Maximum likelihoodbased (ML) receiver structures are derived for the decoding of the downlink control channels in the new longterm evolution (LTE) standard based on multipleinput and multipleoutput (MIMO) antennas and orthogonal frequency division multiplexing (OFDM). The performance of the proposed receiver structures for the physical control format indicator channel (PCFICH) and the physical hybridARQ indicator channel (PHICH) is analyzed for various fadingchannel models and MIMO schemes including space frequency block codes (SFBC). Analytical expressions for the average probability of error are derived for each of these physical channels. The impact of channelestimation error on the orthogonality of the spreading codes applied to users in a PHICH group is investigated, and an expression for the signaltoself interference plus noise ratio is derived for Single Input Multiple Output (SIMO) systems. Finally, a matched filter bound on the probability of error for the PHICH in a multipath fading channel is derived. The analytical results are validated against computer simulations.
1. Introduction
A new standard for broadband wireless communications has emerged as an evolution to the Third Generation Partnership Project (3GPP) wideband codedivision multiple access (CDMA) Universal Mobile Telecommunication System (UMTS), termed long term evolution or LTE (3GPPrelease 8). The main difference between LTE and its predecessors is the use of scalable OFDM (orthogonal frequency division multiplexing, used on the downlink with channel bandwidth of 1.4 all the way up to 20 MHz.) together with MIMO (multiple input multiple output, configurations of up to 4 transmit antennas at the base station and 2 receive antennas at the user equipment.) antenna technology as shown in Table 1. Compared to the use of CDMA in releases 4–7, the LTE system separates users in both the time and frequency domain. OFDM is bandwidth scalable, the symbol structure is resistant to multipath delay spread without the need for equalization, and is more suitable for MIMO transmission and reception. Depending on the antenna configuration, modulation, coding and user category, LTE supports both frequencydivision duplexing (FDD) as well as timedivision duplexing (TDD) with peak data rates of 300 Mbps on the downlink and 75 Mbps on the uplink [1–3]. In this paper, the FDD frame structure is analyzed, but the results also reflect the performance of TDD frame structure.
Table 1. System numerology.
Another fundamental deviation in LTE specification relative to previous standard releases is the control channel design and structure to support the capacity enhancing features such as link adaptation, physical layer hybrid automatic repeat request (ARQ), and MIMO. Correct detection of the control channel is needed before the payload information data can be successfully decoded. Thus, the overall link and system performance are dependent on the successful decoding of these control channels.
The performance of the physical downlink control channels in the typical urban (TU3 km/h) channel was reported in [4] using computer simulations only, without rigorous mathematical analyses. The motivation behind this paper is to describe the analytical aspects of the performance of optimal receiver principles for the decoding of the LTE physical control channels. We develop and analyze the performance of ML receiver structures for the downlink physical control format indicator channel (PCFICH) as well as the physical hybrid ARQ indicator channel (PHICH) in the presence of additive white Gaussian noise, frequency selective fading channel with different transmit and receive antenna configurations, and spacefrequency block codes (SFBC). These analyses provide insight into system performance and can be used to study sensitivity to design parameters, for example, channel models and algorithm designs. Further, it would serve as a reference tool for fixedpoint computer simulation models that are developed for hardware design.
The rest of the paper is organized as follows. A brief description of the LTE control channel specification is given in Section 2. The BER analyses of the physical channels PCFICH and PHICH are given in Sections 3 and 4, respectively. Section 5 contains some concluding remarks.
Notation 1.
, , and denote element by element product, complex conjugate, and conjugate transpose, respectively. is the inner product of the vectors and . denotes the convolution operator.
2. Brief Description of the 3GPPLTE Standard
The downlink physical channels carry information from the higher layers to the user equipment. The physical downlink shared channel (PDSCH) carries the payloadinformation data, physical broadcast channel (PBCH) broadcasts cell specific information for the entire cellcoverage area, physical multicast channel (PMCH) is for multicasting and broadcasting information from multiple cells, physical downlink control channel (PDCCH) carries scheduling information, physical control format indicator channel (PCFICH) conveys the number of OFDM symbols used for PDCCH and physical hybrid ARQ indicator Channel (PHICH) transmits the HARQ acknowledgment from the base station (BS). BS in 3GPPLTE is typically referred to as eNodeB. Downlink control signaling occupies up to 4 OFDM symbols of the first slot of each subframe, followed by data transmission that starts at the next OFDM symbol as the control signaling ends. This enables support for microsleep which provides batterylife savings and reduced buffering and latency [4]. Reference signals transmitted by the BS are used by UE for channel estimation, timing and frequency synchronization, and cell identification.
