Waveform domain framework for capacity analysis of uplink WCDMA systems
 Tan Tai Do^{1},
 Su Min Kim^{2}Email author,
 Tobias J. Oechtering^{1},
 Mikael Skoglund^{1} and
 Gunnar Peters^{3}
https://doi.org/10.1186/s1363801504805
© Do et al. 2015
Received: 23 June 2015
Accepted: 10 November 2015
Published: 2 December 2015
Abstract
This paper investigates the capacity limit of an uplink WCDMA system considering a continuoustime waveform signal. Various realistic assumptions are incorporated into the problem, which make the study valuable for performance assessment of real cellular networks to identify potentials for performance improvements in practical receiver designs. An equivalent discretetime channel model is derived based on sufficient statistics for optimal decoding of the transmitted messages. The capacity regions are then characterized using the equivalent channel considering both finite constellation and Gaussian distributed input signals. The capacity with sampling at the receiver is also provided to exemplify the performance loss due to a typical postprocessing at the receiver. Moreover, we analyze the asymptotic capacity when the signaltonoise ratio goes to infinity. The conditions to simultaneously achieve the individual capacities are derived, which reveal the impacts of signature waveform space, channel frequency selectivity and signal constellation on the system performance.
Keywords
1 Introduction
Code division multiple access (CDMA) has become standard in several wireless communication systems from IS95, UMTS wideband CDMA (WCDMA) to HSPA, and so on [1–3]. Although being introduced more than 50 years ago, CDMA is still largely employed and developed nowadays due to its various advantages such as enabling universal frequency reuse, improving handover performance by softhandover, and mitigating the effects of interference and fading. The performance assessment of such networks is of significant importance. In addition, the architecture of WCDMA systems still has room for improvement, especially at the uplink receiver side (base station) [4, 5].
In the literature, most studies on fundamental limits of multiuser CDMA systems have been done under the assumptions of synchronous, timeinvariant (each user uses the same spreading sequence for all data symbols), and/or random spreading sequences [6–11]. In [6–8], the optimal spreading sequences and capacity limits for synchronous CDMA have been studied with a discretetime signal model. A more theoretical approach on CDMA capacity analysis has been pursued in [9–11] by modeling the spreading sequences with random sequences. However, the assumption of perfect synchronization between users is not realistic, especially for a cellular CDMA uplink. Moreover, in practice, timevariantspreading sequences based on Gold or Kasami codes [1–3] are often used rather than timeinvariant or random spreading sequences.
The capacity limit for a CDMA system with symbolasynchronous transmission (the symbol epochs of the signal are not aligned at the receiver) has also been studied in [12–15]. In [12], Verdú studied the capacity region of an uplink timeinvariant CDMA system with intersymbol interference (ISI) by exploiting the asymptotic properties of Toeplitz matrices. In [13, 14], the authors studied user and sum capacities of a symbolasynchronous CDMA system but with chipsynchronous transmission (the timing of the chip epochs are aligned) assumption, which made the analysis tractable using a discretetime model. In [15], the spectral efficiency of an asynchronous CDMA system has been considered while neglecting the ISI by assuming a large spreading factor.
There have been several studies trying to deal with the continuoustime asynchronous CDMA system setup. However, most of them focus on other performance metrics than capacity (e.g., error probability considering different detection algorithms) [16–20]: timeinvariant CDMA [21] or asynchronous CDMA but with an ISIfree assumption [22]. The capacity analysis for a real cellular network with continuoustime waveform, timevariantspreading, asynchronous CDMA is difficult due to the following reasons. First, an equivalent discretetime signal model is complicated to be expressed due to the asynchronization between symbols and chips. Next, for a timevariant spreading CDMA system, the approach based on the asymptotic properties of a Toeplitz form [23], which is crucial for the capacity analysis with ISI channel [12, 24], cannot be employed since the variation of spreading sequence destroys the Toeplitz structure of the equivalent channel matrix.
