Performance Analysis of the 3GPP-LTE Physical Control Channels

  • S. J. Thiruvengadam1, 2 and

    Affiliated with

    • Louay M. A. Jalloul3Email author

      Affiliated with

      EURASIP Journal on Wireless Communications and Networking20102010:914934

      DOI: 10.1155/2010/914934

      Received: 8 May 2010

      Accepted: 11 November 2010

      Published: 28 November 2010

      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.

      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 [13]. 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) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq1_HTML.gif

      1.4

      3.0

      5.0

      10.0

      15.0

      20.0

      Number of physical resource blocks http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq2_HTML.gif

      6

      15

      25

      50

      75

      100

      FFT size http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq3_HTML.gif

      128

      256

      512

      1024

      1536

      1024

      Sampling frequency (Msps) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq4_HTML.gif

      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.

      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq5_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq6_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq7_HTML.gif denote element by element product, complex conjugate, and conjugate transpose, respectively. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq8_HTML.gif is the inner product of the vectors http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq9_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq10_HTML.gif . http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq11_HTML.gif 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.
      Table 3

      CFI (32,2) Block code [2].

      CFI

      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq12_HTML.gif

      1

      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq13_HTML.gif

      2

      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq14_HTML.gif

      3

      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq15_HTML.gif

      4 (Reserved)

      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq16_HTML.gif

      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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ1_HTML.gif
      (1)
      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq17_HTML.gif is the number of receive antennas at UE, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq18_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq19_HTML.gif received subcarrier vector, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq20_HTML.gif is the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq21_HTML.gif complex QPSK symbol vector corresponding to the 32-bit CFI codewords, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq22_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq23_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq24_HTML.gif complex channel frequency response, and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq25_HTML.gif represents the contribution of thermal noise and interference, modeled as zero-mean circularly symmetric complex Gaussian with covariance http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq26_HTML.gif . 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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq27_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq28_HTML.gif is a complex Gaussian random variable. Maximizing the log-likelihood function of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq29_HTML.gif given http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq30_HTML.gif , results in the following ML decision rule:
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ2_HTML.gif
      (2)
      which simplifies to
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ3_HTML.gif
      (3)
      where the soft outputs are given by
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ4_HTML.gif
      (4)
      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq31_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq32_HTML.gif . Expanding (4) yields
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ5_HTML.gif
      (5)
      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq33_HTML.gif . Without loss of generality, it is assumed that the first CFI codeword is used, that is http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq34_HTML.gif , thus we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ6_HTML.gif
      (6)
      as per the predefined CFI codewords in [1]. Then, the probability of error is well approximated by the union bound as
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ7_HTML.gif
      (7)
      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq35_HTML.gif is the pair-wise error probability (PEP). In the case of a static AWGN channel with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq36_HTML.gif , and single-receive antenna, let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq37_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq38_HTML.gif . Thus, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq39_HTML.gif is Gaussian with mean http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq40_HTML.gif and variance http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq41_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq42_HTML.gif is Gaussian with mean http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq43_HTML.gif and variance http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq44_HTML.gif . Thus, the union bound can be evaluated to be
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ8_HTML.gif
      (8)
      The union bound can be tightened further, by improving the evaluation of the PEP using the joint probability of error due to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq45_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq46_HTML.gif . Then, the union bound becomes
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ9_HTML.gif
      (9)
      Using the bound that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq47_HTML.gif , the joint probability term can be written as,
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ10_HTML.gif
      (10)
      For flat-fading channels, the average pair-wise probability of error, averaged over the channel http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq48_HTML.gif distribution, is given by
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ11_HTML.gif
      (11)
      For a Rayleigh fading channel, (11) reduces to [5]
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ12_HTML.gif
      (12)
      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq49_HTML.gif is evaluated to be
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ13_HTML.gif
      (13)

      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq50_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq51_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq52_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq53_HTML.gif is the SNR per tone per antenna and the scaling factors http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq54_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq55_HTML.gif .

