# Wave Walker DSP

## DSP Algorithms for RF Systems

Foundations of Digital Signal Processing: Complex Numbers is available Amazon now! Great for young engineers looking for a simple explanation of complex numbers.

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.