1.1 Contributions of this work
Motivated by the fact that most existing research on multiuser CDMA capacity have focused on theoretical analysis with simplified system assumptions, in this work, we present a framework for capacity analysis of a WCDMA system with more realistic assumptions, which make the framework and results more valuable for the performance assessment of real cellular networks. Our main contributions are summarized as follows: − We provide a precise channel model reflecting practical operations of the uplink WCDMA physical (PHY) layer based on the 3GPP release 11 specification [1]. Various realistic assumptions are included into the system such as: continuoustime waveformtransmitted signal and timevariant spreading and an asynchronous multicode CDMA system with ISI over frequencyselective channels. It is worth noting that although the signal model is constructed based on a WCDMA system, the approach and framework can be extended or transferred to other wireless standards. − We derive sufficient statistics for decoding the transmitted symbols based on the continuoustimereceived signal, which provides us an equivalent discretetime signal model. A matrix representation of channel model is provided for which the equivalent additional noise is shown to be a Gaussian distributed random vector. − Since sufficient statistics preserve the mutual information ([25], Chap. 2), the capacity is then derived using the equivalent discretetime signal model. In particular, we characterize the capacity region when the input signal is fixed to finite constellations, e.g., PSK, QAM, and so on, with a uniform input distribution, which are widely used in current real cellular networks. Additionally, we provide the capacity region when the input signal follows a Gaussian distribution, which is the optimal input distribution for additive Gaussian noise channels. Accordingly, the Gaussian capacity offers a capacity outer bound for the real WCDMA cellular networks using finite constellation input. − Due to the dataprocessing inequality ([25], Chap. 2), the mutual information between input and output cannot increase through any postprocessing at the receiver. Given the capacity bounds measured directly at the receive antenna of a real system, we can now assess the capacity loss due to a specific postprocessing at the receiver. Therewith, we investigate the capacity loss due to sampling, which is a traditional discretization approach in practical systems. Note that in the real cellular networks, since the sampling window is finite, perfect reconstruction of a bandlimited signal is not guaranteed even if the sampling rate is equal to Nyquist rate ([26], Chap. 8). The assessment of such impact on the capacity is also considered in this work. − We analyze the asymptotic sumcapacity when the signaltonoise ratio (SNR) goes to infinity, for which we derive the conditions on the signature waveform space so that on every link to the base station, the individual capacities are achieved simultaneously. To this end, we first derive the sufficient condition, which holds for all kinds of input signals including signals based on finite and infinite constellations. Next, once again, we motivate our study from a practical perspective by focusing on the finite constellation input signal. Accordingly, a necessary condition to simultaneously achieve the individual capacities with a finite constellation input signal, which takes the signal constellation structure into account, is derived. Those results are particularly useful for spreading sequence design in a real WCDMA cellular network.
The rest of the paper is organized as follows: Section 2 presents the signal model where sufficient statistics and an equivalent matrix representation are derived. In Section 3, the capacity analysis is provided considering finite constellations and Gaussiandistributed input signals. The capacity employing sampling is also investigated in this section. The asymptotic capacity when the SNR goes to infinity is analyzed and discussed in Section 4. Finally, Section 5 concludes the paper.
2 Signal model
Since the physical layer defines the fundamental capacity limit of the uplink WCDMA channel [2], we focus on a signal model reflecting the operations of the uplink WCDMA PHY layer based on the 3GPP release 11 specification [1].
2.1 Waveform signal model
where E _{ k } denotes the transmitted power of user k, N denotes the number of symbols, \(s_{\textit {ki}}^{m}(t)=\frac {1}{\sqrt {N_{\textit {sf}}}}\sum _{n=0}^{N_{\textit {sf}}1} c_{\textit {ki}}^{m}[n]p(t(i1)T_{s}nT_{c})\) is the signature waveform for the ith symbol of the mth stream of user k, p(t) is the chip waveform with unit power and finite bandwidth W ^{1}, T _{ c } is the chip duration and T _{ s }=N _{ sf } T _{ c } is the symbol duration, and \(c_{\textit {ki}}^{m}[n]\) denotes the spreading sequence which satisfies \(\sum _{n=0}^{N_{\textit {sf}}1}c_{\textit {ki}}^{m}[n]^{2}=N_{\textit {sf}}\). In a timevariant CDMA system, a different spreading sequence \(\left \{c_{\textit {ki}}^{m}[n]\right \}_{n}\) is used for each transmitted symbol \(d_{\textit {ki}}^{m}\). This corresponds to a real cellular CDMA network with long scrambling codes, in which the effective spreading sequence will vary between symbols.
