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Cyclic block filtered multitone modulation
EURASIP Journal on Advances in Signal Processing volume 2014, Article number: 109 (2014)
Abstract
A filter bank modulation transceiver is presented. The idea is to obtain good subchannel frequency confinement as it is done by the family of exponentially modulated filter banks that is typically referred to as filtered multitone (FMT) modulation. However, differently from conventional FMT, the linear convolutions are replaced with circular convolutions. Since transmission occurs in blocks, the scheme is referred to as cyclic block FMT (CBFMT). This paper focuses on the principles, design, and implementation of CBFMT. In particular, it is shown that an efficient realization of both the transmitter and the receiver is possible in the frequency domain (FD), and it is based on the concatenation of an inner discrete Fourier transform (DFT) and a bank of outer DFTs. Such an implementation suggests a simple subchannel FD equalizer. The overall required implementation complexity is lower than in FMT. Furthermore, the orthogonal filter bank design is simplified. The subchannel frequency confinement in CBFMT yields compact power spectrum and lower peaktoaverage power ratio than in OFDM. Furthermore, the FD equalization allows the exploitation of the transmission medium time and frequency diversity; thus, it potentially yields lower symbol error rate and higher achievable rate in timevariant frequencyselective fading.
1 Introduction
Filter bank modulation (FBM) systems, also referred to as multicarrier modulation systems, have been successfully applied to a wide variety of digital communication applications over the past several years. The research of improved solutions is still large because of the increasing demand for broadband telecommunication services both over wireline and wireless channels.
Wideband channels are characterized by frequency selectivity which translates in timedispersive impulse responses that cause significant intersymbol interference (ISI) in digital communication systems. FBM transceivers employ a transmission technique where a set of narrow band signals (low data rate sequences) are transmitted simultaneously over a broadband channel [1]. In particular, each low rate data sequence is transmitted through a subchannel that is shaped according to a subchannel pulse. If the subchannels are sufficiently narrowband, they will experience an overall flat frequency response so that the medium frequency selectivity does not introduce significant intercarrier interference (ICI) and ISI. Therefore, the channel equalization task can be simplified. More in general, FBM architectures aim to increase the system spectral efficiency, to enable the agile use of spectrum and the flexible adaptation of available resources.
Two baseline FBM solutions are orthogonal frequency division multiplexing (OFDM) [2] and filtered multitone (FMT) modulation [3]. FMT consists of an exponentially modulated filter bank that privileges the subchannel frequency confinement rather than the time confinement, as for example OFDM does. In FMT, with frequency confined pulses, the subchannels are quasiorthogonal to each other which prevents the system to suffer from ICI, while the ISI introduced by the frequencyselective medium can be mitigated with subchannel equalization [3, 4] or with the use of outer OFDM modules, one per subchannel, as described in the concatenated OFDMFMT scheme in [5]. Clearly, to obtain high subchannel frequency confinement, long prototype pulses are required. In such a case, the implementation complexity may increase significantly. Therefore, the efficient FMT implementation as well as the design of good pulses is an important aspect [6–10].
In this paper, a novel FBM scheme is described. The ambitious goal is to merge the strengths of both OFDM and FMT. We refer to it as cyclic block filtered multitone modulation (CBFMT). Similarly to conventional FMT, CBFMT aims at generating well frequency localized subchannels. However, differently from it, CBFMT transmits data symbols in blocks, and the filter bank does not use linear convolutions but cyclic convolutions. Similarly to OFDM, the block transmission can reduce latency, but the subchannel frequency confinement is much higher, more similarly to FMT. This translates in higher spectral selectivity, more confined power spectral density, and lower peaktoaveragepower ratio (PAPR) than in OFDM given the same target spectral efficiency. Furthermore, the implementation complexity is lower than that in FMT with the same number of subchannels and even with longer pulses. In fact, an efficient realization can be devised if both the synthesis and the analysis filter banks in CBFMT are implemented in the frequency domain (FD) via a concatenation of an inner (with respect to (w.r.t.) the channel) discrete fourier transform (DFT) and a bank of outer DFTs. Such a FD architecture enables the use of a FD subchannel equalizer designed according to the zero forcing (ZF) or the minimum mean square error (MMSE) criterion [11]. In particular, the ZF solution will restore perfect orthogonality if a cyclic prefix (similarly to OFDM) is appended to each block of signal coefficients that are transmitted over a dispersive (frequency selective) channel. In the presence of channel time variations (because of mobility), the ICI is small due to the subchannel frequency confinement so that the subchannel equalizer is sufficient to cope with the ISI experienced by the data symbols transmitted in a block. This equalization scheme is capable to coherently collect the subchannel energies so that frequency and time diversity, offered by the fading channel, can be exploited. Consequently, this can provide better performance, i.e., lower symbol error rate and higher achievable rate, than OFDM.
The CBFMT idea and principles were originally presented in [12]. Some aspects related to the FD implementation were disclosed more recently in [13], while in [14] a preliminary analysis of the robustness of the scheme in fading channels was reported. In this paper, we provide a detailed description of CBFMT with emphasis to the design, implementation, equalization, and performance aspects.
