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  • Abstract

On the Filter Design for a Class of Multimode Transmultiplexers

This paper discusses some issues related to the filter design in a class of multimode transmultiplexers (TMUXs). These TMUXs cover a large set of frequency division multiplexed (FDM) scenarios with simple reconfigurations. The reconfiguration is performed by changing the values of some multipliers. The paper outlines a direct filter design to decrease the level of inter-symbol and inter-carrier interference by the use of time-varying periodic filters. These time-varying periodic filters are derived from the known FDM scenarios and they are included as additional constraints in the filter design. Both minimax and constrained least-squares approaches are treated and it is shown that by including the additional constraints, the level of the TMUX noise can be reduced. This results in a better approximation of perfect reconstruction and makes the filter design direct.

SECTION I

Introduction

TRANSMULTIPLEXERs (TMUXs) convert data between time division multiplexed (TDM) and frequency division multiplexed (FDM) formats and have successfully been utilized for multiuser communications. A TMUX simultaneously transmits different data streams through a single channel using the FDM technique. Specifically, the transmitters (synthesis filters) cover different uniform or nonuniform regions of frequency so that independent user data streams are packed into adjacent frequency bands. This leads to uniform [1] or nonuniform TMUXs [2], [3], [4] and the frequency bands are determined by the passbands of the synthesis filters which are generally bandpass filters. Having packed the data streams into the frequency bands, they are added to obtain a composite signal which is transmitted over a common medium (channel). Specifically, upsamplers compress the frequency spectrum and they are followed by bandpass filters to extract the desired frequency bands. On the receiver side, the analysis filers decompose the composite signal into different frequency bands where a downsampling is needed to derive the reconstructed data streams.

In the case of nonuniform TMUXs, there is a need for different upsampling/downsampling pairs at each branch of the system so that different data streams can be transmitted in different portions of the spectrum. This also necessitates different bandpass filters with different passband widths. If it is desired to have a high degree of freedom in determining the frequency bands for each data stream, there will be a need to either (i) design a large set of filters, or (ii) online design of the filters. In other words, if one decides to change the FDM scenario (or the multimode setup [5], [6], [7]), new sets of filters would be required. This becomes a difficult task if the number of FDM scenarios increases in a time-varying manner. Thus, reconfigurable (or multimode) TMUX structures are required to cover different FDM scenarios with reasonable design and implementation effort.

Due to the fixed structure and parameter sets in uniform (and nonuniform) TMUXs, conventional filter design methods can easily be applied to these TMUXs. For example, using the duality of a filter bank (FB) and a TMUX, a set of filters which has been designed for a FB can easily be adopted for a TMUX. However, multimode TMUXs do not allow for these techniques to be applied and, hence, new formulations must be used in the filter design. Previous work has applied general filter design formulations where the resulting TMUXs can cover any FDM scenario [5], [6], [7]. In other words, the filter design was indirect as it did not consider the FDM scenarios. However, in cases where the FDM scenarios are known, one can utilize direct filter design techniques to reduce the levels of error. It is the aim of this paper to consider these issues.

A set of known FDM scenarios results in known sets of interpolators/decimators and, thus, one could in principle use the conventional nonuniform TMUXs [2], [3], [4]. However, the TMUX, considered in this paper, has a strong advantage over the conventional nonuniform TMUXs as it is fully reconfigurable. Furthermore, the proposed TMUX requires only one set of filters and, consequently, the filter design problem needs to be solved only once and offline. The reconfiguration of the TMUX can easily be performed by modifying some multiplier values.

A. Paper Outline

Section II introduces the TMUX structure based on rational sampling rate conversion (SRC) and formulates its indirect minimax and constrained least-squares (CLS) filter design problems. In Section III, the direct filter design constraints which consider specific (and known) FDM scenarios are discussed. Then, Section IV provides a design example for the different filter design formulations of Sections II and III. Finally, Section V outlines some concluding remarks.

