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Interference mitigation in multiuser communication by faster than symbol rate sampling
Physical Communication 25 (2017) 148–157
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Interference mitigation in multiuser communication by faster than symbol rate sampling Eren Balevi *, A. Özgür Yılmaz Department of Electrical and Electronics Engineering, Middle East Technical University, Ankara, Turkey
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Article history: Received 26 February 2017 Received in revised form 19 July 2017 Accepted 6 October 2017 Available online 16 October 2017 Keywords: Faster than symbol rate sampling Interference mitigation Multiuser detectors Iterative receivers
a b s t r a c t The degrees of freedom (DoF) gain due to excess bandwidth is lost in case of Symbol Rate (SR) sampling where Faster than Symbol Rate (FTSR) sampling emerges as a promising approach to exploit this gain. In this direction, FTSR sampling is studied to increase the total number of users in a network for a given fixed bandwidth. The rank analysis indicates that FTSR sampling brings rank advantage proportional to the excess bandwidth for unequally spaced channel taps. With the help of an iterative FTSR sampled MMSE multiuser receiver, this rank advantage can be employed in the separation of two users’ signals from each other that transmit simultaneously on the same frequency band. © 2017 Elsevier B.V. All rights reserved.
1. Introduction Multiple users can transmit at the same time and frequency by using the same pulse shape or waveform. In this challenging multiuser settings, excess bandwidth can increase the maximum number of simultaneous transmissions, because it creates additional degrees of freedom (DoF) gain [1]. It is important to exploit all available bandwidth considering the fact that current resources are not enough to satisfy multiple users’ demands on future wireless networks. However, excess bandwidth disappears in case of symbol rate (SR) sampling due to aliasing. Faster than Nyquist (FTN) signaling is proposed to gain the benefit of excess bandwidth, which disappears in SR sampling when symbols are orthogonally sent [2–4]. Although it is shown that FTN signaling efficiently exploits the excess bandwidth, it leads to complicated receiver design [5]. Moreover, adopting the FTN signaling to the existing networks requires modification at transmitters as well as receivers. In this paper, Faster than Symbol Rate (FTSR) sampling is proposed as an alternative to FTN signaling in order to benefit from excess bandwidth with low complexity receiver operation and no modification at transmitters. In FTSR sampling, the received signal is sampled at faster than transmission rate forming additional signal space dimension. Indeed, there is a similarity between FTSR sampling and multiple antenna systems when extra samples are regarded as virtual antennas. It is well-known that employing multiple antennas can
* Corresponding author.
E-mail addresses: [email protected] (E. Balevi), [email protected] (A. Özgür Yılmaz). https://doi.org/10.1016/j.phycom.2017.10.001 1874-4907/© 2017 Elsevier B.V. All rights reserved.
reduce the multiple access interference (MAI) and enhance detection [6,7]. Although one may expect that FTSR sampling can become advantageous because of the similarity with multiple antenna structure, FTSR sampling creates correlation among channel taps and correlated noise that may in the end counter-act the gain of additional signal space dimensions. Therefore, a detailed analysis of FTSR sampling is presented here to show its potential for multiuser communication. There are various studies in which FTSR sampling is instrumental for interference suppression in multiuser communication [8–12]. However, those studies are short of fully characterizing FTSR sampling by considering the potential of excess bandwidth and drawback of correlated noise. Moreover, the concept of FTSR sampling was not investigated in detail, and it is unclear when and for which conditions FTSR sampling works better. Those studies did not consider FTSR sampling as an alternative to FTN signaling as well, whose benefit can be exploited with simple receiver architectures. Our goal is to fully characterize the behavior of FTSR sampling for a multiple access transmission over frequency selective wireless channel when MAI, interblock interference (IBI) and intersymbol interference (ISI) occur with the ultimate aim of exploiting excess bandwidth. Low complexity receiver is one of the primary concern while taking the advantage of excess bandwidth with FTSR sampling, and thus simple receiver front-end employing pulse matched filtering matched to the transmitted pulse instead of the channel is employed. The contributions of this paper are two-fold. The first contribution is to show that FTSR sampling can totally exploit DoF gain due to excess bandwidth in a multiple access frequency selective wireless channel. Further results illustrate that FTSR sampling
E. Balevi, A. Özgür Yılmaz / Physical Communication 25 (2017) 148–157
introduces significant rank advantage provided that ISI channel taps due to frequency selective wireless channel are unequally spaced and excess bandwidth exists. The second contribution of this paper is to observe the interference mitigation capability of FTSR sampling in practical applications including MMSE multiuser detectors and canonical MMSE based iterative receivers. It is illustrated that when two users transmit to a single receiver all with single antenna over multiuser frequency selective wireless channel, MAI can be perfectly eliminated with FTSR sampling once ISI channel has unequally spaced taps and excess bandwidth exists. This result trivially refers to the fact that number of subscribers in a TDMA or FDMA network can be doubled. The paper is organized as follows. The efficiency of FTSR sampling in multiuser communication is analyzed in Section 2. The applications of FTSR sampling are presented in Section 3 and the paper concludes with Section 4.
where pic (t) = p(t) ∗ c i (t). The received signal can then be written as r(t) = s˜i (t) + w (t)
C (Γ ) =
1+r 2T
∫
log2 (1 + Γ |Pc (f )|2 )df
where Γ = Es /MN0 and L ∑
2.1. DoF analysis
∑L
C sr (Γ ) =
1 2T
∫
2
log2 (1 + Γ |Pcsr (f )| )df
(7)
Pc (f + k/T ), −1/2T ≤ f ≤ 1/2T .
(8)
0
such that ∞ ∑
Pcsr (f ) =
k=−∞
When Γ → ∞, 1
lim C sr (Γ ) =
We concentrate on a canonical multiple access channel such that there are many transmitters and a single receiver all with single antenna. Transmitters perform linear modulation for all M users (1)
such that i = 1, 2, . . . , M, xin is the nth transmitted symbol of ith user, p(t) is a real transmitter filter and T is the symbol period. It is assumed that the total average transmission power is Es and each user transmits with an average power of Es /M. Users’ channels are modeled as having (L + 1) multipath components such that
C ftsr (Γ ) =
αki δ (t − τki )
(2)
k=0
where αki represents kth path’s complex channel coefficient and τki is its propagation delay for ith user. Channel coefficients are zero-mean complex Gaussian. It is assumed that channel state information of users is perfectly known at the receiver. Symbols are transmitted in blocks with block length N in which (L + 1) < N and the channel is taken as static during each block and changes independently among blocks [13]. Moreover, zeros are padded among blocks at the transmitter to avoid IBI and then removed at the receiver. The symbol block after passing through the channel can be represented as s˜i (t) =
N −1 ∑ n=0
1 +r 2T
∫
xin pic (t − nT )
(3)
log Γ .
(9)
2
log2 (1 + Γ |Pcftsr (f )| )df
(10)
0
where Pcftsr (f ) =
∞ ∑
Pc (f + k(1 + r)/T ),
k=−∞
− (1 + r)/2T ≤ f ≤ (1 + r)/2T
(11)
and lim C ftsr (Γ ) =
Γ →∞
L
2T
On the other hand, faster sampling rates can yield
n
c i (t) =
(6)
since pc (t) = p(t) ∗ ( k=0 αk δ (t − τk )). When the sampling rate is 1/T , Pc (f ) is subject to aliasing due to excess bandwidth and the capacity becomes
Γ →∞
∑
αk exp(−j2π f τk )
k=0
A multiple access channel is considered throughout the paper such that each user sends their symbols to a single receiver all equipped with single antenna over frequency selective wireless channel. FTSR sampling is assessed for this scenario to improve the system performance by making use of excess bandwidth and obtain more efficient interference mitigation. Our analysis is divided into 2 main parts including DoF analysis at infinite signal to noise (SNR) for Gaussian distributed input symbols, and rank analysis for any SNR and input distribution. Since FTSR sampling is proposed as an alternative to FTN signaling, a comparison is made between these two methods as well.
xin p(t − nT )
(5)
0
2. Analysis of FTSR sampling in multiuser communication
∑
(4)
where w (t) is the additive circularly symmetric complex white Gaussian noise with zero mean and power spectral density N0 . To analyze the DoF for this model, the capacity of single user continuous time ISI channel is firstly stated for excess bandwidth, r. Accordingly, the capacity can be expressed for Gaussian input distribution by omitting the superscript i as [14]
Pc (f ) = P(f )
si (t) =
149
1+r 2T
log Γ .
(12) C (Γ )
Multiplexing gain is defined as limΓ →∞ log(Γ ) [15]. The multiplexing gain ratio of C ftsr (Γ ) to C sr (Γ ) can then be found by using (9) and (12) as lim
Γ →∞
C ftsr (Γ ) C sr (Γ )
= 1 + r.
(13)
Generalizing this result from single user channel to multiple access channel when Γ → ∞ yields [1] Ri < C ftsr (Γ ) = (1 + r)C sr (Γ )
.. . M ∑ i=1
Ri < C ftsr (M Γ ) = (1 + r)C sr (M Γ )
(14) (15)
(16)
150
E. Balevi, A. Özgür Yılmaz / Physical Communication 25 (2017) 148–157
where Ri represents the ith user’s achievable rate. (14)–(16) state that there is a (1+r)−fold multiplexing gain improvement for FTSR sampling due to excess bandwidth. Since multiplexing gain refers to DoF [16], there is a DoF advantage for a multiple access channel in case of FTSR sampling once excess bandwidth is employed. To summarize, SR sampled channel cannot exploit the all available DoF in the channel, whereas FTSR sampled channel can make use of it. 2.2. Rank analysis
1 ≤ normalized(rank) ≤ (1 + r).
(17)
To evaluate the inequality in (17) more explicitly, two transmitters are considered without any loss of generality which have identical waveforms and send their signals at the same time on the same frequency band to a single receiver. Assuming that transmitted symbols, {x1n }’s and {x2n }’s are i.i.d., where {x1n } represents the first user’s transmitted symbol sequence and {x2n } denotes the second user’s symbols, pulse matched filtering the signal component in (4) gives d(t) =
x1n g 1 (t
˜
− nT ) +
n=0
N −1 ∑
x2n g 2 (t
˜
− nT )
(18)
n=0
y(t) = d(t) + z(t)
(19)
such that z(t) = w (t) ∗ p(−t).
