Cyclic block filtered multitone modulation is a multicarrier modulation scheme. As such, a high data rate information sequence is split into a series of *K* low data rate sequences. We denote the *k*-th data sequence with {a}^{\left(k\right)}\left(\mathrm{\ell N}\right),\phantom{\rule{0.3em}{0ex}}\ell \in \mathbb{Z}, which corresponds to a stream of complex data symbols belonging to a certain constellation, e.g., *M*-QAM or *M*-PSK, transmitted with symbol period *NT*, where *T* is the sampling period in the system. A normalized sampling period is assumed, i.e., *T*=1. The data sequences are transmitted in parallel sub-channels obtained by partitioning the wideband transmission medium in *K* sub-bands.

The principle of CB-FMT is depicted in Figure 1. In this scheme, the low data rate data sequences are interpolated by a factor *N* and, then, filtered with a prototype pulse that is identical for all sub-channels. Differently from conventional FMT, the convolutions in the filter bank are circular. The filter outputs are multiplied by a complex exponential to obtain a spectrum translation. Finally, the *K* modulated signals are summed together yielding the transmitted discrete time signal.

The circular convolution involves periodic signals, and it can be efficiently realized in the frequency domain via the discrete Fourier transform (DFT). To use the circular convolution, a blockwise transmission is needed. Thus, we gather the low data rate sequences in blocks of *L* symbols *a*^{(k)}(*ℓ* *N*), *ℓ* ∈ {0,…,*L* − 1}, for each sub-channel. Then, we consider the prototype pulse *g*(*n*) to be a causal finite impulse response (FIR) filter, with a number of coefficients equal to *M* = *L* *N*. If the length is lower than *M*, we can extend the pulse length to *M* with zero-padding, without loss of generality. The CB-FMT transmitted signal can be written as

\begin{array}{ll}\phantom{\rule{.8em}{0ex}}x\left(n\right)\hfill & =\sum _{k=0}^{K-1}\left[{a}^{\left(k\right)}\otimes g\right]\left(n\right)\phantom{\rule{2em}{0ex}}\hfill \\ =\sum _{k=0}^{K-1}\sum _{\ell =0}^{L-1}{a}^{\left(k\right)}\left(\mathrm{\ell N}\right)g\left({(n-\mathrm{\ell N})}_{M}\right){W}_{K}^{-\mathit{\text{nk}}},\phantom{\rule{2em}{0ex}}\hfill \\ \phantom{\rule{2em}{0ex}}n\hfill & \in \{0,\dots ,M-1\},\phantom{\rule{2em}{0ex}}\hfill \end{array}

(1)

where ⊗ denotes the circular convolution operator and *g*((*n*)_{
M
}) denotes the cyclic (periodic) repetition of the prototype pulse *g*(*n*) with a period equal to *M*, i.e., *g*((*n*)_{
M
}) = *g*(mod(*n*,*M*)) where mod(·,·) is the integer modulo operator. {W}_{K}^{-\mathit{\text{nk}}}={e}^{j2\mathrm{\pi nk}/K} is the complex exponential function and *j* is the imaginary unit.

The signal *x*(*n*) is digital-to-analog converted and, then, transmitted over the transmission medium. At the receiver, after analog-to-digital conversion, the discrete time received signal, denoted with *y*(*ℓ*), is defined as

y\left(\ell \right)=\sum _{s=0}^{P-1}{h}_{\text{CH}}(s,\ell )x(\ell -s)+\eta \left(\ell \right),\phantom{\rule{1em}{0ex}}\ell \in \mathbb{Z},

(2)

where *h*_{CH}(*s*,*ℓ*) is the time-variant channel impulse response and *η*(*ℓ*) is the background noise. In the following, we assume the channel response to be ideal. The more general case will be discussed in Section 5.

Similarly to the synthesis stage, we can apply the circular convolution to the analysis filter bank. The *k*-th sub-channel output is obtained as follows:

\begin{array}{ll}{z}^{\left(k\right)}\left(\mathit{\text{nN}}\right)\hfill & =\sum _{\ell =0}^{M-1}y\left(\ell \right){W}_{K}^{\mathrm{\ell k}}h\left({(\mathit{\text{nN}}-\ell )}_{M}\right),\phantom{\rule{2em}{0ex}}\hfill \\ \phantom{\rule{4em}{0ex}}k\hfill & \in \{0,\dots ,K-1\},\phantom{\rule{1em}{0ex}}n\in \{0,\dots ,L-1\},\phantom{\rule{2em}{0ex}}\hfill \end{array}

(3)

where *h*((*n*)_{
M
}) denotes the periodic repetition of the prototype analysis pulse *h*(*n*) with period *M*.