where λ _{ k } denotes the time delay of the transmitted signal from user k, the symbol ∗ denotes the convolution operation, and n(t) represents the additive white Gaussian noise with a twosided power spectral density (PSD) N _{0}/2=σ ^{2}.
2.2 Sufficient statistic and equivalent channel
where \(n_{\textit {ki}}^{m}:=\sum _{l=1}^{L}{g_{\textit {kl}}}^{*}\int _{\infty }^{\infty }n(t){s_{\textit {ki}}^{m}(t\lambda _{k}\tau _{\textit {kl}})}^{*}dt\) is the equivalent noise term associated with \(z_{\textit {ki}}^{m}\) after matched filtering.
Moreover, it is shown in Appendix 1 that the equivalent noise vector n is a complex Gaussian random vector with zero mean and covariance matrix σ ^{2} H with \(\mathbf {H}=[\mathbf {H}_{1}, \ldots, \mathbf {H}_{K}] \in \mathbb {C}^{KNM\times KNM}\).
Remark 1.
In this work, the signal model is constructed based on the practical operation of an uplink WCDMA PHY layer. However, the approach and framework can be extended or transferred to other wireless standards. Indeed, the signal model in (1) can be used to describe the continuoustimetransmitted signal of a general system, in which \(s_{\textit {ki}}^{m}(t)\) are the waveforms used for the modulation at the transmitter. For example, in a OFDM system, \(s_{\textit {ki}}^{m}(t)\) can be replaced by the corresponding orthogonal waveforms. Moreover, the resulting equivalent channel in (5) corresponds to a traditional discretetime MIMO multipleaccess channel (MAC), which are used in various research literature.
3 Capacity analysis
In this section, we analyze the capacity of the continuoustime uplink WCDMA channel. Recalling that z is a sufficient statistic for optimal (i.e., capacity preserving) decoding d based on r(t). Any coding scheme which achieves the capacity of the channel with input d and output r(t) can also be employed to the channel with input d and decoding based on output z instead of r(t). Therefore, the channel capacity is preserved when the continuoustime output r(t) is replaced by the sufficient statistic z. Thus, we can focus on the capacity of the equivalent discretetime channel in (5), which is given by the capacity region of a discrete memoryless MAC [32].
for all index subsets \(\mathcal {J} \subseteq \{1,\ldots,K\}\) and some joint pmf \(p(\mathbf {d})=\prod _{k=1}^{K}p(\mathbf {d}_{k})\), where \(\mathcal {J}^{c}\) denotes the complement of \(\mathcal {J}\) and \(\mathbf {d}_{\mathcal {J}}= \{\mathbf {d}_{k} : k\in \mathcal {J}\}\).
We now characterize the uplink WCDMA capacity region considering two specific input signals: finite constellation with uniformly distributed input and Gaussiandistributed input.
3.1 Finite constellation input
Typically, the capacity region of a channel with finite constellation input is numerically characterized via Monte Carlo simulation because a closedform expression does not exist. It is worth noting that in order to calculate the first term of (8), one has to average overall possible \(M_{c}^{\mathcal {J}NM}\) input symbols (up to \(M_{c}^{KNM}\) for sumrate) according to (9). However, when M _{ c } and/or N are too large, this task becomes intractable due to prohibitive computational complexity. In MIMO channels with finite constellation input, a similar problem occurs when the input alphabet set or the number of antennas is too large, e.g., 64QAM or 8 ×8 MIMO [33]. In order to tackle this problem, we have proposed an effective approximation algorithm based on spheredecoding approach to find the approximate capacity for large MIMO system with finite constellation input in [34]. The algorithm to compute the entropy is out of the scope of this work. However, we use it in the numerical results section (Section 3.4) to compute approximations on the capacity curves for large N. The specific details about the algorithm can be found in [34].
