1 “Figures and images used in these lecture notes by permission, copyright 1997 by Alan V. Oppenheim and Alan S. Willsky” Signals and Systems Spring 2003 Lecture #8 Jacob White (Slides thanks to A. Willsky, T. Weiss, Q. Hu, and D. Boning)

2 Fourier Transform System Frequency Response and Unit Sample Response Derivation of CT Fourier Transform pair Examples of Fourier Transforms Fourier Transforms of Periodic Signals Properties of the CT Fourier Transform

3 The Frequency Response of an LTI System

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5 First Order CT Low Pass Filter Direct Solution of Differential Equation

6 Using Impulse Response Note map from unit sample response to frequency response

7 Fourier’s Derivation of the CT Fourier Transform x(t) - an aperiodic signal - view it as the limit of a periodic signal as T ! 1 For a periodic sign, the harmonic components are spaced  0 = 2  /T  apart... as T  and  o  0, then  = k  0  becomes continuous  Fourier series  Fourier integral

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9 Discrete frequency points become denser in  as T increases Square Wave Example

10 “Periodify” a non-periodic signal For simplicity, assume x(t) has a finite duration.

11 Fourier Series For Periodified x(t)

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13 Limit of Large Period

14 a) Finite energy In this case, there is zero energy in the error What Signals have Fourier Transforms? (1) x(t) can be of infinite duration, but must satisfy: c) By allowing impulses in x(t) or in X(j  ), we can represent even more signals b) Dirichlet conditions (including)

15 Fourier Transform Examples (a) (b) Impulses

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17 Fourier Transform of Right-Sided Exponential Even symmetryOdd symmetry

18 Fourier Transform of square pulse Useful facts about CTFT’s Note the inverse relation between the two widths  Uncertainty principle

19 Fourier Transform of a Gaussian (Pulse width in t)(Pulse width in  )  ∆t∆  ~ (1/a 1/2 )(a 1/2 ) = 1

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21 CT Fourier Transforms of Periodic Signals

22 Fourier Transform of Cosine

23 Note: (period in t) T  (period in  ) 2  /T Impulse Train (Sampling Function)

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25 Properties of the CT Fourier Transform FT magnitude unchanged Linear change in FT phase 1) Linearity 2) Time Shifting

26 Properties (continued) 3)Conjugate Symmetry Or When x(t) is real (all the physically measurable signals are real), the negative frequency components do not carry any additional information beyond the positive frequency components:  ≥ 0 will be sufficient. Even Odd Even Odd

27 More Properties 4)Time-Scaling a) x(t) real and even b) x(t) real and odd c)

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29 Conclusions System Frequency Response and Unit Sample Response Derivation of CT Fourier Transform pair CT Fourier Transforms of pulses, exponentials FT of Periodic Signals  Impulses Time shift, Scaling, Linearity