Another FBM scheme referred to as generalized OFDM (GFDM) has been independently presented in [15]. According to [15], GFDM is a FBM scheme that uses a nonorthogonal design where the subchannel spacing is smaller than the subchannel Nyquist band. It can be viewed as an FMT scheme operating beyond the critical sample rate. An extended CP is used to take into account the pulse tails and allow the implementation of a socalled tail biting convolution. The tail biting convolution in turn can be implemented with a circular convolution. Since the design is not orthogonal, ICI is present also in an ideal channel which requires some form of equalization to mitigate it. This may yield performance that is worse than OFDM. However, it is shown in [15] that other benefits are offered by GFDM as spectrum agility and lower PAPR. CBFMT is a more general architecture than GFDM that shares the idea of using circular convolutions instead of linear convolutions. As FMT essentially represents a general exponentially modulated filter bank with linear convolutions, CBFMT represents a general exponentially modulated filter bank with circular convolutions. An orthogonal CBFMT system can be designed (as done in this paper) without requiring the use of a CP unless more robustness is desirable in time dispersive channels. Orthogonal CBFMT can offer lower BER and higher spectral efficiency compared to OFDM.
The specific contributions of this paper can be summarized as follows:

The CBFMT key elements are described in Section 2. These include the derivation of an efficient frequency domain implementation starting from the time domain signal representation (Section 2.1).

The relations between CBFMT and conventional FMT/OFDM are briefly described in Section 2.2 to better understand the differences.

The complexity analysis is carried out in Section 2.3.

The conditions under which the CBFMT scheme is orthogonal are studied in Section 3. Herein, a simple orthogonal pulse design is also proposed.

The analytic derivation of the CBFMT power spectral density (PSD) and the PAPR are discussed in Section 4.

Equalization in timevariant frequencyselective fading is discussed in Section 5. A subchannel FD MMSE equalizer is herein proposed.

Several numerical examples, which include pulse shapes, complexity, PSD and PAPR, as well as symbol error rate (SER) and achievable data rate comparisons with OFDM, are collected in Section 6.
Finally, the conclusions follow.
2 Cyclic block FMT modulation
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 kth 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., MQAM or MPSK, 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 subchannels obtained by partitioning the wideband transmission medium in K subbands.
The principle of CBFMT 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 subchannels. 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 subchannel. 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 zeropadding, without loss of generality. The CBFMT transmitted signal can be written as
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 digitaltoanalog converted and, then, transmitted over the transmission medium. At the receiver, after analogtodigital conversion, the discrete time received signal, denoted with y(ℓ), is defined as
where h_{CH}(s,ℓ) is the timevariant 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 kth subchannel output is obtained as follows:
where h((n)_{ M }) denotes the periodic repetition of the prototype analysis pulse h(n) with period M.
Each subchannel 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
where μ is the cyclic prefix length in samples.
2.1 Frequency domain implementation
One of the goals in CBFMT 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
Then, the Mpoint DFT of the transmitted signal in (1) is computed. We obtain
Under the assumption M=K Q, we can write
We now multiply and divide (7) by ${W}_{M}^{(i\mathit{\text{kQ}})\mathrm{\ell N}}$ to obtain
where in (8) we denoted with G(i) the Mpoint DFT of the pulse g(n).
Since M = L N, it should be noted that the summation with index ℓ in (8) is the Lpoint DFT of the data block a^{(k)}(N n) cyclical shifted by kQ, i.e.,
Finally, substituting (9) in (8), we obtain
This suggests an implementation of the CBFMT synthesis filter bank as shown in Figure 2. For each subchannel block of data, we apply an Lpoint 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 subchannel block of M coefficients is weighted with the M coefficients G(i) of the prototype pulse DFT. Next, each subchannel block is cyclically shifted by a factor k Q, where k is the subchannel index. Finally, the shifted blocks are summed together, and an Mpoint IDFT (referred to as inner IDFT) is applied to obtain the signal to be transmitted.
If we assume the Mpoint 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
This suggests an efficient implementation of the CBFMT synthesis filter bank as shown in Figure 3. For each subchannel block of data, we apply an Lpoint 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 subchannel block of Q coefficients is weighted with the Q nonzero coefficients G(i) of the prototype pulse DFT. Finally, we apply an Mpoint 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:
Assuming M = K Q, we can write
Now, we multiply and divide (13) by ${W}_{M}^{(i\mathit{\text{kQ}})\mathit{\text{nN}}}$ to obtain
where in (14) we can recognize the shifted version of the Mpoint DFT of h(ℓ) defined as
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
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
In (17), we can recognize the Lpoint IDFT of the signal Z^{(k)}(p) that is given by
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 Mpoint DFT. Then, the output coefficients are weighted with the prototype pulse Mpoint DFT coefficients H(i). Next, a periodic repetition with period L is performed for each subchannel block of coefficients to obtain (18), where Y(p) is the Mpoint DFT of the received signal.
