Page 55 - Phase Space Optics Fundamentals and Applications

36 Chapter One

2 2
|S f (x,u;w)| |S f (x,u;w)|
x x

u u
(a) (b)

FIGURE 1.5 Spectrogram of a sinusoidal FM signal
exp{i[2 u 0 x + a 1 sin(2 u 1 x)]} with (a) a medium-sized window, leading to a
space-frequency representation with smearing, and (b) a long window,
leading to a pure frequency representation.

the spectrogram. Although the spectrogram is a quadratic signal rep-
2
resentation |S f (x, u; w)| , the squaring is introduced only in the final
step and therefore does not lead to undesirable cross-terms that are
present in other bilinear signal representations. This freedom from
artifacts, together with simplicity, robustness, and ease of interpre-
tation, has made the spectrogram a popular tool for speech analysis
since its invention in 1946. 91 The price that has to be paid, however,
is that the auto-terms are smeared by the window w(x). Note that for
w(x) = (x), the spectrogram yields the pure space representation
2
2
|S f (x, u; w)| =| f (x)| , whereas for w(x) = 1, it yields the pure fre-
¯
2
2
quency representation |S f (x, u; w)| =| f (u)| . This is illustrated in
Fig. 1.5 on the sinusoidal FM signal
exp{i[2 u 0 x + a 1 sin(2 u 1 x)]}
and a rectangular window w(x) = rect(x/X) of variable width X.
Note in particular the smearing that appears in Fig. 1.5a.
Based on Eq. (1.88), but replacing the signal f (x) by its frac-

tional Fourier transform F (x), the -rotated version P (x, u; w, z)of
f
the smoothed interferogram P f (x, u; w, z) was defined subsequently
as 89,92

f
P (x, u; w, z) = P F (x, u; w, z)

1 1
= S F x, u + t; w z(t) S ∗ x, u − t; w dt (1.89)
2 F 2