Download presentation

Presentation is loading. Please wait.

1
S. Mandayam/ ECOMMS/ECE Dept./Rowan University Electrical Communications Systems ECE.09.331 Spring 2007 Shreekanth Mandayam ECE Department Rowan University http://engineering.rowan.edu/~shreek/spring07/ecomms/ Lecture 2a January 23, 2007

2
S. Mandayam/ ECOMMS/ECE Dept./Rowan UniversityPlan Recall: Intoduction to Information Theory 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

3
S. Mandayam/ ECOMMS/ECE Dept./Rowan University ECOMMS: Topics

4
S. Mandayam/ ECOMMS/ECE Dept./Rowan University Measures of Information Definitions Probability Information Entropy Source Rate Recall: Shannon’s Theorem If R < C = B log 2 (1 + S/N), then we can have error- free transmission in the presence of noise MATLAB DEMO: entropy.m

5
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

6
S. Mandayam/ ECOMMS/ECE Dept./Rowan University ECOMMS: Topics

7
S. Mandayam/ ECOMMS/ECE Dept./Rowan University Signal Properties: Terminology Waveform Time-average operator Periodicity DC value Power RMS Value Normalized Power Normalized Energy

8
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!

9
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

10
S. Mandayam/ ECOMMS/ECE Dept./Rowan University ECOMMS: Topics

11
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

12
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

13
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!

14
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

15
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

16
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

17
S. Mandayam/ ECOMMS/ECE Dept./Rowan UniversitySummary

Similar presentations

© 2020 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google