The downlink OFDM FDD radio frame of 10 ms duration is equally divided into 10 subframes where each subframe consists of two 0.5 ms slots. Each slot has 7 or 6 OFDM symbols depending on the cyclic prefix (CP) duration. Two CP durations are supported: normal and extended. The entire timefrequency grid is divided into physical resource blocks (PRB), wherein each PRB contains 12 resource elements (subcarriers). PRBs are used to describe the mapping of physical channels to resource elements. Resource element groups (REG) are used for defining the control channels to resource element mapping. The size of the REG varies depending on the OFDM symbol number and antenna configuration [1]. The PCFICH is always mapped into the first OFDM symbol of the first slot of each subframe. For the normal CP duration, the PHICH is also mapped into the first OFDM symbol of the first slot of each subframe. On the other hand, for the extended CP duration, the PHICH is mapped to the first 3 OFDM symbols of the first slot of each subframe. All control channels are organized as symbolquadruplets before being mapped to a single REG. In the first OFDM symbol, two REGs per PRB are available. In the third OFDM, there are 3 REGs per PRB. In the second OFDM symbol, the number of REGs available per PRB will be 2 for single or twotransmit antennas, and 3 for fourtransmit antennas.
This paper focuses on the performance analyses of the PCFICH and PHICH between the UE and the BS in three types of channels: (1) static (additive white Gaussian noise (AWGN)), (2) frequency flatfading, and (3) ITU frequency selective channel models. The powerdelay profiles of the ITU models, used in the analyses, are given in Table 2.
Table 2. Power delay profiles for pedestrian B and ITU channel models.
3. Physical Control Format Indicator Channel
The two CFI bits are encoded using a (32,2) block code as shown in Table 3. The 32 encoded bits are QPSK modulated, layer mapped, and, finally, are resource element mapped.
3.1. PCFICH with SIMO Processing
The received signal is processed as follows: the cyclic prefix is removed, then the FFT is taken, followed by resourceelement demapping. The complexvalued output at the kth receive antenna is modeled as
where is the number of receive antennas at UE, is received subcarrier vector, is the complex QPSK symbol vector corresponding to the 32bit CFI codewords, , is complex channel frequency response, and represents the contribution of thermal noise and interference, modeled as zeromean circularly symmetric complex Gaussian with covariance . Modeling the interference as Gaussian is justified, since in a multicell multisector system such as LTE, there are typically between 3 to 6 dominant interferers. These interferers are uncorrelated due to independent largescale propagation, shortterm fading, and uncorrelated scrambling sequences. Therefore, their sum can be well approximated as a Gaussian random variable. Conditioned on , is a complex Gaussian random variable. Maximizing the loglikelihood function of given , results in the following ML decision rule:
which simplifies to
where the soft outputs are given by
where for . Expanding (4) yields
where . Without loss of generality, it is assumed that the first CFI codeword is used, that is , thus we have
as per the predefined CFI codewords in [1]. Then, the probability of error is well approximated by the union bound as
where is the pairwise error probability (PEP). In the case of a static AWGN channel with , and singlereceive antenna, let and . Thus, is Gaussian with mean and variance and is Gaussian with mean and variance . Thus, the union bound can be evaluated to be
The union bound can be tightened further, by improving the evaluation of the PEP using the joint probability of error due to and . Then, the union bound becomes
Using the bound that , the joint probability term can be written as,
For flatfading channels, the average pairwise probability of error, averaged over the channel distribution, is given by
For a Rayleigh fading channel, (11) reduces to [5]
where is evaluated to be
where , , , and is the SNR per tone per antenna and the scaling factors and .
3.2. Analysis of CFI with Repetition Coding
In this section, we compare the performance of the (32,2) block code of Table 3 used for CFI encoding with a simple rate 1/16 repetition code. The repetition code for is represented by a 32bitlength vector , by , and by [1 1 1 1]. When or , the Hamming distance between the other codewords are 32 and 16, otherwise, the Hamming distance is 16. Since the CFI assumes the value between 1 and 3, in an equiprobable manner, the probability of error, in the static AWGN channel, is given by
The expression in (14) is compared to that in (9).
3.3. PCFICH with Transmit Diversity Processing
Transmit diversity with twotransmit antennas or fourtransmit antennas, is achieved using space frequency block code (SFBC) in combination with layer mapping [1]. Assume that there are two transmit antennas at the BS transmitter and receive antennas at the UE. The received signal is processed as follows. The output at the th layer (two consecutive tones), is given by
where , is a receivedsignal vector at the th receive antenna for the th layer, is transmit signal vector corresponds to , where , at the lth layer, and denotes thermalnoise vector. The channel matrix is given by
is the complex channel frequency response between th transmit antenna and th receive antenna, at th symbol layer. The maximal ratio combiner (MRC) output is given as
The decision on the CFI is taken as in (3), and the soft output variable is given by
where .
