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Electrical Communication Systems ECE.09.433 Spring 2019

Explore properties of signals and noise in digital and analog communication systems. Learn about Fourier transforms, signal terminology, power signals, energy signals, and the Decibel measurement. Understand Continuous Fourier Transform (CFT) and Discrete Fourier Transform (DFT) for signal analysis. Delve into the importance and implementation of Fast Fourier Transform (FFT) in communications. MATLAB demonstrations and theoretical concepts covered.

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Electrical Communication Systems ECE.09.433 Spring 2019

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  1. Electrical Communication SystemsECE.09.433Spring 2019 Lecture 2aJanuary 29, 2019 Shreekanth Mandayam ECE Department Rowan University http://engineering.rowan.edu/~shreek/spring19/ecomms/

  2. Plan • Digital and Analog Communications Systems • Properties of Signals and Noise • Terminology • Power and Energy Signals • Recall: Continuous Fourier Transform (CFT) Discrete Fourier Transform (DFT) • Recall: CFT’s (spectra) of common waveforms • Impulse • Sinusoid • Rectangular Pulse

  3. ECOMMS: Topics

  4. 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

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

  6. 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!

  7. 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

  8. Continuous Fourier Transform (CFT) Frequency, [Hz] Phase Spectrum Amplitude Spectrum Inverse Fourier Transform (IFT) Continuous Fourier Transform See p. 46 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. 52 FT Theorems

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

  11. FS: Periodic Signals CFT: Aperiodic Signals CFT for Periodic Signals Recall: • We want to get the CFT for a periodic signal • What is ?

  12. Equal time intervals 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 frequency intervals n = 0, 1, 2,….., N-1 k = 0, 1, 2,….., N-1

  13. Importance of the DFT • Allows time domain / spectral domain transformations using discrete arithmetic operations • Computational Complexity • Raw DFT: N2 complex operations (= 2N2 real operations) • Fast Fourier Transform (FFT): N log2 N real operations • Fast Fourier Transform (FFT) • Cooley and Tukey (1965), ‘Butterfly Algorithm”, exploits the periodicity and symmetry of e-j2pkn/N • VLSI implementations: FFT chips • Modern DSP

  14. n=0 1 2 3 4 n=N f=0 f = fs 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) Need to know fs

  15. Summary

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