Properties of DTFT ¾ Periodicity ¾ Linearity ¾ Time shifting ¾ Frequency

Properties of dtft ¾ periodicity ¾ linearity ¾

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Properties of DTFT ¾ Periodicity ¾ Linearity ¾ Time shifting ¾ Frequency shifting ¾ Parseval’s Relation 20-Mar-19 EIE3001 Sig & Sys, Spring 2019 27
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Conjugation and conjugate symmetry 20-Mar-19 EIE3001 Sig & Sys, Spring 2019 28 ¾ The conjugate property ¾ If x [ n ] is real, then its transform is conjugate symmetric The real part of the transform is even, and the imaginary part is odd Equivalently, the amplitude of the transform is even, the phase is odd.
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Time Expansion 20-Mar-19 EIE3001 Sig & Sys, Spring 2019 29
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Time Expansion 20-Mar-19 EIE3001 Sig & Sys, Spring 2019 30
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Time Expansion 20-Mar-19 EIE3001 Sig & Sys, Spring 2019 31
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Time Expansion 20-Mar-19 EIE3001 Sig & Sys, Spring 2019 32
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Convolution Property 20-Mar-19 EIE3001 Sig & Sys, Spring 2019 33
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Ideal Lowpass Filter 20-Mar-19 EIE3001 Sig & Sys, Spring 2019 34
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Ideal Lowpass Filter and Highpass Filter 20-Mar-19 EIE3001 Sig & Sys, Spring 2019 35
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Ideal Lowpass Filter in Time Domain 20-Mar-19 EIE3001 Sig & Sys, Spring 2019 36
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First-order System Approximating Filtering ¾ Consider a discrete-time differential equation ¾ DTFT: ¾ Solution: ¾ LTI system ¾ Time domain solution 20-Mar-19 EIE3001 Sig & Sys, Spring 2019 37
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First-order System Approximating Filtering 20-Mar-19 EIE3001 Sig & Sys, Spring 2019 38
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Time Shift Filter 20-Mar-19 EIE3001 Sig & Sys, Spring 2019 39
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Multiplication/Modulation Property ¾ Continuous-Time ¾ Discrete-Time 20-Mar-19 EIE3001 Sig & Sys, Spring 2019 40
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Modulation Example ¾ Consider the periodic signal with N = 2 The Fourier series give us Fourier transform: 20-Mar-19 EIE3001 Sig & Sys, Spring 2019 41
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Modulation Example ¾ Consider a second signal x 2 [ n ] with DTFT By computing the convolution of the above two spectrum graphs, the spectrum of the signal y [ n ]= x 1 [ n ] x 2 [ n ] can be determined as 20-Mar-19 EIE3001 Sig & Sys, Spring 2019 42
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Ideal Filter Converted from Modulation ¾ Use of modulation of a system impulse response to convert the system from a lowpass to a highpass filter 20-Mar-19 EIE3001 Sig & Sys, Spring 2019 43
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First-order System Approximation of Ideal Filters 20-Mar-19 EIE3001 Sig & Sys, Spring 2019 44
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A Summary 20-Mar-19 EIE3001 Sig & Sys, Spring 2019 45
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