# Performance Analysis of the 3GPP-LTE Physical Control Channels

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## Abstract

Maximum likelihood-based (ML) receiver structures are derived for the decoding of the downlink control channels in the new long-term evolution (LTE) standard based on multiple-input and multiple-output (MIMO) antennas and orthogonal frequency division multiplexing (OFDM). The performance of the proposed receiver structures for the physical control format indicator channel (PCFICH) and the physical hybrid-ARQ indicator channel (PHICH) is analyzed for various fading-channel models and MIMO schemes including space frequency block codes (SFBC). Analytical expressions for the average probability of error are derived for each of these physical channels. The impact of channel-estimation error on the orthogonality of the spreading codes applied to users in a PHICH group is investigated, and an expression for the signal-to-self interference plus noise ratio is derived for Single Input Multiple Output (SIMO) systems. Finally, a matched filter bound on the probability of error for the PHICH in a multipath fading channel is derived. The analytical results are validated against computer simulations.

### Keywords

Orthogonal Frequency Division Multiplex Cyclic Prefix Orthogonal Frequency Division Multiplex Symbol Spreading Code Physical Resource Block## 1. Introduction

System numerology.

Channel bandwidth (MHz) Open image in new window | 1.4 | 3.0 | 5.0 | 10.0 | 15.0 | 20.0 |
---|---|---|---|---|---|---|

Number of physical resource blocks Open image in new window | 6 | 15 | 25 | 50 | 75 | 100 |

FFT size Open image in new window | 128 | 256 | 512 | 1024 | 1536 | 1024 |

Sampling frequency (Msps) Open image in new window | 1.92 | 3.84 | 7.68 | 15.36 | 23.04 | 30.72 |

Another fundamental deviation in LTE specification relative to previous standard releases is the control channel design and structure to support the capacity enhancing features such as link adaptation, physical layer hybrid automatic repeat request (ARQ), and MIMO. Correct detection of the control channel is needed before the payload information data can be successfully decoded. Thus, the overall link and system performance are dependent on the successful decoding of these control channels.

The performance of the physical downlink control channels in the typical urban (TU-3 km/h) channel was reported in [4] using computer simulations only, without rigorous mathematical analyses. The motivation behind this paper is to describe the analytical aspects of the performance of optimal receiver principles for the decoding of the LTE physical control channels. We develop and analyze the performance of ML receiver structures for the downlink physical control format indicator channel (PCFICH) as well as the physical hybrid ARQ indicator channel (PHICH) in the presence of additive white Gaussian noise, frequency selective fading channel with different transmit and receive antenna configurations, and space-frequency block codes (SFBC). These analyses provide insight into system performance and can be used to study sensitivity to design parameters, for example, channel models and algorithm designs. Further, it would serve as a reference tool for fixed-point computer simulation models that are developed for hardware design.

The rest of the paper is organized as follows. A brief description of the LTE control channel specification is given in Section 2. The BER analyses of the physical channels PCFICH and PHICH are given in Sections 3 and 4, respectively. Section 5 contains some concluding remarks.

Notation 1.

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

## 2. Brief Description of the 3GPP-LTE Standard

The downlink physical channels carry information from the higher layers to the user equipment. The physical downlink shared channel (PDSCH) carries the payload-information data, physical broadcast channel (PBCH) broadcasts cell specific information for the entire cell-coverage area, physical multicast channel (PMCH) is for multicasting and broadcasting information from multiple cells, physical downlink control channel (PDCCH) carries scheduling information, physical control format indicator channel (PCFICH) conveys the number of OFDM symbols used for PDCCH and physical hybrid ARQ indicator Channel (PHICH) transmits the HARQ acknowledgment from the base station (BS). BS in 3GPP-LTE is typically referred to as eNodeB. Downlink control signaling occupies up to 4 OFDM symbols of the first slot of each subframe, followed by data transmission that starts at the next OFDM symbol as the control signaling ends. This enables support for microsleep which provides battery-life savings and reduced buffering and latency [4]. Reference signals transmitted by the BS are used by UE for channel estimation, timing and frequency synchronization, and cell identification.

