The DFT and its Inverse

GUIDE: Mathematics of the Discrete Fourier Transform (DFT). The DFT and its Inverse

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The DFT and its Inverse

Let $x(n), n=0,1,2,\ denote an $n$-sample complex sequence, i.e., $x\. Then the spectrum of $x$ is defined by the Discrete Fourier Transform (DFT):

\

The inverse DFT (IDFT) is defined by

\

Note that for the first time we are not carrying along the sampling interval $T=1/f_s$ in our notation. This is actually the most typical treatment in the digital signal processing literature. It is often said that the sampling frequency is $f_s=1$. However, it can be set to any desired value using the substitution

\

However, for the remainder of this reader, we will adopt the more common (and more mathematical) convention $f_s=1$. In particular, we’ll use the definition $\ for this chapter only. In this case, a radian frequency $\ is in units of “radians per sample.” Elsewhere in this course, $\ always means “radians persecond.” (Of course, there’s no difference when the sampling rateis really $1$.) Another term we use in connection with the $f_s=1$convention is normalized frequency: All normalized radian frequencies lie in the range $[-\, and all normalized frequencies in Hz lie in the range $[-0.5,0.5)$.



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