IEEE Signal Processing - March 2018 - 144

Here, the time is t = nDt and the frequency is given by
f = k/N w Dt, Dt being the time sampling interval. Note that
N w, the dimension of the DFT, is now much larger than the
size of the time window, L w. This will enhance details in the
power spectrum. Reference [38] provides an overview of methods to compute the sliding DFT.
The STFT provides an efficient method to estimate the local
frequency content of a discrete signal. The main disadvantages
are low resolution and spectral leakage from sidelobes [2], [4].
The spectral leakage may be reduced by multiplying with a window function, but that reduces resolution [33].

e m ^nh =

Lc

/ b m ^ j h z m ^n + jh

(11)

where the backward prediction error filter is b m ( j) with
b m (0) = 1. Note that, in (10) and (11), we only use data that are
within the data analysis window. There are no assumptions about
the data outside the window. From now on, all equations refer to
the data in a window with midpoint m. We therefore do not use
the subscript m in the following.
The forward prediction error energy is

CSTAR power spectrum

F=

To obtain higher resolution for certain signals, an AR model is
being estimated for a short time window of the real signal x(n)
in the STAR process [32], [39], [40]. In these methods, the local
frequency spectrum is obtained as the inverse of the amplitude
spectrum of the linear prediction error filter. Here, we extend this
to CSTAR applied to the analytic signal (complex trace)
z ^nh = x ^nh + iH 6x ^nh@,

(6)

x ^nh,
0,

(7)

where H 6x (n)@ is the Hilbert transform x (n) [6].
Following [41], we add N zeros to the real-valued signal of
length N, resulting in
x , ^nh = '

n = 0, 1, f , N - 1
n = N, N + 1, f, 2N - 1,

which has DFT X , (k). Next, we compute

X , ^ k h, k = 0, k = N
Z , (k) = H ^ k h X , (k) = * 2X , ^ k h, k = 1, f, N - 1
(8)
0,
k = N + 1, f, 2N - 1

with inverse DFT z , (n). The analytic signal or complex trace
is now
z ^nh = '

z , ^nh, n = 0, 1, f, N - 1
0,
n = N, N + 1f, 2N - 1.

(9)

The effect of the zero-padding is to avoid wrap-around
effects in the time domain caused by the frequency-domain
product in (8).
From the complex data in a short time window z m (,),
, = 0, 1, f, L w - 1, we use the complex AR model, with L c
AR coefficients,

n = 0, 1, f, L w - 1 - L c,

j= 0

Lw - 1

/

2

f ^nh ,

(12)

n = Lc

and the backward prediction error energy is
E=

Lw - Lc - 1

/

2

e ^nh .

(13)

n=0

We have used three different methods to compute the forward prediction error operator, c m ( j), all of which minimize the
sum of the energies of the forward and backward prediction
errors. They are recursive in the order L c, and they use type
recursion [34] to reduce the number of numerical operations.
In the algorithms in [31] and [35], the backward prediction
error filter is the complex conjugate of the forward prediction
error filter
b ^nh = c) ^nh.

(14)

In this case, the energies of the forward and backward
prediction errors are identical. In [31], the energy of the prediction error is minimized with respect to a reflection coefficient that occurs in the Levinson algorithm. Reference [35]
also uses a Levinson-type recursion, but the prediction error
energy is minimized with respect to the coefficients in the
prediction error filter. Reference [36] minimized the sum of
the prediction error energies (they are now different) with
respect to different forward and backward prediction error
operators. We have used these algorithms with complex data.
We accomplish this by extending the algorithm in [36] from
real to complex signals and complex AR coefficients.
An estimate for the power spectrum of the signal, using the
forward prediction error operator [31], is
P^k h =

F ,
2
C^k h

(15)

where
C^k h =

Lc

nk

/ c^nhe -2 i N
r

w

(16)

n=0

fm ^n h =

Lc

/ c m ^ j h z m ^n - jh

n = L c, L c + 1, f, L w - 1 (10)

j=0

with c m (0) = 1 to compute the forward prediction error fm (n)
in the selected time window with midpoint index m. The backward prediction error is
144

is the DFT of the prediction error operator. The backward prediction error filter gives an estimate for the power spectrum
Q^k h =

E ,
2
B^k h

(17)

where B (k) is the DFT of the backward prediction error filter,

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