EECE 301 Signals & Systems Prof. Mark Fowler1/21EECE 301 Signals & SystemsProf. Mark FowlerNote Set #13• C-T Signals: Fourier Series (for Periodic Signals)• Reading Assignment: Section 3.2 & 3.3 of Kamen and Heck2/21Ch. 1 IntroC-T Signal ModelFunctions on Real LineD-T Signal ModelFunctions on IntegersSystem PropertiesLTICausalEtcCh. 2 Diff EqsC-T System ModelDifferential EquationsD-T Signal ModelDifference EquationsZero-State ResponseZero-Input ResponseCharacteristic Eq.Ch. 2 ConvolutionC-T System ModelConvolution IntegralD-T System ModelConvolution SumCh. 3: CT Fourier SignalModelsFourier SeriesPeriodic SignalsFourier Transform (CTFT)Non-Periodic SignalsNew System ModelNew SignalModelsCh. 5: CT Fourier SystemModelsFrequency ResponseBased on Fourier TransformNew System ModelCh. 4: DT Fourier SignalModelsDTFT(for “Hand” Analysis)DFT & FFT(for Computer Analysis)New SignalModelPowerful Analysis ToolCh. 6 & 8: Laplace Models for CTSignals & SystemsTransfer FunctionNew System ModelCh. 7: Z Trans.Models for DTSignals & SystemsTransfer FunctionNew SystemModelCh. 5: DT Fourier System ModelsFreq. Response for DTBased on DTFTNew System ModelCourse Flow DiagramThe arrows here show conceptual flow between ideas. Note the parallel structure between the pink blocks (C-T Freq. Analysis) and the blue blocks (D-T Freq. Analysis).3/213.2 & 3.3 Fourier SeriesIn the last set of notes we looked at building signals using:∑−==NNktjkkectx0)(ωN = finite integerWe saw that these build periodicsignals.Q: Can we get anyperiodic signal this way?A: No! There are some periodic signals that need an infinitenumber of terms: ∑=++=NkkktkAAtx100)cos()(θω∑∞−∞==ktjkkectx0)(ωFourier Series(Complex Exp. Form)k are integers∑∞=++=100)cos()(kkktkAAtxθωk are integersFourier Series(Trig. Form)These are two different forms of the same tool!!There is a 3rdform that we’ll see later.Sect. 3.3There is a related Form in Sect. 3.24/21Q: Does this now let us get any periodic signal?A: No! Although Fourier thought so!Dirichlet showed that there aresome that can’t be written in terms of a FS! But… those will never show up in practice!See top of p. 155So we can write any practical periodic signal as a FS with infinite # of terms!So what??!! Here is what!!We can now break virtually anyperiodic signal into a sum of simple things…and we already understand how these simple things travel through an LTI system!So, instead of: )(th)(tx)()()( thtxty ∗=We break x(t) into its FS components and find how each component goes through. (See chapter 5)5/21To do this kind of convolution-evading analysis we need to be able to solve the following:Given time-domain signal model x(t)Find the FS coefficients {ck}“Time-domain” model “Frequency-domain model”Converting “time-domain” signal model into a “frequency-domain” signal model6/21Q: How do we find the (Exp. Form) Fourier Series Coefficients?A: Use this formula (it can be proved but we won’t do that!)