1. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Electrical Communication Systems ECE.09.331 Spring 2009 Shreekanth Mandayam ECE Department Rowan University http://engineering.rowan.edu/~shreek/spring09/ecomms/ Lecture 3a February 3, 2009

2. S. Mandayam/ ECOMMS/ECE Dept./Rowan UniversityPlan 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 CFTs of Common Waveforms Impulse (Dirac Delta) Rectangular pulse Sinusoid

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

4. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Fourier Series Fourier Series Applet: http://www.gac.edu/~huber/fourier/ Any periodic power signal Infinite sum of sines and cosines at different frequencies Fourier Series

5. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Recall: Fourier Series Exponential Representation Periodic Waveform w(t) t |W(n)| f -3f 0 -2f 0 -f 0 f 0 2f 0 3f 0 2-Sided Amplitude Spectrum f 0 = 1/T 0 ; T 0 = period T0T0

6. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Fourier Transform Fourier Series of periodic signals finite amplitudes spectral components separated by discrete frequency intervals of f 0 = 1/T 0 We want a spectral representation for aperiodic signals Model an aperiodic signal as a periodic signal with T 0 ----> infinity Then, f 0 -----> 0 The spectrum is continuous!

7. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Continuous Fourier Transform We want a spectral representation for aperiodic signals Model an aperiodic signal as a periodic signal with T 0 ----> infinity Then, f 0 -----> 0 The spectrum is continuous! t T 0 Infinity w(t) Aperiodic Waveform |W(f)| f f 0 0

8. S. Mandayam/ ECOMMS/ECE Dept./Rowan UniversityDefinitions Continuous Fourier Transform (CFT) Frequency, [Hz] Amplitude Spectrum Phase Spectrum Inverse Fourier Transform (IFT) See p. 45 Dirichlet Conditions

9. S. Mandayam/ ECOMMS/ECE Dept./Rowan University 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

10. S. Mandayam/ ECOMMS/ECE Dept./Rowan University CFT’s of Common Waveforms Impulse (Dirac Delta) Sinusoid Rectangular Pulse Matlab Demo: recpulse.m

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