Electrical Communications Systems ECE.09.331 Spring 2010

1. Electrical Communications SystemsECE.09.331Spring 2010 Lab 1: Pre-lab InstructionJanuary 25, 2010 Shreekanth Mandayam ECE Department Rowan University http://engineering.rowan.edu/~shreek/spring10/ecomms/

2. ECOMMS: Topics

3. Plan • 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

4. Lab 1 Agilent Infinium Oscilloscope HP 33120A Arb Fn Gn Computer Computer Matlab code >> >> >> >> Matlab code >> >> >> >> Electrical Signal Speaker Mathematical Waveform Signal Spectrum

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

6. Communications Waveforms Hallelujah chorus “Random” noise

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

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

9. f(x) x m 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

10. Examples of Normal Distribution N(+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')

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. Lab Project 1:Waveform Synthesis and Spectral Analysis Part 1: Digital Waveform Synthesis http://users.rowan.edu/~mease/Ecomms/lab1.html

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

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

16. Equal time intervals Recall: 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 frequency intervals n = 0, 1, 2,….., N-1 k = 0, 1, 2,….., N-1

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

18. n=0 N/2 n=N f=0 fs/2 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?

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

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

21. Part 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.

22. R T wave P wave Q S ECG Signal Components of the Electrocardiogram P-Wave Depolarization of the atria P-R Interval Depolarization of the atria, and delay at AV junction QRS Complex Depolarization of the ventricles S-T Segment Period between ventricular depolarization and repolarization T-Wave Repolarization of the ventricles R-R Interval Time between two ventricular depolarizations A “Normal” ECG Heart Rate 60 - 90 bpm PR Interval 0.12 - 0.20 sec QRS Duration 0.06 - 0.10 sec QT Interval (QTc < 0.40 sec)

23. Lab Project 1:Waveform Synthesis and Spectral Analysis http://users.rowan.edu/~mease/Ecomms/lab1.html

24. Summary