# On performance analysis for optimum combining of DF relaying with fast-fading multiple correlated CCIs, correlated source-relay, and thermal noise

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## Abstract

This paper analyzes the outage probability (OP) and the average symbol error rate (SER) of decode-and-forward (DF) relaying. The paper derives closed-form expressions for the OP and the average SER with optimum combining (OC) considering fast-fading multiple correlated CCIs, the correlated source-relay, and thermal noise. It is shown that the performance of the large distance between the source and the relay is better than that of the small distance, regardless of interference fading speed at the destination. We also show that given the source-relay distance, the performance of slow-fading interference is basically better than that of fast fading, except in the low signal-to-noise-ratio (SNR) regime for the distance being small. In result, the source-relay distance is generally a more dominating factor for the performance than fading CCIs.

## Keywords

Decode-and-forward relaying Fast-fading correlated multiple CCIs Correlated source-relay Optimum combining Rayleigh fading Thermal noise Outage probability Symbol error rate## Abbreviations

- AF
Amplify-and-forward

- AWGN
Additive white Gaussian noise

- BPSK
Binary phase shift keying

- CCIs
Co-channel interferers

- DF
Decode-and-forward

- DSTC
Distributed space-time coding

- FFT
Fast Fourier transform

- MGF
Moment-generating function

- MIMO
Multiple-input multiple-output

- MRC
Maximal-ratio combining

- OC
Optimum combining

- OP
Outage probability

- OR
Opportunistic relaying

Probability density function

- SER
Symbol error rate

- SINR
Signal-to-interference-plus-noise ratio

- SNR
Signal-to-noise-ratio

- STD
Single transmission duration

- TDD
Time division duplex

## 1 Introduction

Cooperative communications have been prominent because of diversity gain [1]. In cooperative networks, there are mainly two methods, such as the amplify-and-forward (AF) relay network and the decode-and-forward (DF) relay network [2]. Then with multiple copies, the destination can achieve cooperative diversity. In order to do so, we can use maximal-ratio combining (MRC) [3] (p., 316) or optimum combining (OC) [4]. MRC maximizes the signal-to-noise-ratio (SNR), while OC maximizes the signal-to-interference-plus-noise ratio (SINR). When co-channel interferers (CCIs) are present at the destination, OC reduces CCIs’ power and increases diversity [5]. Since the analytical expressions for the outage probability (OP) and the average symbol error rate (SER) are complex for derivation, some simplified models have been used [6, 7]. Usually, thermal noise is ignored for the tractability of analytical expressions, assuming CCIs being dense, and the system models are simplified. In this case, when the effect of thermal noise is greater than that of CCIs, the analysis of the simplified model might be incorrect [8]. Therefore thermal noise is considered, but fading is still assumed to be slow so that fading in phase 1 and 2 is unchanged and constant. In this case, the analysis is limited for slow fading with MRC [8] or with OC [9]. In addition, it has been assumed that the source and the relay are always far enough to be uncorrelated, which is not always true. Sometimes they become so close that the correlation between them occurs, with which the performance degrades to some extent.

Recently, there have been many research advances in the DF relay network: the opportunistic relaying (OR) in the presence of CCIs is investigated in [10]; a new transmission scheme for selective DF relaying networks is presented, considering the employment of different modulation levels at the transmitting nodes [11]; and a joint scheme (JS) has been proposed for a multiple-relay multiple-input multiple-output (MIMO) network with a DF relaying strategy [12]. In [13], a novel distributed space-time coding (DSTC) transmission scheme for a two-path successive DF relay network is proposed. The average SER is analyzed for a wireless-powered three-node DF relaying system in Nakagami-*m* fading environment [14].

In this paper, a DF protocol is considered. It is assumed that at the relay symbol-by-symbol decoding is executed, and at the destination, full decoding is carried out [2]. We also assume that multiple correlated CCIs are fast faded, the source and relay are correlated, and thermal noise is present. To the best of our knowledge, the performance analysis for this system has not been reported. First, we derive closed-form expressions for the OP and the average SER with OC considering fast-fading multiple correlated CCIs, the correlated source-relay, and thermal noise. Second, we investigate the effects of the source-relay distance and fast/slow-fading CCIs on the performance.

