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## Discrete Fourier Transform

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In this chapter, the discrete Fourier transform (DFT) will be discussed. The Fourier transform discussed in the previous chapter is quite useful, but the applications will be limited for two reasons. First, the function in the time domain must be representable in closed form so that the Fourier integral can be performed. Thus, unless the input function can be written in closed form, it is impossible to evaluate the integral. Second, even if the time function can be written in closed form, it might also be difficult to find a closed-form solution to the integral.

Chapter Contents:

• 4.1 Introduction
• 4.2 Signal Digitization
• 4.3 Graphical Description of Discrete Fourier Transform (DFT)
• 4.4 Analytic Approach to Discrete Fourier Transform
• 4.5 Properties of Discrete Fourier Transform
• 4.5.1 Limited Frequency Bandwidth
• 4.5.2 Unmatched Time Interval
• 4.5.3 Overlapping Aliasing Effect on Real Data
• 4.6 Window Functions
• 4.6.1 Rectangular Window
• 4.6.2 Gaussian Window
• 4.6.3 Cosine Window Raised to the ath Order
• 4.6.4 Generalized Hamming Window
• 4.7 Fast Fourier Transform (FFT)
• 4.8.1 Initial Data Accumulation
• 4.8.2 Sliding DFT
• 4.9 Periodogram
• 4.10 Averaged Periodogram
• References

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