3.2 Gaussian input
for all \(\mathcal {J}\!\subseteq \{1,\ldots,\!K\}\). Since the Gaussiandistributed input is the optimal input for a given mean power constraint, (10) serves as an outer bound for the capacity region with a practically motivated input, i.e., finite constellation input as discussed in Section 3.1.
3.3 Sampling
Since the matched filtering at the receiver yields a sufficient statistic, the uplink WCDMA capacity achieved by any other receiver structures is upper bounded by the capacity achieved by the sufficient statistic using matched filtering. Regarding the capacity upper bounds in Sections 3.1–3.2 as benchmarks for the performance assessment, we now analyze the capacity achieved by sampling to evaluate the capacity performance loss due to specific postprocessing at the receiver.
for a finite constellation input signal, where \(\mathbf {r}_{\mathcal {J}}^{\text {sp}} \triangleq \sum _{k\in \mathcal {J}}\sqrt {E_{k}}\bar {\mathbf {S}}_{k} \mathbf {d}_{k} + \mathbf {n}_{\text {sp}}\) is a Gaussian mixture random vector.
3.4 Numerical characterization
In this subsection, we numerically characterize the capacity for a twouser uplink WCDMA example. For numerical experiments, we set the parameters which are close to those in a real uplink UMTS system as specified in [1]: timevariant CDMA with orthogonal variable spreading factor (OVSF) codes and Gold sequences, spreading factor N _{ sf }=4, SRRC chip waveform p(t) with rolloff factor 0.22, and uniform power allocation E _{1}/σ ^{2}=E _{2}/σ ^{2}=SNR. In the simulations, we employ a timeinvariant multipath channel with L=3 taps, a relative pathamplitude vector a=[0,−1.5,−3] dB, a relative pathphase vector \(\boldsymbol {\theta }=\left [0,\frac {\pi }{3}, \frac {2\pi }{3}\right ]\), and pathdelay vector \(\boldsymbol {\tau }=\left [0,\frac {T_{c}}{2}, T_{c}\right ]\). Thus, the lth element of the pathcoefficient vector g _{1} is given by \(a_{l} \cdot \mathrm {e}^{j \theta _{l}}/\\mathbf {a}\\) where a _{ l } is the lth element of a, θ _{ l } is the lth element of vector θ, and \(\mathbf {g}_{2}=\sqrt {2}\mathbf {g}_{1}\). In addition, we use fixed user delays which are randomly drawn within a symbol time once at the beginning of simulations, i.e., \(\lambda _{k} \sim \mathcal {U}(0,T_{s})\).
4 Asymptotic analysis
Recalling (7) with \(\mathcal {J} = \{1,\ldots,K\}\), we have \(\sum _{k=1}^{K}R_{k}\leq I(\mathbf {d};\mathbf {z})\leq \sum _{k=1}^{K} H(\mathbf {d}_{k})\), i.e., the sumcapacity is upper bounded by the sum of the sources entropies. However, the results in Fig. 2 show that when SNR→∞, the individual capacities can be simultaneously achieved, i.e., the sumcapacity approaches the sum of the individual source entropies. In this section, we provide a deeper analysis on this observation by characterizing the asymptotic behavior of the sumcapacity. For convenience, we begin with a simple MAC model then extend the result to the uplink WCDMA system in the following.
4.1 Simple Kuser MAC model
Firstly, we start from the asymptotic sumcapacity of a simple Kuser MAC, where each user transmits only one data stream. This setup corresponds to our uplink WCDMA system with M=1 and N=1 in a frequencynonselective channel. The results are mainly based on the following lemma.
