On the DMT of TDD-SIMO Systems with Channel-Dependent Reverse Channel Training

On the DMT of TDD-SIMO Systems with Channel-Dependent Reverse Channel Training

Abstract

This paper investigates the Diversity-Multiplexing gain Trade-off (DMT) of a training based reciprocal Single Input Multiple Output (SIMO) system, with (i) perfect Channel State Information (CSI) at the Receiver (CSIR) and noisy CSI at the Transmitter (CSIT), and (ii) noisy CSIR and noisy CSIT. In both the cases, the CSIT is acquired through Reverse Channel Training (RCT), i.e., by sending a training sequence from the receiver to the transmitter. A channel-dependent fixed-power training scheme is proposed for acquiring CSIT, along with a forward-link data transmit power control scheme. With perfect CSIR, the proposed scheme is shown to achieve a diversity order that is quadratically increasing with the number of receive antennas. This is in contrast with conventional orthogonal RCT schemes, where the diversity order is known to saturate as the number of receive antennas is increased, for a given channel coherence time. Moreover, the proposed scheme can achieve a larger DMT compared to the orthogonal training scheme. With noisy CSIR and noisy CSIT, a three-way training scheme is proposed and its DMT performance is analyzed. It is shown that nearly the same diversity order is achievable as in the perfect CSIR case. The time-overhead in the training schemes is explicitly accounted for in this work, and the results show that the proposed channel-dependent RCT and data power control schemes offer a significant improvement in terms of the DMT, compared to channel-agnostic orthogonal RCT schemes. The outage performance of the proposed scheme is illustrated through Monte-Carlo simulations.

1Introduction

Reliability and system throughput are two fundamental parameters of interest in any wireless communication system, and the inherent tradeoff between the two at high SNR was elegantly captured by the Diversity Multiplexing gain Tradeoff (DMT) proposed in the seminal work of Zheng and Tse [2]. It is known that a significant improvement in the outage performance can be obtained if the Channel State Information (CSI) at the receiver (CSIR) and the transmitter (CSIT) are perfect [3], [4], while [2] considered perfect CSIR and no CSIT.

In a Time Division Duplex (TDD) system, CSI could be estimated at the transmitter and receiver by sending a known training sequence in the forward and reverse-link directions, respectively. This has two consequences. First, the estimation error results in incorrect data rate or power adaptation at the transmitter, in turn leading to higher outage rate. Second, training incurs a time overhead, which could be non-trivial when the training occupies a significant fraction of the channel coherence time, as it affects the pre-log term in the achievable data rate [5]. This paper therefore focuses on the important problem of analytically comparing the DMT performance of different channel estimation techniques and identifying training signals and data power control schemes that result in a good performance in terms of the achievable DMT. We start with a brief survey of related literature.

The impact of imperfect CSIT on the DMT of a multiple antenna system has been a popular area of research, and it is known that even with imperfect CSIR and CSIT, a significant improvement in DMT can be obtained, compared to the no-CSIT case (see, for example, [6]). The effect of imperfect CSIR on the DMT of a MIMO system was first studied in [9]. The DMT analysis of a multiple antenna system with perfect CSIR and when the CSIT is modeled as the CSI plus Gaussian noise whose variance decreases with training SNR was investigated in [10]. In a TDD setup, the achievable DMT improvement using power control based on noisy CSIT was shown in [13]. Other works that study the DMT performance with quantized feedback of CSI and/or target data rate control based on noisy CSIT include [7]. In [17], the DMT of two-way and multi-round training schemes in a TDD system was derived. In these studies, the channel feedback signal on the reverse link is chosen to satisfy an average power constraint, rather than an instantaneous power constraint.

Most of the aforementioned studies of the DMT with imperfect CSI typically ignore the training duration overhead. Hence, they are primarily applicable to slowly varying channels, where the time overhead in training occupies an insignificant fraction of the channel coherence time. An exception is [13], where, taking the training overhead into account, the authors concluded that for nonzero multiplexing gain , the diversity order saturates as increases, where is the number of receive antennas. Hence, for fast varying channels, the authors suggest turning off receive antennas in order to achieve higher multiplexing gains. It is important to account for the training duration overhead in deriving the achievable DMT, because, as the SNR goes to infinity, although the estimation error goes to zero, the training duration overhead remains fixed and has a direct impact on the DMT. Also, by modeling the CSIT as the sum of the true CSI and an additive error, most of the past studies implicitly assume that a channel-agnostic orthogonal training signal is employed for channel estimation. When the training signal is channel-dependent, the imperfect CSI can no longer be modeled as the sum of the true CSI and an additive noise. Due to this, the existing results cannot be directly extended to analyze the DMT performance of channel-dependent training schemes.

