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Outage Probabilities in Infrastructure-Based Single-Frequency Relay Links Taneli Riihonen, Stefan Werner, Risto Wichman, and Jyri H¨am¨al¨ainen Helsinki University of Technology P.O. Box 3000, FI-02015 TKK, Finland Email: {taneli.riihonen, stefan.werner, risto.wichman, jyri.hamalainen} @tkk.fi Abstract—This paper considers usage of a single-frequency mode for cell coverage extension in infrastructure-based relay links. We provide new closed-form expressions for outage probability by taking into account practical constraints such as interference due to frequency reuse and signal leakage between the relay transmitter and receiver. The analysis covers both decode-andforward and amplify-and-forward protocols both in downlink and in uplink. For amplify-and-forward relaying, variable gain and fixed gain methods for transmit power normalization are discussed. Simulations show excellent agreement with theory and confirm that the single-frequency mode can be applied at the cost of tolerable signal degradation.

I. I NTRODUCTION Wireless relays are transceivers that receive, process, and then retransmit radio signals. In cellular networks, relays offer a cost-effective solution for network coverage extension or gap filling [1]. Processing in relays is usually classified into two main protocols [2], [3]: Nonregenerative relays amplifyand-forward (AF) their input signals, while regenerative relays decode-and-forward (DF) their input signals. Furthermore, there are two well-established protocols for normalizing the relay transmit power in AF relaying [4]: the relay can amplify the signal by using either a variable gain (VG) or a fixed gain (FG). Analysis and comparisons presented in this paper cover all of these protocols. We consider a single-cell setup, where a base station (BS) communicates with a mobile user equipment (UE) via a fixed infrastructure-based relay node (RN). Similar setup is studied, for example, by simulations in [5] and by measurements in [6]. The study covers both downlink (DL) and uplink (UL) direction of the communication. Especially, we concentrate on the scenario of cell coverage extension. Thus, it is assumed that the end-to-end communication always consists of two hops, i.e., direct communication between the BS and the UE is not possible. Although direct communication is not feasible, the effect of the direct signal is included in our analysis as an additive interference term. As opposed to the present trend to study mobile relaying and user cooperation as in [2]–[4], [7], we investigate a transmission system employing infrastructure-based relays. A severe constraint for mobile relays is that they cannot receive and transmit concurrently on the same frequency, because the transmitted signal loops back and interferes with the received signal. As a consequence, mobile relays need to resort to a half-duplex mode wherein two orthogonal time or frequency

channels are used for transmission which reduces the spectral efficiency by a factor of two. On the contrary, this constraint can be relaxed for fixed relays by appropriate design. Infrastructure-based RNs can mitigate the loop interference problem by employing two spatially separated antennas [8]– [11]: one for receiving and the other for transmitting. High physical isolation between the antennas has to be guaranteed, e.g., by placing one antenna on rooftop and the other antenna on street level. Another possible setup is outdoor-to-indoor transmission where the backhaul antenna is placed outside of a building and the indoor service antenna fills a BS coverage area gap. Furthermore, the antennas can be directive. In literature [12]–[14], interference cancellation by subtracting an estimated loop signal from the relay input is shown to reduce further the need for physical isolation between the two antennas. In practice, loop interference cancellation is imperfect which may degrade the end-to-end transmission. As loop interference does not impose an insurmountable constraint for infrastructure-based RNs, spectral efficiency can be improved over half-duplex operation by applying frequency reuse. Thus, we propose that the infrastructure-based RN operates in a single-frequency mode, in which the RN forwards a continuous stream of input symbols on the same frequency that it uses for receiving. Each received symbol is forwarded after a processing delay. However, as was discussed earlier, singlefrequency operation will suffer from signal quality degradation due to an interfering first-hop transmission (admittedly weak) and imperfect loop interference cancellation. The main contribution of the paper is the derivation of new closed-form outage probability expressions for the singlefrequency relay link. The outage probability expressions facilitate performance analysis that takes into account the effects of loop interference and frequency reuse. Finally, our discussion indicates that single-frequency operation can be implemented with tolerable signal degradation while spectral efficiency is improved when compared to half-duplex operation. The rest of the paper is organized as follows. In the next section, we introduce the system model of the infrastructurebased single-frequency relay link, and explain the channel models selected for performance analysis. In Section III, we derive closed-form outage probability expressions that are exploited for performance evaluation and for comparison of the relaying protocols in Section IV. Finally, we present paper’s conclusions in Section V.

