Open Access

Performance Analysis of the 3GPP-LTE Physical Control Channels

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) https://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 https://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 https://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) https://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.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq5_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq6_HTML.gif , and https://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. https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq9_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq10_HTML.gif . https://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

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

1

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

2

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

3

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

4 (Reserved)

https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ1_HTML.gif
(1)
where https://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, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq18_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq19_HTML.gif received subcarrier vector, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq20_HTML.gif is the https://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, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq22_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq23_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq24_HTML.gif complex channel frequency response, and https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq27_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq29_HTML.gif given https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ2_HTML.gif
(2)
which simplifies to
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ4_HTML.gif
(4)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq31_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq32_HTML.gif . Expanding (4) yields
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ5_HTML.gif
(5)
where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq34_HTML.gif , thus we have
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ7_HTML.gif
(7)
where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq36_HTML.gif , and single-receive antenna, let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq37_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq38_HTML.gif . Thus, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq39_HTML.gif is Gaussian with mean https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq40_HTML.gif and variance https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq41_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq42_HTML.gif is Gaussian with mean https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq43_HTML.gif and variance https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq45_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq46_HTML.gif . Then, the union bound becomes
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ9_HTML.gif
(9)
Using the bound that https://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,
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq48_HTML.gif distribution, is given by
https://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]
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ12_HTML.gif
(12)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq49_HTML.gif is evaluated to be
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ13_HTML.gif
(13)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq50_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq51_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq52_HTML.gif , and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq54_HTML.gif and https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq57_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq58_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq59_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq60_HTML.gif by [1 1 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq61_HTML.gif 1 1]. When https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq62_HTML.gif or https://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
https://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 https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ15_HTML.gif
(15)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq66_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq67_HTML.gif is a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq68_HTML.gif received-signal vector at the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq69_HTML.gif th receive antenna for the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq70_HTML.gif th layer, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq71_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq72_HTML.gif transmit signal vector corresponds to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq73_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq74_HTML.gif , at the l th layer, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq75_HTML.gif denotes https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq76_HTML.gif thermal-noise vector. The channel matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq77_HTML.gif is given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ16_HTML.gif
(16)
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq79_HTML.gif th transmit antenna and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq80_HTML.gif th receive antenna, at https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq82_HTML.gif is given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ18_HTML.gif
(18)

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

For flat-fading channel, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq84_HTML.gif . Then (18) becomes,
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq85_HTML.gif , where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ20_HTML.gif
(20)
Substituting for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq86_HTML.gif in (19), it becomes
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ21_HTML.gif
(21)

Conditioned on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq87_HTML.gif is Gaussian with mean https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq88_HTML.gif and variance https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq90_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq91_HTML.gif . https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq92_HTML.gif is Gaussian with mean https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq93_HTML.gif and variance https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq94_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq95_HTML.gif is Gaussian with mean https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq96_HTML.gif and variance https://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 https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq99_HTML.gif distribution, is given by (13) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq100_HTML.gif . For MIMO ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq101_HTML.gif ) flat-fading channel, the diversity order https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ23_HTML.gif
(23)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ24_HTML.gif
(24)

where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq104_HTML.gif .
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq105_HTML.gif and the Union Bound becomes
https://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 https://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.
https://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, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq107_HTML.gif is the spreading code for https://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, https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq111_HTML.gif spreading sequences are formed by https://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 https://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.
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq114_HTML.gif th receiver antenna is given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ26_HTML.gif
(26)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq115_HTML.gif is an https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq116_HTML.gif vector, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq117_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq118_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq119_HTML.gif are the power levels of the https://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), https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq122_HTML.gif th user HI, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq123_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq124_HTML.gif is an https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq129_HTML.gif .

The ML decoding is given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ27_HTML.gif
(27)
where https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ28_HTML.gif
(28)
where
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq131_HTML.gif is given by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq132_HTML.gif , https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq135_HTML.gif . By expanding (29), we get that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ30_HTML.gif
(30)
Note that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq136_HTML.gif . Thus (28) becomes
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq138_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq140_HTML.gif is thus given by
https://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, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq141_HTML.gif , the SINR is simply given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ33_HTML.gif
(33)

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

For ideal channel estimation, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq145_HTML.gif is thus given by
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ35_HTML.gif
(35)
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq147_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq148_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq149_HTML.gif  − 6 dB, and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq151_HTML.gif , that is, 0 dB, with https://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.
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq153_HTML.gif , https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ36_HTML.gif
(36)
where https://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]
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ37_HTML.gif
(37)

where https://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.
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq158_HTML.gif th receive antenna and https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ38_HTML.gif
(38)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq160_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq161_HTML.gif is a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq162_HTML.gif received-signal vector, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq163_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq164_HTML.gif transmit-signal vector, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq165_HTML.gif denotes https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq167_HTML.gif . The channel matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq168_HTML.gif is given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ39_HTML.gif
(39)
where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq170_HTML.gif th transmit antenna and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq171_HTML.gif th receive antenna, at https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq172_HTML.gif th symbol layer in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq173_HTML.gif th REG. The transmit-signal vector https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq175_HTML.gif in i th REG. The https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq176_HTML.gif vector https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq177_HTML.gif is given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ40_HTML.gif
(40)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq178_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq179_HTML.gif https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ42_HTML.gif
(42)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ43_HTML.gif
(43)
and where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ44_HTML.gif
(44)
In a flat-fading channel, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq181_HTML.gif . Then the decision statistic https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq182_HTML.gif is given by,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ45_HTML.gif
(45)
The instantaneous SNR of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq183_HTML.gif is evaluated to be
https://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, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq184_HTML.gif , the SNR is given by https://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,
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq186_HTML.gif distribution, is given by [5]
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ48_HTML.gif
(48)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq187_HTML.gif and https://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 ( https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ49_HTML.gif
(49)

where the diversity order https://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.
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ50_HTML.gif
(50)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq191_HTML.gif and https://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 https://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), https://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, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq195_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq196_HTML.gif is the system bandwidth. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq197_HTML.gif be a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq198_HTML.gif complex vector that contains https://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,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ51_HTML.gif
(51)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq200_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq201_HTML.gif tap-locations vector of https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq203_HTML.gif , where https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ52_HTML.gif
(52)
where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq207_HTML.gif is the average power of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq208_HTML.gif th element of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq209_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq210_HTML.gif is constant with respect to https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ53_HTML.gif
(53)
The characteristics function of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq212_HTML.gif is given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ54_HTML.gif
(54)
As https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_Equ55_HTML.gif
(55)
where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq215_HTML.gif [5]. The average probability of error, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq216_HTML.gif is given by
https://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 https://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 https://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 https://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 https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq224_HTML.gif and https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F914934/MediaObjects/13638_2010_Article_2058_IEq227_HTML.gif , at the BER of https://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.
https://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.

https://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.MATH
  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 ArticleMATH
  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.