The Discrete-time Fourier Transform Rui Wang, Assistant professor Dept. of Information and Communication Tongji University it Email: ruiwang@tongji.edu.cn
Outline Representation of Aperiodic signals: The discrete-time Fourier Transform The Fourier transform for periodic signals Properties of the discrete-time Fourier transform The convolution/multiplication property System characterization by linear constant- coefficient difference equations 2
5.1 The discrete-time Fourier transform The difference of the Fourier series representation between continuous-time case and discrete-time case The continuous-time signal needs to represented by infinite number of exponential signals The discrete-time signal needs to be represented by finite number exponential signals 3
5.1 The discrete-time Fourier transform Extend non-periodic signal to periodic signal 4
5.1 The discrete-time Fourier transform Fourier series representation Define We have 5
5.1 The discrete-time Fourier transform Then we express When 6
5.1 The discrete-time Fourier transform The Fourier transform is 7
5.1 The discrete-time Fourier transform 8
5.1 The discrete-time Fourier transform Example 1: consider a signal Determine the Fourier transform. Solution: 9
5.1 The discrete-time Fourier transform 10
5.1 The discrete-time Fourier transform Example 2: consider a signal Determine the Fourier transform. Solution: 11
5.1 The discrete-time Fourier transform Example 3: consider a rectangular pulse Determine the Fourier transform. Solution: 12
5.1 The discrete-time Fourier transform Example 4: consider a signal Determine the Fourier transform. Solution: 13
5.1 The discrete-time Fourier transform The Convergence issue: Finite duration: always convergent infnite duration: or 14
5.2 The Fourier transform for periodic signals The Fourier transform of exponential signal We have 15
5.2 The Fourier transform for periodic signals Why? 16
5.2 The Fourier transform for periodic signals The Fourier transform of periodic signal We have 17
5.2 The Fourier transform for periodic signals Why? W_0 are: 18
5.2 The Fourier transform for periodic signals Example 2.1: consider a periodic signal Determine the Fourier transform Solution: 19
5.2 The Fourier transform for periodic signals Solution: 20
5.2 The Fourier transform for periodic signals Example 2.2: 2: consider a periodic signal Determine the Fourier transform Solution: 21
5.2 The Fourier transform for periodic signals Solution: 22
5.2 The Fourier transform for periodic signal Example 2: Consider the impulse train Determine its Fourier transform Solution: 23
5.3 Properties Linearity: Time shifting: 24
5.3 Properties Example 3.1: Consider the frequency response of the lowpass filter with cutoff frequency An ideal highpass filter is with cutoff frequency 25
5.3 Properties The frequency shifting property p implies 26
5.3 Properties Conjugation &conjugate symmetry If x[n] is real 27
5.3 Properties Differencing and accumulation 28
5.3 Properties Example 3.2: use Fourier transform of delta function to derive the Fourier transform of unit step function 29
5.3 Properties Time reversal Proof: Replace m by -n 30
5.3 Properties Time expansion If n is a multiple of k If n is not a multiple of k 31
5.3 Properties Then we have 32
5.3 Properties When k>1, the signal is spread out, while its Fourier transform is compressed. 33
5.3 Properties Example 3.3: 3: we have signal x[n] 34
5.3 Properties Example 3.3: 3: we have signal x[n] 35
5.3 Properties Now we consider with 36
5.3 Properties Differentiation in frequency 37
5.3 Properties Parselval s s relation 38
5.3 Properties Example 3.4: consider the following signal, determine whether or not periodic in time domain, real, even and of finite energy? 39
5.3 Properties Not periodic in time domain: since periodicity in time domain implies impulses located at various integer multiples of the fundamental frequency. Real in time domain: since the Fourier transform have even magnitude and odd phase function. Not even signal: since the following function is not real-valued function. Finite energy: since the following value is finite 40
5.4 The convolution property For discrete-timetime signals Map the convolution of two signals to the simple algebraic operation of multiple their Fourier transform Facilitate the analysis of signals and systems The frequency response captures the change in complex amplitude of the Fourier transform of the input at each frequency 41
5.4 The convolution property Example 4.1: Consider an LTI system with impulse response 42
5.4 The convolution property Example 4.2: Consider the following frequency response, determine the unit impulse response 43
5.4 The convolution property Example 4.3: Consider the unit impulse response and input are given as follows: Determine the output. 44
5.4 The convolution property Solution: 45
5.4 The convolution property When 46
5.4 The convolution property When 47
5.5 The multiplication property We have Above equation corresponds to a periodic convolution. 48
5.5 The multiplication property Example 4.1: consider a signal x[n] which is the product of two signals, Determine the Fourier transform. 49
5.5 The multiplication property Solution: choosing Let We have 50
5.5 The multiplication property 51
5.5 The multiplication property 52
5.6 Summary of the properties X(t), y(t) Linearity Time shifting Frequency shifting Conjugation Time reversal e Time expansion n is multiple of k not Convolution 53
5.6 Summary of the properties Multiplication Differencing in time Accumulation Differentiation in frequency Conjugate Symmetry for Real Signals real 54
5.6 Summary of the properties Symmetry for Real and Even Signals Real and Odd Signals Even-odd Decomposition of Real Signal [x[n] real] Parseval s Relation for nonperiodic Signals x[n] real and even x[n] real and odd real and even imaginary and odd 55
5.6 Summary of the properties Signal Fourier transform Fourier series coefficients 56
5.6 Summary of the properties Signal Fourier transform Fourier series coefficients 57
5.6 Summary of the properties Signal Fourier transform Fourier series coefficients 58
5.6 Summary of the properties Signal Fourier transform Fourier series coefficients 59
5.7 Duality Duality in the discrete-time time Fourier series The Fourier series coefficients of the periodic sequence a_k are the values of, i.e., are proportional to the values of the original signal reversed in time. 60
5.7 Duality Properties 61
5.7 Duality Example 7.1: determine the Fourier coefficients of following signal with N=9 Solution: 62
5.7 Duality We have 63
5.8 System characterization by Linear constant-coefficient t i t differential equation Describe the LTI system with inputoutput relationship given as Use Fourier transform to derive the frequency enc response 64
5.8 System characterization by Linear constant-coefficient t i t differential equation One-way: assume an input, then the output has a form of We can also use Fourier transform to determine 65
5.8 System characterization by Linear constant-coefficient t i t differential equation Example 8.1: consider a LTI system characterized by Determine the frequency response. Solution: 66
5.8 System characterization by Linear constant-coefficient t i t differential equation Example 8.2: consider a LTI system characterized by Determine the frequency response. Solution: 67
5.8 System characterization by Linear constant-coefficient t i t differential equation We factor the denominator as 68
5.8 System characterization by Linear constant-coefficient t i t differential equation Example 8.2: consider a LTI system characterized by Now we have an input of, determine the output. 69
5.8 System characterization by Linear constant-coefficient t i t differential equation Solution: 70