# Wave Walker DSP

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Fourier Transform of the Boxcar Window
July 13, 2022

#### Introduction

This blog shows the time domain of the boxcar window, derives it’s frequency response as well as the magnitude-squared of the frequency response.

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#### Boxcar Window

The boxcar window of length N is defined as

(1)

Equation (1) is plotted in Figure 1 with N=16.

#### Fourier Transform of Boxcar Window

Since (1) is discrete-time the Discrete-Time Fourier Transform (DTFT) must be taken.

The DTFT is defined as (reference)

(2)

Applying (1) to (2),

(3)

Equation (3) is now a finite geometric series. The sum of a finite geometric series

(4)

can be written as (reference)

(5)

Comparing (3) to (4), a = 1 and . The summation from (5) can therefore be written as

(6)

#### Frequency Response Magnitude

The magnitude-squared of (6) is given by

(7)

Multiplying the terms from (7),

(8)

Gathering like terms,

(9)

(10)

which can be simplified to:

(11)

Equation (11) for N=16 is plotted in Figure 2. The computed magnitude is shown along side to show their equivalence and the correctness of the derivation.

#### Conclusion

The equation for the boxcar window was given and the frequency response and it’s magnitude-squared was derived. The magnitude-squared of the frequency response was plotted in both equation from and being computed using software to show their equivalence.

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