For flatfading channel, . Then (18) becomes,
Without loss of generality, it is assumed that the first CFI codeword is used, that is , where
Substituting for in (19), it becomes
Conditioned on is Gaussian with mean and variance . The probability of error is well approximated by the union bound, as shown in (10).
In the case of singlereceive antenna, let and . is Gaussian with mean and variance and is Gaussian with mean and variance . In the static AWGN channel, conditioned on , the union bound is evaluated to be
For the MISO flatfading channel, the average probability of error, averaged over the channel distribution, is given by (13) with . For MIMO () flatfading channel, the diversity order and the average probability of error is given by
where
where .
The PCFICH performance in the presence of AWGN is shown in Figure 1. It is seen that the Union Bound approximation closely matches with the Monte Carlo simulation results. It is observed that the predefined codes for CFI yields approximately 0.5 dB SNR improvement compared to a repetition code, at the blockerror rate (BLER) of .
Figure 1. PCFICH performance in AWGN.
Currently, the fourth CFI codeword in Table 3 is reserved for future expansion. When all the four codewords are used to convey the CFI, an additional term is introduced in the error probability given as and the Union Bound becomes
Thus, it requires an additional 0.45 dB (approximately) to achieve the BLER of , compared to using the first three codewords. The PCFICH performance in the presence of Rayleigh fading channels is shown in Figure 2.
Figure 2. PCFICH performance in flatfading channel.
4. Physical Hybrid ARQ Indicator Channel
The PHICH carries physical hybrid ARQ ACK/NAK indicator (HI). Data arrives to the coding unit in form of indicators for HARQ acknowledgement. Figure 3 shows the PHICH transport channel and physical channel processing on hybrid ARQ data, is the spreading code for th user in a PHICH group, obtained from an orthogonal set of codes [1]. In LTE, spreading sequences are used in a PHICH group, where for normal CP and 2 for extended CP. The first set of spreading sequences are formed by Hadamard matrix, and the second set of spreading sequences are in quadrature to the first set.
Figure 3. PHICH transmit processing.
4.1. PHICH with SIMO Processing
The received signal is processed as follows. The cyclic prefix is removed, then the FFT is taken, followed by resource element demapping. The output that represents the ith resourceelement group and th receiver antenna is given by
where is an vector, and , are the power levels of the orthogonal codes (for the normal CP case), is the data bit value of the th user HI, and and is an complex channel frequency response vector. Without loss of generality, it is assumed that the desired HI channel to be decoded uses the first orthogonal code denoted as . The second and third terms in (26) denote the remaining spreading codes used for the other HI channels within a PHICH group (in this analytical model, we treat the general case of the normal CP. The extended CP is easily handled as shown in the final errorrate formulas.) The term denotes the thermal noise, which is modeled as circularly symmetric zeromean complex Gaussian with covariance .
The ML decoding is given by
where is the number of antennas at the UE receiver and
where
where the estimated channel frequency response is given by , is the estimation error which is uncorrelated with and zeromean complex Gaussian with covariance . By expanding (29), we get that
Note that . Thus (28) becomes
For ideal channel estimation, then due to the orthogonality property of the spreading codes, no interference is introduced to from the other HI channels within a PHICH group. However, in the presence of channelestimation error, selfinterference and cochannel interference are introduced as seen in the second and third terms, respectively, in (31). Since and , the signal to interference plus noise ratio (SINR) of the decision statistic is thus given by
In the case of a static AWGN channel with a single antenna at the UE receiver, that is, , the SINR is simply given by
where in (33) is the processing gain obtained from the spreading code of length 4, and (3,1) repetition code in the case of normal CP [1, 2]. In case of extended CP, a maximum of 4 HI channels are allowed in a PHICH group, and hence a spreading code of length 2 is used for each HI channel, which results in .
For ideal channel estimation, and the SNR of the decision statistic is thus given by
The average loss in SNR due to channelestimation error is given by
is plotted in Figure 4 as a function of the ratio between the desired power to the interfering signal power , for , − 6 dB, and =− 9 dB. Figure 4 shows that if , that is, 0 dB, with , results in a 3 dB loss in the SNR.
Figure 4. Effect of channel estimation error in PHICH.