Finally, to obtain the kth subchannel output, an Lpoint IDFT is applied to (18), i.e.,
Assuming that the analysis pulse has only Q nonzero 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 subchannel 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
which shows that the output sample is equal to the nth data symbol of the kth subchannel 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 subchannel energy.
Also the FMT system can be implemented in the frequency domain. However, the presence of linear convolutions renders it more complex since an overlapandadd 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 OFDMFMT architecture presented in [5].
2.2 Relation with FMT and OFDM
In the following, we briefly highlight the main differences of CBFMT w.r.t. FMT [3] and OFDM [2]:
2.2.1 Relation with FMT
Similarly to conventional FMT, CBFMT targets the use of frequency confined subchannels. However, while the filter banks in FMT deploy linear convolutions, in CBFMT, the filter banks use cyclic convolutions. Therefore, the FMT transmitted signal does not read as in (1) but as follows:
where g(n) is the prototype pulse. Furthermore, while in FMT the transmission is typically continuous, in CBFMT, data signals are transmitted in blocks, each of the K subchannels transmits a block of L data symbols.
In FMT, the rate equals
The efficient implementation of FMT can exploit a polyphase DFT filter bank architecture [10]. Nevertheless, the FMT complexity is higher than the CBFMT complexity, as shown in Section 2.3 assuming the same number of subchannels and the same prototype pulse length.
In CBFMT, 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, nonorthogonal solutions are often adopted in FMT as for instance the use of a truncated rootraisedcosine prototype pulse [3] or ad hoc frequency localized pulses [7, 8].
When transmission is in time/frequencyselective fading channels, the good subchannel frequency confinement in FMT provides robustness to ICI, while the residual ISI can be mitigated with subchannel linear equalization [3] or maximum a posteriori sequence estimation [4, 17]. In CBFMT, 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 subchannel time domain localization, rather than the frequency domain localization. OFDM can be seen as a particular case of both FMT and CBFMT. 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
Starting from CBFMT, 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
where μ is the cyclic prefix length in samples. It should be noted that cyclic prefix is not necessary equal in CBFMT and in OFDM. The same applies to the number of data subchannels that can be different in the two systems.
2.3 Computational complexity
In this section, we evaluate the computational complexity of CBFMT in terms of number of complex operations [cop] (additions and multiplications) per sample.
Let us assume that an Mpoint 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 Lpoints are used, together with one Mpoint 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 nonzero 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
where S = (2C − 1)M for the transmitter. At the receiver, the periodic repetition increases the complexity by S = 2M C − K L for (Q − L)C < L and by S = (M + K L)C otherwise. When the prototype pulse has only Q nonzero coefficients, i.e., C = 1, S = M for the transmitter and S = 2M − 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
assuming K subchannels, 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 20N while CBFMT with L = K, Q = N resulting in a longer filter length equal to M = 64N coefficients, the receiver complexity will be equal to {45.8,21.7} [cop] respectively for FMT and CBFMT. This shows the gain in complexity of CBFMT yet having a longer pulse. More results about the complexity are reported in Section 6.2.
3 Orthogonality conditions and prototype pulse design
The frequency domain implementation of CBFMT allows us to deduce the system orthogonality conditions. A filter bank system is orthogonal when it has the perfect reconstruction property and the transmitreceive filters are matched, i.e., g(n) = h^{∗}(−n) and H(i) = G^{∗}(i), so that the system exhibits neither ISI nor ICI [18].
When the prototype pulse satisfies the following two conditions, the CBFMT system will be orthogonal [19]:

1.
The Mpoint DFT of the prototype pulse has only Q nonzero coefficients, i.e., G(i) = 0 for i ∈ {Q,…,M − 1} (sufficient condition).

2.
The correlation between g(n) and g ^{∗}(−n), computed with the circular convolution and sampled by a factor N, is the Kronecker delta, i.e.,
$$\begin{array}{ll}\phantom{\rule{14.0pt}{0ex}}\left[g\otimes {g}_{}^{\ast}\right]\left(\mathit{\text{nN}}\right)& =\sum _{\ell =0}^{M1}g\left({\left(\ell \right)}_{M}\right){g}^{\ast}\left({(\mathit{\text{nN}}+\ell )}_{M}\right)=\delta \left(n\right),\phantom{\rule{2em}{0ex}}\end{array}$$(27)
where δ(n) is the Kronecker delta function, i.e., δ(n) is equal to 1 for n = 0 and 0 otherwise.
3.1 Proof of the orthogonality condition
To prove the orthogonality conditions, we proceed in two steps. Firstly, we prove the condition to have orthogonality between different subchannels, i.e., no ICI. Then, we prove the condition to have orthogonality between the data symbols of each subchannel, i.e., no ISI.
3.1.1 Subchannel orthogonality
The subchannels will be orthogonal if the Mpoint DFT of the prototype pulse, G(i), is equal to zero for i∈{Q,…,M1}. In this case, there is no ICI.
Proof.