Each sub-channel conveys a block of *L* data symbols over a time period equal to *LNT*. Therefore, the transmission rate equals *R*=*K*/(*N* *T*) symbols/s. More in general, a cyclic prefix can be added (but it is not mandatory) to the transmitted signal, as explained in Section 5. In this case, the rate equals

R=\frac{\mathit{\text{KL}}}{(M+\mu )T}\phantom{\rule{1em}{0ex}}\text{symbols/s},

(4)

where *μ* is the cyclic prefix length in samples.

### 2.1 Frequency domain implementation

One of the goals in CB-FMT is the reduction of the computational complexity in the filtering operation w.r.t. the conventional FMT scheme. This can be achieved by exploiting a frequency domain implementation of the system as described in the following.

Firstly, we define a constant integer *Q* subject to

M=\mathit{\text{LN}}=\mathrm{KQ.}

(5)

Then, the *M*-point DFT of the transmitted signal in (1) is computed. We obtain

\begin{array}{ll}X\left(i\right)\hfill & =\sum _{n=0}^{M-1}x\left(n\right){W}_{M}^{\mathit{\text{ni}}}\phantom{\rule{2em}{0ex}}\hfill \\ =\sum _{n=0}^{M-1}\sum _{k=0}^{K-1}\sum _{\ell =0}^{L-1}{a}^{\left(k\right)}\left(\mathrm{\ell N}\right)g\left({(n-\mathrm{\ell N})}_{M}\right)\phantom{\rule{2.77626pt}{0ex}}{W}_{K}^{-\mathit{\text{kn}}}{W}_{M}^{\mathit{\text{ni}}},\phantom{\rule{2em}{0ex}}\hfill \\ \phantom{\rule{1.5em}{0ex}}i\hfill & \in \{0,\dots ,M-1\}.\phantom{\rule{2em}{0ex}}\hfill \end{array}

(6)

Under the assumption *M*=*K* *Q*, we can write

\begin{array}{ll}X\left(i\right)& =\sum _{k=0}^{K-1}\sum _{\ell =0}^{L-1}{a}^{\left(k\right)}\left(\mathrm{\ell N}\right)\phantom{\rule{2.77626pt}{0ex}}\sum _{n=0}^{M-1}g\left({(n-\mathrm{\ell N})}_{M}\right){W}_{M}^{(i-\mathit{\text{kQ}})n}.\phantom{\rule{2em}{0ex}}\end{array}

(7)

We now multiply and divide (7) by {W}_{M}^{(i-\mathit{\text{kQ}})\mathrm{\ell N}} to obtain

\begin{array}{ll}X\left(i\right)\hfill & =\sum _{k=0}^{K-1}\sum _{\ell =0}^{L-1}{a}^{\left(k\right)}\left(\mathrm{\ell N}\right)\sum _{n=0}^{M-1}g\left({(n-\mathrm{\ell N})}_{M}\right)\hfill \\ \phantom{\rule{1em}{0ex}}\times {W}_{M}^{(i-\mathit{\text{kQ}})(n-\mathrm{\ell N})}{W}_{M}^{(i-\mathit{\text{kQ}})\mathrm{\ell N}}\hfill \\ =\sum _{k=0}^{K-1}\sum _{\ell =0}^{L-1}{a}^{\left(k\right)}\left(\mathrm{\ell N}\right)G(i-\mathit{\text{kQ}}){W}_{M}^{(i-\mathit{\text{kQ}})\mathrm{\ell N}},\hfill \end{array}

(8)

where in (8) we denoted with *G*(*i*) the *M*-point DFT of the pulse *g*(*n*).

