designs for DFT-based transceivers, a postprocessing matrix that is generally channel dependent, is needed to have a zero-forcing receiver. We show that postprocessing is channel independent if and only if the window itself has the cyclic-prefixed property. We design optimal windows with minimum spectral leakage subject to the cyclic-prefixed condition. Moreover, we analyze how postprocessing affects the signal-to-noise ratio (SNR) at the receiver, which is an aspect that is not considered in most of the earlier works. The resulting SNR can be given in a closed form. Join optimization of spectral leakage and SNR are also considered. Furthermore, examples demonstrate that we can have a significant reduction in spectral leakage at the cost of a small SNR loss. In addition to cyclic-prefixed systems, window designs for zero-padded DFT-based transceivers are considered. For the zero-padded transceivers, windows that minimize spectral leakage can also be designed.
Index Terms—DMT, egress control, multicarrier, PSD mask,
window.
I. INTRODUCTION
D
ISCRETE Fourier transform (DFT) based multicarrier systems have found applications in a wide range of transmission systems, e.g., discrete multitone (DMT) for asym-metric digital subscriber lines (ADSLs) [2], very high speed digital subscriber lines (VDSLs) [3], and orthogonal frequency division multiplexing (OFDM) for wireless local area networks (LANs) [4] and digital video broadcasting (DVB) [5]. The transmitter and receiver perform, respectively, -point IDFT (inverse DFT) and DFT computation, where is the number of subchannels. At the transmitter side, each block is padded with cyclic prefix of length . The number is chosen to be no smaller than the order of the channel, which is usually assumed to be a finite impulse response (FIR) filter. Using redundant cyclic prefix, intersymbol interference (ISI) is can-celed completely. As a result, an FIR channel is converted into subchannels. The subchannel gains are the -point DFT of the FIR channel impulse response.Manuscript received May 14, 2003; revised March 17, 2004. The work was supported in part by the NSC under Grants 009-108 and 90-2213-E-002-097, the Ministry of Education under Contract 89-E-FA06-2-4, Taiwan, R.O.C., and the Lee and MTI Center for Networking Research. The associate editor coordinating the review of this manuscript and approving it for publica-tion was Dr. Ta-Hsin Li.
Y.-P. Lin is with the Department of 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 Communications Engineering, National Taiwan University, Taipei, Taiwan, R.O.C.
Digital Object Identifier 10.1109/TSP.2004.842173
has large spectral sidelobes, there is a large spectral leakage. This could pose a problem in applications where the power spectral density (PSD) of the transmit signal is required to have a large rolloff in certain frequency bands. For example, in some wired transmission application, the PSD of the downstream transmit signal needs to fall below a threshold in the frequency bands of upstream transmission to avoid interference [2], [3]. The PSD should also be attenuated in amateur radio bands to reduce interference or egress emission [3].
Many methods have been proposed to reduce sidelobes by windowing, filtering, or using different pulse-shaping filters. A number of nonrectangular continuous-time pulse shapes have been proposed to improve the spectral rolloff of the transmit signal, e.g., [6]–[9]. It is demonstrated in [10] that OFDM sys-tems with nonrectangular continuous-time pulse shaping filters can be represented by a shaping matrix followed by the usual analog transmitter. Usually, continuous-time pulse shapes are designed based on analog implementation of OFDM transmit-ters, and these pulses usually do not admit a digital implementa-tion [11]. Discrete-time windows that can be easily incorporated in digital transmitters implementation have been considered in [12]–[14]. In [12], overlapping windows of duration longer than one OFDM symbol is proposed to reduce spectral sidelobes. In this case, significant ISI is generated even if the channel does not introduces ISI, and an additional postprocessing equalizer is used to remove ISI. The design of overlapping windows for OFDM with offset quadrature amplitude modulation (QAM) over distortion-less channels are studied fully in [13] and [14]. When the channel is distortion-less, orthogonality among the subchannels is preserved [13], [14], and a better spectral effi-ciency is achieved.
However, for channels with distortion, i.e., ISI channels, the subchannel outputs contain intra- and subchannel inter-ference; additional processing is required to remove interfer-ence in this case. If extra guard time is available, postprocessing can be avoided at the cost of a reduced transmission rate [15]. When there is no extra cyclic prefix, the use of windowing at the transmitter requires postprocessing at the receiver. More re-cently, transmitting windows with the cyclic-prefixed property have been proposed in [16] for egress control. Windows that are the inverse of a raised cosine function are optimized to min-imize spectral leakage and, hence, minmin-imize egress emission. The corresponding zero-forcing receiver also requires postpro-cessing equalization.
