Table 1 Explanation of adopted symbols
From: Parameter estimation using the sliding-correlator’s output for wideband propagation channels
Explanation | |
|---|---|
Symbol | |
u(t) | Transmitted pseudo-noise (PN) binary chip sequence |
L | Number of chips in the sequence of u(t) |
V 0 | Magnitude of the chips in u(t) |
a i | Values of the chips in u(t) with i being chip index. |
a i ∈[−1,1] | |
f c | Chip rate of the sequence u(t) |
u ′(t) | PN sequence correlated with received signals |
f c′ | Chip rate of u ′(t) |
γ | “Sliding factor” of the sliding correlator. γ=f c /(f c −f c′) |
a u (τ) | Auto-correlation function of u(t) in the delay domain τ |
a u (τ/γ) | Time-dilated version of a u (τ) |
B | Bandwidth of a low-pass-filter (LPF) applied to sliding |
correlator’s output | |
h(τ) | Channel impulse response in the delay domain |
\(\hat {h}(\tau)\) | Estimate of h(τ) |
\(\hat {(\cdot)}\) | Estimate of given argument |
n | Half of the width of the LPF normalized by f c /γ |
B n | Width of the LPF B n =[−n f c /γ,n f c /γ] |
N | Number of frequency components of the signals |
r(f) | Baseband representation of received signals in the |
frequency domain | |
M | Total number of propagation paths in a channel |
α ℓ | Complex attenuation coefficient of the ℓth path |
τ ℓ | Delay of the ℓth path |
ν ℓ | Doppler frequency of the ℓth path |
n(f) | White Gaussian noise represented in the frequency |
domain | |
w(f) | White Gaussian noise with spectral height equal to N 0 |
y(f) | Output of sliding correlator |
s(f) | The signal component of the output of sliding correlator |
n ′(f) | The noise component in the output of sliding correlator |
p(f;τ ℓ ,ν ℓ ) | Convolution results between u(f) distorted by a channel |
and the clean sequence u ′(f) | |
y | Concatenated received signals in frequencies, i.e. |
y=[y(f);f∈(f 1,…,f N )] | |
Θ | The parameters of propagation paths in a channel |
\(\hat {\boldsymbol {\Theta }}^{[0]}\) | Initial estimates of Θ |
\(\hat {\boldsymbol {\Theta }}^{[i]}\) | Estimates of Θ obtained in the ith SAGE iteration |
\(\hat {\boldsymbol {\Theta }}_{\text {SAGE}}\) | The estimates of Θ obtained when the SAGE algorithm |
converges. | |
x ℓ (f) | Admissible hidden data defined in the SAGE algorithm |
Λ(θ ℓ ) | Likelihood function of the parameters θ ℓ |
\(\hat {x}^{[i]}_{\ell }(f)\) | Estimated admissible hidden data in the E-step of the ith |
SAGE iteration | |
W | Diagonal matrix with its diagonal elements equal to |
E[|n ′(f)|2],f=f 1,…,f N . | |
η(τ,ν) | Objective function maximized in the M-step |
ϱ | Fraction of total length of SC’s output |
T | The time duration of the CIR |
Notations | |
(·)∗ | Complex conjugate operation |
(·)T | Transpose operation |
(·)H | Hermitian transpose operation |
∗ | Convolution operation |
E[·] | Expectation operation |
\(\arg \min \limits _{a}\) | Minimization operation with respect to a |
\(\arg \max \limits _{a}\) | Maximization operation with respect to a |
(W)−1 | Inverse operation of matrix W |