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 ω S as the result of sampling. Overlap occurs when ω S is not fast enough.

5. Shannon’s Sampling Theorem Let ω S be the sampling frequency Let ω M 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. ω S should be larger than 2 ω M

6. Aliasing ω=0.9π ω S =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 Magnitude frequency spectrum of a period signal Power spectra density Normalize Power within less than 1000 rad/s Weight of impulse function

13. Power Spectral Density

14. Spectral Reshaping