A New Look at DualHop Relaying:
Performance Limits with Hardware Impairments
Abstract
Physical transceivers have hardware impairments that create distortions which degrade the performance of communication systems. The vast majority of technical contributions in the area of relaying neglect hardware impairments and, thus, assumes ideal hardware. Such approximations make sense in lowrate systems, but can lead to very misleading results when analyzing future highrate systems. This paper quantifies the impact of hardware impairments on dualhop relaying, for both amplifyandforward and decodeandforward protocols. The outage probability (OP) in these practical scenarios is a function of the effective endtoend signaltonoiseanddistortion ratio (SNDR). This paper derives new closedform expressions for the exact and asymptotic OPs, accounting for hardware impairments at the source, relay, and destination. A similar analysis for the ergodic capacity is also pursued, resulting in new upper bounds. We assume that both hops are subject to independent but nonidentically distributed Nakagami fading. This paper validates that the performance loss is small at low rates, but otherwise can be very substantial. In particular, it is proved that for high signaltonoise ratio (SNR), the endtoend SNDR converges to a deterministic constant, coined the SNDR ceiling, which is inversely proportional to the level of impairments. This stands in contrast to the ideal hardware case in which the endtoend SNDR grows without bound in the highSNR regime. Finally, we provide fundamental design guidelines for selecting hardware that satisfies the requirements of a practical relaying system.
I Introduction
The use of relay nodes for improving coverage, reliability, and qualityofservice in wireless systems has been a hot research topic over the past decade, both in academia [2, 3, 4] and in industry [5, 6]. This is due to the fact that, unlike macro base stations, relays are lowcost nodes that can be easily deployed and, hence, enhance the network agility. The vast majority of works in the context of relaying systems make the standard assumption of ideal transceiver hardware.
However, in practice, hardware suffers from several types of impairments; for example, phase noise, I/Q imbalance, and high power amplifier (HPA) nonlinearities among others [7, 8, 9]. The impact of hardware impairments on various types of singlehop systems was analyzed in [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. For instance, I/Q imbalance was considered in [12] and it was shown to attenuate the amplitude and rotate the phase of the desired constellation. Moreover, it creates an additional imagesignal from the mirror subcarrier, which leads to a symbol error rate floor. In addition, [13] characterized the effect of nonlinear HPAs as a distortion of the constellation position plus an additive Gaussian noise. The authors therein showed that, in the presence of HPA nonlinearities, the biterrorrate increases compared to linear HPAs; for severe nonlinearities, an irreducible error floor emerges. Hardware impairments are typically mitigated by compensation algorithms, but there are always residual impairments [8, 9, 10]. As a general conclusion, hardware impairments have a deleterious impact on the achievable performance [11, 12, 10, 13, 14, 18, 19, 15, 16, 17]. This effect is more pronounced in highrate systems, especially those employing inexpensive hardware [8]. Recent works in information theory have demonstrated that nonideal hardware severely affects multiantenna systems; more specifically, [18] proved that there is a finite capacity limit at high signaltonoise ratio (SNR), while [19] provided a general resource allocation framework where existing signal processing algorithms are redesigned to account for impairments.
Despite the importance of transceiver hardware impairments, their impact on oneway relaying^{1}^{1}1Analysis of twoway AF relaying was conducted in our recent paper [20]. has only been partially investigated; bit error rate simulations were conducted in [15] for amplifyandforward (AF) relaying, while [16, 17] derived expressions for the bit/symbol error rates considering only nonlinearities or I/Q imbalance, respectively. Most recently, [21, 22] elaborated on the impact of I/Q imbalance on AF relaying and suggested novel digital baseband compensation algorithms. In this paper, we follow a different line of reasoning by providing a detailed performance analysis of dualhop relaying systems in the presence of aggregate transceiver impairments, both for AF and decodeandforward (DF) protocols. To the best of our knowledge, this is the first paper presenting an analytical study of relaying with transceiver impairments under the generalized system model of [8, 9, 10, 11]. The paper makes the following specific contributions:

After obtaining the instantaneous endtoend signaltonoiseplusdistortion ratios (SNDRs) for both AF and DF relaying, we derive new closedform expressions for the exact outage probability (OP) of the system. This enables us to characterize the impact of impairments for any arbitrary SNR value. New upper bounds on the ergodic capacity are also provided. Note that our analysis considers Nakagami fading, which has been extensively used in the performance analysis of communication systems.

