by

for (8)

where is the th subchannel SNR of the

OFDM system.

Proof: Using (1) and (4), we observe that the variance of

the th subchannel is given by

In addition, by using the fact that is unitary with , the columns of have unit energy, i.e., , for all . We have

Similarly, we can show that

. The bounds of SNR follow directly from the bounds of .

These relations hold for any unitary precoder . For a dif-ferent choice of , the noise variances are distributed differ-ently, but they are always bounded between and . For any precoder , the best subchannel is no better than the best subchannel of the OFDM system, and the worst subchannel is no worse than the worst subchannel of the OFDM system. In the next section, we derive the optimal such that the average BER is minimized.

III. OPTIMALPRECODERS

For the convenience of the following discussion, we introduce the function

(9)

Fig. 5. Plot off(y) = Q(1=py) for 0  y  1.

The subchannel BER can be expressed as

. The BER performance is closely related to the behavior of the function . Important properties of are given in the following lemma. A proof is given in Appendix A.

Lemma 2: The function is monotone in-creasing. It is convex when and concave when . A plot of is shown in Fig. 5. Each subchannel is oper-ating in the convex or the concave region of the function , depending on subchannel SNR . In particular, when

, the th subchannel is operating in the convex region of

and . If , the th subchannel

is operating in the convex region, and . We de-fine three useful SNR quantities

By definition, they satisfy . We also define three SNR regions:

When , we have - , i.e., the subchannels of the SC-CP system operating on the boundary between the convex and the concave regions of . For the two SNR regions and , the following can be observed.

• For the SNR region , , and all the subchannels

in the OFDM system have SNR for all .

In addition, using Lemma 1, we know that for any unitary precoder . Therefore, all the subchannels are

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operating in the concave region of for any precoder .

• For the SNR region , . In this case, all the subchannels in the OFDM system have SNR

. Moreover, the results in Lemma 1 imply that for an arbitrary unitary precoder , we always have ; all the subchannels are operating in the convex region of the function for any precoder .

For and , we can establish the following relations among the BER performances of the three systems OFDM, SC-CP, and an OFDM system with an arbitrary unitary precoder

. A proof is given in Appendix B.

Theorem 1: Let be the BER of the OFDM system with a unitary precoder in Fig. 2. Then

- for

- for

Each of the two inequalities relating and - becomes an equality if and only if subchannel noise variances are equalized, i.e., , where is as given in (3).

Channel - Independent Transmitters Achieving - :

Theorem 1 states that we have - if and only if

are equalized, i.e., ,

where is as given in (3). In particular, to have channel independent solutions of , we can choose

(10) In this case, all the subchannel BERs are the same:

- . There are many unitary matrices satisfying (10). Two well-known solutions satisfying (10) are the DFT matrix and the Hadamard matrix [18]. When , the trans-mitting matrix , and the transceiver in Fig. 2 becomes the SC-CP system in [9]. The Hadamard matrices can be generated recursively for , that is, a power of 2. The 2 2 Hadamard matrix is given by

The Hadamard matrix can be given in terms of the Hadamard matrix by

The Hadamard matrix is real with elements equal to . The resulting transmitting matrix will be complex. The implementation of Hadamard matrices requires only addi-tions. The complexity of the transceiver is slightly more than the OFDM system due to the two extra Hadamard matrices.

When we have a unitary that has the equal magnitude prop-erty in (10), we can use to generate other unitary matrices satisfying the equal magnitude property. For example, consider a matrix with

for arbitrary real choices of and . The new matrix is also unitary, and it has the equal magnitude property.

BER of Precoded OFDM Systems in Different SNR Re-gions: The results in Theorem 1 imply that the conventional

OFDM system ( ) is the optimal solution for in . When all the subchannels are operating in the concave region of , the OFDM system has the smallest error rate. For in , it is the worst solution; when all the subchannels are operating in the convex region of , the OFDM system has the largest error rate. However, as we will see next, the SNR region corresponds to a high error rate, whereas corresponds to a more useful range of BER. The error rate behavior can be analyzed by considering the value of in the following three regions.

1) The case : In this range, the OFDM system is the op-timal solution. All the subchannels have , and

hence, . In this range of SNR,

the error rate is at least 0.0416, which is a BER that is too large for many applications. Furthermore, the min-imum error rate 0.0416 can be achieved only when all the subchannels have , which is true only in

the special case .

