Windowing is often applied to multicarrier systems to improve frequency separation among the subcarriers. At the transmitter side better frequency separation leads to a smaller out-of-band spectral leakage and also less interfer-ence to radio frequency transmission. At the receiver side better separation gives more suppression of radio frequency interference. As these are frequency based characteristics, a filterbank representation presents a natural and useful framework for formulating the problem. In this work, we pro-pose a unified filterbank approach to the design of windows for multicarrier systems. The filterbank viewpoint provides an additional insight into the transmitter design for spectral leakage reduction as well as to the receiver design for inter-ference suppression. A better frequency separation among the subchannels can be achieved.
Yuan-Pei Lin, Chien-Chang Li, and See-May Phoong
Abstract
© 1997 DIGITALSTOCK
Feature
* This work was supported in parts by NSC 95-2221-E-009-080, NSC94-2752-E-002-006-PAE, and NSC 95-2221-E-002-197, Taiwan, R.O.C.
Introduction
T
he DFT (discrete Fourier transform) based multi-carrier systems have found important applications in DMT (discrete multitone) systems such as ADSL (asymmetric digital subscriber lines) [1] and VDSL (very high speed digital subscriber lines) [2], and in OFDM (orthogonal frequency division multiplexing) systems such as wireless LAN (local area network) [3] and DVB (digital video broadcasting) [4]. The transmitter and receiver perform M-point IDFT (inverse DFT) and DFT computation, respectively, where M is the number of sub-channels. At the transmitter side, a cyclic prefix of length ν is inserted. The channel can usually be assumed to be an FIR (finite impulse response) filter of order L after proper time-domain equalization. When the prefix length ν ≥ L, there will be no inter-block interference. With the aid of redundant cyclic prefix, an FIR channel is convert-ed into M zero-ISI subchannels. The subchannel gains are the M-point DFT of the FIR channel impulse response.In the conventional DFT based multicarrier system the transmitting and receiving filters come from rectangular windows. Rectangular windows are known to have large spectral sidelobes and the stopband attenuation is insuffi-cient for many applications. At the transmitter side poor frequency separation leads to significant spectral leakage. This could pose a problem in applications where the spec-trum of the transmitted signal is required to have a large roll-off in certain frequency bands. For example, in some wired transmission application, the spectrum of the mitted signal needs to fall below a threshold in the trans-mission bands of the opposite direction to avoid interference [1], [2]. The transmitted spectrum should also be attenuated in amateur radio bands to reduce inter-ference or egress emission [2]. On the other hand, poor frequency separation at the receiver side results in poor out-of-band rejection. In DMT applications such as ADSL
and VDSL, some of the frequency bands are also shared by radio transmission systems, e.g., amplitude-modulation stations and amateur radio. The radio frequency signals can be coupled into the wires and this introduces radio frequency interference (RFI) or ingress [5]. Ill frequency separation means many neighboring tones can be affect-ed. The signal-to-interference-noise ratio of these tones are reduced and the total transmission rate decreased.
Many methods have been proposed to improve the fre-quency characteristics of the transmitter and receiver. To improve the spectral roll-off of the transmitted signal, a number of continuous-time pulse shaping filters have been proposed, [6]–[11]. Usually continuous-time pulse shapes are designed based on an analog implementation of transmitters and a digital implementation is not admit-ted [12]. Discrete-time windows have been considered in [13]–[15]. The design of overlapping windows for OFDM with offset QAM (quadrature amplitude modulation) over ISI free channels are studied fully in [14], [15]. More recently, transmitting windows with the cyclic-prefixed property have been considered in [16], [17] for egress control. Windows that are the inverse of a raised cosine function are optimized in [16], to minimize egress emis-sion. To compensate for the transmitter window, the corresponding receiver requires post-processing equal-ization [16], [17]. A joint consideration of spectral roll-off and SNR degradation due to post-processing is given in [17]. Per-tone windows are proposed in [18] for shaping transmitted spectrum. The shaping of spectrum allows more tones to be used for transmission.
Windowing is also often applied at the receiver side. In [19], Muschallik used Nyquist windows, which have the property that shifts of the window in the time domain add to a constant, to improve the reception of OFDM systems. Optimal Nyquist windows are considered in [20] to miti-gate the effect of additive noise and carrier frequency
off-Yuan-Pei Lin and Chien-Chang Li are with the Dept. Electrical and Control Engr., National Chiao Tung Univ., Hsinchu, Taiwan, R.O.C. and See-May Phoong is with the Dept. of EE & Graduate Inst. of Comm. Engr., National Taiwan Univ., Taipei, Taiwan, R.O.C., smp@cc.ee.ntu.edu.tw
s0(n) s1(n) sM−1(n) IDFT W† DFT W P/S S/P Cyclic Prefix x(n) q(n) r(n) C(z) DiscardPrefix 1/λ0 1/λ1 1/λM−1
sets. To improve RFI suppression, receiver windowing is proposed first in [21] by Spruyt et al. For the suppression of sidelobes without using extra redundant samples, it is suggested in [22] to use windows that introduced con-trolled IBI, later removed using decision feedback. Joint consideration of RFI and channel noise is considered in [23]; the optimal window can be found using the statistics of the RFI and noise. Using statistics of channel noise and RFI, a joint design of the TEQ and the receiving window for maximizing bit rates is given in [24].
