Comm 502: Communication Theory - GUCeee.guc.edu.eg/Courses/Communications/COMM502... · Comm 502:...
Transcript of Comm 502: Communication Theory - GUCeee.guc.edu.eg/Courses/Communications/COMM502... · Comm 502:...
![Page 1: Comm 502: Communication Theory - GUCeee.guc.edu.eg/Courses/Communications/COMM502... · Comm 502: Communication Theory A complex low-pass signal has a bandwidth of 200 kHz. What is](https://reader033.fdocuments.in/reader033/viewer/2022052319/5a7e385e7f8b9a66798e4938/html5/thumbnails/1.jpg)
1
Dr. Ahmed El-Mahdy
Communications Department Comm 502: Communication Theory
Comm 502: Communication Theory
Lecture 2
Transmission of Baseband Signals
- Sampling theorem, Pulse Amplitude Modulation
![Page 2: Comm 502: Communication Theory - GUCeee.guc.edu.eg/Courses/Communications/COMM502... · Comm 502: Communication Theory A complex low-pass signal has a bandwidth of 200 kHz. What is](https://reader033.fdocuments.in/reader033/viewer/2022052319/5a7e385e7f8b9a66798e4938/html5/thumbnails/2.jpg)
2
Dr. Ahmed El-Mahdy
Communications Department Comm 502: Communication Theory
Types of Communication Channels
Wireless Communication
channels Wire Communication
channels
![Page 3: Comm 502: Communication Theory - GUCeee.guc.edu.eg/Courses/Communications/COMM502... · Comm 502: Communication Theory A complex low-pass signal has a bandwidth of 200 kHz. What is](https://reader033.fdocuments.in/reader033/viewer/2022052319/5a7e385e7f8b9a66798e4938/html5/thumbnails/3.jpg)
3
Dr. Ahmed El-Mahdy
Communications Department Comm 502: Communication Theory
Transmission of Baseband Signals
Baseband channel: Coaxial cable or
pair of wires Conversion from a bit stream to a
Sequence of pulse waveform
PAM PCM
![Page 4: Comm 502: Communication Theory - GUCeee.guc.edu.eg/Courses/Communications/COMM502... · Comm 502: Communication Theory A complex low-pass signal has a bandwidth of 200 kHz. What is](https://reader033.fdocuments.in/reader033/viewer/2022052319/5a7e385e7f8b9a66798e4938/html5/thumbnails/4.jpg)
4
Dr. Ahmed El-Mahdy
Communications Department Comm 502: Communication Theory
Transmission of Baseband Signals Block diagram (Cont.)
• Data already in a digital format would bypass the formatting function.
•Textual information is transformed into binary digits by use of a coder (for ex. ASCII code(American Standard Code for Information Interchange). • Analog information is formatted using three separate processes: - Sampling - Quantization - Coding In all cases, the formatting step results in a sequence of binary digits. •These digits are transmitted through a baseband channel after transforming the digits to waveforms (pulses).
![Page 5: Comm 502: Communication Theory - GUCeee.guc.edu.eg/Courses/Communications/COMM502... · Comm 502: Communication Theory A complex low-pass signal has a bandwidth of 200 kHz. What is](https://reader033.fdocuments.in/reader033/viewer/2022052319/5a7e385e7f8b9a66798e4938/html5/thumbnails/5.jpg)
5
Dr. Ahmed El-Mahdy
Communications Department Comm 502: Communication Theory
Example: coding the word: THINK using 6-bit ASCII code
Formatting Text
![Page 6: Comm 502: Communication Theory - GUCeee.guc.edu.eg/Courses/Communications/COMM502... · Comm 502: Communication Theory A complex low-pass signal has a bandwidth of 200 kHz. What is](https://reader033.fdocuments.in/reader033/viewer/2022052319/5a7e385e7f8b9a66798e4938/html5/thumbnails/6.jpg)
6
Dr. Ahmed El-Mahdy
Communications Department Comm 502: Communication Theory
Formatting Text (Cont.)