SECTION II

Rational SRC Multimode TMUX

To perform rational interpolation (decimation) by ratio Formula, integer interpolation (decimation) by Ap comes before (after) downsampling (upsampling) by Bp [1]. Using the Farrow structure, to realize the polyphase components of general lowpass interpolation/decimation filers, different integer interpolation/decimation filters can be obtained [6], [7], [8]. Briefly, a general lowpass filter for integer interpolation/decimation by ratio Ap can be realized using a Farrow structure1 with L subfilters of odd order N1. This results in the overall filter order to be (N1+ 1) Ap. By including these integer interpolation/decimation structures in a rational SRC model, a multimode TMUX as shown in Fig. 1 can be constructed [7]. This P-channel TMUX consists of upsamplers/downsamplers Ap, Bp,p = 0,1,…,P − 1; lowpass interpolation/decimation filters, i.e., Gp(z) for interpolation and Formula for decimation; and adjustable frequency shifters, i.e., frequency shifts by ωp and Formula. Assuming the sampling period of Tp at branch p of the TMUX, we haveFormula TeX Source $$T_0 {B_0 \over A_0} = T_1{B_1 \over A_1} = \cdots = T_y\eqno{\hbox{(1)}}$$where Ty is the sampling period of y(n). In the synthesis FB, the TMUX generates the required bandwidths through rational interpolation by Formula. Having placed the user signals at appropriate positions in the frequency spectrum, they form y(n) for transmission. In the analysis FB, the desired signal is obtained by baseband processing which consists of upsampling by Bp, lowpass filter Formula, and downsampling by Ap.

Figure 1
Fig. 1. Multimode TMUX composed of variable rational SRC and adjustable frequency shifters. (Note: An integer SRC variant of this TMUX is presented in [6]. Furthermore, the rational SRC is implemented using the Farrow structure but the conventional models of rational SRC are only shown for clarification.)

To ensure the approximation of perfect reconstruction (PR), the output of the integer downsampling by Bp, in the synthesis FB, must be bandlimited to [0, π]. Consequently, no aliasing occurs during decimation by Ap. In other words, the values for Ap, Bp, and ρ should be chosen such that2Formula TeX Source $${A_p \over 1 + \rho} \geq B_p\geq 2, \quad \{A_p,B_p\}\in N.\eqno{\hbox{(2)}}$$Thus, to get an SRC by Formula, one can design the filters for integer SRC by Ap and, then, perform the additional integer SRC by Bp through [7]

  • A commutator in case of interpolation by Formula.

  • An upsampler in case of decimation by Formula.

This indirect design method only considers the values of Ap and will be discussed in the next subsection.

A. Indirect Filter Design

In general, there are two important sources of interference in TMUXs. The filters in each branch of the TMUX, between Xp(z) and Formula, cause inter-symbol interference (ISI) whereas the contribution of signals from other branches, i.e., between Xi(z) and Formula, gives rise to inter-carrier interference (ICI). Assuming Vpp and Vip to represent ISI and ICI, respectively, it is desired to have Formula and Formula with δISI and δICI being the allowed ISI and ICI where ηp is the delay at each branch p of the TMUX. Considering the redundancy of the TMUX and the fact that user signals do not overlap, ICI is here controlled by the stopband attenuation of Gp(z) and Formula. Furthermore, ISI can be controlled by appropriate choice of Gp(z) and Formula so that the zeroth polyphase component of Formula approximates an allpass transfer function3.

In a general redundant nonuniform TMUX, different branches have different values of Rp and, thus, ICI becomes time-varying. However, according to the discussion above, the important issue is to control the stopband attenuation of Gp(z) and Formula as well as the zeroth polyphase component of Formula. To approximate PR as close as desired, the filter Formula should approximate an Rp th-band filter as close as desired [1].

Due to the indirect filter design, the values of Bp are not taken into account and it is only ensured that (2) holds. This way, there is no aliasing and, thus, the principle of reducing ISI and ICI can be utilized by just considering the values of Ap in the filter design. Specifically, the filters Gp(z) and Formula are designed such that their cascade approximates Ap th-band filters. Consequently, the filters Gp(z) and Formula should be designed such that (i) they have sufficiently small ripples in their stopbands, and (ii) the zeroth polyphase component of Formula approximates an allpass transfer function. Assuming one set of Farrow subfilters, resulting in Formula the zeroth polyphase component of Formula can be written as4Formula TeX Source $$\eqalignno{F_p(e^{j\omega T}) = [G_p(e^{j\omega T})\hat{G}_p (e^{j\omega T})e^{j {N\omega T \over 2}}]_{\rm zeroth} \cr&= \sum_{n=0}^{A_{p-1}} [G_p(e^{j (\omega T - {2\pi n \over A_p})}) e^{j (\omega T - {2\pi n \over A_p}){N\over 2}}]^2&\hbox{(3)}}$$To solve the indirect filter design problem, one can utilize the minimax method as

