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 , integer interpolation (decimation) by *A*_{p} comes before (after) downsampling (upsampling) by *B*_{p} [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 *A*_{p} can be realized using a Farrow structure^{1} with *L* subfilters of odd order *N*_{1}. This results in the overall filter order to be (*N*_{1}+ 1) *A*_{p}. 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 *A*_{p}, *B*_{p},*p* = 0,1,…,*P* − 1; lowpass interpolation/decimation filters, i.e., *G*_{p}(*z*) for interpolation and for decimation; and adjustable frequency shifters, i.e., frequency shifts by *ω*_{p} and . Assuming the sampling period of *T*_{p} at branch *p* of the TMUX, we have
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$$T_0 {B_0 \over A_0} = T_1{B_1 \over A_1} = \cdots = T_y\eqno{\hbox{(1)}}$$where *T*_{y} is the sampling period of *y*(*n*). In the synthesis FB, the TMUX generates the required bandwidths through rational interpolation by . 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 *B*_{p}, lowpass filter , and downsampling by *A*_{p}.

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

This indirect design method only considers the values of *A*_{p} 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 *X*_{p}(*z*) and , cause inter-symbol interference (ISI) whereas the contribution of signals from other branches, i.e., between *X*_{i}(*z*) and , gives rise to inter-carrier interference (ICI). Assuming *V*_{pp} and *V*_{ip} to represent ISI and ICI, respectively, it is desired to have and with δ_{ISI} and δ_{ICI} being the allowed ISI and ICI where *η*_{p} 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 *G*_{p}(*z*) and . Furthermore, ISI can be controlled by appropriate choice of *G*_{p}(*z*) and so that the zeroth polyphase component of approximates an allpass transfer function^{3}.

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

Due to the indirect filter design, the values of *B*_{p} 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 *A*_{p} in the filter design. Specifically, the filters *G*_{p}(*z*) and are designed such that their cascade approximates *A*_{p} th-band filters. Consequently, the filters *G*_{p}(*z*) and should be designed such that (i) they have sufficiently small ripples in their stopbands, and (ii) the zeroth polyphase component of approximates an allpass transfer function. Assuming one set of Farrow subfilters, resulting in the zeroth polyphase component of can be written as^{4}
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$$\eqalignno{F_p(e^{j\omega T}) = [G_p(e^{j\omega T})\hat{G}_p (e^{j\omega T})e^{j {N\omega T \over 2}}]_{\rm zeroth} \cr&= \sum_{n=0}^{A_{p-1}} [G_p(e^{j (\omega T - {2\pi n \over A_p})}) e^{j (\omega T - {2\pi n \over A_p}){N\over 2}}]^2&\hbox{(3)}}$$To solve the indirect filter design problem, one can utilize the minimax method as

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

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

This design method is indirect as it does not consider the effects of downsampling by *B*_{p}. The presence of the integer downsampling by *B*_{p} 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 *B*_{p}, one needs to use additional constraints which arise due to the presence of upsampling/downsampling by *B*_{p}. In one branch of the TMUX, shown in Fig. 2(a), the input-output relation can be written as^{5}
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$$Y(z) = \sum_{m=0}^{A-1}\sum_{k=0}^{B-1} \hat{G}\left(z^{1\over A} W^m_A\right) G\left(z^{1\over A} W^k_B W^m_A \right)X\left(zW^k_B W^m_A\right)\eqno{\hbox{(7)}}$$with and . As shown in Fig. 2(b), the system in Fig. 2(a) can be modeled as the operation of a time-varying periodic filter *h*_{n}(*k*) on the input signal as [9]
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$$Y (z) = H_n(z) X(z),\eqno{\hbox{(8)}}$$where
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$$H_n(z) = \sum_k h_n(k)z^{-k}.\eqno{\hbox{(9)}}$$In the time domain, (7) can be written as
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$$\eqalignno{y(n) &= \sum_k x(k) \sum_m g(mB - Ak)\hat{g} (nA - mB)\cr&= \sum_k x(n - k) \sum_m g(mB - nA + kA)\hat{g}(nA - mB), \cr&= \sum_k x(n - k)h_n(k).&\hbox{(10)}}$$Thus, we have *B* impulse responses given by
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$$h_n(k) = \sum_m g(mB - nA + kA)\hat{g}(nA - mB),\eqno{\hbox{(11)}}$$where *n* = 0, 1,…, *B* − 1. Alternatively, it is also possible to determine these time-varying periodic filters by feeding the sequences *x*_{m}(*n*) = δ (*n* − *m*),*m* = 0,1,…, to the structure of Fig. 2(a) and computing the set of corresponding outputs *y*_{m}(*n*). The impulse responses *h*_{n}(*k*) can then be computed as *h*_{n}(*n* − *m*) = *y*_{m}(*n*).