ASCII Code
![Page 7: Comm 502: Communication Theory - GUCeee.guc.edu.eg/Courses/Communications/COMM502... · Comm 502: Communication Theory A complex low-pass signal has a bandwidth of 200 kHz. What is](https://reader033.fdocuments.in/reader033/viewer/2022052319/5a7e385e7f8b9a66798e4938/html5/thumbnails/7.jpg)
7
Dr. Ahmed El-Mahdy
Communications Department Comm 502: Communication Theory
Formatting Text (cont.) • Message: Text message consists of a sequence of
alphanumeric Characters.
• Bit stream: When digitally transmitted, the characters are
first encoded to a sequence of bits called bit
stream.
• Symbols: group of k bits can be combined to form a
symbols, from a finite symbol set of
symbols.
A system using symbol set of size M is called M-ary system
For k=1 (i.e. M=2), the system is called Binary (only two
waveforms is used). For k=2, (i.e. M=4), the system is
called 4-ary system.
kM 2
![Page 8: Comm 502: Communication Theory - GUCeee.guc.edu.eg/Courses/Communications/COMM502... · Comm 502: Communication Theory A complex low-pass signal has a bandwidth of 200 kHz. What is](https://reader033.fdocuments.in/reader033/viewer/2022052319/5a7e385e7f8b9a66798e4938/html5/thumbnails/8.jpg)
8
Dr. Ahmed El-Mahdy
Communications Department Comm 502: Communication Theory
Formatting Analog Information
• Analog information is formatted using three separate
processes:
- Sampling
- Quantization
- Coding
The process of transforming an analog waveform into a form that is compatible with a digital communication system starts with sampling the waveform to produce a discrete pulse amplitude modulated (PAM) waveform.
![Page 9: Comm 502: Communication Theory - GUCeee.guc.edu.eg/Courses/Communications/COMM502... · Comm 502: Communication Theory A complex low-pass signal has a bandwidth of 200 kHz. What is](https://reader033.fdocuments.in/reader033/viewer/2022052319/5a7e385e7f8b9a66798e4938/html5/thumbnails/9.jpg)
9
Dr. Ahmed El-Mahdy
Communications Department Comm 502: Communication Theory
Pulse Amplitude Modulation • PAM is an engineering term that is used to describe the
conversion of the analog signal to a pulse type signal in which the amplitude of the pulse represents the analog information.
• Because it uses pulses, the BW of the PAM is wider than of the analog waveform.
Pulse Amplitude Modulation
Pulse Amplitude modulation that
uses flat-top sampling
Pulse Amplitude modulation that
uses natural sampling
![Page 10: Comm 502: Communication Theory - GUCeee.guc.edu.eg/Courses/Communications/COMM502... · Comm 502: Communication Theory A complex low-pass signal has a bandwidth of 200 kHz. What is](https://reader033.fdocuments.in/reader033/viewer/2022052319/5a7e385e7f8b9a66798e4938/html5/thumbnails/10.jpg)
10
Dr. Ahmed El-Mahdy
Communications Department Comm 502: Communication Theory
Sampling • Analog signal is sampled every TS seconds.
• Ts is referred to as the sampling interval.
• fs = 1/Ts is called the sampling rate or sampling frequency.
• There are 3 sampling methods:
– Impulse (ideal) sampling - an impulse at each sampling instant
– Natural sampling - a pulse of short width with varying amplitude
– Flat-top sampling - sample and hold, like natural but with single amplitude value.
• The process is referred to as pulse amplitude modulation PAM .
![Page 11: Comm 502: Communication Theory - GUCeee.guc.edu.eg/Courses/Communications/COMM502... · Comm 502: Communication Theory A complex low-pass signal has a bandwidth of 200 kHz. What is](https://reader033.fdocuments.in/reader033/viewer/2022052319/5a7e385e7f8b9a66798e4938/html5/thumbnails/11.jpg)
11
Dr. Ahmed El-Mahdy
Communications Department Comm 502: Communication Theory
Impulse (Ideal) Sampling • Impulse sampling is performed by multiplying the
waveform with a unit-weight impulse train.
