Electrical Communications Systems ECE.09.331 Spring 2007

1. Electrical Communications SystemsECE.09.331Spring 2007 Lecture 2aJanuary 23, 2007 Shreekanth Mandayam ECE Department Rowan University http://engineering.rowan.edu/~shreek/spring07/ecomms/

2. Plan • Recall: • Intoduction to Information Theory • Properties of Signals and Noise • Terminology • Power and Energy Signals • Recall: Fourier Analysis • Fourier Series of Periodic Signals • Continuous Fourier Transform (CFT) and Inverse Fourier Transform (IFT) • Amplitude and Phase Spectrum • Properties of Fourier Transforms

3. ECOMMS: Topics

4. Measures of Information • Definitions • Probability • Information • Entropy • Source Rate • Recall: Shannon’s Theorem • If R < C = B log2(1 + S/N), then we can have error-free transmission in the presence of noise MATLAB DEMO: entropy.m

5. Digital Finite set of messages (signals) inexpensive/expensive privacy & security data fusion error detection and correction More bandwidth More overhead (hw/sw) Analog Continuous set of messages (signals) Legacy Predominant Inexpensive Communications Systems

6. ECOMMS: Topics

7. Signal Properties: Terminology • Waveform • Time-average operator • Periodicity • DC value • Power • RMS Value • Normalized Power • Normalized Energy

8. Power Signal Infinite duration Normalized power is finite and non-zero Normalized energy averaged over infinite time is infinite Mathematically tractable Energy Signal Finite duration Normalized energy is finite and non-zero Normalized power averaged over infinite time is zero Physically realizable Power and Energy Signals • Although “real” signals are energy signals, we analyze them pretending they are power signals!

9. The Decibel (dB) • Measure of power transfer • 1 dB = 10 log10 (Pout / Pin) • 1 dBm = 10 log10 (P / 10-3) where P is in Watts • 1 dBmV = 20 log10 (V / 10-3) where V is in Volts

10. ECOMMS: Topics

11. Fourier Series Infinite sum of sines and cosines at different frequencies Any periodic power signal Fourier Series Fourier Series Applet: http://www.gac.edu/~huber/fourier/

12. |W(n)| -3f0 -2f0 -f0 f0 2f0 3f0 f Fourier Series Exponential Representation Periodic Waveform w(t) t T0 2-Sided Amplitude Spectrum f0 = 1/T0; T0 = period

13. Fourier Transform • Fourier Series of periodic signals • finite amplitudes • spectral components separated by discrete frequency intervals of f0 = 1/T0 • We want a spectral representation for aperiodic signals • Model an aperiodic signal as a periodic signal with T0 ----> infinity Then, f0 -----> 0 The spectrum is continuous!

14. Continuous Fourier Transform Aperiodic Waveform • We want a spectral representation for aperiodic signals • Model an aperiodic signal as a periodic signal with T0 ----> infinity Then, f0 -----> 0 The spectrum is continuous! w(t) t T0 Infinity |W(f)| f f0 0

15. Continuous Fourier Transform (CFT) Frequency, [Hz] Phase Spectrum Amplitude Spectrum Inverse Fourier Transform (IFT) Definitions See p. 45 Dirichlet Conditions

16. Properties of FT’s • If w(t) is real, then W(f) = W*(f) • If W(f) is real, then w(t) is even • If W(f) is imaginary, then w(t) is odd • Linearity • Time delay • Scaling • Duality See p. 50 FT Theorems

17. Summary