3 Introduction to Image Processing Dr. Kourosh Kiani Email: firstname.lastname@example.org@yahoo.com Email: Kourosh.email@example.comKourosh.firstname.lastname@example.org Email: Kourosh.email@example.comKourosh.firstname.lastname@example.org Web: www.kouroshkiani.comwww.kouroshkiani.com Present to: Amirkabir University of Technology (Tehran Polytechnic) & Semnan University
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