Solution 12 EIE3001 Sig Sys Spring 2019 Example 412 Given the FT of

# Solution 12 eie3001 sig sys spring 2019 example 412

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Solution: 12-Mar-19 39

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EIE3001 Sig & Sys, Spring 2019 Example 4.12 Given the FT of the square pulse x 0 ( t ) as , find the FT of x ( t ). Solution: Consider to differentiate x ( t ). We get 12-Mar-19 40 x 0 ( t ) t 1
EIE3001 Sig & Sys, Spring 2019 Time and Frequency Scaling Time reversal translates to frequency reversal. Time compression translates to frequency dilation, and vice versa. Proof: 12-Mar-19 41

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EIE3001 Sig & Sys, Spring 2019 Duality If then 12-Mar-19 42
EIE3001 Sig & Sys, Spring 2019 Example 12-Mar-19 43

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EIE3001 Sig & Sys, Spring 2019 Properties Implied by Duality 12-Mar-19 44
EIE3001 Sig & Sys, Spring 2019 Parseval’s Relation Total energy in time domain = total energy in frequency domain (up to a factor of ) 12-Mar-19 45 Proof:

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EIE3001 Sig & Sys, Spring 2019 Outline 1. From Fourier series to Fourier transform: aperiodic continuous time signals 2. Periodic signals 3. Properties of CT Fourier transform 4. Convolution Property and LTI Systems 5. Multiplication Property and Application in Modulation 12-Mar-19 46
EIE3001 Sig & Sys, Spring 2019 Convolution Convolution in time domain implies multiplication in frequency domain Conversely, multiplication in time domain implies convolution in frequency domain (will be discussed later) 12-Mar-19 47

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EIE3001 Sig & Sys, Spring 2019 Convolution: Intuition from LTI Systems Recall that is the eigenfunction of LTI systems. Take 12-Mar-19 48
EIE3001 Sig & Sys, Spring 2019 Convolution: the Proof 12-Mar-19 49

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EIE3001 Sig & Sys, Spring 2019 Associative and Commutative Properties of Convolution Revisited 12-Mar-19 50
EIE3001 Sig & Sys, Spring 2019 Example: Delay System 12-Mar-19 51

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EIE3001 Sig & Sys, Spring 2019 Example: Ideal Low Pass Filter 12-Mar-19 52
EIE3001 Sig & Sys, Spring 2019 Ideal Low Pass Filter is Non-causal 12-Mar-19 53 Approximate low pass filter:

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EIE3001 Sig & Sys, Spring 2019 Outline 1. From Fourier series to Fourier transform: aperiodic continuous time signals 2. Periodic signals 3. Properties of CT Fourier transform 4. Convolution Property and LTI Systems 5. Multiplication Property and Application in Modulation 12-Mar-19 54
EIE3001 Sig & Sys, Spring 2019 Multiplication Property Example: Modulation 12-Mar-19 55

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EIE3001 Sig & Sys, Spring 2019 Modulation (cont’d) 12-Mar-19 56
EIE3001 Sig & Sys, Spring 2019 Modulation 12-Mar-19 57

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EIE3001 Sig & Sys, Spring 2019 Demodulation 12-Mar-19 58
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