So far we have introduced the structures required to transmit and receive FBMC signals and how the filter bank theory can be used to improve the efficiency of such structures. Furthermore, we have shown that different types of FBMC signals can be obtained from a generic signal model characterized by a set of four signal parameters. In this section, we introduce a unified framework that connects all these elements in the form of a systematic architecture derivation for flexible FBMC transmitters with arbitrary signal parameters. A conceptual representation of the architectures that will be presented herein is schematically depicted in Figure 7. The proposed framework allows the particularization of the different parts of the system by means of a proper design of a *polyphase network* and a *matching network*. The main purpose of the matching network is to adapt the sampling rates of the signals delivered by the IDFT block to the ones required by the filters that make up the polyphase network.

Apart from the quadruple of key signal parameters described in Section 2, FBMC architectures are determined by an additional parameter: the order of the polyphase network *B*. Different choices of *B* can be adopted at either the transmitter or the receiver end, so the subscripts _{t} and _{r} are adopted herein to indicate transmission and reception, respectively. Typically, the following values of *B* have been adopted in the literature: *B*_{
x
}={lcm(*P*_{
x
},*N*_{ss}),*L*_{
g
}} where *x*={*t*,*r*} and lcm stands for *least common multiple*. However, a more general approach suggests a wider range of possibilities. In particular, we consider the following set of values for our study:

{B}_{x}=\left\{{P}_{x},{N}_{\text{ss}},\text{lcm}({P}_{x},{N}_{\text{ss}})\right\}.

(9)

Such values represent the most significant examples from an architectural point of view. Architectures for other pairs of values (*B*_{t},*B*_{r}) can be easily derived following the steps presented in this section. That is the case of (*B*_{t},*B*_{r})=*L*_{
g
} for instance, for which the followed methodology would yield architectures based on polyphase subfilters of unitary length. It is important to mention that the matching network module in Figure 7 usually requires interpolation operations by a factor *Q*, which can be easily accommodated as long as *Q* is an integer number. The implementation for rational values of *Q* becomes challenging and it can be shown that the role of these interpolation modules would lead to time-variant input-output responses of the polyphase network when conventional methods are used [14, 33, 34]. In this work though, we show that it is possible to obtain efficient and time-invariant architectures for any value of *Q*, thus enabling a much simpler and cheaper implementation of FBMC user and network terminals. Addressing the extra complexity entailed by rational values of *Q*, which are usually avoided in practice, is one of the main contributions of this paper.

### 4.1 Efficient transmitter architectures for integer values of Q

First of all, we introduce the transmitter architectures obtained for integer values of *Q*, which are conceptually simpler in terms of implementation. The advantages of polyphase structures become apparent in this case since it is possible to obtain a polyphase network of order *P*_{t} (i.e., the very same number of transmitted subcarriers) that minimizes the required amount of hardware. For an arbitrary quadruple of design parameters, we can express the reference transmission signal model (2) as follows:

x\left[m\right]=\sum _{l=-\infty}^{\infty}\sum _{n=0}^{N-1}{s}_{n}\left[l\right]\phantom{\rule{0.3em}{0ex}}g[m-{\mathit{\text{lQP}}}_{\mathrm{t}}]\phantom{\rule{0.3em}{0ex}}{e}^{j2\mathrm{\pi n}\frac{m}{{P}_{\mathrm{t}}}}.

(10)

At this point, it can be observed in (10) that a *P*_{t}-point IDFT operation over the source symbols appears naturally. We are assuming here that typically *P*_{t≥}*N* and we proceed to arrange the source symbols in an (*N*×1) vector: \mathbf{\text{s}}\left[l\right]\doteq \phantom{\rule{1em}{0ex}}{\left[{s}_{0}\left[l\right],{s}_{1}\left[l\right],\dots ,{s}_{N-1}\left[l\right]\right]}^{\text{T}}, where the superscript ^{T} denotes the transpose operator. Likewise, we define the following notation for the IDFT operation: {S}_{\text{mod}(m,{P}_{\mathrm{t}})}^{{P}_{\mathrm{t}}}\left[l\right]\doteq \phantom{\rule{1em}{0ex}}{\text{IDFT}}_{m,{P}_{\mathrm{t}}}\left(\mathbf{\text{s}}\right[l\left]\right)={\sum}_{n=0}^{N-1}{s}_{n}\left[l\right]{e}^{j2\mathrm{\pi n}\frac{m}{{P}_{\mathrm{t}}}}, which leads to the following compact expression of the signal model:

x\left[m\right]=\sum _{l=-\infty}^{\infty}{S}_{\text{mod}(m,{P}_{\mathrm{t}})}^{{P}_{\mathrm{t}}}\left[l\right]g\left[m-l{\mathit{\text{QP}}}_{\mathrm{t}}\right].

