Upsampling with an emphasis on the frequency domain
By: Michael Deufel
- Introduction
- Derivation
- Examples
- Conclusion
1. Introduction
The purpose of Upsampling is to manipulate a signal in order to artificially increase the sampling rate. This is done by...
- Discretize the signal
- Pad original signal with zeros
- Take the DTFT
- Send through a LPF (low pass filter)
- Take the inverse DTFT to return to the time domain
We will overview the whole process but focus on the effect upsampling has in the frequency domain
2. Derivation
We will start with discrete signal $ x_1[n] $
now we "pad with zeros" to define $ x_2[n] $
$ x_2[n] = \begin{cases}x[\frac{n}{D}], & \text{if} \frac{n}{D} \in \mathbb{Z} \\0, &\text{else} \end{cases} f $
$ x_2[n] $ can also be defined by
$ x_2[n] = \sum_{n = -\inf}^{\inf} $
