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IMAGING WITH QUASI-MONOCHROMATIC WAVES |
The analytic signal can be further written as |
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sðtÞ ¼ jsðtÞje jfðtÞ |
ð10:3-5Þ |
where |
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jsðtÞj2 ¼ u2ðtÞ þ v2ðtÞ |
ð10:3-6Þ |
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Þ ¼ |
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vðtÞ |
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ð |
10:3-7 |
Þ |
fð |
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u t |
Þ |
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ð |
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The analytic signal is often used with narrowband waveforms such as quasimonochromatic wave fields with central frequency fc. Then, fðtÞ can be written as
fðtÞ ¼ 2pfct þ f0ðtÞ |
ð10:3-8Þ |
Equation (10.2-13) can be written as |
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sðtÞ ¼ mðtÞe j2pfct |
ð10:3-9Þ |
where |
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mðtÞ ¼ jsðtÞje jf0ðtÞ |
ð10:3-10Þ |
mðtÞ is called the complex envelope. In optics, it is also referred to as the phasor amplitude.
EXAMPLE 10.4 Find the analytic signal corresponding to uðtÞ ¼ cosð2pftþ fðf ÞÞ. Solution: In Example 1.25, we found that
vðtÞ ¼ sinð2pft þ fð f ÞÞ |
ð10:3-11Þ |
Hence, |
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sðtÞ ¼ cosð2pftÞ þ j sinð2pftÞ ¼ e j2pft |
ð10:3-12Þ |
EXAMPLE 10.5 Find the energy in the analytic signal.
Solution: The energy of the analytic signal is
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jsðtÞj2dt ¼ |
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u2ðtÞdt þ |
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v2ðtÞdt ¼ 2 |
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u2ðtÞdt ¼ 2 ð |
jUð f Þj2df |
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Figure 10.1. The spectrum of a lowpass signal.
It is observed that the analytic signal has twice the energy of the corresponding real signal.
EXAMPLE 10.6 The amplitude spectrum of a lowpass signal uðtÞ is shown in Figure 10.1.
It is modulated by cosð2pf0tÞ to generate gðtÞ ¼ uðtÞ cosð2pf0tÞ. (a) Find and draw the amplitude spectrum of gðtÞ, and (b) Draw the amplitude spectrum of the analytic signal generated from uðtÞ, and (c) Find and draw the amplitude spectrum of pðtÞ ¼ uðtÞ cosð2pf0tÞ vðtÞ sinð2pf0tÞ.
Solution:
(a) gðtÞ can be written as
g t |
Þ ¼ |
uðtÞ |
e j2pf0t |
þ |
uðtÞ |
e j2pf0t |
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ð |
10:3-14 |
Þ |
ð |
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The FT of gðtÞ is given by |
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Gð f Þ ¼ |
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Uð f f0 |
Þ þ |
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Uð f þ f0 |
Þ |
ð10:3-15Þ |
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The amplitude spectrum is given by |
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jUð f f0Þ þ Uð f þ f0Þj |
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jGð f Þj ¼ |
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ð10:3-16Þ |
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jGð f Þj is shown in Figure 10.2. |
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and f0 fmax |
The frequency components in the range f0 f f0 þ fmax |
f f0 are known as the upper sideband. The frequency components in the range
f0 f f0 þ fmax and f0 fmax f f0 are known as the lower sideband.
(b) The analytic signal generated from u(t) is given by
sðtÞ ¼ uðtÞ þ jvðtÞ |
ð10:3-17Þ |
160 |
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IMAGING WITH QUASI-MONOCHROMATIC WAVES |
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Figure 10.2. The amplitude spectrum of Gðf Þ.
Its FT is
Sð f Þ ¼ Uð f Þ þ jVð f Þ
¼ Uð f Þ½1 þ jHð f Þ&
(
The amplitude spectrum of s(t) is shown in Figure 10.3.
(c) pðtÞ ¼ uðtÞ cosð2pf0tÞ vðtÞ sinð2pf0tÞ can be written as
pðtÞ ¼ Re½ðuðtÞ þ jvðtÞÞe j2p f0t&
¼Re½sðtÞe j2p f0t&
¼1 ½sðtÞe j2p f0t þ s ðtÞe j2p f0t&
2
The FT of pðtÞ can be written as
Pð f Þ ¼ 12 Sð f f0Þ þ 12 S ð f f0Þ
8
> Uð f f0Þ f0 f f0 þ fmax
>
<
¼> U ð f f0Þ f0 fmax f f0
>
: 0 otherwise
ð10:3-18Þ
ð10:3-19Þ
ð10:3-20Þ
Figure 10.3. The amplitude spectrum of the analytic signal sðtÞ.
ANALYTIC SIGNAL REPRESENTATION OF A NONMONOCHROMATIC WAVE FIELD 161
Figure 10.4. The amplitude spectrum of the single sideband signal pðtÞ.
The amplitude spectrum of pðtÞ is shown in Figure 10.4.
pðtÞ is known as the single sideband signal because it contains only the upper sideband of uðtÞ. If it was chosen instead as uðtÞ cosð2pf0tÞ þ vðtÞ sinð2pf0tÞ, it would contain the lower sideband only. In this way, the frequency bandwidth required to transmit the signal over a channel is reduced by a factor of 2.
10.4 ANALYTIC SIGNAL REPRESENTATION OF A NONMONOCHROMATIC WAVE FIELD
Let uðr; tÞ represent the real representation of a nonmonochromatic wave field. As discussed in Section 2.7, uðr; tÞ can be written in Fourier representation as
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uðr; tÞ ¼ 2 |
ð Uðr; f Þ cosð2p ft þ fð f Þd f |
ð10:4-1Þ |
0
where Uðr; f Þ and fð f Þ are the amplitude and phase spectra of uðr; tÞ with respect to t, respectively.
The analytic signal corresponding to uðr; tÞ is given by
uAðr; tÞ ¼ uðr; tÞ þ jvðr; tÞ |
ð10:4-2Þ |
where vðr; tÞ is the Hilbert transform of uðr; tÞ. As the Hilbert transform of cosð2pft þ fð f ÞÞ equals sinð2pft þ fð f ÞÞ by Example 10.1, vðr; tÞ can be written as
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vðr; tÞ ¼ 2 |
ð0 |
Uðr; f Þ sinð2p ft þ fð f ÞÞdf |
ð10:4-3Þ |
Hence, the analytic signal uAðr; tÞ is given by |
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ð |
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uAðr; tÞ ¼ 2 Uðr; f Þejð2pftþfð f ÞÞdf |
ð10:4-4Þ |