2) The case : For this range, the OFDM system has the largest BER, and the BERs of all precoded OFDM systems are lower bounded by - . All subchannels are operating in the convex region of , and . The subchannel error rate is less than , and the average

. Notice that when , the worst subchannel

of the OFDM system has an error rate ,

and the average BER is at least . Therefore, is also the minimum SNR to have an error rate lower than

in the OFDM system. For example, for , is the smallest SNR for the OFDM system, to achieve an

error rate . For , is the

smallest SNR for achieving a BER

. The SNR region corresponds to a more useful range of BER.

3) The case : We can plot - and as functions of . The curves of and - cross in this range as is smaller than - for and larger than - for . In most of our experiments, the crossing of the two curves happens at an SNR close to , i.e., the SNR for which the subchannels of the SC-CP system fall in the convex region of the function .

Remarks: When the channel has a spectral null, say , the subchannel noise variances in the SC-CP system given in (3) go to infinity. The average probability of error is half in all subchannels, regardless of the value of SNR. In this case, goes to infinity, and the SC-CP system is not an optimal solution for any SNR. Such cases can be avoided by using an MMSE receiver, to be discussed in Section V.

IV. OTHERMODULATIONSCHEMES

The derivations in Sections II and III are carried out for QPSK modulation. Using approximations of BER obtained from symbol error rate (SER), we can extend the results to

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PAM, QAM, and PSK with slight modifications. Optimal pre-coders are obtained based on BER approximation formulae. We will take QAM modulation as an example. Suppose the inputs are ary QAM symbols with variance . The subchannel SER can be approximated by [19]

(11)

where is the SNR in the th subchannel. When Gray code is used, BER can be approximated from SER

as [19]. Therefore, we have

where

and (12)

The subchannel SNRs observe the same bounds given in (8), . When SNR is large enough such that all the subchannels satisfy

, equalizing the subchannel noise variances will minimize the approximated BER given in (12). The condition for this is

where (13)

On the other hand, when for all , the conven-tional OFDM system is the optimal transceiver. The condition

for this is , where is now . The

conditions now depend on the QAM constellation. For a large constellation, i.e., larger , both and also become larger. Similarly, the above technique is valid for any modulation scheme in which the subchannel symbol error probability can be either approximated or expressed as

for some constants and that are independent of subchannels. Examples of such a case include PAM, QAM, and PSK modu-lation schemes. Once the error probability is in such a form, we can invoke the convexity and concavity of to obtain the SNR ranges for which the OFDM system or the SC-CP system is optimal.

Remark: For real modulation symbols, e.g., PAM, the noise

relevant for symbol detection of the th subchannel is only the real part of but not the imaginary part. The subchannel noise has equal variance in real and imaginary parts. Therefore, the relevant noise variance is , which should be used in

the evaluation of , i.e., .

V. MMSE TRANSCEIVERS

In this section, we consider the case that the receiver is one that has MMSE. We will show how to derive the optimal

precoder for an MMSE receiver. We will see that using an MMSE receiver improves the system performance, especially when the channel has spectral nulls. The following lemma gives the MMSE receiving matrix for a given unitary precoder (the proof is given in Appendix C).

Lemma 3: Consider the precoded OFDM transceiver in

Fig. 2. Suppose the inputs have zero mean and variance , with real and imaginary parts having equal variances . The noise is circular complex Gaussian with variance . Let be the receiver output, and let the error vector . For a given unitary transmitting matrix , the optimal receiving matrix that minimizes is given by

where diag

(14)

The real and imaginary parts of have equal variance. The

average MSE is

. In this case, the th receiver output can be expressed as

where

(15) For the th subchannel, the variance of interference plus noise is . The subchannel signal-to-interference-noise-ratio

(SINR) is given by

(16) From the above lemma, we see that the average MSE is also independent of the choice of like the zero forcing case. The MMSE receiver can be easily obtained from the zero forcing re-ceiver by replacing the channel-dependent scalars from to given in (14).

When an MMSE receiver is used, the system is not ISI free, and the error does not come from channel noise alone. The term is a mixture of channel noise and signals from all the other subchannels. However, Gaussian tail renders a very nice approx-imation of BER [20]–[22], as we will see later in examples. The approximation is extremely good for a reasonably large , e.g., . Throughout the rest of the paper, we will use the Gaussian assumption.

The computation of BER depends on the modulation scheme used. We will use QPSK as an example. The results in Lemma 3 tell us that subchannel errors have equal variances in real and imaginary parts; therefore, the real and imaginary parts of the QPSK symbols have equal probability of error. The BER of the

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th subchannel is . To simplify derivations, let us de-fine

Then, the subchannel BER is

Using (16), we can verify that the argument of in the above equation can be expressed as

(17)

Therefore, we have

where

(18) For the conventional OFDM system, the precoder is the identity matrix. We can see from (16) that the subchannel SINR is : the same as that in the zero-forcing case (5); an OFDM system with a zero-forcing receiver has the same performance as that with an MMSE receiver. For the SC-CP system, is the DFT matrix with . Using the definition of , the BERs of OFDM and SC-CP systems are given, respectively, by

- (19)

Lemma 4: The function defined for

is convex with and .