In this paper, we will consider window designs for DFT-based multicarrier system without using extra cyclic prefix. Windowed transceivers with both cyclic prefix and zero padding will be considered for ISI channels. For the cyclic-prefixed case, post-1053-587X/$20.00 © 2005 IEEE
Fig. 1. Cyclic-prefixed DFT-based multicarrier system over a channelC(z) with additive noise (n).
processing is, in general, channel dependent. We will derive the explicit dependency on the channel and show that the postpro-cessing matrix that cancels ISI at the output of the receiver is channel independent if the window itself has the cyclic-prefixed property. In this case, the output of the transmitter has the usual cyclic-prefixed property. Techniques that exploit cyclic prefix for synchronization can still be used. We will design windows that minimize spectral leakage subject to the cyclic-prefixed constraint.
Moreover, we show that postprocessing can affect the SNR at the receiver, which is an aspect mostly overlooked in ear-lier window designs for DFT-based transceivers. We will see that the resulting SNR can be given in a closed form in terms of the transmit window if the channel noise is additive white Gaussian noise (AWGN). Furthermore, joint optimization of spectral leakage and SNR can be achieved by using an objective function that incorporates both terms. Examples will be given to demonstrate that a good tradeoff between spectral rolloff and SNR can be obtained through such an optimization. For the zero padding case, we will see that the postprocessing matrix de-pends on the window only but not on the channel. The output SNR, unlike the cyclic prefix case, depends on the window as well as the channel. A lower bound and an upper bound can be found for the SNR at the receiver. As in the cyclic prefix case, we design optimal windows that minimize the spectral leakage. The window can also be designed by jointly optimizing the SNR and spectral leakage.
The sections are organized as follows. In Section II, we con-sider windowed transceivers with cyclic prefix and derive the re-ceiver for a given window. The zero padding case is considered in Section III. The design method and examples of windowed systems will be given in Section IV.
Notations and Preliminaries:
1) Boldfaced lowercase letters are used to represent vec-tors, and boldfaced uppercase letters are reserved for matrices. The notation denotes transpose-conju-gate of .
2) The function denotes the expected value of the random variable .
3) The notation is used to represent the
iden-tity matrix. The subscript is omitted whenever the size is clear from the context.
4) The notation diag denotes an
diagonal matrix with the th diagonal element equal
to .
5) The notation is used to represent the
or-thogonal DFT matrix given by for
6) An matrix is called a circulant matrix if it is of the form
..
. . .. ...
(1)
It is known that an circulant matrix can be di-agonalized using DFT matrices. In particular, the matrix can be expressed as
(2)
where is a diagonal matrix, whose
diag-onal elements are the -point DFT of the
se-quence . That is,
. Conversely, any matrix of the form in (2) is a circulant matrix.
II. DFT-BASEDTRANSCEIVERSWITHCYCLICPREFIX In this section, we consider windowed DFT-based trans-ceivers with cyclic prefix. The block diagram of the DFT-based transceivers with cyclic prefix 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.
The input symbols are passed through an by IDFT
matrix, followed by the parallel-to-serial (P/S) operation and the insertion of cyclic prefix. The length of the cyclic prefix is chosen to be equal to or larger than the order of the channel
. At the receiver, the cyclic prefix is discarded, and the samples are again blocked into by 1 vectors for -point DFT computation. The scalar multipliers are also called
frequency domain equalizers, where are the
-point DFT of the channel impulse response . The prefix is discarded at the receiver to remove interblock interference. The transceiver is ISI free and the receiver is a zero-forcing receiver.
Fig. 3. Windowed DFT-based transceiver.
A. System Model
The transceiver in Fig. 1 can be redrawn as in Fig. 2. The matrices and shown in Fig. 2 are of dimensions
and , where . They are given, respectively,
by
and (3)
where we have used the subscript “cp” to highlight that these matrices are for the case of cyclic prefix. The matrix indicated in Fig. 2 is diagonal and given by
diag
We can obtain a windowed system by applying a window to each transmitter output block, as shown in Fig. 3. The length of the window is the same as the block length . The window has
coefficients , with z-transform . The
con-ventional system in Fig. 2 can be viewed as having a rectangular window with length . Due to the nonrectangular window at the transmitter, the receiver needs an additional postprocessing matrix to cancel intersubchannel interference. As there is no constraint on the matrix , there is no loss of generality in con-sidering the receiver of the form shown in Fig. 3. The transmit-ting matrix can be written as , where is the diag-onal matrix
diag (4)
We partition as
where and are of dimensions , and is of
di-mensions . For a given window, we now
derive the condition on so that the overall system is ISI free.