In this work, we propose a unified filterbank framework for the design of windows for multicarrier systems. The approach used here will be more general: we will introduce the so-called subfilters. The use of sub-filters will enhance the frequency selectivity of the transmitting/receiving filters while maintaining the orthogonality among the subchannels. Correspondingly, for the transmitter side spectral leakage can be reduced and for the receiver side RFI can be further suppressed. When the subfilters form a DFT bank, they can be tied nicely to the conventional windowing. The windows can be optimized through the design of subfilters and fre-quency separation among the subchannels can be con-siderably improved.
DMT Systems
The block diagram of the DMT system is as shown in Figure 1. After proper time-domain equalization (if
necessary), the channel is modeled as an FIR filter C(z) of order L with additive noise q(n). The modulation symbols to be transmitted are first blocked into vectors of size M, where M is the number of subchannels, usually much larg-er than the channel ordlarg-er L. The inputs of the transmittlarg-er are modulation symbols. They are passed through an M by M IDFT matrix. The outputs are converted to a block of M serial samples by the parallel to serial operation (P/S). Then a cyclic prefix of length ν is inserted by copying the last ν samples of the block to the beginning. The length of the cyclic prefix ν is chosen so that ν ≥ L, which ensures that inter-block-interference (IBI) can be removed easily by discarding the prefix at the receiver.
At the receiver, after prefix removal the samples are blocked into M× 1 vectors (‘serial to parallel’ or S/P operation) for M-point DFT computation. The scalar mul-tipliers 1/λkare also called frequency domain equalizers (FEQ), where λk are the M-point DFT of the channel impulse response. The transceiver is ISI free and the receiver is a zero-forcing receiver. The receiver outputs are identical to the inputs of the transmitter in the absence of channel noise.
Filterbank Representation
Let us derive the filterbank representation of a DMT sys-tem, which will be useful for later discussion. In Figure 1, the operation ‘P/S’ followed by the insertion of a cyclic prefix can be viewed as the interconnection of the matrix
1. The notation A†denotes transpose-conjugate of A.
2. The notation W is used to represent the M× M normalized DFT matrix given by
[W]kn=√1 MW
kn, for 0 ≤ k, n ≤ M − 1, where W = e−j2π M.
3. The notation [X(z)]↓Ndenotes the N-fold decimation of X(z). In the time domain Y(z) = [X(z)]↓Nmeans y(n) = x(Nn). 4. Polyphase identity [25]. The following interconnection is known to be an LTI system with transfer function T(z) = [C(z)]↓N.
5. Noble identities [25]. Identities for exchanging an LTI filter and an expander/decimator. Table 1.
Notation and multirate identities.
x(n) y(n) C(z) x(n) T(z) y(n) N N (a) x(n) y(n) x(n) C(z N) y(n) N C(z) N (b) x(n) y(n) x(n) C(z) C(z N) y(n) N N
0 Iν
IM
followed by ‘parallel to serial’ for every N = M + ν parallel samples as shown in Figure 2(a). The ‘P/S’ opera-tion is represented using expanders and a delay chain in the figure. On the other hand, the operation ‘discard pre-fix’ followed by ‘serial to parallel’ and M-point DFT for every M samples in Figure 1 can be viewed as ‘serial to parallel’ for every N samples followed by the matrix ( 0 W ) as shown in Figure 2(a), where W is the normal-ized DFT matrix defined in Table 1. Thus the transmitter and receiver can be redrawn as in Figure 2(a), where we have combined the two matrices at the transmitter as one matrix G.
