Performance Analysis of the 3GPP-LTE Physical Control Channels

Open Access
Research Article

Abstract

Maximum likelihood-based (ML) receiver structures are derived for the decoding of the downlink control channels in the new long-term evolution (LTE) standard based on multiple-input and multiple-output (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 hybrid-ARQ indicator channel (PHICH) is analyzed for various fading-channel 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 channel-estimation error on the orthogonality of the spreading codes applied to users in a PHICH group is investigated, and an expression for the signal-to-self 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.

Keywords

Orthogonal Frequency Division Multiplex Cyclic Prefix Orthogonal Frequency Division Multiplex Symbol Spreading Code Physical Resource Block 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1. Introduction

A new standard for broadband wireless communications has emerged as an evolution to the Third Generation Partnership Project (3GPP) wideband code-division multiple access (CDMA) Universal Mobile Telecommunication System (UMTS), termed long term evolution or LTE (3GPP-release 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 frequency-division duplexing (FDD) as well as time-division duplexing (TDD) with peak data rates of 300 Mbps on the downlink and 75 Mbps on the uplink [1, 2, 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.

Channel bandwidth (MHz) Open image in new window

1.4

3.0

5.0

10.0

15.0

20.0

Number of physical resource blocks Open image in new window

6

15

25

50

75

100

FFT size Open image in new window

128

256

512

1024

1536

1024

Sampling frequency (Msps) Open image in new window

1.92

3.84

7.68

15.36

23.04

30.72

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 (TU-3 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 space-frequency 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 fixed-point 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.

Open image in new window , Open image in new window , and Open image in new window denote element by element product, complex conjugate, and conjugate transpose, respectively. Open image in new window is the inner product of the vectors Open image in new window and Open image in new window . Open image in new window denotes the convolution operator.

2. Brief Description of the 3GPP-LTE Standard

The downlink physical channels carry information from the higher layers to the user equipment. The physical downlink shared channel (PDSCH) carries the payload-information data, physical broadcast channel (PBCH) broadcasts cell specific information for the entire cell-coverage 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 3GPP-LTE 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 battery-life 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 time-frequency 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 symbol-quadruplets 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 two-transmit antennas, and 3 for four-transmit 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 flat-fading, and (3) ITU frequency selective channel models. The power-delay 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.

Ped-B channel model

TU channel model

Delay

(nsec)

Average power

(dB)

Delay

(μ sec)

Average power

(dB)

0

0

0

1.000

200

− 0.9

0.813

0.669

800

− 4.9

1.626

0.448

1200

− 8.0

2.439

0.300

2300

− 7.8

3.252

0.200

3700

− 23.9

4.056

0.134

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 resource-element demapping. The complex-valued output at the k-th receive antenna is modeled as
where Open image in new window is the number of receive antennas at UE, Open image in new window is Open image in new window received subcarrier vector, Open image in new window is the Open image in new window complex QPSK symbol vector corresponding to the 32-bit CFI codewords, Open image in new window , Open image in new window is Open image in new window complex channel frequency response, and Open image in new window represents the contribution of thermal noise and interference, modeled as zero-mean circularly symmetric complex Gaussian with covariance Open image in new window . 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 large-scale propagation, short-term fading, and uncorrelated scrambling sequences. Therefore, their sum can be well approximated as a Gaussian random variable. Conditioned on Open image in new window , Open image in new window is a complex Gaussian random variable. Maximizing the log-likelihood function of Open image in new window given Open image in new window , results in the following ML decision rule:
which simplifies to
where the soft outputs are given by
where Open image in new window . Without loss of generality, it is assumed that the first CFI codeword is used, that is Open image in new window , 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 Open image in new window is the pair-wise error probability (PEP). In the case of a static AWGN channel with Open image in new window , and single-receive antenna, let Open image in new window and Open image in new window . Thus, Open image in new window is Gaussian with mean Open image in new window and variance Open image in new window and Open image in new window is Gaussian with mean Open image in new window and variance Open image in new window . 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 Open image in new window and Open image in new window . Then, the union bound becomes
Using the bound that Open image in new window , the joint probability term can be written as,
For flat-fading channels, the average pair-wise probability of error, averaged over the channel Open image in new window distribution, is given by
For a Rayleigh fading channel, (11) reduces to [5]

where Open image in new window , Open image in new window , Open image in new window , and Open image in new window is the SNR per tone per antenna and the scaling factors Open image in new window and Open image in new window .