The lemma is proved in Appendix D. A plot of for is given in Fig. 6. We only need to consider that the interval as the argument of in (18) is in this range. Using the convexity of , we can show the following theorem.

Theorem 2: Let , as given in (18), be the BER of the MMSE-equalized precoded OFDM systems with a unitary precoder . Then

-The first inequality becomes an equality if and only if sub-channel SINRs are equalized.

Proof: Using (17) and the fact that the SINR

, we can see that

Fig. 6. Plot ofh(y) = Q(py 0 1) for 0  y  1.

for

As in the proof in Theorem 1, we can use the convexity of to show that

The lower and upper bounds are, respectively, - and , which are given in (19).

As we mentioned in the previous section for the zero forcing receiver, the output noise becomes infinitely large in the pres-ence of spectral null. In the MMSE case, the expression of sub-channel SINR in (16) indicates that even if the channel has spectral nulls, is not zero.

Optimal Precoders: Theorem 2 states that the minimum

- is achieved, if and only if are equalized. Observing (16), we see that can be equalized by choosing , which has the equal magnitude property in (10). The same class of achieving - in the zero forcing case is also optimal for the MMSE case. Again, the Hadamard matrix along with the DFT matrix are examples of such solutions. On the other hand, the conventional OFDM system, although optimal for low SNR in zero-forcing case, is the worst solution in MMSE case for all SNR .

Other Modulation Schemes: For modulations other than QPSK, we can use approximations of BER from SER as in Section IV. The results will be stated without proofs. We will use -ary QAM as an example. By (11), , where is as given in (16). Let us define

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Similar to the QPSK case, we can use (16) to express the argument of in the above equation as

. Therefore, we have

where

(20) To exploit the convexity and the concavity of , we define

and where

and

We can show that is convex over and , and it is

concave over . Notice that when , the th

subchannel is operating in the convex region of . For the conventional OFDM system, the condition for this is

(21) Similarly, when and are such that

(22) the th subchannel of the conventional OFDM system is oper-ating in the convex region . When SNR is high enough so that (21) is true for all or when the SNR is low enough so that (22) is true for all , all the subchannels of the conventional OFDM system will operate in one of the convex regions of . In this case, we can invoke the convexity of to show that the class of unitary matrices satisfying the equal magnitude property in (10) are optimal. It can be verified that the case corresponds to a very high error rate, whereas corresponds to a more useful range. To have all subchannels operating in , we need

where

It can be further verified that is less than the value of given in (13). This means that - becomes the min-imum BER at a smaller SNR than - .

VI. SIMULATIONEXAMPLES

We will assume that the noise is AWGN with variance . The modulation symbols are QPSK with values equal

to and SNR . The number

of subchannels is 64. The length of cyclic prefix is 3. Two channels with four coefficients will be

used in the first three examples: ,

, , ,

and , ,

, . The magnitude

responses of the two channels and are shown in

Fig. 7. Frequency responses of the two channelsp (n) and p (n).

Fig. 8. Example 1. Performance comparison ofP ,P , P , P - , and P - for the channelp (n).

Fig. 7. The BER performance is obtained through Monte Carlo simulation, unless otherwise mentioned.

Example 1: We will use in this example. We compute

the values of , ,

and , respectively, as , 8.85, and 14.74

dB.

Fig. 8 shows and - as functions of SNR . We also show the BER for the case when the transmitting matrix is a unitary type II DCT matrix, which is denoted as . In this case, the precoder given by does not have the unit magnitude property in (10). Whenever SNR

is larger than dB, - given in (7) becomes the minimum BER for any unitary precoder . For , the conventional OFDM system is the optimal solution. For , we observe that . In this case, the OFDM system is optimal only for BER larger than 0.2. For either SNR range, or , and the performance of is in between

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Fig. 9. Example 2. Performance comparison ofP ,P , P , P - , and P - for the channelp (n).

and - . The BER performances of MMSE receivers and - are also shown in the same plot. In each case, the BER of the MMSE receiver is lower than the zero-forcing receiver for all SNR. For the OFDM system, an

MMSE receiver does not improve BER: .

In addition, observe that the crossing of - and occurs around BER and dB, which is a value very close

to dB.