As G is a constant matrix, we can exchange G and the expanders; the resulting transmitter is as shown in Figure 2(b). Similarly, we can exchange ( 0 W ) and the decima-tors to yield the receiver shown in Figure 2(b). Note that the 1× M system from p(n) to x(n) is LTI. Let’s call the 1× M transmitting bank f(z), then f(z) is a row vector given by ( 1 z−1 · · · z−(N−1)) G. Each element of the row vector can be obtained by multiplying out this expression. Suppose that the k-th element is Fk(z) (k-th transmitting filter), we have
Fk(z) 1 √ M N−1 i=0
W−(i−ν)kz−i, where W = e−j 2π/M. (1)
Then the transmitter in Figure 2(b) can be redrawn as in Figure 3. Now consider the receiver side. Denote the M× 1 system from r(n) to v(n) in Figure 2(b) as h(z). We can write h(z) as h(z) = ( 0 W ) ( 1 z · · · zN−1)T. Suppose that the k-th element is Hk(z) (the k-th receiving filter), then we have
Hk(z) = zν √ M M−1 i=0 Wikzi. (2)
We can redraw the receiver as the receiving bank struc-ture in Figure 3.
Having the filterbank representation, we can now obtain the overall transfer matrix. Using the polyphase identity described in Table 1, we observe that the trans-fer function Tk,i(z) from the i-th transmitter input si(n) to the k-th signal yk(n) at the receiver is given by
Tk,i(z) = [Hk(z)C (z)Fi(z)]↓N, C(z) s0(n) s1(n) sM−1(n) W 0 IνIM G N N N P/S x(n) q(n) r (n) N N N S/P [0 W] 1/λ0 1/λ1 1/λM−1 z−1 z−1 z z (a) C(z) N N N G z−1 z−1 s0(n) s1(n) sM−1(n) p(n) x(n) q(n) r (n) v(n) N N N 1/λ0 1/λ1 1/λM−1 Receiving Filterbank f(z) h(z) Transmitting Filterbank [0 W] z z (b)
where the notation [ A(z)]↓N denotes the N -fold decimated version of A(z) as defined in Table 1. Note that the DMT sys-tem is ISI free, meaning that there is zero inter-block and inter-sub-channel ISI, and the subinter-sub-channel gain from the transmitter input sk(n) to the receiver output ˆsk(n) is one. That is, in the absence of channel noise, ˆsk(n) = sk(n). The system from si(n) to ˆsk(n) is LTI with transfer function δ(k − i). As ˆsk(n) differs from yk(n) only in the scalar 1/λk, we can con-clude that Tk,i(z) = λkδ(k − i).
Summarizing, we can obtain the following lemma.
Lemma 1 For the system in Figure 3, the transfer function
Tk,i(z) from the i-th transmitter input si(n) to the k-th signal yk(n) at the receiver is given by
Tk,i(z) = λkδ(k − i ), 0 ≤ k, i ≤ M − 1. (3) The constants λkare the M-point DFT of c(n). The result holds for any FIR filter C(z) of order L ≤ ν.
So long as the order of C(z) is not larger than ν, the system is free from inter-block interference and inter-subchannel interference. Thisimplies that, if we cascade another filter before or after the channel, as long as the product of this extra filter and C(z) has order no larger than $\nu$ the overall system remains ISI free. We will use this observa-tion later to design transmitters and receivers.
From (1) and (2), we see that the transmitting and receiving filters are derived from rectangular windows. In particular, the first transmitting filter F0(z) is a
rectangu-lar window of length N. All the other transmitting filters are scaled and frequency-shifted versions of the first transmitting filter (prototype filter),
Fk(z) = WνkF0(zWk). (4) Similarly, the first receiving filter is also a rectangular window, but of length M . All the other receiving filters are scaled and frequency-shifted versions of the first receiving filter, Hk(z) = W−νkH0(zWk). As the prototype filters are rectangular windows, the frequency selectivi-ty is not good. The first sidelobe has an attenuation of around 13 dB only and the stopband decays slowly at a rate inversely proportional to the frequency. Poor fre-quency selectivity leads to bad frefre-quency separation. This results in spectral leakage in the transmitted spectrum and poor RFI suppression at the receiver side.
Receivers with Subfilters
To improve the frequency selectivity of the receiving filters, we introduce additional FIR Qk(z) to the receiving bank, as
1/λ0 1/λ1 1/λM−1 N N N y0(n) y1(n) yM−1(n) u0(n) u1(n) uM−1(n) H0(z) H1(z) HM−1(z) Q0(z ) Q1(z ) QM−1(z) h'(z)
Figure 4. The receiving bank with subfilters.
s0(n) s1(n) sM−1(n) N N N FM−1(z) wM−1(n) w1(n) w0(n) x(n) r(n) q(n) F1(z) F0(z) C(z) H0(z) H1(z) HM−1(z) yM−1(n) 1/λ0 1/λ1 1/λM−1 s0(n) s1(n) sM−1(n) y1(n) y0(n) N N N
shown in Figure 4. These additional filters will be called sub-filters as their order α is generally much smaller than M. With the subfilters, the k-th effective receiving filter becomes Hk(z) = Hk(z)Qk(z) and the frequency responses of the receiving filters are further shaped by the subfilters. The transfer function from the i-th transmitter input si(n) to the k-th signal yk(n) at the receiver in Figure 3 becomes Tk,i(z) = [Hk(z)(Qk(z)C (z))Fi(z)]↓N; it is the same expres-sion except that the channel is replaced by the composite channel Qk(z)C (z). From Lemma 1, we know that the sys-tem is free from ISI as long as the order of the composite channel is not larger than ν. In particular, Tk,i(z) is the same as in (3) except that the coefficients λkare now the M-point DFT of the composite channel.
