Lecture-02(1).pdf - ECE 3337 Signals Systems Analysis...

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ECE 3337: Signals & Systems AnalysisLecture 2: Basic WaveformsReference: Chapter 1(Sections 1.3 – 1.4.3, Appendix B)
RecapSystemInputSignalOutputSignalRepresenting a signal is the first step to signalprocessing:Basic Signal Transformations:Analog continuous-time: x(t)Discrete-time signal: x[n]Digital signal: x[n], but xis quantizedIf a<1, then x(t) is expandedIf a>1, then x(t) is compressedShift:y(t)=x(tT)Scaling:y(t)=x(at)Reversal : y(t)=x(t)Combined: y(t)=x(atb)
We will explore some waveform propertiesEven and Odd SymmetryPeriodic and Non-periodic waveformsWe will introduce some common periodicand non-periodic waveformsSinusoidsStep functionRamp functionWe will briefly review some math that wewill need laterComplex variablesToday
For some signals,This is a case ofeven symmetryFor some other signals,Symmetryx(t)=x(t)x(t)=x(t)x(t)tt010-5-1055-510-1015-15x(t)Odd symmetry
What kinds of symmetry do the followingsignals exhibit?Questionx(t)=sin(t)x(t)=cos(t)x(t)=tan(t)x(t)=t
For any signal, it is easy to verify thatis always even symmetric, because:Now, verify that the following signal is alwaysodd symmetric:Interesting Propertiesxe(t)=x(t)+x(t)2xe(t)=x(t)+x(t)2=xe(t)xo(t)=x(t)x(t)2
Easy to see that any signal can berepresented as a sum of an even symmetricsignal and an odd symmetric signalWe will take advantage of the symmetryproperties of signals to simplify our analysis(Chapter 5)Cool Observationx(t)=xe(t)+xo(t)xe(t)=x(t)+x(t)2xo(t)=x(t)x(t)2
ExampleGiven SignalEven ComponentOdd Componentxe(t)=x(t)+x(t)2xo(t)=x(t)x(t)2
Signals that keep onrepeating, everysecondsPeriodic SignalsT0x(t)=x(t+nT0)integerTime Period(seconds)f0=1T0Frequency(repetitions per sec)
Sine and Cosine Functionssin(θ)cos(θ)θ(radians)Period = 2πradians = 360oLater in this course (Chapter 5) we will learn thatany periodic waveform can be constructed fromsines and cosinesThe big reason for studying them carefully!
Notation & Termsx(t)=Asin2πtT0+θAmplitudeTime Periodω0=2πT0Angular frequency(radians/second)f0=1T0Frequency(per second)Phase Shift(radians)

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