(20)
(18) can be written based on (2) and (3) as d(t) =
x1n αk1 p˜ (t − nT − τk1 )
N −1 L ∑ ∑
x2n αk2 p˜ (t − nT − τk2 )
(21)
n=0 k=0
where p˜ (t) = p(t) ∗ p(−t). Sampling the signal for l = 0, . . . , N − 1 and g = 0, 1, . . . , G − 1 yields samples at {d(lT − gT /G)} where G is the sampling rate, which is 2 here for ease of explanation. In accordance with that, when (21) is sampled at {lT } for g = 0, this gives the SR samples that can be expressed as dsrs = d(lT ). l
For ease of presentation, the summations in (21) can be written for t = lT and by defining m = l − n so that l ∑
L ∑
x1l−m αk1 p˜ (mT − τk1 )
m=l−N +1 k=0 l ∑
+
L ∑
x2l−m αk2 p˜ (mT − τk2 ).
(24)
m=l−N +1 k=0
Similarly, (21) can be stated for oversamples, i.e., t = lT − T /2 and by defining m = l − n as l ∑
dos l =
L ∑
x1l−m αk1 p˜ (mT − T /2 − τk1 )
m=l−N +1 k=0 l ∑
+
L ∑
x2l−m αk2 p˜ (mT − T /2 − τk2 ).
(25)
m=l−N +1 k=0
In order to write (24) and (25) in matrix notation, we define
γmi =
L ∑
αki p˜ (mT − τki )
(26)
αki p˜ (mT − T /2 − τki ).
(27)
k=0
and
βmi =
L ∑ k=0
The overall vector–matrix representation is obtained as ysrs 0
⎞
⎜ ysrs ⎟ ⎜ 1 ⎟ ⎜ ⎟ ⎜ . ⎟ ⎜ .. ⎟ ⎜ ⎟ ⎜ srs ⎟ ⎜y ⎟ ⎜ N −1 ⎟ ⎜ os ⎟ ⎜ y0 ⎟ ⎜ os ⎟ ⎜ y1 ⎟ ⎜ ⎟ ⎜ . ⎟ ⎜ . ⎟ ⎝ . ⎠ ⎛
n=0 k=0
+
(23)
yos N −1
L
∑∑
dos l = d(lT − T /2).
⎛
where g˜ i (t) = pic (t) ∗ p(−t) and the overall signal is
N −1
ples, which can be represented as
= dsrs l
The question of whether the additional DoF gain due to excess bandwidth exploited by FTSR sampling at infinite SNR, shown in the previous Section, can be exploited as well at finite SNR is analyzed in terms of the FTSR sampled channel matrix rank, which refers to the used DoF in the channel [17]. Here, the normalized rank is utilized as a performance measure to depict the impact of FTSR sampling, which is defined as the rank of the FTSR sampled channel matrix normalized with transmission block length. The normalized rank analysis will be performed for both orthogonal and non-orthogonal signaling considering finite and infinite N. Here, orthogonal signaling refers to the case of equally spaced channel taps that are spaced with symbol period and nonorthogonal signaling means transmission over a channel that has unequally spaced taps. One can state that the normalized rank can become maximum (1 + r) for N → ∞ depending on the DoF analysis, i.e., the normalized rank for strictly bandlimited channel lies in the interval of
N −1 ∑
On the other hand, sampling (21) at {lT } for g = 1 forms oversam-
(22)
γ01
γ−11 · · · γ−1N +1
⎜ 1 ⎜ γ1 γ01 ⎜ ⎜ . .. ⎜ . ⎜ . . ⎜ ⎜ 1 1 ⎜ γN −1 γN −2 =⎜ ⎜ 1 ⎜ β0 β−1 1 ⎜ ⎜ β1 β01 ⎜ 1 ⎜ ⎜ . .. ⎜ .. . ⎝
· · · γ−1N +2 .. .
.. .
···
γ01
γ02
γ−21 · · · γ−2N +1
γ12
γ02
.. .
.. .
γN2−1 γN2−2
· · · β−1 N +1
β02
β−2 1
· · · β−1 N +2
β12
β02
.. .
.. .
.. .
βN1 −1 βN1 −2 · · ·
.. . β01
⎞
⎟ · · · γ−2N +2 ⎟ ⎟ .. .. ⎟ ⎟ . . ⎟ ⎟ ⎟ · · · γ02 ⎟ ⎟ ⎟ · · · β−2 N +1 ⎟ ⎟ · · · β−2 N +2 ⎟ ⎟ ⎟ .. .. ⎟ . . ⎟ ⎠
βN2 −1 βN2 −2 · · ·
β02
E. Balevi, A. Özgür Yılmaz / Physical Communication 25 (2017) 148–157
⎛ srs ⎞ x10 z0 ⎜ 1 ⎟ ⎜ srs ⎟ ⎜ x1 ⎟ ⎜ z 1 ⎟ ⎜ ⎟ ⎟ ⎜ . ⎟ ⎜ .. ⎟ ⎜ . ⎟ ⎜ ⎟ ⎜ . ⎟ ⎜ . ⎟ ⎜ ⎟ ⎜ ⎜ ⎜ 1 ⎟ ⎜ srs ⎟ ⎜xN −1 ⎟ ⎜zN −1 ⎟ ⎟ ⎟ ×⎜ ⎟ ⎜ 2 ⎟+⎜ os ⎟ ⎜ z ⎜ x0 ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ os ⎟ ⎜ x2 ⎟ ⎜ z ⎟ ⎜ 1 ⎟ ⎜ 1 ⎟ ⎜ ⎟ ⎜ . ⎟ ⎜ .. ⎟ ⎟ ⎜ .. ⎟ ⎜ . ⎠ ⎝ ⎝ ⎠ ⎛
(28)
zNos−1
x2N −1
srs where ysrs represent the SR samples of the received signal l and zl os and the noise respectively. Similarly, yos l and zl denote the additional samples. Then, the discrete-time model can be expressed for a total transmission power Es as
√ y=
Es 2
Hx + z
Lemma 2.1. Equally spaced channel taps, i.e., τk = kT for integer k, cannot create extra rank and normalized rank remains the same in bandlimited multiple access channels despite excess bandwidth and FTSR sampling. Proof. Consider a 2N × 2N FTSR sampled channel matrix for 2 transmitters Hsyn =
α01
⎜ ⎜ 0 ⎜ ⎜ . ⎜ . ⎜ . ⎜ ⎜ 0 ⎜ ⎜ 1 ⎜ β0 ⎜ ⎜ 1 ⎜ β−1 ⎜ ⎜ . ⎜ .. ⎝ β−1 N +1
···
αL1
0
α02
···
αL2
α01
···
αL1
0
α02
···
..
..
..
.
.. .
..
..
1 0
0
0
0
β
β
2 1
···
.
0
β
Fig. 1. The cdf of FTSR sampled channel matrix rank for 10 equally spaced taps and 0.3 excess bandwidth.
(29)
srs os os T where y = [ysrs = [x10 · · · x1N −1 0 · · · yN −1 y0 · · · yN −1 ] , x 2 2 T x0 · · · xN −1 ] , and z is the additive Gaussian noise such that z = [z0srs · · · zNsrs−1 z0os · · · zNos−1 ]T . The channel matrix H is of size 2N × 2N due to FTSR sampling, whereas it is a N × 2N matrix in case of SR sampling. The rank of H will state whether one system can achieve the benefit of excess bandwidth by FTSR sampling or not. Although there is a DoF gain in the channel due to excess bandwidth exploited by FTSR sampling, it may not necessarily mean that all FTSR sampled channels employing excess bandwidth can achieve a rank enhancement. We evaluate this for orthogonal signaling when channel taps are equally spaced with τk = kT and non-orthogonal signaling in case of unequally spaced channel taps with τk ̸ = kT .
⎛
151
⎞
.
0
α
.
.
1 1
···
β
β01 .. .
··· .. .
βN1 −2 .. .
β−2 1 .. .
β02 .. .
··· .. .
···
β−1 1
β01
β−2 N +1
···
β−2 1
1 N −1
2 0
0
⎞
⎟ αL2 ⎟ ⎟ .. ⎟ . ⎟ ⎟ ⎟ α02 ⎟ ⎟ ⎟ βN2 −1 ⎟ ⎟ ⎟ βN2 −2 ⎟ ⎟ .. ⎟ ⎟ . ⎠ β02
(30)
where βki is a linear function of α0i , α1i , · · · , αLi for k = −N + 1, . . . , 0, . . . , N − 1. The (N + 1)th row of Hsyn cannot be written in terms of the linear combination of the first N rows. That is, (N +1)th row is linearly independent from the first N rows. To illustrate, the first term of (N + 1)th row is composed of α01 , α11 , · · · , αL1 , whereas only α01 is available in the first N rows corresponding to the first term. The same discussion is valid for the (N + 2)th row such that it cannot be written by using the first (N + 1) rows. This fact can be explained by considering the first term in the (N + 2)th row, which is composed of L terms, α01 , α11 , · · · , α21 , however only 2 terms from
the 1st and (N + 1)th rows are available. On the other hand, once the first N + L rows are given, one can attain the remaining rows. This emphasizes that the rank of Hsyn is equal to N + L. Then, for a bandlimited channel, i.e., N → ∞, the normalized rank becomes lim
N +L N
N →∞
=1
(31)
showing that the equally spaced channel taps cannot benefit from excess bandwidth. □ This result can be supported by a numerical example. When the channel has 10 equally spaced complex Gaussian channel taps, the empirical cumulative distribution function (cdf) of the channel matrix rank is attained by generating 10,000 random channel realizations for 0.3 excess bandwidth. The cdf of rank, F(rank), is plotted for different block lengths and rank is normalized with respect to the block length N such that a rank of 1 expresses that there is no rank advantage of FTSR sampling. As can be seen in Fig. 1, normalized rank goes to 1 for bandlimited channels, i.e., N → ∞. In practice, channel taps are randomly distributed in between 0 and the maximum channel delay spread, i.e., they are unequally spaced or asynchronous implying that there is an inherent nonorthogonal signaling in the transmission. In this case, one cannot obtain a regular, known matrix structure for FTSR sampling such as in (30), and thus it is not so obvious to determine the rank of the channel matrix by elementary matrix operations, especially for higher values of N. However, it is obvious that asynchronism plays a key role in the channel matrix rank that brings linear independence among the matrix rows or columns that leads to a rank increase, which is formally proven in [17]. This increase can be upper bounded when N → ∞ considering the result of DoF analysis as specified in (17). That is, the normalized rank converges to (1 + r) when N → ∞. On the other hand, the normalized rank can be larger for finite N as will be proven in the following Lemma. Lemma 2.2. Normalized rank of the FTSR sampled channel matrix decreases with incremental and finite N when channel taps are unequally spaced. Proof. The 2N × 2N channel matrix H in (28) can be decomposed as
( H=
H1srs H1os
H2srs H2os
) (32)
where H1srs , H2srs , H1os , H2os can be taken as banded N × N nonHermitian Toeplitz matrices and denote the matrix due to SR and
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E. Balevi, A. Özgür Yılmaz / Physical Communication 25 (2017) 148–157
Table 1 Example channels of both users. 1th user channel
2nd user channel
τ01 = 0 τ11 = 3T /4 α01 = 0.8 α02 = 0.6
τ02 = 0 τ12 = 5T /4 α02 = 0.8 α02 = 0.6
FTSR sampling for the first and second user respectively. The band of these matrices increases for higher N and some eigenvalues exponentially go to zero implying rank degradation [18]. Note that H can be viewed as a block Toeplitz matrix by elementary matrix operations with changing the order of rows and columns. Since the same property is still valid for block Toeplitz matrix, normalized rank decreases for higher block lengths [18]. □ To support Lemma 2.2, a simple example can be beneficial assuming that N = 4 and pulse shape filter is a raised cosine with a roll off factor of 0.3. When each user has a channel with 2 taps whose coefficients are equal but propagation delays are different as given in Table 1, the resultant channel matrix becomes
Fig. 2. The cdf of FTSR sampled channel matrix rank for 10 unequally spaced taps and 0.3 excess bandwidth.
