Applications of Fourier Transform

Outline Sampling Bandwidth Energy density Power spectral density

Putting Everything Together

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.

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

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

Rectangular Pulses and their Frequency Spectra (Figure 5.6)

Bandwidth of a Rectangular Pulse (Figure 6.23)

Energy Spectral Density of a Rectangular Pulse

Time Truncation of a Power Signal (Figure 5.34)

Calculation of Power Spectral Denstiy

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

Power Spectral Density

Spectral Reshaping