信号与系统英文版教学课件：ch3 Fourier Series Representation of Periodic Signals

1、1INTRODUCTIONRepresentation of continuous-time and discrete-time periodic signals Fourier series.Use Fourier methods to analyze and understand signals and LTI systems. 21. The Response of LTI Systems to Complex Exponentials1) Important concept signal decomposition(1) basic signals: possess two prope

2、rties A. The set of basic signals can be used to construct a broad and useful class of signals. B. It should be convenient for us to represent the response of an LTI system to any signal constructed as a linear combination of the basic signals. (2) complex exponential signals in discrete time: in co

3、ntinuous time:32) The Response of an LTI System to a Complex Exponential input is the same Complex Exponential with only a change in amplitudein discrete time: in continuous time:Complex amplitude facts H(s) or H(z) is in general a function of the complex variable s or z .Why?4in a continuoustime LT

4、I system with impulse response h(t) and an input So the outputin a discretetime LTI system with impulse response hn and an input So the output53) Eigenfunction(特征函数) Eigenfunction and Eigenvalue( of the system): A signal for which the system output is a (possibly complex) constant times the input is

5、 referred to as an eigenfunction of the system,and the amplitude factor is referred to as the systems eigenvalue(特征值).For a specific value of sk or zk , or : eigenvalue .is the eigenfunction of continuoustime systemsis the eigenfunction of discretetime systems6If the input to a continuous (discrete)

6、 time LTI system is represented as a linear combination of complex exponentials: then from the eigenfunction property and the superposition property, the output will be: 4) Decomposing general signals in terms of eigenfunctions72. Fourier Series Representation of Continuous-Time Periodic Signals1) C

7、omplex Exponential Fourier Series Given periodic x(t) with fundamental period T , its complex exponential Fourier series is :The signals in the set are harmonically related complex exponentials. : fundamental components or the first harmonic components the Nth harmonic components where the coefficie

8、nts ak is generally a complex function of . 8 the Fourier series coefficients are determined by equation:synthesis equation: (综合公式)analysis equation: (分析公式)Fourier series coefficients or spectrum of x(t): phase spectrum:magnitude spectrum:constant component or dc of x(t): 92) Trigonometric Fourier S

9、eriesSuppose x(t) is real, thatWe obtainReplacing k by k , we havecomplex exponential Fourier Series:So or 10Since are complex conjugates ,soIf ,thenIf ,then11Example 3.1 Consider a real periodic signal x(t), with fundamental frequency 2, that is expressed in the complex exponential Fourier series a

10、swhere Use the trigonometric form to express the signal x(t).12Construction of the signal x(t) as a linear combination of the harmonically related sinusoidal signals13Example 3.2 Consider the signal Plot the magnitude spectrum and phase spectrum of x(t). Thus, the Fourier series coefficients for thi

11、s example are :1411/2 -3 -2 -1 0 1 2 3k/4karctan(1/2) -2 1 -3 -1 0 2 3 Plots of the magnitude spectrum and phase spectrum of the signal x(t) 15Example 3.3 The periodic square wave is defined over one period as: -T -T/2 -T1 T1 T/2 Ttx(t)Represent it in Fourier series. 16For T = 4T1, the coefficients

12、are: For T = 8T1, the coefficients are: -2 0 2kk -4 0 4k -8 0 8 Plots of the Fourier Series coefficients for the periodic square wave with T1 fixed and for several values of T: (a) T=4 T1; (b) T=8 T1; (c) T=16 T1. 17 spectrum of periodic square wave18193) Convergence of the Fourier Series(1) The que

13、stion of the validity of Fourier series representionsAn approximation of x(t) isThe approximation error:The energy in the error over one period:When , EN is minimize.There exist error between the original signal x(t) and the approximation xN(t) and with the N increases, the error decreases. 2022Cond

14、ition 1: Over any period, x(t) must be absolutely integrable Condition 2: In any finite interval of time, x(t) is of bounded variation; that is, there are no more than a finite number of maxima and minima during any single period of the signal.Condition 3: In any finite interval of time, there are o