(11)

The above model, which extensively relies on the use of IDFT, was originally introduced by [51] and has been considered one of the catalysts in the success and widespread deployment of MC systems due to its efficient implementation through FFT processors. In that sense, one of the interesting features of polyphase structures is the exploitation of the cyclic nature of the IDFT/DFT exponentials, which leads us to introduce the modulo operation in the generated signal sample index in (11). Since any integer *m* can always be expressed as m=\text{mod}(m,{P}_{\mathrm{t}})+\lfloor \frac{m}{{P}_{\mathrm{t}}}\rfloor {P}_{\mathrm{t}}, we can rewrite (11) as follows:

\begin{array}{l}x\left[m\right]=\sum _{l=-\infty}^{\infty}{S}_{\text{mod}(m,{P}_{\mathrm{t}})}^{{P}_{\mathrm{t}}}\left[l\right]\phantom{\rule{0.3em}{0ex}}g\left[\text{mod}(m,{P}_{\mathrm{t}})\right.\\ \phantom{\rule{4em}{0ex}}+\left(\right)close="]">\left(\u230a\frac{m}{{P}_{\mathrm{t}}}\u230b-\mathit{\text{lQ}}\right){P}_{\mathrm{t}}& .\end{array}\n

(12)

Given that mod(*m*,*P*_{t}) takes the values {0,1,…,*P*_{t}−1}, we can consider that the signal in (12) implies a total of *P*_{t} different discrete-time convolutions, one associated to each value of mod(*m*,*P*_{t}). In terms of a polyphase decomposition, each of those convolutions will be associated to a different subfilter and consequently, to a different row in the polyphase network of Figure 7. Hence, we can regard the term mod(*m*,*P*_{t}) as a branch (or row) index that identifies the specific subfilter involved in the generation of the *m* th sample. Moreover, since each subfilter operates at a sampling rate *P*_{t} times lower than the serial signal *x*[*m*], we ought to apply a subfilter decimation by a factor of *P*_{t} over the prototype filter *g*[*m*]. We will henceforth make use of the notation introduced in (5) to rewrite (12) so that it explicitly reflects the mentioned manipulations:

\begin{array}{l}\phantom{\rule{-13.0pt}{0ex}}x\left[m\right]=\phantom{\rule{0.3em}{0ex}}\sum _{l=-\infty}^{\infty}{S}_{\text{mod}(m,{P}_{\mathrm{t}})}^{{P}_{\mathrm{t}}}\left[l\right]{g}_{\text{mod}(m,{P}_{\mathrm{t}})}^{\left({P}_{\mathrm{t}}\right)}\left[\u230a\frac{m}{{P}_{\mathrm{t}}}\u230b-\mathit{\text{lQ}}\right]\\ \phantom{\rule{2em}{0ex}}={\mathcal{I}}_{Q}\left\{{\text{IDFT}}_{(m,{P}_{\mathrm{t}})}\right(\mathbf{\text{s}}\left[\phantom{\rule{0.3em}{0ex}}k\right]\left)\right\}\ast \phantom{\rule{0.3em}{0ex}}{g}_{\text{mod}(m,{P}_{\mathrm{t}})}^{\left({P}_{\mathrm{t}}\right)}\left[\phantom{\rule{0.3em}{0ex}}k\right]\phantom{\rule{0.3em}{0ex}}\left|{}_{k=\lfloor \frac{m}{{P}_{\mathrm{t}}}\rfloor}\right.\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}},\end{array}

(13)

where {g}_{\text{mod}(m,{P}_{\mathrm{t}})}^{\left({P}_{\mathrm{t}}\right)}\left[k\right] denotes the polyphase subfilter resulting from a {P}_{\mathrm{t}}^{\text{th}}-order decimation of the prototype filter *g*[*m*] with an offset of mod(*m*,*P*_{t}) samples:

{g}_{\text{mod}(m,{P}_{\mathrm{t}})}^{\left({P}_{\mathrm{t}}\right)}\left[k\right]\doteq \phantom{\rule{1em}{0ex}}g[{\mathit{\text{kP}}}_{\mathrm{t}}+\text{mod}(m,{P}_{\mathrm{t}}\left)\right].

(14)

The advantage of (13) is that it clearly outlines the series of operations that needs to be carried out for generating *x*[*m*] in an efficient manner. In particular, *x*[*m*] is the result of the convolutions of each IDFT output {S}_{\text{mod}(m,{P}_{\mathrm{t}})}^{{P}_{\mathrm{t}}}\left[k\right] (upsampled by *Q*) with a downsampled version of the prototype filter {g}_{\text{mod}(m,{P}_{\mathrm{t}})}^{\left({P}_{\mathrm{t}}\right)}\left[k\right] followed by an upsampling operation by *P*_{t}. Therefore, there is a correspondence among the subfilters indexes defined in (14), the sample index of *x*[*m*], and the phase index on the IDFT output, thus leading to a rather intuitive architecture as depicted in Figure 8.

