S. Mandayam/ ECOMMS/ECE Dept./Rowan University
Electrical Electrical Communications SystemsCommunications Systems
ECE.09.331ECE.09.331 Spring 2011Spring 2011
Shreekanth MandayamECE Department
Rowan University
http://engineering.rowan.edu/~shreek/spring11/ecomms/
Lab 1: Pre-lab InstructionLab 1: Pre-lab InstructionJanuary 24, 2011January 24, 2011
S. Mandayam/ ECOMMS/ECE Dept./Rowan University
ECOMMS: TopicsECOMMS: Topics
Probability
Inform ation
Entropy
Channel Capacity
Discrete
Pow er & Energy Signals
Continuous Fourier Transform
Discrete Fourier Transform
Baseband and Bandpass Signals
Com plex Envelope
Gaussian Noise & SNR
Random VariablesNoise Calculations
Continuous
Signals
AMSw itching M odulator
Envelop Detector
DSB-S CProduct M odulatorCoherent Detector
Costas Loop
SSBW eaver's MethodPhasing M ethod
Frequency M ethod
Frequency & Phase M odulationNarrowband/WidebandVCO & Slope Detector
PLL
Analog
Source EncodingHuffm an codes
Error-control EncodingHam m ing Codes
Sam plingPAM
QuantizationPCM
Line Encoding
Tim e Division M uxT1 (DS1) Standards
Packet Sw itchingEthernet
ISO 7-Layer Protocol
BasebandCODEC
ASKPSKFSK
BPSK
QPSK
M -ary PSK
QAM
BandpassM ODEM
DigitalDigital Com m Transceiver
Systems
Electrical Comm unication Systems
S. Mandayam/ ECOMMS/ECE Dept./Rowan University
PlanPlan• Recall:
• Deterministic and Stochastic Waveforms
• Random Variables• PDF and CDF• Gaussian PDF
• Noise model
• Lab Project 1• Part 1: Digital synthesis of arbitrary waveforms with specified
SNR
• Recall: • How to generate frequency axis in DFT
• Lab Project 1• Part 2: CFT, Sampling and DFT (Homework!!!)• Part 3: Spectral analysis of AM and FM signals• Part 4: Spectral analysis of an ECG signal
S. Mandayam/ ECOMMS/ECE Dept./Rowan University
Lab 1Lab 1
Matlab code>>>>>>>>
0 0.1 0.2 0.3 0.4-10
-5
0
5
10
time in s
Am
plitu
de in
Vol
ts
3400 3600 3800 4000 4200 44000
2
4
6
8x 10
-3
f in Hz
Am
plitu
de in
Vol
ts
Matlab code>>>>>>>>
HP 33120AArb Fn Gn
MathematicalWaveform
ElectricalSignal
Speaker
Agilent InfiniumOscilloscope
Signal Spectrum
Computer Computer
S. Mandayam/ ECOMMS/ECE Dept./Rowan University
RecallRecallWaveforms
Deterministic Stochastic
Signal(desired)
Noise(undesired)
• Probability
nn
AP A
nlim)(
Random Experiment
Random Event
outcome
S. Mandayam/ ECOMMS/ECE Dept./Rowan University
Communications WaveformsCommunications Waveforms
0 200 400 600 800 1000-4
-2
0
2
4
time in ms
Am
plitu
de in
Vol
ts
0 2 4 6 8 10-1
-0.5
0
0.5
1
time in s
Am
plitu
de in
Vol
ts
“Random” noise Hallelujah chorus
0 0.1 0.2 0.3 0.4-10
-5
0
5
10
time in s
Am
plitu
de in
Vol
ts
6102cos10)( ttw
S. Mandayam/ ECOMMS/ECE Dept./Rowan University
Random VariableRandom Variable
Random Event,
s
Real Number,
a
Random Variable,
X
• Definition: Let E be an experiment and S be the set of all possible outcomes associated with the experiment. A function, X, assigning to every element s S, a real number, a, is called a random variable.