• The resulting sampled
waveform is given by:
Time Domain Frequency Domain
![Page 12: Comm 502: Communication Theory - GUCeee.guc.edu.eg/Courses/Communications/COMM502... · Comm 502: Communication Theory A complex low-pass signal has a bandwidth of 200 kHz. What is](https://reader033.fdocuments.in/reader033/viewer/2022052319/5a7e385e7f8b9a66798e4938/html5/thumbnails/12.jpg)
12
Dr. Ahmed El-Mahdy
Communications Department Comm 502: Communication Theory
Spectrum of Impulse Sampling
The spectrum of the impulse
sampled waveform
![Page 13: Comm 502: Communication Theory - GUCeee.guc.edu.eg/Courses/Communications/COMM502... · Comm 502: Communication Theory A complex low-pass signal has a bandwidth of 200 kHz. What is](https://reader033.fdocuments.in/reader033/viewer/2022052319/5a7e385e7f8b9a66798e4938/html5/thumbnails/13.jpg)
13
Dr. Ahmed El-Mahdy
Communications Department Comm 502: Communication Theory
Impulse Sampling with increasing sampling frequency
Sampled waveform
0
1 201
Sampled waveform
0
1 201
Sampled waveform
0
1 201
Sampled waveform
0
1 201
![Page 14: Comm 502: Communication Theory - GUCeee.guc.edu.eg/Courses/Communications/COMM502... · Comm 502: Communication Theory A complex low-pass signal has a bandwidth of 200 kHz. What is](https://reader033.fdocuments.in/reader033/viewer/2022052319/5a7e385e7f8b9a66798e4938/html5/thumbnails/14.jpg)
14
Dr. Ahmed El-Mahdy
Communications Department Comm 502: Communication Theory
Natural sampling • Natural sampling is performed by
multiplying w(t) by a train of pulses:
where
• Natural sampling takes a slice
of the waveform and the top of
the slice preserves the shape of
the waveform.
PAM using Natural sampling
k
skTtPts
)(
)()()( tstwtws
![Page 15: Comm 502: Communication Theory - GUCeee.guc.edu.eg/Courses/Communications/COMM502... · Comm 502: Communication Theory A complex low-pass signal has a bandwidth of 200 kHz. What is](https://reader033.fdocuments.in/reader033/viewer/2022052319/5a7e385e7f8b9a66798e4938/html5/thumbnails/15.jpg)
15
Dr. Ahmed El-Mahdy
Communications Department Comm 502: Communication Theory
Spectrum of Natural Sampling
• The spectrum (F.T) of the natural sampled signal can be derived as follows:
k
ss
s
s
τ
kTtPF.T*W(f)(f)W
tsF.T*W(f)(f)W
:Transform rierTaking Fou
tw(t)s(t)w
Proof Reading
ONLY
![Page 16: Comm 502: Communication Theory - GUCeee.guc.edu.eg/Courses/Communications/COMM502... · Comm 502: Communication Theory A complex low-pass signal has a bandwidth of 200 kHz. What is](https://reader033.fdocuments.in/reader033/viewer/2022052319/5a7e385e7f8b9a66798e4938/html5/thumbnails/16.jpg)
16
Dr. Ahmed El-Mahdy
Communications Department Comm 502: Communication Theory
Spectrum of Natural Sampling (Cont.)
