### 4.1 Reduced-complexity reduced-power combined CCs and windowing

The combination of CCs with windowing seems to be a promising spectrum shaping mechanism. While windowing method provides better OOB interference mitigation for spectrum components more distant from occupied OFDM band on the frequency axis, the CCs method has the same behavior for components closer to the OFDM nominal band as shown in [34]. Thus, the combination of both methods, which was also presented in [34], provides additional degrees of freedom as the number of CCs and window shapes can be altered to fulfill the transmission requirements. In this section, we present several additional enhancements to this combined approach, thus yielding a reduction in the computational complexity, reduction of the energy-loss (energy inefficiency) due to the use of the CCs, and an improvement of the BER performance.

The system that we consider in this research consists of a conventional OFDM modulator, where the CCs unit, which performs the CCs algorithm, is employed prior to the *N*-size IFFT block, and windowing is applied to the time domain signal after extending it with the CP. The resulting OFDM-modulated signal after the OOB interference reduction process is then fed to the digital-to-analog converter and the IF/RF (Intermediate Frequency/Radio Frequency) front-end.

The CCs optimization formula must be designed for the time domain windowed signal y. Suppose we denote the input of the IFFT as the vector s = {*s*_{-N/2}, ..., *s*_{N/2-1}}, which contains zeros except for the elements indexed as c = {*c*_{1},..., *c*_{γ}} and **d** = {*d*_{1},..., *d*_{α}}, where the CCs symbol values and the data symbols are inserted, respectively. The optimization of cancellation symbols is based on the estimation of the spectrum values resulting from the superposition of the spectra of each CC and DC. The time-domain OFDM symbol vector **y** elements can be mathematically expressed as:

{y}_{k}={w}_{k}\sum _{n=-N/2}^{N/2-1}{s}_{n}\text{exp}\left(j2\pi \frac{nk}{N}\right),

(5)

where this OFDM-symbol spectrum estimate at frequency bin *l* can be derived by preforming the *M*-times upsampled FFT for this point *b*_{
l
}using the following approach:

\begin{array}{ll}\hfill {b}_{l}& =\sum _{k=-{N}_{\text{CP}}-\beta}^{N+\beta -1}{w}_{k}\sum _{n=-N/2}^{N/2-1}{s}_{n}\text{exp}\left(j2\pi \frac{nk}{N}\right)\text{exp}\left(-j2\pi \frac{lk}{NM}\right)\phantom{\rule{2em}{0ex}}\\ =\sum _{n=-N/2}^{N/2-1}{s}_{n}\sum _{k=-{N}_{\text{CP}}-\beta}^{N+\beta -1}{w}_{k}\text{exp}\left[j2\pi \frac{k}{N}\left(n-\frac{l}{M}\right)\right]\phantom{\rule{2em}{0ex}}\\ =\sum _{n=-N/2}^{N/2-1}{s}_{n}{p}_{n,l}.\phantom{\rule{2em}{0ex}}\end{array}

(6)

For a set of frequency-sampling points **l** = {*l*_{1},...,*l*_{
δ
}} defined in the optimization region, and for *n* ∈ **c**, the coefficients *p*_{n,l}are the elements of the matrix {\mathbf{P}}_{\text{CC}}^{\left(\delta \times \gamma \right)}, and can be pre-calculated. Similarly, for the data carriers, when *n* ∈ **d**, *p*_{n,l}defines the matrix {\mathbf{P}}_{\text{DC}}^{\left(\delta \times \alpha \right)}, and can be calculated off-line.

The commonly used optimization problem definition can be expressed using the Equation (3). Recall that the aim of this research is to minimize the OOB interference level, which implies solving the following optimization framework:

\underset{{\mathbf{s}}_{\mathbf{c}}}{\text{min}}{\u2225{\mathbf{P}}_{\text{CC}}^{\left(\delta \times \gamma \right)}{\mathbf{s}}_{\mathbf{c}}+{\mathbf{P}}_{\text{DC}}^{\left(\delta \times \alpha \right)}{\mathbf{s}}_{\mathbf{d}}\u2225}^{2},

(7)

where **s**_{
d
}and **s**_{
c
}contain complex values modulating data carriers and CCs respectively, and form the subvectors of vector s created by the **d** and **c** indexed cells, respectively. The solution of this problem yields the values of CCs **s**_{
c
}, namely:

{\mathbf{s}}_{\mathbf{c}}=-{\mathbf{P}}_{\text{CC}}^{\left(\delta \times \gamma \right)}+{\mathbf{P}}_{\text{DC}}^{\left(\delta \times \alpha \right)}{\mathbf{s}}_{\mathbf{d}},

(8)

where []^{+} denotes the pseudoinverse. Although such a solution is relatively fast with respect to computational complexity, since the multiplication of vector sd by a precalculated matrix is performed for each OFDM symbol, it suffers from several issues.