Lemma 1: Consider the DFT-based transceiver with cyclic
prefix in Fig. 3. The receiver is zero forcing if and only if the postprocessing matrix is given by
(5) where is an by lower triangle Toeplitz matrix with the
first column given by .
A proof is given in the Appendix.
From the above lemma, we see that the solution of the post-processing matrix depends on the window as well as the channel. This channel dependency means that needs to be computed along with other channel-dependent parameters. To remove such a dependency, we observe that is channel in-dependent if . That is, the window itself has the cyclic-prefixed property. In this case, the postprocessing matrix is given by
diag (6)
Notice that to have a channel-independent for any channel, the condition is not only sufficient but also necessary.
Spectral Leakage: Let denote the Fourier transform of the window function. The stopband energy of the window is
Fig. 4. Noise path at the receiver for (a) the cyclic prefix case and (b) the zero padding case.
where . We define the spectral leakage as
(8) where is the stopband energy of the rectangular window, and is the stopband energy of the window . In Section IV, we will see how to design windows that improve the spectral leakage of the window subject to the cyclic prefix condition.
B. Noise Analysis
We assume that the window has the cyclic-prefixed property and the postprocessing matrix is channel independent as given in (6). Suppose the channel noise is AWGN with vari-ance . We constrain the transmission power to be the same as the conventional system with a rectangular window. That is, the window satisfies the condition
(9) Fig. 4(a) shows the noise path for the windowed system. The input noise vector is a block of Gaussian random variables drawn from the channel noise process . The autocorrelation matrix of the noise vector is given by
where diag
From the above equation, we know that is a circulant matrix; all the elements in has the same variance equal to
Therefore, , as indicated in Fig. 4(a), has variance . As the DFT matrix preserves power, the total output noise power is the sum of all . That is
(10)
where we have used the cyclic-prefixed property of the window. In the conventional system, the window is rectangular with
, for . The total output noise power
in this case is simply
(11)
To compare the output noise with the conventional system with rectangular window for the same signal variance, we define the
quantity SNR loss . Using (10) and (11), we
have
(12)
III. DFT-BASEDTRANSCEIVERSWITHZEROPADDING In a DFT-based transceiver with zero padding, for each block of size to be transmitted, zeros, instead of cyclic prefix, are padded. The zero-padded system can also be represented using the block-based transceiver in Fig. 2. Now, the matrices and
in Fig. 2 are given by
We can obtain a windowed system with zero padding from Fig. 3 by setting the last window coefficients to zero. In this case, the transmitting matrix can be expressed as
, where is an diagonal matrix with
diag . The window has
coeffi-cients, . The window is contrained to satisfy
so that it has the same transmission power as the zero-padded system with a rectangular window.
Lemma 2: Consider the DFT-based transceiver with zero
padding. For a given transmit window, the receiver is zero forcing if and only if the postprocessing matrix is given by
the above equation, is circulant, as in (1). Using
in (2) and , we arrive at
The receiver is zero-forcing solution, i.e., , if and only if .
Similar to the cyclic-prefixed case, we define the spectral
leakage as
where is the stopband energy of the rectangular window with length . The expression is exactly as in (7). The only dif-ference is that the window is now a window of coefficients. For a given window, we can also compute the total output noise power , as in the cyclic prefix case. We can also define the SNR loss as the ratio of the total output noise power of the win-dowed system over that of the zero-padded DFT-based trans-ceiver with a rectangular window
The total output noise power for a given window is given in the following lemma.
Lemma 3: Consider the windowed DFT-based transceiver
system with zero padding in Fig. 3. The channel noise is AWGN with variance . The total output noise power is given by
where (13)
Proof: Fig. 4(b) shows the noise path at the receiver for the
zero-padded system with windowing. The input noise vector
is of dimensions . Let
Then, the noise vector indicated in Fig. 4(b) is . The output noise vector can be expressed as
where , as given in (13). As and are
uncorrelated, the two noise vectors and are uncorrelated as
well; the output noise power can be computed
by adding together and . The noise vector is
correlated. Let be an column vector whose th ele-ment is equal to , and all the other elements are zero for
. That is, for is
given by , for . Then,
, and therefore, can be expressed as
Note that has only one nonzero element, and hence, . As the elements of are uncorrelated,
we have . Therefore
As , we arrive at the expression in
(13).
The expression in (13) implies is an increasing function of ; as increases, more diagonal terms are added.
The value of attains the maximum when ( is
assumed to be ). The output noise power for the
rectangular window case can be obtained by setting
in the expression of in (13). In particular, we can verify that is given by
From (13), we see that unlike the cyclic prefix case, is a quantity that depends on the channel. The following lemma gives a channel-independent lower and upper bound on . The gap between the lower bound and the upper bound is 3 dB.
