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S. Mandayam/ ECOMMS/ECE Dept./Rowan University Electrical Communication Systems ECE.09.331 Spring 2008 Shreekanth Mandayam ECE Department Rowan University http://engineering.rowan.edu/~shreek/spring08/ecomms/ Lecture 2b January 30, 2008

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S. Mandayam/ ECOMMS/ECE Dept./Rowan UniversityPlan 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

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

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S. Mandayam/ ECOMMS/ECE Dept./Rowan University Signal Properties: Terminology Waveform Time-average operator Periodicity DC value Power RMS Value Normalized Power Normalized Energy

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S. Mandayam/ ECOMMS/ECE Dept./Rowan University Power and Energy Signals 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 Although “real” signals are energy signals, we analyze them pretending they are power signals!

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S. Mandayam/ ECOMMS/ECE Dept./Rowan University The Decibel (dB) Measure of power transfer 1 dB = 10 log 10 (P out / P in ) 1 dBm = 10 log 10 (P / 10 -3 ) where P is in Watts 1 dBmV = 20 log 10 (V / 10 -3 ) where V is in Volts

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

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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

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

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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!

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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

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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

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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

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

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

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