Difference Equation of FIR Filter
Definition
An FIR (Finite Impulse Response) filter is a type of digital filter whose impulse response lasts for a finite duration of time. The FIR filter is characterized by its impulse response coefficients.
Difference Equation
The general form of the difference equation for an FIR filter of length M, input x(n), and output y(n) can be expressed as:
y(n) = b(0)x(n) + b(1)x(n-1) + b(2)x(n-2) + ... + b(M-1)x(n-M+1)
where:
- y(n) is the output signal at time n
- x(n) is the input signal at time n
- M is the length of the FIR filter
- b(0), b(1), ..., b(M-1) are the filter coefficients
The difference equation represents the linear combination of the input signal samples with the filter coefficients to produce the output signal.
Explanation
The difference equation provides a mathematical representation of how the input signal is processed by the FIR filter to produce the output signal. Each input sample is multiplied by a corresponding filter coefficient, and the results are summed to generate the output sample. The filter coefficients determine the frequency response and behavior of the FIR filter.
In practice, the filter coefficients are designed based on the desired frequency response characteristics such as low-pass, high-pass, band-pass, or band-stop. By adjusting the coefficients, the FIR filter can effectively filter out unwanted frequencies from the input signal.
Overall, the difference equation of an FIR filter serves as a fundamental tool in digital signal processing for designing and implementing filters with specific frequency response properties.