From signals and systems, the Fourier transform of
the periodic pulse is
Then we have:
n
s
ss
n
s
ss
s
nffδ*W(f)T
nτsinc
T
τ
nffδT
nτsinc
T
τ*W(f)(f)W
][
![Page 17: Comm 502: Communication Theory - GUCeee.guc.edu.eg/Courses/Communications/COMM502... · Comm 502: Communication Theory A complex low-pass signal has a bandwidth of 200 kHz. What is](https://reader033.fdocuments.in/reader033/viewer/2022052319/5a7e385e7f8b9a66798e4938/html5/thumbnails/17.jpg)
17
Dr. Ahmed El-Mahdy
Communications Department Comm 502: Communication Theory
Spectrum of Natural Sampling (Cont.)
n
s
ss
s nffWT
nτsinc
T
τ(f)W
tionDelta Functhe ofproperty shifting the Using
Constant for each n
(not depend on the f)&
discrete
Periodic
W(f)
![Page 18: Comm 502: Communication Theory - GUCeee.guc.edu.eg/Courses/Communications/COMM502... · Comm 502: Communication Theory A complex low-pass signal has a bandwidth of 200 kHz. What is](https://reader033.fdocuments.in/reader033/viewer/2022052319/5a7e385e7f8b9a66798e4938/html5/thumbnails/18.jpg)
18
Dr. Ahmed El-Mahdy
Communications Department Comm 502: Communication Theory
Spectrum of Natural Sampling (Cont.)
-10 0 10 20 30 40 50 60 70 80 90 1000
0.2
0.4
0.6
0.8
1
f
Mag
nit
ud
e o
f W
(f)
MAGNITUDE SPECTRUM of analog signal
-10 0 10 20 30 40 50 60 70 80 90 1000
0.05
0.1
0.15
0.2
0.25
0.3
f
Mag
nit
ud
e o
f W
s(f
)
MAGNITUDE SPECTRUM of NATURAL SAMPLED PAM signal
n
s
ss
s nffWT
nτsinc
T
τ(f)W
![Page 19: Comm 502: Communication Theory - GUCeee.guc.edu.eg/Courses/Communications/COMM502... · Comm 502: Communication Theory A complex low-pass signal has a bandwidth of 200 kHz. What is](https://reader033.fdocuments.in/reader033/viewer/2022052319/5a7e385e7f8b9a66798e4938/html5/thumbnails/19.jpg)
19
Dr. Ahmed El-Mahdy
Communications Department Comm 502: Communication Theory
Top Sampling-Flat
• Easy to implement using sample and hold operation. • Flat top sampling takes a slice of the waveform but cuts off the top of the slice horizontally. The top of the slice does not preserve the shape of the waveform.
Pulse Amplitude Modulation using
Flat Top sampling
![Page 20: Comm 502: Communication Theory - GUCeee.guc.edu.eg/Courses/Communications/COMM502... · Comm 502: Communication Theory A complex low-pass signal has a bandwidth of 200 kHz. What is](https://reader033.fdocuments.in/reader033/viewer/2022052319/5a7e385e7f8b9a66798e4938/html5/thumbnails/20.jpg)
20
Dr. Ahmed El-Mahdy
Communications Department Comm 502: Communication Theory
top Sampling Waveform-Proof of Spectrum of Flat
The output of the sample and hold circuit generates the sampled
signal:
:
2/||,0
2/||,1)(
:
)(
)
issignalresultingtheofTransformFourier
t
ttpth
isshapepulsethesamplingtopflatfor
andshapepulsesamplingthedenotesthwhere
k
sss kT-h(t )w(kT(t)w
Proof Reading
ONLY
![Page 21: Comm 502: Communication Theory - GUCeee.guc.edu.eg/Courses/Communications/COMM502... · Comm 502: Communication Theory A complex low-pass signal has a bandwidth of 200 kHz. What is](https://reader033.fdocuments.in/reader033/viewer/2022052319/5a7e385e7f8b9a66798e4938/html5/thumbnails/21.jpg)
21
Dr. Ahmed El-Mahdy
Communications Department Comm 502: Communication Theory
