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1、 5 The discrete-time Fourier transform5.0 Introduction5. The Discrete time Fourier TransformCTFS ( The Continuous-Time Fourier Series ): DTFS ( The Discrete-Time Fourier Series ): CTFT ( The Continuous-Time Fourier Transform ): DTFT ( The Discrete-Time Fourier Transform ): xnak5.1 Representation of
2、Aperiodic Signals: The Discrete-time Fourier Thransform 5 The discrete-time Fourier transform1. Development of DTFT from DTFS :Discrete-time Fourier series (periodic signal)Fixed value of N1 Several values of N 5 The discrete-time Fourier transformNote: As N, 0=(2/N)0, the envelope is sampled with a
3、 closer and closer spacing. 5 The discrete-time Fourier transformWith =k0 thought of as a continuous variable, the set of Fourier series coefficients approaches the envelope function as N tends to infinity.As , let So we haveDTFTDTFS: is periodic with period From As 5 The discrete-time Fourier trans
4、formand We getandA representation of xn as a linear combination of complex exponentials. 5 The discrete-time Fourier transformFourier series (periodic signal)Fourier transform (aperiodic signal)DTFS:DTFT:5.1.2. Examples of Discrete-Time Fourier Transforms1. 5 The discrete-time Fourier transformMagni
5、tudePhase 5 The discrete-time Fourier transformLowpassHighpass2. 5 The discrete-time Fourier transformreal evenreal even3.Square wave: 5 The discrete-time Fourier transformreal evenreal evenComparisons:(1).With discrete-time periodic signalIt is easy to see that 5 The discrete-time Fourier transform
6、(DTFS)(DTFT)(Sampling) 5 The discrete-time Fourier transform(2).With continuous-time aperiodic signal(DTFT)(CTFT)4. 5 The discrete-time Fourier transform5.1.3 Convergence Issue Associated with DTFT1. The sequence has finite energy, that is2. The sequence is absolutely summable, that is 5 The discret
7、e-time Fourier transformThe frequency of the oscillations increases as W , and disappear entirely for W=. No any behavior like the Gibbs phenomenon.Then, the DTFT for is as follows: 5 The discrete-time Fourier transform5.2 The Fourier Transform for Periodic Signals(continuous-time periodic signal)(d
8、iscrete-time periodic signal)From DTFS we know 5 The discrete-time Fourier transformComparison: 5 The discrete-time Fourier transform(CTFT)(DTFT)Example 1.Example 1.xn is periodic if 5 The discrete-time Fourier transform(rational) 5 The discrete-time Fourier transformExample 2.Example 2.Impulse trai
9、n(DTFS)(DTFT)thenIf1. PeriodicityNote: DTFT is always periodic in with period 2 5 The discrete-time Fourier transform5.3 Properties of the Discrete-Time Fourier Transform2. LinearityIfthen3. Time Shifting and Frequency Shifting Time ShiftingFrequency Shifting4. Reflection 5 The discrete-time Fourier
10、 transformIfthenIfthen5.5. Conjugation and Conjugate SymmetryFrom this, it follows :i.e.(1) Ifis real valued，then 5 The discrete-time Fourier transformIfthenis real and even，thenSo we geti.e.,is real and even. 5 The discrete-time Fourier transform(2) If(3) Ifis real and odd，thenSo we geti.e.,is imag
11、inary and odd.then(4) If6. Differencing and Accumulation 5 The discrete-time Fourier transformExample:(CTFT)(DTFT)7. Time Expansion (Interplation ): 5 The discrete-time Fourier transformDefineif n is a multiple of kif n is not a multiple of k(DTFT)(CTFT) 5 The discrete-time Fourier transform8. Diffe
12、rentiation in Frequency 9. 9. Parsevals Relation : Energy-density spectrum of xn.(DTFS) 5 The discrete-time Fourier transform(DTFT) : Average power in the kth harmonic component of xn.If then:Frequency response of a DT LTI systemNote: This property maps the convolution of two signals to the simple a
13、lgebraic operation of multiplying their Fourier transforms. 5 The discrete-time Fourier transform5.4 The Convolution PropertyExample: 5 The discrete-time Fourier transformPeriodic convolution of and If then 5 The discrete-time Fourier transform5.5 The Multiplication PropertyExample: 5 The discrete-t
14、ime Fourier transform 5 The discrete-time Fourier transform5.6 Tables of Fourier Transform Properties and Basic Fourier Transform Pairs Table 5.1 Table 5.2Since ak are a periodic sequence, we can expand the sequence ak in a Fourier series5.7.1.Duality in DTFSor 5 The discrete-time Fourier transform5
15、.7 Duality(CTFT)Example 1: Time shift Frequency shiftFrom the Time shift property we getFrom the duality we have(Frequency shift) 5 The discrete-time Fourier transformFrom the convolution property we get(Duality)(Multiplication) 5 The discrete-time Fourier transformExample 2: Convolution Multiplicat
16、ionLet be a periodic signal with period 2, then it can be represented using CTFS as follows:is a continuous and periodic functionof with period 2 and we have 5 The discrete-time Fourier transform5.7.2 Duality between DTFT and CTFSIf then 5 The discrete-time Fourier transformExample: Differentiation
17、in Time (CTFS) Differentiation in Frequency(DTFT)Differentiation in Time (CTFS)AsDifferentiation in Frequency (DTFT)AsUsing the duality we getMultiplication (DTFT) 5 The discrete-time Fourier transformExample: Convolution(CTFS) Multiplication(DTFT)Convolution (CTFS) i.e.,i.e.,The discrete-time LTI s
18、ystem can be represented as1. Frequency Response H(ej) 5 The discrete-time Fourier transform5.8 Systems Characterized by Linear Constant-Coefficient Difference Equations(LCCDE)A:DTFTLCCDEB:DTFTStable System2. Block Diagram Representation:D DD D 5 The discrete-time Fourier transformPartial-Fraction E
19、xpansion1 1) ) X(s)为有理真分式有理真分式( m n)，极点为一阶极点一阶极点 5 The discrete-time Fourier transform2 2) ) X(s)为有理真分式有理真分式( m n)，极点为r r重阶极点重阶极点 5 The discrete-time Fourier transformPartial-Fraction Expansion3 3) ) X(s)为有理假分式有理假分式( m n)为真分式真分式，根据极点情况按1 1) )或2 2) )展开。 5 The discrete-time Fourier transformPartial-Fr
20、action ExpansionDualityContinuous & PeriodicContinuous & AperiodicDiscrete & PeriodicDiscrete & Aperiodic时域时域采样采样DualityDuality频域采样频域采样频域采样频域采样时域时域采样采样 5 The discrete-time Fourier transformProblems: 5.2 5.3 5.6 5.9 5.12 5.13 5.19 5.25 5.29 5.48 5.51 5.55 4 The continuous time Fourier transform 4 The continuous time Fourier transform
