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Electrical Communication Systems ECE.09.331 Spring 2008

Explore Fourier series, transforms, spectral components, and common waveforms in ECE. Learn about amplitude and phase spectrum, properties of Fourier transforms, and practical applications in signal processing. MATLAB demo included.

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Electrical Communication Systems ECE.09.331 Spring 2008

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  1. Electrical Communication SystemsECE.09.331Spring 2008 Lecture 3aFebruary 5, 2008 Shreekanth Mandayam ECE Department Rowan University http://engineering.rowan.edu/~shreek/spring08/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. |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

  5. 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!

  6. 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

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

  8. 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

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

  10. Summary

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