top Sampling Waveform-Proof of Spectrum of Flat
Proof Reading
ONLY
![Page 22: Comm 502: Communication Theory - GUCeee.guc.edu.eg/Courses/Communications/COMM502... · Comm 502: Communication Theory A complex low-pass signal has a bandwidth of 200 kHz. What is](https://reader033.fdocuments.in/reader033/viewer/2022052319/5a7e385e7f8b9a66798e4938/html5/thumbnails/22.jpg)
22
Dr. Ahmed El-Mahdy
Communications Department Comm 502: Communication Theory
Spectrum of Flat Top Sampling
k
s
s
s )kf-W(fT
(f)W
fsinc
Variable
continuously
with f
Periodic
W(f)
![Page 23: Comm 502: Communication Theory - GUCeee.guc.edu.eg/Courses/Communications/COMM502... · Comm 502: Communication Theory A complex low-pass signal has a bandwidth of 200 kHz. What is](https://reader033.fdocuments.in/reader033/viewer/2022052319/5a7e385e7f8b9a66798e4938/html5/thumbnails/23.jpg)
23
Dr. Ahmed El-Mahdy
Communications Department Comm 502: Communication Theory
-10 0 10 20 30 40 50 60 70 80 90 1000
0.2
0.4
0.6
0.8
1
f
Mag
nit
ud
e o
f W
(f)
MAGNITUDE SPECTRUM of analog signal
-10 0 10 20 30 40 50 60 70 80 90 1000
5
10
15
20
f
Mag
nit
ud
e o
f W
s(f
)
MAGNITUDE SPECTRUM of Flat-topped SAMPLED PAM signal
Spectrum of Flat Top Sampling
k
s
s
s )kf-W(fT
(f)W
fsinc
![Page 24: Comm 502: Communication Theory - GUCeee.guc.edu.eg/Courses/Communications/COMM502... · Comm 502: Communication Theory A complex low-pass signal has a bandwidth of 200 kHz. What is](https://reader033.fdocuments.in/reader033/viewer/2022052319/5a7e385e7f8b9a66798e4938/html5/thumbnails/24.jpg)
24
Dr. Ahmed El-Mahdy
Communications Department Comm 502: Communication Theory
Sampling Top Sampling and Natural-Difference between Ideal, Flat
![Page 25: Comm 502: Communication Theory - GUCeee.guc.edu.eg/Courses/Communications/COMM502... · Comm 502: Communication Theory A complex low-pass signal has a bandwidth of 200 kHz. What is](https://reader033.fdocuments.in/reader033/viewer/2022052319/5a7e385e7f8b9a66798e4938/html5/thumbnails/25.jpg)
25
Dr. Ahmed El-Mahdy
Communications Department Comm 502: Communication Theory
Top Sampling-Comparison between Natural Sampling and Flat
- In flat top sampling the electronic circuit required for sampling are less complicated as compared to the one used in natural sampling because it is easier to generate rectangular pulses with fixed amplitude.
- if message reconstruction by lowpass filtering: Natural samples results in no distortion but there is a distortion when flat top pulses are involved because is multiplied by another function of frequency( )
)kf-W(f s
ff sinc)Q(
![Page 26: Comm 502: Communication Theory - GUCeee.guc.edu.eg/Courses/Communications/COMM502... · Comm 502: Communication Theory A complex low-pass signal has a bandwidth of 200 kHz. What is](https://reader033.fdocuments.in/reader033/viewer/2022052319/5a7e385e7f8b9a66798e4938/html5/thumbnails/26.jpg)
26
Dr. Ahmed El-Mahdy
Communications Department Comm 502: Communication Theory
top sampling is so common in communication -Why Flatalthough it generates spectral distortion
The reasons for using flat top sampling in communications are:
(1) The rectangular pulse (in Time Domain) is an in easy
shape to generate.