15、nly a finite number of discontinuities. Furthermore, each of these discontinuities is finite. (2) The Convergence conditions of the Fourier Series -The Dirichlet conditions 23For a periodic signal that has no discontinuities, the Fourier series representation converges and equals the original signal

16、 at every value of t. For a periodic signal with a finite number of discontinuities in each period, the Fourier series representation equals the signal everywhere except at the isolated points of discontinuity, at which the series converges to the average value of the signal on either side of the di

17、scontinuity. Gibbs phenomenon: the truncated Fourier series approximation xN(t) of a discontinuous signal x(t) will in general exhibit high-frequency ripples and overshoot x(t) near the discontinuities.24Convergence of the Fourier series representation of a square wave: an illustration of the Gibbs

18、phenomenon. 253. Properties of Continuous-time Fourier SeriesWe generally use a shorthand notation to indicate the relationship between a periodic signal and its Fourier series coefficients, that is1) LinearityIf then 2) Time ShiftingIf then When a periodic signal is shifted in time, the magnitude s

19、pectrum remains unaltered. 263) Time ReversalIf then Time reversal applied to a continuous-time signal results in a time reversal of the corresponding sequence of Fourier series coefficients. If x(t) is even :If x(t) is odd: 4) Time ScalingIf then The Fourier coefficients for each of those component

20、s remain the same. However, the harmonic components change with the change in the fundamental frequency. 275) MultiplicationIf then hk is the convolution sum of the sequence representing the Fourier coefficients of x(t) and the sequence representing the Fourier coefficients of y(t). 6) Conjugation a

21、nd Conjugate SymmetryIf then if x(t) is real, a0 is realif x(t) is real and even, then so are its Fourier series coefficients.if x(t) is real and odd, then its Fourier series coefficients are purely imaginary and odd. 28Parsevals relation states that the total average power in a periodic signal equa

22、ls the sum of the average powers in all of its harmonic components.because7) Parsevals Relation for Continuous-Time Periodic SignalsIf then 29Example 3.4 Determine the Fourier series representation of g(t) which is shown in the right figure: -2 -1 1 2-1/2g(t)t1/2 g(t)= x(t-1)1/2, where x(t) is the p

23、eriodic square wave in Example 3.3, and T=4 and T1=1. time shifting property : the Fourier coefficients of x(t-1) is , where ak is the Fourier coefficients of x(t). -1/2 is the dc offset in g(t). ( supposing that the constant component in x(t) is a0 , then the constant component in g(t) is a0 1/2. )

24、linear property :30Example 3.5 Consider the triangular wave signal x(t) which is shown in the right figure.x(t)t -2 21The derivative of x(t) is the signal g(t) in last example we just considered. Denoting the coefficients of g(t) by dk and those of x(t) by ek, then we have:( differentiation property

25、: )Thus, For k = 0, e0 can be determined by finding the area under one period of x(t) and dividing by the length of the period:31Example 3.6 Determine the Fourier series representation of the impulse train, which is periodic with period T and is expressed as :1 -T Ttx(t)1 -T -T/2 -T1 T1 T/2 Ttg(t)-1

26、1 -T/2 -T1 T1 T/2 Ttg(t)g(t)= x(t+T1)x(tT1). 32Example 3.7 Giving the following facts about a signal x(t): 1. x(t) is a real signal; 2. x(t) is periodic with period T=4, and it has Fourier series coefficients ; 3. for ; 4. The signal with Fourier coefficients is odd; 5. . Determine the signal. From

27、2 and 3, we obtain:From 1 and Conjugate Symmetry for Real Signals:33soFrom Time Reversal and Time Shifting, we know:From 1 and 4, the Fourier coefficients of x(-t+1) must be purely imaginary and odd. ThusFrom 5, we have:From Parsevals Relation:soThusso344. Fourier Series Representation of Discrete-T

28、ime Periodic Signals1) Linear Combination of Harmonically Related Complex ExponentialsGiven periodic xn with fundamental period N , its Fourier series has the form:Since the sequences are distinct only over a range of N successive values of k, the summation need only include terms over this range. W

