1. Electrical Communications Systems ECE.09.433 Spring 2019
Lab 1: Pre-lab Instruction January 29, 2019
Shreekanth Mandayam
ECE Department
Rowan University

2. ECOMMS: Topics

3. Plan Recall: Random Variables Lab Project 1
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
How to generate frequency axis in DFT
Part 2: CFT, Sampling and DFT (Homework!!!)
Part 3: Spectral analysis of AM and FM signals
Part 4: Spectral analysis of an ECG signal

4. Lab 1 Oscilloscope Computer Computer Arb Fn Gn Electrical Signal
Matlab code
>>
Matlab code
>>
Electrical
Signal
Speaker
Mathematical
Waveform
Signal Spectrum

5. Recall Probability Random Experiment Random Event Waveforms
Deterministic
Stochastic
Signal
(desired)
Noise
(undesired)
Probability
Random Experiment
outcome
Random Event

6. Communications Waveforms
“Random” noise
Hallelujah chorus

7. Random Variable Real Random Number, Event, Variable, a s 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
Random
Variable
Real
Number
Appendix B
Prob & RV
Random
Event

8. The Probability Density Function (PDF) of a Random Variablex
f(x) a b

9. PDF Model: The Gaussian Random Variable
The most important pdf model
Used to model signal, noise……..
m: mean; s2: variance
Also called a Normal Distribution
Central limit theorem x
f(x) m

10. Examples of Normal Distribution
>> 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')

11. Examples of Normal Distribution
>> plot(randn(1,1000))
N(0,1)
>> plot(x,pdf('Normal',x,0,1),'b',
x,pdf('Normal',x,0,4),'r' )
N(0,4)
>> plot(2*randn(1,1000),'r')

12. Generating Normally Distributed Random Variables
Most math software provides you functions to generate -
N(0,1): zero-mean, unit-variance, Gaussian RV
Theorem:
N(0,s2) = sN(0,1)
Use this for generating normally distributed r.v.’s of any variance
Matlab function:
randn
Variance Power (how?)

13. Why are we doing this? Transfer Characteristic Input pdf Output pdf
h(x)
Input pdf
fx(x)
Output pdf
fy(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

14. Lab Project 1: Waveform Synthesis and Spectral Analysis

15. Recall: CFT Continuous Fourier Transform (CFT)
Frequency, [Hz]
Amplitude
Spectrum
Phase
Inverse Fourier Transform (IFT)

16. Recall: DFT Discrete Domains Discrete Fourier Transform Inverse DFT
Equal time intervals
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 frequency intervals
n = 0, 1, 2,….., N-1
k = 0, 1, 2,….., N-1

17. How 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?
(N-point FFT)
Need to
know fs
n= n=N
f= f = fs

18. DFT 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= N/ n=N
f= fs/ f = fs

19. Part 2: CFT, DFT and Sampling
This is homework!!! t
in ms
w(t) 1V 0V

20. Part 3: AM and FM Spectra AM FM s(t) = Ac[1 + Amcos(2pfmt)]cos(2pfct)
s(t) = Accos[2pfct + bf Amsin(2pfmt)] t
s(t) t
s(t)

21. Summary