Frequency Domain View of Upsampling
Why Interpolator needs a LPF after Upsampling
A slecture by ECE student Chloe Kauffman
Partly based on the ECE438 Fall 2014 lecture material of Prof. Mireille Boutin.
Contents
Outline
- Background
- Introduction
- Derivation
- Example
- Conclusion
Background
$ {f}_{s} $ = sampling frequency (number of samples/second) Hz
$ {T}_{s} $ = sampling period (number of seconds/sample) seconds
$ {f}_{s} = {\frac{1}{{T}_{s}}} $
Sampling above Nyquist frequency guarantees a bandlimited sampled CT signal's reconstruction. **add source**
Define Nyquist Sampling rate as $ {f}_{s} = 2{f}_{M} $
$ {f}_{M} $ is max frequency of CT signal
Introduction
Sampling at frequencies much larger than Nyquist requires a filter for reconstruction with a less sharp cutoff. A digital LPF can be used to then obtain the reconstructed signal.
**add picture & source**
Assume $ {x}_{c}(t) $ is a bandlimited CT signal,
$ {x}_{1}[n] $ is a DT sampled signal of $ {x}_{c}(t) $ with sampling period $ {T}_{1} $
This leads to the question, can you use
$ {x}_{1}[n] = x_{c}(n{T}_{1}) $
to obtain
$ {x}_{u}[n] = {x}_{c}(n{T}_{u}) $, a signal sampled at a HIGHER sampling frequency than $ {x}_{1}[n] $, without having to fully reconstruct $ {x}_{c}(t) $
Derivation
