(9-10) Fourier Analysis

(9-10) Fourier Analysis

1. Examples

2. Fourier Series

   • Periodic signals
   • Composition of equation (cosine)
   • Fourier coefficients
   • Amplitude and Phase Spectrum
     • Synthesis of the signal
   • Complex numbers
   • Exponential Fourier series
     • How to derive this
     • Purpose of negative frequencies

3. Fourier Transform

   • Generalisation of Fourier Series for non-periodic signals
     • Effect on time and fundamental frequencies
   • Forward and inverse FT
   • FT of a Guassian
     • Time-frequency uncertainty

4. Sampling & Discrete Signals

   • Process of sampling in a circuit
   • Sampling theorem
     • Consequence of sample at low frequency
   • Anti-aliasing
     • Why required
     • How performed

5. Discrete FT

   • Time and band-limited
   • Can be represented by discrete spectrum
   • Want to fit Fourier to the sample
   • Picket fence effect
   • How to get amplitude and phase spectrum
   • Aim is to convert from Xr to xn and the inverse

6. Short-time FT

   • Spectrum within certain time window
   • Use of windowing function
     • Sidelobe suppression
   • Trade off of window width
   • Spectrogram
   • Time-frequency analysis applications

7. Fourier Analysis of 2D images

   • Applications