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II. S YSTEM M ODEL In this section, we introduce a signal model for the singlefrequency relay link and derive an expression for the end-toend signal-to-interference and noise ratio (SINR) in amplifyand-forward relaying. As illustrated in Fig. 1, a source (S) transmits continuous stream of symbols to a destination (D) via a relay (R) while the communication is degraded by interfering first-hop and loop echo signals. Depending on the direction of the communication (DL/UL), the source and the destination nodes are either the BS or the UE. hLI

hSR

R

hRD

hSD

S

D

and DF protocols differently. By recursive substitution of (1) and (2) we see that the output signal is t[i] = β

∞ 

(hLI β)

j−1

(hSR x[i − jτ ] + nR [i − jτ ])

(3)

j=1

with AF. It is not meaningful to derive similar expression for DF, because the feedback loop just causes symbol or bit errors in t[i] instead of accumulation of a signal. To allow fair performance comparison, the transmit powers of all considered relaying protocols are normalized to unity. With DF, it is straightforward to fix the instantaneous transmit power (Ex {|t[i]|2 } = 1). But with AF, there are two common protocols for transmit power normalization. The relay can either normalize the instantaneous transmit power by using a variable gain (VG), or it can normalize the average transmit power (Eh {Ex {|t[i]|2 }} = 1) by using a fixed gain (FG). These conditions result in the amplification factors    1 2 −2 |hSR |2 + |hLI |2 + σR , with VG β= (4)   1 2 2 2 −2 Eh {|hSR | } + |hLI | + σR , with FG. Finally, the received signal in the destination is given by

Fig. 1.

System model of a two-hop single-frequency relay link.

y[i] = hRD t[i] + hSD x[i] + nD [i],

The system consists of four frequency flat channels, namely source–relay (SR), loop interference (LI), relay–destination (RD) and source–destination (SD) channels. We first present the signal model with fixed instantaneous channel representations, and the fading models selected for the channels are explained in the end of the section. A. Signal Model The source transmits signal x[i] to the relay with a normalized transmit power Ex {|x[i]|2 } = 1 while the transmission is also overheard by the destination. The relay operates in a single-frequency mode, i.e., it concurrently receives signal r[i] and transmits signal t[i]. Thus, the relay receives a combination of the signal transmitted by the source, loop interference and receiver noise: r[i] = hSR x[i] + hLI t[i] + nR [i],

(1)

where hSR is the flat-fading source–relay channel, hLI is the loop interference channel between the transmit antenna and the receive antenna of the relay, and the power of the noise term 2 . If the relay exploits any loop nR [i] is Ex {|nR [i]|2 } = σR interference cancellation algorithm, hLI denotes the virtual channel remaining after imperfect cancellation. The relay introduces a processing delay of τ ≥ 1 symbols. The transmit signal of the relay is then  βr[i − τ ], with AF t[i] = (2) x ˜[i − τ ], with DF, where β > 0 is an amplification factor and x ˜[i − τ ] is a regenerated version of the relay input signal. The feedback loop between the relay transmission and reception affects AF

