1. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Electrical Communications Systems ECE.09.331 Spring 2011 Shreekanth Mandayam ECE Department Rowan University http://engineering.rowan.edu/~shreek/spring11/ecomms/ Lab 1: Pre-lab Instruction January 24, 2011

2. S. Mandayam/ ECOMMS/ECE Dept./Rowan University ECOMMS: Topics

3. S. Mandayam/ ECOMMS/ECE Dept./Rowan UniversityPlan 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. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Lab 1 Matlab code >> Matlab code >> HP 33120A Arb Fn Gn Mathematical Waveform Electrical Signal Speaker Agilent Infinium Oscilloscope Signal Spectrum Computer

5. S. Mandayam/ ECOMMS/ECE Dept./Rowan UniversityRecall Waveforms Deterministic Stochastic Signal (desired) Noise (undesired) Probability Random Experiment Random Event outcome

6. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Communications Waveforms “Random” noise Hallelujah chorus

7. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Random 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 Random Variable Random Event Real Number Appendix B Prob & RV

8. S. Mandayam/ ECOMMS/ECE Dept./Rowan University The Probability Density Function (PDF) of a Random Variable x f(x) a b a b

9. S. Mandayam/ ECOMMS/ECE Dept./Rowan University PDF Model: The 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 x f(x) m

10. S. Mandayam/ ECOMMS/ECE Dept./Rowan University 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. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Examples of Normal Distribution 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')

12. S. Mandayam/ ECOMMS/ECE Dept./Rowan University 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,  2 ) =  N(0,1) Use this for generating normally distributed r.v.’s of any variance Matlab function: randn Variance Power (how?)

13. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Why are we doing this? Transfer Characteristic h(x) Input pdf f x (x) Output pdf f y (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. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Lab Project 1: Waveform Synthesis and Spectral Analysis Part 1: Digital Waveform Synthesis http://users.rowan.edu/~shreek/spring11/ecomms/ lab1.html

15. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Recall: CFT Continuous Fourier Transform (CFT) Frequency, [Hz] Amplitude Spectrum Phase Spectrum Inverse Fourier Transform (IFT)

16. S. Mandayam/ ECOMMS/ECE Dept./Rowan University 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 time intervals Equal frequency intervals n = 0, 1, 2,….., N-1 k = 0, 1, 2,….., N-1

17. S. Mandayam/ ECOMMS/ECE Dept./Rowan University 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) n=0 1 2 3 4 n=N f=0 f = f s Need to know f s

18. S. Mandayam/ ECOMMS/ECE Dept./Rowan University 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 f s /2 f = f s

19. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Part 2: CFT, DFT and Sampling This is homework!!! t in ms w(t) 0.6 0.7 1.0 1V 0V

20. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Part 3: AM and FM Spectra AM s(t) = A c [1 + A m cos(2  f m t)]cos(2  f c t) FM s(t) = A c cos[2  f c t +  f A m sin(2  f m t)] t s(t) t

21. S. Mandayam/ ECOMMS/ECE Dept./Rowan University 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. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Components of the Electrocardiogram P-WaveDepolarization of the atria P-R IntervalDepolarization of the atria, and delay at AV junction QRS ComplexDepolarization of the ventricles S-T SegmentPeriod between ventricular depolarization and repolarization T-WaveRepolarization of the ventricles R-R IntervalTime between two ventricular depolarizations A “Normal” ECG Heart Rate60 - 90 bpm PR Interval0.12 - 0.20 sec QRS Duration0.06 - 0.10 sec QT Interval (QTc < 0.40 sec) ECG Signal P wave T wave Q R S

23. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Lab Project 1: Waveform Synthesis and Spectral Analysis http://users.rowan.edu/~shreek/spring11/ecomms/ lab1.html

24. S. Mandayam/ ECOMMS/ECE Dept./Rowan UniversitySummary