(2) The distortion in the spectrum can be corrected in the
receiver by using a low pass filter followed by an
equalizing filter as shown in the fig:
(f) W(t)
![Page 27: Comm 502: Communication Theory - GUCeee.guc.edu.eg/Courses/Communications/COMM502... · Comm 502: Communication Theory A complex low-pass signal has a bandwidth of 200 kHz. What is](https://reader033.fdocuments.in/reader033/viewer/2022052319/5a7e385e7f8b9a66798e4938/html5/thumbnails/27.jpg)
27
Dr. Ahmed El-Mahdy
Communications Department Comm 502: Communication Theory
• Questions
– What should be the sampling rate?
– Can we reconstruct the original signal after the sampling process?
![Page 28: Comm 502: Communication Theory - GUCeee.guc.edu.eg/Courses/Communications/COMM502... · Comm 502: Communication Theory A complex low-pass signal has a bandwidth of 200 kHz. What is](https://reader033.fdocuments.in/reader033/viewer/2022052319/5a7e385e7f8b9a66798e4938/html5/thumbnails/28.jpg)
28
Dr. Ahmed El-Mahdy
Communications Department Comm 502: Communication Theory
Effect of Sampling on Frequency Content of Signals
m(t)
t (sec.)
Representation of analog signal m(t) in
time domain
f (Hz)
M(f)
0 W -W
• Let assume that the frequency content of analog signal in the frequency domain is confined with W
• Define TS as the sampling interval
• Define fS as the sampling frequency
![Page 29: Comm 502: Communication Theory - GUCeee.guc.edu.eg/Courses/Communications/COMM502... · Comm 502: Communication Theory A complex low-pass signal has a bandwidth of 200 kHz. What is](https://reader033.fdocuments.in/reader033/viewer/2022052319/5a7e385e7f8b9a66798e4938/html5/thumbnails/29.jpg)
29
Dr. Ahmed El-Mahdy
Communications Department Comm 502: Communication Theory
fS>2W m(t)
f (Hz)
M(f)
W -W
TS=1/fS t (sec)
0 fS -fS fS-W fS+W -fS+W -fS-W
• By using a LPF at the receiver, it is possible to reconstruct the original signal from received samples
LPF
fcutoff
![Page 30: Comm 502: Communication Theory - GUCeee.guc.edu.eg/Courses/Communications/COMM502... · Comm 502: Communication Theory A complex low-pass signal has a bandwidth of 200 kHz. What is](https://reader033.fdocuments.in/reader033/viewer/2022052319/5a7e385e7f8b9a66798e4938/html5/thumbnails/30.jpg)
30
Dr. Ahmed El-Mahdy
Communications Department Comm 502: Communication Theory
fS=2W m(t)
f (Hz)
M(f)
W -W
TS=1/f
S
t (sec)
0 fS=2W -fS=-2W 3W -3W
• By using a LPF with fcutoff=W at the receiver, it is possible to reconstruct the original signal from received samples
LPF
fcutoff
![Page 31: Comm 502: Communication Theory - GUCeee.guc.edu.eg/Courses/Communications/COMM502... · Comm 502: Communication Theory A complex low-pass signal has a bandwidth of 200 kHz. What is](https://reader033.fdocuments.in/reader033/viewer/2022052319/5a7e385e7f8b9a66798e4938/html5/thumbnails/31.jpg)
31
Dr. Ahmed El-Mahdy
Communications Department Comm 502: Communication Theory
fS<2W m(t)
f (Hz)
M(f)
W -W
TS=1/f
S
t (sec)
0 fS -fS fS+W -fS-W
• It is no longer possible to reconstruct the original signal from received samples
-fS+W fS-W
![Page 32: Comm 502: Communication Theory - GUCeee.guc.edu.eg/Courses/Communications/COMM502... · Comm 502: Communication Theory A complex low-pass signal has a bandwidth of 200 kHz. What is](https://reader033.fdocuments.in/reader033/viewer/2022052319/5a7e385e7f8b9a66798e4938/html5/thumbnails/32.jpg)
32
Dr. Ahmed El-Mahdy
Communications Department Comm 502: Communication Theory
Sampling Theorem
• Sampling Theorem states that
A band-limited signal of finite energy which has no frequency components higher than W Hz may be completely recovered from knowledge of its samples taken at the rate of fs=2W samples per second.
