3.2 Time-delay estimation for the AWGN channel
3.2.1 Maximum likelihood estimation
j2πnm N
, (3.1.3)
and the discrete received signal is y (m) .
= yc(mTs) for m = 0, 1, . . . , N− 1 (thanks to the circular convolution of N samples), whereas the discrete CIR is h (m) .
= hc(mTs). After applying an N-point discrete Fourier transform (DFT) to y (m), we have
r (n) = √
2C· b (n) · H (n) + w (n) , (3.1.4) where n is the index of the subcarriers, H (n) = F {h (m)} is the channel frequency response, being F {·} the discrete Fourier transform operator, and w (n) are the noise frequency samples, which are statistically uncorrelated with w (n) ∼ N (0, σw2).
3.2 Time-delay estimation for the AWGN channel
The performance of the time-delay estimation is first assessed considering the impact of the AWGN channel only. Thus, let us define the propagation channel model as
hc(t) = h0 · δ (t − tǫ) , (3.2.1) where h0 is the complex channel coe cient, δ (t) is the Dirac delta, and tǫ is the time delay introduced by the channel. Using a bandlimited representation for the channel, the discrete CIR of this model is
h (m) = h0· sinc (m − τ) , (3.2.2) where sinc (x) .
= sin(π·x)π·x is the sinc function, and τ .
= tǫ/Ts is the discrete-time symbol-timing error, which is the time delay to estimate for positioning purposes. For sake of simplicity, and without loss of generality, the channel coe cient h0 is assumed equal to one.
3.2.1 Maximum likelihood estimation
The maximum likelihood estimation (MLE) of the time delay in AWGN channels is based on the matched filter, being widely studied in the literature, such as in [Kay98, p.192].
Considering a pilot signal, the MLE is based on the maximization of the correlation between the received signal and the transmitted pilots (i.e. the \matched" filter in this case), resulting in the time-delay estimation
^
τ = arg max
τ |Ryx(τ )|2 , (3.2.3)
where Ryx(τ ) is the correlation function. Since the cyclic prefix introduces a circular symmetry in the signal at the output of the channel, the correlation between the received signal and the pilots is a circular correlation, which is defined as
Ryx(τ ) .
=
N −1
X
m=0
y (m)· x∗s(m− τ) , (3.2.4)
where x∗s(m) is a circular shifted and conjugate version of the original x (m). This op-eration results in the matched filter of the received signal. The MLE can be e ciently implemented by using the FFT operation, as it is shown in Figure 3.2. This implementa-tion is typically adopted for the bi-dimensional search of time-delay and carrier-frequency errors in GNSS receivers [Bor07, p.79].
As it has been described in (3.2.3), the MLE is obtained by measuring the time delay corresponding to the maximum of the correlation function of (3.2.4), or correlation peak. Let us analyse the correlation function by considering only the transmitted signal, resulting on the auto-correlation function (ACF) of the discrete signal x (m), defined as
Rxx(τ ) .
=
N −1
X
m=0
x (m)· x∗s(m− τ) . (3.2.5)
Assuming an equipowered data and pilot structure, i.e. power is uniformly distributed among all the subcarriers, the relative power weight at the n-th subcarrier is p (n) =
FFT IFFT | · |2 max( )
( )∗ FFT
Received signal
Reference signal
Estimated timing y(m)
x(m)
ˆ τ
Figure 3.2: FFT implementation of the correlation at the receiver.
p1/N. Thus, the circular auto-correlation of (3.2.5) results in
where d (n)· d∗(n) = 1. Let us consider the allocation of LTE pilot signals, which defines the following subcarriers allocation sets:
• Synchronisation signals:
The number of subcarriers equivalent to the e ective bandwidth, i.e. bandwidth occupied by the active subcarriers, when only transmitting the synchronisation signals is NSS= 63, and when only transmitting the reference signals is NRS = 12· NRB−4. Thus, the circular auto-correlation function of the synchronisation signals results in
Rxx,SS(τ ) = 2C
where the geometric progression, i.e.