We can choose the subfilters so that λkremain the same after the subfilters are included. To have this property, we need Qk(ej 2πk/M) = 1, i.e., the k-th DFT coefficient of Qk(z) normalized to one. In the special case that the subfilters are chosen as shifted versions of the first subfilter, Qk(z) = Q0(zWk), then Qk(ej 2πk/M) is equal to Q0(ej 0). That is, we only need the DC value of the first subfilter to be one. This translates to the time-domain condition that the sum of the coefficients is one. Suppose that Q0(z) is a causal FIR filter of order β, then the condition is
β
n=0
q0(n) = 1. (5)
This condition can be easily sat-isfied by a simple normalization after q0(n) is designed without constraint. The normalization in (5) will be assumed in the fol-lowing discussions. Further-more when the other subfilters are shifted versions of the first subfilter, the new receiving filter becomes Hk(z) = W−νkH0(zWk). They are also shifted versions of the new prototype filter H0(z) except for some scalars. We will see below that these receiving filters form a DFT bank and can be implemented efficiently. The complexity is almost the same as the conventional DMT sys-tem without subfilters.
Implementation of Receiving Bank with Subfilters
The new prototype filter is the product of Q0(z) and the rectan-gular window H0(z) given in (2). Let the coefficients of H0(z) be bi/ √ M and we write it as H 0(z) = z√ν−β M M+β−1
i=0 bizi. We will call bireceiver window coefficients for reasons that will become clear later. Using the relation Hk(z) = W−νkH0(zWk), we can write the new k-th receiving filter as
Hk(z) = z√ν−β M M+β−1 i=0 biWk(i−β)zi =z√ν−β M ( 1 W k · · · Wk(M−1)) g(z), where g(z) 0 Iβ IM diag(b0b1· · · bM+β−1) 1 z .. . zM+β−1 .
Then the new receiving bank h(z) as indicated in Figure 4 can be written as h(z) = zν−βWg(z). This expression
gives rise to the implementation of the receiver in Figure 5, where we have moved the decimators to the left by using the Noble identity for decimators in Table 1. Note that the first ν − β samples are discarded due to the advance zν−β. N N N N N N N N z z z z z z z 1/λ0 1/λ1 1/λM−1 W β Many + β Many + ν−β Many Window b0 bβ−1 bM bM+β −1 d(n)
Window Coefficients bk The new prototype filter H0(z) is the convolution of h0(n) and a much shorter q0(n). As h0(z) is a rectangular window, each coeffi-cient bk is a partial sum of the coefficients of q0(n). With the nor-malization in (5), most of the win-dow coefficients are equal to one, except for those on the two ends.
The middle M− β coefficients are equal to one, and the remaining coefficients, β coefficients on each side, have non-unity values. Figure 6(a) gives an example of window coefficients. Furthermore, we can verify that the time shifts of bkadd up to one, in particular,
∞ =−∞
bk−M= 1. (6)
This is known as the time-domain Nyquist property [19], [20]. The subfilter viewpoint allows the time-domain Nyquist property to be satisfied inherently in the receiver design.
Connection with the Usual Receiver Windowing
If we observe the implementation in Figure 5, we see that the samples are first multiplied by bk(i.e., windowed by
bk). The matrix I0
β IM
performs the operation of folding the first β samples and add to the last β samples as shown in Figure 6(b). Then the resulting last M samples are passed over for DFT computation, followed by FEQ. This is the same as the usual receiver windowing described in [21].
Transmitters with Subfilters
Similar to the case of the receiving end, we can also intro-duce subfilters to the transmitter side to improve the fre-quency selectivity of the transmitting filters. Figure 7 shows the transmitting bank with subfilters. Suppose the subfilters are FIR filters Pk(z) with order α. The k-th new transmitting filter is Fk(z) = Fk(z)Pk(z). The new trans-mitting filters are of length N+ α, as Fk(z) are of length N. Now the transfer function from the i-th transmitter input si(n) to the k-th signal yk(n) at the receiver (Figure 3(b)) becomes Tk,i(z) = [Hk(z)(Pi(z)C (z))Fi(z)]↓N. We can also apply the result in Lemma 1 here. The overall system remains ISI free if the order of the subfilters α
sat-isfies α + L ≤ ν. The transfer function Tk,i(z) is the same as in (3), except that now the coefficients λkare the M-point DFT of pk(n) ∗ c(n). As in the receiver case, we can choose the subfilters to be shifted versions of the first subfilter, i.e., Pk(z) = P0(zWk). In this case we can have λk remain the same after subfilters are included by normal-izing the DC value of P0(z) like that in (5). (Without loss of generality, such a normalization will be assumed in the following discussion.) Furthermore, as we will derive next, the resulting transmitting filters form a DFT bank, which can be implemented very efficiently.