Note that the case of minimum frequency separation (*Q*=1) leads to the simplest possible polyphase architecture, where no upsampling operation would be required prior to the subfiltering operations. This example might correspond to the case of an OFDM modulation where no cyclic prefix has been added.

### 4.2 Efficient transmitter architectures for non-integer values of *Q*

We now move one step further by considering the more general case of any rational value of *Q* (i.e., when the subcarrier period and the MC symbol period do not share any common link). In that case, it follows from (13) that rational upsampling operations would be required prior to the subfilter convolution, thus complicating the design of time-invariant structures. The main implementation obstacle here is set by the rate imbalance between the symbol rate and the polyphase network output rate, which is given by the order of the polyphase network. In particular, a {P}_{\mathrm{t}}^{\text{th}}-order polyphase transmitter network generates blocks of *P*_{t} samples at its output (one for each subfilter). However, the number of samples per symbol generated should be *N*_{ss} in order to meet the desired output rate of the digital communication signal being transmitted. In other words, if *Q* is not an integer, the symbol period in samples, *N*_{ss}, does not account for an integer number of periods of the fundamental subcarrier frequency, thus making it hard to exploit the cyclic nature of the IDFT. Furthermore, since the duration of the symbol in samples is not a multiple of the order of the polyphase network, it would be necessary to apply a different set of filter coefficients to every symbol delivered by the IDFT block.

For these reasons, the implementation issues of this type of MC signals have been ignored in the literature or solved by means of time-variant schemes [33, 34]. In spite of these obstacles, we show in this work that if the polyphase and matching networks are properly designed, it is certainly possible to obtain a time-invariant structure for any rational value of *Q*. This clearly provides significant advantages, enabling complete freedom in the choice of the MC signal parameters that best suit the requirements of the application under consideration.

Regarding the architectures to be presented next, it should be noted that they are essentially equivalent in the extent that they generate the same signal, while merely differing in the layout of the polyphase and matching networks. The flexibility of the framework provided in this work is clearly highlighted by this fact, since any of these schemes can be used indistinctly depending on the specific constraints of the application of interest. Hereunder we present a derivation of such structures for the proposed polyphase orders *B*_{t}={*P*_{t},*N*_{ss},lcm(*P*_{t},*N*_{ss})}. This set of values of *B*_{t} will let us show the necessary steps required to derive any other architecture.

#### 4.2.1 Order of the polyphase network *B*_{t}=*P*_{t}

For the sake of clarity, let us express the index of the convolution in (2) as a function of two subindexes: *l*=*l*_{
b
}*P*_{t}+*l*_{
r
}, being {l}_{b}\phantom{\rule{1em}{0ex}}\doteq \phantom{\rule{1em}{0ex}}\lfloor \frac{l}{{P}_{\mathrm{t}}}\rfloor and {l}_{r}\phantom{\rule{1em}{0ex}}\doteq \phantom{\rule{1em}{0ex}}\text{mod}(l,{P}_{\mathrm{t}}). This decomposition is motivated by the order of the polyphase network *P*_{t} and allows us to introduce the term mod(*l*,*P*_{t}), which will serve as a row (or branch) index in the resulting polyphase network. Besides, there is a multiple-of- *P*_{t} term that acts as a sample index of the convolution operation for each subfilter. Then, we can rewrite (2) according to the notation introduced in (11) as follows:

\begin{array}{l}x\left[m\right]=\sum _{{l}_{r}=0}^{{P}_{\mathrm{t}}-1}\sum _{{l}_{b}=-\infty}^{\infty}{S}_{\text{mod}(m,{P}_{\mathrm{t}})}^{{P}_{\mathrm{t}}}[{l}_{b}{P}_{\mathrm{t}}+{l}_{r}]g[m-{l}_{r}{N}_{\text{ss}}\\ \phantom{\rule{4em}{0ex}}-{l}_{b}{P}_{\mathrm{t}}{N}_{\text{ss}}].\end{array}

(15)

Additionally, we can further decompose the term *m*−*l*_{
r
}*N*_{ss} according to the dual indexing that we will be permanently seeking throughout this paper, which consists in expressing the sample index as the sum of a multiple-of- *B*_{t} term plus a modulus-of- *B*_{t} residual. That leads us to

m-{l}_{r}{N}_{\text{ss}}=\u230a\frac{m-{l}_{r}{N}_{\text{ss}}}{{P}_{\mathrm{t}}}\u230b{P}_{\mathrm{t}}+\text{mod}(m-{l}_{r}{N}_{\text{ss}},{P}_{\mathrm{t}}).

