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Signals & Waves
A. Eskandarian
7/11/2017
1
Signals & Waves
We have learned how to construct wave
packets for 3-dimensional waves similar
to 1-dimensional waves.
We’ll revisit the frequency spectrum
We’ll learn about analog Vs digital
signals
We will use the Sampling theorem for
digitization of analog signals
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Signals & Waves
Fourier series allows us to construct (represent)
any periodic function in terms of sums of Sine
and Cosine functions (possibly infinite series)
with differing amplitudes and arguments that
are multiples of a fundamental frequency.
It also allows us to represent any general signal
being composed of frequencies with continuous
values in a given bandwidth.
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Signals & Waves
In the first case the frequency spectrum is
discrete. (i.e., the graph of Amplitude VS
Frequency)
In the second case the curve is a
continuous curve.
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Signals & Waves
Amplitude
f0
3f0
5f0
7f0
frequency
Frequency spectrum for the discrete case (maybe
infinite number of component frequencies)
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Signals & Waves
Bandwidth = (f2- f1)
Amplitude
f1
f2
frequency
Frequency spectrum for the continuous case (infinitely
many frequencies between f1 and f2, f1< f < f2)
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Signals & Waves
 The following general observations may be
made in conjunction with the frequency
spectrum of a given function:
1. A discrete frequency spectrum belongs to a
signal that is periodic and can be represented
by a Fourier series (either finite or infinite)
2. A continuous frequency spectrum mostly
limited to a finite bandwidth is that of a
general localized signal that does not
necessarily appear to exhibit periodic
behavior.
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Signals & Waves
If a general analog signal of the second type is
given, and sent by a transmitter. It must be
reproduced exactly at the receiver site to convey
the information it carries. Distortion,
attenuation, and noise will transform its profile.
Is it possible to reproduce this signal without
having to know the amplitude at every given
time?
YES
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Signals & Waves
Sampling Theorem (Shannon) tells us that
if the frequency spectrum of the signal is
limited to Bandwidth = (f2- f1) then a
sampling rate of 2(f2- f1) will faithfully
reproduce the original signal. In other
words, all the information present in the
original signal can be reconstructed by
having knowledge of only those sample
points.
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(a)
7D/2
5D/2
3D/2
D/2
-D/2
-3D/2
-5D/2
-7D/2
(b)
7D/2
5D/2
3D/2
D/2
-D/2
-3D/2
-5D/2
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Signals & Waves
In the previous graph the vertical lines
designate the number of times the
function is sampled.
The horizontal lines designate how many
quantized levels are needed to represent
the value of the amplitude accurately.
Here 8 levels are needed corresponding
to 3 bits (23=8).
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Signals & Waves
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Signals & Waves
Sent
Received
• e.g. AM, FM, TV transmission
Sent
Received
• e.g digital telephone, CD Audio
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Transmission segment
Source
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Repeater
Repeater
Destination
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Information
1
0
1
1
0
1
+1
Amplitude
Shift
Keying
0
T
2T
3T
4T
5T
6T
0
T
2T
3T
4T
5T
6T
0
T
2T
3T
4T
5T
6T
-1
+1
Frequency
Shift
Keying
-1
+1
Phase
Shift
Keying
-1
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t
Frequency (Hz)
102
10
10-2
10-6
x rays
10-4
gamma rays
1012 1014 1016 1018 1020 1022 1024
ultraviolet light
broadcast
radio
104
1010
visible light
108
infrared light
106
106
microwave
radio
104
power &
telephone
102
10-8 10-10
10-12 10-14
Wavelength (meters)
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Frequency (Hz)
104
105
106
108
107
109
1012
1011
1010
FM radio & TV
Wireless cable
AM radio
Cellular
& PCS
satellite & terrestrial
microwave
LF
10
4
MF
103
HF
102
VHF
101
UHF
1
SHF
10-1
EHF
10-2
10-3
Wavelength (meters)
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Multiplexing
(a)
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(b)
A
A
A
B
B
B
C
C
C
A
Trunk
group
MUX
MUX
B
C
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(a) Individual signals occupy W Hz
A
f
W
0
B
0
f
W
C
f
0
W
(b) Combined signal fits into channel bandwidth
A
B
C
f
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Frequency Division Multiplexing
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Signals & Waves
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FDM was introduced in
the telephone networks in
the 1930s. Each voice
signal requires 4kHz of
bandwidth. The basic
analog multiplexer
combines 12 voice
channels in one line. The
multiplexer modulates
each voice signal so that it
occupies a 4kHz slot in the
band between 60 and 108
kHz. A hierarchy of such
groupings has been
20
Time Division Multiplexing
(a) Each signal transmits 1 unit every 3T seconds
A1
0T
A2
6T
3T
B1
B2
C1
0T
t
6T
3T
0T
t
C2
t
6T
3T
(b) Combined signal transmits 1 unit every T seconds
A1 B1
0T 1T 2T
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C1
A2
3T 4T
B2
C2
t
5T 6T
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Signals & Waves
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TDM was introduced in the
telephone network in early
1960s. The T-1 carrier system
was designed to carry 24
digital connections. It uses a
transmission frame that
consists of 24 slots of 8 bits
each. The beginning of each
frame is indicated by a single
bit that follows a periodic
pattern. The speed is then (1+
24*8)bits/frame *
8000frames/sec=1.544Mbps
22
100
50
10
Loss (dB/km)
5
Infrared absorption
1
0.5
Rayleigh scattering
0.1
0.05
0.01
0.8
1.0
1.2
1.4
1.6
1.8
Wavelength (m)
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Signals & Waves
The previous Figure shows that a range
of low attenuation wavelengths about
100nm wide is available in the 1300nm
range. This range corresponds to a
bandwidth of 18teraherz (THz). Another
band of about 100nm in the 1550nm
range provides another 19 THz of
bandwidth.
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Signals & Waves
Wavelength-division multiplexing (WDM) can
be viewed as an optical domain version of FDM
in which multiple information signals modulate
optical signals at different optical wavelengths
(colors). The resulting signals are combined
and transmitted simultaneously over the same
optical fiber. Prisms and diffraction gratings
can be used to combine and split color signals
(i.e, it is not necessary to change the optical
signals to electrical signals first for switching
purposes).
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Wavelength-division multiplexing (WDM)
1
2
m
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Optical
MUX
Optical
deMUX
1  2 .
m
1
2
Optical
fiber
m
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Signals & Waves
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Signals & Waves
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Signals & Waves
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Signals & Waves
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