Lemma 1.
where d _{1},…,d _{ K } are the unit power transmitted symbols, which are independent and transmitted using Knormalized signature waveforms s _{1}(t),…,s _{ K }(t) and n(t) denotes the Gaussian noise process with PSD \(\frac {1}{\text {SNR}}\). When SNR→∞, the asymptotic sumcapacity of channel (15), \(C_{\text {sum}}^{\text {as}}\,=\,\sum _{k=1}^{K}\! H(d_{k})\), is achieved if the vector space \(\mathcal {S}_{K}=\text {span}\{s_{1}(t),\dots,s_{K}(t)\}\) has the dimension K.
The proof of Lemma 1 is given in Appendix 2. The idea for the proof is that we first show that the received signal passing through a bank of matched filters, which are matched to the signature waveforms, yields a sufficient statistic for decoding d= [d _{1},⋯,d _{ K }] based on y(t). Then, we show that d can be uniquely decoded, i.e., the decoder is able to decode the messages correctly from this sufficient statistic when SNR→∞ if \(\text {dim}(\mathcal {S}_{K})=K\). Based on the uniquely decodable property, we then prove that the asymptotic sumcapacity \(C_{\text {sum}}^{\text {as}}\) approaches the sum of source entropies if the signature waveforms are linearly independent of each other, i.e., \(\text {dim}(\mathcal {S}_{K})=K\).
4.2 Uplink WCDMA system
Next, we extend the results from the above simple Kuser MAC to the asymptotic sumcapacity of the uplink WCDMA system. Let us recall the transmitted signal from user k of the uplink WCDMA system in (1), and take into account all K users. The transmitted signal can be considered as one of equivalent KNMuser MACs in (15) using KNM signature waveforms \(s_{\textit {ki}}^{m}(t\lambda _{k})\), k=1,…,K,i=1,…,N, and m=1,…,M. The following propositions, which can be derived from Lemma 1, specify the asymptotic sumcapacity of the uplink WCDMA system in different channel environments considering frequencynonselective (L=1) and frequencyselective (L≥2) channels.
Proposition 1.
The asymptotic sumcapacity of the frequencynonselective uplink WCDMA channel as described in (2) with L=1 is \(C_{\text {sum}, \text {nsec}}^{\text {as}}= \sum _{k=1}^{K} H(\mathbf {d}_{k})\) if the dimension of the signature waveforms space \(\mathcal {S}_{T}=\text {span}\left \{s_{11}^{1}(t\lambda _{1}),\ldots, s_{\textit {KN}}^{M}(t\lambda _{K})\right \}\) is KNM.
The proof for Proposition 1 is given in Appendix 3. The intuition behind Proposition 1 can be expressed as: K users transmit KNM symbols and the receiver performs matched filtering with KNM fingers. Although the uplink WCDMA multiuser channel implies Kuser SISO MACs, the matching process virtually converts this to an equivalent K N M×K N M MIMO channel. Thus, by appropriately choosing the signature waveforms and matched fingers, which yield a fullrank equivalent channel matrix H, the transmitted symbols, d _{1},…,d _{ K }, can be perfectly (i.e., errorfree) recovered from z as SNR goes to infinity. In other words, a Kuser uplink WCDMA channel can asymptotically achieve the capacity of KNM parallel channels as long as \(\text {dim}(\mathcal {S}_{T})=KNM\).
Proposition 2.
The asymptotic sumcapacity of the frequencyselective uplink WCDMA channel as described in (2) is \(C_{\text {sum}, \text {sec}}^{\text {as}}~~~~=\sum _{k=1}^{K}\! H(\mathbf {d}_{k})\) if \(\text {dim}(\mathcal {\overline {S}})=KNM\), where \(\mathcal {\overline {S}}=\text {span}\left \{\bar {s}_{11}^{1}(t),\cdots, \bar {s}_{\textit {KN}}^{M}(t)\right \}\) is the vector space spanned by the effective signature waveform \(\bar {s}_{\textit {ki}}^{m}(t)=\sum _{l=1}^{L}g_{\textit {kl}}s_{\textit {ki}}^{m}\) (t−λ _{ k }−τ _{ kl }).