Implementation of the Transmitting Bank with Subfilters
When the subfilters are frequency shifted versions of the first subfilter, the new transmitting filters are also frequency shifted versions of the new prototype except for some scalars. In particular, Fk(z) = WνkF
0(zWk). Let the coefficients of the proto-type be ai/
√
M and F0(z) =√1 M
N+α−1
i=0 aiz−i. Like the case of receiver windowing, we call these ai window coefficients. As there is a frequency shifting relation among the transmitting filters, given the coefficients of the prototype, we can obtain the coefficients of all
ν 0 ν M−β β 1 β N Discarded ν−β β M + N (a) (b) 0
Figure 6. (a) An example of receiver window; (b) receiver windowing.
s0(n) s1(n) sM−1(n) F0(z) F1(z) P0(z) P1(z) FM−1(z) PM−1(z) N N N x(n) f'(z)
Figure 7. The transmitting bank with subfilters.
Better separation among the transmitting filters translates to less spectral leakage
in the transmitted spectrum while better separation among the receiving filters leads
to improved RFI suppression.
the other transmitting filters. The new transmitting bank f(z) = ( F0(z) F1(z) · · · FM−1(z) ) as indiciat-ed in Figure 7 can be expressindiciat-ed as
f(z) = ( 1 z−1 · · · z−N+1) G(zN), where G(z) =D0 D1z−1 0 0IMIν Iα 0 W†. (7)
The matrices D0 and D1 are diagonal matrices, respectively, diag( a0 a1 · · · aN−1), and diag( aN aN+1 · · · aN+α−1).
Such an expression gives rise to the implementa-tion in Figure 8, where we have used the Noble identi-ty for exchanging LTI filters and expanders in Table 1 to move G(zN) to the left of the expanders. The coeffi-cients aicome from the convolution of an N -point
rec-tangular window with a much shorter p0(n) of length α. When the sum of the coefficients of p0(n) is normalized to one, most of the coefficients ai are equal to one. The middle N− α coeffi-cients are equal to one. Only the remaining 2α coefficients, α on each end, can have non-unity values and only for these coeffi-cients multiplications are need-ed.
Connection with the Usual Transmitter Windowing
Observing the DFT bank imple-mentation in Figure 8, we see that for each input block, M -point IDFT is performed, fol-lowed by the insertion of cyclic prefix of length ν and also the insertion of suffix of length α. The resulting vector p(n), as shown in Figure 8, is of size N+ α. The window coefficients are applied to each vector. Then the last α samples of the previ-ous block are added to the first α samples of the current block, as shown in Figure 9. This is the same as the usual transmitter windowing [2].
Transmitted Power Spectrum
The filterbank representation allows us to express the power spectrum of the transmitted signal x(n) in terms of the transmitting filters and thus in terms of the subfilters to be optimized. No assumption will be made on the length of the transmitting filters and the result is also applicable to the cases with subfilters.
For OFDM systems in wireless applications, the inputs sk(n) can be assumed to be uncorrelated and the trans-mitted power spectrum has been derived in [12]. The assumption of uncorrelated input symbols is not valid for DMT systems in wired applications. This is because the DMT system uses baseband transmission and the signal to be transmitted is real. This requires that the inputs of the IDFT matrix have the conjugate symmetric property, 0 0 s0(n) s1(n) sM−1(n) Iv IM I α W α many α many N N N N + + + + Window a0 aα−1 aN aN+α−1 z−1 z−1 z−1 z−1 z−1 x(n) G(z) p(n)
Figure 8. Efficient DFT implementation of the transmitting bank.
The filterbank representation allows us to express the power spectrum of the
transmitted signal
x
(n)
in terms of the transmitting filters and thus in terms of the
subfilters to be optimized.
Output Due to the (i−1)-th Block Output Due to the
i-th Block (i−1)N (i−1)N+ν ν M α iN iN ν iN+ν M α (i+1)N Prefix Suffix
sk(n) = sM∗−k(n), k = 1, 2, · · · , M − 1, and s0(n) is real. For even M, usually the case in practice, sM/2(n) is also real. This conjugate symmetric property means that the sym-bols assigned to the second half and the first half of the subchannels are strongly correlated and thus we can no longer assume that the inputs are uncorrelated.