(16)

Therefore, we can rewrite *x*[*m*] applying the notation in (13) to reflect the {P}_{\mathrm{t}}^{\mathit{\text{th}}}-order subfilter decimation:

\phantom{\rule{-15.0pt}{0ex}}\begin{array}{l}x\left[m\right]=\sum _{{l}_{r}=0}^{{P}_{\mathrm{t}}-1}\sum _{{l}_{b}=-\infty}^{\infty}{S}_{\text{mod}(m,{P}_{\mathrm{t}})}^{{P}_{\mathrm{t}}}[{l}_{b}{P}_{\mathrm{t}}+{l}_{r}]\phantom{\rule{0.3em}{0ex}}{g}_{\text{mod}(m-{l}_{r}{N}_{\text{ss}},{P}_{\mathrm{t}})}^{\left({P}_{\mathrm{t}}\right)}\\ \phantom{\rule{4em}{0ex}}\times \left[\u230a\frac{m-{l}_{r}{N}_{\text{ss}}}{{P}_{\mathrm{t}}}\u230b-{l}_{b}{N}_{\text{ss}}\right]\\ \phantom{\rule{2em}{0ex}}=\sum _{{l}_{r}=0}^{{P}_{\mathrm{t}}-1}{\mathcal{I}}_{{N}_{\text{ss}}}\left\{{\mathcal{D}}_{{P}_{\mathrm{t}}}\right\{{S}_{\text{mod}(m,{P}_{\mathrm{t}})}^{{P}_{\mathrm{t}}}[k+{l}_{r}]\left\}\right\}\ast {g}_{\text{mod}(m-{l}_{r}{N}_{\text{ss}},{P}_{\mathrm{t}})}^{\left({P}_{\mathrm{t}}\right)}\left[k\right]\\ \phantom{\rule{4em}{0ex}}\times \left|{}_{k=\u230a\frac{m-{l}_{r}{N}_{\text{ss}}}{{P}_{\mathrm{t}}}\u230b}\right..\end{array}

(17)

A careful analysis of (17) reveals some similarities with the transmitted signal expression in the case of integer *Q* shown in (13). In this case though, there appears an additional delay term of *l*_{
r
}*N*_{ss} samples that affects each subfilter output as well as the subfilter indexes. Therefore, it is not possible to generate the transmit signal *x*[*m*] with a single {P}_{\mathrm{t}}^{\text{th}}-order polyphase structure like the one shown in Figure 8. However, it is actually possible to consider separately the architecture defined by each value of *l*_{
r
} and deal with them as different parts of a bigger structure. These parts are actually polyphase networks of order *P*_{t} themselves that we will refer to as *subnetworks*. Therefore, the resulting scheme employs a total of *P*_{t} polyphase subnetworks of order *P*_{t}.

Moreover, the IDFT output must be downsampled by *P*_{t} and it is also subject to a variable sampling offset of *l*_{
r
} samples that is constant for each subnetwork. Therefore, the samples delivered by the IDFT will be processed separately by different subnetworks within the entire polyphase network. This fact is reflected in the architecture through what we call a block-wise serial-to-parallel converter of order *P*_{t}. This module vertically concatenates *P*_{t} blocks of *P*_{t} samples as they are sequentially output by the IDFT. In addition, it should be noticed that in (17), the index of the polyphase subfilters mod(*m*−*l*_{
r
}*N*_{ss},*P*_{t}) and the index of the IDFT output mod(*m*,*P*_{t}) will not coincide, as opposed to what happened in the case of integer *Q*. Then, in order to achieve a proper matching between the IDFT output and the polyphase network rows, it is necessary to compensate the unbalance of *l*_{
r
}*N*_{ss} samples between the subscript terms in (17). One possible way to do it is through the introduction of a phase rotation over the input source symbols *s*_{
n
}[*l*]. Such a rotation will take place at the input of the IDFT and will produce a delay of the same amount of samples at its output. With this slight modification and by virtue of the Fourier transform properties, we are able to compensate the mentioned unbalance and we also make sure we are not altering the generated signal. Hence, let us define {\stackrel{~}{s}}_{n}\left[l\right]\doteq \phantom{\rule{1em}{0ex}}{s}_{n}\left[l\right]{e}^{j2\mathrm{\pi n}\frac{{l}_{r}{N}_{\text{ss}}}{{P}_{\mathrm{t}}}}={s}_{n}\left[l\right]{e}^{j2\mathrm{\pi n}{l}_{r}Q} so that we obtain

\begin{array}{lcr}{\text{IDFT}}_{m-{l}_{r}{N}_{\text{ss}},{P}_{\mathrm{t}}}\left(\stackrel{~}{\mathbf{\text{s}}}\right[l\left]\right)& =& \sum _{n=0}^{N-1}{\stackrel{~}{s}}_{n}\left[l\right]{e}^{\phantom{\rule{0.3em}{0ex}}j2\mathrm{\pi n}\frac{m-{l}_{r}{N}_{\text{ss}}}{{P}_{\mathrm{t}}}}\end{array}

