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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 Complex attenuation coefficient of the th path
τ Delay of the th path
ν Doppler frequency of the th 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 ,ν ) 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 (f) Admissible hidden data defined in the SAGE algorithm
Λ(θ ) Likelihood function of the parameters θ
\(\hat {x}^{[i]}_{\ell }(f)\) 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