Hasy = 0.97 0.53 −0.09 0.03 0.70 0.53 0.17 −0.05 0 0.43 0.67 0.39 −0.03⎟ ⎜ 0.40 1.03 0.03 ⎜−0.05 0.97 0.53 −0.09 0.03 0.70 0.53 0.17 ⎟ ⎜ ⎟ ⎜−0.10 0.40 1.03 0.03 −0.11 0.43 0.67 0.39 ⎟ ⎜ ⎟ ⎜ 0.02 −0.05 0.97 0.53 −0.01 0.03 0.70 0.53 ⎟ . (33) ⎜ ⎟ ⎜ 0.04 −0.10 0.40 1.03 0.04 −0.11 0.43 0.67 ⎟ ⎝ 0 ⎠ 0.02 −0.05 0.97 0 −0.01 0.03 0.70 −0.01 0.04 −0.10 0.40 −0.02 0.04 −0.11 0.43
⎛
⎞
Relying on (33), the rank becomes 8 even if channel coefficients are the same due to asynchronism among the matrix entries. Thus, the normalized rank becomes 2, i.e., FTSR sampled channel matrix doubles the rank of SR sampled channel matrix for a small value of N, although it is upper bounded by 1.3 for N → ∞. By this is meant that, the normalized rank gradually decreases for increasing N. Empirical cdf of the channel matrix rank can be obtained by generating 10,000 random channel realizations for a root raised cosine pulse shape of 0.3 excess bandwidth and 10 unequally spaced complex Gaussian channel taps that are distributed uniformly between 0 and 10 symbol intervals. There is an asynchronism between users’ channels as well as the channel taps. Rank is normalized with respect to the block length N and plotted for different block lengths as given in Fig. 2. Accordingly, as the block length becomes larger, the rank converges to 1.3 as predicted by the expression in (13). That is, the maximum simultaneous transmissions or the total number of users in a multiple access ISI channel can be increased in direct proportion to excess bandwidth by FTSR sampling. This result is generalized for 5 users which simultaneously transmit their symbols to a receiver. Similarly, there are 10 unequally spaced complex Gaussian channel taps that are distributed uniformly between 0 and 10 symbol intervals with a root raised cosine pulse shape of 0.3 excess bandwidth and G = 2. The same result is observed with the previous case as presented in Fig. 3. It shows that the increase in the normalized rank due to FTSR sampling is independent of the number of users provided that excess bandwidth exists and channel taps are unequally spaced. Note that increasing the sampling rate beyond 2 gives the same gain, since excess bandwidth has already been exploited and there is nothing to gain. Users can experience similar multipath fading profiles in rural areas or urban areas when they are geographically close to each other. Even if the channel taps are unequally spaced, users may have nearly identical channels. This case is evaluated in Fig. 4
Fig. 3. The cdf of FTSR sampled channel matrix rank for 10 unequally spaced taps and 0.3 excess bandwidth for 5 users.
showing that normalized rank becomes 1 despite FTSR sampling. Especially, it is not hard to predict this situation by observing the channel matrix in (28). Therefore, it is important to have nonidentical channels among users as well as unequally spaced channel taps to exploit excess bandwidth advantage by FTSR sampling. To summarize, additional DoF due to excess bandwidth cannot be exploited for synchronous transmission, whereas it is possible to attain this DoF for asynchronous multiuser transmission by FTSR sampling. Since the rank of H does not depend on SNR [19], FTSR sampling can take advantage of the excess bandwidth at reasonable SNR for carefully designed systems. Moreover, our results are valid for any input distribution. Notice that the rank difference between the equally and unequally spaced taps can be explained as well based on a study that discusses the capacity of synchronous and asynchronous transmission which supports our results [17]. 2.3. Comparison with FTN signaling An alternative view of FTSR sampling is FTN signaling. Although FTN signaling is based on the idea of packing the orthogonal pulses closer than the Nyquist rate resulting in reduced symbol times without any loss in minimum Euclidean distance [20,21], it has recently been studied in a rather different concept to exploit the available capacity due to excess bandwidth [2–4]. Analogous to FTN signaling which aims to increase the number of bits for a given
E. Balevi, A. Özgür Yılmaz / Physical Communication 25 (2017) 148–157
153
such that there are two transmitters and a single receiver all with single antenna and our aim is to separate these two users’ symbols at the receiver. 3.1. FTSR sampled MMSE multiuser detector
Fig. 4. The cdf of FTSR sampled channel’s matrix rank for 10 unequally spaced taps and 0.3 excess bandwidth when the user’s channels are identical.
One of the widely used multiuser detection technique to suppress interference is based on MMSE criterion that has reasonable complexity with acceptable performance [22]. The conventional MMSE multiuser detector for 2 users is comprised of 2 channel matched filters, each of which is matched to a single user [22] and has to become adaptive so that they change in each block because of varying channel conditions that results in high computational complexity. Instead of this structure, we will study a FTSR sampled multiuser detector consisting of a single pulse matched filter whose output is sampled at FTSR and processed by an MMSE filter. Notice that the noise becomes colored due to pulse matched filtering and FTSR sampling, whose auto-correlation function can be specified by using the (20) as E [z(t)z ∗ (t + τ )] = σ 2 p˜ (τ ).
bandwidth, FTSR sampling can be used to increase the number of users. Although both methods can fully exploit the available DoF in the channel due to excess bandwidth, FTSR sampling is a more practical technique than FTN signaling. FTSR sampling requires modification only at the receiver side, while FTN needs modification at the transmitter side as well. Hence, FTSR sampling can be easily adopted in existing networks. Moreover, FTN complicates the receiver design that necessitates alternative methods [5]. The major impediment in FTSR sampling occurs when channel taps are equally spaced in which transmission become orthogonal. In this case, FTSR sampling cannot exploit the advantage of excess bandwidth, however, this is a hypothetical situation not present in actual physical channels and practically unimportant.
where σ is the variance of z(t). The noise auto-correlation matrix, Rz = E [zzH ] can then be written in terms of (34) in which the (q, r) element of Rz becomes
[Rz ]q,r = σ 2 p˜ ((q − r)T /2).
(35) H
The transmitted symbols are i.i.d., that is, E [xx ] = I2N such that I2N represents the 2N × 2N identity matrix, which yields the following MMSE filter [23]
√ Wmmse =
Es 2
HH
(
Es 2
HHH + Rz
)−1
.
(36)
The MMSE matrix, M, which is equal to E [(Wmmse y − x)(Wmmse y − x)H ], can then be written as
3. Applications of FTSR sampling
√ M = I2N −
FTSR sampling has certain rank advantage in multiuser communication provided that excess bandwidth exists and channel taps are unequally spaced, which are inherent in today’s communication systems. However, rank is independent of noise statistics. The basic goal of this part is to determine the benefits of FTSR sampling by considering the effect of correlated noise stemming from pulse matched filtering and FTSR sampling. Pulse matched filtering is preferred regarding practical concerns due to the fact that it can be implemented with non-adaptive analog filter rather than channel matched filtering requiring adaptive analog filter. Within this scope, a multiuser detector is firstly investigated designed as the cascade of an FTSR sampled pulse matched filter and an MMSE filter. Although this detector has a well known structure, it is not analyzed in detail under different superimposed ISI channel conditions for practical system designs, i.e., for different pulse shaping filters, channel characteristics and sampling rates. To better explain the FTSR sampled detector, a virtual multiple input multiple output (MIMO) model is set up by FTSR sampling and analyzed for multiple access ISI channels with this equivalent MIMO structure in mind. Secondly, this FTSR sampled MMSE detector will be adapted to a canonical iterative receiver to increase the number of users in a multiple access network. What we propose is to add a new transmitter corresponding to each user that utilizes the same resources with this user, i.e., they transmit at the same time and frequency and use the same pulse shape for transmitter filter. If these two users’ symbols are separated at the receiver, this will obviously double the number of users in a given TDMA or FDMA network. Therefore, throughout the paper we focus on a scenario
(34)
2
Es 2
( + Gmmse
√ Gmmse H − Es 2
HHH + Rz
Es 2
)
HH GH mmse
GH mmse .
(37)
The mean square error (MSE) of the mth transmitted symbol depending on (37) for m = 0, 1, . . . , (2N − 1) is equal to MSEm = Mm,m
(38)
where Mm,m represents the mth diagonal term of the matrix M. In order to determine the signal to interference plus noise ratio (SINR), a well known relation between the SINR and MSE is used [8], which is SINRm =
1 MSEm
− 1.
