1 DREAM PLAN IDEA IMPLEMENTATION

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3 Introduction to Image Processing Dr. Kourosh Kiani Web: Present to: Amirkabir University of Technology (Tehran Polytechnic) & Semnan University

4 Lecture 08 Fourier Transform Lecture 08 Fourier Transform

5 Development of the Fourier Transform Representation of an Aperiodic Signal In the last lecture we saw how a periodic signal could be represented as a linear combination of cos(nω) and sin(nω). In fact, these results can be extended to develop a representation of aperiodic signals as a linear combination of cos(nω) and sin(nω).

6 Continuous Fourier transform T 1 T0T0 2T 0 T 1 -T 0 -2T 0 -3T 0 T0T0 T 1 T0T0

7 Since And also since x(t)=0 outside this interval, equation (2) can be rewritten as: Therefore, defining the envelope X(ω) of T 0 a k as: We have that the coefficients a k can be expressed as: Combining (1) and (4), can be expressed in the term X(ω) as:

8 Or equivalently, since As, approaches x(t), and consequenetly eq. (7) becomes a representation of x(t). Furthermore, as and the right-hand side of eq (7) passes to an integral. Each term in the summation on the right-hand side of eq. (7) is the area of a rectangle of height and width (here t is regarded as fixed). As this by definition converges to the integral of. Therefore, using the fact that as eq. (7) and (4) become

9 Fourier Transform

10 Comments

11 Example

12 Samples of Fourier Transforms of Aperiodic Signals Spectrum 0 f3f5f 0 f 3f5f

13 CTFT Properties

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26 Example: the Fourier Transform of a rectangle function: rect(t) Imaginary Component = 0 F()F() 

27 Example: the Fourier Transform of a decaying exponential: exp(-at) (t > 0) A complex Lorentzian!

Questions? Discussion? Suggestions ?

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