To design a 5-tap FIR (Finite Impulse Response) High Pass Filter (HPF) using the Fourier transform method, you can follow these steps:
1. Determine the desired filter specifications: In this case, the cut-off frequency is 1000 Hz, and the sampling rate is 9000 Hz.
2. Calculate the normalized cut-off frequency: Divide the desired cut-off frequency by half of the sampling rate to get the normalized cut-off frequency. In this case, the normalized cut-off frequency is 1000 Hz / (9000 Hz / 2) = 0.2222.
3. Determine the ideal frequency response: For a high-pass filter, the ideal frequency response is a passband of 1 above the normalized cut-off frequency and a stopband of 0 below the normalized cut-off frequency.
4. Take the inverse Fourier transform of the desired ideal frequency response: This will give you the impulse response of the ideal filter in the time domain.
5. Determine the filter length: In this case, you want a 5-tap filter, so the impulse response should have a length of 5.
6. Calculate the filter coefficients: Use any suitable method, such as the windowing method or the frequency-sampling method, to determine the filter coefficients from the impulse response. In this case, you can use the Fourier transform method.
Using the Fourier transform method, you can calculate the filter coefficients by taking the inverse Fourier transform of the desired ideal frequency response. This can be done using software tools like MATLAB or Python.
Here are the filter coefficients for a 5-tap FIR HPF with a cut-off frequency of 1000 Hz and a sampling rate of 9000 Hz:
b_0 coefficient = -0.1567
b_1 coefficient = -0.2046
b_2 coefficient = 0.7778
b_3 coefficient = -0.2046
b_4 coefficient = -0.1567
These coefficients represent the weights assigned to the input samples in the filter's impulse response. To implement the filter, you would multiply the current and previous input samples by these coefficients, sum the results, and output the final value.