(39)
The performance of detector is determined in terms of (39) when two single antenna transmitters send their symbols to a receiver which has also single antenna assuming that there are 10 complex Gaussian channel taps whose propagation delays are uniformly distributed in [0, 10T ], the block length is equal to 100 and G = 2. The influence of excess bandwidth due to root raised cosine pulse shaping of transmitter filter for these settings is observed in Fig. 5. As can be seen, excess bandwidth has a major effect on the SINR increase in FTSR sampling, which supports the acquired expression in (13). The impact of FTSR sampling with G = 2 for equally T spaced channel taps as opposed to unequally spaced taps in Fig. 5 is depicted for 10 complex Gaussian channel taps when the block
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Fig. 5. Excess bandwidth effect on the FTSR sampled MMSE detector for 10 unequally spaced complex Gaussian channel taps.
Fig. 7. The effect of sampling rate for 10 unequally spaced complex Gaussian channel taps with a roll off factor 0.3 on the FTSR sampled MMSE detector.
to MIMO. On the other hand, there is a gain provided that dsrs l is not srs fully correlated with dos and l . Indeed, the correlation between dl dos depends on the correlation among channel taps which can be l found as ∗
E [γmi βnj ] =
L ∑
2
E [|αki | ]˜pm p˜ n δ[i − j]
(40)
k=0
where p˜ m = p˜ (mT − τki ) and p˜ n = p˜ (nT − T /2 − τki ). In case of normalized channel taps, (40) becomes i j∗ m n
E [γ β ] =
{ < p˜ 1 , p˜ 2 > 0
if i = j if i ̸ = j
(41)
where p˜ 1 and p˜ 2 are 1 × (L + 1) vectors consisting of (L + 1) points on pulse shape p˜ (t). When L → ∞, the correlation between consecutive channel taps can be represented as Fig. 6. The effect of 10 equally spaced complex Gaussian channel taps on the FTSR sampled MMSE detector when roll off factor is 0.6.
length becomes 100 and the transmitter pulse shaping filter is a root raised cosine with a roll off factor of 0.6 in Fig. 6. It can be observed that FTSR sampling does not lead to any significant advantage for equally spaced channel taps even for a large excess bandwidth. The result is in parallel with Fig. 1, which emphasizes the disappearance of rank advantage for equally spaced ISI taps. To analyze the performance enhancement depending on the increase in sampling rate in a more rigorous manner, single antenna transmitters are regarded as a single transmitter with multiple antennas and extra samples in time domain are modeled as virtual antennas in spatial domain. This approach converts the multiple access channel into a virtual MIMO channel that has correlated channel taps and correlated noise due to FTSR sampling. Furthermore, separate coding has to be performed in virtual MIMO instead of joint coding due to independence of transmitted symbols from each other, and power is shared equally among users, because channel state information at the transmitter is not available. It is well known from MIMO systems that correlation among antennas leads to diminishing performance [19,24]. In particular, when channel taps are fully correlated, the advantage of multiple antennas at receiver disappears [19,24]. Since additional samples are considered as virtual antennas, there will be no gain after FTSR os sampling if dsrs l in (22) is fully correlated with dl in (23) analogous
∗
E [γmi βni ] =
∫
∞
p˜ (t)p˜ (t − T /2)dt .
(42)
−∞
When the offset between functions is T , they become orthogonal resulting in uncorrelated samples. On the other hand, they become fully correlated if there is no offset. The smaller the offset between functions, the higher the correlation among channel taps are. Since the offset is equal to gT /G, the biggest improvement of integer FTSR sampling occurs when G = 2 and the impact of FTSR sampling decreases by increasing G. To observe this case numerically, we assume that there are 10 complex Gaussian channel taps which have uniformly distributed propagation delays in [0, 10T ] when the block length is equal to 100 and the transmitter filter is root raised cosine with 0.3 roll off factor. As shown in Fig. 7, the highest change due to FTSR sampling is attained for G = 2. Further increases in the sampling rate has diminishing returns. In particular, there is no difference between G = 3 and G = 5. 3.2. An iterative receiver based on FTSR sampling It is appealing to share resources among users to increase the total number of users. Within this scope, two users transmit their symbols at the same time and frequency to a single receiver all with single antenna when they have the same pulse shape but different channels. The rationale behind the selection of two users stems from the fact that normalized rank doubles at most for G = 2. Although it is a difficult task to distinguish the symbols of
E. Balevi, A. Özgür Yılmaz / Physical Communication 25 (2017) 148–157
155
Fig. 8. An iterative receiver based on FTSR sampling for multiple users.
(a). Fig. 10. Performance of the FTSR sampled iterative receiver with G = 3 for 3 transmitters and a single receiver.
(b).
Fig. 9. There are 2 transmitters for 2 taps ISI channel (a) Performance of the SR sampled iterative receiver (b) Performance of the FTSR sampled iterative receiver with G = 2.
multiple users for this scenario, the problem may be solved using both FTSR sampling and iterative receiver. We study a receiver for FTSR sampling in case of two users based on the canonical iterative receiver [25], which is given in Fig. 8. It comprises of an FTSR sampled MMSE detector and channel decoders working on the maximum a posteriori probability criterion. Note that pulse matched filter is used instead of channel matched filter making the receiver more practical. According to the output of the channel decoder soft estimation and interference cancellation is performed at the detector. Iterative receivers with FTSR sampled front-ends are mentioned in the literature [26,27], however, they did not consider MAI, and performance comparisons between FTSR and SR sampled iterative receivers are not available.
Information sequence belonging to the first user, {In1 } and the second user, {In2 } are convolutionally encoded, interleaved with distinct interleaves and then BPSK modulated. The resultant N × 1 symbol vectors are represented as a1 = [a1 a3 · · · a2N −1 ] and a2 = [a0 a2 · · · a2N −2 ] for the first and the second user that are transmitted in blocks with block length N. After passing through the ISI channels that are specified in (2), the received signal is low pass filtered with pulse matched filter p(−t) and FTSR sampled with period T ′ such that T ′ < T . Since there are two users, T ′ = T /2 is a sufficiently good choice. The system model becomes as (29), where x = [a1 a2 ] with the only difference that the transmitted symbols are now convolutionally coded. Throughout the process, extrinsic information between the detector and decoder is exchanged in an iterative manner [28], and MMSE filtering is applied to the received signal in (29) which leads to v = Wmmse y
(43)
where v is a 1 × 2N vector such that odd terms of the v are utilized to decode the first user symbols v1 = [v1 v3 · · · v2N −1 ]T
(44)
and the remaining terms of v are for the second user v2 = [v0 v2 · · · v2N −2 ]T .
(45)
After MMSE filtering, the a posteriori log likelihood ratio of the channel decoder is found, from which the extrinsic information generated by the channel decoder is attained [25]. In the last iteration, information bit log likelihood ratios are determined and the decision is made. The FTSR sampled iterative receiver’s performance is evaluated when the modulation format is BPSK and the symbols are transmitted in blocks with block length 100 for a 0.3 roll off factor. It is assumed that the channel becomes constant within one block,
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with iterations between the SR sampled iterative receiver and FTSR sampled iterative receiver is compatible with this interpretation. The performance of the FTSR sampled iterative receiver is investigated for 3 users and G = 3. The average error performance for 3-fold FTSR sampled iterative receiver is depicted in Fig. 10. Although there is a degradation with respect to the 2-user receiver performance with G = 2, it has satisfactory performance in the 3th iteration. It can be inferred that increasing the number of users in parallel with the FTSR sampling ratio leads to a performance loss. The channel may have more than 2 taps, therefore, the simulation results are repeated for 5 taps. According to that, both users have 5 complex Gaussian taps whose propagation delays are uniformly distributed in [0, 5T ] and their variances are normalized to unity. The SR and FTSR sampled iterative receiver performance is presented in Figs. 11(a) and 11(b), respectively. Similar to the previous case, FTSR sampled iterative receiver gives reasonable results, while SR sampling cannot. (a).
4. Conclusions FTSR sampling is studied in multiuser communication within the purpose of better interference mitigation to increase the number of users. DoF and rank analysis show that FTSR sampling can better exploit the excess bandwidth depending on unequally spaced channel taps and asynchronism among users for a multiple access channel. The level of improvement due to unequally spaced channel taps and excess bandwidth is quantified with an FTSR sampled MMSE detector. The results show that there is an advantage provided that excess bandwidth exists and channel taps are unequally spaced. These results are instrumental in designing an iterative receiver based on FTSR sampling. Simulation results illustrate that FTSR sampled iterative receiver better mitigates the interference among multiple users, and it makes possible to increase the number of users in a given network. (b).
Fig. 11. There are 2 transmitters for 5 taps ISI channel (a) Performance of the SR sampled iterative receiver (b) Performance of the FTSR sampled iterative receiver with G = 2.
and changes from block to block. Moreover, convolutional coding is employed at the transmitters with coding rate 1/2, constraint length 5, and generators are (23, 35) in terms of octal notation. Suppose that both users have 2 complex Gaussian taps whose propagation delays are uniformly distributed in [0, 5T ] and their variances are normalized to unity. In this case, SR sampled iterative receiver’s bit error rate (BER) is presented in Fig. 9(a), which includes the single user non-iterative receiver performance as a comparison to clarify the multiuser interference cancellation capability of the receiver. Indeed, a genie-aided interference cancellation scheme would perform the same as the single user receiver. As can be observed, the SR sampled iterative structure does not reach the single user receiver performance. Hence, we can conclude that even an iterative receiver could not perform very well with SR sampling. On the other hand, the FTSR sampled iterative receiver with G = 2 can greatly reduce the interference between users as can observed in Fig. 9(b). The error performance difference between the single user receiver and the 3th iteration of the FTSR sampled iterative receiver is very small. One critical point about the iterative receivers is the initial BER. In the context of iterative receivers, it is shown that iterative structures with enhanced initial BER yield better results [29]. The increasing performance difference
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Eren Balevi received the B.S., M.S., and Ph.D. degrees in electrical and electronics engineering from Middle East Technical University, Ankara, Turkey, in 2008, 2010 and 2016 respectively. His current research interests are in the 5G wireless systems and the Internet of Things in addition to the general areas of nanoscale communications, multiuser wireless communications and signal processing.