      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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq56_HTML.gif is represented by a 32-bit-length vector http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq57_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq58_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq59_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq60_HTML.gif by [1 1 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq61_HTML.gif 1 1]. When http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq62_HTML.gif or http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq63_HTML.gif , 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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ14_HTML.gif
      (14)

      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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq64_HTML.gif receive antennas at the UE. The received signal is processed as follows. The output at the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq65_HTML.gif th layer (two consecutive tones), is given by
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ15_HTML.gif
      (15)
      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq66_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq67_HTML.gif is a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq68_HTML.gif received-signal vector at the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq69_HTML.gif th receive antenna for the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq70_HTML.gif th layer, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq71_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq72_HTML.gif transmit signal vector corresponds to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq73_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq74_HTML.gif , at the l th layer, and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq75_HTML.gif denotes http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq76_HTML.gif thermal-noise vector. The channel matrix http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq77_HTML.gif is given by
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ16_HTML.gif
      (16)
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq78_HTML.gif is the complex channel frequency response between http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq79_HTML.gif th transmit antenna and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq80_HTML.gif th receive antenna, at http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq81_HTML.gif th symbol layer. The maximal ratio combiner (MRC) output is given as
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ17_HTML.gif
      (17)
      The decision on the CFI is taken as in (3), and the soft output variable http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq82_HTML.gif is given by
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ18_HTML.gif
      (18)

      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq83_HTML.gif .

      For flat-fading channel, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq84_HTML.gif . Then (18) becomes,
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ19_HTML.gif
      (19)
      Without loss of generality, it is assumed that the first CFI codeword is used, that is http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq85_HTML.gif , where
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ20_HTML.gif
      (20)
      Substituting for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq86_HTML.gif in (19), it becomes
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ21_HTML.gif
      (21)

      Conditioned on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq87_HTML.gif is Gaussian with mean http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq88_HTML.gif and variance http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq89_HTML.gif . The probability of error is well approximated by the union bound, as shown in (10).

      In the case of single-receive antenna, let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq90_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq91_HTML.gif . http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq92_HTML.gif is Gaussian with mean http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq93_HTML.gif and variance http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq94_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq95_HTML.gif is Gaussian with mean http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq96_HTML.gif and variance http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq97_HTML.gif . In the static AWGN channel, conditioned on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq98_HTML.gif , the union bound is evaluated to be
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ22_HTML.gif
      (22)
      For the MISO flat-fading channel, the average probability of error, averaged over the channel http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq99_HTML.gif distribution, is given by (13) with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq100_HTML.gif . For MIMO ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq101_HTML.gif ) flat-fading channel, the diversity order http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq102_HTML.gif and the average probability of error is given by
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ23_HTML.gif
      (23)
      where
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ24_HTML.gif
      (24)

      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq103_HTML.gif .

      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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq104_HTML.gif .
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Fig1_HTML.jpg
      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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq105_HTML.gif and the Union Bound becomes
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ25_HTML.gif
      (25)
      Thus, it requires an additional 0.45 dB (approximately) to achieve the BLER of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq106_HTML.gif , compared to using the first three codewords. The PCFICH performance in the presence of Rayleigh fading channels is shown in Figure 2.
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Fig2_HTML.jpg
      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, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq107_HTML.gif is the spreading code for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq108_HTML.gif th user in a PHICH group, obtained from an orthogonal set of codes [1]. In LTE, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq109_HTML.gif spreading sequences are used in a PHICH group, where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq110_HTML.gif for normal CP and 2 for extended CP. The first set of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq111_HTML.gif spreading sequences are formed by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq112_HTML.gif Hadamard matrix, and the second set of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq113_HTML.gif spreading sequences are in quadrature to the first set.
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Fig3_HTML.jpg
      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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq114_HTML.gif th receiver antenna is given by
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ26_HTML.gif
      (26)

      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq115_HTML.gif is an http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq116_HTML.gif vector, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq117_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq118_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq119_HTML.gif are the power levels of the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq120_HTML.gif orthogonal codes (for the normal CP case), http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq121_HTML.gif is the data bit value of the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq122_HTML.gif th user HI, and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq123_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq124_HTML.gif is an http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq125_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq126_HTML.gif . The second and third terms in (26) denote the remaining http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq127_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq128_HTML.gif denotes the thermal noise, which is modeled as circularly symmetric zero-mean complex Gaussian with covariance http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq129_HTML.gif .