When the channel is reciprocal and block-fading, e.g., in a TDD system, the receiver could exploit its channel knowledge (acquired through an initial forward-link training phase) in designing its reverse-training sequence, not only to reduce the channel estimation error at the transmitter, but also to reduce the required training duration overhead. Hence, the goals of this paper are two-fold: (a) to analyze the DMT performance of a channel dependent training scheme for acquiring CSIT and an associated power control mechanism for data transmission; and (b) to contrast the DMT performance of the proposed training and power control schemes with that achieved by conventional channel agnostic training schemes. Our study focuses on point-to-point Single Input Multiple Output (SIMO) systems. This is of practical importance, since it applies, for example, to the uplink of wireless networks where the base station has multiple antennas, the mobile users have a single antenna, and orthogonal access is used (e.g., OFDM/TDMA) as in WLANs and 4G/LTE systems. The channel dependent training sequence employed here was first proposed by us in [20] and [1] in a MIMO and SIMO context, respectively, and was independently explored in [21], although not in a DMT context.

In this paper, for analytical simplicity and clarity of presentation, we start by assuming that perfect CSI is available at the receiver, as in [10]. We propose a fixed-power RCT sequence, using which, the CSI can be estimated at the transmitter using a minimum duration of only one symbol, i.e., with a factor of reduction in training duration compared to orthogonal RCT. For data transmission, we propose a modified truncated channel inversion-type power control scheme based on the noisy CSIT. For this system, we show that a diversity of is achievable. Here, is the multiplexing gain, is the coherence time, is the reverse training duration, and is a parameter in the data power control scheme. (See Section 3.)

Next, we consider the more practical case where noisy CSIR is acquired via a forward link training sequence, and propose a three-way training scheme followed by data transmission. We show that a DMT of is achievable, where is the total training overhead from all three training phases, which is again an improvement over conventional orthogonal training schemes. For example, a nonzero diversity order can be achieved with , which is not possible with orthogonal training schemes without switching off receive antennas and incurring an associated reduction in diversity order. (See Section 4.)

Note that although the perfect CSIR case is a special case of the three-way training scheme with infinite forward-link training power, we briefly present the perfect CSIR case also, as it provides insights into the impact of the reverse-training and data power control mechanisms on the DMT. Moreover, it is useful as an upper bound on the performance with imperfect CSIR. Also, we assume that power control is employed only at the transmitter and focus on fixed-power RCT in the sequel. Using power controlled RCT significantly changes the problem; we analyze this case in our follow up work [22].

An important implication of our work is that it shows that by exploiting the receiver’s knowledge of the CSI in designing the reverse channel training (RCT) sequence and using our proposed data power control scheme, one can achieve a higher diversity order than conventional RCT for all values of . Somewhat surprisingly, we also demonstrate that although the DMT analysis corresponds to taking the SNR to infinity, it can nonetheless be used to discriminate between different training schemes both in terms of the estimation error as well as the training overhead. At finite SNR, this translates to an improvement in the outage probability performance and the achievable data rate, as will be illustrated through Monte-Carlo simulations in Section 6.

We use the following notation. Bold face letters are used for vectors and normal font letters are used for scalars. We use to denote the expected value of . We use to represent the norm of . The transpose conjugate, absolute value, and real part are denoted by , and , respectively. We write to mean Similarly, we define to mean .

2System Model

The system model consists of two communicating nodes, node A with a single antenna and node B with antennas, with node A attempting to send data to node B over a wireless channel. The forward channel from node A to node B, denoted by , is modeled as a Rayleigh flat fading channel whose entries are i.i.d. Circularly Symmetric Complex Gaussian (CSCG) random variables with zero mean and unit variance, i.e., . The channel is assumed to be block-fading, i.e., it remains constant for a duration of the coherence time , and evolve in an i.i.d. fashion across coherence times. We assume a TDD system with perfect reciprocity, and hence, taking the complex conjugate of the received signal at node A, the reverse link channel is . We let , where is the singular value and is the singular vector of . Since our goal is to study the achievable DMT performance with channel training, we first explain the two-way training protocol used for acquiring CSI at node B and node A. Later, in Section 4, an additional phase of forward link training is introduced, which is not presented here for simplicity.