A. Özgür Yılmaz received the B.S. degree in electrical engineering from the University of Michigan, Ann Arbor in 1999. He received the MS and Ph.D. degrees at the same school in 2001 and 2003. He is a professor in the Department of Electrical and Electronics Engineering, Middle East Technical University, Turkey. His research interests include coding and information theory in wireless communication systems, cooperative communications, multiantenna systems, and radar signal processing.
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Physical Communication journal homepage: www.elsevier.com/locate/phycom
Full length article
Interference mitigation in multiuser communication by faster than symbol rate sampling Eren Balevi *, A. Özgür Yılmaz Department of Electrical and Electronics Engineering, Middle East Technical University, Ankara, Turkey
article
info
Article history: Received 26 February 2017 Received in revised form 19 July 2017 Accepted 6 October 2017 Available online 16 October 2017 Keywords: Faster than symbol rate sampling Interference mitigation Multiuser detectors Iterative receivers
a b s t r a c t The degrees of freedom (DoF) gain due to excess bandwidth is lost in case of Symbol Rate (SR) sampling where Faster than Symbol Rate (FTSR) sampling emerges as a promising approach to exploit this gain. In this direction, FTSR sampling is studied to increase the total number of users in a network for a given fixed bandwidth. The rank analysis indicates that FTSR sampling brings rank advantage proportional to the excess bandwidth for unequally spaced channel taps. With the help of an iterative FTSR sampled MMSE multiuser receiver, this rank advantage can be employed in the separation of two users’ signals from each other that transmit simultaneously on the same frequency band. © 2017 Elsevier B.V. All rights reserved.
1. Introduction Multiple users can transmit at the same time and frequency by using the same pulse shape or waveform. In this challenging multiuser settings, excess bandwidth can increase the maximum number of simultaneous transmissions, because it creates additional degrees of freedom (DoF) gain [1]. It is important to exploit all available bandwidth considering the fact that current resources are not enough to satisfy multiple users’ demands on future wireless networks. However, excess bandwidth disappears in case of symbol rate (SR) sampling due to aliasing. Faster than Nyquist (FTN) signaling is proposed to gain the benefit of excess bandwidth, which disappears in SR sampling when symbols are orthogonally sent [2–4]. Although it is shown that FTN signaling efficiently exploits the excess bandwidth, it leads to complicated receiver design [5]. Moreover, adopting the FTN signaling to the existing networks requires modification at transmitters as well as receivers. In this paper, Faster than Symbol Rate (FTSR) sampling is proposed as an alternative to FTN signaling in order to benefit from excess bandwidth with low complexity receiver operation and no modification at transmitters. In FTSR sampling, the received signal is sampled at faster than transmission rate forming additional signal space dimension. Indeed, there is a similarity between FTSR sampling and multiple antenna systems when extra samples are regarded as virtual antennas. It is well-known that employing multiple antennas can
* Corresponding author.
E-mail addresses: [email protected] (E. Balevi), [email protected] (A. Özgür Yılmaz). https://doi.org/10.1016/j.phycom.2017.10.001 1874-4907/© 2017 Elsevier B.V. All rights reserved.
reduce the multiple access interference (MAI) and enhance detection [6,7]. Although one may expect that FTSR sampling can become advantageous because of the similarity with multiple antenna structure, FTSR sampling creates correlation among channel taps and correlated noise that may in the end counter-act the gain of additional signal space dimensions. Therefore, a detailed analysis of FTSR sampling is presented here to show its potential for multiuser communication. There are various studies in which FTSR sampling is instrumental for interference suppression in multiuser communication [8–12]. However, those studies are short of fully characterizing FTSR sampling by considering the potential of excess bandwidth and drawback of correlated noise. Moreover, the concept of FTSR sampling was not investigated in detail, and it is unclear when and for which conditions FTSR sampling works better. Those studies did not consider FTSR sampling as an alternative to FTN signaling as well, whose benefit can be exploited with simple receiver architectures. Our goal is to fully characterize the behavior of FTSR sampling for a multiple access transmission over frequency selective wireless channel when MAI, interblock interference (IBI) and intersymbol interference (ISI) occur with the ultimate aim of exploiting excess bandwidth. Low complexity receiver is one of the primary concern while taking the advantage of excess bandwidth with FTSR sampling, and thus simple receiver front-end employing pulse matched filtering matched to the transmitted pulse instead of the channel is employed. The contributions of this paper are two-fold. The first contribution is to show that FTSR sampling can totally exploit DoF gain due to excess bandwidth in a multiple access frequency selective wireless channel. Further results illustrate that FTSR sampling
E. Balevi, A. Özgür Yılmaz / Physical Communication 25 (2017) 148–157
introduces significant rank advantage provided that ISI channel taps due to frequency selective wireless channel are unequally spaced and excess bandwidth exists. The second contribution of this paper is to observe the interference mitigation capability of FTSR sampling in practical applications including MMSE multiuser detectors and canonical MMSE based iterative receivers. It is illustrated that when two users transmit to a single receiver all with single antenna over multiuser frequency selective wireless channel, MAI can be perfectly eliminated with FTSR sampling once ISI channel has unequally spaced taps and excess bandwidth exists. This result trivially refers to the fact that number of subscribers in a TDMA or FDMA network can be doubled. The paper is organized as follows. The efficiency of FTSR sampling in multiuser communication is analyzed in Section 2. The applications of FTSR sampling are presented in Section 3 and the paper concludes with Section 4.
where pic (t) = p(t) ∗ c i (t). The received signal can then be written as r(t) = s˜i (t) + w (t)
C (Γ ) =
1+r 2T
∫
log2 (1 + Γ |Pc (f )|2 )df
where Γ = Es /MN0 and L ∑
2.1. DoF analysis
∑L
C sr (Γ ) =
1 2T
∫
2
log2 (1 + Γ |Pcsr (f )| )df
(7)
Pc (f + k/T ), −1/2T ≤ f ≤ 1/2T .
(8)
0
such that ∞ ∑
Pcsr (f ) =
k=−∞
When Γ → ∞, 1
lim C sr (Γ ) =
We concentrate on a canonical multiple access channel such that there are many transmitters and a single receiver all with single antenna. Transmitters perform linear modulation for all M users (1)
such that i = 1, 2, . . . , M, xin is the nth transmitted symbol of ith user, p(t) is a real transmitter filter and T is the symbol period. It is assumed that the total average transmission power is Es and each user transmits with an average power of Es /M. Users’ channels are modeled as having (L + 1) multipath components such that
C ftsr (Γ ) =
αki δ (t − τki )
(2)
k=0
where αki represents kth path’s complex channel coefficient and τki is its propagation delay for ith user. Channel coefficients are zero-mean complex Gaussian. It is assumed that channel state information of users is perfectly known at the receiver. Symbols are transmitted in blocks with block length N in which (L + 1) < N and the channel is taken as static during each block and changes independently among blocks [13]. Moreover, zeros are padded among blocks at the transmitter to avoid IBI and then removed at the receiver. The symbol block after passing through the channel can be represented as s˜i (t) =
N −1 ∑ n=0
1 +r 2T
∫
xin pic (t − nT )
(3)
log Γ .
(9)
2
log2 (1 + Γ |Pcftsr (f )| )df
(10)
0
where Pcftsr (f ) =
∞ ∑
Pc (f + k(1 + r)/T ),
k=−∞
− (1 + r)/2T ≤ f ≤ (1 + r)/2T
(11)
and lim C ftsr (Γ ) =
Γ →∞
L
2T
On the other hand, faster sampling rates can yield
n
c i (t) =
(6)
since pc (t) = p(t) ∗ ( k=0 αk δ (t − τk )). When the sampling rate is 1/T , Pc (f ) is subject to aliasing due to excess bandwidth and the capacity becomes
Γ →∞
∑
αk exp(−j2π f τk )
k=0
A multiple access channel is considered throughout the paper such that each user sends their symbols to a single receiver all equipped with single antenna over frequency selective wireless channel. FTSR sampling is assessed for this scenario to improve the system performance by making use of excess bandwidth and obtain more efficient interference mitigation. Our analysis is divided into 2 main parts including DoF analysis at infinite signal to noise (SNR) for Gaussian distributed input symbols, and rank analysis for any SNR and input distribution. Since FTSR sampling is proposed as an alternative to FTN signaling, a comparison is made between these two methods as well.
xin p(t − nT )
(5)
0
2. Analysis of FTSR sampling in multiuser communication
∑
(4)
where w (t) is the additive circularly symmetric complex white Gaussian noise with zero mean and power spectral density N0 . To analyze the DoF for this model, the capacity of single user continuous time ISI channel is firstly stated for excess bandwidth, r. Accordingly, the capacity can be expressed for Gaussian input distribution by omitting the superscript i as [14]
Pc (f ) = P(f )
si (t) =
149
1+r 2T
log Γ .
(12) C (Γ )
Multiplexing gain is defined as limΓ →∞ log(Γ ) [15]. The multiplexing gain ratio of C ftsr (Γ ) to C sr (Γ ) can then be found by using (9) and (12) as lim
Γ →∞
C ftsr (Γ ) C sr (Γ )
= 1 + r.
(13)
Generalizing this result from single user channel to multiple access channel when Γ → ∞ yields [1] Ri < C ftsr (Γ ) = (1 + r)C sr (Γ )
.. . M ∑ i=1
Ri < C ftsr (M Γ ) = (1 + r)C sr (M Γ )
(14) (15)
(16)
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where Ri represents the ith user’s achievable rate. (14)–(16) state that there is a (1+r)−fold multiplexing gain improvement for FTSR sampling due to excess bandwidth. Since multiplexing gain refers to DoF [16], there is a DoF advantage for a multiple access channel in case of FTSR sampling once excess bandwidth is employed. To summarize, SR sampled channel cannot exploit the all available DoF in the channel, whereas FTSR sampled channel can make use of it. 2.2. Rank analysis
1 ≤ normalized(rank) ≤ (1 + r).
(17)
To evaluate the inequality in (17) more explicitly, two transmitters are considered without any loss of generality which have identical waveforms and send their signals at the same time on the same frequency band to a single receiver. Assuming that transmitted symbols, {x1n }’s and {x2n }’s are i.i.d., where {x1n } represents the first user’s transmitted symbol sequence and {x2n } denotes the second user’s symbols, pulse matched filtering the signal component in (4) gives d(t) =
x1n g 1 (t
˜
− nT ) +
n=0
N −1 ∑
x2n g 2 (t
˜
− nT )
(18)
n=0
y(t) = d(t) + z(t)
(19)
such that z(t) = w (t) ∗ p(−t).
