华工信号分析与处理课件俞祝良老师1

1、1 Chapter 5 The Discrete-time Fourier Transform Yu Zhuliang College of Automation Science and Engineering South China University of Technology 2 3 Determination of the Fourier Series Representation of a Continous-time Periopdic Signal Periodic vs. Discrete 4 Fourier Series Representation of Discrete

2、- time Periodic Signals Periodic vs. Discrete 5 Fourier Series Representation of Discrete- time Periodic Signals nImportant property nSo we select k from 0 to N-1 nIt is sometimes convenient to think of ak as a sequence define for all values of k, but where only N successive elements in the sequence

3、 will be used in the Fourier Series representation. 6 Representation of aperiodic signals: The continous-time Fourier transform Fourier Transform Pair 7 The Fourier Transform for Periodic Signals 8 9 Question? nHow do we get the Fourier Transform for aperiodic continuous-time signal? 10 Representati

4、on of aperiodic signals: The continous-time Fourier transform nOutside the range, x(t)=0 11 Representation of aperiodic signals: The continous-time Fourier transform nWe have 12 Representation of aperiodic signals: The continous-time Fourier transform nSince nWhen T approaches infinity 13 The proble

5、m we study today nSimilar to the derivation of Fourier Transform of continous-time signal, what is the Fourier Transform of a discrete-time signal? 14 Representation of Aperiodic Signals nN approaches infinity, 15 Representation of Aperiodic Signals 16 Representation of Aperiodic Signals ? ? 17 Repr

6、esentation of Aperiodic Signals nLets define nWe have 18 Representation of Aperiodic Signals 19 Representation of Aperiodic Signals 20 Representation of Aperiodic Signals Discrete-time Fourier Transform Pair 21 Representation of Aperiodic Signals Finite integral interval Periodicity 22 nFourier Tran

7、sform of aperiodic continuous- time signal does not have periodicity. nWhy Fourier Transform of Discrete-time signal has periodicity? 23 Periodicity 24 Example of Discrete-time Fourier Transforms 25 26 Example of Discrete-time Fourier Transforms 27 28 Example of Discrete-time Fourier Transforms 29 C

8、onvergence of Discrete-time Fourier Transform nSince the synthesis equation involves integral over finite interval, there are generally no convergence problem. 30 Fourier Transform for Periodic Signals For continuous-time periodic signal 31 Fourier Transform for Periodic Signals 32 33 Fourier Transf

9、orm for Periodic Signals 34 The Fourier Transform for Periodic Signals (Comparison) 35 Fourier Transform for Periodic Signals nExample 5.5 36 Fourier Transform for Periodic Signals nExample 5.6 37 Properties of the Discrete-time Fourier Transform nPeriodicity of DTFT 38 Properties of the Discrete-ti

10、me Fourier Transform nLinearity of DTFT 39 Properties of the Discrete-time Fourier Transform nTime shifting and frequency shifting 40 Properties of the Discrete-time Fourier Transform nApplication () j hp He 41 Properties of the Discrete-time Fourier Transform nConjugation and Conjugate Symmetry 42

11、Properties of the Discrete-time Fourier Transform nDifferencing and Accumulation 43 Properties of the Discrete-time Fourier Transform nTime Reversal 44 Properties of the Discrete-time Fourier Transform nTime Expansion 45 46 Properties of the Discrete-time Fourier Transform nDifferentiation in Freque

12、ncy 47 Properties of the Discrete-time Fourier Transform nParsevals Relation 10.31 48 49 Properties of the Discrete-time Fourier Transform nConvolution Property 50 Properties of the Discrete-time Fourier Transform nExample 5.13 51 Properties of the Discrete-time Fourier Transform nExample 5.14 52 Pr

13、operties of the Discrete-time Fourier Transform nMultiplication Property 53 54 Determination of the Fourier Series Representation of a Continous-time Periopdic Signal Periodic vs. Discrete 55 Fourier Series Representation of Discrete- time Periodic Signals Periodic vs. Discrete 56 Fourier Series Rep

14、resentation of Discrete- time Periodic Signals nImportant property nSo we select k from 0 to N-1 nIt is sometimes convenient to think of ak as a sequence define for all values of k, but where only N successive elements in the sequence will be used in the Fourier Series representation. 57 Representat

15、ion of aperiodic signals: The continous-time Fourier transform Fourier Transform Pair 58 The Fourier Transform for Periodic Signals 59 60 Representation of Aperiodic Signals Discrete-time Fourier Transform Pair 61 Representation of Aperiodic Signals Finite integral interval Periodicity 62 Fourier Transform for Periodic Signals 63 The Fourier Transform for Periodic Signals (Comparison) 64 Properties of the Discrete-time Fourier Transform nSummary in Table 5.1 nDTFT of S