The downlink OFDM FDD radio frame of 10 ms duration is equally divided into 10 subframes where each subframe consists of two 0.5 ms slots. Each slot has 7 or 6 OFDM symbols depending on the cyclic prefix (CP) duration. Two CP durations are supported: normal and extended. The entire time-frequency grid is divided into physical resource blocks (PRB), wherein each PRB contains 12 resource elements (subcarriers). PRBs are used to describe the mapping of physical channels to resource elements. Resource element groups (REG) are used for defining the control channels to resource element mapping. The size of the REG varies depending on the OFDM symbol number and antenna configuration [1]. The PCFICH is always mapped into the first OFDM symbol of the first slot of each subframe. For the normal CP duration, the PHICH is also mapped into the first OFDM symbol of the first slot of each subframe. On the other hand, for the extended CP duration, the PHICH is mapped to the first 3 OFDM symbols of the first slot of each subframe. All control channels are organized as symbol-quadruplets before being mapped to a single REG. In the first OFDM symbol, two REGs per PRB are available. In the third OFDM, there are 3 REGs per PRB. In the second OFDM symbol, the number of REGs available per PRB will be 2 for single- or two-transmit antennas, and 3 for four-transmit antennas.

Power delay profiles for pedestrian B and ITU channel models.

Ped-B channel model | TU channel model | ||
---|---|---|---|

Delay (nsec) | Average power (dB) | Delay ( | 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

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

### 3.1. PCFICH with SIMO Processing

*k*-th receive antenna is modeled as

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

### 3.2. Analysis of CFI with Repetition Coding

The expression in (14) is compared to that in (9).

### 3.3. PCFICH with Transmit Diversity Processing

*l*th layer, and Open image in new window denotes Open image in new window thermal-noise vector. The channel matrix Open image in new window is given by

where Open image in new window .

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

where Open image in new window .

## 4. Physical Hybrid ARQ Indicator Channel

### 4.1. PHICH with SIMO Processing

*i*th resource-element group and Open image in new window th receiver antenna is given by

where Open image in new window is an Open image in new window vector, Open image in new window and Open image in new window , Open image in new window are the power levels of the Open image in new window orthogonal codes (for the normal CP case), Open image in new window is the data bit value of the Open image in new window th user HI, and Open image in new window and Open image in new window is an Open image in new window complex channel frequency response vector. Without loss of generality, it is assumed that the desired HI channel to be decoded uses the first orthogonal code denoted as Open image in new window . The second and third terms in (26) denote the remaining Open image in new window spreading codes used for the other HI channels within a PHICH group (in this analytical model, we treat the general case of the normal CP. The extended CP is easily handled as shown in the final error-rate formulas.) The term Open image in new window denotes the thermal noise, which is modeled as circularly symmetric zero-mean complex Gaussian with covariance Open image in new window .

where Open image in new window in (33) is the processing gain obtained from the spreading code of length 4, and (3,1) repetition code in the case of normal CP [1, 2]. In case of extended CP, a maximum of 4 HI channels are allowed in a PHICH group, and hence a spreading code of length 2 is used for each HI channel, which results in Open image in new window .

where Open image in new window .

### 4.2. PHICH with Transmit Diversity Processing

*i*th REG. The Open image in new window vector Open image in new window is given by

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

where the diversity order Open image in new window .

### 4.3. Matched Filter Bound for ITU Channel Models

where Open image in new window is Open image in new window tap-locations vector of Open image in new window at which the tap coefficient is nonzero.

In case of transmit diversity using SFBC, MFB of PHICH is the function of Open image in new window . For a MIMO system, the channels are assumed to be independent and have the same statistical behavior [7]. For single-receive antenna, the MFB is a function of 12 independent chi-square distributed random variables with 2 degrees of freedom, and it is written as Open image in new window as in (54).

## 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.

### References

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