In order to obtain more engineering insights, we elaborate on the highSNR regime and demonstrate the presence of a socalled SNDR ceiling. This fundamental ceiling is explicitly quantified and its value is shown to be inversely proportional to the level of impairments. This observation manifests that both AF and DF relaying systems are intimately limited by hardware impairments—especially at high SNRs and when high rates are desirable. On a similar note, the ergodic capacity exhibits a socalled capacity ceiling.

In the last part of the paper, we provide some design guidelines for optimizing the performance of hardwareconstrained relaying systems. These results are of particular importance when it comes down to finding the lowest hardware quality (i.e., highest level of impairments) that can theoretically meet stipulated requirements.
The remainder of the paper is organized as follows: In Section II, the signal and system models, for both ideal and impaired hardware, are outlined. For the sake of generality, we consider both dualhop AF and DF relaying and assume that both hops are subject to independent and nonidentically distributed fading. In Section III, an OP analysis is pursued that can be applied for any type of fading and is specialized to the cases of Nakagami and Rayleigh fading. A similar analysis for the ergodic capacity is performed in Section IV, which results in new upper bounds. The performance limits of hardwareconstrained relaying systems in the highSNR regime are examined in Section V and some fundamental design guidelines are also obtained. Our numerical results are provided in Section VI, while Section VII concludes the paper.
Ia Notation
Circularlysymmetric complex Gaussian distributed variables are denoted as where is the mean value and is the variance. Gamma distributed variables are denoted as , where is the shape parameter and is the scale parameter. The expectation operator is denoted and is the probability of an event . The operator denotes a definition. The gamma function of an integer satisfies .
Ii Signal and System Model
This paper revisits classical dualhop relaying where a source communicates with a destination through a relay; see Fig. 1(a). There is no direct link between the source and the destination (e.g., due to heavy shadowing), but the results herein can be extended to that scenario as well. Contrary to most prior works, we consider a generalized system model that accounts for transceiver hardware impairments. This model is described in the following subsections and the block model is shown in Fig. 1(b).
Iia Preliminaries on Distortion Noise from Impairments
We first describe a generalized system model for singlehop transmission that originates from [8, 9, 10, 11]. Suppose an information signal is conveyed over a flatfading wireless channel with additive noise . This channel can, for example, be one of the subcarriers in a multicarrier system based on orthogonal frequencydivision multiplexing (OFDM) [23]. The received signal is conventionally modeled as
(1) 
where , , and are statistically independent. However, physical radiofrequency (RF) transceivers suffer from impairments that are not accurately captured in this way. Informally speaking, such impairments 1) create a mismatch between the intended signal and what is actually generated and emitted; and 2) distort the received signal during the reception processing. This calls for the inclusion of additional distortion noise sources that are statistically dependent on the signal power and channel gain.
Detailed distortion models are available for different sources of impairments (e.g., I/Q imbalance, HPA nonlinearities, and phasenoise); see [8] for a detailed description of hardware impairments in OFDM systems and related compensation algorithms. However, the combined influence at a given flatfading subcarrier is often wellmodeled by a generalized channel model [8], where the received signal becomes
(2) 
while are distortion noises from impairments in the transmitter and receiver, respectively [8]. The distortion noises are defined as
(3) 
which is a model that has been supported and validated by many theoretical investigations and measurements (see e.g., [13, 24, 9, 10, 11] and references therein). The design parameters are described below. The joint Gaussianity in (3) is explained by the aggregate effect of many impairments.^{2}^{2}2Note that the Gaussian assumption holds particularly well for the residual distortion when compensation algorithms are applied to mitigate hardware impairments [9]. For a given channel realization , the aggregate distortion seen at the receiver has power
(4) 
Thus, it depends on the average signal power and the instantaneous channel gain . Note that this dependence is not supported by the classical channel model in (1), because the effective distortion noise is correlated with the channel and is not Gaussian distributed.^{3}^{3}3The effective distortion noise can be seen as two independent jointly Gaussian variables and that are multiplied with the fading channel . The effective distortion noise is thus only complex Gaussian distributed when conditioning on a channel realization, while the true distribution is the product of the complex Gaussian distribution of the distortion noise and the channel fading distribution. This becomes a complex double Gaussian distribution under Rayleigh fading [25], while the distribution under Nakagami fading does not admit any known statistical characterization.