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 Open image in new window is represented by a 32-bit-length vector Open image in new window , Open image in new window by Open image in new window , and Open image in new window by [1 1 Open image in new window 1 1]. When Open image in new window or Open image in new window , 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 two-transmit antennas or four-transmit 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 Open image in new window receive antennas at the UE. The received signal is processed as follows. The output at the Open image in new window th layer (two consecutive tones), is given by
where Open image in new window , Open image in new window is a Open image in new window received-signal vector at the Open image in new window th receive antenna for the Open image in new window th layer, Open image in new window is Open image in new window transmit signal vector corresponds to Open image in new window , where Open image in new window , at the l th layer, and Open image in new window denotes Open image in new window thermal-noise vector. The channel matrix Open image in new window is given by
Open image in new window is the complex channel frequency response between Open image in new window th transmit antenna and Open image in new window th receive antenna, at Open image in new window 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 Open image in new window is given by

where Open image in new window .

For flat-fading channel, Open image in new window . Then (18) becomes,
Without loss of generality, it is assumed that the first CFI codeword is used, that is Open image in new window , where
Substituting for Open image in new window in (19), it becomes

Conditioned on Open image in new window is Gaussian with mean Open image in new window and variance Open image in new window . The probability of error is well approximated by the union bound, as shown in (10).

In the case of single-receive antenna, let Open image in new window and Open image in new window . Open image in new window is Gaussian with mean Open image in new window and variance Open image in new window and Open image in new window is Gaussian with mean Open image in new window and variance Open image in new window . In the static AWGN channel, conditioned on Open image in new window , the union bound is evaluated to be
For the MISO flat-fading channel, the average probability of error, averaged over the channel Open image in new window distribution, is given by (13) with Open image in new window . For MIMO ( Open image in new window ) flat-fading channel, the diversity order Open image in new window and the average probability of error is given by

where Open image in new window .

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 block-error rate (BLER) of Open image in new window .
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 Open image in new window and the Union Bound becomes
Thus, it requires an additional 0.45 dB (approximately) to achieve the BLER of Open image in new window , 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 flat-fading 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, Open image in new window is the spreading code for Open image in new window th user in a PHICH group, obtained from an orthogonal set of codes [1]. In LTE, Open image in new window spreading sequences are used in a PHICH group, where Open image in new window for normal CP and 2 for extended CP. The first set of Open image in new window spreading sequences are formed by Open image in new window Hadamard matrix, and the second set of Open image in new window 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 i th resource-element group and Open image in new window th receiver antenna is given by

where Open image in new window is an Open image in new window vector, Open image in new window and Open image in new window , Open image in new window are the power levels of the Open image in new window orthogonal codes (for the normal CP case), Open image in new window is the data bit value of the Open image in new window th user HI, and Open image in new window and Open image in new window is an Open image in new window 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 Open image in new window . The second and third terms in (26) denote the remaining Open image in new window 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 error-rate formulas.) The term Open image in new window denotes the thermal noise, which is modeled as circularly symmetric zero-mean complex Gaussian with covariance Open image in new window .

The ML decoding is given by
where Open image in new window is the number of antennas at the UE receiver and
where the estimated channel frequency response Open image in new window is given by Open image in new window , Open image in new window is the estimation error which is uncorrelated with Open image in new window and zero-mean complex Gaussian with covariance Open image in new window . By expanding (29), we get that
Note that Open image in new window . Thus (28) becomes
For ideal channel estimation, then due to the orthogonality property of the spreading codes, no interference is introduced to Open image in new window from the other HI channels within a PHICH group. However, in the presence of channel-estimation error, self-interference and cochannel interference are introduced as seen in the second and third terms, respectively, in (31). Since Open image in new window and Open image in new window , the signal to interference plus noise ratio (SINR) of the decision statistic Open image in new window is thus given by
In the case of a static AWGN channel with a single antenna at the UE receiver, that is, Open image in new window , the SINR is simply given by

where Open image in new window 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 Open image in new window .

For ideal channel estimation, Open image in new window and the SNR of the decision statistic Open image in new window is thus given by
The average loss in SNR due to channel-estimation error is given by
Open image in new window is plotted in Figure 4 as a function of the ratio between the desired power to the interfering signal power Open image in new window , for Open image in new window , Open image in new window  − 6 dB, and Open image in new window =− 9 dB. Figure 4 shows that if Open image in new window , that is, 0 dB, with Open image in new window , 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 single-receive antenna is simply Open image in new window , Open image in new window 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 Open image in new window . For a frequency-flat Rayleigh fading channel, (36) reduces to [5]

where Open image in new window .