Since *M* = *L* *N*, it should be noted that the summation with index *ℓ* in (8) is the *L*-point DFT of the data block *a*^{(k)}(*N* *n*) cyclical shifted by *kQ*, i.e.,

\begin{array}{ll}{A}^{\left(k\right)}(i-\mathit{\text{kQ}})& =\sum _{\ell =0}^{L-1}{a}^{\left(k\right)}\left(\mathrm{\ell N}\right){W}_{L}^{(i-\mathit{\text{kQ}})\ell}.\phantom{\rule{2em}{0ex}}\end{array}

(9)

Finally, substituting (9) in (8), we obtain

X\left(i\right)=\sum _{k=0}^{K-1}{A}^{\left(k\right)}(i-\mathit{\text{kQ}})G(i-\mathit{\text{kQ}}),\phantom{\rule{1em}{0ex}}i\in \{0,\dots ,M-1\}.

(10)

This suggests an implementation of the CB-FMT synthesis filter bank as shown in Figure 2. For each sub-channel block of data, we apply an *L*-point DFT (referred to as outer DFT). We extend cyclically the block of coefficients at the output of the outer DFT from *L* points to *M* points. Then, each sub-channel block of *M* coefficients is weighted with the *M* coefficients *G*(*i*) of the prototype pulse DFT. Next, each sub-channel block is cyclically shifted by a factor -*k* *Q*, where *k* is the sub-channel index. Finally, the shifted blocks are summed together, and an *M*-point IDFT (referred to as inner IDFT) is applied to obtain the signal to be transmitted.

If we assume the *M*-point DFT of the prototype pulse to be confined in *Q* points, i.e., *G*(*i*) ≠ 0 for *i* ∈ {0,…,*Q* − 1} and *G*(*i*) = 0 for *i* ∈ {*Q*,…,*M* − 1}, we can simplify (10) into

\phantom{\rule{-16.0pt}{0ex}}\begin{array}{ll}X\left(i\right)\hfill & ={A}^{\left(k\right)}(i-\mathit{\text{kQ}})G(i-\mathit{\text{kQ}}),\hfill \\ \phantom{\rule{1.3em}{0ex}}i\hfill & \in \{\mathit{\text{kQ}},\dots ,(k+1)Q-1\},\phantom{\rule{1em}{0ex}}k\in \{0,\dots ,K-1\}.\hfill \end{array}

(11)

This suggests an efficient implementation of the CB-FMT synthesis filter bank as shown in Figure 3. For each sub-channel block of data, we apply an *L*-point DFT (referred to as outer DFT). We extend cyclically the block of coefficients at the output of the outer DFT from *L* points to *Q* points. Then, each sub-channel block of *Q* coefficients is weighted with the *Q* non-zero coefficients *G*(*i*) of the prototype pulse DFT. Finally, we apply an *M*-point IDFT (referred to as inner IDFT) to obtain the signal to be transmitted.

The analysis filter bank can also be implemented in the frequency domain. We start from (3) and we substitute the signal *y*(*ℓ*) with the IDFT of its frequency response as follows:

\begin{array}{ll}{z}^{\left(k\right)}\left(\mathit{\text{nN}}\right)\hfill & =\sum _{\ell =0}^{M-1}\sum _{i=0}^{M-1}Y\left(i\right){W}_{M}^{-\mathrm{\ell i}}{W}_{K}^{\mathrm{\ell k}}h\left({(\mathit{\text{nN}}-\ell )}_{M}\right),\hfill \\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}k\hfill & \in \{0,\dots ,K-1\},\hfill \\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}n\hfill & \in \{0,\dots ,L-1\}.\hfill \end{array}

(12)

Assuming *M* = *K* *Q*, we can write

\begin{array}{ll}{z}^{\left(k\right)}\left(\mathit{\text{nN}}\right)& =\sum _{i=0}^{M-1}Y\left(i\right)\sum _{\ell =0}^{M-1}h\left({(\mathit{\text{nN}}-\ell )}_{M}\right)\phantom{\rule{2.22144pt}{0ex}}{W}_{M}^{-(i-\mathit{\text{kQ}})\ell}.\phantom{\rule{2em}{0ex}}\end{array}

(13)

Now, we multiply and divide (13) by {W}_{M}^{-(i-\mathit{\text{kQ}})\mathit{\text{nN}}} to obtain

\begin{array}{ll}{z}^{\left(k\right)}\left(\mathit{\text{nN}}\right)=\hfill & \sum _{i=0}^{M-1}Y\left(i\right)\sum _{\ell =0}^{M-1}h\left({(\mathit{\text{nN}}-\ell )}_{M}\right)\hfill \\ \times {W}_{M}^{-(i-\mathit{\text{kQ}})(\ell -\mathit{\text{nN}})}{W}_{M}^{-(i-\mathit{\text{kQ}})\mathit{\text{nN}}},\hfill \end{array}

