Signals and Systems

							          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
            Two Key Questions

• What are the eigenfunctions of a general LTI
  system?



• What kinds of signals can be expressed as
  superpositions of these eigenfunctions?
Obtaining the Fourier series coefficients
        Orthogonality in the Hilbert space:
   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)
               MRI (Magnetic Resonance Imaging) NMR (Nuclear Magnetic Imaging)
               Outline:


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