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The transmitter output of the OFDM sys-tem is known to have large spectral side-lobes. Windowing and pulse shaping have been proposed to reduce sidelobes in the lit-erature. When these methods are used, addi-tional post-processing equalization is needed at the receiver. The post processing computa-tions are often channel dependent and the co-efficients need to be computed when the chan-nel changes. In this report we consider win-dow designs for the OFDM system. We will design windows that minimize out-of-band en-ergy with the condition that post processing is channel independent.
Keywords: OFDM system, windowed
OFDM, out-of-band energy
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The OFDM system has found applications in a wide range of wireless systems such as wireless local area network, and digital au-dio broadcasting [1][2]. In the conventional OFDM system the pulse shaping filter is a rectangular window. As the rectangular win-dow has large spectral sidelobes, the trans-mitter output has large out-of-band energy. Many methods have been proposed to re-duce sidelobes by windowing, filtering or us-ing different pulse shapus-ing filters. A num-ber of non-rectangular continuous-time pulse shapes have been proposed to improve the spectral roll-off of the transmitted signal, e.g., [3]-[7]. Usually continuous-time pulse shapes are designed based on analog implementation of OFDM transmitters and these pulses usu-ally do not admit a digital implementation [8]. Discrete-time windows that can be eas-ily incorporated in digital implementation of OFDM transmitters have been considered in
[9][10]. In [9], overlapping windows of dura-tion longer than one OFDM symbol is pro-posed to reduce spectral sidelobes but signif-icant ISI is generated even if the channel is AWGN. A post-processing equalizer is used to remove ISI. Overlapping windows that pre-serve the orthogonality of the OFDM system for AWGN channels are designed in [10]. If ex-tra guard time is available, post processing can be avoided at the cost of a reduced transmis-sion rate [11]. When there is no extra cyclic prefix, the use of windowing at the transmit-ter requires post processing at the receiver. In an ISI free system, if the transmitter is given, the receiver is determined accordingly. For an arbitrary window, the post processing is chan-nel dependent in general and the post process-ing coefficients need to be computed when the channel changes. For wireless applications, it is important to have post processing that is channel independent.
In this report we will consider window de-signs for the OFDM system without using ex-tra cyclic prefix. Windowed OFDM system with both cyclic prefix (WOFDM-CP) and zero padding (WOFDM-ZP) will be consid-ered for frequency selective channels. We will design windows so that the transmitter put has good spectral roll-off or small out-of-band energy. For the cyclic-prefixed case, post-processing is in general channel depen-dent. We will consider the explicit depen-dency on the channel and show that the post-processing matrix can be made channel in-dependent. In fact, the post processing ma-trix is channel independent if the window it-self has cyclic-prefixed property. In this case, the output of the transmitter has the usual cyclic-prefixed property. Techniques that ex-ploit cyclic prefix for synchronization can still be used. We will design windows that mini-mize the out-of-band energy of the transmit-ter output subject to the cyclic-prefixed con-straint.
For the WOFDM-ZP case, we will see that the post-processing matrix depends on the window only but not on the channel. As in the WOFDM-CP case, we will design optimal windows that minimize the out-of-band energy of the transmitter outputs.
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The block diagram of the conventional OFDM system with cyclic prefix is as shown in Fig. 1. The modulation symbols to be transmitted are first blocked into M by 1 vectors, where M is the number of subchannels. The input sym-bols sk are passed through an M -point IDFT,
followed by the parallel to serial (P/S) oper-ation and the insertion of cyclic prefix. The length of the cyclic prefix L is chosen to be equal to or larger than the order of the chan-nel C(z). At the receiver, the cyclic prefix is discarded and the samples are again blocked into M by 1 vectors and passed through an M × M DFT matrix W. The scalar multi-pliers 1/Ck are also called frequency domain
equalizers, where C0, C1, · · · , CM −1are the M
-point DFT of the channel impulse response cn. The prefix is discarded at the receiver
to remove inter-block ISI. The transceiver is ISI free and the receiver is a zero-forcing re-ceiver. The only channel dependent part of the transceiver design is the set of scalars 1/Ck, for
k = 0, 1, · · · M − 1.
The conventional OFDM system with cyclic prefix can be redrawn as in Fig. 2. The matri-ces G and S shown in Fig. 2 are of dimensions N × M and M × N , where N = M + L. They are given respectively by
Gcp = 0 IL IM W†, and Scp = W 0 I . (1) The matrix Λ indicated in Fig. 2 is diagonal,
given by
Λ = diagC11 C12 · · · 1 CM−1
.
