time_frequency_plane_sampling.md

Time-frequency plane sampling

Signal can be represented as a composition of sinusoid pulses in time-frequency plane.

Decompositions:

• time domain - energy as a function of time
  • energy from all frequencies for each time bin
  • vertical stripes of TF plane
  • disjoint coverage
• frequency domain - energy as a function of frequency
  • energy from all time for each frequency bin
  • obtained from time domain by Fourier transform
  • horizontal stripes of TF plane
  • disjoint coverage
  • basis - sinusoids
• Short-time Fourier Transform (STFT)
  • time divided into frames and windowed
  • regular rectangular grid of samples in TF plane
  • width/height proportion depends on the window size
  • disjoint coverage
  • basis - sinusoid pulses
• Wavelet transform (WT)
  • irregular sampling of the TF plane - hyperbolic grid
  • longer time interval, shorter frequency interval for lower frequencies
  • shorter time interval, longer frequency interval for higher frequencies
  • intervals sizes are scaled by two
  • disjoint coverage
• Constant-Q transform (CQT)
  • frequency bands dependent on their central frequency (constant quotient)
  • frequencies sampled logarithmically
  • time sampler regularly, but the window size is not constant
  • not disjoint coverage
• Nonstationary Gabor Transform (NSGT)
  • ??? - needs studing the papers
  • not disjoint
• Reassigned STFT
  • allows to compute precise location of TF plane sampled
  • energy can be then requantized in an arbitrary way
  • still dependent on window size
  • possible to go multi-scale
• Sparse FT
  • ???
  • http://groups.csail.mit.edu/netmit/sFFT/
• other possible approaches to sampling the TF plane:
  • stochastic sampling
    • eg. using samples with blue-noise distribution
  • adaptive sampling
  • iterative sampling
• general TF plane transformations
  • Fractional Fourier Transform
    • rotation
    • basis - chirps
  • Linear canonical transformation
    • linear transformation

image credit: http://dsp.stackexchange.com/questions/651/which-time-frequency-coefficients-does-the-wavelet-transform-compute

Desirable properties:

• good time and frequency localisation
• efficient computation
• efficient storage
• invertibility

References

• Time-Frequency and Time-Scale Methods: Adaptive Decompositions, Uncertainty Principles, and Sampling