- Deriving Filter specifications
- Deriving the Kaiser Window parameters
- Polyphase Filter Implementation
- Filter Evaluation
- Deriving the Ideal Passband Gain, Passband, and Stopband Edge frequencies for the Anti-Imaging filters to be designed
- Designing Anti-Imaging filters with two different Stopband attenuations, 𝐻30(𝑧) and 𝐻60(𝑧)
- Implementing the designed Anti-Imaging filters in Polyphase structure and the M-fold interpolators using the Efficient structure
- Evaluating the performance of the M-fold interpolators in-terms of the ability to re-sample the original signal and the computational complexities.
The passband gains, passband and stopband edges with widest possible transition width are derived in this section. To derive them, I employed the concepts of the sampling theory. For the complete derivation, check the final report.
Parameter | Symbol | Value | Units |
---|---|---|---|
Upsampling factor | M | 4 | - |
Fundamental frequency | Ω0 | 60π | rad/s |
Sampling frequency | Ωs | 200π | rad/s |
Passband Gain | Gp | 4 | - |
Stopband Gain | Gs | 0 | - |
Passband Edge Frequency | Ωp | 188.496 | rad/s |
Cut-off Frequency | Ωc | 314.159 | rad/s |
Stopband Edge Frequency | Ωss | 439.823 | rad/s |
Transition width | BT | 80 | rad/s |
Passband Ripple | Ap | 0.1 | dB |
Stopband Attenuation for filter 1 | As,30 | 30 | dB |
Stopband Attenuation for filter 2 | As, 60 | 60 | dB |
Parameter | Value | Units | Parameter | Value | Units | |
---|---|---|---|---|---|---|
𝛿̃𝑝 | 0.00576 | - | 𝛿̃𝑝 | 0.00576 | - | |
𝛿̃a, 30 | 0.03162 | - | 𝛿̃a, 30 | 0.001 | - | |
𝛿30 | 0.00576 | - | 𝛿30 | 0.001 | - | |
Aa, 30 | 44.79 | dB | Aa, 30 | 60 | dB | |
Ap, 30 | 0.1 | dB | Ap, 30 | 0.1 | dB | |
𝑁𝑇, 30 | 22 | - | 𝑁𝑇, 30 | 30 | - | |
𝑁𝑃, 30 | 26 | - | 𝑁𝑃, 30 | 38 | - |
Based on the polyphase decomposition the two filters 𝐻30(𝑧) and 𝐻60(𝑧) were implemented following a Type-I design which is shown below.
Before taking the RMSE, a correction to the output sequences should be done by removing the group delays of each filter from the corresponding output sequence. The group delays obtained were 13 and 19 samples for 𝐻30(𝑧) and 𝐻60(𝑧) respectively. Even though the polyphase filters may have their own group delays, the final group delay affecting the output of the LTI system is same as the overall filter implemented as it is.
After making the adjustment, samples from 1000 ≤ 𝑛 ≤ 3000 from both the 𝑥𝑢[𝑛] and 𝑦𝑖[𝑛] are taken to measure the RMSE.
- Evaluating the computational complexity between original implementation and the efficient implementation for each of the filter
- Comparing the computational complexity between the two filters 𝐻30(𝑧) and 𝐻60(𝑧) in their efficient implementation.
Further, the results depicts that the M-fold Interpolator is not only computationally efficient but also provides a very close representation of a sequence sampled at a sampling frequency M times higher than the sampling frequency used to obtain input sequence.
When designing the Anti-Imaging filters, the flexibility of the Kaiser Window has been incorporated. Since the ideal filters cannot be practically implemented, it is advantageous to be able to make a flexible filter of which the limitations can be controlled. This is a practical approach since small imperfections such as pass band ripple will not cause an observable difference in the filtered output. This means that the parameters of the filter can be adjusted until the differences between the filter output and an ideal output become indistinguishable for all the practical purposes.
[1] Alan V. Oppenheim and Ronald W. Schafer. 2009. Discrete-Time Signal Processing (3rd. ed.). Prentice Hall Press, USA.
[2] Adams, M. D. (2013). Multiresolution signal and geometry processing: filter banks, wavelets, and subdivision (version: 2013-09-26). Michael Adams.