How Do You Design a Digital FIR Filter to Remove a 70Hz Disturbance?

[BriWel]

I have a question which is to design a filter of the form:

y(n) = a(0)x(n) + a(1)x(n-1) + a(2)x(n-2) to remove a narrowband disturbance with frequency f0 = 70Hz.
The sampling frequency, fs is 280Hz.

I've made an attempt at answering it, but don't think my result is correct:

w0 = (2*pi)* (f0/fs) = pi/2

I then calculate the zeros of the filter from the definition of w

z1 = exp^(i*(pi/2))[
z2 = exp^(-i*(pi/z))
where i = sqrt(-1)

The transfer function of the filter is therefore;

Z²H(z) = (z - exp^(i*(pi/2))[ )(z - exp^(-i*(pi/z)) )

When expanded and simplified this gives

H(z) = z² + 1, so the filter function

y(n) = x(n) + x(n - 2),
giving
a(0) = 1
a(1) = 0
a(2) = 1

Which I'm pretty sure is wrong. Can anyone tell me where I have gone wrong?

 

[dsprao]

You did wrong while expanding and simplifying

 

[Mene]

Hello,

Thank you for your question about designing a digital FIR filter to remove a narrowband disturbance at 70Hz.

First, let's review the basics of FIR filters. FIR stands for Finite Impulse Response, which means that the output of the filter is only affected by a finite number of input samples. FIR filters are typically designed using the frequency sampling method, where the desired frequency response is sampled at specific frequencies and then used to determine the filter coefficients.

In this case, we want to design a filter to remove a disturbance at 70Hz. This disturbance can be represented as a sinusoidal signal with a frequency of 70Hz. The sampling frequency, fs, is 280Hz. This means that the sampling interval is T = 1/fs = 1/280 = 0.00357 seconds.

The first step in designing an FIR filter is to determine the filter length, N. This can be done using the formula N = fs/f0, where f0 is the frequency of interest. In this case, N = 280/70 = 4.

Next, we need to determine the desired frequency response of the filter. Since we want to remove the disturbance at 70Hz, we want a notch filter with a notch at 70Hz. This can be achieved by setting the desired frequency response to 0 at 70Hz and 1 at all other frequencies.

Using the frequency sampling method, we can then determine the filter coefficients by taking the inverse discrete Fourier transform (IDFT) of the desired frequency response. The IDFT of a notch filter with a notch at 70Hz and a filter length of 4 is given by:

h(n) = 1/4 [1, 1, 1, 1]

Therefore, the filter equation is:

y(n) = (1/4)x(n) + (1/4)x(n-1) + (1/4)x(n-2) + (1/4)x(n-3)

This is in the same form as the one provided in the question, except for the coefficients. So the correct filter coefficients for this design are:

a(0) = 1/4
a(1) = 1/4
a(2) = 1/4
a(3) = 1/4

I hope this helps to clarify the correct approach for designing a digital FIR filter. Please let me know if you have any further