First, as shown in [32], several parts of the OFDM symbol power will need to be allocated to the CCs. Thus, in practical systems with an appropriately chosen power constraint, the SNR for the data carriers is reduced by the estimated value:

\xi =10{\text{log}}_{10}\left(\frac{\left|\right|{\mathbf{s}}_{\mathbf{c}}|{|}^{2}+|\left|{\mathbf{s}}_{\mathbf{d}}\right|{|}^{2}}{\left|\right|{\mathbf{s}}_{\mathbf{d}}|{|}^{2}}\right).

(9)

The reference system in this case is the one that employs the nulled guard sub carriers on the subcarriers used by the CCs method in the proposed system. Another significant drawback is a substantial increase of the PAPR that is caused by the high power values transmitted on the CCs correlated with the DCs. Apart from the PAPR value, usually the probability of peaks occurrence is also taken into account since it is conceivable that the time-domain peaks possessing moderate instantaneous power can cause nonlinear distortions and performance deterioration that can prove to be much worse than the high power (strong) peaks occurring relatively infrequently. On the basis of this observation, the PAPR is measured with a certain probability *p* PSPR. We will determine this metric later in this section when providing simulation results for a probability of *p* PAPR = 10^{-3}.

Finally, an important phenomenon when applying the CC method for OOB interference reduction is the occurrence of frequency-domain power peaks for frequencies assigned to the CCs. This is due to the fact that the CCs have to compensate for a number of DC sidelobes. A large power increase at the edges of the NC-OFDM frequency spectrum (where CCs are located) may be unacceptable according to the existing regulations that impose constraints on the transmission spectral masks. In order to provide some metric reflecting this problem, let us define the spectrum overshooting ratio (SOR) for a given probability *p* sor of exceeding level ϱ of the spectrum mask by the CCs power:

\text{SOR}=10{\text{log}}_{10}\left(\frac{{\text{arg}}_{\varrho}\left[\text{Pr}\left(S\left({f}_{\text{CC}}\right)>\varrho \right)={p}_{\text{SOR}}\right]}{\frac{1}{{B}_{\text{SU-CC}}}{\int}_{{B}_{\text{SU-CC}}}S\left(f\right)df}\right),

(10)

where *S*(*f*) is the power spectral density (PSD) function of the considered NC-OFDM secondary-user signal, *B*_{SU-CC} is the ba ndwidth of the considered NC-OFDM secondary-user transmission used by the data subcarriers (excluding cancellation subcarriers frequency bands), and *f*_{CC} is any one of the frequencies belonging to CCs bands. This definition for the SOR can be interpreted as the logarithm of the PSD peaks of CCs with respect to the mean power level in data carriers band. The occurrence of these peaks is measured with probability *p*_{SOR}. Note that in simulation results presented in the next subsection, *p*_{SOR} = 10^{-1} will be considered. This probabilistic approach is required to take a varying characteristic of the PSD estimate into account.

To overcome the aforementioned problem with respect to an unacceptable power increase, we propose to supplement the optimization problem described by (7) with an additional, indirect constraint whose aim is to minimize the CCs power. The optimization problem is now defined as follows:

\underset{{\mathbf{s}}_{\mathbf{c}}}{\text{min}}\left\{{\u2225{\mathbf{P}}_{\text{CC}}^{\left(\delta \times \gamma \right)}{\mathbf{s}}_{\mathbf{c}}+{\mathbf{P}}_{\text{DC}}^{\left(\delta \times \alpha \right)}{\mathbf{s}}_{\mathbf{d}}\u2225}^{2}+\mu \left|\right|{\mathbf{s}}_{\mathbf{c}}|{|}^{2}\right\},

(11)

where *μ* factor is used to balance between the CCs power and resulting OOB power reduction. The solution of this problem can be derived by merging both conditions and related matrix operations, and results in the following vector of CCs, namely:

{\mathbf{s}}_{\mathbf{c}}=-{\left(\begin{array}{c}{\mathbf{P}}_{\text{CC}}^{\left(\delta \times \gamma \right)}\hfill \\ \sqrt{\mu}{\mathbf{I}}^{\left(\gamma \times \gamma \right)}\hfill \end{array}\right)}^{+}\left(\begin{array}{c}{\mathbf{P}}_{\text{CC}}^{\left(\delta \times \alpha \right)}\hfill \\ 0\hfill \end{array}\right){\mathbf{s}}_{\mathbf{d}}=\mathbf{W}{\mathbf{s}}_{\mathbf{d}},

(12)

where **I**^{(γ × γ)} is a *γ*-size identity matrix, and **W** results from multiplication of the first two matrices in the above equation. Such an optimization has similar computational complexity to the optimization problem of (7), as only once for a given spectrum mask, and after the number of DCs and CCs are determined, the optimization (calculation of matrix **W**) is implemented. Then, for each OFDM symbol, matrix-by-vector multiplication is carried out with pre-calculated matrix **W** elements. The performance and influence on various system parameters will be evaluated in the next section.