The design parameters characterize the level of impairments in the transmitter and receiver hardware, respectively. These parameters are interpreted as the error vector magnitudes (EVMs). EVM is a common quality measure of RF transceivers and is the ratio of the average distortion magnitude to the average signal magnitude.^{4}^{4}4The EVM at the transmitter is defined as [26]. 3GPP LTE has EVM requirements in the range , where smaller values are needed to support the highest spectral efficiencies [27, Sec. 14.3.4]. Since the EVM measures the joint impact of different hardware impairments and compensation algorithms, it can be measured directly in practice (see e.g., [26]). As seen from (4) it is sufficient to characterize the aggregate level of impairments of the channel, without specifying the exact contribution from the transmitter hardware () and the receiver hardware (). This observation is now formalized.
Lemma 1
The generalized channel in (2) is equivalent to
(5) 
where the independent distortion noise describes contributions from hardware impairments at both the transmitter and the receiver, such that .
IiB System Model: Relaying with NonIdeal Hardware
Consider the dualhop relaying scenario in Fig. 1. Let the transmission parameters between the source and the relay have subscript 1 and between relay and destination have subscript 2. Using the generalized system model in Lemma 1, the received signals at the relay and destination are
(6) 
where are the transmitted signals from the source and relay, respectively, with average signal power . In addition, represents the complex Gaussian receiver noise and is the distortion noise (introduced in Section IIA), for . The distortion noise from hardware impairments (after conventional compensation algorithms have been applied) acts as an unknown noiselike interfering signal that goes through the same channel as the intended signal, thus making (6) fundamentally different from a conventional multipleaccess channel, where each user signal experiences independent channel fading.
The channel magnitudes are modeled as independent but nonidentically distributed Nakagami variates, such that the channel gains . These are Gamma distributed with integer^{5}^{5}5The assumption of integer shape parameters is made to facilitate the, otherwise tedious, algebraic manipulations for the Nakagami fading case. shape parameters and arbitrary scale parameters .^{6}^{6}6We recall that Nakagami fading reduces to the classical Rayleigh fading with variance when and ; thus, Nakagami fading brings more degreesoffreedom for describing practical propagation environments and has been shown to provide better fit with real measurement results in various multipath channels [28]. In this case, the cumulative distribution functions (cdfs) and probability distribution functions (pdfs) of the channel gains, , are
(7)  
(8) 
for . Note that most of the analysis in this paper is generic and applies for any fading distribution. The choice of Nakagami fading is only exploited for deriving closedform expressions for quantities such as the OP and ergodic capacity. For any fading distribution, the quantity
(9) 
is referred to as the average SNR, for . The average fading power is under Nakagami fading.
Remark 1 (High SNR)
The level of impairment depends on the SNR [11, 29, 19]. In most of our analysis, we consider an arbitrary fixed and thus can be taken as a constant. However, some remarks are in order for our highSNR analysis in Section V. As seen from (9), a high SNR can be achieved by having high signal power and/or high fading power . If we increase the signal power to operate outside the dynamic range of the power amplifier, then the level of impairments increases as well due to the HPA nonlinearities [29]. Advanced dynamic power adaptation is then required to maximize the performance [14]. If we, on the other hand, increase the fading power (e.g., by decreasing the propagation loss) then it has no impact on . For brevity, we keep the analysis clean by assuming that any change in SNR is achieved by a change in the average fading power, while the signal power is fixed. We stress that the upper bounds and necessary conditions derived in Section V are also valid when the signal power is increased, but then they will be optimistic and no longer tight in the highSNR regime.
In the next subsections, we derive the endtoend SNDRs for AF and DF relaying, respectively.