(20)
(18) can be written based on (2) and (3) as d(t) =
x1n αk1 p˜ (t − nT − τk1 )
N −1 L ∑ ∑
x2n αk2 p˜ (t − nT − τk2 )
(21)
n=0 k=0
where p˜ (t) = p(t) ∗ p(−t). Sampling the signal for l = 0, . . . , N − 1 and g = 0, 1, . . . , G − 1 yields samples at {d(lT − gT /G)} where G is the sampling rate, which is 2 here for ease of explanation. In accordance with that, when (21) is sampled at {lT } for g = 0, this gives the SR samples that can be expressed as dsrs = d(lT ). l
For ease of presentation, the summations in (21) can be written for t = lT and by defining m = l − n so that l ∑
L ∑
x1l−m αk1 p˜ (mT − τk1 )
m=l−N +1 k=0 l ∑
+
L ∑
x2l−m αk2 p˜ (mT − τk2 ).
(24)
m=l−N +1 k=0
Similarly, (21) can be stated for oversamples, i.e., t = lT − T /2 and by defining m = l − n as l ∑
dos l =
L ∑
x1l−m αk1 p˜ (mT − T /2 − τk1 )
m=l−N +1 k=0 l ∑
+
L ∑
x2l−m αk2 p˜ (mT − T /2 − τk2 ).
(25)
m=l−N +1 k=0
In order to write (24) and (25) in matrix notation, we define
γmi =
L ∑
αki p˜ (mT − τki )
(26)
αki p˜ (mT − T /2 − τki ).
(27)
k=0
and
βmi =
L ∑ k=0
The overall vector–matrix representation is obtained as ysrs 0
⎞
⎜ ysrs ⎟ ⎜ 1 ⎟ ⎜ ⎟ ⎜ . ⎟ ⎜ .. ⎟ ⎜ ⎟ ⎜ srs ⎟ ⎜y ⎟ ⎜ N −1 ⎟ ⎜ os ⎟ ⎜ y0 ⎟ ⎜ os ⎟ ⎜ y1 ⎟ ⎜ ⎟ ⎜ . ⎟ ⎜ . ⎟ ⎝ . ⎠ ⎛
n=0 k=0
+
(23)
yos N −1
L
∑∑
dos l = d(lT − T /2).
⎛
where g˜ i (t) = pic (t) ∗ p(−t) and the overall signal is
N −1
ples, which can be represented as
= dsrs l
The question of whether the additional DoF gain due to excess bandwidth exploited by FTSR sampling at infinite SNR, shown in the previous Section, can be exploited as well at finite SNR is analyzed in terms of the FTSR sampled channel matrix rank, which refers to the used DoF in the channel [17]. Here, the normalized rank is utilized as a performance measure to depict the impact of FTSR sampling, which is defined as the rank of the FTSR sampled channel matrix normalized with transmission block length. The normalized rank analysis will be performed for both orthogonal and non-orthogonal signaling considering finite and infinite N. Here, orthogonal signaling refers to the case of equally spaced channel taps that are spaced with symbol period and nonorthogonal signaling means transmission over a channel that has unequally spaced taps. One can state that the normalized rank can become maximum (1 + r) for N → ∞ depending on the DoF analysis, i.e., the normalized rank for strictly bandlimited channel lies in the interval of
N −1 ∑
On the other hand, sampling (21) at {lT } for g = 1 forms oversam-
(22)
γ01
γ−11 · · · γ−1N +1
⎜ 1 ⎜ γ1 γ01 ⎜ ⎜ . .. ⎜ . ⎜ . . ⎜ ⎜ 1 1 ⎜ γN −1 γN −2 =⎜ ⎜ 1 ⎜ β0 β−1 1 ⎜ ⎜ β1 β01 ⎜ 1 ⎜ ⎜ . .. ⎜ .. . ⎝
· · · γ−1N +2 .. .
.. .
···
γ01
γ02
γ−21 · · · γ−2N +1
γ12
γ02
.. .
.. .
γN2−1 γN2−2
· · · β−1 N +1
β02
β−2 1
· · · β−1 N +2
β12
β02
.. .
.. .
.. .
βN1 −1 βN1 −2 · · ·
.. . β01
⎞
⎟ · · · γ−2N +2 ⎟ ⎟ .. .. ⎟ ⎟ . . ⎟ ⎟ ⎟ · · · γ02 ⎟ ⎟ ⎟ · · · β−2 N +1 ⎟ ⎟ · · · β−2 N +2 ⎟ ⎟ ⎟ .. .. ⎟ . . ⎟ ⎠
βN2 −1 βN2 −2 · · ·
β02
E. Balevi, A. Özgür Yılmaz / Physical Communication 25 (2017) 148–157
⎛ srs ⎞ x10 z0 ⎜ 1 ⎟ ⎜ srs ⎟ ⎜ x1 ⎟ ⎜ z 1 ⎟ ⎜ ⎟ ⎟ ⎜ . ⎟ ⎜ .. ⎟ ⎜ . ⎟ ⎜ ⎟ ⎜ . ⎟ ⎜ . ⎟ ⎜ ⎟ ⎜ ⎜ ⎜ 1 ⎟ ⎜ srs ⎟ ⎜xN −1 ⎟ ⎜zN −1 ⎟ ⎟ ⎟ ×⎜ ⎟ ⎜ 2 ⎟+⎜ os ⎟ ⎜ z ⎜ x0 ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ os ⎟ ⎜ x2 ⎟ ⎜ z ⎟ ⎜ 1 ⎟ ⎜ 1 ⎟ ⎜ ⎟ ⎜ . ⎟ ⎜ .. ⎟ ⎟ ⎜ .. ⎟ ⎜ . ⎠ ⎝ ⎝ ⎠ ⎛
(28)
zNos−1
x2N −1
srs where ysrs represent the SR samples of the received signal l and zl os and the noise respectively. Similarly, yos l and zl denote the additional samples. Then, the discrete-time model can be expressed for a total transmission power Es as
√ y=
Es 2
Hx + z
Lemma 2.1. Equally spaced channel taps, i.e., τk = kT for integer k, cannot create extra rank and normalized rank remains the same in bandlimited multiple access channels despite excess bandwidth and FTSR sampling. Proof. Consider a 2N × 2N FTSR sampled channel matrix for 2 transmitters Hsyn =
α01
⎜ ⎜ 0 ⎜ ⎜ . ⎜ . ⎜ . ⎜ ⎜ 0 ⎜ ⎜ 1 ⎜ β0 ⎜ ⎜ 1 ⎜ β−1 ⎜ ⎜ . ⎜ .. ⎝ β−1 N +1
···
αL1
0
α02
···
αL2
α01
···
αL1
0
α02
···
..
..
..
.
.. .
..
..
1 0
0
0
0
β
β
2 1
···
.
0
β
Fig. 1. The cdf of FTSR sampled channel matrix rank for 10 equally spaced taps and 0.3 excess bandwidth.
(29)
srs os os T where y = [ysrs = [x10 · · · x1N −1 0 · · · yN −1 y0 · · · yN −1 ] , x 2 2 T x0 · · · xN −1 ] , and z is the additive Gaussian noise such that z = [z0srs · · · zNsrs−1 z0os · · · zNos−1 ]T . The channel matrix H is of size 2N × 2N due to FTSR sampling, whereas it is a N × 2N matrix in case of SR sampling. The rank of H will state whether one system can achieve the benefit of excess bandwidth by FTSR sampling or not. Although there is a DoF gain in the channel due to excess bandwidth exploited by FTSR sampling, it may not necessarily mean that all FTSR sampled channels employing excess bandwidth can achieve a rank enhancement. We evaluate this for orthogonal signaling when channel taps are equally spaced with τk = kT and non-orthogonal signaling in case of unequally spaced channel taps with τk ̸ = kT .
⎛
151
⎞
.
0
α
.
.
1 1
···
β
β01 .. .
··· .. .
βN1 −2 .. .
β−2 1 .. .
β02 .. .
··· .. .
···
β−1 1
β01
β−2 N +1
···
β−2 1
1 N −1
2 0
0
⎞
⎟ αL2 ⎟ ⎟ .. ⎟ . ⎟ ⎟ ⎟ α02 ⎟ ⎟ ⎟ βN2 −1 ⎟ ⎟ ⎟ βN2 −2 ⎟ ⎟ .. ⎟ ⎟ . ⎠ β02
(30)
where βki is a linear function of α0i , α1i , · · · , αLi for k = −N + 1, . . . , 0, . . . , N − 1. The (N + 1)th row of Hsyn cannot be written in terms of the linear combination of the first N rows. That is, (N +1)th row is linearly independent from the first N rows. To illustrate, the first term of (N + 1)th row is composed of α01 , α11 , · · · , αL1 , whereas only α01 is available in the first N rows corresponding to the first term. The same discussion is valid for the (N + 2)th row such that it cannot be written by using the first (N + 1) rows. This fact can be explained by considering the first term in the (N + 2)th row, which is composed of L terms, α01 , α11 , · · · , α21 , however only 2 terms from
the 1st and (N + 1)th rows are available. On the other hand, once the first N + L rows are given, one can attain the remaining rows. This emphasizes that the rank of Hsyn is equal to N + L. Then, for a bandlimited channel, i.e., N → ∞, the normalized rank becomes lim
N +L N
N →∞
=1
(31)
showing that the equally spaced channel taps cannot benefit from excess bandwidth. □ This result can be supported by a numerical example. When the channel has 10 equally spaced complex Gaussian channel taps, the empirical cumulative distribution function (cdf) of the channel matrix rank is attained by generating 10,000 random channel realizations for 0.3 excess bandwidth. The cdf of rank, F(rank), is plotted for different block lengths and rank is normalized with respect to the block length N such that a rank of 1 expresses that there is no rank advantage of FTSR sampling. As can be seen in Fig. 1, normalized rank goes to 1 for bandlimited channels, i.e., N → ∞. In practice, channel taps are randomly distributed in between 0 and the maximum channel delay spread, i.e., they are unequally spaced or asynchronous implying that there is an inherent nonorthogonal signaling in the transmission. In this case, one cannot obtain a regular, known matrix structure for FTSR sampling such as in (30), and thus it is not so obvious to determine the rank of the channel matrix by elementary matrix operations, especially for higher values of N. However, it is obvious that asynchronism plays a key role in the channel matrix rank that brings linear independence among the matrix rows or columns that leads to a rank increase, which is formally proven in [17]. This increase can be upper bounded when N → ∞ considering the result of DoF analysis as specified in (17). That is, the normalized rank converges to (1 + r) when N → ∞. On the other hand, the normalized rank can be larger for finite N as will be proven in the following Lemma. Lemma 2.2. Normalized rank of the FTSR sampled channel matrix decreases with incremental and finite N when channel taps are unequally spaced. Proof. The 2N × 2N channel matrix H in (28) can be decomposed as
( H=
H1srs H1os
H2srs H2os
) (32)
where H1srs , H2srs , H1os , H2os can be taken as banded N × N nonHermitian Toeplitz matrices and denote the matrix due to SR and
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Table 1 Example channels of both users. 1th user channel
2nd user channel
τ01 = 0 τ11 = 3T /4 α01 = 0.8 α02 = 0.6
τ02 = 0 τ12 = 5T /4 α02 = 0.8 α02 = 0.6
FTSR sampling for the first and second user respectively. The band of these matrices increases for higher N and some eigenvalues exponentially go to zero implying rank degradation [18]. Note that H can be viewed as a block Toeplitz matrix by elementary matrix operations with changing the order of rows and columns. Since the same property is still valid for block Toeplitz matrix, normalized rank decreases for higher block lengths [18]. □ To support Lemma 2.2, a simple example can be beneficial assuming that N = 4 and pulse shape filter is a raised cosine with a roll off factor of 0.3. When each user has a channel with 2 taps whose coefficients are equal but propagation delays are different as given in Table 1, the resultant channel matrix becomes
Fig. 2. The cdf of FTSR sampled channel matrix rank for 10 unequally spaced taps and 0.3 excess bandwidth.