Lemma 4: Consider the windowed DFT-based transceiver
system with zero padding in Fig. 3. The channel noise is AWGN with variance . The total output noise power is bounded as follows:
(14) The SNR loss is bounded by
(15)
bounds of given in (14), we can obtain the bounds
of in (15).
IV. WINDOWDESIGNS
In this section, we will design optimal windows for cyclic-prefixed and zero-padded systems to minimize spectral leakage. The use of windows improves the spectral rolloff of the trans-mitter outputs significantly. We will also optimize both spectral leakage and SNR by considering a joint objective function con-sisting of both terms. The joint optimization yields a tradeoff between spectral leakage and SNR.
A. Windows for Cyclic-Prefixed Systems
We have shown in Section II that a cyclic-prefixed window yields channel-independent postprocessing. We will design win-dows for cyclic-prefixed systems subject to this constraint. Let
be the by 1 window vector and
be a column vector containing only the last coefficients of the window. The cyclic-prefixed property means that can be written as
where
The Fourier transform of the window can be expressed as where
(16) It follows that the squared magnitude response of the window is
where . The stopband energy of the
window in (7) can be expressed as
(17)
The matrix is given by
otherwise. It is real, symmetric, and positive semi definite.
Using (17), we can see that the minimization of spectral
leakage becomes the minimization of . As the
product matrix is positive semi definite, the objective
function can be minimized by choosing to be
the eigen vector corresponding to the smallest eigen value of
mild assumption. To see that, let the smallest eigen value of be and the associated eigen vector be , i.e.,
(18) We assume the multiplicity of is one and . Let denote the reversal matrix. For example, the 3 by 3 reversal matrix is given by
We observe that has the property that . In addition, by direct multiplication, we can verify that
By multiplying the both sides of (18) by , we obtain
(19) This means that is the eigen vector of associated
with . Multiplying (19) by and using , we can
obtain . If we use the property
, then we arrive at the following equation:
This means that is also an eigen vector of
asso-ciated with . Because the multiplicity of the eigen value is one and the energy of is the same as that of , we
observe that and are related by .
No-tice that is the flipped version of the window , i.e.,
for . The relation
implies that is symmetric or antisymmetric. Note that the antisymmetric property leads to a zero at [17]. As our desired window is a lowpass filter, we can con-clude that the window is symmetric. Therefore, in this case, the window has both cyclic-prefixed property and symmetry prop-erty. Combining these two properties, we can further deduce that
the first coefficients of the window satisfy for
.
To incorporate SNR in the optimization, we can form the ob-jective function
(20) where . In this case, the objective function is no longer in quadratic form. Nonlinear optimization packages can be used to design the window, e.g., [19]. The parameter gives a tradeoff between spectral leakage and SNR.
Example 1—Windowed DFT-Based Transceiver With Cyclic Prefix: The block size , and prefix length . We form the positive semi definite matrix and compute
Fig. 5. Example 1. Windows for cyclic-prefixed transceivers. (a) Time-domain plot of the window with minimum spectral leakage. (b) Magnitude response of the window in (a). (c) Power spectral density of the transmit signal.
the eigen vector corresponding to the smallest eigen value to obtain . The resulting window is as shown in Fig. 5(a). The magnitude response of is shown in Fig. 5(b). Fig. 5(c) shows the spectrum of the transmitter output using the window in Fig. 5(a). The subcarriers used are 38 to 99 and 111 to 255, as in [16]. The subcarriers with indices smaller than 38 are re-served for voice band and upstream transmission, and those with indices between 99 and 111 are for egress control. We see that
Fig. 6. Example 2. Joint optimization for cyclic-prefixed transceivers. (a) Win-dows obtained by joint optimization of spectral leakage and SNR. (b) Spectral leakage and SNR loss versus the trade-off factor c. (c) Performances of windowed transceivers.
the spectrum of the windowed output has a much smaller spec-tral leakage in unused bands.
Example 2—Joint Optimization, Cyclic Prefix Case: Using
the nonlinear optimization package in [19], we optimize the window to minimize the objective function in (20) that incorpo-rates both spectral leakage and SNR. The resulting windows for , 0.3, and 0.5 are as shown in Fig. 6(a). Fig. 6(b) shows
and versus the trade-off factor . For example, when , spectral leakage is 0.4, and is 1.1 (0.41 dB).
bols, and subchannel SNRs are used to determined bit allocation. We have used the same set of subcarriers in Fig. 5(c). The win-dows are designed using , 0.3, and 0.5. The channel used in the simulation is Loop 6, which is a typical channel in car-rier serving area [2], and the channel noise is AWGN. Fig. 6(c) shows the number of bits transmitted per block. For example, when SNR dB, 2233 bits, and 2204 bits per block are
trans-mitted, respectively, for and . Compared with
2268 bits per block for the rectangular window, the rate losses are only around 1.6% and 2.8%, respectively. For , the result is virtually the same as that of the rectangular window.