(14)

where in (14) we can recognize the shifted version of the *M*-point DFT of *h*(*ℓ*) defined as

\begin{array}{ll}\phantom{\rule{-10.0pt}{0ex}}H(i-\mathit{\text{kQ}})\hfill & =\sum _{\ell =0}^{M-1}h\left({(\mathit{\text{nN}}-\ell )}_{M}\right)\phantom{\rule{2.22144pt}{0ex}}{W}_{M}^{-(i-\mathit{\text{kQ}})(\ell -\mathit{\text{nN}})},\phantom{\rule{2em}{0ex}}\hfill \\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}i\hfill & \in \{0,\dots ,M-1\},\phantom{\rule{2em}{0ex}}\hfill \\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}k\hfill & \in \{0,\dots ,K-1\},\phantom{\rule{2em}{0ex}}\hfill \end{array}

(15)

where the second equality holds since the signals are periodic of *M*. Considering that *M* = *L* *N*, we have that {W}_{M}^{-(i-\mathit{\text{kQ}})\mathit{\text{nN}}}={W}_{L}^{-(i-\mathit{\text{kQ}})n}, and we can write

\begin{array}{ll}{z}^{\left(k\right)}\left(\mathit{\text{nN}}\right)\hfill & =\sum _{i=0}^{M-1}Y\left(i\right)H(i-\mathit{\text{kQ}}){W}_{L}^{-(i-\mathit{\text{kQ}})n}\phantom{\rule{2em}{0ex}}\hfill \\ =\sum _{i=0}^{M-1}Y(i+\mathit{\text{kQ}})H\left(i\right){W}_{L}^{-\mathit{\text{in}}},\phantom{\rule{2em}{0ex}}\hfill \\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}i\hfill & \in \{0,\dots ,M-1\},\phantom{\rule{2em}{0ex}}\hfill \\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}k\hfill & \in \{0,\dots ,K-1\}.\phantom{\rule{2em}{0ex}}\hfill \end{array}

(16)

The index *i* can be decomposed into two indexes *p* and *q* as *i* = *p*+*L* *q*, with *p* ∈ {0,…,*L*−1} and *q* ∈ {0,…,*N*−1}, to obtain

\begin{array}{ll}{z}^{\left(k\right)}\left(\mathit{\text{nN}}\right)& =\sum _{p=0}^{L-1}\sum _{q=0}^{N-1}Y(p+\mathit{\text{qL}}+\mathit{\text{kQ}})\phantom{\rule{2.77626pt}{0ex}}H(p+\mathit{\text{qL}}){W}_{L}^{-\mathit{\text{pn}}}.\phantom{\rule{2em}{0ex}}\end{array}

(17)

In (17), we can recognize the *L*-point IDFT of the signal *Z*^{(k)}(*p*) that is given by

\begin{array}{ll}{Z}^{\left(k\right)}\left(p\right)\hfill & =\sum _{q=0}^{N-1}Y(p+\mathit{\text{qL}}+\mathit{\text{kQ}})H(p+\mathit{\text{qL}}),\phantom{\rule{2em}{0ex}}\hfill \\ \phantom{\rule{2.7em}{0ex}}p\hfill & \in \{0,\dots ,L-1\},\phantom{\rule{2em}{0ex}}\hfill \\ \phantom{\rule{2.7em}{0ex}}k\hfill & \in \{0,\dots ,K-1\}.\phantom{\rule{2em}{0ex}}\hfill \end{array}

(18)

This is the periodic repetition with period *L* of the signal *Y*(*i*)*H*(*i* − *k* *Q*). Therefore, the receiver can be summarized as follows (Figure 2): Firstly, the received signal *y*(*n*) is processed with an *M*-point DFT. Then, the output coefficients are weighted with the prototype pulse *M*-point DFT coefficients *H*(*i*). Next, a periodic repetition with period *L* is performed for each sub-channel block of coefficients to obtain (18), where *Y*(*p*) is the *M*-point DFT of the received signal.