IiC EndtoEnd SNDR: AmplifyandForward Relaying
The information signal should be acquired at the destination. In the AF relaying protocol, the transmitted signal at the relay is simply an amplified version of the signal received at the relay: for some amplification factor . With nonideal () hardware, as described by (6), the received signal at the destination is now obtained as
(10)  
where the amplification factor is selected at the relay to satisfy its power constraint. The source needs no channel knowledge. If the relay has instantaneous knowledge of the fading channel, , it can apply variable gain relaying with [30]. Otherwise, fixed gain relaying with can be applied using only statistical channel information [3].^{7}^{7}7The relay then has a longterm power constraint where expectation is taken over signal, noise, and channel fading realizations. For fixed and variable gain relaying, reads respectively as
(11)  
(12) 
where for Nakagami fading.
Note that variable gain relaying has always an output power of at the relay, whilst for fixed gain relaying this varies with the channel gain of the first hop. This, in turn, affects the variance of the distortion noise for the second hop, which by definition is for AF relaying. This reduces to the simple expression for variable gain relaying, while it becomes for fixed gain relaying.
After some algebraic manipulations (e.g., using the expressions for ), the endtoend SNDRs for fixed and variable gain relaying are obtained as
(13)  
(14) 
respectively, assuming that the destination knows the two channels and the statistics of the receiver and distortion noises. Note that the parameter that appears in (13)–(14) plays a key role in this paper.
Remark 2 (Ideal Hardware)
The endtoend SNRs for AF relaying with ideal () hardware were derived in [30, 3]. The results of this section reduce to that special case when setting . The amplification factors then become
(15) 
and the endtoend SNRs become
(16) 
for fixed and variable gain relaying, respectively. Comparing the SNDRs in (13)–(14) with the ideal hardware case in (16), the mathematical form of the former is more complicated, since the product appears in the denominator. It is, therefore, nontrivial to generalize prior works on AF relaying with Nakagami fading (e.g., [31, 4, 32]) to the general case of nonideal hardware. This generalization is done in Section III and is a main contribution of this paper.
IiD EndtoEnd SNDR: DecodeandForward Relaying
In the DF relaying protocol, the transmitted signal at the relay should equal the original intended signal . This is only possible if the relay is able to decode the signal (otherwise the relayed signal is useless); thus, the effective SNDR is the minimum of the SNDRs between 1) the source and relay; and 2) the relay and destination. We assume that the relay knows and the destination knows , along with the statistics of the receiver and distortion noises.
With nonideal hardware as described by (6), the effective endtoend SNDR becomes
(17) 
and does not require any channel knowledge at the source. In the special case of ideal hardware (i.e., ), (17) reduces to the classical result from [2]; that is
(18) 
Just as for AF relaying, the SNDR expression with DF relaying is more complicated in the general case with hardware impairments. This is manifested in (17) by the statistical dependence between numerators and denominators, which is different from the ideal case in (18).
Iii Outage Probability Analysis
This section derives new closedform expressions for the exact OPs under the presence of transceiver impairments. These results generalize the well known results in the literature, such as [2, 3, 4, 31, 32], which rely on the assumption of ideal hardware. The OP is denoted by and is the probability that the channel fading makes the effective endtoend SNDR fall below a certain threshold, , of acceptable communication quality. Mathematically speaking, this means that
(19) 
where is the effective endtoend SNDR.
Iiia Arbitrary Channel Fading Distributions
This subsection derives general expressions for the OP that hold true for any distributions of the channel gains . These offer useful tools, which later will allow us to derive closedform expressions for the cases of Nakagami and Rayleigh fading. Note that appear in both numerators and denominators of the SNDRs in (13)–(14) and (17). The following lemma enable us to characterize this structure.
Lemma 2
Let be strictly positive constants and let be a nonnegative random variable with cdf . Then,
(20) 
Suppose instead, then (20) simplifies to
(21) 
Proof:
The lefthand side of (20) is equal to
(22) 
after some basic algebra. The last expression is exactly . If , then the inequality is satisfied for any realization of the nonnegative variable . \qed
Based on Lemma 2, we can derive integral expressions for the OPs with AF relaying.