(5)

where hRD and hSD are the respective flat-fading relay– destination and source–destination channels, and the power of 2 . the noise term nD [i] is Ex {|nD [i]|2 } = σD In the performance analysis, we parametrize the system with channel signal-to-noise ratio (SNR) values to simplify notations. The instantaneous channel SNRs are defined as 2 2 2 , γRD = |hRD |2 /σD , γSD = |hSD |2 /σD , γSR = |hSR |2 /σR 2 2 and γLI = |hLI | /σR . Similarly, the average channel SNRs are 2 2 , γ¯RD = Eh {|hRD |2 }/σD , defined as γ¯SR = Eh {|hSR |2 }/σR 2 2 2 2 γ¯SD = Eh {|hSD | }/σD , and γ¯LI = Eh {|hLI | }/σR . B. End-to-End SINR in Amplify-and-Forward Outage probabilities for the DF protocol are calculated based on the SINRs of the two hops. But in order to calculate outage probabilities for the AF protocol, an expression for the end-to-end SINR is needed. By assuming that all signal and noise samples are mutually independent, the instantaneous relay transmit power is calculated from (3) as 2

Ex {|t[i]| }

=

β

2

∞  

|hLI |2 β 2

j−1 

2 |hSR |2 + σR



(6)

j=1

=

β2

2 |hSR |2 + σR . 1 − |hLI |2 β 2

(7)

The sum in (6) converges, because with both VG and FG pro1 tocols, the relay gain is limited by β 2 < |hLI |2 . This condition prevents oscillation and guarantees finite relay transmit power. Again by requiring signal and noise independence, the receive power in the destination is calculated from (5) as 2 Ex {|y[i]|2 } = |hRD |2 Ex {|t[i]|2 }+|hSD |2 Ex {|x[i]|2 }+σD . This expression with substitution of (7) can be further reorganized

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into a sum of desired signal power, loop interference power, source–destination interference power, and noise power: Ex {|y[i]|2 }

|hSR |2 β 2 |hRD |2  2  2 β |hRD |2 + |hSR |2 + σR =

|hLI |2 β 2 1 − |hLI |2 β 2 2 2 2 2 2 + |hSD | + β |hRD | σR + σD . (8)

As |hSD |2 is small, when the relay is employed for cell coverage extension, it is reasonable to embed the source– destination interference in the destination receiver noise. Thus, it is assumed that the destination does not exploit any equalization technique for separating the two-hop transmission from the direct transmission. By dividing the desired signal power by the interference and noise power, the instantaneous end-to-end SINR can be expressed from (8) with simplification as |hSR |2 |hRD |2  γ =  (|h |2 +σ2 )|h |2 SR LI 2 |h R 2 + σ 2 2 RD | + R 1/β −|hLI |

2 |hSD |2 +σD β2

.

(9)

Finally by substituting the amplification factors from (4), the end-to-end SINR for the AF protocol is given by  γR γD with VG γR +γD +1 , (10) γ= γR γD , with FG, γ ¯R +γD +1 where the instantaneous and average receive SINRs in the relay (first hop SINR) and the instantaneous receive SINR in the destination (second hop SINR) are given by γR = γγLISR +1 , ¯SR γRD and γ = , respectively. γ¯R = γγLI D +1 γSD +1 C. Channel Models In order to calculate the outage probabilities, we have to adopt certain models for the channels in the system. The channel models then define the probability distributions of the channel SNRs. First of all, we assume that all fading channels are mutually independent and subject to block-fading, i.e., they remain stationary during a transmission block and change independently from block to block. The BS and the RN are fixed nodes in the infrastructurebased relay link. Thus, we can reasonably approximate that BS–RN (DL), RN–BS (UL) and loop interference channels are non-fading. In practice, some small channel fluctuation would happen due to movement of surrounding objects, but most of the channel components would be static. Thus, the most realistic results would be obtained from simulations with Rice-fading channels (high K-factor) as in [15]. An accurate approximation of such a Rician channel, that still allows us to avoid resorting to simulations, is the additive white Gaussian noise channel assumed in our analysis. Furthermore, in a slow fading environment, the following analysis represents the instantaneous performance within a channel coherence time. The UE is assumed to be a mobile terminal without lineof-sight connection to the RN or to the BS. Thus, we model the BS–UE (DL), UE–BS (UL), RN–UE (DL), and UE–RN (UL) channels as Rayleigh fading. The corresponding channel SNRs become exponential random variables.

