S. Mandayam/ ECOMMS/ECE Dept./Rowan University Electrical Communication Systems ECE Spring 2010 Shreekanth Mandayam ECE Department Rowan University Lecture 2a January 27, 2010

S. Mandayam/ ECOMMS/ECE Dept./Rowan UniversityPlan Digital and Analog Communications Systems 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

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

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

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

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!

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 / ) where P is in Watts 1 dBmV = 20 log 10 (V / ) where V is in Volts

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

S. Mandayam/ ECOMMS/ECE Dept./Rowan University Fourier Series Fourier Series Applet: Any periodic power signal Infinite sum of sines and cosines at different frequencies Fourier Series

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

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 > infinity Then, f > 0 The spectrum is continuous!

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 > infinity Then, f > 0 The spectrum is continuous! t T 0 Infinity w(t) Aperiodic Waveform |W(f)| f f 0 0

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

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

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