Lecture_2.pdf - ECE 3337 Signals Systems Analysis Lecture 2...

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ECE 3337: Signals & Systems Analysis Lecture 2: Basic Waveforms Reference: Chapter 1 (Sections 1.3 – 1.4.3, Appendix B)
Recap System Input Signal Output Signal Representing a signal is the first step to signal processing Basic Signal Transformations: If a < 1, then x ( t ) is expanded If a > 1, then x ( t ) is compressed Shift : y ( t ) = x ( t T ) Scaling : y ( t ) = x ( at ) Reversal : y ( t ) = x ( t ) Combined: y ( t ) = x ( at b ) “PLAYBACK SPEED” ANALOGY
Think through your strategy before attempting problems ! Recap: Combined Transformations y ( t ) = x ( at b ) x ( t ) x ( at ) x ( a ( t b a )) x ( t ) x ( t b ) x ( at b ) HOW DO WE TELL THAT A STRATEGY IS VALID?? #1 #2
Check on key “landmark points” Another Sanity Check Method y ( t ) = x ( 2 t + 6) A B C A’ B’ C’ t A = 3 t A ' = 4.5 t A = 2 t A ' + 6 t A ' = (6 t A ) 2
We will explore some waveform properties Even and Odd Symmetry Periodic and Non-periodic waveforms We will introduce some common periodic and non-periodic waveforms Sinusoids Step function Ramp function We will briefly review some math that we will need later Complex variables Today
For some signals, This is a case of even symmetry For some other signals, Symmetry x ( t ) = x ( t ) x ( t ) = x ( t ) x ( t ) t t 0 10 - 5 - 10 5 5 - 5 10 - 10 15 - 15 x ( t ) Odd symmetry
What kinds of symmetry do the following signals exhibit? Question x ( t ) = sin( t ) x ( t ) = cos( t ) x ( t ) = tan( t ) x ( t ) = t WE WILL LEARN LATER THAT SYMMETRY PROPERTIES CAN BE USED TO SIMPLIFY OUR CALCULATIONS
For any signal, it is easy to verify that is always even symmetric, because: Now, verify that the following signal is always odd symmetric: Interesting Properties x e ( t ) = x ( t ) + x ( t ) 2 x e ( t ) = x ( t ) + x ( t ) 2 = x e ( t ) x o ( t ) = x ( t ) x ( t ) 2
Easy to see that any signal can be represented as a sum of an even symmetric signal and an odd symmetric signal We will take advantage of the symmetry properties of signals to simplify our analysis in Chapter 5 Cool Observation x ( t ) = x e ( t ) + x o ( t ) x e ( t ) = x ( t ) + x ( t ) 2 x o ( t ) = x ( t ) x ( t ) 2
Example Given Signal Even Component Odd Component x e ( t ) = x ( t ) + x ( t ) 2 x o ( t ) = x ( t ) x ( t ) 2
Signals that keep on repeating, every seconds Periodic Signals T 0 x ( t ) = x ( t + nT 0 ) integer Time Period (seconds) f 0 = 1 T 0 Frequency (repetitions per sec) x ( t ) = A cos(2 π t / T 0 )
Sine and Cosine Functions sin( θ ) cos( θ ) θ (radians) Period = 2 π radians = 360 o Later in this course (Chapter 5) we will learn that any periodic waveform can be constructed from sines and cosines The big reason for studying them carefully!
Sine and Cosine sin θ = cos( θ π 2 ) sin( θ ) cos( θ ) θ (radians) Period = 2 π radians = 360 o y ( t ) = x ( t T ) Remember: Shift
Sine and Cosine cos θ = sin( θ + π 2 ) sin( θ ) cos( θ ) θ (radians) Period = 2 π radians = 360 o y ( t ) = x ( t T ) Remember: Shift
Notation & Terms x ( t ) = A sin 2 π t T 0 + θ

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