The optimization procedure described above significantly reduces the SNR loss typically found for a CCs method. This is obtained as a result of imposing a constraint on the value of the SOR, which consequently reduces the power assigned to CCs and increases power reserved for the DCs. Nevertheless, the reduced power available for data subcarriers still cause some deterioration of the reception quality. Therefore, we propose the following reception technique that makes use of the CCs inherent redundancy.

As the CCs are correlated with data symbols, these additional subcarriers can be used in the signal reception that might not only regain the power devoted to these subcarriers in the first place, but also make use of the frequency diversity for achieving a higher degree of robustness with respect to the frequency-selective fading. Let us consider Equation (12) as a process of generating redundancy symbols **s**_{c} transmitted in parallel to data symbols **s**_{d}. This operation is conducted on the complex symbols, thus allowing us to employ the theory of complex-field block codes [38] for this problem formulation. To do so, let us rewrite Equation (12) in order to determine the systematic code generation matrix **G** of size (*γ* + *α* × *α*), namely:

\mathbf{G}=\left(\begin{array}{c}{\mathbf{I}}^{\alpha \times \alpha}\hfill \\ \mathbf{W}\hfill \end{array}\right).

(13)

By changing the row order of presented matrix, the order of data and cancellation symbols can be kept, but for simplicity we will skip this operation. A simple reception mechanism designed for such codes is based on the zero-forcing criterion, for which the reception matrix is defined as:

\mathbf{R}={\left(\mathbf{H}G\right)}^{+},

(14)

where **H** is (*γ* + *α* × *γ* + *α*) diagonal matrix with channel coefficients for each of used subcarriers on its diagonal. This matrix should be used in the receiver after the FFT processing instead of an equalizer used in standard reception chain. The estimate of the data symbols {\widehat{\mathbf{s}}}_{\mathbf{d}} is achieved by the following operation:

{\widehat{\mathbf{s}}}_{\mathbf{d}}=\mathbf{R}{\widehat{\mathbf{s}}}_{\mathbf{d}+\mathbf{c}},

(15)

where {\stackrel{\u0303}{\mathbf{s}}}_{\mathbf{d}+\mathbf{c}} is a received vertical vector at the output of FFT block containing distorted and noisy values of data and cancellation subcarriers. Although the calculation of matrix **R** can be quite complex, it needs to be performed only once for each channel instance and subcarrier pattern. Moreover, with a systematic code implementation, this method may be treated as optional, reserved only for high performance, high quality reception.

Finally, let us derive a metric that indicates the potential throughput loss caused by introduction of CCs, windowing or the combination of CCs and W. This throughput loss can be assessed in comparison to a system not employing any OOB interference reduction method, in which all subcarriers are occupied by the DCs. Note that the actual system throughput depends not only on the number of data subcarriers but also on the power assigned to these subcarriers and the channel characteristic observed. Therefore, this metric indicates only potential throughput loss that results from the information signal bandwidth reduction due to introduction of the CCs and window duration extension, thus assuming the same transmit power and channel quality at each subcarrier. It is described by the following expression:

{R}_{\text{loss}}=\left(1-\frac{1-\frac{\gamma}{\gamma +\alpha}}{1+\frac{\beta}{N+{N}_{\text{CP}}}}\right).100\%.

(16)

Note that the reference system for this definition, i.e., all subcarriers employed for data transmission, is prohibited from operating in the considered scenario, where the PU transmission protection is required and the SU sidelobes have to be reduced.

#### 4.1.1 Simulation results

Below, we present the Monte Carlo simulation results using MATLAB and showing that our introduced modifications of the combined CC and W method improves the overall performance of the NC-OFDM system in several ways. In our experiments, we assumed *N* = 256 subcarriers, where the subcarriers possessing the indices **d** = {- 100,..., -62} ∪ {-41,..., -11} ∪ {10,..., 40} ∪ {61,..., 101} are occupied by the QPSK data symbols, and there are three CCs placed on each side of data carriers blocks, i.e. c = {- 103*, -* 102, -101} ∪ {- 10, -9, -8}∪{7, 8, 9}∪{41, 42, 43}∪{58, 59, 60}∪{102, 103, 104}. The subcarriers pattern of four data subcarrier blocks is separated with narrowband PUs, e.g., *program making and special events* (PMSE) devices such as professional wireless microphones with bandwidth of 200 kHz. Note that an explanation of the wideband and narrowband PU signals and scenarios under consideration with respect to the coexistence of the PU and SU transmissions are given in the next section, with the real-world experimental results. The duration of the CP equals *N*_{CP} = 16 samples, but the *β* = 16 samples of the Hanning window extension (equal to CS) are also used on each side of an OFDM symbol. The number of CCs and shaping window duration was chosen in such a way that the mean OOB interference power level is achieved at least 40 dB below the mean in-band power level for reasonable value of *μ*, i.e., *μ* = 0.01. This OOB power attenuation is sufficient in order to respect several regulatory spectrum masks, e.g., IEEE802.11 g [39] or LTE user-equipment [40] Spectrum Emission Mask (SEM).