We substitute (10) in (18) assuming Y(i) = X(i). The inputoutput relation between the Lpoint DFT of the data block is given by
where H^{(q)}(p) = H(p + q L), G^{(q)}(p) = G(p + q L), and ${\Delta}_{k{k}^{\prime}}={k}^{\prime}k$. To have orthogonality between the K subchannels, the terms in the second summation must be zero for k^{′} ≠ k. This is possible when ${G}^{\left(q\right)}(p{\Delta}_{k{k}^{\prime}}Q)$ is equal to zero for ${\Delta}_{k{k}^{\prime}}\ne 0$ and ∀ p ∈ {0,…,L − 1},∀ q ∈ {0,…,N − 1}. This condition is satisfied when $G(p+\mathit{\text{qL}}{\Delta}_{k{k}^{\prime}}Q)=0$, only for p+q L ∈ [0,…,Q−1].
3.1.2 Block orthogonality
No ISI will be present between the L data symbols in each subchannel block, when the prototype pulses are matched and (27) is satisfied.
Proof.
Under the subchannels orthogonality (previous condition) and with matched filters, (28) becomes
To have perfect reconstruction, we need to have [19]
In (30), the summation represents a periodic repetition of G(p+q L)G^{∗}(p+q L). In the time domain, this translates in sampling by a factor M/L = N. Thus, if we apply an Lpoint IDFT to (30), we will obtain (27).
3.2 Orthogonal prototype pulse design
The frequency domain implementation and the orthogonality conditions suggest to synthesize the pulse in the frequency domain with a finite number of frequency components.
We start by choosing a pulse that belongs to the Nyquist class with rolloff β, Nyquist frequency 1/(2N T), total bandwidth 1/(K T), and frequency response $\u011c\left(\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}f\right)$. Then, we set M. The coefficients in the frequency domain of the CBFMT prototype pulse are obtained by sampling the response $\sqrt{\u011c\left(\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}f\right)}$, i.e., $G\left(i\right)=\sqrt{\u011c(i/(\mathit{\text{MT}}\left)\right)}$ with i∈{0,…,M − 1}. To satisfy the orthogonality conditions, only Q out of M coefficients are nonzero, while L < Q coefficients fall in the the band 1/(N T). It should also be noted that there is a limiting condition on the choice of the rolloff β. In fact, the maximum rolloff to prevent the pulse tails from exceeding the bandwidth 1/(K T) of Qpoints is β_{max} = (Q − L)/Q. Therefore, we must choose β ≤ β_{max}.
We recall that the parameters of CBFMT are related to each other through the relation M = L N = K Q, which can be rearranged as
where p and q are relative prime integers. Therefore, once we have chosen M, the number of subchannels K, and the ratio N/K, we obtain the rest of the parameters N, Q, L.
Some examples of pulse responses are reported in Section 6.1. An optimal pulse design method that targets the maximization of the inbandtooutofband pulse energy has been recently presented in [19]. In particular, complex asymetric pulses are also considered. It is also interesting to note that a trivial orthogonal solution is obtained by using a rectangular FD window of Q nonzero coefficients [20]. In such a case, the CBFMT scheme becomes the dual of the OFDM system that uses, instead, a rectangular window in the time domain.
4 PSD and PAPRrelated aspects
4.1 PSDrelated aspects
In this section, we study the power spectral density (PSD) of the transmitted CBFMT signal. The PSD is an important aspect to evaluate the confinement of the transmitted spectrum. The objective is to limit the outofband emissions. More in general, the PSD must comply to regulatory aspects that typically set an upper limit, also known as spectrum mask, e.g., as in the IEEE 802.11 WLAN standard [21] or in the HomePlug PLC system [22].
To derive an analytic expression for the PSD of CBFMT, we can start by expressing the (continuous) transmission of samples x(n) as follows:
where the subchannel symbol period M_{1} = M + μ takes into account the fact that a cyclic prefix (CP) of length μ samples can be used for the equalization, as it will be explained in Section 5. The CP is appended to the block of coefficients at the inner IDFT output. Furthermore, X^{(k)}(m M_{1}) are the coefficients at the input of the inner IDFT in the transmitter at time instant m M_{1}, and g_{ P }(n) is the rectangular window that is equal to 1 for n ∈ {0,…,M_{1} − 1} and zero otherwise. Essentially, X^{(k)}(m M_{1}) represents the block of coefficients defined in (11) that is transmitted in the mth CBFMT block. The data symbols a^{(k)}(ℓ N) are assumed to be independent, identically distributed with zero mean and power equal to M_{ a } = E[ a^{(k)}(ℓ N)^{2}].
To convert the signal in (32) from discrete time to continuous time, an interpolation filter with response g_{ I }(t) is needed. The interpolated signal can be expressed as
To obtain the signal PSD, the correlation of x(t), namely r_{ x }(t,τ) = E[x^{∗}(t + τ)x(t)] where E[·] is the expectation operator, needs to be computed. Since the interpolated signal is cyclostationary, the correlation is periodic [23], i.e., r_{ x }(t + T,τ) = r_{ x }(t,τ). To remove the dependency on the variable t, the mean correlation $\overline{{r}_{x}\left(\tau \right)}=\frac{1}{T}{\int}_{0}^{T}{r}_{x}(t,\tau )\mathrm{d}t$ has to be computed, from which the mean PSD is obtained via a Fourier transform. The mathematical expressions involved in the PSD computation are convoluted. In the following, we report the main steps to obtain the PSD. The final result is given in (43) to (45).
