Gibb’s Phenomenon Rectangular and Hamming Window Implementation
What is Gibb’s Phenomenon?
Gibb’s Phenomenon in Digital Filter’s refer to the manner how a Fourier series of periodic functions behave near jump discontinuation, the partial sum of the Fourier series has large oscillations near the discontinuation, which might increase the maximum of the partial sum above that of the function itself. Thus, there is an overshoot as the terms are getting added of the Fourier series at the discontinuity.
In the signal processing terms, we can translate Gibb’s Phenomenon as the step response of the filter & the truncation of our signal via Fourier series can be termed as filtering out the signal.
We will be understanding this via example problem:
Specifications of the Filter
- Digital Lowpass FIR filter with 41 taps.
- Cutoff frequency of the filter is pi/4
We will be implementing the solution using both the Rectangular & Hamming Window Technique.
What is Rectangular Window Technique?
In the Rectangular window, the Fourier Transform converges to sinc function, giving a rectangular shape at the edges thus the name Rectangular Window. The Rectangular is not widely used in the industry for filtering signals.
What is the Hamming Window Technique?
Hamming Window is a far more optimized approach to filter signals as it cuts off signal points on either side for us to see the more clear picture of the signal’s frequency spectrum.
Step 1: Importing all the necessary libraries.
Step 2: Define variables with the given specifications of the filter.
Step 3: Computations to calculate the magnitude, phase response to get Rectangular Window coefficients
Step 4: Plotting the Truncated Impulse Response, Frequency Response, Frequency Response of the Filter using The Rectangular Window Method
Step 5: Computations to calculate the magnitude, phase response to get Hamming Window coefficients
Step 6: Plotting the Truncated Impulse Response, Frequency Response, Frequency Response of the Filter using The Hamming Window Method