III. O UTAGE P ROBABILITY A NALYSIS In this section, we evaluate the performance of the system with the different relaying protocols in terms of the outage probability Pout [16]. In other words, we consider the probability that the end-to-end relay link cannot support a desired performance level defined by a threshold SINR γth > 0. With AF, the outage probability is given by the cumulative distribution function (CDF), Fγ , of the end-to-end SINR [7]. A DF relay link experiences an outage, if either of the hops is in outage, i.e., the performance is always determined by the weaker hop [7]. Thus, outage probability is expressed as  P (γ < γth ) = Fγ (γth ), with AF (11) Pout (γth ) = P (min{γR , γD } < γth ) with DF, where P (Z) denotes the probability of event Z. In the following, we study downlink and uplink separately. A. Downlink In DL, the source node is the BS and the destination node is the UE. Following the reasoning presented in Section II-C, the channel SNRs are modeled as γSR = γ¯SR , γLI = γ¯LI , γRD ), and γSD ∼ Exp(1/¯ γSD ). Now X ∼ γRD ∼ Exp(1/¯ ¯ denotes that X is an exponential random variExp(1/X) ¯ probability distribution function (PDF) able with average X, ¯ ¯ −s/X ¯ , and CDF FX (s) = 1 − e−s/X . fX (s) = (1/X)e ¯SR The receive SINR in the relay is fixed: γR = γ¯R = γ¯γLI +1 . Thus, the CDF is simply  0, s < γ¯R (12) FγR (s) = 1, s ≥ γ¯R . The receive SINR in the destination is given by γD = i.e., its CDF is calculated as ∞ FγRD (s u + s) fγSD (u)du FγD (s) =

γRD γSD +1 ,

0

γ¯RD − s e γ¯RD . (13) γ¯RD + γ¯SD s 1) Amplify-and-forward: Due to the channel models adopted for the infrastructure-based relay link, VG and FG protocols are equivalent in DL. By noting γD = (¯γγ¯RR+1)γ −γ from (10), we can use a transform of random variables:  1,  γth > γ¯R  (14) Pout (γth ) = [¯ γR +1]γth FγD γ¯R −γth , otherwise. =

1−

After substitution, the outage probability for the AF protocol in DL is given by (¯ γR +1)γth 1 − e (¯γR −γth )¯γRD (15) Pout (γth ) = 1 − (¯ γR +1)γth 1 + (¯γR −γth )¯γRD γ¯SD for γth < γ¯R , and Pout (γth ) = 1 for γth ≥ γ¯R . 2) Decode-and-forward: By substituting (12) and (13) into (11), the outage probability in DL with the DF protocol can be calculated as Pout (γth )

1 − [1 − FγR (γth )] [1 − FγD (γth )] (16)  1, γth > γ¯R γ − γ¯ th (17) = 1 RD , 1 − 1+ γth e otherwise. γ ¯

=

γ ¯RD

SD

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IV. D ISCUSSION

B. Uplink In UL, the source node is the UE and the destination node is the BS. Following the reasoning presented in Section II-C, the γSR ), γLI = γ¯LI , channel SNRs are modeled as γSR ∼ Exp(1/¯ γSD ). γRD = γ¯RD , and γSD ∼ Exp(1/¯ The receive SINR in the relay is given by γR = γ¯γLISR +1 ∼ +1 ), i.e., its CDF is Exp( γ¯γLI ¯SR −

FγR (s) = 1 − e

γ ¯LI +1 γ ¯SR s

− γ¯s

=1−e

R

.