The proof for Proposition 2 is given in Appendix 4. Unlike the frequencynonselective channel case, the sufficient condition for achieving the asymptotic sumcapacity in a frequencyselective channel case is based on the effective signature waveforms, which include the impact of the channel gains {g _{ k }}_{ k } and delays {τ _{ kl }}_{ k,l }. This implies that the multipath channel may help the equivalent channel matrix H to achieve fullrank. For instance, if \(\text {dim}(\mathcal {S}_{T}) < KNM\), H is obviously singular when L=1, while H can be still invertible when L>1 since \(\text {dim}(\overline {\mathcal {S}})\) is possible to be equal to KNM according to the channel selectivity and the potential offset in the multipath environment^{5}. This is particularly helpful in an overloaded CDMA system [8, 21], where the number of users exceed the spreading factor.
4.3 Necessary condition
Propositions 1 and 2 state sufficient conditions that the transmitted messages can be uniquely decoded when SNR→∞, which holds for all kinds of input signals including both finite and infinite constellation signals. However, the conditions in Propositions 1 and 2 can be relaxed in certain scenarios with finite constellation inputs. In this subsection, we first consider a simple example where such conditions can be relaxed. The necessary condition for the unique decoding with finite constellation input is then studied in the following.
where Y _{ a } and Y _{ b } denote the sufficient statistics and N _{ a } and N _{ b } are the equivalent noises. We can see that when the noise power becomes zero (or SNR→∞), d _{ a } (and so d _{ a1} and d _{ a2})) cannot be uniquely decoded from Y _{ a } since the conditional entropy H(d _{ a1},d _{ a2}Y _{ a })=0.5>0 when SNR→∞. However, (d _{ b1} and d _{ b2}) can be uniquely decoded from Y _{ b } since H(d _{ b1},d _{ b2}Y _{ b })=0 when SNR→∞ even though \(\text {dim}(\mathcal {S}_{2})=1<2\). It shows that the condition \(\text {dim}(\mathcal {S}_{K})=K\) can be relaxed for certain signal constellation structures. Therefore, it is expected that necessary conditions for achieving the unique decoding with finite constellation input have to take both the signature waveforms and the structure of the signal constellation into account.
In other words, the necessary condition for the unique decoding is that any vector v _{ ij } with i≠j is not in the null space of matrix H E. This necessary condition includes the impact of signal constellation reflected via v _{ ij }.
Remark 2.
This result is consistent with the sufficient conditions in Propositions 1 and 2. Indeed, when the (effective) signature waveform space has dimension KNM and H is invertible, the null space of H E is empty. Thus, the condition in (17) holds for any set of vector v _{ ij }, and the unique decoding is achieved for any kind of input signal.
5 Conclusions
This paper studies the capacity limit of the uplink WCDMA system whose setup has been chosen to be close to real CDMA cellular networks. We present a theoretical framework, which can be used to evaluate how close the maximal performance of a practical system design is to the theoretical fundamental limit. To this end, sufficient statistics for decoding the transmitted messages were derived using a bank of matched filters, each of which is matched to the signature waveforms. An equivalent discretetime channel model based on the derived sufficient statistics was provided which can be used to analyze the capacity of the system. The capacity regions for finite constellation input and Gaussiandistributed input signals have been both analytically and numerically characterized. The comparison with the sampling capacity showed that sampling within the transmission time window might cause a capacity loss even if the sampling was performed at Nyquist rate. Fortunately, this loss could be significantly diminished by extending the sampling window by only two symbol durations. Moreover, the asymptotic analysis shows that for proper choices of the (effective) signature waveforms, a Kuser uplink WCDMA channel can be decoupled so that each user achieves a pointtopoint channel capacity when SNR goes to infinity. The presented framework and results provide valuable insights for the design and further development of not only WCDMA but also other wireless standard networks.