The paper is organized as follows: Section 2 defines the system and channel model. In Section 3, the exact analytical expressions are derived for the OP and the average SER. Section 4 presents the analytical and simulation results, which we discuss in detail. The paper is concluded in Section 5.

## 2 System and channel model

*t*

_{1}and

*t*

_{2}. One time interval

*t*

_{1}is for phase 1 and the other

*t*

_{2}is for phase 2. Therefore, a single transmission duration (STD)

*t*

_{STD}becomes

*t*

_{1}+

*t*

_{2}. Note that under fast fading assumption, channel states change and are not constant over

*t*

_{STD}. The relay system consists of a source (

*S*), a relay (

*R*), a destination (

*D*), interferers (\( {I}_R^{(j)} \), \( j=1,2,\cdots, {N}_{I_R} \)) at the relay, and interferers (\( {I}_D^{(i)} \), \( i=1,2,\cdots, {N}_{I_D} \)) at the destination. We model thermal noise as circularly symmetric additive white Gaussian noise (AWGN). Each channel is affected by AWGN. The system and channel model is depicted in Fig. 1. (The source-relay channel correlation coefficient

*r*

_{SR}and the destination interferer channel correlation coefficient \( {r}_{I_D} \) are defined in the following sections.)

*t*

_{1}, i.e., in phase 1, the source transmits its data symbols. The received signal at the destination is expressed by:

*b*

_{0}, and

*b*

_{i}are the power transmitted by the source over the time slot

*t*

_{1}, the power transmitted by each interferer over the time slot

*t*

_{1}, the source data symbol with unit average power, and each interferer data symbol with unit average power for \( i=1,\cdots, {N}_{I_D} \), respectively, and \( {N}_{I_D} \) is the number of interferers. Furthermore, the channel propagation parameters \( {g}_0^{\left(S,D,{t}_1\right)} \) and \( {g}_i^{\left({I}_D,D,{t}_1\right)} \), \( i=1,2,\cdots, {N}_{I_D} \), \( \sim \mathbf{\mathcal{CN}}\left(0,{1}^2\right) \) are Rayleigh faded, and thermal noise \( {n}^{\left(S,D\right)}\sim \mathbf{\mathcal{CN}}\left(0,{N}_0\right) \) is complex AWGN, where the notation \( \mathbf{\mathcal{CN}}\left(\mu, \Sigma \right) \) denotes the complex circularly symmetric normal distribution with mean

*μ*and variance Σ. The received signal at the relay is expressed by:

where \( {E}_{I_R}^{t_1} \) is each interferer power over the time slot *t*_{1} and *r*_{i}, and \( j=1,\cdots, {N}_{I_R} \) are the interferer data symbols each with unit average power. The channel parameters \( {g}_i^{\left({I}_R,R,{t}_1\right)} \), \( i=1,2,\cdots, {N}_{I_R} \), \( \sim \mathbf{\mathcal{CN}}\left(0,{1}^2\right) \) are Rayleigh faded, and \( {n}^{\left(S,R,{t}_1\right)}\sim \mathbf{\mathcal{CN}}\left(0,{N}_0\right) \) is complex AWGN.

*t*

_{2}, i.e., in phase 2, if the relay correctly decodes the symbol, then it forwards the symbol to the destination. In this case, the signal at the destination is expressed by:

where \( {E}_R^{t_2} \) is the transmitter power and \( {E}_{I_D}^{t_2} \) is each interferer power. The channel parameters \( {g}_0^{\left(R,D,{t}_2\right)} \) and \( {g}_i^{\left(I,D,{t}_2\right)} \), \( i=1,2\cdots, {N}_{I_D,} \), \( \sim \mathbf{\mathcal{CN}}\left(0,{1}^2\right) \) are Rayleigh faded, and \( {n}^{\left(R,D,{t}_2\right)}\sim \mathbf{\mathcal{CN}}\left(0,{N}_0\right) \) is complex AWGN.