B. Windows for Zero-Padded Systems
For the zero-padded system, Lemma 2 shows that the postpro-cessing matrix is channel independent for any choice of window . Similar to the cyclic-prefixed case, the Fourier transform of the window function can be expressed as
where and the vector is as in (16).
Using derivations similar to those for the cyclic-prefixed case, the stopband energy of the window can be written as
where is as given in (17). As the matrix is real, symmetric, and positive semi definite, we can find the optimal window by computing the eigen vector corresponding to the smallest eigen value. One can show that the optimal window is real and symmetric by following a procedure similar to the cyclic prefix case. We can also use the objective function
(21) to incorporate SNR in the design. When , the window is
optimized to minimize the SNR loss . Notice
that the function is convex, which implies
(22) Using the above inequality, we see that the optimal solution when becomes the rectangular window.
Example 3—Windowed DFT-based Transceiver With Zero Padding: As in earlier examples, we choose and . Like the cyclic-prefixed case, we use the nonlinear optimization package in [19] to optimize the window so that the objective function given in (21) is minimized. The resulting windows for , 0.2 and 0.5 are as shown in Fig. 7(a). No-tice that the window corresponding to is the rectangular window, as we have expected. The magnitude responses of the windows in Fig. 7(a) are shown in Fig. 7(b). The windows
Fig. 7. Example 3. Windows for zero-padded transceivers. (a) Windows obtained by joint optimization of spectral leakage and SNR. (b) Magnitude response of the window in (a). (c) Spectral leakage , , lower bound and upper bound of versus the tradeoff factor c.
correspond to larger have better stopband attenuation and hence smaller spectral leakage.
Fig. 7(c) shows for different trade-off factor . For the zero-padded transceiver, is channel dependent. To plot , we use a random channel of 32 taps, where each tap is an in-dependent Gaussian random variable with unit variance, and a total of 10 000 channel realizations are used. The channel noise is AWGN. The average from all the channel realizations is
carrier systems with cyclic prefix and with zero padding. The spectral leakage of the transmit signal can be reduced signif-icantly by using windows. Like earlier works of windowed transceivers, the use of windows at the transmitter side requires postprocessing at the receiver side, which is usually channel dependent. We show that for the cyclic-prefixed system with a zero-forcing receiver, postprocessing is channel independent if the window itself has cyclic-prefixed property. For both cyclic-prefixed and zero-padded systems, the optimal window that minimizes spectral leakage of the transmit signal can be given in closed forms. We also show that postprocessing affects the output SNR. The output SNR can be given in terms of the window and the channel. Furthermore, for both cyclic-prefixed and zero-padded systems with windowing, we can jointly optimize spectral leakage and SNR. The results show that these window designs provide a good tradeoff between SNR and spectral leakage.
APPENDIX PROOF OFLEMMA1
As there is no interblock interference, we only need to con-sider intrablock interference, and the transfer function of the system from the transmitter input to the receiver output is a constant matrix . Using and given in (3), the overall transfer matrix is
where the matrix is of dimensions . It is Toeplitz
with the first row given by . With the
partition of in (4), we have
Let us partition as
where and are of dimensions , and is
. The matrix can be written as
(23)
(24)
following expression for :
For a zero-forcing solution, is the inverse of the matrix in the bracket.
ACKNOWLEDGMENT
The authors would like to thank Y.-Y. Jian, C.-C. Su, and P.-J. Chung for performing the simulations.
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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.
logical University, Singapore, from September 1996 to September 1997. In September 1997, he joined the Graduate Institute of Communication Engineering and the Department of Elec-trical Engineering, NTU, as an Assistant Professor, and since August 2001, he has been an Associate Professor. His interests include multirate signal pro-cessing and filterbanks and their application to communications.
Dr. Phoong is currently an Associate Editor for the IEEE SIGNALPROCESSING
LETTERS. He served as an Associate Editor for the EEE TRANSACTIONS ON
CIRCUITS ANDSYSTEMSII from January 2002 to December 2003. He received the Charles H. Wilts Prize in 1997 for outstanding independent research in elec-trical engineering at Caltech.