Phase I (Forward-link training)

Here, the training sequence is transmitted from node A to node B, where denotes the training duration and is the training power1. Throughout this paper, we use as the average power constraint during both training and data transmission. The corresponding received training signal is given by,

The entries of are assumed to be distributed as i.i.d. . From the received training signal , node B computes an MMSE estimate of , denoted . The error in the estimate, denoted , has i.i.d. distributed entries.

In a TDD-SIMO system, node A only requires knowledge of to perform power control, which in turn improves the diversity order compared to the no-CSIT case. Therefore, in phase II, we estimate only at node A, using a channel dependent training sequence.

Phase II (Reverse-link training)

Since node B has an estimate (say, ) of the channel, in this phase, it exploits its CSI to transmit the following training sequence [1]:

where is the reverse training duration. Using the corresponding received signal, where the AWGN is distributed as , node A computes an estimate of the singular value as follows:

where . Note that the estimate could be negative; this is taken care of by the power control proposed in Section 3, which uses only when it is greater than a positive threshold. Since a low or negative is likely to be inaccurate, the thresholding technique helps to avoid the poor DMT performance due to such estimates. The RCT scheme employed above is different from existing channel agnostic methods in that the minimum training length in the proposed scheme is only symbol. This represents a factor of reduction compared to orthogonal RCT schemes, where the minimum training length increases linearly with , and this difference in overhead could be significant when is small. Also, if is error-free, it is the optimal beamforming vector for estimating at node A.

Multiplexing Gain and Diversity Order

We recall the definitions of the multiplexing gain, , and the diversity order from [2]:

where is the target data rate when the average data power constraint is , and is the corresponding outage probability, i.e., the probability that exceeds the channel capacity. In this work, the target data rate is fixed and is independent of the CSIT; the extension of our proposed methods to joint rate and power adaptation is relegated to future work. The rate of data transmission is increased with by increasing the cardinality of the signal set, keeping the symbol duration fixed. We ignore the effect of spectral leakage, and assume that the signal bandwidth remains fixed as goes to infinity. Also, we use outage probability as a proxy for the probability of error at high SNR with finite-length codes; this is because the probability of error can be made to decrease as fast as the outage probability using finite-length approximately universal codes [23].

In the next section, we assume perfect CSI at node B and derive the achievable DMT performance of our proposed training and data transmission schemes.

3DMT Analysis with Perfect CSIR

When the CSIR is perfect, we have , and in this case, it is easy to see that is optimal for estimating given a power constraint on the training signal. This is because, in general, the training signal can be expressed as the linear combination , where is orthogonal to and and are some constants. Then, the received training signal at node A is , i.e., the power in does not help in estimating . From , an unbiased estimator of the singular value at node A is given by

Note that since the channel is assumed to be Rayleigh fading, is chi-square distributed with degrees of freedom. Also, we employ this estimator primarily because we are interested in deriving the achievable DMT performance, and for this purpose, this simple unbiased estimator is sufficient.

3.1Power-Controlled Data Transmission from Node A to Node B

Given the CSIT in , node A uses a power in the forward link data transmission phase, to avoid outages while satisfying the average data power constraint . The corresponding data signal received at node B is given by,

where , and with appropriate power normalization, the entries of the AWGN are assumed to be i.i.d. . Also, is chosen independent of such that , where the expectation is with respect to given in , taken across all coherence blocks. Since within a block, this ensures that the average data power constraint at node A is satisfied.

We now present the data power control function considered in this paper. Our proposed power control function is motivated as follows. The capacity of a fading channel with mismatched CSIT and CSIR is not known in closed form [25]. Since the outage probability computation requires a closed form expression for the capacity, we consider a genie-aided receiver as in [26], where node B is assumed to know . This is schematically illustrated in Fig. ?. Then, the achievable data rate conditioned on the knowledge of is given by [25]

=2.2in =2.8in System model for reverse channel training with perfect CSIR used in Section .

