#### Table of Contents

#### Introduction

This blog derives the frequency response of a complex sinusoid using the discrete Fourier transform (DFT). The derivation starts by apply the discrete time Fourier transform (DTFT) of the complex sinusoid, simplifying the summation as a geometric series, and evaluating the DTFT at specific frequencies of the DFT.

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#### Discrete Time Fourier Transform

The discrete-time Fourier transform (DTFT) is defined as [Oppenheim1999, p.48]

(1)

A realizable (ex: can be built in the real world) digital system must deal with signals which are finite length. A finite length complex sinusoid is defined according to

(2)

(3)

#### Geometric Series

#### Applying L'Hospital's Rule

Equation (9) is undefined when therefore L’Hospital’s rule is applied,

(10)

Simplifying (10),

(11)

The frequency response can therefore be written as

(12)

#### Discrete Fourier Transform (DFT) of Complex Sinusoid

The DFT is a simplified case of the DTFT where the frequencies are only evaluated at

(13)

where , [Oppenheim1999, p.542]. Substituting (13) into (12),

(14)

The DFT (14) can be further simplified as

(15)

#### Comparing to Simulation

Figures 1-3 compare equation (15) to NumPy’s DFT function for sinusoids with frequencies and N=4096.

You may have noticed that all of the sinusoid’s energy is contained within a single bin when is of the form as in Figures 1 and 2. For Figure 3, the energy is spread across multiple frequency bins. Figure 4 shows a zoomed in version of Figure 3 for more clarity.

The sinusoid’s energy is spread across multiple bins because the sinusoid does not complete an integer number of cycles of frequency in N samples. This effect is related to the frequency resolution of the DFT.

#### Conclusion

The discrete time Fourier transform (DTFT) and discrete Fourier transform (DFT) of a complex sinusoid were derived. The analytic equation was compared for correctness against a numerical DFT function.

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