Proposition 1
Suppose is an independent nonnegative random variable with cdf and pdf for . The OP with AF relaying and nonideal hardware is
(23) 
for and for . Recall that . The choice of AF protocol determines :
In the special case of ideal hardware, (23) reduces to
(24) 
where the parameters depend on the AF protocol:
Proof:
The result in Lemma 2 also allows explicit expressions for the OPs with DF relaying.
Proposition 2
Suppose is an independent nonnegative random variable with cdf for . The OP with DF relaying and nonideal hardware is
(25) 
with . In the special case of ideal hardware, (25) reduces to
(26) 
Proof:
Note that the OP expressions in Propositions 1 and 2 allow the straightforward computation of the OP for any channel fading distribution, either directly (for DF) or by a simple numerical integration (for AF). In Section IIIB, we particularize these expressions to the cases of Nakagami and Rayleigh fading to obtain closedform results.
Interestingly, Propositions 1 and 2 show that the OP, , is always 1 for when using AF and 1 for when using DF. Note that these results hold for any channel fading distribution and SNR; hence, there are certain SNDR thresholds that can never be crossed. This has an intuitive explanation since the SNDRs derived in Section II are upper bounded as and . We elaborate further on this fundamental property in Section V.
IiiB Nakagami and Rayleigh Fading Channels
Under ideal hardware, the OPs with fixed and variable gain AF relaying were obtained in [3, Eq. (9)] and [30, Eq. (14)], respectively. These prior works considered Rayleigh fading, while closedform expression for the case of Nakagami fading were obtained in [31, 4, 32] under ideal hardware. Unfortunately, the OP in the general AF relaying case with nonideal hardware cannot be deduced from these prior results; for example, the general analysis in [32] does not handle cases when appears in the denominator of the SNDR expression, which is the case in (13)–(14).
The following key theorem provides new and tractable closedform OP expressions in the presence of transceiver hardware impairments.
Theorem 1
Suppose are independent and where is an integer and for . The OP with AF relaying and nonideal hardware is
(27) 
for and for . The thorder modified Bessel function of the second kind is denoted by , while
(28) 
The parameters depend on the choice of the AF protocol and are given in Proposition 1, while . In the special case of Rayleigh fading (, ), the OP becomes
(29) 
for and for .
Theorem 1 generalizes the prior works mentioned above, which all assume ideal hardware. Note that OP expressions equivalent to those in prior works, can be obtained by setting in Theorem 1, which effectively removes the possibility of since .
Next, closedform OP expressions for DF relaying are obtained in the general case of nonideal hardware.
Theorem 2
Suppose are independent and where is an integer and for . The OP with DF relaying and nonideal hardware is
(30) 
for where and for . In the special case of Rayleigh fading (, ), the OP becomes
(31) 
Proof:
By plugging the respective cdfs of Nakagami and Rayleigh fading into Proposition 2, we obtain the desired results. \qed
We stress that Theorem 2 generalizes the classical results of [33, Eq. (21)] and [2, 34], which were reported for the case of DF relaying with ideal hardware. We also note that Theorem 2 can be straightforwardly extended to multihop relaying scenarios with hops. The only difference would be to let the index account for all hops.
Iv Ergodic Capacity Analysis
In the case of ergodic channels, the ultimate performance measure is the ergodic channel capacity, expressed in bits/channel use. Similar to [35, 36, 37], the term channel refers to the endtoend channel with a fixed relaying protocol (e.g., AF or DF). When compared to the ergodic capacity with arbitrary relaying protocols, as in [38], the results for the AF and DF relaying channels should be interpreted as ergodic achievable rates. This section provides tractable bounds and approximations for the ergodic capacities of AF and DF relaying.
Iva Capacity of AF Relaying
While the capacity of the AF relaying channel with ideal hardware has been well investigated in prior works (see e.g., [35, 36, 37] and references therein), the case of AF relaying with hardware impairments has been scarcely addressed. In the latter case, the channel capacity can be expressed as
(32) 
where the factor accounts for the fact that the entire communication occupies two time slots. The ergodic capacity can be computed by numerical integration, using the fact that the pdf of can be deduced by differentiating the cdf in Theorem 1. However, an exact evaluation of (32) is tedious, if not impossible, to obtain in closedform.