(18)

Furthermore, the receive SINR in the destination is γD = γ ¯RD γ ¯RD −γD . Therefore, the CDF can be γSD +1 , i.e., γSD = γD calculated with a transform of random variables:  1, s > γ¯RD   FγD (s) = 1 − FγSD γ¯RDs −s , otherwise  1, s > γ¯RD γ ¯ −s = (19) − γ¯RD s SD e , otherwise. 1) Amplify-and-forward: In UL, the VG and FG protocols result in different performance due to the adopted channel models. With the VG protocol, solving γR from (10) results in   γ ¯RD + 1 γ γSD +1 γR (γSD , γ) = . (20) γ ¯RD γSD +1 − γ Thus, the outage probability can be calculated as γ¯RDγ −γth th {1 − FγR [γR (u, γth )]} fγSD (u)du. Pout (γth ) = 1 −

In this section, we apply the closed-form performance expressions derived in the previous section for evaluating the effects of loop interference and frequency reuse, and for comparing the different relaying protocols. For the discussion, we select a test setup, where the average BS–RN/RN–BS channel SNRs (¯ γSR in DL, γ¯RD in UL) are 15dB and the average RN–UE/UE–RN channel SNRs (¯ γRD in DL, γ¯SR in UL) are 20dB. In all figures, we have also simulated the performance to verify the analytical expressions. Basically, our simulator calculates the same outage probabilities using the Monte Carlo method, and thus we see perfect agreement between analytical and simulation results. The outage probability is illustrated in Fig. 2 for the case when the average powers of the loop interference and the interference due to frequency reuse are at the same level as the receiver noise power (¯ γSD = γ¯LI = 0dB). We notice that the behavior in DL and in UL is quite different. In DL, the endto-end SINR is highly limited by the first-hop SINR, which is shown, for example, by the sudden increase to total outage at γth = 12dB with the DF protocol. On the other hand, the end-to-end SINR limitation due to the RN-BS channel SNR is less critical in UL. The FG and VG protocols are equivalent in DL, but the FG protocol is worse than the VG protocol in UL, except when the threshold SINR is large. The DF protocol is better than the AF protocol both in DL and in UL, but it requires more complex processing in the relay. 1

0

for γth < γ¯RD and Pout (γth ) = 1 for γth ≥ γ¯RD . This expression exploits the untabulated integral I(a, b) that can be computed as discussed in Appendix, see (25). With the FG protocol, solving γR from (10) results in γR = γR +1)+¯ γRD )γ/¯ γRD . Thus, the outage probability ((γSD +1)(¯ is calculated as

 ∞ [¯ γR + 1]γth + γth fγSD (u)du Pout (γth ) = F γR γ¯RD /[u + 1] 0 γ ¯ +¯ γ +1 1 − Rγ¯ γ¯RD γth R RD = 1− e . (23) th 1 + (¯γγ¯RR+1)γ ¯SD γ ¯RD γ 2) Decode-and-forward: With the DF protocol, the outage probability expression is obtained by substituting (18) and (19) into (16), which results in −

Pout (γth ) = 1 − (1 − e

γ ¯RD −γth γ ¯SD γth

γ

− γ¯th

)e

R

for γth ≤ γ¯RD and Pout (γth ) = 1 for γth > γ¯RD .

(24)

0.8 0.7

Pout (γth )

(21) for γth < γ¯RD . By changing the integration variable, the outage probability of the VG protocol can be finally expressed as th γ¯RD − γth γ¯1 − γ¯γRD γ−γ th ¯SD e R Pout (γth ) = 1 − γth γ¯SD 

1 [¯ γRD + 1]γth γth − γ¯RD (22) × I + , γ¯R [¯ γRD − γth ]¯ γR γth γ¯SD

AF in DL DF in DL AF with VG in UL AF with FG in UL DF in UL

0.9

0.6 0.5 0.4 0.3 0.2 0.1 0

0

2

4

6

8

10

12

14

16

18

γth [dB] Fig. 2. Outage probability at the mid-SNR range. The average BS–RN/RN– BS channel SNRs are 15dB, the average RN–UE/UE–RN channel SNRs are ¯LI = 0dB. The markers illustrate the simulated values. 20dB, and γ ¯SD = γ

The system suffers from two interfering signals due to the single-frequency mode. The effect of loop interference on outage probability is illustrated in Fig. 3 when γ¯SD = 0dB. Similarly, the effect of interference from frequency reuse is illustrated in Fig. 4 when γ¯LI = 0dB. In terms of the outage probability, the FG protocol is clearly inferior to the other protocols in UL. We see that the loop interference power limits