∫+−=TtttjkkdtetxTc000)(1ωSlightly different than book…It uses t0= 0.Integrate over anycomplete periodwhere: T = fundamental period of x(t) (in seconds) ω0= fundamental frequency of x(t) (in rad/second)= 2π/Tt0 = any time point (you pick t0to ease calculations) k ∈ all integersComment:Note that for k = 0 this gives∫+=TttdttxTc00)(10c0is the “DC offset”, which is the time-average over one period7/21Summarizing rules for converting between the Time-Domain Model & the Exponential Form FS Model“Synthesis”∑∞−∞==ktjkkectx0)(ωUse FS Coefficients to “Build”the Signal “Read recipe and cook food”“Analysis”∫+−=TtttjkketxTc000)(1ωUse signal to figure out the FS Coefficients“Eat food and figure out recipe”Time-Domain Model: The Periodic Signal ItselfFrequency-Domain Model: The FS CoefficientsThere are similar equations for finding the FS coefficients for the other equivalent forms… But we won’t worry about them because once you have the ckyou can get all the others easily…8/21…,3,2,1212100=⎪⎭⎪⎬⎫===−−keAceAcAckkjkkjkkθθ()()…,3,2,1212100=⎪⎭⎪⎬⎫+=−==−kjbacjbacackkkkkkExponential Form∑∞−∞==ktjkkectx0)(ωTrig Form: Amplitude & Phase∑∞=++=100)cos()(kkktkAAtxθωTrig Form: Sine-Cosine[]∑∞=++=1000)sin()cos()(kkktkbtkaatxωω{}{}……,3,2,1,Im2,3,2,1,Re200=−====kcbkcacakkkk…,3,2,1200=⎪⎭⎪⎬⎫∠===kccAcAkkkkθ⎟⎟⎠⎞⎜⎜⎝⎛−=+==−kkkkkkabbaAaA12200tanθ)sin()cos(00kkkkkkAbAacaθθ−===Three (Equivalent) Forms of FS and Their Relationships9/21Example of Using FS AnalysisIn electronics you have seen (or will see) how to use diodes and an RC filter circuit to create a DC power supply:60Hz Sine wave with around 110V RMSmsT 67.166011==x(t)RC60 Hztms67.16x(t)T = T1/2 = 8.33mst1. This signal goes into the RC filter… what comes out? (Assume zero ICs)2. How do we choose the desired RC values?10/21t)(th)(tx?)(x(t)=tyAFor now we will just find the FS of x(t)…Later(Ch. 5) we will use it to analyze what y(t) looks likeThe equation for the FS coefficients is:∫−=TtjkkdtetxTc00)(1ωTπω20=Choose t0= 0 here to make things easier:tx(t)At0t0+TThis kind of choice would make things harder:tx(t)At0t0+T11/21Now what is the equation for x(t) over t ∈ [0,T]?TttTAtx ≤≤⎟⎠⎞⎜⎝⎛=⇒ 0sin)(πSo using this we get:∫∫⎟⎠⎞⎜⎝⎛−−⎟⎠⎞⎜⎝⎛==TtTjkTtjkkdtetTATdtetxTc020sin1)(10πωπTπω20=i.e., over the range of integrationDetermined by looking at the plottx(t)At0t0+TSo… now we “just” have to evaluate this integral as a function of k…12/21…we do a Change of Variables. There are three steps:1. Identify the new variable and sub it into the integrand2. Determine its impact on the differential3. Determine its impact on the limits of integration∫⎟⎠⎞⎜⎝⎛−⎟⎠⎞⎜⎝⎛=TtTjkkdtetTATc02sin1ππTo evaluate the integral:dtTdπτ=τπdTdt =⇒Step 2: when t = 0 ⇒when t = T ⇒Step 3: 00 ==Tπτππτ== TTtTπτ=Step 1: ττ2)sin(jke−()()∫∫∫−−⎟⎠⎞⎜⎝⎛−=⎟⎠⎞⎜⎝⎛=⎟⎠⎞⎜⎝⎛=πτπτπττπτπτπ020202sinsin1sin1deAdTeATdtetTATcjkjkTtTjkk13/21… use your favorite Table of Integrals (a short one is available on the course web site):∫+−=22)]cos()sin([)sin(babxbbxaedxbxeaxaxWe get our case with: a = -j2k b = 1So…πτττπ02241)]cos()sin(2[⎥⎦⎤⎢⎣⎡−−−=−kkjeAckjkSo…[]πττπ022)cos()41(kjkekAc−−−=Recall: sin(0) =