Applications of Fourier Transform

1. Applications of Fourier Transform

2. Outline • Sampling • Bandwidth • Energy density • Power spectral density

3. Putting Everything Together

4. Frequency Spectrum of Sampled Data Signal F(ω) is replicated at integers of ωS as the result of sampling. Overlap occurs when ωS is not fast enough.

5. Shannon’s Sampling Theorem • Let ωS be the sampling frequency • Let ωM be the highest frequency in the frequency spectrum of the signal to be sampled. • If we want to avoid aliasing, F(ω) needs to be bandlimited. • ωS should be larger than 2 ωM

6. Aliasing ω=0.9π ωS=0.8π Aliasing as a result of sampling.

7. Rectangular Pulses and their Frequency Spectra (Figure 5.6)

8. Bandwidth of a Rectangular Pulse (Figure 6.23)

9. Energy Spectral Density of a Rectangular Pulse

10. Time Truncation of a Power Signal (Figure 5.34)

11. Calculation of Power Spectral Denstiy

12. Power Spectral Density of Period Signal Weight of impulse function Magnitude frequency spectrum of a period signal Normalize Power within less than 1000 rad/s Power spectra density

13. Power Spectral Density

14. Spectral Reshaping