N
Substituting (3.2.10) in (3.2.9), we obtain
Rxx,SS(τ ) = 2C
NSS2 · ejπ32τ /NSS + e−jπ32τ /NSS · sin (π31τ /NSS) sin (πτ /NSS) =
= 4C
NSS2 · cos (π32τ/NSS)· sincd(31; τ /NSS) , (3.2.11) where sincd(N; x) .
= sin(π·N ·x)
sin(π·x) is the discrete sinc function, which is a periodic function, unlike the traditional sinc function, and corresponds to the Fourier transform of a rect-angular pulse of N samples when x = f , being f the frequency. Following the same procedure, the ACF of the reference signals can be obtained as
Rxx,RS(τ ) = 2C NRS2
X
n∈NRS
ej2πnτ /NRS =
= 2C
NRS2 · ej2πθτ /NRS·
NRB
X
n=1
e−j2π6nτ /NRS+
NRB−1
X
n=0
ej2π(6n+1)τ /NRS
!
, (3.2.12)
where
NRB
X
n=1
e−j2π6nτ /NRS = e−j2π6τ /NRS − e−j2π6(NRB+1)τ /NRS 1− e−j2π6τ /NRS =
= e−jπ6(NRB+1)τ /NRS · sincd(NRB; 6τ /NRS) , (3.2.13) and
NRB−1
X
n=0
ej2π(6n+1)τ /NRS = ej2πτ /NRS− ej2π(6NRB+1)τ /NRS 1− ej2π6τ /NRS =
= ejπ(6NRB−4)τ /NRS · sincd(NRB; 6τ /NRS) . (3.2.14) Substituting (3.2.13) and (3.2.14) in (3.2.12), we obtain
Rxx,RS(τ ) = 2C
NRS2 · ej2πθτ /NRS · ejπ(6NRB−4)τ /NRS + e−jπ6(NRB+1)τ /NRS ·
· sincd(NRB; 6τ /NRS) =
= 4C
NRS2 · ej(5+2θ)πτ /NRS · cos (π (6NRB+ 1) τ /NRS)·
· sincd(NRB; 6τ /NRS) . (3.2.15) From (3.2.11) and (3.2.15), the ACF is described by the number and distribution of the
−30 −20 −10 0 10 20 30
Correlation lag (Ts units)
Auto−correlation function
Figure 3.3: Auto-correlation function of the LTE synchronization signals and reference signals for the di erent signal bandwidths.
pilots subcarriers. Focusing on the reference signals, the equispacing between pilot subcar-riers mainly determines the ACF. For instance, the spacing of six subcarsubcar-riers introduces a multiplicative factor of six in the discrete sinc of (3.2.15). This factor introduces periodic peaks on the ACF, approximately around τ = 2k· NRB for k ∈ Z, as it is shown in Figure 3.3(a). The equispaced pilots of the reference signal also a ect the zeros of the ACF, as it can be seen by finding the values of τ that make the cosine and the discrete sinc of (3.2.15) equal to zero:
τ = k· NRS
Moreover, any variation on the uniform pattern (i.e. contiguous or equispaced) modifies the ACF, as it is produced by the avoidance of the DC subcarrier that adds the cosine term in (3.2.11) and (3.2.15). However, a shift of the subcarrier allocation, such as the term θRS in the reference signals, only adds a phase shift in the ACF. In Figure3.3(b), the ACF Rxx(τ ) is shown for the SS and the RS with di erent LTE bandwidth configurations using only one OFDM symbol. As it can be noticed, the RS-bandwidth is denoted according to the number of resource blocks allocated in the frequency domain (i.e. 180 kHz per RB). As it could be expected from (3.2.16), the main lobe of the ACF is narrower as the bandwidth increases.
−600 −400 −200 0 200 400 600 Bandwidth (kHz)
2π·n·τ N
Power Spectral DensityPhase
Figure 3.4: Time delay in the frequency domain.