信号与系统英文版课件：Chap4 The Continuous-Time Fourier Transform

1、The Continuous Time Fourier TransformChapter 4 Lecture 1Signals and Systems Spring 2015Homework # 44.1(a) 4.2(a)(b) 4.3(a) 4.4(a) 4.5 4.6(a)(b)(c) 4.7(a) 4.10 4.11 4.13(a)(b) 4.14 4.15 4.16(a)(b) 4.21(g) 4.22(a)(b)(d) 4.24(a) 4.25 4.27 4.30 4.32(a)(b) 4.34(a)(b)(c) 4.35 4.36Representation of CT Aper

2、iodic SigsPeriodic signals can be represented as a summation of weighted complex exponentials. Can aperiodic signals be represented by a summation of weighted complex exponentials as well? To proceed, we consider an aperiodic signal as a periodic signal with an infinite period. From Periodic to Aper

3、iodic SignalsPulse trains fT(t)Single pulse f (t) T From Periodic to Aperiodic SignalsFrom an aperiodic finite duration signal f (t), we can construct a periodic signal fT(t), T2T1.Fourier series coeffs of fT(t) are Define continuous function wrt jThe Continuous Time Fourier TransformChapter 4 Lectu

4、re 2Signals and Systems Spring Inverse Fourier TransformFourier TransformFourier Transform (FT)Inverse Fourier TransformOften use notationone-to-one mappingFourier Transformf(t): time domain representationF(j): frequency domain representationReformulating the inverse FT formulaF() called frequency s

5、pectrum density funcmagnitude spectrumphase spectrumFourier Transform Example 1Single-sided exponential The signal contains more low frequency components than high frequency components.Fourier Transform Example 2Double-sided exponential Fourier Transform Example 3Rectangular pulse (gate signal) Four

6、ier Transform Example 4Unit impulse signal (t)Unite impulse signal contains all frequency components with equal magnitudes. Fourier Transform Example 5Unit signal f(t) = 1Define unit impulse signal in freq domain and take the inverse FTFourier Transform Example 6Signal Define gate function in freq d

7、omain Convergence of Fourier TransformsNot all aperiodic signals have a Fourier transform representation. The aperiodic signal f(t) must meet certain conditions, so as to guarantee the existence of finite F(j) and the convergence of the inverse Fourier transform to the original signal. Dirichlet Con

8、ditions f(t) is absolutely integrable f(t) has a finite number of maxima and minima in any finite interval f(t) has a finite number of discontinuous within any finite interval Gibbs Phenomenon Practically, the bandwidth is limited Gibbs phenomenon appears Properties of CT Fourier TransformLinearityT

9、ime shifting Frequency shiftingTime and frequency scalingConjugation and conjugate symmetryDuality The convolution propertyDifferentiation and integration Parsevals relationLinearityIf and then where a1 and a2 are constants. Ex: Time ShiftingIf , then Proof: Note that shifting in time does not alter

10、 the magnitude of its Fourier transform, which only introduces a phase shift t0 into its transform. Ex: time shiftingtime shiftingEx: Frequency Shifting If , then Proof: Example Determine the Fourier Transform of (1) (2) (3)Solution: (1)frequency shifting(2) (3)linearitylinearityThe Continuous Time

11、Fourier TransformChapter 4 Lecture 3Signals and Systems Spring Time and Frequency ScalingIf , then Proof: Ex1:Ex 2: scalingTime ReversalUse the time scaling property and let a = -1Ex: Example Given the Fourier transform of f(t), F(j), determine the FT of .Solution: time shiftingtime scalingConjugati

12、onIf , then Proof: Ex: Conjugate Symmetry of Real SignalsIf and f(t) is real, since and The Fourier transform of a real signal is conjugate symmetric. orEx: Recall the FT of the sinusoidal signal which does have conjugate symmetry. Conjugate Symmetry of Real SignalsevenoddThe real part of the FT of

13、a real signal is an even function of frequency, and the imaginary part is an odd function of frequency. Conjugate Symmetry of Real SignalsEx: evenoddConjugate Symmetry of Real Even SigIf and f(t) is real and even The Fourier transform of a real even signal is real and even. Ex: , gate signal , , etc

14、. have real even Fourier transformsConjugate Symmetry of Real Odd SigIf and f(t) is real and odd The Fourier transform of a real odd signal is purely imaginary and odd. Ex: has a pure imaginary and odd Fourier transformFT of Real and Odd Parts of Real Sigreal signal f (t)even partodd partEx: For a r