Example 2: The channel in this example has a spec-tral null around (see Fig. 7). The DFT coefficients around are very small. The values of , , and are, respectively, 1.4, 33.8, and 51.9 dB. Fig. 9 shows the five BER performance

curves as in the previous example, , , ,

- , and - . Due to the zero close to the unit circle, the BERs of the three zero forcing systems , , and - become small only for large SNR. However, there is no serious performance degradation in the SC-CP system with an MMSE receiver. Notice that the crossing of - and occurs around dB, which is a value closer to

dB than to dB or dB. The BER

corre-sponding to the crossing is .

Example 3: In this example, the channel is , like in Example 1. We plot the actual BER and the approximation computed from (18) (see Fig. 10). For the SNR grid considered in the plot, the actual BER is obtained by Monte Carlo simulation. Two cases are shown: the SC-CP system and the case that the transmitting matrix is a DCT matrix. We can see that in both cases, the approximations are indistinguishable from the actual BER. This example demonstrates that even though output errors consist of ISI terms and channel noise, the BER is well approximated by Gaussian tail for all SNR. We will use (18) in the next example to compute BER over a fading channel.

Example 4: We use a multipath fading channel with four

coefficients. The coefficients are obtained from independent circular complex Gaussian random variables with zero mean and variances given, respectively, by 8/15, 4/15, 2/15, and

Fig. 10. Example 3. Comparison of the actual BER and the BERP computed from (18). For the DCT case, the actual BER is the dotted line, and P is the dotted line marked with “2.” For the SC-CP system, the actual BER is the solid line, andP - is the solid line marked with “.”

Fig. 11. Example 4. BER performances P , P , and P - over a four-tap fading channel.

1/15. We compute the BER performances , ,

and using (18) and average the results for

20 000 random channels (see Fig. 11). For high SNR range, - requires a significantly smaller transmission power than for the same BER. The performance of

is in between - and for all SNR.

VII. CONCLUSIONS

In the context of transceiver designs, the optimality addressed in most of the earlier works is in the sense of mean squared error minimization. In the paper, we consider directly the min-imization of uncoded BER for the class of OFDM transceivers with unitary precoders. For QPSK signaling and MMSE recep-tion, there exist channel-dependent optimal precoders. This is

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the class of unitary matrices with the equal magnitude

prop-erty, i.e., , for . One of the

solutions is the DFT matrix, and the resulting transceiver is the SC-CP system. In the case of zero-forcing receivers, the solution of optimal precoders depends on SNR. For higher SNR associ-ated with a practical range of BER, the optimal precoders that are channel independent are also the set of unitary matrices with the equal magnitude property. In the presence of channel spec-tral nulls, the performance of zero-forcing receivers exhibits se-rious degradation. Robustness against channel spectral nulls can be achieved by using MMSE receivers.

APPENDIX A PROOF OFLEMMA2

Let for ; then, . The

lemma can be proved by computing the first and second

deriva-tive of . The function for is convex

with first derivative and second

deriva-tive . The function for is also

convex with and

The first derivative is given by

, which means that is strictly monotone increasing. We can verify that can be expressed as

Therefore, for and for ,

which proves the lemma.

APPENDIX B PROOF OFTHEOREM1

We will use the concavity and convexity of to prove

Theorem 1. Given a set of numbers with

, the strictly monotone increasing property and the convexity of imply

where and

Similarly, given with , the concave

property of for implies

where and

Let us first consider the case . For this range, the

subchannel SNR of the OFDM system . For a

general unitary precoder , we can use (8) to see that whenever

a subchannel has SNR satisfying , it is operating in the convex region of . We have

(23) On the other hand, using (2), we have

The inequality follows from the fact that is in the convex region of for . Therefore

(24) where we have used the fact that for any unitary , its rows have

unit energy for all . Combining (23) and

(24), we obtain - for . Similarly,

when , we can use the concavity of to show that - .

APPENDIX C PROOF OFLEMMA3

Proof: Without loss of generality, we can consider as the

interconnection , where is a general

non-singular matrix, and is a diagonal matrix with th diagonal element . Let be the output vector of the matrix . If we choose , then becomes the zero forcing so-lution. In the absence of channel noise, we have .