An outage occurs when , the target data rate, exceeds . Its probability is upper bounded by

Note that the exact outage probability is obtained by minimizing the right hand side above over all satisfying . Hence, using our proposed data power control scheme leads to an upper bound on the outage probability, which is sufficient for obtaining the achievable DMT performance. If the CSIT is perfect (i.e., ), it is shown in [3] that the power control that minimizes the outage probability is given by

Note that since and , satisfies for large enough , provided . With inaccurate CSIT, due to the estimation error in , the natural extension of using a transmission power of could result in allocating insufficient power or more power than required, which could lead to suboptimal performance. Also, inverting the channel for all values of results in an infinite average power since the Gaussian noise can make the estimate arbitrarily small with a non-zero probability. One solution is to use a transmit power of when and a zero power otherwise, where is chosen such that . The drawback of this method is that it results in an outage probability of when , leading to a zero diversity order. To overcome this problem, we choose the threshold such that as . Moreover, when , we do not necessarily want to use zero power, since the small value of could be due to the estimation error. This motivates the following modified power control:

where is a parameter, and we use , , for mathematical tractability. The parameters , and are chosen such that . Although similar power control schemes have been employed in the literature with perfect CSIT [3] or orthogonal RCT [12], the form in is new. Specifically, the power control scheme in [3] can be obtained from by setting , and ; while that in [19] can be obtained by setting , and .

Power constraint

The description of the power control would be complete if the parameters , and can be chosen such that , which is the essence of the following Lemma.

Proof

: See Appendix Section 7.2.

Due to Lemma ?, in the rest of this paper, we consider . Also, in Section 4, we show that a minor modification of the above data power control scheme can be employed even with imperfect CSIR. The next subsection presents the achievable DMT of the proposed training and power control schemes.

3.2Achievable DMT Analysis

Proof:

See Appendix Section 7.3.

Remark

: From a DMT perspective, it is clear from Theorem ? that is superior to . On the other hand, when , could be much greater than . Thus, in practical systems with a peak power per transmitted codeword constraint, could be preferable over . In the sequel, for convenience, we associate with and with , and drop the explicit dependence of the diversity order on . Further remarks and discussions on the result obtained here are deferred to Section 5.

4Three Way Training

Table 1: Three Way Training in a TDD-SIMO System
Phase Description Input-Output Equation

I

Fixed power training (Node A Node B)

II

Fixed power training (Node B Node A)

III

Power controlled training (Node A Node B)

IV

Power controlled data (Node A Node B)

In this section, we consider the more practical scenario where training is performed in both directions. We show that with fixed power training, one can achieve nearly the same DMT as derived in Section 3 for the perfect CSIR case. Unlike in the previous section, the analysis presented here is exact, in the sense that it does not require the assumption of a genie aided receiver, and hence, the DMT derived here is indeed achievable in practice. The transmission protocol now consists of four phases, as shown in Table 1. The CSIR and CSIT are obtained by transmitting a fixed power training sequence in both directions, as explained in Section 2. However, even a small mismatch in the CSI knowledge at node A and node B can potentially lead to a large mismatch in their estimate of the data transmit power [13]. Thus, it is essential to train node B about node A’s knowledge of . This leads to a third phase of training, which is an additional power-controlled forward-link training phase. First, in the following subsection, we explain the power control scheme that is employed here.

4.1Power Control Scheme

The power control scheme we propose to employ in this section is as given by , due to the following. Let denote the MMSE estimate of the channel at node B, and consider in . We have

where . Note that and are independent Gaussian random variables2. Since is uniformly distributed on the unit sphere and is independent of , is Gaussian distributed. This implies that the effective noise, , is Gaussian distributed with and independent of . Therefore, the estimate of the singular value at node A is statistically similar to the estimate given by in the perfect CSIR case. Thus, we use a similar power control, in , where is given by . Also, with a slight abuse of notation, , where is the training duration in the third phase of training (phase III), which is in the forward-link direction.

In this section, without loss of generality, we move the power scaling into the data symbol transmitted by node A, so that (see below), where the expectation is taken with respect to the distribution of in . Now, in the proof of Lemma ?, using the probability density function (pdf) of in place of the pdf of , and noting that the effective noise variance , we get and the constraint to satisfy at high SNR. In the next subsection, we explain the third round of training that alleviates the mismatch in the knowledge of the data transmit power.