The correlation can be written as
where E[x^{∗}(n)x(m)] represents the correlation of the discrete time transmitted signal before the analog interpolation filter. To compute (34), we rewrite (32) as
where ${g}_{P}^{\left(k\right)}\left(n\right)={g}_{P}\left(n\right){e}^{j2\mathrm{\pi nk}/M}$. The correlation of (35) is given by
where the term ${r}_{X}({k}_{1},\phantom{\rule{0.3em}{0ex}}{p}_{1},\phantom{\rule{0.3em}{0ex}}{k}_{2},\phantom{\rule{0.3em}{0ex}}{p}_{2})=E\left[{\left({X}^{\left({k}_{1}\right)}\left({p}_{1}{M}_{1}\right)\right)}^{\ast}\right.\phantom{\rule{0.3em}{0ex}}\left(\right)close="]">\phantom{\rule{0.3em}{0ex}}{X}^{\left({k}_{2}\right)}\left({p}_{2}{M}_{1}\right)$ represents the correlation between the signal coefficients at the input of the inner IDFT of the transmitter. We obtain
The correlation in (36) is cyclostationary, i.e., r(n,m) = r(n + M_{1},m). Thus, we compute the mean correlation as
Consequently, the mean PSD is
where $\overline{R\left(\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}f\right)}$ is the discrete Fourier transform of (38).
Assuming that the prototype pulse DFT has only Q nonzero coefficients, (37) can be rewritten as
Then, the PSD can be written as
where f_{k,q} = (q + k Q)/(M T),S = {(x,y)x,y ∈ {0,…,Q − 1}, x−y∈{−L,0,L}} and G_{ P }(f) is the periodic sinc function, defined as
In (41), we can split the summation with indexes (q_{1},q_{2}) in the two sums and thus in two resulting terms:
For q_{1} = q_{2}, we obtain the term
while for q_{1} ≠ q_{2}, we obtain the term
where G_{ I }(f) is the Fourier transform of the analog interpolation filter. The first term, P_{1}(f), is a sum of sinc functions, each centered in f_{k,q} and weighted by the prototype pulse DFT coefficients. The main lobe of the sinc function has a bandwidth equal to 1/((M + μ)T). The second term, P_{2}(f), is related to the correlation between the signal coefficients X(i) at the input of the inner IDFT, defined in (11). In detail, we may reconsider (11). In fact, such coefficients can be written as
The correlation between (46) and (47) will not be null if Q > L. This is due to the fact that the block of coefficients A^{(k)}(i), at the output of the outer DFT, is cyclically extended. Equation 45 takes this correlation into account.
4.2 PAPRrelated aspects
The peaktoaverage power ratio (PAPR) is a measure of the transmitted signal x(t), defined as
The PAPR indicates how much the signal peak power is higher than the mean power value. A signal with high PAPR exhibits high dynamic range. Consequently, this poses a challenge to the analog components of the front end which may introduce distortions. For example, if the signal exceeds the power amplifier dynamic range, the output signal will be clipped to the supply voltage level. In turn, unintentional outband interference due to spurious emissions is generated as well as the signal distortion may cause a performance loss in the receiver stage. In OFDM, the high PAPR is a known drawback. It grows exponentially with the subchannel number [24]. Generally, the PAPR cannot be expressed in closedform.
In OFDM, x(t) can be approximately modeled as a Rayleigh process as shown in [25] so that pseudoclosed expressions for the distribution of the PAPR can be derived. In CBFMT, the problem is more complex, so that we have to resort to a numerical approach to evaluate the PAPR as it will be discussed in Section 6.3.
5 Equalization in timevariant frequencyselective channels
The orthogonality conditions, described in Section 3, render CBFMT free from interference when the channel is static and has a flat frequency response. When the channel is frequency selective and/or time variant, some interference may be present. However, orthogonality can be restored with an equalizer. From the frequency domain implementation of CBFMT (Figure 3), we note that the chain comprising the Mpoint inner IDFT at the transmitter, the transmission medium, and the Mpoint inner DFT at the receiver is similar to the OFDM system. This suggests to append a CP of μ samples to the transmitted block of samples, as shown in Figure 4.