(18)

\begin{array}{lc}=& \sum _{n=0}^{N-1}{s}_{n}\left[l\right]{e}^{\phantom{\rule{0.3em}{0ex}}j2\mathrm{\pi n}\frac{{l}_{r}{N}_{\text{ss}}}{{P}_{\mathrm{t}}}}{e}^{\phantom{\rule{0.3em}{0ex}}j2\mathrm{\pi n}\frac{m-{l}_{r}{N}_{\text{ss}}}{{P}_{\mathrm{t}}}}\end{array}

(19)

\begin{array}{lc}=& {\text{IDFT}}_{m,{P}_{\mathrm{t}}}\left(\mathbf{\text{s}}\right[l\left]\right).\end{array}

(20)

Finally, we are left with the following expression for the transmit signal:

\begin{array}{l}x\left[m\right]=\sum _{{l}_{r}=0}^{{P}_{\mathrm{t}}-1}{\mathcal{I}}_{{N}_{\text{ss}}}\left\{{\mathcal{D}}_{{P}_{\mathrm{t}}}\left\{{\text{IDFT}}_{m-{l}_{r}{N}_{\text{ss}},{P}_{\mathrm{t}}}\left(\stackrel{~}{\mathbf{\text{s}}}\right[k+{l}_{r}\left]\right)\right\}\right\}\\ \phantom{\rule{4em}{0ex}}\ast {g}_{\text{mod}(m-{l}_{r}{N}_{\text{ss}},{P}_{\mathrm{t}})}^{\left({P}_{\mathrm{t}}\right)}\left[k\right]\left|{}_{k=\u230a\frac{m-{l}_{r}{N}_{\text{ss}}}{{P}_{\mathrm{t}}}\u230b}\right..\end{array}

(21)

The final transmitter architecture shown in Figure 9 follows directly from (21). It is worth to observe that the phase rotation over the source symbols remains constant within each subnetwork because it is a function of the subnetwork index *l*_{
r
}. Besides, according to the properties of the convolution, the delay of *l*_{
r
}*N*_{ss} samples in (21) has been readily moved to the subfilter outputs with no loss of generality.

#### 4.2.2 Order of the polyphase network *B*_{t}=*N*_{ss}

The approach adopted in the previous case would lead to time-variant architectures for the present case of *B*_{t}=*N*_{ss}. Since time-varying schemes is indeed what we intend to avoid in this work, a slightly different approach is required herein. To do so, let us decompose the output index of the convolution *m* into both a multiple-of- *N*_{ss} term (*m*_{
b
}) and a modulus-of- *N*_{ss} residual (*m*_{
r
}) according to the desired polyphase order. Therefore,

m={m}_{b}{N}_{\text{ss}}+{m}_{r}=\u230a\frac{m}{{N}_{\text{ss}}}\u230b{N}_{\text{ss}}+\text{mod}(m,{N}_{\text{ss}}).

(22)

This decomposition by itself does not lead to the derivation of an efficient architecture, so we need to apply a further decomposition of the index *m*_{
b
} as follows:

{m}_{b}={m}_{b1}{P}_{\text{to}}+{m}_{b2},

(23)

where {m}_{b1}\doteq \phantom{\rule{1em}{0ex}}\lfloor \frac{{m}_{b}}{{P}_{\text{to}}}\rfloor and {m}_{b2}\phantom{\rule{1pt}{0ex}}\doteq \phantom{\rule{1pt}{0ex}}\text{mod}({m}_{b},{P}_{\text{to}}). We have also assumed that {\text{lcm(P}}_{\mathrm{t}},{N}_{\text{ss}}\text{)}\doteq \phantom{\rule{1pt}{0ex}}{P}_{\text{to}}{N}_{\text{ss}}, being *P*_{to} an integer number as well. Replacing (22) and (23) in (11) we are left with:

\begin{array}{lcr}x\left[m\right]& =& x[{m}_{b1},{m}_{b2},{m}_{r}]\end{array}

(24)

\begin{array}{l}=\sum _{l=-\infty}^{\infty}{S}_{\text{mod}({m}_{b2}{N}_{\text{ss}}+{m}_{r},{P}_{\mathrm{t}})}^{{P}_{\mathrm{t}}}\left[l\right]\phantom{\rule{0.3em}{0ex}}{g}_{\text{mod}({m}_{r},{N}_{\text{ss}})}^{\left({N}_{\text{ss}}\right)}[{m}_{b1}{P}_{\text{to}}\\ \phantom{\rule{1.6em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{m}_{b2}-l].\end{array}

(25)

Note that we have applied a subfilter decimation by *N*_{ss} in order to obtain an {N}_{\text{ss}}^{\text{th}}-order polyphase structure. The associated subfilters are defined as {g}_{\text{mod}({m}_{r},{N}_{\text{ss}})}^{\left({N}_{\text{ss}}\right)}\left[k\right]\doteq \phantom{\rule{1pt}{0ex}}g[{\mathit{\text{kN}}}_{\text{ss}}+\text{mod}({m}_{r},{N}_{\text{ss}}\left)\right].

