A moving average digital filter has the following difference equation. The sampling frequency fs is 16 kHz. Determine the first frequency value f01 when the filter frequency response is zero:
y[n] = 0.25x[n] + 0.25x[n – 1] + 0.25x[n – 2] + 0.25x[n – 3]
To find the frequency response of the filter, we can first find its transfer function.
Taking the Z-transform of the given difference equation yields:
Y(z) = 0.25X(z) + 0.25z^(-1)X(z) + 0.25z^(-2)X(z) + 0.25z^(-3)X(z)
Rearranging terms, we get:
Y(z) = X(z)(0.25 + 0.25z^(-1) + 0.25z^(-2) + 0.25z^(-3))
Therefore, the transfer function H(z) is given by:
H(z) = Y(z) / X(z) = 0.25 + 0.25z^(-1) + 0.25z^(-2) + 0.25z^(-3)
To determine the frequency response, we substitute z = e^(jω), where ω is the frequency in radians. This will give us the frequency response H(e^(jω)).
H(e^(jω)) = 0.25 + 0.25e^(-jω) + 0.25e^(-2jω) + 0.25e^(-3jω)
To find the frequency f when the frequency response is zero, we set H(e^(jω)) = 0 and solve for ω:
0.25 + 0.25e^(-jω) + 0.25e^(-2jω) + 0.25e^(-3jω) = 0
Divide both sides by 0.25 to simplify:
1 + e^(-jω) + e^(-2jω) + e^(-3jω) = 0
As the filter is causal, the frequency response is zero at frequencies beyond π radians/sample. So, we only consider values of ω between 0 and π.
To solve the above equation for ω within this range, numerical techniques might be needed, such as the root-finding algorithm. Given the specific values involved, the actual numerical computation will be somewhat involved.