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.