Journal of Computers - Academy Publisher

JOURNAL OF COMPUTERS, VOL. 6, NO. 9, SEPTEMBER 2011 1979
where K (, tu)
is kernel function of FRFT, is defined as
α
2 2
⎧ 1−jcotαt + u ut
⎪ exp( j cot α − j ), α ≠nπ
⎪ 2π 2 sinα
⎪
Kα(, t u) = δ( t− u), α = 2nπ
(14)
⎨
⎪ δ( t+ u), α = (2n+ 1) π
⎪
⎪⎩
where α is rotation angle. With a view of timefrequency
plane rotation to explain, then the following
equations are established
0
R ( u) = y ( t)
yi π
i
yii R ( u) = y ( − t)
π /2
R ( u) = FT( y ( t))
(15)
yii π /2
where Ry( u)
corresponds to the FT of signal y ()
i
i t .
The traditional FT can be seen as the time-frequency
distribution of signal in the projection of frequency axis,
while FRFT can be seen as the time-frequency
distribution of signal in the projection of the rotated
frequency axis. The representation of signal in the
fractional Fourier domain includes both the time domain
and frequency domain information, so FRFT is also
considered a generalized time-frequency analysis[13,14].
By the definition of FRFT, a LFM signal only at the
appropriate fractional domain is an impulse function.
Therefore, FRFT in a fractional Fourier domain has the
best gathering characteristics for the given LFM signal. In
the time-frequency plane, a limited length of the LFM
signal appears as the distribution of dorsal fin shape of
diagonal line, but FRFT is essentially the "rotating" of
signal. If choose the appropriate rotation angle, it will
show the energy aggregation and apparent peak in the
fractional Fourier domain of signal. It was shown in Fig.3.
| Xp( u)|
Figure 3 .The distribution of time-frequency and in the projection of
fractional Fourier domain of LFM signals
The bandwidth of signal in time domain and frequency
domain can be estimated by the second-order central
moments[15], and the bandwidth of signal in the
fractional Fourier domain can be obtained by the secondorder
central moments of FRFT[16]. The second-order
central moments(SCM) of FRFT Pα is defined as
∞
2
α
2
Pα = ∫ Ry( u) ( u−m ) du
−∞ i
α
(16)
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where
α
Ry( u)
i
is FRFT of yi() t , mα is first-order
moments of FRFT
m
∞
= ∫
2
α
R ( u) udu
(17)
α
−∞
yi
As FRFT is a periodic function with the period of
α+ π α
2π about α , and meet R ( u) = R ( − u)
, so the
yi yi
second-order central moments of FRFT Pα has a
maximum or minimum value in the range of α ∈ [0, π ) .
As Pα represents the bandwidth of signal in the fractional
Fourier domain, when the rotation angle of timefrequency
planeα = αe
, the bandwidth has a minimum.
We can find spindle direction of time-frequency
distribution α by searching the minimum point of Pα ,
namely, the best fractional Fourier transform domain. The
bandwidth of noise is wide in the fractional Fourier
transform domain, α = αe
corresponds to the minimum of
bandwidth (the minimum of FRFT) also very large,
namely, Pα = α corresponds to the minimum. So we can
e
determine signal or noise by the bandwidth of fractional
Fourier transform domain.
The noise can be removed from the separated signals
after the LFM signal and noise discrimination method
based on second-order central moments of FRFT. Then
only detect the remaining LFM signal.
C. FRFT detection for LFM signal
As shown in Fig.3, the observed signal was
continuously proceed FRFT for rotation angle variable
α , the two-dimensional distribution of signal energy was
formed in the parameter ( α , u)
plane [14]. And the
detection and parameter estimation of LFM signals can
be realized by two-dimensional search of peak point
threshold in this plane. For type (1), the process of this
model can be described as[16]
∧ ∧
2
{ α0, u0} = arg max X ( u)
α
α , u
(18)
⎧
⎪
⎪
∧ ∧
⎪
µ 0 =−cot
α 0,
⎪
⎪ ∧ ∧ ∧
⎨ f0 = u0
csc α 0,
⎪
∧
⎪ X ∧ ( u0
)
∧
⎪
α0
⎪
Ai
=
⎪ ∆tA
∧
⎩
α0
(19)
IV. SIMULATION VERIFICATION
Select a group of mixed-signal in order to verify the
validity of the method proposed in this paper. Mixedsignal
composed of two-component LMF signal and
noise, the first LFM signal is
2
− j5π t
x1= e (initial
f = 0 and chirp rate k 10 =− 10 ), the second is
frequency 10