15、eal causal signal , Thus, the signal can be determined by knowing either the real or imaginary part of its FT. orExampleThe FT of a real causal signal f (t) is . Given , determine f (t). Solution: Since the signal is casual, DualityIf thenProof: Example Calculate the FT of . Solution: We know . Use

16、duality . Letting , after manipulation, we get difficult Signal having a gate/sampling function in the time domain has a sampling/gate function in the freq domain.ExampleCalculate the FT of . Solution: Again, difficult to compute the FT by definition. Recall Convolution PropertyIf , thenProof: Convo

17、lution PropertyThe convolution of two signals is mapped into the product of their Fourier transforms. It is an importance property.can simplify calculationExampleGive an LTI system with impulse response Determine the response to input .Solution: Calculating the convolution directly is difficult. Con

18、sider the convolution property of FT.ExampleCompute the FT of the triangular signal as shown below. Solution: The triangular signal is the convolution of two gate signals with time interval .convolution propertyDifferentiationIf , thenProof: The differentiation operation in the time domain implies a

19、 multiplication by j in the frequency domain. Similarly, for the n-th order derivative ExampleCompute the FT of (1) Sign signal (2) Unite step signalSolution: (1) Taking derivative of the sign signal Let . Using differentiation property Further, using duality and noting sgn() is odd (2) The unite st

20、ep signal can be expressed as Since and using linearity The Continuous Time Fourier TransformChapter 4 Lecture 4Signals and Systems Spring IntegrationIf , thenProof: ExampleCompute the FT of the given triangular signal. Solution: Taking derivatives of , we get Using integration property,we getParsev

21、als RelationIf , thenProof: instantaneous power (energy per a unit of time)energy spectrum density (energy per a unit of frequency)Parsavels relation is also called Energy Theorem.Ex: The energy of is which is difficult to compute. Since using Parsavels relation, we obtainAmplitude ModulationIf , th

22、enProof: The multiplication of two signals in the time domain corresponds to the convolution of their FTs in the frequency domain.The multiplication of two signals can be viewed as one signal modulating the amplitude of the other amplitude modulation. ExampleThe FT of f(t) is shown below. Determine

23、the FT of Solution: Let . Thus, Based on the amplitude modulation property, the FT of isThe amplitude modulation moves the freq spectrum of f(t), halves the amplitude of it, but does not change the shape of it. ExampleDetermine the FT of .Solution: amp mod proUsing the differentiation and integratio

24、n property of the convolution operation, we getDifferentiation in FrequencyIf , thenProof: Differentiation in FrequencySimilarly, we can prove Integration in FrequencyIf , thenDetermine the FT of .Solution: Using differentiation property in frequency ExampleUsing the differentiation property in freq

25、uency again, we getMore generally, we haveFourier Transform of Periodic Signals Can we compute the FT of a periodic signal using ?No - Dirichlet condition (absolutely integrable) can not be satisfied. Next we show an alternative way to proceed. Fourier Transform of Periodic Signals A periodic signal

26、 can be expressed asSince using the linear property of the FT, we getFT of a periodic signal is an impulse train, where the impulse appears at each of the harmonic components with frequency The magnitude of the k-th impulse is proportional to the Fourier coefficient ak. Two steps for computing the F

27、T of a periodic signal: calculate the Fourier coef and find the Fourier representationtake Fourier transform ExampleGiven impulse train signal , compute its FT P().Solution: The Fourier coefficient of p(t) istake FTTables of FT Properties and FT PairsTable of Fourier Transform Properties.P 328, Tabl

28、e 4.1 Table of Fourier Transform Pairs.P 329, Table 4.2Freq Domain Analysis of CT LTI SysA CT LTI system can be described asLTI differential equationfrequency responseDefine frequency response (system function) of a CT LTI system asmagnitude frequency responsephase frequency responseLTI differential

29、 equationfrequency responseimpulse responseThe impulse response of a CT LTI system is . Given an inputdetermineFrequency response of the system Differential equation of the systemSystem response ExampleSolution: (1) From impulse response , we can get the freq response(2) Since , we have Differential

30、 equation is(3) FT of the input isExampleThe differential equation of a CT LTI system is Given an input signal , determine the frequency response, impulse response, and system response of the system. Determining System Response Via FT Solution: From the given differential eqn, we haveFreq response(2)Impulse response(3)System responseThe Continuous Time Fourier TransformChapter 4 Lecture 5Signals and Systems Spring Filtering is the signal processing process that changes the shape of , or eliminates s