6 Appendices
7 Appendix 1 Derivation of equivalent noise statistic
Moreover, we have the (a,b)th element of H expressed as
As a result, n is a complex Gaussian random vector with zero mean and covariance matrix σ ^{2} H.
8 Appendix 2 Proof of Lemma 1

Part 1: We first show that the received signal passed through a bank of matched filters, which match to the signature waveforms, yields a sufficient statistic for decoding d= [d _{1},⋯,d _{ K }]^{ T } based on y(t). Moreover, when SNR→∞, d can be uniquely decoded if \(\text {dim}(\mathcal {S}_{K})=K\).

Part 2: Based on the uniquely decodable property, we then derive the asymptotic sumcapacity.
Part 1: Part 1 is a result of the following claim
Claim.
Then y =[y _{1},⋯,y _{ K }]^{ T } is a sufficient statistic for decoding d based on y(t). Moreover, if the vector space \(\mathcal {S}_{K}=\text {span}\{s_{1}(t),\cdots,s_{K}(t)\}\) has a dimension of K, d can be uniquely decoded from the sufficient statistic y as SNR→∞.
Proof.
Following similar steps as in Section 2.2, it can be shown that y is a sufficient statistic for decoding d based on y(t).
Therefore, when \(\text {dim}(\mathcal {S}_{K})=K\), R _{ s } is invertible and d can be uniquely decoded from y when SNR→∞ since \(\underset {\text {SNR}\rightarrow \infty }{\lim }\mathbf {R}_{s}^{1}\mathbf {y}=\mathbf {d}\).
Part 2: We next derive the asymptotic sumcapacity based on the sufficient statistic from Part 1.
This completes the proof for Lemma 1. ■
9 Appendix 3 Proof of Proposition 1

Step 1: We first reformulate (5) by an equivalent inputoutput model, in which H is decomposed into a multiplication of multiple matrices including a matrix that depends only on the signature waveform correlation coefficients. To this end, we rewrite the equivalent channel H as follows:$$\begin{array}{@{}rcl@{}} \mathbf{H}&=&[\mathbf{H}_{1}, \ldots, \mathbf{H}_{K}]\\ &=&\left[ \begin{array}{ccccccccccccc} \mathbf{G}_{1}^{\dagger}\mathbf{R}_{1}^{1}\mathbf{G}_{1} & \mathbf{G}_{1}^{\dagger}\mathbf{R}_{1}^{2}\mathbf{G}_{2} & \cdots &\mathbf{G}_{1}^{\dagger}\mathbf{R}_{1}^{K}\mathbf{G}_{K} \\ \mathbf{G}_{2}^{\dagger}\mathbf{R}_{2}^{1}\mathbf{G}_{1} & \mathbf{G}_{2}^{\dagger}\mathbf{R}_{2}^{2}\mathbf{G}_{2} & \cdots &\mathbf{G}_{2}^{\dagger}\mathbf{R}_{2}^{K}\mathbf{G}_{K} \\ \vdots & \vdots & \vdots & \vdots \\ \mathbf{G}_{K}^{\dagger}\mathbf{R}_{K}^{1}\mathbf{G}_{1} & \mathbf{G}_{K}^{\dagger}\mathbf{R}_{K}^{2}\mathbf{G}_{2} & \cdots &\mathbf{G}_{K}^{\dagger}\mathbf{R}_{K}^{K}\mathbf{G}_{K} \\ \end{array} \right]. \end{array} $$
Therefore, the equivalent channel H can be expressed as
$$ \mathbf{H}=\mathbf{G}^{\dagger}\mathbf{R}\mathbf{G}, $$(C.1)whereand$$\mathbf{G}=\text{blkdiag} \left(\mathbf{G}_{1},\ldots,\mathbf{G}_{K} \right) \in \mathbb{C}^{KNML\times KNM}, $$$$ \mathbf{R}=\left[ \begin{array}{ccccccccccccc} \mathbf{R}_{1}^{1}& \mathbf{R}_{1}^{2}& \cdots &\mathbf{R}_{1}^{K}\\ \mathbf{R}_{2}^{1}& \mathbf{R}_{2}^{2} & \cdots &\mathbf{R}_{2}^{K}\\ \vdots & \vdots & \vdots & \vdots \\ \mathbf{R}_{K}^{1}&\mathbf{R}_{K}^{2}& \cdots &\mathbf{R}_{K}^{K}\\ \end{array} \right] \in\mathbb{C}^{KNML\times KNML}. $$Thus, we have the equivalent inputoutput model as$$ \mathbf{z}=\mathbf{HEd}+\mathbf{n}, $$(C.2)where$$\mathbf{E}=\text{diag} \left(\underbrace{\sqrt{E_{1}},\ldots,\sqrt{E_{1}}}_{NM ~ \text{elements}},\ldots,\underbrace{\sqrt{E_{K}},\ldots,\sqrt{E_{K}}}_{NM ~ \text{elements}} \right). $$ 
Step 2: We show that H is invertible when L=1 and \(\text {dim}(\mathcal {S}_{T})=KNM\).
