Fourier transform

• Analysis

  • from time domain to frequency domain

  • find the contribution of different frequencies

  • discover "hidden" signal properties

• Synthesis

  • from frequency domain to time domain

  • create signals with known frequency content

  • fit signals to specific frequency regions.

• Simple example: finite-length signals -> change basis-> change of perspective->reveal things.

• Analysis formula:

• Synthesis formula:

• Change of basis in matrix form

• Analysis formula:

• Synthesis formula:

• Analysis formula:

  • My thought: Here the minus on the exponential comes from the inner product definitions

• Synthesis formula:

• Short-Time Fourier Transform

  • Since the DFT does not include the time information, take small signal pieces of length L.

  • Spectrogram

    • Long window: narrowband spectrogram

      • long window (Large L) => more DFT poonts => more frequency resolution

      • long window => more "things can happen" => less precision in time.

    • Short window: wideband spectrogram

      • short window (small L0 => many time slides => precise location of transitions

      • short window => fewer DFT points => poor frequency resolution.

    • time "resolution" delta t = L

    • frequency "resolution" delta f = 2 pi / L

    • delta t * delta f = 2 pi