**g**_{0},

**g**_{i}, \( i=1,2\cdots, {N}_{I_D} \), and

*n*are (2 × 1) zero-mean complex symmetric Gaussian random vectors. For \( {P}_{\mathrm{e}}^{(R)}=1 \), the received signal at the destination is expressed by:

## 3 OP and SER derivation

_{full − ∞}, for \( {P}_{\mathrm{e}}^{(R)}=0 \) and later derive the OP, \( {P}_{\mathrm{out}}^{\left(\mathrm{non}\hbox{-} \mathrm{co}\right)}\left({\gamma}_{Th}^{\left(\mathrm{non}\hbox{-} \mathrm{co}\right)}\right) \), and the average SER, \( {\mathrm{SER}}_{\mathrm{non}\hbox{-} \infty } \), for \( {P}_{\mathrm{e}}^{(R)}=1 \). In order to obtain the decision \( {x}_{\mathrm{full}\hbox{-} \mathrm{co}}={\mathrm{w}}_{\mathrm{full}\hbox{-} \mathrm{co}}^{\dagger}\;{\mathrm{y}}_{\mathrm{full}\hbox{-} \mathrm{co}} \), the weight vector w

_{full ‐ co}is expressed by w

_{full ‐ co}=

**R**

^{−1}

**g**_{0}with the interference-plus-noise correlation matrix

**R**= N

_{0}

**I**

_{2}+

*E*

_{I}

**GG**

^{†}and \( \mathrm{G}=\left[{\boldsymbol{g}}_1{\boldsymbol{g}}_2\cdots {\boldsymbol{g}}_{N_{I_D}}\right] \) [4]. The notation

**I**

_{2}is the (2 × 2) identity matrix, and the notation (•)

^{†}is the conjugation and transposition. The instantaneous maximum output SINR at the destination is expressed as

*γ*

_{full ‐ co}is given by:

**Σ**

_{SR}, the power ratio Γ

_{0}≜

*E*

_{S}/

*N*

_{0}is the SNR over each time slot, i.e.,

*t*

_{1}or

*t*

_{2}, and the power ratio \( {\Gamma}_1\triangleq {E}_{I_D}/{N}_0 \) is the interference-to-noise ratio (INR) over each time slot, i.e.,

*t*

_{1}or

*t*

_{2}. We express the (2 × 2) Hermitian matrix

**GG**

^{†}as the eigenvalue decomposition [16].

*β*

_{1}and

*β*

_{2}with

*β*

_{1}≥

*β*

_{2}are the non-zero ordered real eigenvalues of

**GG**

^{†}and

**U**is the (2 × 2) unitary matrix. The MGF \( {M}_{\gamma_{\mathrm{full}\hbox{-} \mathrm{co}}}(s) \) is given by:

*θ*is uniformly distributed in the interval [−

*π*,

*π*) [17]. The MGF \( {M}_{\gamma_{\mathrm{co}}}(s) \) is simplified by the integration over the RV

*θ*as:

*a*| is the absolute value of a scalar

*a*. The expectation over

**Β**is obtained using the probability density function (PDF)

*f*

_{Β}(

**Β**) [18] as:

*a*

_{i, j},

*i*,

*j*= 1, 2, and the constant

*K*is given by:

*α*

_{1}and

*α*

_{2}with

*α*

_{1}≥

*α*

_{2}are the eigenvalues of the destination interferer channel parameter (2 × 2) correlation matrix \( {\boldsymbol{\Sigma}}_{I_D}\triangleq \mathbb{E}\left[{\boldsymbol{g}}_i{{\boldsymbol{g}}_i}^{\dagger}\right] \), \( i=1,2,\cdots, {N}_{I_D} \). Using the (2 × 2) determinant formula, the MGF \( {M}_{\gamma_{\mathrm{full}\hbox{-} \mathrm{co}}}(s) \) is given by:

*λ*

_{1}[19] and the RV

*λ*

_{1}be the random eigenvalue of

**R**(the other eigenvalue of

**R**is the constant

*N*

_{0}). (Note that

*λ*

_{1}and

*N*

_{0}are the eigenvalues of

**R**=

*N*

_{0}

**I**

_{2}+

*E*

_{I}

**GG**

^{†}, and

*β*

_{1}and

*β*

_{2}are the eigenvalues of

**GG**

^{†}.) Then the MGF \( {M}_{\gamma_{\mathrm{full}\hbox{-} \mathrm{co}}}(s) \) of

*γ*

_{full ‐ co}is derived as:

*λ*

_{1}≥

*N*

_{0}. Now we have derived \( {M}_{\gamma_{\mathrm{full}\hbox{-} \mathrm{co}}}(s) \) for all \( {N}_{I_D}\ge 1 \). From \( {M}_{\gamma_{\mathrm{full}\hbox{-} \mathrm{co}}}(s) \), we obtain the characteristic function (CF) \( {\phi}_{\gamma_{\mathrm{full}\hbox{-} \mathrm{co}}}(t)={M}_{\gamma_{\mathrm{full}\hbox{-} \mathrm{co}}}\left(\sqrt{-1}\;t\right) \). The PDF \( {f}_{\gamma_{\mathrm{full}\hbox{-} \mathrm{co}}}\left({\gamma}_{\mathrm{full}\hbox{-} \mathrm{co}}\right) \) of

*γ*

_{full ‐ co}is obtained from \( {\phi}_{\gamma_{\mathrm{full}\hbox{-} \mathrm{co}}}(t) \) by the Fourier transform, which is easily calculated using the fast Fourier transform (FFT). Then, \( {\mathtt{SER}}_{\mathrm{full}\hbox{-} \mathrm{co}} \) with the coherent binary phase shift keying (BPSK) is calculated as [19]:

_{full ‐ co}can be alternatively calculated by Eq. (10.20) in [19]. For a given threshold \( {\gamma}_{Th}^{\left(\mathrm{full}\hbox{-} \mathrm{co}\right)} \), the OP \( {P}_{\mathrm{out}}^{\left(\mathrm{full}\hbox{-} \mathrm{co}\right)}\left({\gamma}_{Th}^{\left(\mathrm{full}\hbox{-} \mathrm{co}\right)}\right) \) is defined and is calculated as:

_{non ‐ co}for \( {P}_{\mathrm{e}}^{(R)}=1 \). To obtain the decision \( {x}_{\mathrm{non}\hbox{-} \mathrm{co}}={\mathrm{w}}_{\mathrm{non}\hbox{-} \mathrm{co}}^{\dagger }{y}^{\left(S,D,{t}_1\right)} \), the weight w

_{non ‐ co}is expressed by \( {\mathrm{w}}_{\mathrm{non}\hbox{-} \mathrm{co}}={\mathrm{R}}_{\mathrm{non}\hbox{-} \mathrm{co}}^{-1}{g}_0^{\left(S,D,{t}_1\right)} \) with \( {\mathrm{R}}_{\mathrm{non}\hbox{-} \mathrm{co}}={N}_0\kern0.36em +{E}_{\mathrm{I}}{\left\langle {g}_i^{\left({I}_D,D,{t}_1\right)}\right\rangle}_{i=1}^{N_{I_D}}{\left({\left\langle {g}_i^{\left({I}_D,D,{t}_1\right)}\right\rangle}_{i=1}^{N_{I_D}}\right)}^{\dagger } \). The notation \( {\left\langle {a}_i\right\rangle}_{i=1}^N \) is a (1 ×

*N*) matrix with elements

*a*

_{i}, \( i=1,2,\cdots, {N}_{I_D} \). The maximum instantaneous output SINR at the destination can be expressed as:

*f*

_{X}(

*x*) ≜

*e*

^{−x},

*x*≥ 0. The chi-squared-distributed RV \( W\triangleq {\left\langle {g}_i^{\left({I}_D,D,{t}_1\right)}\right\rangle}_{i=1}^{N_{I_D}}{\left({\left\langle {g}_i^{\left({I}_D,D,{t}_1\right)}\right\rangle}_{i=1}^{N_{I_D}}\right)}^{\dagger}\sim {\chi}_{N_{I_D}}^2 \) has the PDF \( {f}_W(w)\triangleq 1/\left({N}_{I_D}-1\right)!\kern0.5em \cdot {w}^{N_{I_D}-1}{e}^{-w} \),

*w*≥ 0, with \( {N}_{I_D} \) complex degree of freedom. The RV

*Y*≜ 1/Γ

_{0}+ Γ

_{1}/Γ

_{0}

*W*has the PDF

*y*≥ 1/Γ

_{0}. Then, the RV

*γ*

_{non ‐ co}=

*X*/

*Y*is ratio distributed, and the \( {f}_{\gamma_{\mathrm{non}\hbox{-} \mathrm{co}}}\left({\gamma}_{\mathrm{non}\hbox{-} \mathrm{co}}\right) \) is derived as:

Similarly as in the \( {P}_{\mathrm{e}}^{(R)}=0 \) case, with \( {f}_{\gamma_{\mathrm{non}\hbox{-} \mathrm{co}}}\left({\gamma}_{\mathrm{non}\hbox{-} \mathrm{co}}\right) \), we calculate \( {P}_{\mathrm{out}}^{\left(\mathrm{non}\hbox{-} \mathrm{co}\right)}\left({\gamma}_{Th}^{\left(\mathrm{non}\hbox{-} \mathrm{co}\right)}\right) \) and SER_{non ‐ co} for \( {P}_{\mathrm{e}}^{(R)}=1 \).

*P*

_{out}(

*γ*

_{Th}) at the destination as:

With these exact analytical expressions, we can investigate the effects of the distance between the source and the relay, i.e., **Σ**_{SR} and fast/slow-fading interference at the destination, i.e., \( {\boldsymbol{\Sigma}}_{I_D} \).

## 4 Results and discussion

*r*

_{SR}∈ [0, 1):

We define the total SNR as \( {\Gamma}_0^{\mathrm{total}}\triangleq {E}_S^{\mathrm{total}}/{N}_0 \) and the total INR \( {\Gamma}_1^{\mathrm{total}}\triangleq {E}_{I_D}^{\mathrm{total}}/{N}_0 \), where \( {E}_S^{\mathrm{total}}={E}_S^{t_1}+{E}_R^{t_2}=2{E}_S \) and \( {E}_{I_D}^{total}=\left(2{N}_{I_D}\right){E}_{I_D} \), where the factor 2 represents two time slots. The correlation coefficients are explained as follows: the smaller the *r*_{SR} is, the larger the distance between the source node and the relay node is. On the other hand, the smaller the \( {r}_{I_D} \) is, the more independent, i.e., the less correlated, the two channel coefficients \( {g}_i^{\left({I}_D,D,{t}_1\right)} \) and \( {g}_i^{\left({I}_D,D,{t}_2\right)} \) are, for a given *i* among \( i=1,2,\cdots, {N}_{I_D} \). This means that the maximum Doppler spread is larger so that the coherence time is smaller, i.e., fast fading [21]. Thinking in the opposite direction, i.e., slow fading, is also true.

*P*

_{out}(

*γ*

_{Th}) at the destination. We assume that with \( {N}_{I_D}=2 \), \( {\Gamma}_1^{total}=\left(2{N}_{I_D}\right){\Gamma}_1=3\kern0.5em \mathrm{dB}+3\kern0.5em \mathrm{dB}+4\kern0.5em \mathrm{dB}=10\kern0.5em \mathrm{dB} \) is fixed. We also assume that there are almost uncorrelated users (

*r*

_{SR}= 0.01) and slow-fading multiple correlated CCIs (\( {r}_{I_D}=0.99 \)), which are the assumptions of the previous researches, i.e., independent users and flat-fading interference over

*t*

_{STD}. In Fig. 2, the OP performance is shown for various \( {P}_{\mathrm{e}}^{(R)} \) values. We observe in Fig. 2 that the OP performance with \( {P}_{\mathrm{e}}^{(R)}\le 0.001 \) reaches that with the full cooperative case \( {P}_{\mathrm{e}}^{(R)}=0 \). Since this paper focuses on the performance analysis for the source-relay distance and fast/slow fading CCIs, from now on, we set \( {P}_{\mathrm{e}}^{(R)}=0.001 \). In Fig. 2, we also show the analytical and simulation results, which are in good agreement, so that the following analyses are based on the analytical expressions.

