信号与系统英文版课件：Chap3 Fourier Series Representation of Periodic Sigs

1、Fourier Series Representation of Periodic SignalsChapter 3 Lecture 1Signals and Systems Spring 2015Homework # 3 3.1 3.2 3.3 3.5 3.8 3.13 3.15 3.22 3.34 3.35 3.37 3.40 3.43Representation of LTI SystemsRepresent a DT signal fn as linear combinations of shifted impulsesThe response of a DT LTI system t

2、o any input fn is the same linear combination of the individual responses to each of the shifted impulses basic signalsRepresent a CT signal f(t) as linear combinations of shifted impulsesKEY: Representing signals as linear combinations of a set of basic signals.basic signalsRepresentation of LTI Sy

3、stemsAre there any other sets of basic signals that can be used to construct a broad and useful class of signal? The response of an LTI system to each of these signals should be simple enough in structure to provide a convenient representation for the response of the system to any signal constructed

4、 as a linear combination of the basic signals?KEY: Representing signals as linear combinations of a set of basic signals.Response of LTI Sys to Complex Exph(t)s0 : a complex constantH(s0): a constant determined by s0 and h(t) is an eigenfunction of the LTI systemH(s0) is the corresponding eigenvalue

5、 Response of LTI Sys to Complex ExpIf f(t) can be represented as linear combinations of complex exponentials, thenh(t)h(t)sk : complex constantsH(sk): constants determined by sk and h(t)?Fourier Series Representation of Periodic SignalsThe complex exponential is periodic with period T=2/0. is called

6、 the kth (k integer) harmonically related complex exponential of , which is also periodic with period T. must be periodic with period T=2/0.Can a periodic signal f(t) with period T=2/0 be represented as ?If yes, how to determine the coefficients ak ?Assume the above equation holds and try to solve f

7、or ak. Consider a CT signal f(t) with period T=2/0 that can be represented as k and m both integersFourier series coefficientFourier Series Representation of Periodic SignalsChapter 3 Lecture 2Signals and Systems Spring Consider a CT signal f(t) with period T=2/0 that can be represented as It is a w

8、eighted sum of complex exponentials. average of f(t) over a period: direct current (DC) component : kth harmonic component : angular frequency of the kth harmo comFourier series representationIt is a weighted sum of complex exponentials. complex amplitude of :Let , then magnitudephasefrequency spect

9、rumExample 1 For a CT signal , determine whether it is periodic? If yes, please find its Fourier series representation and Fourier coefficients. Solution: It is periodic with period T=2/0. Fourier series representation ?Eulers formulaNote that is a summation of sinusoidal signals. Fourier series coe

10、fficients Fourier series representationExample 2 Given a CT periodic signal as shown below. Find the Fourier series representation of it.Solution: Period of f(t) is T. Fourier series representation: where .Fourier coefficients: sampling functionSa(x) = sinx / xx : , : ,spectrum spacing and magnitude

11、 differ as T variesFourier Series CoefficientsComplex signal f(t) has Fourier coeffs Fourier coeffs of f*(t) is Fourier Series CoefficientsFor a real signal f(t) = f*(t) orak conjugate symmetricFourier Series CoefficientsFor a real even signal f(t) = f*(t) = f(-t)Let , then ak real and evenFourier S

12、eries CoefficientsFor a real odd signal f(t) = f*(t) = -f(-t)Let , then ak imaginary and oddTrigonometric Function Representation of Real Periodic SignalsFourier series representation for periodic signalFor real signals, Let Trigonometric Function Representation of Real Periodic Signals Example Give

13、n a real periodic signal . Determine its Fourier coeffs and trigonometric function representation.Solution: Period T, fund angular freq , Fourier coeffsThe magnitude and phase of arePlugging Ak and k into we getFourier Series Representation of Periodic SignalsChapter 3 Lecture 3Signals and Systems S

14、pring Convergence of Fourier Series Not all periodic signals have a Fourier series representation. Periodic signal f(t) must meet certain conditions, so as to ensure the existence of ak and the convergence of the corresponding Fourier series representation. Dirichlet ConditionsOver any period, f(t)

15、is absolutely integrableIn any finite time interval, f(t) has a finite number of maxima and minima. In any finite time interval, f(t) has a finite number of discontinuous points, where each of these discontinuities is finite. signal violates cond 1signal violates cond 2signal violates cond 3Gibbs Ph

16、enomenonTheoreticallyIn the real world, the number of harmonic components k cannot be infinite. Use an approximation (partial sum) such that Gibbs PhenomenonAs N increases, the ripples in the partial sums becomes compressed towards the discontinuity, but for any finite value of N, the peak value of

17、the ripples remains constant. For any t other than the discontinuity, the partial sums will converge to the correct value (as N increases), and at the discontinuity they will converge to one-half the sum of the values of the signal on either side of the discontinuity. Fourier Series and LTI Systemsh

18、(t)Recall that eigenvalueh(t)eigenvalueKey steps for computing the response of a LTI system to a periodic signal: Compute Fourier series coefs of the signal Compute the eigenvalue of the LTI systemh(t)Example Given an LTI system with impulse response . Determine the response of this system to the periodic signal f(t) as shown below.Solution: First compute the Fourier coefficients of the input periodic signal Then, compute the eigenvalue of the LTI systemThus, the system response is Exam