#### Table of Contents

#### Introduction

This blog shows how to approximate the 3 dB cutoff frequency of the moving average filter.

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#### Moving Average Filter Impulse Response

The boxcar filter or boxcar window is an FIR filter whose weights are all 1:

(1)

The filter weights in (1) are normalized in the time domain to create a moving average (MA) filter

(2)

#### Magnitude-Squared Frequency Response

The magnitude-squared of the frequency response for the boxcar window was derived in a previous blog,

(3)

The moving average is normalized by a factor of 1/N therefore the magnitude-squared of the moving average is normalized by a factor of ,

(4)

#### Gain at Zero-Frequency (omega=0)

There are multiple bandwidth definitions (see Bandwidth Measures). The focus of this blog is the half-power or 3 dB measure. The 3 dB bandwidth in decibels is

(5)

The moving average filter is a low-pass filter whose magnitude is the maximum value at . Therefore to measure the half-energy point the maximum value has to be calculated, which is (4) at ,

(6)

#### Finding the 3 dB Bandwidth

The 3 dB bandwidth is the frequency at which the magnitude-squared is 1/2 the maximum value,

(7)

After substituting (4) equation (7) can now be written as

(8)

which is simplified to

(9)

Finding a closed-form solution for (9) is not trivial (see Wolfram’s solution when N=5), but the bandwidth can be approximated as *approximately*

(10)

Figures 1-3 show the cutoff frequency approximation (10) for N=5, 10 and 16.

The cutoff approximation (10) is not an exact value for the frequency but it is a reasonable approximation.

#### Conclusion

The moving average filter is a normalized boxcar window or boxcar filter. The 3 dB point is where the magnitude-squared is 0.5. The 3 dB bandwidth of a moving average filter length N can be approximated by .

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