• fS=2W is called the Nyquist Rate
• Ts=1/2W is called the Nyquist interval
![Page 33: Comm 502: Communication Theory - GUCeee.guc.edu.eg/Courses/Communications/COMM502... · Comm 502: Communication Theory A complex low-pass signal has a bandwidth of 200 kHz. What is](https://reader033.fdocuments.in/reader033/viewer/2022052319/5a7e385e7f8b9a66798e4938/html5/thumbnails/33.jpg)
33
Dr. Ahmed El-Mahdy
Communications Department Comm 502: Communication Theory
According to the Nyquist theorem, the
sampling rate must be
at least 2 times the highest frequency
contained in the signal.
Then
![Page 34: Comm 502: Communication Theory - GUCeee.guc.edu.eg/Courses/Communications/COMM502... · Comm 502: Communication Theory A complex low-pass signal has a bandwidth of 200 kHz. What is](https://reader033.fdocuments.in/reader033/viewer/2022052319/5a7e385e7f8b9a66798e4938/html5/thumbnails/34.jpg)
34
Dr. Ahmed El-Mahdy
Communications Department Comm 502: Communication Theory
Practical Sampling Rates
• Speech: » Telephone quality speech has a bandwidth of 4
kHz
» Most digital telephone systems sample at 8000 samples/sec
• Audio: » The highest frequency the human ear can hear
is approximately 15 kHz
» CDs sample at rate 44,000 samples/sec
• Video: » The human eye requires samples at a rate of at
least 20 frames/sec to achieve smooth motion
![Page 35: Comm 502: Communication Theory - GUCeee.guc.edu.eg/Courses/Communications/COMM502... · Comm 502: Communication Theory A complex low-pass signal has a bandwidth of 200 kHz. What is](https://reader033.fdocuments.in/reader033/viewer/2022052319/5a7e385e7f8b9a66798e4938/html5/thumbnails/35.jpg)
35
Dr. Ahmed El-Mahdy
Communications Department Comm 502: Communication Theory
A complex low-pass signal has a bandwidth of 200 kHz. What is the minimum
sampling frequency for this signal? If the signal is multiplied by a a cos with
frequency fc=1 MHz What is the minimum sampling frequency in this case?
Solution The bandwidth of a low-pass signal is between 0 and f,
where f is the maximum frequency in the signal.
Therefore, we can sample this signal at 2 times the
highest frequency (200 kHz). The sampling rate is
therefore 400 KHz.
-The maximum frequency in the signal now is 1200 kHz,
then the minimum sampling rate is: 2400 kHz
Example
![Page 36: Comm 502: Communication Theory - GUCeee.guc.edu.eg/Courses/Communications/COMM502... · Comm 502: Communication Theory A complex low-pass signal has a bandwidth of 200 kHz. What is](https://reader033.fdocuments.in/reader033/viewer/2022052319/5a7e385e7f8b9a66798e4938/html5/thumbnails/36.jpg)
36
Dr. Ahmed El-Mahdy
Communications Department Comm 502: Communication Theory
-200 kHz 200 kHz
1200 kHz 800 kHz -1200 kHz
-1000 kHz
-800 kHz
1000 kHz
Low Pass signal
![Page 37: Comm 502: Communication Theory - GUCeee.guc.edu.eg/Courses/Communications/COMM502... · Comm 502: Communication Theory A complex low-pass signal has a bandwidth of 200 kHz. What is](https://reader033.fdocuments.in/reader033/viewer/2022052319/5a7e385e7f8b9a66798e4938/html5/thumbnails/37.jpg)
37
Dr. Ahmed El-Mahdy
Communications Department Comm 502: Communication Theory
Thank You