Case I: minimize δ, over all Ap, subject toFormula TeX Source $$\eqalignno{\vert F_p(e^{j\omega T}) - 1\vert &\leq \delta, \quad \omega T \in [0, \pi] \cr\vert G_p(e^{j\omega T})\vert&=\delta s, \quad \omega T\in [\omega_s T, \pi],&\hbox{(4)}}$$where δs is the desired stopband attenuation andFormula TeX Source $$\omega_s T = {\pi (1 + \rho) \over A_p}\eqno{\hbox{(5)}}$$Similarly, in the CLS formulation, we have

Case II: minimize δ, over all Ap, subject toFormula TeX Source $$\eqalignno{\int_0^{\pi} \vert F_p (e^{j\omega T}) - 1\vert^2 &\leq \delta,\quad \omega T \in [0, \pi] \cr\vert G_p(e^{j\omega T})\vert &\leq \delta s, \quad \omega T\in [\omega_s T, \pi].&\hbox{(6)}}$$By choosing a fixed value for δs, one can compare these two design methods. As noted before, the deviation of Fp (ejω T) from an allpass transfer function (or a pure delay) controls the ISI whereas the stopband attenuation of Formula controls the ICI.

This design method is indirect as it does not consider the effects of downsampling by Bp. The presence of the integer downsampling by Bp introduces some undesired terms which can be taken into consideration in a direct filter design. This direct filter design can further decrease the noise introduced by the TMUX.

SECTION III

Direct Filter Design

To consider the effects of SRC by Bp, one needs to use additional constraints which arise due to the presence of upsampling/downsampling by Bp. In one branch of the TMUX, shown in Fig. 2(a), the input-output relation can be written as5Formula TeX Source $$Y(z) = \sum_{m=0}^{A-1}\sum_{k=0}^{B-1} \hat{G}\left(z^{1\over A} W^m_A\right) G\left(z^{1\over A} W^k_B W^m_A \right)X\left(zW^k_B W^m_A\right)\eqno{\hbox{(7)}}$$with Formula and Formula. As shown in Fig. 2(b), the system in Fig. 2(a) can be modeled as the operation of a time-varying periodic filter hn(k) on the input signal as [9]Formula TeX Source $$Y (z) = H_n(z) X(z),\eqno{\hbox{(8)}}$$whereFormula TeX Source $$H_n(z) = \sum_k h_n(k)z^{-k}.\eqno{\hbox{(9)}}$$In the time domain, (7) can be written asFormula TeX Source $$\eqalignno{y(n) &= \sum_k x(k) \sum_m g(mB - Ak)\hat{g} (nA - mB)\cr&= \sum_k x(n - k) \sum_m g(mB - nA + kA)\hat{g}(nA - mB), \cr&= \sum_k x(n - k)h_n(k).&\hbox{(10)}}$$Thus, we have B impulse responses given byFormula TeX Source $$h_n(k) = \sum_m g(mB - nA + kA)\hat{g}(nA - mB),\eqno{\hbox{(11)}}$$where n = 0, 1,…, B − 1. Alternatively, it is also possible to determine these time-varying periodic filters by feeding the sequences xm(n) = δ (nm),m = 0,1,…, to the structure of Fig. 2(a) and computing the set of corresponding outputs ym(n). The impulse responses hn(k) can then be computed as hn(nm) = ym(n).

Figure 2
Fig. 2. General and its equivalent model (using periodic time-varying FIR filter) for each branch in the TMUX of Fig. 1.