X(s) = a
RandomVariable
RandomEvent
RealNumber
Appendix BProb & RV
S. Mandayam/ ECOMMS/ECE Dept./Rowan University
The Probability Density Function The Probability Density Function (PDF) of a Random Variable (PDF) of a Random Variable
b
adxxfaFbFbxaP )()()()(
x
f(x)
a b0 200 400 600 800 1000
-4
-2
0
2
4
time in ms
Am
plitu
de in
Vol
ts
a
b
1)( dxxf
S. Mandayam/ ECOMMS/ECE Dept./Rowan University
PDF Model: PDF Model: The Gaussian Random VariableThe Gaussian Random Variable
• The most important pdf model• Used to model signal, noise……..
• m: mean; 2: variance
• Also called a Normal Distribution
• Central limit theorem
),(21
)( 22 2
2
mNexf
mx
x
f(x)
m
2
1
S. Mandayam/ ECOMMS/ECE Dept./Rowan University
Examples of Normal DistributionExamples of Normal Distribution
-10 -5 0 5 100
0.1
0.2
0.3
0.4
x in Volts
f(x)
0 200 400 600 800 1000-10
-5
0
5
10
time in ms
Am
plitu
de in
Vol
tsN(+3,1)
N(-3,1)
>> plot(x,pdf('Normal',x,-3,1),'b', x,pdf('Normal',x,3,1),'r' ) >> t=[0:999]';
>> plot(t,randn(1,1000)-3,'b',t,randn(1,1000)+3,'r')
S. Mandayam/ ECOMMS/ECE Dept./Rowan University
Examples of Normal DistributionExamples of Normal Distribution
-10 -5 0 5 100
0.1
0.2
0.3
0.4
x in Volts
f(x)
0 200 400 600 800 1000-4
-2
0
2
4
time in ms
Am
plitu
de in
Vol
ts
0 200 400 600 800 1000-10
-5
0
5
10
time in ms
Am
plitu
de in
Vol
ts
N(0,1)
N(0,4)>> plot(x,pdf('Normal',x,0,1),'b', x,pdf('Normal',x,0,4),'r' )
>> plot(randn(1,1000))
>> plot(2*randn(1,1000),'r')
S. Mandayam/ ECOMMS/ECE Dept./Rowan University
Generating Normally Distributed Generating Normally Distributed Random VariablesRandom Variables
• Most math software provides you functions to generate - • N(0,1): zero-mean, unit-variance, Gaussian RV
• Theorem:• N(0,2) = N(0,1)• Use this for generating normally distributed r.v.’s of any
variance
• Matlab function:• randn
• Variance Power (how?)
S. Mandayam/ ECOMMS/ECE Dept./Rowan University
Why are we doing this?Why are we doing this?
Transfer Characteristic
h(x)
Input pdffx(x)
Output pdffy(y)
• For many situations, we can “model” the pdf using standard functions
• By studying the functional forms, we can predict the expected values of the random variable (mean, variance, etc.)
• We can predict what happens when the r.v. passes through a system
S. Mandayam/ ECOMMS/ECE Dept./Rowan University
Lab Project 1:Lab Project 1:Waveform Synthesis and Waveform Synthesis and
Spectral Analysis Spectral Analysis
Part 1: Digital Waveform SynthesisPart 1: Digital Waveform Synthesis
http://users.rowan.edu/~shreek/spring11/ecomms/lab1.html
S. Mandayam/ ECOMMS/ECE Dept./Rowan University
Recall: CFTRecall: CFT
)f(j
ft2j
e )f(W)f(W
)f(Y j)f(X)f(W
dte )t(w)t(w)f(W
F
Continuous Fourier Transform (CFT)
Frequency, [Hz]
AmplitudeSpectrum
PhaseSpectrum
dfe )f(W)f(W)t(w ft2j1-
F
Inverse Fourier Transform (IFT)
S. Mandayam/ ECOMMS/ECE Dept./Rowan University
Recall: DFTRecall: DFT• Discrete Domains
• Discrete Time: k = 0, 1, 2, 3, …………, N-1• Discrete Frequency: n = 0, 1, 2, 3, …………, N-1
• Discrete Fourier Transform
• Inverse DFT
Equal time intervals
Equal frequency intervals
1N
0k
nkN2
j;e ]k[x]n[X
1N
0n
nkN2
j;e ]n[X
N1
]k[x
n = 0, 1, 2,….., N-1
k = 0, 1, 2,….., N-1
S. Mandayam/ ECOMMS/ECE Dept./Rowan University
How to get the frequency axis in the DFTHow to get the frequency axis in the DFT
• The DFT operation just converts one set of number, x[k] into another set of numbers X[n] - there is no explicit definition of time or frequency
• How can we relate the DFT to the CFT and obtain spectral amplitudes for discrete frequencies?
