The PHICH carries physical hybrid ARQ ACK/NAK indicator (HI). Data arrives to the coding unit in form of indicators for HARQ acknowledgement. Figure 3 shows the PHICH transport channel and physical channel processing on hybrid ARQ data, is the spreading code for th user in a PHICH group, obtained from an orthogonal set of codes [1]. In LTE, spreading sequences are used in a PHICH group, where for normal CP and 2 for extended CP. The first set of spreading sequences are formed by Hadamard matrix, and the second set of spreading sequences are in quadrature to the first set.

### 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 th receiver antenna is given by

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

The ML decoding is given by

where is the number of antennas at the UE receiver and

where

where the estimated channel frequency response is given by , is the estimation error which is uncorrelated with and zero-mean complex Gaussian with covariance . By expanding (29), we get that

Note that . Thus (28) becomes

For ideal channel estimation, then due to the orthogonality property of the spreading codes, no interference is introduced to from the other HI channels within a PHICH group. However, in the presence of channel-estimation error, self-interference and cochannel interference are introduced as seen in the second and third terms, respectively, in (31). Since and , the signal to interference plus noise ratio (SINR) of the decision statistic is thus given by

In the case of a static AWGN channel with a single antenna at the UE receiver, that is, , the SINR is simply given by

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

For ideal channel estimation, and the SNR of the decision statistic is thus given by

The average loss in SNR due to channel-estimation error is given by

is plotted in Figure 4 as a function of the ratio between the desired power to the interfering signal power , for , − 6 dB, and =− 9 dB. Figure 4 shows that if , that is, 0 dB, with , results in a 3 dB loss in the SNR.

The probability of error in the AWGN case with a single-receive antenna is simply , is the per tone per antenna SNR as shown in (33) and (34). The probability of error averaged over the channel realization is given by

where . For a frequency-flat Rayleigh fading channel, (36) reduces to [5]

where .

The PHICH performance for static AWGN and frequency-flat Rayleigh fading channels is shown in Figure 5, for ideal channel estimation.

### 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 th layer (consecutive two tones) on the th receive antenna and th resource element group (REG) is given by

where , is a received-signal vector, is transmit-signal vector, and denotes thermal-noise vector, and each of its elements is modeled as circularly symmetric zero-mean complex Gaussian with covariance . The channel matrix is given by

where is a complex channel-frequency response between th transmit antenna and th receive antenna, at th symbol layer in th REG. The transmit-signal vector is generated by layer mapping and precoding the HI data vector in *i* th REG. The vector is given by

and are the power levels of the 8 spreading codes. The soft output from each layer is given by

The ML decision statistic, is given by

where

and where

In a flat-fading channel, . Then the decision statistic is given by,

The instantaneous SNR of is evaluated to be

In the case of a static AWGN channel with a single antenna at the UE receiver, that is, , the SNR is given by . The probability of error is given by,

For the MISO Rayleigh flat-fading channel, the average probability of error, averaged over the channel distribution, is given by [5]

where and , is the SNR per antenna.

For a MIMO () flat-fading channel, the average probability of error is given by

where the diversity order .

Figure 6 shows the PHICH performance in MIMO systems in the presence of AWGN and Rayleigh flat-fading channels. The analytical results match well with the computer simulations.

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

The objective of this section is to analyze the performance of the LTE downlink control channel PHICH, in general, using matched filter bounds for various practical channel models. The base band channel impulse response can be represented as

where and are the amplitude and delay of the th path which define power delay profile (PDP), is a zero-mean, unit-variance complex Gaussian random variable, , and is the system bandwidth. Let be a complex vector that contains nonzero taps which depends on the sampling frequency, and its corresponding system bandwidth is as shown in Table 1. The channel frequency response is given by,

where is tap-locations vector of at which the tap coefficient is nonzero.

The decision statistic SNR or matched filter bound (MFB) of PHICH is a function of , where . Thus, the MFB is a function of independent chi-square distributed random variables with 2 degrees of freedom. For single-receive antenna

where is independent chi-square distributed random variable with 2 degrees of freedom and is the average power of th element of . Since is constant with respect to for the given PDP, MFB can be simply written as

The characteristics function of is given by

As 's are distinct, the probability density function is given by

where . Then, the bit-error probability for the matched-filter outputs is given by [5]. The average probability of error, is given by

In case of transmit diversity using SFBC, MFB of PHICH is the function of . For a MIMO system, the channels are assumed to be independent and have the same statistical behavior [7]. For 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 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 , corresponds to the system bandwidth of 1.4 MHz, where is the number of PRBs used for downlink transmission. For , the average powers of resolvable taps of each channel coefficient are [0.1883, 0.1849, 0.1197, 0.1806, 0.1131, 0.1741] for a TU channel and [0.3298, 0.0643, 0.0673, 0.0017] for a Ped-B channel. The average powers of resolvable taps for , 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 for MISO and MIMO systems and a Ped-B channel with and are shown in Figures 7 and 8, respectively. It is also observed that the performance of Ped-B channels at has approximately 4.7 dB SNR gain with , at the BER of , and a TU channel has 3 dB SNR gain.