Digital Signal Processing Reference
In-Depth Information
x
−
x
,p,
0
dx
W
ψ
(
x, p, t
)
=
δ(
x
+
2
apt
)
W
ψ
(
)
(4.144)
=
W
ψ
(
x
−
2
apt, p,
0
).
(4.145)
Therefore, the phase space Green's functions defined by
x, x
,p,p
,t
x
,p
,
0
dx
dp
,
W
ψ
(
x, p, t
)
=
G
W
ψ
(
)
W
ψ
(
)
(4.146)
x, x
,p,p
,t
x
,p
,
0
dx
dp
W
u
(
x, p, t
)
=
G
W
u
(
)
W
u
(
)
(4.147)
are
x, x
,p,p
,t
x
−
p
−
G
W
ψ
(
)
=
δ(
x
+
2
apt
)δ(
p
)
,
(4.148)
exp
e
−
2
Dp
2
t
√
2
x
)
2
−
(
x
−
x, x
,p,p
,t
p
−
G
W
u
(
)
=
δ(
p
).
(4.149)
2
Dt
π
Dt
References
1.
N.F. Barber and F. Ursell, The response of a resonant system to a gliding tone,
Philos. Mag.,
39, 345-361, 1948.
2.
G. Hok, Response of linear resonant systems to excitation of a frequency varying
linearly with time,
J. Appl. Phys.,
19, 242-250, 1948.
3.
M.C. Wang and G.E. Uhlenbeck, On the theory of the Brownian motion. II,
Rev.
Mod. Phys.,
17 (2 and 3), 1945.
4.
E.P. Wigner, On the quantum correction for thermodynamic equilibrium,
Phys.
Rev.,
40, 749-759, 1932.
5.
J.G. Kirkwood, Quantum statistics of almost classical ensembles,
Phys. Rev.,
44,
31-37, 1933.
6.
L. Cohen, Time-frequency distributions—a review,
Proc. IEEE,
77, 941-981, 1989.
7.
L. Cohen,
Time-Frequency Analysis,
Englewood Cliffs, NJ: Prentice-Hall, 1995.
8.
L. Cohen, Generalized phase-space distribution functions,
J. Math. Phys.,
7, 781-
786, 1966.
9.
M.G. Amin, Time-varying spectrum estimation of a general class of nonstationary
processes,
Proc. IEEE,
74, 1800-1802, 1986.
10.
J. Pitton, The statistics of time-frequency analysis,
J. Franklin Inst.,
337, 379-388,
2000.
11.
Special Issue on Applications of Time-Frequency Analysis, P. Loughlin (Ed.),
Proc.
IEEE,
84(9), 1996.
12.
J.E. Moyal, Quantum mechanics as a statistical theory,
Proc. Cambridge Philos. Soc.,
45, 99, 1949.
13.
W. Martin and P. Flandrin, Wigner-Ville spectral analysis of nonstationary pro-
cesses,
IEEE Trans. Acoust. Speech Signal Process.,
33, 1461-1470, Dec. 1985.
Search WWH ::
Custom Search