2025-L04su10

# 2025-L04su10 - 1 1 ECE2025 Summer 2010 Lecture 4 Periodic...

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Unformatted text preview: 1 5/25/2010 1 ECE2025 Summer 2010 Lecture 4 Periodic Signals, Harmonics & Time-Varying Sinusoids 26 May 10 5/25/2010 3 ANNOUNCEMENTS Â¡ Special case: HW #1 due June 3, Wed, at start of lecture Â¡ But pretend itâ€™s due sooner! Â¡ No lecture Monday May 31 (or Friday May 28) Â¡ Enjoy your Memorial Day weekend! Â¡ Go to first lab today (Wed) or Thursday Â¡ Lab due the following week at the start of your lab (June 2 or 3) 5/25/2010 4 READING ASSIGNMENTS Â¡ This Lecture: Â¡ Chapter 3, Sections 3-2 and 3-3 Â¡ Chapter 3, Sections 3-7 and 3-8 Â¡ Next Lecture: Â¡ Fourier Series ANALYSIS Fourier Series ANALYSIS Â¡ Sections 3-4, 3-5 and 3-6 5/25/2010 5 Review of the SPECTRUM Â¡ Add the spectrum components: What is the formula for the signal x(t)? 100 250 â€“100 â€“250 f (in Hz) 3 / 7 Ï€ j e 3 / 7 Ï€ j e âˆ’ 2 / 4 Ï€ j e âˆ’ 2 / 4 Ï€ j e 10 2 5/25/2010 6 Add Spectral Components t j j t j j t j j t j j e e e e e e e e t x ) 250 ( 2 2 / ) 250 ( 2 2 / ) 100 ( 2 3 / ) 100 ( 2 3 / 4 4 7 7 10 ) ( Ï€ Ï€ Ï€ Ï€ Ï€ Ï€ Ï€ Ï€ âˆ’ âˆ’ âˆ’ âˆ’ + + + = 100 250 â€“100 â€“250 f (in Hz) 3 / 7 Ï€ j e 3 / 7 Ï€ j e âˆ’ 2 / 4 Ï€ j e âˆ’ 2 / 4 Ï€ j e 10 5/25/2010 7 Use Eulerâ€™s Formula to get REAL sinusoids: Simplify Components t j j t j j t j j t j j e e e e e e e e t x ) 250 ( 2 2 / ) 250 ( 2 2 / ) 100 ( 2 3 / ) 100 ( 2 3 / 4 4 7 7 10 ) ( Ï€ Ï€ Ï€ Ï€ Ï€ Ï€ Ï€ Ï€ âˆ’ âˆ’ âˆ’ âˆ’ + + + = t j j t j j e Ae e Ae t A Ï‰ Ï• Ï‰ Ï• Ï• Ï‰ âˆ’ âˆ’ âˆ’ + = + 2 1 2 1 ) cos( 5/25/2010 8 FINAL ANSWER So, we get the general form: âˆ‘ = + + = N k k k k t f A A t x 1 ) 2 cos( ) ( Ï•...
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2025-L04su10 - 1 1 ECE2025 Summer 2010 Lecture 4 Periodic...

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