To characterize the ergodic capacity of the AF relaying channel with fixed or variable gain, an upper bound is derived by the following theorem.
Theorem 3
For Nakagami fading channels, the ergodic capacity (in bits/channel use) with AF relaying and nonideal hardware is upper bounded as
(33) 
with
(34) 
where denotes the Whittaker function [39, Ch. 9.22]. The parameters take different values for fixed and variable gain relaying and are given in Proposition 1.
Proof:
For brevity, the proof is given in Appendix B. \qed
Although the expression in (34) is complicated, we note that analytical expressions for the derivatives of arbitrary order are known for the Whittaker function [32]; thus, the upper bound in Theorem 3 can be analytically evaluated in an efficient way. For the purpose of numerical illustrations in Section VI, we implemented the upper bound in Theorem 3 using the Symbolic Math Toolbox in MATLAB [40].
Nevertheless, a simpler closedform expression for the ergodic capacity is achieved by applying the approximation
(35) 
to (32). For Nakagami fading channels, we obtain
(36) 
where the parameters were defined in Proposition 1 for fixed and variable gain relaying. Despite the approximative nature of this result, we show numerically in Section VI that (36) is an upper bound that is almost as tight as the one in Theorem 3. In addition, both expressions are asymptotically exact in the highSNR regime.
IvB Capacity of DF Relaying
Next, we consider the ergodic capacity of the DF relaying channel which is more complicated to analyze than the AF relaying channel; the decoding and reencoding at the relay gives additional constraints and degreesoffreedom to take into account [38]. For example, an information symbol must be correctly decoded at the relay before reencoding, and different symbol lengths and transmit powers can then be allocated to the two hops to account for asymmetric fading/hardware conditions.
For brevity, we consider a strict DF protocol with fixed power and equal time allocation. Based on [38, Eq. (45)], [35, Eq. (11a)], and the effective SNDR expression in (17), the ergodic channel capacity under hardware impairments can be upper bounded as
(37) 
The intuition behind this expression is that the information that can be sent from the source to the destination is upper bounded by the minimum of the capacities of the individual channels. A closedform upper bound, which holds for any channel fading distributions, is derived in the new theorem.
Theorem 4
The ergodic capacity (in bits/channel use) with DF relaying and nonideal hardware is upper bounded as
(38) 
Proof:
For brevity, the proof is given in Appendix B. \qed
This theorem shows clearly the impact of hardware impairments on the channel capacity: the distortion noise shows up as an interference term that is proportional to the SNR. The upper bound will therefore not grow unboundedly with the SNR, as would be the case for ideal hardware [38, 35]. The next section elaborates further on the highSNR regime.
V Fundamental Limits: Asymptotic SNR Analysis
To obtain some insights on the fundamental impact of impairments, we now elaborate on the highSNR regime. Recall the SNR definition, for , in (9) and the corresponding Remark 1 on the SNR scaling.
For the ease of presentation, we assume that grow large with for some fixed ratio , such that the relay gain remains finite and strictly positive.
Corollary 1
Suppose grow large with a finite nonzero ratio and consider any independent fading distributions on that are strictly positive (with probability one).
The OP with AF relaying and nonideal hardware satisfies
(39) 
while the OP with DF relaying and nonideal hardware satisfies
(40) 
Proof:
Referring back to (14), observe that we can rewrite the SNDR in terms of by extracting out the average fading power as (where represents a normalized channel gain). By taking the limit (with ), we can easily see that the endtoend SNDR, for variable gain AF relaying, converges to
(41) 
for any nonzero realization of . Since this happens with probability one, the OP in (39) is obtained in this case. The proofs for the cases of fixed gain AF relaying and DF relaying follow a similar line of reasoning. \qed
A number of conclusions can be drawn from Corollary 1. First, an SNDR ceiling effect appears in the highSNR regime, which significantly limits the performance of both AF and DF relaying systems. This means that for smaller than the ceiling, goes to zero with increasing SNR (at the same rate as with ideal hardware; see Section VI) while the OP always equals one for larger than the ceiling. This phenomenon is fundamentally different from the ideal hardware case, in which an increasing SNR makes the endtoend SNDR grow without bound and for any . Note that this ceiling effect is independent of the fading distributions of the two hops. Similar behaviors have been observed for twoway relaying in [20], although the exact characterization is different in that configuration.