In a PR system, we have y(n) = x(n − Δ) meaning that the output signal is a delayed (by Δ) version of the input signal. Defining the error e(n) as e(n) = y(n) − x(n), we have (assuming Δ = 0)Formula TeX Source $$e(n) = {1 \over 2\pi} \int^\pi_{-\pi} (H_n(e^{j\omega T}) - 1) X(e^{j\omega T}) e^{jn\omega T} d(\omega T).\eqno{\hbox{(12)}}$$Ideally, e(n) = 0 and, then, Hn(ejω T) = 1. However, in practice, Hn(ejω T) can generally only approximate one in the frequency range of interest. Consequently, the aim of the filter design problem is to determine the coefficients hn(k) so that the error e(n) is minimized according to some criterion. A problem is that e(n) depends on the impulse responses of the filters as well as the spectrum of X(ejω T). This means that one generally requires knowledge about the spectrum of the input signal to determine optimum filters. In practice, there is no complete knowledge about X (ejω T) and, thus, one has to accept a suboptimum solution instead. In this regard, it can be convenient to make use of the Lp-norm of a general signal S(ejω T) given byFormula TeX Source $$\Vert S(e^{j\omega T})\Vert_p = \root{p}\of {{1 \over 2\pi} \int^{\pi}_{-\pi} \vert S(e^{j\omega T})\vert^p d(\omega T)}.\eqno{\hbox{(13)}}$$Using the triangle inequality of integrals in (12), we haveFormula TeX Source $$\vert e(n)\vert \leq {1 \over 2\pi} \int^{\pi}_{-\pi} \vert H_n(e^{j\omega T}) - 1\Vert X(e^{j\omega T})\vert d(\omega T)\eqno{\hbox{(14)}}$$which in terms of L-norm can be written asFormula TeX Source $$\vert e(n)\vert \leq \Vert H_n(e^{j\omega T}) - 1\Vert_{\infty}\Vert X(e^{j\omega T})\Vert_{\infty}.\eqno{\hbox{(15)}}$$AsFormula TeX Source $$\Vert S(e^{j\omega T}) - 1\Vert_{\infty} = \max_{\omega T} \{\vert S(e^{j\omega T}) - 1\vert\},\eqno{\hbox{(16)}}$$minimizing the maximum of Formula corresponds to minimizing the maximum of the error function e(n) resulting from that particular hn(k). However, every branch of the TMUX in Fig. 1 results in Bp periodic functions and, consequently, one can express the direct minimax filter design problem as

Case III: minimize δ, over all Ap and Bp, subject toFormula TeX Source $$\eqalignno{{1\over B_p} \sum^{B_{p-1}}_{n=0} \vert H_n(e^{j\omega T}) - 1\vert &\leq \delta, \quad \omega T \in[0, \pi] \cr\vert G_p(e^{j\omega T})\vert &\leq \delta_s, \quad \omega T\in [\omega_s T, \pi].&\hbox{(17)}}$$Similarly, (14) can also be written in terms of L2-norm to minimize the energy of the error function leading to the direct CLS design problem as

Case IV: minimize δ, over all Ap and Bp, subject toFormula TeX Source $$\eqalignno{{1\over B_p} \sum^{B_{p-1}}_{n=0} \int_0^{\pi} \vert H_n(e^{j\omega T}) - 1\vert^2 &\leq \delta, \quad \omega T \in[0, \pi] \cr\vert G_p(e^{j\omega T})\vert &\leq \delta_s, \quad \omega T\in [\omega_s T, \pi].&\hbox{(18)}}$$

SECTION IV

Design Example

In this section, some design examples for different cases of direct and indirect filter design will be considered. To compare the performance of the TMUX, we use the error vector magnitude (EVM) which is a metric of transmitter signal quality. EVM provides a statistical estimate of the error vector normalized by the magnitude of the ideal signal and is defined asFormula TeX Source $$EVM_{rms} = \sqrt{\sum^{N_s-1}_{k=0} \vert s(k) - s_{ref} (k)\vert^2 \over \sum^{N_s-1}_{k=0} \vert s_{ref} (k)\vert^2},\eqno{\hbox{(19)}}$$where s(k) and sref(k) represent the length- Ns measured and ideal sequences, respectively.

In all the filter design problems, the free optimization parameters are the coefficients of the Farrow subfilters, i.e., Sk(z), and Gp,0(z). In other words, the values of N1, L, ρ, Rp and the values of the fractional delays in the Farrow structure are fixed during the optimization. The filter design procedure only determines the coefficients of Sk (z) and Gp,0(z). To solve these design problems, the M-file function fminimax from the optimization toolbox of MATLAB is used.