For those inputs sk(n) that are in conjugate pairs, let the real part be sk(r)(n) and the imaginary part be sk(i)(n). We can treat these real parts and imaginary parts as random processes and assume, reasonably, that these random processes are white, uncorrelated, jointly wide-sense stationary with zero mean and variance Es,k/2. (The scalar 1/2 is included so that the variance of sk(n) is Es,k.) For the k-th and (M − k)-th subchannels, the inputs are a complex conjugate pair. When the transmitting filters are shifted versions of the proto-type filter as in (4) and the protoproto-type has real coeffi-cients, the coefficients of the transmitting filters are also in conjugate pairs, fM−k(n) = fk∗(n). As the result, the outputs of each pair are also the conjugates of each other. Now instead of considering the output of an individual subchannel, let us consider the sum of the outputs of each pair. Let the output of the k-th transmitting filter be wk(n) as indicated in Figure 3 and define wk(n) = wk(n) + wM−k(n). Then wk(n) can be written as wk(n) = 2 s(r) k ()f (r) k (n − N) − s (i) k ()f (i) k (n − N) , where fk(r)(n) and fk(i)(n) are, respectively, the real and imaginary part of fk(n). As the real and imaginary parts of the transmitter inputs are uncorrelated, the power spec-trum of wk(n) is Sw k(e jω) = 2Es,k N F(r) k (ejω) 2 +Fk(i)(ejω) 2 , where Fk(r)(ejω) and Fk(i)(ejω) are respectively the Fourier transforms of fk(r)(n) and fk(i)(n). It turns out that 2 F(r) k (ejω) 2 +Fk(i)(ejω) 2 =|Fk(ejω)|2+|FM −k(ejω)|2). We can obtain the transmitted power spectrum by sum-ming up the contributions from wk(n), plus w0(n) and wM/2(n) (if M is even). We arrive at the following simple expression for the transmitted spectrum
Sx(ejω) = 1 N
M−1
k=0
Es,k|Fk(ejω)|2. (8)
We can further observe that if an equal power allocation is used, the inputs of all the subchannels have the same
vari-ance Esand the transmitted power spectrum becomes the same as that of the OFDM system derived in [12].
Now let us consider the transmitted spectrum when there are subfilters. Assume that the other subfilters are frequency shifted versions of the first subfilter. When the first subfilter has real coefficients, the coefficients of the k-th and (M − k)-th new transmitting filters also form a conjugate pair. The above derivation can also be carried out for the case with subfilters. The transmitted power spectrum can be easily obtained by replacing the trans-mitting filters in (8) by Fk(ejω).
Design of Receiver Subfilters
The frequency selectivity of the receiving filters are important for RFI suppression. The radio interference is known to be of a narrowband nature. For the duration of one DMT symbol, it can be considered as a sum of sinu-soids. To analyze the effect of interference, we can apply an interference-only signal v(n) to the receiver in Figure 4. Suppose that there are J interference sources, and the interference is modelled as v(n) ==0J−1µcos(ωn+ θ). The interference term at the output of the k-th receiving filter Hk(z) is
uk(n) =1 2 J−1 =0 µ[Hk(ejω)ej(ωn+θ) + Hk(e−jω)e−j(ωn+θ)].
Minimization of interference terms requires the knowl-edge of µ, ωand θ.
First let us consider the case when the information of the interference is not available. In this case, we can alle-viate the effect of interference in the k-th subchannel by minimizing the stopband of the receiving filters. When the receiving filters are frequency shifted versions of the pro-totype, we only need to consider the stopband energy of the prototype,
φh=
ω∈Oh
|H0(ejω)|2dω, (9) where Oh denotes the stopband of the prototype filter. Note that H0(z) is the product of Q0(z) and H0(z). We can write its Fourier transform as H0(ejω) = H0(ejω)τβ(ω)q0, where τβ(ω) is the 1× β row vector ( 1 e−jω · · · e−jβω) and q
0 is the column vector ( q0(0) q0(1) · · · q0(β) )T. The stopband energy is
φh = q†0Bq0, where
B =
ω∈Oh
|H0(ejω)|2τβ†(ω)τβ(ω)dω. (10) To avoid a trivial solution, we can fix the energy of the
first subfilter to be one, q†0q0= 1.The matrix B is pos-itive definite because the objective function repre-sents the stopband energy of the prototype filter, which is always positive. To minimize φh, we can choose q0 as the eigenvector associated with the smallest eigenvalue of B. Such an approach does not depend on the RFI statistics or the channel; it has the advantage that the subfilters need to be designed only once. The subfilters need not be redesigned when the interference changes.