4.2Phase III (Power-Controlled Forward Link Training)

In this phase, node A transmits the training sequence: , where is the training duration. The corresponding received training signal at node B is given by,

where is the AWGN with entries. The goal at node B is to estimate the composite channel . Dividing by , we get

From , node B computes an MMSE estimate of , denoted by . Let . Although a closed form expression for is hard to find, the error in the MMSE estimate has the following interesting property, which facilitates the calculation of the outage probability in Section 4.4. An analogous result has been shown in [27] for the scalar case.

Proof:

See Appendix Section 7.4.

4.3Phase IV (Data Transmission)

Using , node A sends the data signal , where is distributed as and is independent of . Note that by construction, where the expectation is taken with respect to both and . The corresponding signal received at node B is

Since is an MMSE estimate, using the worst case noise theorem [5], we have the following lower bound on the mutual information, , where

and is the fractional data transmit duration after accounting for the time overheads in all three training phases.

4.4DMT Analysis With Three-Way Training

Proof:

See Appendix Section 7.5.

Remark

: The above three way training scheme can be generalized to training rounds to improve the diversity order, as in [17]. However, this is mathematically cumbersome and out of the scope of our work.

5Discussion

Recall that with perfect CSIR and imperfect CSIT, with , and for a genie aided channel, it was shown in Theorem ? that the following DMT is achievable:

where , . In contrast, for the same genie aided channel, it was shown in [26] that a diversity order of

is achievable using orthogonal reverse channel training. Note that saturates as gets large, as opposed to , which is monotonically increasing in . In order to achieve a , in [13], the authors suggest turning off one receive antenna at a time to reduce the training burden until . For example, turning off one antenna, is achievable at a reduced diversity order of . This is in contrast to our result, which can accommodate a larger multiplexing gain, irrespective of , while simultaneously achieving a higher diversity order at each . We note that for a SIMO channel, a diversity order of for was obtained in [12], using channel-independent training, and without accounting for the training duration overhead. This corresponds to taking in . The performance of the proposed scheme is schematically contrasted with orthogonal RCT in Figure 1 for a SIMO system with , , and symbol.

Figure 1: The achievable DMT with the training and power control scheme proposed in Sec. , compared with the performance with orthogonal RCT and the data power control proposed in  (and appropriately accounting for the training duration overhead and switching off antennas to achieve higher values of g_m). The plot corresponds to a SIMO system with r=5 antennas, with coherence time L_c= 20 symbols, and reverse training duration of L_{B,\tau}= 1 symbol.
Figure 1: The achievable DMT with the training and power control scheme proposed in Sec. , compared with the performance with orthogonal RCT and the data power control proposed in (and appropriately accounting for the training duration overhead and switching off antennas to achieve higher values of ). The plot corresponds to a SIMO system with antennas, with coherence time symbols, and reverse training duration of symbol.

The advantage of the proposed scheme at higher values of the multiplexing gain is clear from the plot. The proposed training scheme thus results in a factor -reduction in the training duration, which, along with the proposed data power control scheme, translates to an increase in the range of achievable multiplexing gains, while simultaneously offering a better diversity order compared to orthogonal RCT schemes.

Comparing Theorems ? and ?, we see that the DMT performance of a genie aided receiver with perfect CSIR is an upper bound on the performance of the system with imperfect CSIR and CSIT, as expected. Also, the performance of the two systems is similar, except that in the latter case, the factor captures the loss in data transmission time due to all three training phases. Similar observations as the above regarding the improvement in DMT can be made for the three way training scheme compared to orthogonal RCT schemes.

6Simulation Results

We now briefly present Monte-Carlo simulation results to illustrate the outage probability performance of our proposed RCT and forward-link data power control schemes. We consider a Rayleigh fading channel with three receive antennas. We calculate the outage probability by averaging over i.i.d. channel and training noise instantiations. We set the channel coherence time and reverse training duration as and , respectively.

Figure 2: Outage probability versus the average data power \bar{P} for the fixed-power training scheme proposed in Sec. , with the data power control scheme given by  with s=1. Here, r=3, L_c= 40 and L_{B,\tau}= 1. With g_m=0.8, the target data rate was set as R_{\bar{P}} = 4 + g_m \log \bar{P} to facilitate the comparison of the curves.
Figure 2: Outage probability versus the average data power for the fixed-power training scheme proposed in Sec. , with the data power control scheme given by with . Here, and . With , the target data rate was set as to facilitate the comparison of the curves.
Figure 3: Outage probability versus the average data power \bar{P} for the fixed-power training scheme proposed in Sec. , with the data power control scheme given by  with s=r. Here, r=3, L_c= 40 and L_{B,\tau}= 1.
Figure 3: Outage probability versus the average data power for the fixed-power training scheme proposed in Sec. , with the data power control scheme given by with . Here, and .