Finally, to obtain the *k*-th sub-channel output, an *L*-point IDFT is applied to (18), i.e.,

\begin{array}{ll}{z}^{\left(k\right)}\left(\mathit{\text{nN}}\right)\hfill & =\sum _{p=0}^{L-1}{Z}^{\left(k\right)}\left(p\right){W}_{L}^{-\mathit{\text{pn}}},\phantom{\rule{2em}{0ex}}\hfill \\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}n\hfill & \in \{0,\dots ,L-1\},\phantom{\rule{1em}{0ex}}k\in \{0,\dots ,K-1\}.\phantom{\rule{2em}{0ex}}\hfill \end{array}

(19)

Assuming that the analysis pulse has only *Q* non-zero coefficients, i.e., *H*(*i*) = 0 for *i* ∈ {*Q*,…,*M* − 1}, as when the pulse is matched to the synthesis pulse and equal to *G*^{∗}(*i*), the pulse weighting and the periodic repetition take place on each sub-channel as graphically depicted in Figure 3. In particular, assuming *Q* > *L*, (18) corresponds to adding at the beginning of the block of coefficients *Y*(*i*+*k* *Q*)*H*(*i*), *i* ∈ {0,…,*Q* − 1} the last *Q* − *L* coefficients of the same block.

With the orthogonal design that we describe in Section 3, i.e., when *H*(*i*) = *G*^{∗}(*i*) and assuming that orthogonality conditions are satisfied, we obtain

\begin{array}{ll}{z}^{\left(k\right)}\left(\mathit{\text{nN}}\right)& =\sum _{q=0}^{Q-1}\left|G\right(q\left){|}^{2}{a}^{\left(k\right)}\right(\mathit{\text{nN}})+{N}^{\left(k\right)}(\mathit{\text{nN}}),\phantom{\rule{2em}{0ex}}\end{array}

(20)

which shows that the output sample is equal to the *n*-th data symbol of the *k*-th sub-channel weighted by the pulse energy, plus a noise contribution. More in general, when the transmission medium is not ideal, equalization can be performed (see Section 5) so that in (20) the coefficient that weights the data symbol is given by the sub-channel energy.

Also the FMT system can be implemented in the frequency domain. However, the presence of linear convolutions renders it more complex since an overlap-and-add operation has to be carried out as shown in [16].

As a final remark, the use of inner and outer DFTs appeared also in the concatenated OFDM-FMT architecture presented in [5].

### 2.2 Relation with FMT and OFDM

In the following, we briefly highlight the main differences of CB-FMT w.r.t. FMT [3] and OFDM [2]:

#### 2.2.1 Relation with FMT

Similarly to conventional FMT, CB-FMT targets the use of frequency confined sub-channels. However, while the filter banks in FMT deploy linear convolutions, in CB-FMT, the filter banks use cyclic convolutions. Therefore, the FMT transmitted signal does not read as in (1) but as follows:

\begin{array}{ll}x\left(n\right)& =\sum _{k=0}^{K-1}\sum _{\ell \in \mathbb{Z}}{a}^{\left(k\right)}\left(\mathrm{\ell N}\right)g(n-\mathrm{\ell N}){W}_{K}^{-\mathit{\text{nk}}},\phantom{\rule{1em}{0ex}}n\in \mathbb{Z},\phantom{\rule{2em}{0ex}}\end{array}

(21)

where *g*(*n*) is the prototype pulse. Furthermore, while in FMT the transmission is typically continuous, in CB-FMT, data signals are transmitted in blocks, each of the *K* sub-channels transmits a block of *L* data symbols.

In FMT, the rate equals

R=\frac{K}{\mathit{\text{NT}}}\phantom{\rule{1em}{0ex}}\text{symbols/s}.

(22)

The efficient implementation of FMT can exploit a polyphase DFT filter bank architecture [10]. Nevertheless, the FMT complexity is higher than the CB-FMT complexity, as shown in Section 2.3 assuming the same number of sub-channels and the same prototype pulse length.

In CB-FMT, very simple frequency domain design can be followed to obtain an orthogonal solution as shown in Section 3. In FMT, the orthogonal design is more convoluted [9, 10]. Thus, non-orthogonal solutions are often adopted in FMT as for instance the use of a truncated root-raised-cosine prototype pulse [3] or *ad hoc* frequency localized pulses [7, 8].

When transmission is in time/frequency-selective fading channels, the good sub-channel frequency confinement in FMT provides robustness to ICI, while the residual ISI can be mitigated with sub-channel linear equalization [3] or maximum *a posteriori* sequence estimation [4, 17]. In CB-FMT, instead, simple frequency domain equalization can be adopted as described in Section 5.