      The ML decoding is given by
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ27_HTML.gif
      (27)
      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq130_HTML.gif is the number of antennas at the UE receiver and
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ28_HTML.gif
      (28)
      where
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ29_HTML.gif
      (29)
      where the estimated channel frequency response http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq131_HTML.gif is given by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq132_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq133_HTML.gif is the estimation error which is uncorrelated with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq134_HTML.gif and zero-mean complex Gaussian with covariance http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq135_HTML.gif . By expanding (29), we get that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ30_HTML.gif
      (30)
      Note that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq136_HTML.gif . Thus (28) becomes
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ31_HTML.gif
      (31)
      For ideal channel estimation, then due to the orthogonality property of the spreading codes, no interference is introduced to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq137_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq138_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq139_HTML.gif , the signal to interference plus noise ratio (SINR) of the decision statistic http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq140_HTML.gif is thus given by
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ32_HTML.gif
      (32)
      In the case of a static AWGN channel with a single antenna at the UE receiver, that is, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq141_HTML.gif , the SINR is simply given by
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ33_HTML.gif
      (33)

      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq142_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq143_HTML.gif .

      For ideal channel estimation, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq144_HTML.gif and the SNR of the decision statistic http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq145_HTML.gif is thus given by
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ34_HTML.gif
      (34)
      The average loss in SNR due to channel-estimation error is given by
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ35_HTML.gif
      (35)
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq146_HTML.gif is plotted in Figure 4 as a function of the ratio between the desired power to the interfering signal power http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq147_HTML.gif , for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq148_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq149_HTML.gif  − 6 dB, and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq150_HTML.gif =− 9 dB. Figure 4 shows that if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq151_HTML.gif , that is, 0 dB, with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq152_HTML.gif , results in a 3 dB loss in the SNR.
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Fig4_HTML.jpg
      Figure 4

      Effect of channel estimation error in PHICH.

      The probability of error in the AWGN case with a single-receive antenna is simply http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq153_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq154_HTML.gif 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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ36_HTML.gif
      (36)
      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq155_HTML.gif . For a frequency-flat Rayleigh fading channel, (36) reduces to [5]
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ37_HTML.gif
      (37)

      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq156_HTML.gif .

      The PHICH performance for static AWGN and frequency-flat Rayleigh fading channels is shown in Figure 5, for ideal channel estimation.
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Fig5_HTML.jpg
      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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq157_HTML.gif th layer (consecutive two tones) on the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq158_HTML.gif th receive antenna and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq159_HTML.gif th resource element group (REG) is given by
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ38_HTML.gif
      (38)
      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq160_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq161_HTML.gif is a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq162_HTML.gif received-signal vector, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq163_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq164_HTML.gif transmit-signal vector, and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq165_HTML.gif denotes http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq166_HTML.gif thermal-noise vector, and each of its elements is modeled as circularly symmetric zero-mean complex Gaussian with covariance http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq167_HTML.gif . The channel matrix http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq168_HTML.gif is given by
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ39_HTML.gif
      (39)
      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq169_HTML.gif is a complex channel-frequency response between http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq170_HTML.gif th transmit antenna and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq171_HTML.gif th receive antenna, at http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq172_HTML.gif th symbol layer in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq173_HTML.gif th REG. The transmit-signal vector http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq174_HTML.gif is generated by layer mapping and precoding the HI data vector http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq175_HTML.gif in i th REG. The http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq176_HTML.gif vector http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq177_HTML.gif is given by
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ40_HTML.gif
      (40)
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq178_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq179_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq180_HTML.gif are the power levels of the 8 spreading codes. The soft output from each layer is given by
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ41_HTML.gif
      (41)
      The ML decision statistic, is given by
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ42_HTML.gif
      (42)
      where
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ43_HTML.gif
      (43)
      and where
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ44_HTML.gif
      (44)
      In a flat-fading channel, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq181_HTML.gif . Then the decision statistic http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq182_HTML.gif is given by,
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ45_HTML.gif
      (45)
      The instantaneous SNR of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq183_HTML.gif is evaluated to be
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ46_HTML.gif
      (46)
      In the case of a static AWGN channel with a single antenna at the UE receiver, that is, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq184_HTML.gif , the SNR is given by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq185_HTML.gif . The probability of error is given by,
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ47_HTML.gif
      (47)
      For the MISO Rayleigh flat-fading channel, the average probability of error, averaged over the channel http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq186_HTML.gif distribution, is given by [5]
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ48_HTML.gif
      (48)

      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq187_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq188_HTML.gif , is the SNR per antenna.