Figures Figure 2 and Figure 3 show the outage probability of the proposed fixed-power training scheme and the data power control scheme in with and , respectively, as a function of , with and bits/channel use ( and bits/channel use in case of Figure 3), and with . Although the slopes of the curves do not match with the theoretical diversity order because the latter requires infinite SNR, the improved performance of the proposed schemes is clear from the graphs. Also, in Figure 2, since the proposed scheme uses only training symbol while the orthogonal RCT scheme uses training symbols, the former shows a higher outage than the latter at lower SNRs. Note that, we have not plotted the outage performance of the three-way training scheme in Section 4. This is because the outage probability is hard to compute, since a closed-form expression for is not available.

7Conclusions

This paper proposed reverse training and data power control schemes for a TDD-SIMO system with perfect/imperfect CSIR and investigated its DMT performance. It was shown that a diversity order of is achievable for , and , where represents the fractional data transmit duration. In contrast to channel agnostic orthogonal training schemes, the diversity order was shown to increase monotonically with at nonzero multiplexing gain, which is a significant improvement. The DMT analysis was extended to a more practical situation where the training is done in both directions. In this case also, it was shown that the DMT performance can improve quadratically with the number of receive antennas, and nearly the same DMT can be achieved as that with perfect CSIR and a genie-aided receiver. In terms of system design for reciprocal SIMO systems, the key messages from this work are that it is important to (a) exploit the CSI at the receiver in designing the RCT and (b) use a modified channel-inversion type power control scheme that transmits data at some non-zero power even when the estimated singular value at the transmitter is poor. For fast varying channels, these ingredients can lead to a significant advantage in DMT performance, which, at finite SNR, can translate to a large improvement in outage probability performance compared to orthogonal training schemes. Future work could extend the DMT analysis to a time-selective block fading reciprocal channel, where the channel is correlated within a block [28].

7.1Useful Lemmas

Proof

: The result follows from

Proof

: We upper bound the absolute value of as follows:

where (a) follows from the triangle inequality and (b) follows since .

7.2Proof of Lemma

Consider the following constraint on the data power

where is the pdf of . Substituting in , we get

where is the target data rate and the data transmit power is ,

The proof is complete by choosing

and showing that and that is bounded for large when and . From , can be bounded as,

where (a) follows from Lemma ? above, and the expectation is with respect to the distribution of , (b) follows from the binomial expansion and the fact that when is odd, (c) follows from and , and (d) follows by substituting in the left hand side. From , clearly, for large if and . When and , we have , and therefore we can ensure that for large by scaling by an appropriately chosen constant scaling factor.

Next, we show that is bounded. Note that

Now, it is sufficient to show that the first integral in is bounded, since the second integral is clearly . To this end, we need the distribution of , i.e., , where , and . Consider

where is the pdf of , which is chi distributed with degrees of freedom. Taking the derivative of with respect to , we get

where is the constant term in the standard chi pdf, , and . Let and using the binomial expansion, it can be shown that

Now, using , we can upper bound the first term in as

where , and is some constant that does not scale with . Now, the behavior of the terms above with is governed by

where , and . The exponent corresponding to the first term above is for all . Also, when , the exponent corresponding to the second term above is for all , and hence the integral is bounded for .

Finally, let . Since and are bounded when , using in , we get , where . This completes the proof of Lemma ?.

7.3Proof of Theorem

Using the power control in , the outage probability in can be written as

where , and . In the above, we have used to mean . Using , we have for large and from Lemma ?. From Lemma ? in Appendix ?, we have, . Next, substituting for from , can be written as, Using from Lemma ? in Appendix ? with , we get

It is straightforward to show that provided is strictly increasing with , the first term in the above goes to zero exponentially with for . This implies that , since from Lemma ?. The second term in is upper-bounded as

where (a) follows from Lemma ? in Appendix ?, and the in (b) uses the fact that and . Hence, we have which implies . Using this and