#### 2.2.2 Relation with OFDM

In OFDM, the filter bank privileges the sub-channel time domain localization, rather than the frequency domain localization. OFDM can be seen as a particular case of both FMT and CB-FMT. In fact it corresponds to an FMT system with *N* = *K* and the prototype pulse being a rectangular window, i.e., *g*(*n*) is equal to 1 for *n* ∈ {0,…,*N*−1} and 0 otherwise. It follows that the transmitted signal can be expressed as

\begin{array}{l}\phantom{\rule{-14.0pt}{0ex}}x\left(n\right)=\sum _{k=0}^{K-1}{a}^{\left(k\right)}\left(\mathrm{\ell N}\right){W}_{K}^{-\mathit{\text{nk}}},\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{0.3em}{0ex}}n\in \left\{\mathrm{\ell N},\dots ,(\ell +1)N-1\right\}.\end{array}

(23)

Starting from CB-FMT, we obtain OFDM by setting *N* = *K*, *L* = *Q* = 1, and *G*(0) = 1, so that we also have *K* = *M*.

In OFDM, the rate equals

R=\frac{K}{(K+\mu )T}\phantom{\rule{1em}{0ex}}\text{symbols/s},

(24)

where *μ* is the cyclic prefix length in samples. It should be noted that cyclic prefix is not necessary equal in CB-FMT and in OFDM. The same applies to the number of data sub-channels that can be different in the two systems.

### 2.3 Computational complexity

In this section, we evaluate the computational complexity of CB-FMT in terms of number of complex operations [cop] (additions and multiplications) per sample.

Let us assume that an *M*-point DFT (or IDFT) block has complexity equal to *α* *M* log2(*M*) [cop] where, for instance, *α* = 1.2.

At the transmitter, *K* outer DFTs of *L*-points are used, together with one *M*-point inner IDFT. Furthermore, the inner IDFT input signals are weighted by the DFT components of the prototype pulse. Similarly, this applies at the receiver. The operations performed by the cyclic extension is negligible. Let us assume the number of non-zero DFT coeffients of the prototype pulse to be equal to *Q*_{2} = *Q* *C*, *C* ∈ {1,…,*K*}. Since the transmitted block comprises *LN* coefficients, the number of complex operations per sample is equal to

\begin{array}{l}\phantom{\rule{-15.0pt}{0ex}}{C}_{\text{CB-FMT}}=\frac{\mathrm{K\alpha L}\underset{2}{log}\left(L\right)+\mathrm{\alpha M}\underset{2}{log}\left(M\right)+S}{\mathit{\text{LN}}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\left[\frac{\text{cop}}{\text{sample}}\right],\end{array}

(25)

where *S* = (2*C* − 1)*M* for the transmitter. At the receiver, the periodic repetition increases the complexity by *S* = 2*M* *C* − *K* *L* for (*Q* − *L*)*C* < *L* and by *S* = (*M* + *K* *L*)*C* otherwise. When the prototype pulse has only *Q* non-zero coefficients, i.e., *C* = 1, *S* = *M* for the transmitter and *S* = 2*M* − *K* *L* for the receiver.

As a comparison, we consider the complexity of FMT efficiently implemented with a polyphase DFT filter bank as described in [10]. This is equal to

\begin{array}{l}{C}_{\text{FMT}}=\frac{\mathrm{K\alpha}\underset{2}{log}\left(K\right)+2{L}_{g}}{N}\phantom{\rule{1em}{0ex}}\left[\frac{\text{cop}}{\text{sample}}\right],\end{array}

(26)

assuming *K* sub-channels, an interpolation factor *N*, and a prototype pulse with length *L*_{
g
} coefficients. This complexity does not take into account the operations required by the equalization stage.

If, for example, we assume *K* = 64 and *N* = 80 for both systems and furthermore FMT with a pulse length equal to 20*N* while CB-FMT with *L* = *K*, *Q* = *N* resulting in a longer filter length equal to *M* = 64*N* coefficients, the receiver complexity will be equal to {45.8,21.7} [cop] respectively for FMT and CB-FMT. This shows the gain in complexity of CB-FMT yet having a longer pulse. More results about the complexity are reported in Section 6.2.