Electrical Communication Systems ECE.09.331 Spring 2009

1. Electrical Communication SystemsECE.09.331Spring 2009 Lecture 3aFebruary 3, 2009 Shreekanth Mandayam ECE Department Rowan University http://engineering.rowan.edu/~shreek/spring09/ecomms/

2. Plan • 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. ECOMMS: Topics

4. 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/

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

6. 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!

7. 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

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

9. 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. CFT’s of Common Waveforms • Impulse (Dirac Delta) • Sinusoid • Rectangular Pulse Matlab Demo: recpulse.m

11. Summary