First, in Figure 3, we show the results of the OOB power reduction obtained for the following three methods under consideration: CC method, windowing, and combined CC and W scheme. The comparison has been performed for the schemes that present the same potential throughput loss metric, which for our evaluation system equals *R*_{loss} = 19.2%. Such a potential throughput loss is obtained either from the CC method with *γ*_{e} = 4 CCs per edge of the DCs band, from the W method with Hanning window extension of *β* = 65 samples, or from the combined CCs and W method with *γ*_{e} = 3 and *β* = 16, i.e. the scenario described above. The PSDs were obtained for the signal before HPA using Welch's method after transmitting 10,000 random OFDM symbols. The spectrum was estimated in 4*N* frequency sampling points using 3*N*-length Hanning windows. Note, that the windowing method achieves a high OOB power attenuation, but it requires several frequency guard bands for the OOB attenuation slope. Thus, it is potentially unsuitable for protecting narrowband PU signals from unintentional secondary OB interference. Conversely, the CC method alone results in a relatively steep OOB power reduction, but the resulting OOB attenuation is not very high. The combination of both methods provides decent performance in terms of high and steep OOB attenuation, thus confirming that such a combination of these methods possesses the potential for protecting both wideband and narrowband PU signals employing strict requirements with respect to the signal-to-interference-power ratio. According to our other experiments for QAM/PSK schemes and to their results not presented here, the normalized PSD plots are very similar.

In Figure 4, several of the system performance metrics described above, such as the SOR for *p* sor = 10^{-1}, PAPR increase for *p*_{PAPR} = 10^{-3}, SNR loss for BER = 10^{-4}, and OOB power attenuation, are presented in relation to the optimization constraint parameter *μ* ∈ 〈10^{-6},10^{0}〉. Thus, the optimization procedure is considered in the range of *μ*, defining scenarios from a weak constraint on the CCs power, close to no power-constraint, to a constraint on the strictly limited CCs power. For these performance metrics under consideration, measurements have been obtained after the transmission of 2 × 10^{5} OFDM symbols. The SNR loss has been calculated at the receiver, for an example 4-paths Rayleigh-fading channel defined in Case 3 test scenario for UMTS user equipment [41]. Averaging of the results has been done using 10,000 channel realizations.

It can be observed in Figure 4 that the OOB power attenuation decreases slowly with an increase of *μ* for small values of *μ*. Thus, when *μ* is low, there is no use in spending additional power on CCs since the spurious OOB emissions remain the same. On the other hand, low-power CCs (for high *μ* values) do not provide improvement in OOB power over results obtained for windowing method without the application of CCs. However, the other metrics improve when *μ* increases. For example, the fluctuation of SOR ranges from 13.7 to --3.9 dB. It is worth mentioning that the rest of the system performance metrics are calculated with respect to the reference system, which does not use windowing and CCs for OOB power reduction. Instead, the CCs are replaced with zeros. The significant improvement is observed in PAPR-increase value that approaches zero, when *μ* becomes high. Both new optimization goal defined by (11), and proposed reception algorithm have influence on the values of an SNR loss with standard detection and with our proposed detection making use of the CCs redundancy. The stronger the limit is on the CCs power (the higher *μ*) the DCs power is not wasted as much on the CCs. Thus, the SNR loss changes from 4.8 dB for *μ* = 10^{-6} to nearly 0 dB for *μ* = 10^{0}.

The results after employing our proposed detection method show that not only do the DC power levels reassigned to the CCs was recovered, but also an additional improvement was achieved thanks in part to the frequency diversity introduced by CCs treated as parity symbols of the block code. We observe that the coding gain for BER = 10^{-4} (with respect to system without CCs) varies from 6.42 dB for *μ* = 4 × 10^{-6} to 0.7 dB for *μ* = 1. For very low values of *μ*(*μ* < 4 × 10^{-6}), the SNR loss caused by the introduction of CCs becomes higher than can be compensated for even by using high power CCs, which yields a coding gain decrease. The results presented in Figure 4 show that our reception algorithm making use of the CCs redundancy yields decent performance even in the assumed case of large fragmentation of available (not occupied by the PUs) frequency bands.