Lecture 4

# Lecture 4 - Lecture 4 Discrete-Time Fourier Transform...

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1 Lecture 4 – Discrete-Time Fourier Transform ECSE304 Signals and Systems II ECSE 304 Signals and Systems II Lecture 4: Discrete-Time Fourier Transform Reading: O and W, Secs. 5.1 through 5.7 Boulet, Chapter 12 Richard Rose McGill University Dept. of Electrical and Computer Engineering 2 Lecture 4 – Discrete-Time Fourier Transform ECSE304 Signals and Systems II Review DTFS pair for periodic signal x[n] with period N over arbitrary interval <N> 2 [] j kn N k kN xn ae π = = 2 1 j N k nN ax n e N = = – Discrete Coefficients are periodic with period N – Are scaled versions of the DFT coefficients – Finite sum so DTFS always exists for x[n] periodic a k lq = 1 0 2 e x p N k n kn Xx n j N = ⎛⎞ =− ⎜⎟ ⎝⎠ DTFT pair for an aperiodic signal x[n] X e e d jj n ( ) = z 1 2 2 ω ωω Xe xne n n () [ ] = =−∞ +∞ – Continuous valued function is periodic with period – The DTFT exists if x[n] has finite energy or is absolutely summable j 2

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3 Lecture 4 – Discrete-Time Fourier Transform ECSE304 Signals and Systems II Show your work on all problems Hand in both your code and your plots for the MatLab Exercises MatLab Intro tutorial notes posted under Module 1 You can work in groups of 2 (hand in with cover page) • Problem M1: Note that you need to compute the DTFT Magnitude and Phase analytically for Part a) Discussion of Matlab Exercises 4 Lecture 4 – Discrete-Time Fourier Transform ECSE304 Signals and Systems II The fast Fourier transform (FFT) used in MatLab is a more efficient way for computing the DFT: It reorganizes the computation so exponential terms are not evaluated multiple times – You will see this in ECSE-412 Computational complexity: – DFT: operations – FFT: operations This was important back in 1965 when it was discovered by Cooley and Tukey Importance of the Fast Fourier Transform 1 0 2 [] e x p N k n kn Xx n j N π = ⎛⎞ =− ⎜⎟ ⎝⎠ 2 ~ N log NN PDP8 Computer FFT Computation
5 Lecture 4 – Discrete-Time Fourier Transform ECSE304 Signals and Systems II Problem M2: Sampling Continuous Time Signals Sinusoid with frequency 100 Hz ( rad/sec) Samples of sinusoid with sampling freq. 2000 s fH z = 200 π Samples of sinusoid with sampling freq. 500 s z = 20 samples per period 5 samples per period 6 Lecture 4 – Discrete-Time Fourier Transform ECSE304 Signals and Systems II • Discrete Time Fourier Transform of Periodic Signals • Examples of DTFT of Periodic Signals • Duality • DTFT Examples: – Multiplication Property – Data Windows and Spectrum Estimation Outline

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7 Lecture 4 – Discrete-Time Fourier Transform ECSE304 Signals and Systems II • We can interpret the DTFT of a periodic signal as an impulse train in the frequency domain (?!?!) • To do this, start with the complex exponential: which is periodic.
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Lecture 4 - Lecture 4 Discrete-Time Fourier Transform...

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