      For a MIMO ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq189_HTML.gif ) flat-fading channel, the average probability of error is given by
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ49_HTML.gif
      (49)

      where the diversity order http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq190_HTML.gif .

      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.
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Fig6_HTML.jpg
      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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ50_HTML.gif
      (50)
      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq191_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq192_HTML.gif are the amplitude and delay of the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq193_HTML.gif th path which define power delay profile (PDP), http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq194_HTML.gif is a zero-mean, unit-variance complex Gaussian random variable, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq195_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq196_HTML.gif is the system bandwidth. Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq197_HTML.gif be a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq198_HTML.gif complex vector that contains http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq199_HTML.gif 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,
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ51_HTML.gif
      (51)

      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq200_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq201_HTML.gif tap-locations vector of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq202_HTML.gif at which the tap coefficient is nonzero.

      The decision statistic SNR or matched filter bound (MFB) of PHICH is a function of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq203_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq204_HTML.gif . Thus, the MFB is a function of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq205_HTML.gif independent chi-square distributed random variables with 2 degrees of freedom. For single-receive antenna
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ52_HTML.gif
      (52)
      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq206_HTML.gif is independent chi-square distributed random variable with 2 degrees of freedom and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq207_HTML.gif is the average power of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq208_HTML.gif th element of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq209_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq210_HTML.gif is constant with respect to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq211_HTML.gif for the given PDP, MFB can be simply written as
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ53_HTML.gif
      (53)
      The characteristics function of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq212_HTML.gif is given by
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ54_HTML.gif
      (54)
      As http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq213_HTML.gif 's are distinct, the probability density function is given by
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ55_HTML.gif
      (55)
      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq214_HTML.gif . Then, the bit-error probability for the matched-filter outputs is given by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq215_HTML.gif [5]. The average probability of error, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq216_HTML.gif is given by
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ56_HTML.gif
      (56)

      In case of transmit diversity using SFBC, MFB of PHICH is the function of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq217_HTML.gif . 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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq218_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq219_HTML.gif , corresponds to the system bandwidth of 1.4 MHz, where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq220_HTML.gif is the number of PRBs used for downlink transmission. For http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq221_HTML.gif , 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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq222_HTML.gif , 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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq223_HTML.gif for MISO and MIMO systems and a Ped-B channel with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq224_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq225_HTML.gif are shown in Figures 7 and 8, respectively. It is also observed that the performance of Ped-B channels at http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq226_HTML.gif has approximately 4.7 dB SNR gain with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq227_HTML.gif , at the BER of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq228_HTML.gif , and a TU channel has 3 dB SNR gain.
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Fig7_HTML.jpg
      Figure 7

      PHICH performance in TU channel.

      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Fig8_HTML.jpg
      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.

      Authors’ Affiliations

      (1)
      Smart Antenna Research Group, Department of Electrical Engineering, Stanford University
      (2)
      TIFAC CORE in Wireless Technologies, Thiagarajar College of Engineering
      (3)
      Beceem Communications Inc.

      References

      1. 3GPP TS 36.211 : Evolved Universal Terrestrial Radio Access (E-UTRA); Physical Channels and Modulation (Release 8).
      2. 3GPP TS 36.212 : Evolved Universal Terrestrial Radio Access (E-UTRA); Multiplexing and Channel Coding (Release 8).
      3. 3GPP TS 36.306 : Evolved Universal Terrestrial Radio Access (E-UTRA); User Equipment (UE) radio access capabilities (Release 8).
      4. Love R, Kuchibhotla R, Ghosh A, Ratasuk R, Xiao W, Classon B, Blankenship Y: Downlink control channel design for 3GPP LTE. Proceedings of IEEE Wireless Communications and Networking Conference (WCNC '08), April 2008, Las Vegas, Nev, USA 813-818.
      5. Proakis J: Digital Communications. 3rd edition. McGraw-Hill, Boston, Mass, USA; 1995.
      6. Ling F: Matched filter-bound for time-discrete multipath Rayleigh fading channels. IEEE Transactions on Communications 1995, 43(2):710-713. 10.1109/26.380095View Article
      7. Naguib AF: On the matched filter bound of transmit diversity techniques. Proceedings of the International Conference on Communications (ICC '01), June 2000, Helsinki, Finland 596-603.

      Copyright

      © 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.