1N
0
x
.
x
]k[x
1N
0
X
.
X
]n[X
(N-point FFT)
n=0 1 2 3 4 n=N
f=0 f = fs
N
fs
Need to know fs
S. Mandayam/ ECOMMS/ECE Dept./Rowan University
DFT PropertiesDFT Properties• DFT is periodic
X[n] = X[n+N] = X[n+2N] = ………
• I-DFT is also periodic!
x[k] = x[k+N] = x[k+2N] = ……….
• Where are the “low” and “high” frequencies on the DFT spectrum?
n=0 N/2 n=N
f=0 fs/2 f = fs
S. Mandayam/ ECOMMS/ECE Dept./Rowan University
Part 2: CFT, DFT and Part 2: CFT, DFT and SamplingSampling
• This is homework!!!
tin ms
w(t)
0.6 0.7 1.0
1V
0V
S. Mandayam/ ECOMMS/ECE Dept./Rowan University
Part 3: AM and FM SpectraPart 3: AM and FM Spectra
AMs(t) = Ac[1 + Amcos(2fmt)]cos(2fct)
FMs(t) = Accos[2fct + f Amsin(2fmt)]
t
s(t)
t
s(t)
S. Mandayam/ ECOMMS/ECE Dept./Rowan University
Part 4: ECG SignalsPart 4: ECG Signals• This experiment must be conducted with the instructor present at all times
when you are obtaining the ECG readings.• The procedure that has been outlined below has been determined to be safe
for this laboratory. • You must use an isolated power supply for powering the instrumentation
amplifier.• You must use a 1-X scope probe for recording the amplifier output on the
oscilloscope. • This objective of this experiment is compute the amplitude-frequency spectrum
of real data - this experiment does not represent a true medical study; reading an ECG effectively takes considerable medical training. Therefore, do not be alarmed if your data appears"different" from those of your partners.
• If you observe any allergic reactions when you attach the electrodes (burning sensation, discomfort), remove them and rinse the area with water.
• If, for any reason, you do not want to participate in this experiment, obtain recorded ECG data from your instructor.
S. Mandayam/ ECOMMS/ECE Dept./Rowan University
Components of the Electrocardiogram
P-Wave Depolarization of the atriaP-R Interval Depolarization of the atria, and delay at AV junctionQRS Complex Depolarization of the ventriclesS-T Segment Period between ventricular depolarization and repolarizationT-Wave Repolarization of the ventriclesR-R Interval Time between two ventricular depolarizations
A “Normal” ECGHeart Rate 60 - 90 bpm PR Interval 0.12 - 0.20 sec QRS Duration 0.06 - 0.10 sec QT Interval (QTc < 0.40 sec)
ECG SignalECG Signal
P wave
T wave
Q
R
S
S. Mandayam/ ECOMMS/ECE Dept./Rowan University
Lab Project 1:Lab Project 1:Waveform Synthesis and Waveform Synthesis and
Spectral Analysis Spectral Analysis
http://users.rowan.edu/~shreek/spring11/ecomms/lab1.html
S. Mandayam/ ECOMMS/ECE Dept./Rowan University
SummarySummary
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