DFT Definition

Using our knowledge of complex signals and correlation, we can understand how the DFT helps us observe the spectrum of any signal.

Complex signals

Let’s start with a discrete-time complex sinusoid $e^{j2\pi \frac{k}{N}n}$ that has a discrete frequency of $k/N$. For a time domain signal $x[n]$, the DFT is defined as follows:

$$X[k]=\sum_{n=0}^{N-1}x[n]e^{-j2\pi\frac{k}{N}n}$$

While this might look intimidating, the process becomes easier to understand if the following conventions with regard to the variables are kept in mind:

• $X[k]$ is the $k$-th DFT output.
• $k$ is the DFT output index in frequency domain, and it ranges from $0$ to $N-1$ or $-N/2$ to $N/2-1$. Both are appropriate ranges since the frequency samples after $N/2$ are simply repeated on the left side too.
• The time domain index is represented, as before, by $n$

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