There-fore, can be expressed as , where is

a noise vector from the channel noise alone, and . By the orthogonality principle, should be orthogonal to the

observation vector , i.e., . This yields

. Solving this equation, we get

(25)

where is a diagonal matrix with the th diagonal element

equal to . Therefore, the optimal is

. Letting , we obtain the expression of given in (14). Using as in (25), we can further verify that

(11)

The input symbols are assumed to be uncorrelated with real and imaginary parts having the same variance ; the vector has the same statistics, which are implied by the unitary property of

. Therefore, we have

Therefore, the MSE is as given in

the lemma. We also observe that the error vector consists of two

parts and . With the assumption that the

input have equal variance in real and imaginary parts, we can verify that the vector also has the property that the real and imaginary parts have the same variance. Similarly, the noise vector has the same property, and also has the same property. Therefore, we conclude that has equal variance in real and imaginary parts.

As , the th element has the

expression in (15). The variance of is

The above expression means that and that

. The th subchannel SINR

Using the expression of in (15), we obtain , as given in (16).

APPENDIX D PROOF OFLEMMA4 We will prove the lemma by showing that

and . The function can be written as

, where . The first and second

derivatives of are, respectively,

and , which are both larger than zero

for . The first and second derivatives of are

computed in Appendix A. As and , the

first derivative . We can verify that

the second derivative can be rearranged as

which is larger than or equal to zero.

ACKNOWLEDGMENT

The authors would like to thank the anonymous reviewers for many constructive comments and suggestions that have consid-erably improved this paper.

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[16] Y.-P. Lin and S.-M. Phoong, “Optimal ISI free DMT transceivers for distorted channels with colored noise,” IEEE Trans. Signal Processing, vol. 49, pp. 2702–2712, Nov. 2001.

[17] Y. Ding, T. N. Davidson, S.-K. Zhang, Z.-Q. Luo, and K. M. Wong, “Minimum ber block precoders for zero-forcing equalization,” in Proc.

IEEE International Conf. Acoust., Speech, Signal Processing, 2002.

[18] N. Ahmed and K. R. Rao, Orthogonal Transforms for Digital Signal

Processing: Springer Verlag, 1975.

[19] J. G. Proakis, Digital Communications. New York: McGraw-Hill, 1995.

[20] H. V. Poor and S. Verdu, “Probability of error in MMSE multiuser de-tection,” IEEE Trans. Inform. Theory, vol. 43, pp. 847–857, May 1997. [21] J. Zhang, E. K. P. Chong, and D. N. C. Tse, “Output MAI distribution of linear MMSE multiuser receivers in DS-CDMA system,” IEEE Trans.

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(12)

Yuan-Pei Lin (S’93–M’97) was born in Taipei,

Taiwan, R.O.C., in 1970. She received the B.S. degree in control engineering from the National Chiao-Tung University, Hsinchu, Taiwan, in 1992, and the M.S. and Ph.D. degrees, both in electrical engineering, from the California Institute of Tech-nology, Pasadena, in 1993 and 1997, respectively.

She joined the Department of Electrical and Control Engineering, National Chiao-Tung Univer-sity, in 1997. Her research interests include digital signal processing, multirate filterbanks, and digital communication systems with emphasis on multicarrier transmission.

Dr. Lin is currently an associate editor for IEEE TRANSACTIONS ONSIGNAL

PROCESSINGand an associate editor for the Academic Press journal

Multidimen-sional Systems and Signal Processing.

See-May Phoong (M’96) was born in Johor,

Malaysia, in 1968. He received the B.S. degree in electrical engineering from the National Taiwan University (NTU), Taipei, Taiwan, R.O.C., in 1991 and the M.S. and Ph.D. degrees in electrical engi-neering from the California Institute of Technology (Caltech), Pasadena, in 1992 and 1996, respectively. He was with the Faculty of the Department of Elec-tronic and Electrical Engineering, Nanyang Techno-logical University, Singapore, from September 1996 to September 1997. In September 1997, he joined the Graduate Institute of Communication Engineering, NTU, as an Assistant Pro-fessor, and since August 2001, he has been an Associate Professor. His interests include multirate signal processing and filterbanks and their application to com-munications.

Dr. Phoong is currently an Associate Editor for the IEEE TRANSACTIONS ON

CIRCUITS ANDSYSTEMSII: ANALOG ANDDIGINALSIGNALPROCESSING, as well as for the IEEE SIGNALPROCESSINGLETTERS. He received the Charles H. Wilts Prize in 1997 for outstanding independent research in electrical engineering at Caltech.

數據

Fig. 1. Block diagram of the OFDM system over a channel P (z) with additive noise (n).
Fig. 3. Illustration of noise path at a zero-forcing receiver.
Fig. 4. Block diagram of the SC-CP system over a channel P (z) with additive noise (n).
Fig. 6. Plot of h(y) = Q( p y 0 1) for 0  y  1.
+3

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