It is important to highlight that the order of the polyphase network *N*_{ss} is higher than the duration (in samples) of the subcarrier fundamental period *P*_{t}. That means that the number of polyphase rows is larger than the length of the IDFT output in the architecture. This asymmetry can be easily compensated by extending the length of the IDFT output to match the order of the polyphase network. In particular, we propose a solution based on the addition of the initial part of the symbol at the end of the first *P*_{t} samples, creating a *cyclic extension* of the IDFT output. These *N*_{ss}−*P*_{t} extra samples can be seen as a cyclic prefix appended to the actual symbol that otherwise would have a duration of *P*_{t} samples (e.g., as if no redundancy was introduced). Indeed, there is a degree of freedom from a design point of view to fill up these samples at the last part of the symbol. Note that this clarification was not necessary in the previous case (subsection 4.2.1), since the length of the IDFT output and the order of the polyphase network coincided. Finally, it has to be considered that the values adopted here for the samples in the final part of the symbol are not unique. Other solutions like zero-padding or pilot signaling would be also valid and would not have any meaningful impact on the obtained architectures.

Additionally, due to the imbalance between the order of the polyphase network and the size of the IDFT, the phase continuity over time of the different subcarriers in (2) cannot be ensured with a single {N}_{\text{ss}}^{\text{th}}-order structure. This fact is highlighted in (25) where the subscripts of the IDFT output, mod(*m*_{b 2}*N*_{ss}+*m*_{
r
},*P*_{t}), and the prototype filter, mod(*m*_{
r
},*N*_{ss}), do not match. Therefore, it is convenient to resort once more to a phase rotation over the input source symbols to ensure the signal phase continuity at every symbol transition. Let us then define the following equivalent IDFT output:

\begin{array}{lcr}{\stackrel{~}{S}}_{\text{mod}({m}_{r}-{m}_{b2}{N}_{\text{ss}},{P}_{\mathrm{t}})}^{{P}_{\mathrm{t}}}\left[l\right]& \doteq & {\text{IDFT}}_{{m}_{r}-{m}_{b2}{N}_{\text{ss}},{P}_{\mathrm{t}}}\left(\stackrel{~}{\mathbf{\text{s}}}\right[l\left]\right)\end{array}

(26)

\begin{array}{lc}=& \sum _{n=0}^{N-1}{\stackrel{~}{s}}_{n}\left[l\right]{e}^{j2\mathrm{\pi n}\frac{{m}_{r}-{m}_{b2}{N}_{\text{ss}}}{{P}_{\mathrm{t}}}}\end{array}

(27)

\begin{array}{l}=\sum _{n=0}^{N-1}{s}_{n}\left[l\right]{e}^{j2\mathrm{\pi n}\frac{{m}_{b2}{N}_{\text{ss}}}{{P}_{\mathrm{t}}}}{e}^{j2\mathrm{\pi n}\frac{{m}_{r}-{m}_{b2}{N}_{\text{ss}}}{{P}_{\mathrm{t}}}}\\ ={\text{IDFT}}_{{m}_{r},{P}_{\mathrm{t}}}\left(\mathbf{\text{s}}\right[l\left]\right),\end{array}

(28)

where {\stackrel{~}{s}}_{n}\left[\phantom{\rule{0.3em}{0ex}}l\right]\doteq \phantom{\rule{1em}{0ex}}{s}_{n}\left[\phantom{\rule{0.3em}{0ex}}l\right]{e}^{j2\mathrm{\pi n}\frac{{m}_{b2}{N}_{\text{ss}}}{{P}_{\mathrm{t}}}}={s}_{n}\left[l\right]{e}^{j2\mathrm{\pi n}\text{mod}(\u230am/{N}_{\text{ss}}\u230b,{P}_{\text{to}})Q} represents a phase-rotated version of the source symbols. Using the results of (28) in (25) and expressing *x*[*m*] as a function of the output sample index *m*, we obtain