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)
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$$e(n) = {1 \over 2\pi} \int^\pi_{-\pi} (H_n(e^{j\omega T}) - 1) X(e^{j\omega T}) e^{jn\omega T} d(\omega T).\eqno{\hbox{(12)}}$$Ideally, *e*(*n*) = 0 and, then, *H*_{n}(*e*^{jω T}) = 1. However, in practice, *H*_{n}(*e*^{jω 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 *h*_{n}(*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*(*e*^{jω 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* (*e*^{jω T}) and, thus, one has to accept a suboptimum solution instead. In this regard, it can be convenient to make use of the *L*_{p}-norm of a general signal *S*(*e*^{jω T}) given by
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$$\Vert S(e^{j\omega T})\Vert_p = \root{p}\of {{1 \over 2\pi} \int^{\pi}_{-\pi} \vert S(e^{j\omega T})\vert^p d(\omega T)}.\eqno{\hbox{(13)}}$$Using the triangle inequality of integrals in (12), we have
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$$\vert e(n)\vert \leq {1 \over 2\pi} \int^{\pi}_{-\pi} \vert H_n(e^{j\omega T}) - 1\Vert X(e^{j\omega T})\vert d(\omega T)\eqno{\hbox{(14)}}$$which in terms of *L*_{∞}-norm can be written as
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$$\vert e(n)\vert \leq \Vert H_n(e^{j\omega T}) - 1\Vert_{\infty}\Vert X(e^{j\omega T})\Vert_{\infty}.\eqno{\hbox{(15)}}$$As
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$$\Vert S(e^{j\omega T}) - 1\Vert_{\infty} = \max_{\omega T} \{\vert S(e^{j\omega T}) - 1\vert\},\eqno{\hbox{(16)}}$$minimizing the maximum of corresponds to minimizing the maximum of the error function *e*(*n*) resulting from that particular *h*_{n}(*k*). However, every branch of the TMUX in Fig. 1 results in *B*_{p} periodic functions and, consequently, one can express the direct minimax filter design problem as

Case III: minimize δ, over all *A*_{p} and *B*_{p}, subject to
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$$\eqalignno{{1\over B_p} \sum^{B_{p-1}}_{n=0} \vert H_n(e^{j\omega T}) - 1\vert &\leq \delta, \quad \omega T \in[0, \pi] \cr\vert G_p(e^{j\omega T})\vert &\leq \delta_s, \quad \omega T\in [\omega_s T, \pi].&\hbox{(17)}}$$Similarly, (14) can also be written in terms of *L*_{2}-norm to minimize the energy of the error function leading to the direct CLS design problem as

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

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 as
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$$EVM_{rms} = \sqrt{\sum^{N_s-1}_{k=0} \vert s(k) - s_{ref} (k)\vert^2 \over \sum^{N_s-1}_{k=0} \vert s_{ref} (k)\vert^2},\eqno{\hbox{(19)}}$$where *s*(*k*) and *s*_{ref}(*k*) represent the length- *N*_{s} measured and ideal sequences, respectively.

In all the filter design problems, the free optimization parameters are the coefficients of the Farrow subfilters, i.e., *S*_{k}(*z*), and *G*_{p,0}(*z*). In other words, the values of *N*_{1}, *L*, ρ, *R*_{p} 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 *S*_{k} (*z*) and *G*_{p,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., *R*_{p}.

In this design example, the values of *L*, *N*_{1}, and δ_{s} 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 *B*_{p} in the optimization or not.

Fig. 3 shows the average EVM for 16-QAM signals according to the multimode setups^{6} 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.

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 δ_{s} = 0.01, the EVM values hover around −40 dB. In other words, ICI is controlled by δ_{s} 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 δ_{s} and δ are correlated and for the same filter orders, decreasing δ_{s} would increase δ resulting in a larger EVM. For example, with the filter orders as in Fig. 3, having δ_{s} = 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 *N*_{1}.

Irrespective of the values for δ_{s}, *L*, and *N*_{1}, 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.

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 . Otherwise, one would require to add constraints which result from as well.