1. Signals and Systems Lecture #5 1. Complex Exponentials as Eigenfunctions of LTI Systems 2. Fourier Series representation of CT periodic signals 3. How do we calculate the Fourier coefficients? 4. Convergence and Gibbs’ Phenomenon

2. Two Key Questions What are the eigenfunctions of a general LTI system?What are the eigenfunctions of a general LTI system? What kinds of signals can be expressed asWhat kinds of signals can be expressed as superpositions of these eigenfunctions?

7. Obtaining the Fourier series coefficients Orthogonality in the Hilbert space:

8. Continuous time Fourier transform (CTFT) Motivation: Extends the notion of the frequency response of a system to the frequency content of a signal Widely used tool in many areas – –Traditional EECS areas — communications, control, signal processing – –X-ray diffraction – –Medical imaging — CAT & PET scan ( Computed Axial Tomography ) ( Positron Emission Tomography ) ( Computed Axial Tomography ) ( Positron Emission Tomography ) MRI (Magnetic Resonance Imaging) NMR (Nuclear Magnetic Imaging) MRI (Magnetic Resonance Imaging) NMR (Nuclear Magnetic Imaging)

9. Outline: Example — How to filter the ECG? The continuous time Fourier transform (CTFT) Properties of the CTFT Simple CTFT pairs Conclusion

10. Example — How to filter the ECG? The recorded activity from the surface of the Chest includes the electrical activity of the heart plus extraneous signals or “noise.” How can we design a filter that will reduce the noise?

11. Motivation It is most effective to compute the frequency content of the recorded signal and to identify those components that are due to the electrical activity of the heart and those that are noise. Then the filter can be designed rationally. This is one of many motivations for understanding the Fourier Transform.

12. The continuous time Fourier transform (CTFT) Definition The continuous time Fourier Transform of x(t) is defined as: and the inverse transform is defined as: