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]
f01 = fs/N = 4kHz
To find the first frequency value where the filter frequency response is zero, we need to analyze the frequency response of the moving average filter.
The output of the filter in the frequency domain can be represented by the following equation:
Y(e^jω) = X(e^jω) * H(e^jω)
Where Y(e^jω) is the Fourier Transform of the output signal y[n], X(e^jω) is the Fourier Transform of the input signal x[n], and H(e^jω) is the frequency response of the filter.
Substitute the given difference equation into the Fourier Transform equation:
Y(e^jω) = 0.25X(e^jω) + 0.25X(e^jω)e^(-jω) + 0.25X(e^jω)e^(-j2ω) + 0.25X(e^jω)e^(-j3ω)
Simplify the equation:
Y(e^jω) = X(e^jω) * (0.25 + 0.25e^(-jω) + 0.25e^(-j2ω) + 0.25e^(-j3ω))
To find the frequencies where the filter frequency response is zero, we need to find the values of ω that make the expression inside the parentheses equal to zero.
0.25 + 0.25e^(-jω) + 0.25e^(-j2ω) + 0.25e^(-j3ω) = 0
This equation can be solved by setting it equal to zero and finding the values of the angular frequencies ω that satisfy this condition.
Therefore, the first frequency value f01 when the filter frequency response is zero would be when ω = π.
f01 = π * fs / (2π) = (π * 16 kHz) / (2π) = 8 kHz