The probability of error in the AWGN case with a singlereceive antenna is simply , is the per tone per antenna SNR as shown in (33) and (34). The probability of error averaged over the channel realization is given by
where . For a frequencyflat Rayleigh fading channel, (36) reduces to [5]
where .
The PHICH performance for static AWGN and frequencyflat Rayleigh fading channels is shown in Figure 5, for ideal channel estimation.
Figure 5. PHICH performance in SISO and SIMO systems.
4.2. PHICH with Transmit Diversity Processing
The received signal is processed as follows. The cyclic prefix is removed, then the FFT is taken, followed by resourceelement demapping. The output at the th layer (consecutive two tones) on the th receive antenna and th resource element group (REG) is given by
where , is a receivedsignal vector, is transmitsignal vector, and denotes thermalnoise vector, and each of its elements is modeled as circularly symmetric zeromean complex Gaussian with covariance . The channel matrix is given by
where is a complex channelfrequency response between th transmit antenna and th receive antenna, at th symbol layer in th REG. The transmitsignal vector is generated by layer mapping and precoding the HI data vector in ith REG. The vector is given by
and are the power levels of the 8 spreading codes. The soft output from each layer is given by
The ML decision statistic, is given by
where
and where
In a flatfading channel, . Then the decision statistic is given by,
The instantaneous SNR of is evaluated to be
In the case of a static AWGN channel with a single antenna at the UE receiver, that is, , the SNR is given by . The probability of error is given by,
For the MISO Rayleigh flatfading channel, the average probability of error, averaged over the channel distribution, is given by [5]
where and , is the SNR per antenna.
For a MIMO () flatfading channel, the average probability of error is given by
where the diversity order .
Figure 6 shows the PHICH performance in MIMO systems in the presence of AWGN and Rayleigh flatfading channels. The analytical results match well with the computer simulations.
Figure 6. PHICH performance in MIMO systems.
4.3. Matched Filter Bound for ITU Channel Models
The objective of this section is to analyze the performance of the LTE downlink control channel PHICH, in general, using matched filter bounds for various practical channel models. The base band channel impulse response can be represented as
where and are the amplitude and delay of the th path which define power delay profile (PDP), is a zeromean, unitvariance complex Gaussian random variable, , and is the system bandwidth. Let be a complex vector that contains nonzero taps which depends on the sampling frequency, and its corresponding system bandwidth is as shown in Table 1. The channel frequency response is given by,
where is taplocations vector of at which the tap coefficient is nonzero.
The decision statistic SNR or matched filter bound (MFB) of PHICH is a function of , where . Thus, the MFB is a function of independent chisquare distributed random variables with 2 degrees of freedom. For singlereceive antenna
where is independent chisquare distributed random variable with 2 degrees of freedom and is the average power of th element of . Since is constant with respect to for the given PDP, MFB can be simply written as
The characteristics function of is given by
As 's are distinct, the probability density function is given by
where . Then, the biterror probability for the matchedfilter outputs is given by [5]. The average probability of error, is given by
In case of transmit diversity using SFBC, MFB of PHICH is the function of . For a MIMO system, the channels are assumed to be independent and have the same statistical behavior [7]. For singlereceive antenna, the MFB is a function of 12 independent chisquare distributed random variables with 2 degrees of freedom, and it is written as as in (54).
It is observed that in TU channel, all the six paths are resolvable for the system bandwidths specified in Table 1, and in a PedB channel, only 4 paths are resolvable for , corresponds to the system bandwidth of 1.4 MHz, where is the number of PRBs used for downlink transmission. For , the average powers of resolvable taps of each channel coefficient are [0.1883, 0.1849, 0.1197, 0.1806, 0.1131, 0.1741] for a TU channel and [0.3298, 0.0643, 0.0673, 0.0017] for a PedB channel. The average powers of resolvable taps for , and in a PedB channel are [0.4057, 0.3665, 0.1269, 0.0663, 0.0688, 0.0017]. The performances of PHICH for a TU channel with for MISO and MIMO systems and a PedB channel with and are shown in Figures 7 and 8, respectively. It is also observed that the performance of PedB channels at has approximately 4.7 dB SNR gain with , at the BER of , and a TU channel has 3 dB SNR gain.
5. Conclusion
In this paper, the performance of maximumlikelihoodmethodbased receiver structures for PCFICH and PHICH was evaluated for different types of fading channels and antenna configurations. The effect of channelestimation error on the orthogonality of spreading codes used in a PHICH group was studied. These analytical results provide a bound on the channelestimationerror variance and thus, ultimately decide the channelestimation algorithm and parameters needed to meet such a performance bound.
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