Lecture 11

Lecture 11 - Lecture 11- Review ECSE304 Signals and Systems...

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1 Lecture 11- Review ECSE304 Signals and Systems II ECSE 304 Signals and Systems II Lecture 11: Review Richard Rose McGill University Dept. of Electrical and Computer Engineering 2 Lecture 11- Review ECSE304 Signals and Systems II • Discrete-Time Fourier Series (DTFS) • Discrete-Time Fourier Transform (DTFT) – DTFT of Periodic Sequences • Bilateral Z-Transform – Properties, Region of Convergence, Inverse Z-Transform • Z-Transforms and DLTI Systems – Causality, Stability, Transfer Function Characterizations, Block Diagram Representations • Unilateral Z-Transform – Solution to linear constant coefficient difference equations • Time and Frequency Analysis of DLTI Systems – Approximations to Ideal Filters, FIR Filter Design Topics
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3 Lecture 11- Review ECSE304 Signals and Systems II • Examination “Rules” – Closed-book – A one page (2-sided) hand written crib sheet permitted – Faculty-approved calculators only • Provided with Exam: – Discrete-Time Fourier Series Properties – Discrete-Time Fourier Transform Properties – Discrete-Time Fourier Transform Pairs – Properties of the Unilateral Z-Transform – Z-transform properties – Z-transform pairs Midterm I 4 Lecture 11- Review ECSE304 Signals and Systems II • D-T Fourier Series Pair with Fourier series coefficients for the periodic signal is: • The coefficients are periodic with period N • Convergence is not an issue: All the summations are finite D-T Fourier Series a k kN lq = x n [] xn ae k jk n k jk N n == ∑∑ ω π 0 2 a N xne N k jk n nN jk N n = = 11 0 2 a k =
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5 Lecture 11- Review ECSE304 Signals and Systems II into harmonically related complex exponentials: [] x n • Determine the DTFS coefficients for the signal: – Fundamental Frequency: – Period: – Find DTFS by inspection: – Pick a range of N consecutive integers: – DTFS over <N>: – Expand D-T Fourier Series – Example 13 [] 1 c o s ( ) 12 8 xn n π =+ + 0 j kn k kN ae ω = = 0 /12 = 0 2/ N πω = 24 = { } 11,. .., 1,0,1,. ..,12 N =− 0 12 11 j k k = ∑ 33 12 8 12 8 1 [] 1 2 nn jj e e ππ ⎛⎞ ⎛⎞ +− + ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ ⎡⎤ + ⎢⎥ ⎣⎦ 00 0 (0) (1) ( 1) 88 11 1 22 j nj n ee e ωω + 6 Lecture 11- Review ECSE304 Signals and Systems II • Determine the DTFS coefficients for the signal: – Collect terms: – DTFS: D-T Fourier Series – Example o s ( ) 12 8 n + 0 (1) ( 1) j n e e e + 0 k = 1 k = 1 k {} 0 3 8 1 3 8 1 1, 1 , 2 1 , 2 0, j k j k a a ao t h e r w i s e = = = = = =
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7 Lecture 11- Review ECSE304 Signals and Systems II • Given a finite duration a periodic sequence, , such that outside of the range • Construct a periodic signal, , of period that is equal to for one period … • We develop the Fourier Transform based on the Fourier Series of a D-T sequence whose period tends to infinity The D-T Fourier Transform [] x n ± x n xn [] 0 = 12 Nx nN ≤≤ ± N 8 Lecture 11- Review ECSE304 Signals and Systems II • Discrete-Time Fourier Transform Pair: is periodic with period Xe xne e jj n n jn j n n n j ( ) ( ) () ωπ ω π +− + =−∞ +∞ −− +∞ +∞ == = = ∑∑ 22 2 The D-T Fourier Transform X e e d n ( ) =
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Lecture 11 - Lecture 11- Review ECSE304 Signals and Systems...

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