Re: Piecewise constant approximation

Hi,

I am not interested in reconstructing the original signal. I wish to
find the approximation rate - how the MSE decays with n ? I could prove
that with lipschitz assumption the error decays as 1/n^2. Any ideas for
continuous bandlimited assumption ?

Thanks

john2 wrote:
erudite wrote:
Let s(t) be a continuous bandlimited signal. Let p(t) be its best
piecewise constant approximation on the uniformly partitioned unit
interval [0,1],

i.e p(t) = \sum_{i=1}^n c_i I_i(t)

where I_i(t) is the indicator function which takes 1 on the interval
[(i-1)/n , i/n) and 0 elsewhere.

Now how does the MSE ||s(t)-p(t)||^2 vary with n ?

Thanks,
er


Sounds like a book exercise ? The piecewise constant signal is not BL
but, if filtered properly, the original signal can be reconstructed
without error.

As a first approximation, calculate the spectral energy outside the
upper band limit of the original signal.


john2

.