A crucial point is that the filter design problem is solved only once and offline. Then, a large set of SRC ratios (and user bandwidths) can be supported with any desired minimum ISI and ICI. Specifically, implementation of different bandwidths (and SRC ratios) is obtained by choosing proper values of fractional delays and using appropriate number of polyphase components, i.e., Rp.

In this design example, the values of L, N1, and δs are 5, 17, and 0.01 respectively. Furthermore, the optimization problems of (4), (6), (17), and (18) result, respectively, in the values of δ, to be 6.21 × 10−4, 1.28 × 10−7, 2.70 × 10−4, and 2.49 × 10−8. Note that in all of these filter design problems, (2) holds and their difference lies in the fact whether they include the values of Bp in the optimization or not.

Fig. 3 shows the average EVM for 16-QAM signals according to the multimode setups6 of Table I. Generally, the CLS approach results in a smaller EVM than the minimax method. Furthermore, the direct filter design in Case III (or IV) reduces the EVM compared to the indirect filter design of Case I (or II). This shows that the direct filter design has a superiority over the indirect filter design. Compared to the direct minimax filter design, the direct CLS method brings a larger improvement in system performance.

Figure 3
Fig. 3. EVM values for the multimode setups in Table I.
Table 1
TABLE I SRC Ratios for the Multimode Setups Used in Fig. 3

Although the CLS approach shows a superiority according to the performance measure considered here, in some systems there may be a restriction on the maximum allowable ripples. In such systems, the more appropriate option would be to use the minimax approach to ensure that the ripples are in the allowable ranges.

With δs = 0.01, the EVM values hover around −40 dB. In other words, ICI is controlled by δs and the four filter design problems try to decrease ISI either directly or indirectly. However, the final value for EVM is determined by both ISI and ICI. Furthermore, the values of δs and δ are correlated and for the same filter orders, decreasing δs would increase δ resulting in a larger EVM. For example, with the filter orders as in Fig. 3, having δs = 0.001 results in about 4 dB larger EVM than that depicted in Fig. 3. In this case, EVM is mainly determined by δ and the difference between different filter design problems gets less pronounced. Any desired EVM can be achieved by choosing appropriate values for L and N1.

Irrespective of the values for δs, L, and N1, the direct filter design results in a smaller EVM than the indirect filter design method. Furthermore, the CLS approach has a smaller EVM than the minimax method.

SECTION V

Conclusion

In this paper, some issues related to the filter design for multimode TMUXs are discussed. Four different direct and indirect filter design methods are considered and compared. It is shown that different filter design formulations can result in different system performances. However, the direct filter design has a better control over the TMUX noise. Thus, the system designer must consider all design formulations to get a reduced level of error. It is also stressed that the class of TMUXs, discussed here, requires that the filter design problems of (4), (6), (17), and (18) are solved only once and offline. In these design problems, it is assumed that Formula. Otherwise, one would require to add constraints which result from Formula as well.

Footnotes

Manuscript received ; revised ; accepted . Date of publication ; date of current version .

Amir Eghbali, Håkan Johansson, and Per Löwenborg are with the Division of Electronics Systems, Department of Electrical Engineering, Linköping University, SE-581 83, SWEDEN Email: (amire@isy.liu.se, hakanj@isy.liu.se, perl@isy.liu.se).

1. The zeroth polyphase component of these general lowpass filters is realized by a Type I linear-phase filter Gp,0(z) of order N0 = N1+ 1.

2. The roll-off factor ρ is in the range (0, 1] and throughout this paper, we assume ρ = 0.2.

3. If a filter f(n) is sandwiched between an upsampler and a downsampler of ratio M, the overall system is equivalent to its zeroth polyphase component f(nM) [1].

4. For convenience in the filter design, adding the term Formula results in a noncausal filter to include the center tap in the optimization.

5. All the scaling factors resulting from the interpolation and decimation are omitted for simplicity. These factors do not restrict the filter design problem.

6. For illustration purposes, the values of Ap and Bp are chosen so that y(n), in Fig. 1, occupies between 90–99% of the frequency range [0, 2 π].

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Amir Eghbali

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Håkan Johansson

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Per Löwenborg

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