204 6. CONTINUOUS-TIME FOURIER TRANSFORM
Table 6.2: Alphabet letters encoded with Morse code
A .‒ H .... O ‒ ‒ ‒ V ... ‒
B ‒… I .. P . ‒ ‒. W . ‒ ‒
C ‒. ‒. J .‒ ‒ ‒ Q ‒ ‒. ‒ X ‒.. ‒
D ‒.. K ‒. ‒ R . ‒. Y ‒. ‒ ‒
E . L . ‒.. S ... Z ‒ ‒..
F .. ‒. M ‒ ‒ T ‒
G ‒ ‒. N ‒. U .. ‒
Find the prototype signals d(t) and dash(t) in the supplied file morse.mat. After load-
ing the file morse.mat:
>>load morse
e signals d(t) and dash(t) can be located in the vectors dot and dash, respectively. e
hidden signal, which is encoded, per Equation (6.9), containing the letters of the name, in the
vector xt . Let the three modulation frequencies f
1
, f
2
, and f
3
be 20, 40, and 80 Hz, respectively.
1. Using the amplitude modulation property of the CTFT, determine the three possible
letters and the hidden name. (Hint: Plot the CTFT of xt. Use the values of T and
contained in the file.)
2. Explain the strategy used to decode the message. Is the coding technique ambiguous? at
is, is there a one-to-one mapping between the message waveforms .x
m1
.t/; x
m2
.t/; x
m3
.t//
and the alphabet letters? Or can you find multiple letters that correspond to the same
message waveform?
6.4.2 THE DOPPLER EFFECT
e Doppler Effect phenomenon was covered in the previous chapter. In this exercise, let us
examine the Doppler Effect with a real sound wave rather than a periodic signal. e wave file
firetrucksiren.wav provided in the book software package contains a firetruck siren. Read the
file using the MATLAB function audioread and produce its upscale and downscale versions.
Show the waves in the time and frequency domains (find the CTFT). Figure 6.32 shows the
original sound, the sound as the vehicle approaches, and the sound after the vehicle passes by in
both the time-domain and frequency domains.
6.4. LAB EXERCISES 205
Original Sound
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0 0.5 4.51 1.5 2 2.5 3 3.5 4
FT of Original Sound
0.016
0.014
0.012
0.01
0.008
0.006
0.004
0.002
0
-4000 -3000 -2000 2000-1000 1000 400030000
ig
ig
ig
ig
ig
Sound as Vehicle Approaches
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
0 0.5 4.51 1.5 2 2.5 3 3.5 4
0.014
0.012
0.01
0.008
0.006
0.004
0.002
0
-5000 -4000 -3000 2000-1000-2000 1000 4000 500030000
FT of Sound as Vehicle Approaches
Sound after Vehicle Passes
0 0.5 4.51 1.5 2 2.5 3 3.5 4
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
FT of Sound after Vehicle Passes
-4000 -3000 -2000 2000-1000 1000 4000
30000
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
x 10
-3
Figure 6.32: Doppler Effect system, original sound, sound as vehicle approaches, and sound
after vehicle passes in both time-domain and frequency domain.
206 6. CONTINUOUS-TIME FOURIER TRANSFORM
6.4.3 DIFFRACTION OF LIGHT
e diffraction of light can be described via a Fourier transform [5]. Consider an opaque screen
with a small slit being illuminated by a normally incident uniform light wave, as shown in Fig-
ure 6.33.
Diffraction
Screen
Diffraction
Pattern
Diffraction
Pattern
Diffraction
Pattern
Viewing
Screen
Viewing
Screen
Viewing
Screen
Waveforms
Propagation
Direction
Slit
l
1
l
0
d
Figure 6.33: Diffraction of light.
Considering that d >> l
2
1
= provides a good approximation for any l
1
in the slit, the electric
field strength of the light striking the viewing screen can be expressed as [5]
E
0
.l
0
/ D K
e
j.2d=/
jd
e
j.=d /l
0
2
1
Z
1
E
1
.l
1
/e
j.2=d /l
0
l
1
d l
1
; (6.10)
where
E
1
D field strength at diffraction screen
E
0
D field strength at viewing screen,
K D constant of proportionality, and
D wavelength of light.
e above integral is in fact Fourier transformation in a different notation. One can write
the field strength at the viewing screen as [5]
E
0
.l
0
/ D K
e
j.2d=/
jd
e
j.=d /l
0
2
CTFTfE
1
.t/g
f !l
0
=d
: (6.11)
e intensity I.l
0
/ of the light at the viewing screen is the square of the magnitude of the
field strength. at is,
I.l
0
/ D
j
E
0
.l
0
/
j
2
: (6.12)
Plot the intensity of the light at the viewing screen. Set the slit width to this range (0.5–
5 mm), the wavelength of light to this range (300–800 nm), and the distance of the viewing
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