Finite Impulse Response - Moving Average Example

Moving Average Example

Fig. (b) Pole-Zero Diagram Fig. (c) Amplitude and phase responses

A moving average filter is a very simple FIR filter. It is sometimes called a boxcar filter, especially when followed by decimation. The filter coefficients, are found via the following equation:

To provide a more specific example, we select the filter order:

The impulse response of the resulting filter is:

The Fig. (a) on the right shows the block diagram of a 2nd-order moving-average filter discussed below. To discuss stability and spectral topics we take the z-transform of the impulse response:

Fig. (b) on the right shows the pole-zero diagram of the filter. Zero frequency (DC) corresponds to (1,0), positive frequencies advancing counterclockwise around the circle to (-1,0) at half the sample frequency. Two poles are located at the origin, and two zeros are located at, .

The frequency response, for frequency ω in radians per sample, is:

Fig. (c) on the right shows the magnitude and phase plots of the frequency response. Clearly, the moving-average filter passes low frequencies with a gain near 1, and attenuates high frequencies. This is a typical low-pass filter characteristic. Frequencies above π are aliases of the frequencies below π, and are generally ignored or filtered out if reconstructing a continuous-time signal. The following figure shows the phase response. Since the phase always follows a straight line except where it has been reduced modulo π radians (should be 2π), the linear phase property is demonstrated.