If the information of the interference sources is avail-able to the receiver, the subfilters can be individually opti-mized. The amplitude of the k-th interference signal uk(n) is a nonlinear function of the k-th subfilter coefficients. To simplify the problem, note that the interference due to the -th source will be small if µ2
(|Hk(ejω)|2+ |Hk(e−jω)|2) is
small. The k-th subchannel interference can be mitigated
by designing Qk(z) to minimize φk,h, φk,h= J−1 =0 µ2 (|Hk(ejω)|2+ |Hk(e−jω)|2. (11)
We can write φk,hin a quadratic form similar to that in (10) and find the optimal subfilters. Such an optimization requires only the amplitudes and frequencies, but not the phases, of the interference sources. When the subfilters are so designed, the receiving bank does not have the DFT bank structure in Figure 5. Nonetheless, the receiver can be implemented with a much reduced complexity using the sliding window approach in [26]. When the sub-filters Qk(z) are shifted versions of Q0(z), we can design Q0(z) to minimize the total interference kφk,h[27].
Example 1.
Receiver Subfiltering
In this example, we design the subfilters for RFI reduction at the receiver. The DFT size is M= 512 and cyclic prefix length is ν = 40. The order of the subfilters is β = 10. The channel used in this example is VDSL loop#1 (4500 ft) [2] and the channel noise is AWGN of −140 dBm. Model 1 dif-ferential mode RFI interference is considered [2]. Four RFI sources are assumed in the simulations, at respectively 660, 710, 770 and 1050 KHz, of strength −60, −40, −70, and −55 dBm, respectively.
We will consider two different subfilter designs. In the first design, the subfilters Qk(z) are shifted versions of Q0(z) and only Q0(z) needs to be designed. The subfilter Q0(z) is the solution to the minimization problem in (10). In this case the receiving filters form a DFT bank and can be implemented as in Figure 5. In the second design, the RFI source is known to the receiver and the subfilters Qk(z) are individually optimized by minimizing the objective function φk,hin (11). The SINRs (signal-to-noise-interference ratio) of the subchannels are as shown in Figure 10. The first case is labelled ‘subfilter (DFT bank)’ while the second case ‘sub-filter (RFI known)’. For comparison, we have also shown the subchannel SINRs for the cases of rectangular, Hanning win-dows, and also the window from [23]. The receivers with subfilters enjoy higher SINRs for the tones that are close to the RFI frequencies, especially when the statistics of the RFI source is known and the subfilters are optimized individu-ally. As a result, higher transmission rates can be achieved. The transmission rate of the first case is 7.44 Mbits/sec, and that of the second case is 8.54 Mbits/sec. The transmission rates for the cases of rectangular, Hanning windows, and [23] are 6.84, 7.16, and 7.27 Mbits/sec, respectively.
Design of Transmitter Subfilters
For the transmitter side, let us first consider the case when the transmitting filters are constrained to be
shift-0 0.2 0.4 0.6 0.8 –50 –40 –30 –20 –10 0 Frequency Normalized by π Normalized T
rasmitted Signal Spectrum (dB)
With Subfilters Without Subfilters [16]
Figure 11. The power spectrum of the transmitted signal.
0 50 100 150 200 250 0 10 20 30 40 50 Tone Index SINR (dB)
Subfilter (RFI Known) Subfilter (DFT Bank) [23]
Hanning Window Rectangular Window
ed versions of one prototype. From the expression in (8), we see that spectral leakage can be minimized by mini-mizing the stopband energy of the prototype filter F0(z). Following a procedure similar to the design of receiver subfilters, we can write the stopband energy φfof the pro-totype F0(z) as φf = p†0Ap0, where A= ω∈Of |F0(ejω)|2τα†(ω)τα(ω)dω. (12) We can see that φfcan be minimized by choosing p0to be the eigenvector associated with the minimum eigenvalue of A.
Now consider the case when the subfilters are not con-strained. The total spectral leakage is
ω∈Ou
Sx(ejω)dω, (13)
where Oudenotes the band in which leakage is undesired. The total leakage can be minimized if we can minimize the individual contribution φk,ffrom each subchannel,
φk,f=
ω∈Ou
|Fk(ejω)|2dω.
We can write φk,fin a quadratic form like that in (12) and find the optimal subfilters. In this case the subfilters do not form a DFT bank, and neither do the new transmitting filters. An efficient implementation of the resulting trans-mitting bank can be found in [18].