The SNDR ceiling for dualhop relaying is
(42) 
which is inversely proportional to the squares of . This validates that transceiver hardware impairments dramatically affect the performance of relaying channels and should be taken into account when evaluating relaying systems. The ceiling is, roughly speaking, twice as large for DF relaying as for AF relaying;^{8}^{8}8This is easy to see when have the same value , which gives for DF relaying and for AF relaying. this implies that the DF protocol can handle practical applications with twice as large SNDR constraints without running into a definitive outage state. Apart from this, the impact of and on the SNDR ceiling is similar for both relaying protocols, since is a symmetric function of .
We now turn our attention to the ergodic capacity in the highSNR regime. In this case, the following result is of particular importance.
Corollary 2
Suppose grow large with a finite nonzero ratio and consider any independent fading distributions on that are strictly positive (with probability one).
The ergodic capacity with AF relaying and nonideal hardware satisfies
(43) 
The ergodic capacity with DF relaying and nonideal hardware satisfies
(44) 
Proof:
For AF relaying, the instantaneous SNDR is upper bounded as for any realizations of . The dominated convergence theorem therefore allows us to move the limit in (43) inside the expectation operator of the ergodic capacity expression in (32). The righthand side of (43) now follows directly from (41). For DF relaying, we see directly from Theorem 4 that , as , which gives (44). \qed
Similar to the asymptotic OP analysis, Corollary 2 demonstrates the presence of a capacity ceiling in the highSNR regime. This implies that transceiver hardware impairments make the ergodic capacity saturate, thereby limiting the performance of highrate systems. Similar capacity ceilings have previously been observed for singlehop multiantenna systems in [9, 10, 18]. We finally point out that the approximate capacity expression in (36) becomes asymptotically exact and equal to (43), for the case of Nakagami fading.
Va Design Guidelines for Relaying Systems
Recall from Lemma 1 that is the aggregate level of impairments of the th hop, for . The parameter can be decomposed as
(45) 
where are the levels of impairments (in terms of EVM) in the transmitter and receiver hardware, respectively. The hardware cost is a decreasing function of the EVMs, because lowcost hardware has lower quality and thus higher EVMs. Hence, it is of practical interest to find the EVM combination that maximizes the performance for a fixed cost.
To provide explicit guidelines, we define the hardware cost as , where is a continuously decreasing, twice differentiable, and convex function. The convexity is motivated by diminishing returns; that is, highquality hardware is more expensive to improve than lowquality hardware. The following corollary provides insights for hardware design.
Corollary 3
Suppose for some given cost . The SNDR ceilings in (42) are both maximized by .
Proof:
The proof goes by contradiction. Assume is the optimal solution and that these EVMs are not all equal. The hardware cost is a Schurconvex function (since it is convex and symmetric [41, Proposition 2.7]), thus the alternative solution reduces the cost [41, Theorem 2.21]. To show that the alternative solution also improves the SNDR ceilings, we first note that is a Schurconvex function, thus it is maximized by for any fixed value on [41, Theorem 2.21]. In addition, for any fixed value , in (42) is maximized by , which is easily seen from the structure of for AF and for DF. The alternative solution decreases cost and increases (42), thus the EVMs must be equal at the optimum. \qed
Corollary 3 shows that it is better to have the same level of impairments at every^{9}^{9}9There are four transceiver chains: transmitter hardware at the source, receiver and transmitter hardware at the relay, and receiver hardware at the destination. transceiver chain, than mixing highquality and lowquality transceiver chains. In particular, this tells us that the relay hardware should ideally be of the same quality as the source and destination hardware.
As a consequence, we provide the following design guidelines on the highest level of impairments that can theoretically meet stipulated requirements.
Corollary 4
Consider a relaying system optimize