Following the same arguments as for the proof of Lemma 1, we have R as full rank since \(\text {dim}(\mathcal {S}_{T})=KNM\). Since H=G ^{ † } R G, where G,R are the square matrices with full rank, it follows that H is invertible.

Step 3: Lastly, we can conclude on the asymptotic capacity of the channel (5). Since z in (C.2) is a sufficient statistic for decoding d based on y(t) (from Section 2.2) and H is invertible, similarly to the proof in Part 2 of Lemma 1, it follows that$$ C_{\text{sum},\text{nsec}}^{\text{as}}=H(\mathbf{d})=\sum\limits_{k=1}^{K} H(\mathbf{d}_{k}). $$(C.3)
This completes the proof for Proposition 1. ■
10 Appendix 4 Proof of Proposition 2
Let us define \(\bar {\rho }_{\textit {kim}}^{k'i'm'}\) as the inner product between \(\bar {s}_{\textit {ki}}^{m}(t)\) and \(\bar {s}_{k'i'}^{m'}(t)\), i.e.,
Since \(\mathbf {R}_{\overline {s}}\) is the correlation matrix of \(\overline {\mathcal {S}}\), following similar steps as for the proof in Part 1 of Lemma 1, we arrive at \(\text {rank}(R_{\overline {s}})=KNM\) if \(\text {dim}(\overline {\mathcal {S}})=KNM\).
Thus, rank(H)=K N M and H is invertible.
This completes the proof for Proposition 2. ■
11 Endnotes
^{1} In theory, a bandlimited signal requires infinite time to transmit. However, in practical WCDMA systems, the chip waveforms with fast decaying sidelobes (e.g., root raised cosine (RRC) and squaredroot raised cosine (SRRC) pulses) are used and truncated by the length of several chip intervals.
^{2} It is worth noting that even though the channel impulse response is assumed to be timeinvariant as similar to [12, 24], the Toeplitz structure of the equivalent channel matrix is not maintained because of the variation of the spreading sequences over symbols in a timevariant CDMA system.
^{3} In [27], [Proposition 3.2], the hypotheses are equiprobable and the optimal decision is based on maximum of Φ(d) (ML criterion). In general, the optimal decision is based on MAP criterion, which includes log(p(d)) into μ(t;d). However, this additional term is independent of r(t). Therefore, we do not have to include it in the sufficient statistic.
^{4} Similarly to the capacity achieved by sufficient statistic, the quantities in the righthand sides of (13) and (14) describe the number of bits that can be reliably transmitted per block of N symbols. One can express the capacity in bit/second by normalizing with 1/T, T=N T _{ s }, or in bit/second/Hertz by normalizing with 1/T W.
^{5} When L>1, the channel gain matrix G is not a square matrix anymore. The invertible property of H does not depend only on rank of the correlation matrix R but also on G.
Notes
Declarations
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Authors’ Affiliations
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