Lecture_Notes_04 - University of Toronto ECE316 September...

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University of Toronto ECE316 Fall 2011 CommunicaƟon Systems September 16, 2011 L±²ãçÙ± NÊã±Ý #4 Instructor: Ivo Maljević The Unit Step FuncƟon The unit step funcƟon is usually de³ned as: u ( t ) = 1 , t > 0 1 2 , t = 0 0 , t < 0 , a de³niƟon which is consistent with the value of the Fourier Transform of u ( t ) . The step funcƟon is shown in Figure 1. Figure 1: Unit step funcƟon Note that an alternaƟve de³niƟon is used in the textbook: u ( t ) = { 1 , t 0 0 , t < 0 , The so called causal signals saƟsfy g ( t ) = 0 for t < 0 , so u ( t ) is a causal signal. RelaƟonship to signum (sign) and delta signals The signum signal is de³ned as: sgn ( t ) = 1 , t > 0 0 , t = 0 - 1 , t < 0 , from which follows that u ( t ) = 1 2 [ sgn ( t ) + 1] . If we integrate the δ ( t ) with an upper limit set to be a Ɵme variable, we can establish the following relaƟonship: t -∞ δ ( τ ) d τ = { 1 , t > 0 0 , t < 0 , = u ( t ) Page 1 of 4
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University of Toronto ECE316 Fall 2011 Consequently, we can write: d u ( t ) d t = δ ( t ) Signals as Vectors So far, we have hinted that there is a relaƟonship between the signals and vectors. Without a formal proof, we will ±rst draw parallels between the vectors and signals, and then claim that signals are vectors. Let us start with a vector example where we want to approximate a vector
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This note was uploaded on 01/30/2012 for the course ECE ECE316 taught by Professor Sousa during the Winter '11 term at University of Toronto.

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Lecture_Notes_04 - University of Toronto ECE316 September...

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