*P*

_{out}(

*γ*

_{Th}) at the destination for various

*r*

_{SR}and \( {r}_{I_D} \) values. In Fig. 3, for the fixed \( {r}_{I_D}=0.99 \), i.e., slow fading CCIs (which is the previous research assumption), the OP

*P*

_{out}(

*γ*

_{Th}) at the destination is shown for various

*r*

_{SR}. We observe in Fig. 3 that as the correlation between the source and the relay becomes larger, the OP performance degrades severely and cooperative diversity decreases. It is shown in Fig. 4 that for the fixed \( {r}_{I_D}=0.01 \), i.e., fast fading CCIs (which is considered in this paper), the OP

*P*

_{out}(

*γ*

_{Th}) at the destination is shown for various

*r*

_{SR}. The results in Fig. 4 are similar with those in Fig. 3, but the patterns of the OP performance degradation are different. In order to investigate the difference, we plot the combination of Fig. 3 and Fig. 4 in Fig. 5. It is investigated in Fig. 5 that the performance of the large distance between the source and the relay is better than that of the small distance, regardless of interference fading speed at the destination. We define the impact of fast-fading CCIs on the performance as the SNR Γ

_{0}loss in decibel compared with slow-fading CCIs. We observe in Fig. 5 that given the distance between the source and the relay (

*r*

_{SR}= 0.01 or 0.99), the performance of fast-fading interference at the destination is basically worse than that of slow fading, except in the low SNR regime for the distance being small (

*r*

_{SR}= 0.99

**)**. The exception is explained as follows: since the small distance results in lost diversity, in the low SNR regime, the weak power correlated signals transmitted by the source and the relay are more vulnerable to highly correlated CCIs (\( {r}_{I_D}=0.99 \)) than almost uncorrelated CCIs (\( {r}_{I_D}=0.01 \)). (Note that if we ignored thermal noise, we could not observe the exception in the low SNR regimes.) In other words, besides the exception, DF-relaying OC more easily cancels out almost flat-fading CCIs (\( {r}_{I_D}=0.99 \)) than fast-fading CCIs (\( {r}_{I_D}=0.01 \)). Slow-fading CCIs represent highly correlated CCIs, and fast-fading CCIs represent weakly correlated CCIs. In result, the source-relay distance (

*r*

_{SR}) is generally a more dominating factor than the fading CCIs (\( {r}_{I_D} \)) at the destination, when the performance of OC for these systems is analyzed. It is also shown in Fig. 5 that the previous research assumption (\( {r}_{I_D}=0.01 \), \( {r}_{I_D}=0.99 \)) is the most optimistic. In order to further investigate the effects of various

*r*

_{SR}and \( {r}_{I_D} \) on the average SER for the DF-relaying OC system, Fig. 6 shows the SER of the BPSK modulation versus the SNR Γ

_{0}. It is clearly shown in Fig. 6 that there is the gap between the most optimistic case (\( {r}_{I_D}=0.01 \), \( {r}_{I_D}=0.99 \)) and the most conservative case (

*r*

_{SR}= 0.99, \( {r}_{I_D}=0.01 \)). The gap is about 8 dB in the SNR Γ

_{0}at the SER of 10

^{−4}. We also observe that the results in Fig. 6 are consistent with those in Fig. 5.

Now, we discuss the difference between OC and non-OC. In order to achieve cooperative diversity, OC maximizes the SINR, reduces CCIs’ power, and increases diversity. On the other hand, non-OC, such as MRC, maximizes only the SNR so that a smaller output SINR is produced and the performance is degraded severely in the presence of CCIs.

## 5 Conclusion

In this paper, we investigated the effects of the source-relay distance and fast/slow-fading CCIs on the performance of the DF-relaying OC system. Conditioned on the probability of symbol errors at the relay, we first developed the MGF of the instantaneous maximum output SINR. Using the total probability theorem, we then derived closed-form expressions for the OP and the average SER at the destination. With these analytical expressions, it was shown that the performance of the large distance between the source and the relay is better than that of the small distance, regardless of interference fading speed at the destination. Furthermore, we also showed that given the distance, the performance of slow-fading interference is basically better than that of fast fading, except in the low SNR regime for the distance being small. In result, the source-relay distance is generally a more dominating factor than the fading CCIs. Finally, we presented the average SER performance, which showed the gap between the most optimistic case and the most conservative case.

## Notes

### Acknowledgements

Not applicable.

### Funding

Not applicable.

### Availability of data and materials

Not applicable.

### Authors’ contributions

KC analyzed the outage probability (OP) and the average symbol error rate (SER) of decode-and-forward (DF) relaying. KC derived closed-form expressions for the OP and the average SER with optimum combining (OC) considering fast fading multiple correlated CCIs, the correlated source-relay, and thermal noise. KC read and approved the final manuscript.

### Competing interests

The authors declare that they have no competing interests.

### Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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