To proceed, let us denote the timevariant channel response as follows
where P < μ is the channel impulse response length in samples, and α_{ s }(n) is the sth channel tap at time instant n. Then, the received signal can be written as
To simplify the notation, we focus on the first received block, without loss of generality. After CP removal, under the assumption that the channel duration (in samples) is shorter than the CP, (50) becomes a circular convolution between the transmitted signal and the channel impulse response. In matrix form, we can write
where y =[y(μ) y(μ + 1) … y(M + μ − 1)]^{T}, x = [x(0) x(1) … x(M − 1)]^{T} and H_{CH} is the circulant channel matrix of size M × M. The ith element of the jth channel matrix column is defined as
If we apply an Mpoint DFT to (51), which is what the receiver does through the inner DFT, we will obtain
where F_{ M } is the Mpoint DFT matrix, defined as ${\left\{{\mathbf{F}}_{M}\right\}}_{\mathit{\text{ij}}}={W}_{M}^{\mathit{\text{ij}}}/\sqrt{M},\phantom{\rule{0.3em}{0ex}}i,j\in \{0,\dots ,M1\}$, and X is the vector of coefficients at the input of the inner IDFT at the transmitter side. Essentially, (53) describes the relation that exists between the coefficients at the input of the inner IDFT at the transmitter side and the coefficients at the output of the inner DFT at the receiver side. Such a relation suggests the use of a frequency domain equalizer applied at the output of the receiver inner DFT as shown in Figure 4. To proceed, we need to derive an expression for the elements of the matrix ${\widehat{\mathbf{H}}}_{\text{CH}}$.
We start from (50). Without loss of generality, we can extend the sum from P to M by zeropadding the channel impulse response. The block of received samples can be written as
where H_{1}(p,n) is the Mpoint DFT of the channel impulse response at time instant n, i.e., computed along the ℓ variable. By computing the Mpoint DFT of (54), we obtain the elements of the vector (53):
where H_{2}(p,q) is the twodimensional Mpoint DFT of h_{CH}(ℓ,n), defined as
Finally, it follows that the elements of ${\widehat{\mathbf{H}}}_{\text{CH}}$ are defined as
To derive the FD equalizer, we distinguish between the case of having a timeinvariant channel and the case of having a timevariant channel.
5.1 Timeinvariant channel
When the channel is timeinvariant, h_{CH}(ℓ,n) does not depend on the time instant n. Thus, H_{2}(p,q) is not null only for q = 0. Consequently, the channel matrix ${\widehat{\mathbf{H}}}_{\text{CH}}$ is a diagonal matrix. The Mpoint DFT output, at the receiver stage, can be simply written as follows:
where N(i), i ∈ {0,…,M − 1}, are the Mpoint DFT coefficients of the background noise samples. This shows that there is absence of ICI, i.e., interference among the subchannels. Therefore, the application of a simple 1tap frequency domain equalizer is enabled [11]. In particular, with zero forcing, the equalizer output is given by
where H_{EQ,ZF}(i) is the ith coefficient of the FD zero forcing equalizer. In Figure 4, the matrix H_{EQ,k}, associated to the kth subchannel equalizer, is diagonal. Its pth diagonal element is equal to H_{EQ,ZF}(p + k Q). In such a case, perfect orthogonality is achieved in the system. That is, after zero forcing equalization, the subchannel signal is multiplied with the conjugate of the pulse frequency response G^{∗}(p), and it is finally processed by the other stages depicted in Figure 3. Then, the output reads as in (20).
Alternatively, the equalizer coefficients can be designed according to the MMSE principle and they read
where σ^{2} is the noise variance. This solution provides better performance at low signaltonoise ratios than the zero forcing solution.
5.2 Timevariant channel
When the channel is time variant, the channel matrix ${\widehat{\mathbf{H}}}_{\text{CH}}$ has nonzero elements outside the main diagonal. The number of nonzero elements off the diagonal grows with the channel Doppler spread. The qth inner DFT output coefficient at the receiver can be written as
Relation (61) shows that a simple 1tap equalizer cannot fully remove the interference introduced by the timevariant channel and represented by the second additive term in (61). Thus, we propose to use a subchannel block equalizer that mitigates the interference between the L symbols transmitted in each of the Kth subchannels considering the fact the the ICI between distinct subchannels is small due to their good frequency response confinement.
We start by splitting the matrix ${\widehat{\mathbf{H}}}_{\text{CH}}$ in blocks of Q × Q elements, so that (53) can be written as
where X_{ k } and Y_{ k } are Q × 1 vectors whose elements are the Q coefficients associated to the kth subchannel and N is the background noise vector. B_{i,j} is a Q × Q matrix defined as
The kth subvector in (62) can be written in order to separate the term of interest from the interference as follows
Now, the kth subchannel block equalizer output vector is given by
where the subchannel equalizer matrix is computed according to the MMSE criterion. Such a matrix is obtained as
where I_{ Q } is the Q × Q identity matrix, ${\mathbf{R}}_{{X}_{k}{Y}_{k}}=E\left[{\mathbf{X}}_{k}{\mathbf{Y}}_{k}^{H}\right]$ is the Q × Q correlation matrix between the vector X_{ k } and the vector Y_{ k }, and ${\mathbf{R}}_{{Y}_{k}{Y}_{k}}=E\left[{\mathbf{Y}}_{k}{\mathbf{Y}}_{k}^{H}\right]$ is the Q × Q autocorrelation matrix of the vector Y_{ k }. It should be noted that the signals X_{ i }, i ≠ k, are treated as noise by the kth subchannel block equalizer since the interference that they generate is small due to the subchannel spectral confinement.