The PHICH performance for static AWGN and frequency-flat 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 resource-element demapping. The output at the Open image in new window th layer (consecutive two tones) on the Open image in new window th receive antenna and Open image in new window th resource element group (REG) is given by
where Open image in new window , Open image in new window is a Open image in new window received-signal vector, Open image in new window is Open image in new window transmit-signal vector, and Open image in new window denotes Open image in new window thermal-noise vector, and each of its elements is modeled as circularly symmetric zero-mean complex Gaussian with covariance Open image in new window . The channel matrix Open image in new window is given by
where Open image in new window is a complex channel-frequency response between Open image in new window th transmit antenna and Open image in new window th receive antenna, at Open image in new window th symbol layer in Open image in new window th REG. The transmit-signal vector Open image in new window is generated by layer mapping and precoding the HI data vector Open image in new window in i th REG. The Open image in new window vector Open image in new window is given by
Open image in new window and Open image in new window Open image in new window 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
In a flat-fading channel, Open image in new window . Then the decision statistic Open image in new window is given by,
The instantaneous SNR of Open image in new window is evaluated to be
In the case of a static AWGN channel with a single antenna at the UE receiver, that is, Open image in new window , the SNR is given by Open image in new window . The probability of error is given by,
For the MISO Rayleigh flat-fading channel, the average probability of error, averaged over the channel Open image in new window distribution, is given by [5]

where Open image in new window and Open image in new window , is the SNR per antenna.

For a MIMO ( Open image in new window ) flat-fading channel, the average probability of error is given by

where the diversity order Open image in new window .

Figure 6 shows the PHICH performance in MIMO systems in the presence of AWGN and Rayleigh flat-fading 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 Open image in new window and Open image in new window are the amplitude and delay of the Open image in new window th path which define power delay profile (PDP), Open image in new window is a zero-mean, unit-variance complex Gaussian random variable, Open image in new window , and Open image in new window is the system bandwidth. Let Open image in new window be a Open image in new window complex vector that contains Open image in new window 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 Open image in new window is Open image in new window tap-locations vector of Open image in new window at which the tap coefficient is nonzero.

The decision statistic SNR or matched filter bound (MFB) of PHICH is a function of Open image in new window , where Open image in new window . Thus, the MFB is a function of Open image in new window independent chi-square distributed random variables with 2 degrees of freedom. For single-receive antenna
where Open image in new window is independent chi-square distributed random variable with 2 degrees of freedom and Open image in new window is the average power of Open image in new window th element of Open image in new window . Since Open image in new window is constant with respect to Open image in new window for the given PDP, MFB can be simply written as
The characteristics function of Open image in new window is given by
As Open image in new window 's are distinct, the probability density function is given by
where Open image in new window . Then, the bit-error probability for the matched-filter outputs is given by Open image in new window [5]. The average probability of error, Open image in new window is given by

In case of transmit diversity using SFBC, MFB of PHICH is the function of Open image in new window . For a MIMO system, the channels are assumed to be independent and have the same statistical behavior [7]. For single-receive antenna, the MFB is a function of 12 independent chi-square distributed random variables with 2 degrees of freedom, and it is written as Open image in new window 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 Ped-B channel, only 4 paths are resolvable for Open image in new window , corresponds to the system bandwidth of 1.4 MHz, where Open image in new window is the number of PRBs used for downlink transmission. For Open image in new window , 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 Ped-B channel. The average powers of resolvable taps for Open image in new window , and in a Ped-B channel are [0.4057, 0.3665, 0.1269, 0.0663, 0.0688, 0.0017]. The performances of PHICH for a TU channel with Open image in new window for MISO and MIMO systems and a Ped-B channel with Open image in new window and Open image in new window are shown in Figures 7 and 8, respectively. It is also observed that the performance of Ped-B channels at Open image in new window has approximately 4.7 dB SNR gain with Open image in new window , at the BER of Open image in new window , and a TU channel has 3 dB SNR gain.
Figure 7

PHICH performance in TU channel.

Figure 8

PHICH performance in Ped-B channel.

5. Conclusion

In this paper, the performance of maximum-likelihood-method-based receiver structures for PCFICH and PHICH was evaluated for different types of fading channels and antenna configurations. The effect of channel-estimation error on the orthogonality of spreading codes used in a PHICH group was studied. These analytical results provide a bound on the channel-estimation-error variance and thus, ultimately decide the channel-estimation algorithm and parameters needed to meet such a performance bound.

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Copyright information

© S. J. Thiruvengadam and L. M. A. Jalloul. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Smart Antenna Research Group, Department of Electrical EngineeringStanford UniversityUSA
  2. 2.TIFAC CORE in Wireless TechnologiesThiagarajar College of EngineeringMaduraiIndia
  3. 3.Beceem Communications Inc.Santa ClaraUSA

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