29、e use to indicate this, then, we have: finite series 352) Determination of the Fourier Series Representation of a Periodic SignalMultiplying both sides of the discrete-time Fourier series equation by and summing over N terms, we obtain Interchanging the order of summation on the right-hand side, we

30、have From the identity:36synthesis equation: analysis equation: periodic the Fourier series coefficients are determined by equation:Since there are only N distinct complex exponentials that are periodic with period N, the discrete-time Fourier series representation is a finite series with N terms.37

31、Example 3.8 Consider the signal xn = sin3(2/5)n, draw the graph of coefficients.This signal is periodic with period N = 5. -1/2j1/2j -7 -2 3 8 -8 -3 2 7 12 kFourier coefficients for xn=sin3(2/5)n. 38k1/2 -9-8-7 -5-4-3 0 1 2 3 4 5 6 78 91011 -6 -2-1 12 Magnitude of the coefficients. (magnitude spectr

32、um)-/2/2 -9 -8 -6-5-4-3 -1 0 1 2 4 5 6 7 9 10 11 -7 -2 3 8 kPhase of the coefficients. (phase spectrum)39Example 3.9 Consider the discrete-time periodic square wave: n1 N N1 0 N1 N There are no convergence issues with the discrete-time Fourier series in general, because any discrete-time periodic se

33、quence xn is completely specified by a finite number N of parameters. 40Fourier series coefficients for the periodic square wave of Example 3.9; plots of Nak for 2N1 +1 = 5 and (a) N = 10; (b) N = 20; (c) N = 40 415. Properties of Discrete-time Fourier Series1) MultiplicationIf then periodic convolu

34、tion 2) First DifferenceIf then 423) Parsevals Relation for Discrete-Time Periodic Signals is the average power in the kth harmonic component of xn. Parsevals relation states that the average power in a periodic signal equals the sum of the average powers in all of its harmonic components. Different

35、 from the continuous time case, in discrete time, there are only N distinct harmonic components. 43Example 3.10 Find the Fourier series coefficients of the sequence xn shown in the figure: -5 0 5xn 2 1n -5 0 5x2n 1n -5 0 5x1n 1nThis sequence has a fundamental period of 5.xn=x1n+x2nRepresenting xn as

36、 a sum of the square wave x1n and the dc sequence x2n 44If then For x1n: N1=1,N=5, From Example 3.9 45Example 3.11 Giving the following facts about a sequence xn: 1. xn is periodic with period N = 6. 2. 3. 4. xn has the minimum power per period among the set of signals satisfying the preceding three

37、 conditions.Determine the sequence xn. From analysis equation:Denote the Fourier series coefficients of xn by ak. .46Noting thatsoFrom Parsevals Relation, the average power in xn isSince each nonzero coefficent contributes a positive amount to P , and since the values of a0 and a3 are prespecified ,

38、 the value of P is minimized by choosing a1=a2=a4=a5=0 .It then follows that476. Fourier Series and LTI Systems1) The system functions of LTI systemsIn continuous time:In discrete time:When s or z are general complex numbers , H(s) and H(z) are referred to as the system functions of corresponding sy

39、stems.system function 48If Res = 0, s = j. If | z| = 1, . frequency response 2) The frequence response of LTI systemsIn continuous-time systems:In discrete-time systems:The response of an LTI system to a complex exponential signal of the form (in continuous time) or (in discrete time) is particularl

40、y simple to express in terms of the frequency response of the system. Furthermore, as a result of the superposition property for LTI systems, we can express the response of an LTI system to a linear combination of complex exponentials with equal ease. 493) The response of LTI systems to Periodic Sig

41、nalsIn continuous time, let x(t) be a periodic signal with Fourier series representation given byThen, the output is That is, the effect of the LTI system is to modify individually each of the Fourier coefficients of the input through multiplication by the value of the frequency response at the corresponding frequency. 50In discrete time, let xn be a periodic signal with Fourier series representation given byThen, the output is Thus, yn is also periodic with the same period as xn, and the kth Fourier coefficient of yn is the product of the kth Fourier coefficient of the i