After the subchannel equalization, the output coefficients are weighted with the prototype pulse FD coefficients G^{∗}(i) and, finally, processed by the others stages, as shown in Figures 3 and 4.
6 Numerical results
6.1 Pulse design examples
In Figure 5, we report two examples of pulses obtained with the method described in Section 3.2. Several combinations of parameters are considered. The pulses have been obtained starting from a rootraisedcosine spectrum with rolloff equal to 0.2. The pulses are designed for M = 320 and M = 640. Furthermore, K = 8 and N = 10 or K = 16 and N = 20 are considered, respectively.
In Table 1, we report the ratio between the inband and the outofband energy of the interpolated prototype pulse for several choices of the parameters. Despite the simple design method, the pulses exhibit good frequency confinement which increases for larger values of M.
In all numerical results that will follow, a common configuration is related to the case M = 320 and K ∈ {8,16,32,64}.
6.2 Complexity comparisons
In Figure 6, we show the complexity of OFDM, FMT, and CBFMT as a function of the prototype pulse length (in samples) and assuming it has Q nonzero DFT coefficients. In all FMT, OFDM, and CBFMT, the pulse length L_{ g } is set equal to M. It should be noted that OFDM uses a rectangular window of length M equal to the number of subchannels. The complexity is presented in terms of cop/sample for different combinations of N,K. In CBFMT, we show the complexity at the receiver side when a 1tap equalizer is used.
The figure shows that CBFMT has significant lower complexity than conventional FMT. Clearly, OFDM is the simplest solution. However, CBFMT and OFDM have a more comparable complexity, i.e., CBFMT is more complex than OFDM by a factor of about 1.5. As it will be shown in the next sections, this extra complexity pays back since CBFMT can offer better PSD confinement, lower PAPR, and better performance in fading channels.
6.3 Power spectral density and PAPR
In this section, we consider the PSD and the PAPR of CBFMT. For the OFDM system, the PSD derivation is reported in [26]. In Figure 7, we report an example of PSD of CBFMT assuming the parameters equal to K = 8, N = 10,Q = 40,L = 32 (therefore M = 320), β = 0.2 and the cyclic prefix equal to μ = 8 samples. The interpolation filter is a rootraisedcosine (RRC) pulse with rolloff equal to 0.1. If the interpolation pulse were ideal, i.e., the filter was a perfect lowpass filter, the outband emissions would be null. However, a real interpolation filter introduces outband emissions. For the parameters specified, the ratio between the useful signal power and outband emissions power is equal to 25.48 dB. In OFDM, assuming the number of subchannels equal to K = 320, this ratio is equal to 22.80 dB. CBFMT has slightly better inband/outband power ratio due to a higher subchannel frequency selectivity w.r.t OFDM, under comparable complexity assumption. If we set the number of subchannels in OFDM equal to K = 8, then its inband/outofband power ratio will decrease even further to 20.1 dB.
We now consider the PAPR. The complementary cumulative distribution function (CCDF) of the PAPR for CBFMT and OFDM is shown in Figure 8. The PAPR is influenced by the inner IDFT block size. In Figure 8a, we perform a comparison under similar complexities, i.e., the number of subchannels in OFDM is set equal to K = 320, and in CBFMT, we set M = 320 and K ∈ {4,8,16,32}. In CBFMT, the PAPR is significantly lower than in OFDM for low values of K,N. In Figure 8b, we perform a comparison under an equal number of subchannels. In CBFMT, we keep the IDFT block size equal to M = 320. In this case, OFDM outperforms CBFMT due to the smaller inner IDFT size. In the simulations, a 4PSK constellation is used for both systems. As it is shown in the next section, CBFMT can offer higher spectral efficiency than OFDM with a smaller number of subchannels. In turn, this allows to obtain a lower PAPR.
In Table 2, the mean PAPR for CBFMT is shown when M = 320 and for several combinations of parameters. In OFDM, the mean PAPR is equal to 11.28 dB, i.e., higher than in CBFMT for all parameter combinations herein considered.
6.4 Performance in fading channels
In order to evaluate the performance of CBFMT, we consider the transmission over a wireless fading channel. We consider both static and timevariant channels. In particular, the channel coefficients α_{ ℓ }(n) in (49) are modeled according to Clarke’s isotropic scattering model [27]. Therefore, they are assumed to be independent stationary zeromean complex Gaussian processes with correlation defined as
where f_{ D } and J_{0}(·) are the maximum Doppler frequency and the zeroorder Bessel function of the first kind, respectively. We assume an exponential power decay profile, i.e., Ω_{ ℓ } = Ω_{0}e^{−ℓ/γ}, where Ω_{0} is a normalization constant to obtain unit average power, and γ is the normalized, w.r.t. the sampling period, delay spread. The channel impulse response is truncated at 10 dB.
