Electrical Communications Systems ECE.09.433 Spring 2019 Lab 1: Pre-lab Instruction January 29, 2019 Shreekanth Mandayam ECE Department Rowan University http://engineering.rowan.edu/~shreek/spring19/ecomms/

ECOMMS: Topics

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

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

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

Communications Waveforms “Random” noise Hallelujah chorus

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

The Probability Density Function (PDF) of a Random Variable x f(x) a b

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

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')

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')

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?)

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

Lab Project 1: Waveform Synthesis and Spectral Analysis http://users.rowan.edu/~shreek/spring19/ecomms/labs/lab1.html

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

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

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=0 1 2 3 4 n=N f=0 f = fs

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=0 N/2 n=N f=0 fs/2 f = fs

Part 2: CFT, DFT and Sampling This is homework!!! t in ms w(t) 0.6 0.7 1.0 1V 0V

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)

Summary