Example 2. Transmitter Subfiltering
The block size M= 512 and prefix length ν = 40. The sub-filters are shifted versions of the first subfilter and thus the transmitting filters form a DFT bank. The order α of the subfilters is 20. We form the positive definite matrix A and compute the eigenvector corresponding to the smallest eigenvalue to obtain p0. Figure 11 shows the spectrum of the transmitter output. The subcarriers used are 38 to 90 and 111 to 255. The subcarriers with indices smaller than 38 are reserved for voice band and upstream transmis-sion, and those with indices between 91 and 110 are for egress (interference of DMT signals to wireless radio fre-quency transmission) control. Also shown in the figure is the output spectrum when the transmitter window of [16] is used, which requires no extra cyclic prefix but addi-tional post-processing is needed at the receiver. We see that the spectrum with the subfilters has a much smaller spectral leakage in unused bands.
Implementation and Complexity
For the conventional DMT system in Figure 1, the main computations of the transceiver are those of the IDFT
and DFT matrices, for which fast algorithms can be applied. The complexity of the transmitter is simply that of an IDFT matrix and the complexity of the receiv-er is that of a DFT matrix plus M multiplications for FEQs. Moreover, except for the FEQs, the computations are channel independent. For a system with receiver subfilters, we can observe the implementation com-plexity from Figure 5. Compared with the conventional case, the new receiver needs only 2β more multiplica-tions (due to the non-unity window coefficients) and β more additions for every block of outputs. Similarly, the complexity of the transmitter with subfilters in Fig-ure 8 requires 2α more multiplications and α more additions per output block. As α and β are usually much smaller than M , in either case the overhead of subfiltering is very small.
Conclusions
In this work, we have presented a filterbank approach to the design of transmitter/receiver by introducing subfilters. The frequency separation among the sub-channels can be considerably improved. Better separa-tion among the transmitting filters translates to less spectral leakage in the transmitted spectrum while bet-ter separation among the receiving filbet-ters leads to improved RFI suppression. As these are frequency based characteristics, the filterbank transceiver repre-sentation provides a natural and useful framework for formulating the problem. The transmitter/receiver designs are converted to simple eigen-problems and closed form solutions can be obtained.
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Yuan-Pei Lin (S’93-M’97-SM’03) was
born in Taipei, Taiwan, 1970. She received the B.S. degree in control engi-neering from the National Chiao-Tung University, Taiwan, in 1992, and the M.S. degree and the Ph.D. degree, both in electrical engineering from California Institute of Technology, in 1993 and 1997, respectively. She joined the Department of Electrical and Control Engi-neering of National Chiao-Tung University, Taiwan, in 1997. Her research interests include digital signal pro-cessing, multirate filter banks, and signal processing for digital communication, particularly the area of multicarri-er transmission.
She is a recipient of 2004 Ta-You Wu Memorial Award. She served as an associate editor for IEEE Transaction on Signal Processing 2002–2006. She is cur-rently an associate editor for IEEE Signal Processing Letters, IEEE Transaction on Circuits and Systems II, EURASIP Journal on Applied Signal Processing, and Multidimensional Systems and Signal Processing, Aca-demic Press. She is also a Distinquished Lecturer of the IEEE Circuits and Systems Society for 2006–2007.
Chien-Chang Li was born in Penghu,
Tai-wan, R.O.C., in 1980. He received the B.S. degree in power mechanical engineering from the National Tsing-Hua University, Hsinchu, Taiwan in 2003. He is now pur-suing the Ph.D. degree in the Dept. of Elec-trical and Control Engineering, National Chiao-Tung University, Taiwan. His research interests include digital signal processing, multirate systems and digital communication systems
See-May Phoong (M’96-SM’04) 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 M.S. and Ph.D. degrees in electrical engi-neering from the California Institute of Technology (Caltech), Pasadena, California, in 1992 and 1996, respectively.
He was with the Faculty of Electronic and Electrical Engineering, Nanyang Technological University, Singa-pore, from September 1996 to September 1997. In Sep-tember 1997, he joined the Graduate Institute of Communication Engineering and the Department of Electrical Engineering, NTU, as an Assistant Professor, and since August 2006, he has been a Professor.
Dr. Phoong is currently an Associate Editor for the IEEE Transactions on Circuits and Systems I. He has previ-ously served as an Associate Editor for Transactions on Circuits and Systems II: Analog and Diginal Signal Process-ing (Jan. 2002–Dec. 2003) and IEEE Signal ProcessProcess-ing Let-ters (March 2002–Feb. 2005). His interests include multirate signal processing, filter banks and their appli-cation to communiappli-cations. He received the Charles H. Wilts Prize (1997) for outstanding independent research in electrical engineering at Caltech. He was also a recipi-ent of the Chinese Institute of Electrical Engineering’s Outstanding Youth Electrical Engineer Award (2005).
