[DSP] W07 - Sampling and Interpolation

central topics of signal processing
- sampling: continuous-time to discrete-time
- interpolation: discrete-time tp continuous-time
sampling and interpolation establish an interaction between the continuous-time world and the discrete-time world

interpolation describes the process of building a continuous-time signal (x(t)) from a sequence of samples (x[n])
- allows moving from the discrete-time world to the continuous-time world
interpolation raises two questions:
- how to interpolate between samples?
  - two samples: a straight line that goes between these two samples
  - three samples: a parabola goes through these 3 samples
  - many samples: go through all samples but this is trickier compared to two or three samples
- is there a minimum set of values to measure function to perfectly reconstruct it

sampling is the process of moving from a continuous-time signal to a sequence of samples
- allows moving from the continuous-time world to the discrete-time world
consider equally-spaced samples of a function (x(t))
- the question that comes up is when is there a one-to-one relationship between the continuous-time function and its samples?
- i.e. when do the samples form a unique representation of the continuous-time function
to answer this, tools such as hilbert spaces, projections, filtering, sinc functions, and so on are used
- these tools come together in the sampling theorem

the shannon sampling theorem’s history goes back well before shannon
numerical analysts were concerned about interpolating tables of functions
the first (propaganda) one to prove a version of the sampling theorem was whittaker in England in 1915
harry nyquist at bell labs came up with the nyqvist criterion
- namely that a function that has a maximum frequency (F_0) could be sampled at (2F_0)
in the soviet union, kotelnikov proved a sampling theorem
the son of the first whittaker further proved results on the sampling theorem
- then herbert raabe in berlin wrote his phd thesis about a sampling theorem
- denis gabor worked on a version of the sampling theorem in the mid 1940s
then claude shannon, the inventor of information theory, wrote a paper that is to be explored later
in 1949 someya in japan also proved the sampling theorem
so sampling theorem is a fundamental result where many people independently came up with it

references