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it can be exactly reconstructed from its samples. | it can be exactly reconstructed from its samples. | ||
| + | To recover the signal, the sampled function is simply multiplied by a low-pass filter with height equal to the sampling period T, <math>{H_r}(f)</math>, to isolate the original signal. | ||
| + | |||
| + | <math>{X_r}(f) = {H_r}(f){X_s}(f)</math> | ||
| + | |||
| + | [[Image:ReconstructFilter.png|500px|thumb|center|Sampled function with lowpass filter]] | ||
| + | |||
| + | '''Mathematical Process for finding <math>{X_r}(f)</math>:''' | ||
| + | |||
| + | <math>{X_r}(f) = {H_r}(f) \cdot {X_s}(f)</math> where <math>{H_r}(f) = T {rect}(Tf)</math> | ||
| + | |||
| + | <math>x_r(t) = h_r(t) * x_s(t) = sinc(t/T) * \sum_{n=-\infty}^{\infty} x(nT)\cdot \delta(t - nT)</math> | ||
Revision as of 18:08, 22 September 2009
HW3_Signal Reconstruction_Interpolation (Band-limited)
After having creating a sampled version of your original function, $ X_{s} $, we need to reconstruct the original function $ x(t) $. To do this, the Whittaker-Shannon interpolation formula is utilized.
The sampling theorem says that given a function that meets two requirements:
- 1) It is band-limited. This means that the Fourier transform of the original signal, also known as the spectrum, is 0 for |f| > B, where B is the bandwidth.
- 2) It is sampled at the Nyquist frequency, $ f_s > 2B $
it can be exactly reconstructed from its samples.
To recover the signal, the sampled function is simply multiplied by a low-pass filter with height equal to the sampling period T, $ {H_r}(f) $, to isolate the original signal.
$ {X_r}(f) = {H_r}(f){X_s}(f) $
Mathematical Process for finding $ {X_r}(f) $:
$ {X_r}(f) = {H_r}(f) \cdot {X_s}(f) $ where $ {H_r}(f) = T {rect}(Tf) $
$ x_r(t) = h_r(t) * x_s(t) = sinc(t/T) * \sum_{n=-\infty}^{\infty} x(nT)\cdot \delta(t - nT) $