Hasy = 0.97 0.53 −0.09 0.03 0.70 0.53 0.17 −0.05 0 0.43 0.67 0.39 −0.03⎟ ⎜ 0.40 1.03 0.03 ⎜−0.05 0.97 0.53 −0.09 0.03 0.70 0.53 0.17 ⎟ ⎜ ⎟ ⎜−0.10 0.40 1.03 0.03 −0.11 0.43 0.67 0.39 ⎟ ⎜ ⎟ ⎜ 0.02 −0.05 0.97 0.53 −0.01 0.03 0.70 0.53 ⎟ . (33) ⎜ ⎟ ⎜ 0.04 −0.10 0.40 1.03 0.04 −0.11 0.43 0.67 ⎟ ⎝ 0 ⎠ 0.02 −0.05 0.97 0 −0.01 0.03 0.70 −0.01 0.04 −0.10 0.40 −0.02 0.04 −0.11 0.43
⎛
⎞
Relying on (33), the rank becomes 8 even if channel coefficients are the same due to asynchronism among the matrix entries. Thus, the normalized rank becomes 2, i.e., FTSR sampled channel matrix doubles the rank of SR sampled channel matrix for a small value of N, although it is upper bounded by 1.3 for N → ∞. By this is meant that, the normalized rank gradually decreases for increasing N. Empirical cdf of the channel matrix rank can be obtained by generating 10,000 random channel realizations for a root raised cosine pulse shape of 0.3 excess bandwidth and 10 unequally spaced complex Gaussian channel taps that are distributed uniformly between 0 and 10 symbol intervals. There is an asynchronism between users’ channels as well as the channel taps. Rank is normalized with respect to the block length N and plotted for different block lengths as given in Fig. 2. Accordingly, as the block length becomes larger, the rank converges to 1.3 as predicted by the expression in (13). That is, the maximum simultaneous transmissions or the total number of users in a multiple access ISI channel can be increased in direct proportion to excess bandwidth by FTSR sampling. This result is generalized for 5 users which simultaneously transmit their symbols to a receiver. Similarly, there are 10 unequally spaced complex Gaussian channel taps that are distributed uniformly between 0 and 10 symbol intervals with a root raised cosine pulse shape of 0.3 excess bandwidth and G = 2. The same result is observed with the previous case as presented in Fig. 3. It shows that the increase in the normalized rank due to FTSR sampling is independent of the number of users provided that excess bandwidth exists and channel taps are unequally spaced. Note that increasing the sampling rate beyond 2 gives the same gain, since excess bandwidth has already been exploited and there is nothing to gain. Users can experience similar multipath fading profiles in rural areas or urban areas when they are geographically close to each other. Even if the channel taps are unequally spaced, users may have nearly identical channels. This case is evaluated in Fig. 4
Fig. 3. The cdf of FTSR sampled channel matrix rank for 10 unequally spaced taps and 0.3 excess bandwidth for 5 users.
showing that normalized rank becomes 1 despite FTSR sampling. Especially, it is not hard to predict this situation by observing the channel matrix in (28). Therefore, it is important to have nonidentical channels among users as well as unequally spaced channel taps to exploit excess bandwidth advantage by FTSR sampling. To summarize, additional DoF due to excess bandwidth cannot be exploited for synchronous transmission, whereas it is possible to attain this DoF for asynchronous multiuser transmission by FTSR sampling. Since the rank of H does not depend on SNR [19], FTSR sampling can take advantage of the excess bandwidth at reasonable SNR for carefully designed systems. Moreover, our results are valid for any input distribution. Notice that the rank difference between the equally and unequally spaced taps can be explained as well based on a study that discusses the capacity of synchronous and asynchronous transmission which supports our results [17]. 2.3. Comparison with FTN signaling An alternative view of FTSR sampling is FTN signaling. Although FTN signaling is based on the idea of packing the orthogonal pulses closer than the Nyquist rate resulting in reduced symbol times without any loss in minimum Euclidean distance [20,21], it has recently been studied in a rather different concept to exploit the available capacity due to excess bandwidth [2–4]. Analogous to FTN signaling which aims to increase the number of bits for a given
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153
such that there are two transmitters and a single receiver all with single antenna and our aim is to separate these two users’ symbols at the receiver. 3.1. FTSR sampled MMSE multiuser detector
Fig. 4. The cdf of FTSR sampled channel’s matrix rank for 10 unequally spaced taps and 0.3 excess bandwidth when the user’s channels are identical.
One of the widely used multiuser detection technique to suppress interference is based on MMSE criterion that has reasonable complexity with acceptable performance [22]. The conventional MMSE multiuser detector for 2 users is comprised of 2 channel matched filters, each of which is matched to a single user [22] and has to become adaptive so that they change in each block because of varying channel conditions that results in high computational complexity. Instead of this structure, we will study a FTSR sampled multiuser detector consisting of a single pulse matched filter whose output is sampled at FTSR and processed by an MMSE filter. Notice that the noise becomes colored due to pulse matched filtering and FTSR sampling, whose auto-correlation function can be specified by using the (20) as E [z(t)z ∗ (t + τ )] = σ 2 p˜ (τ ).
bandwidth, FTSR sampling can be used to increase the number of users. Although both methods can fully exploit the available DoF in the channel due to excess bandwidth, FTSR sampling is a more practical technique than FTN signaling. FTSR sampling requires modification only at the receiver side, while FTN needs modification at the transmitter side as well. Hence, FTSR sampling can be easily adopted in existing networks. Moreover, FTN complicates the receiver design that necessitates alternative methods [5]. The major impediment in FTSR sampling occurs when channel taps are equally spaced in which transmission become orthogonal. In this case, FTSR sampling cannot exploit the advantage of excess bandwidth, however, this is a hypothetical situation not present in actual physical channels and practically unimportant.
where σ is the variance of z(t). The noise auto-correlation matrix, Rz = E [zzH ] can then be written in terms of (34) in which the (q, r) element of Rz becomes
[Rz ]q,r = σ 2 p˜ ((q − r)T /2).
(35) H
The transmitted symbols are i.i.d., that is, E [xx ] = I2N such that I2N represents the 2N × 2N identity matrix, which yields the following MMSE filter [23]
√ Wmmse =
Es 2
HH
(
Es 2
HHH + Rz
)−1
.
(36)
The MMSE matrix, M, which is equal to E [(Wmmse y − x)(Wmmse y − x)H ], can then be written as
3. Applications of FTSR sampling
√ M = I2N −
FTSR sampling has certain rank advantage in multiuser communication provided that excess bandwidth exists and channel taps are unequally spaced, which are inherent in today’s communication systems. However, rank is independent of noise statistics. The basic goal of this part is to determine the benefits of FTSR sampling by considering the effect of correlated noise stemming from pulse matched filtering and FTSR sampling. Pulse matched filtering is preferred regarding practical concerns due to the fact that it can be implemented with non-adaptive analog filter rather than channel matched filtering requiring adaptive analog filter. Within this scope, a multiuser detector is firstly investigated designed as the cascade of an FTSR sampled pulse matched filter and an MMSE filter. Although this detector has a well known structure, it is not analyzed in detail under different superimposed ISI channel conditions for practical system designs, i.e., for different pulse shaping filters, channel characteristics and sampling rates. To better explain the FTSR sampled detector, a virtual multiple input multiple output (MIMO) model is set up by FTSR sampling and analyzed for multiple access ISI channels with this equivalent MIMO structure in mind. Secondly, this FTSR sampled MMSE detector will be adapted to a canonical iterative receiver to increase the number of users in a multiple access network. What we propose is to add a new transmitter corresponding to each user that utilizes the same resources with this user, i.e., they transmit at the same time and frequency and use the same pulse shape for transmitter filter. If these two users’ symbols are separated at the receiver, this will obviously double the number of users in a given TDMA or FDMA network. Therefore, throughout the paper we focus on a scenario
(34)
2
Es 2
( + Gmmse
√ Gmmse H − Es 2
HHH + Rz
Es 2
)
HH GH mmse
GH mmse .
(37)
The mean square error (MSE) of the mth transmitted symbol depending on (37) for m = 0, 1, . . . , (2N − 1) is equal to MSEm = Mm,m
(38)
where Mm,m represents the mth diagonal term of the matrix M. In order to determine the signal to interference plus noise ratio (SINR), a well known relation between the SINR and MSE is used [8], which is SINRm =
1 MSEm
− 1.