First, we show the performance in terms of average SER versus signaltonoise ratio (SNR) varying the delay spread γ, considering CBFMT and OFDM both using a CP and a 1tap MMSE equalizer at the receiver. 4PSK modulation is assumed. Then, we show the maximum achievable data rate as a function of Doppler spread for different SNR values.
In Figure 9a, the CBFMT system has parameters K = 8, N = 10, M = 320, and CP with length 8 samples. For the OFDM system, we consider the number of subchannels equal to K = 64, as in the IEEE 802.11 WLAN standard. In OFDM, the CP length is set equal to 18 samples so that the two systems have identical transmission rate assuming an identical transmission bandwidth. We consider channels with normalized delay spread equal to γ = {1,2,3} and no Doppler spread (static channels). The results reveal that CBFMT can significantly lower the SER, especially for high values of delay spread γ. A 10dB SNR gain is found at SER = 10^{4}.
This is due to the fact that CBFMT in conjunction with the MMSE equalizer can exploit the frequency diversity introduced by the channel, and thus, the more dispersive the channel, the higher the gain is for CBFMT. In OFDM, the performance is identical for all values of γ considered, since the subchannels see flat Rayleigh fading [17].
In Figure 9b, we show the SER for several combinations of parameters in CBFMT. In all cases, CBFMT has M = 320, K ∈ {8,16,32,64}, N ∈ {10,20,40,80}, and a CP equal to 8 samples so that it has the same data rate of OFDM. The normalized delay spread is set to γ = 2. The SER grows with the number of subchannels K, and it approaches that of OFDM, i.e., the performance of 4PSK in flat Rayleigh fading. This is because when K increases, Q = M/K decreases and, consequentially, the ability of coherently capturing the subchannel energy (thus exploiting diversity) with the MMSE equalizer is reduced.
In Figure 10, we show the average maximum achievable rate (Shannon capacity [23]) assuming timevariant frequencyselective fading and additive white Gaussian noise. The system parameters for OFDM and CBFMT are equal to those assumed for the SER analysis in Figure 9a. Furthermore, we assume the transmission bandwidth W = 1/T = 20 MHz and the normalized delay spread γ = 2. For the OFDM system, we use a 1tap MMSE subchannel equalizer. For CBFMT, we apply two different equalizers. Firstly, we use a 1tap MMSE equalizer, similar to the OFDM equalizer. Then, we use the subchannel block equalizer described in Section 5.2. For low SNRs (equal to 15 dB in the considered case), the 1tap equalizer in CBFMT is sufficient and provides good performance which is better than that offered by OFDM also in the presence of high Doppler spreads. For high SNRs (25 dB in the considered case), CBFMT with singletap equalization provides higher achievable rate than OFDM for a Doppler below 400 Hz. For higher Doppler, the performance is dominated by the interference. Therefore, the MMSE subchannel block equalizer provides significantly higher performance than the singletap equalizer. In particular, at the maximum Doppler considered that is equal to 4 kHz, the gain in achievable rate of CBFMT over OFDM is 6%, 20% for an SNR equal to 15 and 25 dB, respectively. This shows that CBFMT has the potentiality of bettering the performance of OFDM also in the presence of channel time variations introduced by mobility of the nodes. The gains in Figure 10 are due to the fact that CBFMT is more robust to the channel time variations due to the use of frequency confined pulses that allow to lower the ICI compared to OFDM. Furthermore, as shown also in the BER curves, CBFMT with the FD equalizer can exploit the subchannel frequency diversity.
7 Conclusions
In this paper, a filter bank architecture referred to as cyclic block filtered multitone (CBFMT) modulation is presented. This scheme can be derived from the FMT architecture philosophy. However, linear convolutions are substituted with circular convolutions, and data are processed in blocks, which justifies the acronym CBFMT. The efficient implementation of CBFMT in the frequency domain has been discussed, and the performance analysis has been carried out. The main conclusions can be summarized as follows:

The computational complexity analysis shows that CBFMT can significantly lower the complexity compared to conventional FMT with even longer pulses.

The orthogonal CBFMT design can be done in the frequency domain, and a simple pulse design procedure can be followed by sampling in the FD a bandlimited Nyquist pulse. Optimal frequency localized orthogonal pulses for CBFMT can also be designed in the frequency domain as recently shown in [19].

The orthogonal CBFMT transmitted signal shows high frequency compactness and potentially lower PAPR than OFDM if a lower number of data subchannels is used (still offering the same or higher spectral efficiency).

Subchannel FD MMSE equalization provides good performance in doubleselective fading channels. In particular, lower symbol error rate and higher spectral efficiency than OFDM in multipath timevariant fading channels has been found depending on the choice of parameters.
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Tonello, A.M., Girotto, M. Cyclic block filtered multitone modulation. EURASIP J. Adv. Signal Process. 2014, 109 (2014). https://doi.org/10.1186/168761802014109
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Keywords
 Filter bank modulation
 Multicarrier modulation
 FMT
 OFDM
 Power spectral density
 PAPR
 Fading channels
 Equalization