1. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Electrical Communications Systems ECE.09.331 Spring 2009 Shreekanth Mandayam ECE Department Rowan University http://engineering.rowan.edu/~shreek/spring09/ecomms/ Lecture 4a February 10, 2009

2. S. Mandayam/ ECOMMS/ECE Dept./Rowan UniversityPlan CFT’s for periodic waveforms Sampling Time-limited and Band-limited waveforms Nyquist Sampling Impulse Sampling Dimensionality Theorem Discrete Fourier Transform (DFT) Fast Fourier Transform (FFT) Relation between CFT and DFT Baseband and Bandpass Signals Modulation Battle Plan for Analyzing Comm Systems

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

4. S. Mandayam/ ECOMMS/ECE Dept./Rowan University CFT for Periodic Signals Recall: CFT: Aperiodic Signals FS: Periodic Signals We want to get the CFT for a periodic signal What is ?

5. S. Mandayam/ ECOMMS/ECE Dept./Rowan University CFT for Periodic Signals Sine Wave w(t) = A sin (2  f 0 t) Square Wave A -A T 0 /2 T 0 Instrument Demo

6. S. Mandayam/ ECOMMS/ECE Dept./Rowan UniversitySampling Time-limited waveform w(t) = 0; |t| > T Band-limited waveform W(f)= F {(w(t)}=0; |f| > B -T T w(t) t -B B W(f) f Can a waveform be both time-limited and band-limited?

7. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Nyquist Sampling Theorem Any physical waveform can be represented by where If w ( t ) is band-limited to B Hz and

8. S. Mandayam/ ECOMMS/ECE Dept./Rowan University What does this mean? 1/f s 2/f s 3/f s 4/f s 5/f s w(t) t a 3 = w(3/f s ) If then we can reconstruct w(t) without error by summing weighted, delayed sinc pulses weight = w(n/f s ) delay = n/f s We need to store only “samples” of w(t), i.e., w(n/f s ) The sinc pulses can be generated as needed (How?) Matlab Demo: sampling.m

9. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Impulse Sampling How do we mathematically represent a sampled waveform in the Time Domain? Frequency Domain?

10. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Sampling: Spectral Effect w(t) t w s (t ) t f -B 0 B |W(f)| f |W s (f) | -2f s -f s 0 f s 2 f s (-f s -B) -(f s +B) -B B (f s -B) (f s +B) F F Original Sampled

11. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Spectral Effect of Sampling Spectrum of a “sampled” waveform Spectrum of the “original” waveform replicated every f s Hz =

12. S. Mandayam/ ECOMMS/ECE Dept./Rowan UniversityAliasing If f s < 2B, the waveform is “undersampled” “aliasing” or “spectral folding” How can we avoid aliasing? Increase f s “Pre-filter” the signal so that it is bandlimited to 2B < f s

13. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Dimensionality Theorem A real waveform can be completely specified by N = 2BT 0 independent pieces of information over a time interval T 0 N: Dimension of the waveform B: Bandwidth BT 0 : Time-Bandwidth Product Memory calculation for storing the waveform f s >= 2B At least N numbers must be stored over the time interval T0 = n/f s

14. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Discrete Fourier Transform (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

15. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Importance of the DFT Allows time domain / spectral domain transformations using discrete arithmetic operations Computational Complexity Raw DFT: N 2 complex operations (= 2N 2 real operations) Fast Fourier Transform (FFT): N log 2 N real operations Fast Fourier Transform (FFT) Cooley and Tukey (1965), ‘Butterfly Algorithm”, exploits the periodicity and symmetry of e -j2  kn/N VLSI implementations: FFT chips Modern DSP

16. 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

17. 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

18. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Relation between CFT and DFT Windowing Sampling Generation of Periodic Samples

19. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Baseband and Bandpass Signals Baseband signals: spectral magnitude is non-zero only near the origin and is zero (or negligible) elsewhere Bandpass signals: spectral magnitude is non-zero only near the vicinity of f = ± f c, were f c >> 0 - f m 0 +f m W(f) f - f c 0 +f c W(f) f AF Signals RF Signals Carrier Frequency

20. S. Mandayam/ ECOMMS/ECE Dept./Rowan UniversityModulation Modulating Signal Message Signal m(t) (Baseband) Modulated Signal s(t) (Bandpass) Information Modulation Frequency Translation What is it? How is it done? Modulator message: m(t) carrier: c(t) s(t): radio signal

21. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Why Modulate? Antenna size considerations Narrow banding Frequency multiplexing Common processing

22. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Battle Plan for Analyzing any Comm. Sys. Signals Systems Complex Envelope Time Domain Spectrum Power Performance Transmitters Receivers Standards Modulation Index Efficiency Bandwidth Noise

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