We can obtain a WOFDM system by apply-ing a window to each output block as shown in Fig. 3. The length of the window is the same as the block length N . The window has coefficients d0, d1, · · · , dN −1. The
conven-tional OFDM system in Fig. 2 can be viewed as having a rectangular window with length N . Due to the non-rectangular window at the transmitter, the receiver needs an ad-ditional post processing matrix P to cancel intra-subchannel ISI. As there is no constraint on the matrix P, there is no loss of generality in considering the receiver of the form shown in Fig. 3. The transmitting matrix can be writ-ten as DcpGcp, where Dcp is the diagonal
ma-trix Dcp = diag d0 d1 · · · dN −1 . (2) We partition D as Dcp= D0 0 0 0 D1 0 0 0 D2 ,
where D0and D2 are of dimensions L×L, and
D1 is of dimensions (M − L) × (M − L). For
a given window, we now derive the condition on P so that the overall system is ISI free.
Lemma 1 Consider the WOFDM system
with cyclic prefix in Fig. 3. The system is ISI free if and only if the post processing matrix P is given by Pcp= W WD1 0 0 D2 + ΛW0 C2(D0− D2) 0 0 −1 , (3)
where C2 is an L by L lower triangle
Toeplitz matrix with the first column given by c0 c1 · · · cL−1
T . 2
1: The cyclic-prefixed OFDM system over a channel C(z) with additive noise ν(n).
2: The block based representation of the OFDM system with transmitting matrix G and receiving matrix S.
A proof can be found in [12].
From the above lemma, we see that the solution of the post processing matrix de-pends on the window D as well as the chan-nel. This channel dependency means that Pcp
needs to be updated along with other channel dependent parameters as soon as the chan-nel changes. To remove such a dependency, we observe that Pcp is channel independent if
D0 = D2. That is, the window itself has the
cyclic-prefix property. In this case, the post processing matrix is given by
Pcp = Wdiag 1/dL 1/dL+1 · · · 1/dN −1 W†.
(4) Notice that to have a channel independent Pcp
for any channel, the condition D0 = D2 is
not only sufficient but also necessary. In sec-tion 4, we will see how to design windows that improve the spectral roll-off of the WOFDM transmitter output, subject to the cyclic pre-fix condition.
Zero-padding case. In an OFDM system
with zero padding, for each block of size M to be transmitted, L zeros, instead of cyclic prefix, are padded. The zero-padded OFDM system can also be represented using the block based transceiver in Fig. 2. Now, the matrices G and S in Fig. 3 are given by
Gzp = W† 0 , Szp = W IM IL 0 . We can obtain a WOFDM system with zero padding from Fig. 2 by setting the last L window coefficients to zero. In this case the transmitting matrix can be expressed as DzpW
†
0
, where Dzp is an
M × M diagonal matrix with Dzp =
diag d0 d1 · · · dM −1 . The window has M
coefficients, d0, d1, · · · dM −1. The window is
contrained to satisfy 1/MPM −1
k=0 |dk|2 = 1 so
that it has the same transmission power as the zero-padded OFDM system with a rectangular window.
Lemma 2 Consider the WOFDM system
3: A windowed OFDM system.
with zero padding in Fig. 3. The system is ISI free if and only if the post processing matrix P is given by
Pzp= WD−1zpW †
. A proof can be found in [12]
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[5] K. Matheus, and K.-D. Kammeyer,
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[7] S. B. Slimane, “Performance of OFDM systems with time-limited waveforms over multipath radio channels,” Global Telecommunications Conference, 1998. [8] Yuan-Pei Lin and See-May Phoong,
“OFDM Transmitters: Analog Represen-tation and DFT Based ImplemenRepresen-tation,” to be published in IEEE Trans. Signal Processing.
[9] R. W. Lowdermilk, “Design and perfor-mance of fading insensitive orthogonal frequency division multiplexing (OFDM) using polyphase filtering techniques, Thirtieth Asilomar Conference on Sig-nals, Systems and Computers, Nov. 1996. [10] H. Bolcskei, P. Duhamel, and R. Hleiss, “Design of pulse shaping OFDM/OQAM systems for high data-rate transmission over wireless channels,” Proc. IEEE ICC, 1999.
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