(39)
The performance of detector is determined in terms of (39) when two single antenna transmitters send their symbols to a receiver which has also single antenna assuming that there are 10 complex Gaussian channel taps whose propagation delays are uniformly distributed in [0, 10T ], the block length is equal to 100 and G = 2. The influence of excess bandwidth due to root raised cosine pulse shaping of transmitter filter for these settings is observed in Fig. 5. As can be seen, excess bandwidth has a major effect on the SINR increase in FTSR sampling, which supports the acquired expression in (13). The impact of FTSR sampling with G = 2 for equally T spaced channel taps as opposed to unequally spaced taps in Fig. 5 is depicted for 10 complex Gaussian channel taps when the block
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Fig. 5. Excess bandwidth effect on the FTSR sampled MMSE detector for 10 unequally spaced complex Gaussian channel taps.
Fig. 7. The effect of sampling rate for 10 unequally spaced complex Gaussian channel taps with a roll off factor 0.3 on the FTSR sampled MMSE detector.
to MIMO. On the other hand, there is a gain provided that dsrs l is not srs fully correlated with dos and l . Indeed, the correlation between dl dos depends on the correlation among channel taps which can be l found as ∗
E [γmi βnj ] =
L ∑
2
E [|αki | ]˜pm p˜ n δ[i − j]
(40)
k=0
where p˜ m = p˜ (mT − τki ) and p˜ n = p˜ (nT − T /2 − τki ). In case of normalized channel taps, (40) becomes i j∗ m n
E [γ β ] =
{ < p˜ 1 , p˜ 2 > 0
if i = j if i ̸ = j
(41)
where p˜ 1 and p˜ 2 are 1 × (L + 1) vectors consisting of (L + 1) points on pulse shape p˜ (t). When L → ∞, the correlation between consecutive channel taps can be represented as Fig. 6. The effect of 10 equally spaced complex Gaussian channel taps on the FTSR sampled MMSE detector when roll off factor is 0.6.
length becomes 100 and the transmitter pulse shaping filter is a root raised cosine with a roll off factor of 0.6 in Fig. 6. It can be observed that FTSR sampling does not lead to any significant advantage for equally spaced channel taps even for a large excess bandwidth. The result is in parallel with Fig. 1, which emphasizes the disappearance of rank advantage for equally spaced ISI taps. To analyze the performance enhancement depending on the increase in sampling rate in a more rigorous manner, single antenna transmitters are regarded as a single transmitter with multiple antennas and extra samples in time domain are modeled as virtual antennas in spatial domain. This approach converts the multiple access channel into a virtual MIMO channel that has correlated channel taps and correlated noise due to FTSR sampling. Furthermore, separate coding has to be performed in virtual MIMO instead of joint coding due to independence of transmitted symbols from each other, and power is shared equally among users, because channel state information at the transmitter is not available. It is well known from MIMO systems that correlation among antennas leads to diminishing performance [19,24]. In particular, when channel taps are fully correlated, the advantage of multiple antennas at receiver disappears [19,24]. Since additional samples are considered as virtual antennas, there will be no gain after FTSR os sampling if dsrs l in (22) is fully correlated with dl in (23) analogous
∗
E [γmi βni ] =
∫
∞
p˜ (t)p˜ (t − T /2)dt .
(42)
−∞
When the offset between functions is T , they become orthogonal resulting in uncorrelated samples. On the other hand, they become fully correlated if there is no offset. The smaller the offset between functions, the higher the correlation among channel taps are. Since the offset is equal to gT /G, the biggest improvement of integer FTSR sampling occurs when G = 2 and the impact of FTSR sampling decreases by increasing G. To observe this case numerically, we assume that there are 10 complex Gaussian channel taps which have uniformly distributed propagation delays in [0, 10T ] when the block length is equal to 100 and the transmitter filter is root raised cosine with 0.3 roll off factor. As shown in Fig. 7, the highest change due to FTSR sampling is attained for G = 2. Further increases in the sampling rate has diminishing returns. In particular, there is no difference between G = 3 and G = 5. 3.2. An iterative receiver based on FTSR sampling It is appealing to share resources among users to increase the total number of users. Within this scope, two users transmit their symbols at the same time and frequency to a single receiver all with single antenna when they have the same pulse shape but different channels. The rationale behind the selection of two users stems from the fact that normalized rank doubles at most for G = 2. Although it is a difficult task to distinguish the symbols of
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Fig. 8. An iterative receiver based on FTSR sampling for multiple users.
(a). Fig. 10. Performance of the FTSR sampled iterative receiver with G = 3 for 3 transmitters and a single receiver.
(b).
Fig. 9. There are 2 transmitters for 2 taps ISI channel (a) Performance of the SR sampled iterative receiver (b) Performance of the FTSR sampled iterative receiver with G = 2.
multiple users for this scenario, the problem may be solved using both FTSR sampling and iterative receiver. We study a receiver for FTSR sampling in case of two users based on the canonical iterative receiver [25], which is given in Fig. 8. It comprises of an FTSR sampled MMSE detector and channel decoders working on the maximum a posteriori probability criterion. Note that pulse matched filter is used instead of channel matched filter making the receiver more practical. According to the output of the channel decoder soft estimation and interference cancellation is performed at the detector. Iterative receivers with FTSR sampled front-ends are mentioned in the literature [26,27], however, they did not consider MAI, and performance comparisons between FTSR and SR sampled iterative receivers are not available.
Information sequence belonging to the first user, {In1 } and the second user, {In2 } are convolutionally encoded, interleaved with distinct interleaves and then BPSK modulated. The resultant N × 1 symbol vectors are represented as a1 = [a1 a3 · · · a2N −1 ] and a2 = [a0 a2 · · · a2N −2 ] for the first and the second user that are transmitted in blocks with block length N. After passing through the ISI channels that are specified in (2), the received signal is low pass filtered with pulse matched filter p(−t) and FTSR sampled with period T ′ such that T ′ < T . Since there are two users, T ′ = T /2 is a sufficiently good choice. The system model becomes as (29), where x = [a1 a2 ] with the only difference that the transmitted symbols are now convolutionally coded. Throughout the process, extrinsic information between the detector and decoder is exchanged in an iterative manner [28], and MMSE filtering is applied to the received signal in (29) which leads to v = Wmmse y
(43)
where v is a 1 × 2N vector such that odd terms of the v are utilized to decode the first user symbols v1 = [v1 v3 · · · v2N −1 ]T
(44)
and the remaining terms of v are for the second user v2 = [v0 v2 · · · v2N −2 ]T .
(45)
After MMSE filtering, the a posteriori log likelihood ratio of the channel decoder is found, from which the extrinsic information generated by the channel decoder is attained [25]. In the last iteration, information bit log likelihood ratios are determined and the decision is made. The FTSR sampled iterative receiver’s performance is evaluated when the modulation format is BPSK and the symbols are transmitted in blocks with block length 100 for a 0.3 roll off factor. It is assumed that the channel becomes constant within one block,
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with iterations between the SR sampled iterative receiver and FTSR sampled iterative receiver is compatible with this interpretation. The performance of the FTSR sampled iterative receiver is investigated for 3 users and G = 3. The average error performance for 3-fold FTSR sampled iterative receiver is depicted in Fig. 10. Although there is a degradation with respect to the 2-user receiver performance with G = 2, it has satisfactory performance in the 3th iteration. It can be inferred that increasing the number of users in parallel with the FTSR sampling ratio leads to a performance loss. The channel may have more than 2 taps, therefore, the simulation results are repeated for 5 taps. According to that, both users have 5 complex Gaussian taps whose propagation delays are uniformly distributed in [0, 5T ] and their variances are normalized to unity. The SR and FTSR sampled iterative receiver performance is presented in Figs. 11(a) and 11(b), respectively. Similar to the previous case, FTSR sampled iterative receiver gives reasonable results, while SR sampling cannot. (a).
4. Conclusions FTSR sampling is studied in multiuser communication within the purpose of better interference mitigation to increase the number of users. DoF and rank analysis show that FTSR sampling can better exploit the excess bandwidth depending on unequally spaced channel taps and asynchronism among users for a multiple access channel. The level of improvement due to unequally spaced channel taps and excess bandwidth is quantified with an FTSR sampled MMSE detector. The results show that there is an advantage provided that excess bandwidth exists and channel taps are unequally spaced. These results are instrumental in designing an iterative receiver based on FTSR sampling. Simulation results illustrate that FTSR sampled iterative receiver better mitigates the interference among multiple users, and it makes possible to increase the number of users in a given network. (b).
Fig. 11. There are 2 transmitters for 5 taps ISI channel (a) Performance of the SR sampled iterative receiver (b) Performance of the FTSR sampled iterative receiver with G = 2.
and changes from block to block. Moreover, convolutional coding is employed at the transmitters with coding rate 1/2, constraint length 5, and generators are (23, 35) in terms of octal notation. Suppose that both users have 2 complex Gaussian taps whose propagation delays are uniformly distributed in [0, 5T ] and their variances are normalized to unity. In this case, SR sampled iterative receiver’s bit error rate (BER) is presented in Fig. 9(a), which includes the single user non-iterative receiver performance as a comparison to clarify the multiuser interference cancellation capability of the receiver. Indeed, a genie-aided interference cancellation scheme would perform the same as the single user receiver. As can be observed, the SR sampled iterative structure does not reach the single user receiver performance. Hence, we can conclude that even an iterative receiver could not perform very well with SR sampling. On the other hand, the FTSR sampled iterative receiver with G = 2 can greatly reduce the interference between users as can observed in Fig. 9(b). The error performance difference between the single user receiver and the 3th iteration of the FTSR sampled iterative receiver is very small. One critical point about the iterative receivers is the initial BER. In the context of iterative receivers, it is shown that iterative structures with enhanced initial BER yield better results [29]. The increasing performance difference
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Eren Balevi received the B.S., M.S., and Ph.D. degrees in electrical and electronics engineering from Middle East Technical University, Ankara, Turkey, in 2008, 2010 and 2016 respectively. His current research interests are in the 5G wireless systems and the Internet of Things in addition to the general areas of nanoscale communications, multiuser wireless communications and signal processing.
A. Özgür Yılmaz received the B.S. degree in electrical engineering from the University of Michigan, Ann Arbor in 1999. He received the MS and Ph.D. degrees at the same school in 2001 and 2003. He is a professor in the Department of Electrical and Electronics Engineering, Middle East Technical University, Turkey. His research interests include coding and information theory in wireless communication systems, cooperative communications, multiantenna systems, and radar signal processing.