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

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

Aliasing

ω=0.9π

ωS=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 frequencyspectrum of a period signal

Power spectra density

Normalize Power withinless than 1000 rad/s

Weight of impulsefunction

Power Spectral Density

Spectral Reshaping