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0.5

0.5

AF in DL DF in DL AF with VG in UL AF with FG in UL DF in UL

0.45 0.4

0.4 0.35

Pout (5dB)

Pout (5dB)

0.35 0.3 0.25 0.2

0.3 0.25 0.2

0.15

0.15

0.1

0.1

0.05

0.05

0 −10

AF in DL DF in DL AF with VG in UL AF with FG in UL DF in UL

0.45

−8

−6

−4

−2

0

2

4

6

8

10

0 −10

−8

−6

−4

−2

Fig. 3. Outage probability for threshold SINR γth = 5dB with varying loop interference power. The average BS–RN/RN–BS channel SNRs are 15dB, the average RN–UE/UE–RN channel SNRs are 20dB, and γ ¯SD = 0dB. The markers illustrate the simulated values.

the performance more strictly in DL than in UL, and, vice versa, the high-power interference due to frequency reuse has more deteriorating effect on UL than on DL. Furthermore, the effect of both interfering signals is small, if the interference power does not exceed the corresponding receive noise power with more than three decibels. Then the effect of interference cannot be distinguished from that of the total combined relay and destination input noise, while spectral efficiency is doubled when comparing to the half-duplex mode.

0

2

4

6

A PPENDIX : C OMPUTATION OF I NTEGRAL I(a, b) In this appendix, we briefly explain the computation of the integral ∞ 1 a 1 −(au+ b ) u du e−( u +bu) du = e (25) I(a, b) = u2 0 1 required for evaluating (22). We limit the study to the case a > 0, b < 0, because only this case is needed in the analysis. We start by writing b

This paper analyzed the performance of an infrastructurebased relay link used for cell coverage extension. Our analysis included all of the common relaying protocols (amplify-andforward with fixed or variable gain, and decode-and-forward), and covered both downlink and uplink directions of the communication link. Due to the fixed nature of the relay, an attractive, and more spectrally efficient, alternative is to let the relay operate in a single-frequency mode instead of a half-duplex mode required for mobile relays. The increase in spectral efficiency comes at the expense of additional interference due to frequency reuse and nonideal isolation between relay transmitter and receiver. This extra interference was taken into account when setting up the system model and deriving new closed-form expressions for the outage probability. The performance expressions were exploited for comparing the different relaying protocols, and for studying the effect of the various sources of interference associated with the single-frequency mode. Our results showed that the single-frequency mode can be implemented with only minor increase in outage probability, if the interfering signals are not considerably stronger than the receiver noise.

10

Fig. 4. Outage probability for threshold SINR γth = 5dB with varying interference power due to frequency reuse. The average BS–RN/RN–BS channel SNRs are 15dB, the average RN–UE/UE–RN channel SNRs are 20dB, and γ ¯LI = 0dB. The markers illustrate the simulated values.

1

e−(au+ u ) = e−b(u+ u ) · e−(a−b)u . V. C ONCLUSIONS

8

γ¯SD [dB]

γ¯LI [dB]

(26)

By using [17, Eq. 9.6.33], we find that the first term in the product admits the expression ∞ 

1

e−b(u+ u ) =

uk Ik (2|b|),

u = 0,

(27)

k=−∞

where Ik is the modified Bessel function. After combining (25) and (27), we obtain ∞ ∞  Ik (2|b|) uk−2 e−(a−b)u du. (28) I(a, b) = k=−∞

1

If k ≤ 1 then the integral in the sum can be expressed in terms the exponential integral, see [17, Eq. 5.1.4]. On the other hand, if k > 1 then incomplete gamma function can be used, see [17, Eq. 6.5.3]. Let us divide sum (28) into two parts, namely to S1 and S2 , depending on the special function that is applied. Then, after some elementary manipulations we get S1 S2

= =

∞  k=1 ∞  k=0

Ik−2 (2|b|)Ek (a − b), Ik+2 (2|b|)

Γ(k + 1, a − b) . (a − b)k+1

(29) (30)

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