\phantom{\rule{-13.0pt}{0ex}}\begin{array}{l}x\left[m\right]=\sum _{l=-\infty}^{\infty}{\stackrel{~}{S}}_{\text{mod}\left(\text{mod}\right(m,{N}_{\text{ss}}),{P}_{\mathrm{t}})}^{{P}_{\mathrm{t}}}\left[l\right]\phantom{\rule{0.3em}{0ex}}{g}_{\text{mod}(m,{N}_{\text{ss}})}^{\left({N}_{\text{ss}}\right)}\\ \phantom{\rule{4em}{0ex}}\times \left[\u230a\frac{\lfloor m/{N}_{\text{ss}}\rfloor}{{P}_{\text{to}}}\u230b{P}_{\text{to}}+\text{mod}\left(\u230a\frac{m}{{N}_{\text{ss}}}\u230b,{P}_{\text{to}}\right)-l\right].\end{array}

(29)

Now the indexes of the IDFT output and the subfilter coincide, although the range of variation of the IDFT indexes is restricted to *P*_{t}, which is precisely the motivation for the cyclic extension. The expression of the transmitted signal is

\begin{array}{l}x\left[m\right]=\phantom{\rule{0.3em}{0ex}}{\mathcal{I}}_{{P}_{\text{to}}}\left\{{\mathcal{D}}_{{P}_{\text{to}}}\left\{{\text{IDFT}}_{\text{mod}(m,{N}_{\text{ss}}),{P}_{\mathrm{t}}}\left(\stackrel{~}{\mathbf{\text{s}}}\right[k\left]\right)\ast {g}_{\text{mod}(m,{N}_{\text{ss}})}^{\left({N}_{\text{ss}}\right)}\right.\right.\\ \phantom{\rule{4em}{0ex}}\left(\right)close="\}">\left(\right)close="\}">\times \left[k+\text{mod}\left(\u230a\frac{m}{{N}_{\text{ss}}}\u230b,{P}_{\text{to}}\right)\right]\\ \left|{}_{k=\u230a\frac{m}{{N}_{\text{ss}}}\u230b}\right..\end{array}\n

(30)

Finally, the resulting architecture can be built upon *P*_{to} polyphase subnetworks of order *N*_{ss} as it is illustrated in Figure 10.

#### 4.2.3 Order of the polyphase network *B*_{t}=lcm(*P*_{t},*N*_{ss})

Let us rewrite the polyphase order *B*_{t} as *P*_{to}*N*_{ss}=*N*_{sso}*P*_{t}=lcm(*P*_{t},*N*_{ss}), where both *P*_{to} and *N*_{sso} are integer numbers. Given that the order of the polyphase network is a multiple of the subcarrier period in samples *P*_{t}, we can proceed in this case as we did in subsection 4.2.1. Hence, we can conveniently decompose the convolution index as *l*=*l*_{
b
}*P*_{to}+*l*_{r} with {l}_{b}\doteq \phantom{\rule{1em}{0ex}}\lfloor \frac{l}{{P}_{\text{to}}}\rfloor and {l}_{r}\doteq \phantom{\rule{1em}{0ex}}\text{mod}(l,{P}_{\text{to}}). Replacing (11) we obtain

\begin{array}{l}x\left[m\right]=\sum _{{l}_{r}=0}^{{P}_{\text{to}}-1}\sum _{{l}_{b}=-\infty}^{\infty}{S}_{\text{mod}(m,{P}_{\mathrm{t}})}^{{P}_{\mathrm{t}}}[{l}_{b}{P}_{\text{to}}+{l}_{r}]g[m-{l}_{r}{N}_{\text{ss}}\\ \phantom{\rule{4em}{0ex}}-{l}_{b}{P}_{\text{to}}{N}_{\text{ss}}].\end{array}

(31)

As it was done in (16), we work with a decomposition of the term *m*−*l*_{
r
}*N*_{ss} according to the desired polyphase structure order *P*_{to}*N*_{ss}:

\begin{array}{l}m-{l}_{r}{N}_{\text{ss}}=\u230a\frac{m-{l}_{r}{N}_{\text{ss}}}{{P}_{\text{to}}{N}_{\text{ss}}}\u230b{P}_{\text{to}}{N}_{\text{ss}}\\ \phantom{\rule{6em}{0ex}}+\text{mod}(m-{l}_{r}{N}_{\text{ss}},{P}_{\text{to}}{N}_{\text{ss}}).\end{array}

(32)

Then replacing (32) in (31) and applying a {P}_{\text{to}}{N}_{\text{ss}}^{\text{th}}-order subfilter decimation, we obtain:

\phantom{\rule{-15.0pt}{0ex}}\begin{array}{l}x\left[m\right]\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\sum _{{l}_{r}=0}^{{P}_{\text{to}}-1}\sum _{{l}_{b}=-\infty}^{\infty}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{S}_{{\text{mod(m,P}}_{\mathrm{t}}\text{)}}^{{P}_{\mathrm{t}}}[{l}_{b}{P}_{\text{to}}+{l}_{r}]\phantom{\rule{0.3em}{0ex}}{g}_{\text{mod}(m-{l}_{r}{N}_{\text{ss}},{P}_{\text{to}}{N}_{\text{ss}})}^{\left(\text{lcm}\right)}\\ \phantom{\rule{5em}{0ex}}\left[\u230a\frac{m-{l}_{r}{N}_{\text{ss}}}{{P}_{\text{to}}{N}_{\text{ss}}}\u230b-{l}_{b}\right],\end{array}

(33)

where {g}_{\text{mod}({m}_{r},{P}_{\text{to}}{N}_{\text{ss}})}^{\left(\text{lcm}\right)}\left[k\right]\doteq \phantom{\rule{1em}{0ex}}g[{\mathit{\text{kP}}}_{\text{to}}{N}_{\text{ss}}+\text{mod}({m}_{r},{P}_{\text{to}}{N}_{\text{ss}}\left)\right]. We can see that there appears again a shift of *l*_{
r
}*N*_{ss} samples in (33) at the subscripts of the prototype filter with respect to the IDFT output subscript. Following an analogous reasoning to subsection 4.2.1), we can write

\begin{array}{l}x\left[m\right]=\sum _{{l}_{r}=0}^{{P}_{\text{to}}-1}\sum _{{l}_{b}=-\infty}^{\infty}{\stackrel{~}{S}}_{\text{mod}(m-{l}_{r}{N}_{\text{ss}},{P}_{t})}^{{P}_{\mathrm{t}}}[{l}_{b}{P}_{\text{to}}+{l}_{r}]\ast \phantom{\rule{2em}{0ex}}\\ \phantom{\rule{3.5em}{0ex}}\times {g}_{\text{mod}(m-{l}_{r}{N}_{\text{ss}},{P}_{\text{to}}{N}_{\text{ss}})}^{\left(\text{lcm}\right)}\left[\u230a\frac{m-{l}_{r}{N}_{\text{ss}}}{{P}_{\text{to}}{N}_{\text{ss}}}\u230b-{l}_{b}\right]\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{2em}{0ex}}=\sum _{{l}_{r}=0}^{{P}_{\text{to}}-1}{\mathcal{D}}_{{P}_{\text{to}}}\left\{{\text{IDFT}}_{m-{l}_{r}{N}_{\text{ss}},{P}_{\mathrm{t}}}\right(\stackrel{~}{\mathbf{\text{s}}}[k+{l}_{r}]\left)\right\}\ast \phantom{\rule{2em}{0ex}}\\ \phantom{\rule{3em}{0ex}}\times {g}_{\text{mod}(m-{l}_{r}{N}_{\text{ss}},{P}_{\text{to}}{N}_{\text{ss}})}^{\left(\text{lcm}\right)}\left[k\right]\left|{}_{k=\u230a\frac{m-{l}_{r}{N}_{\text{ss}}}{{P}_{\text{to}}{N}_{\text{ss}}}\u230b}\right..\phantom{\rule{2em}{0ex}}\end{array}

(34)

Note that we have applied the same phase rotation over the source symbols as in (18). Besides, the block-wise serial-to-parallel converter now concatenates *P*_{to} blocks of size lcm(*P*_{t},*N*_{ss}) as it is shown in the resulting transmitter architecture depicted in Figure 11.

Analogously to the previous cases, the final architecture is made up of several polyphase subnetworks of order lcm(*P*_{t},*N*_{ss}), where the subindex *l*_{
r
} can be seen as a subnetwork index for a total of *P*_{to} identical polyphase structures of order *P*_{to}*N*_{ss}. Note that although the ranges of variation of the subscripts in (34) do not coincide, the order of the polyphase networks is an integer number of fundamental carrier cycles. Hence, there is no need to include further phase corrections inside each network block. In other words, intra-block phase continuity is guaranteed by the design of the polyphase layout, whereas inter-block phase continuity is easily achieved by the above-mentioned phase rotation over the source symbols.

To conclude this section, it is important to highlight that we have presented a set of time-invariant FBMC transmitter architectures together with the necessary steps for their derivation starting from the unified signal model introduced in Section 2. These architectures are computationally efficient since they are based on polyphase decompositions of the prototype filter. Additionally, they allow us to implement FBMC transmitters for any configuration of signal parameters (i.e., for arbitrary subcarrier period *P*, symbol period *N*_{ss}, pulse shape length *L*_{
g
}, and normalized subcarrier spacing *Q*) just by using simple digital signal processing blocks such as up/down-sampling converters, filters, and sample delays. As already mentioned, no complicated circular shifts, temporary buffers, or memory swapping operations are required, which means a considerable simplification of those FBMC implementations where *Q* is rational, which have been commonly ignored by the research community.