Research Note ~ Deciphering Python’s FFT

In Python, the forward Discrete Fourier Transform (DFT) for a time signal $a(m)$ using both numpy and scipy is represented as

$$A(k) = \sum_{m=0}^{n-1} a(m) \exp \left(-2\pi i \frac{mk}{n}\right)$$

and the inverse DFT is

$$a(m) = \frac{1}{n} \sum_{k=0}^{n-1} A(k) \exp \left( 2\pi i \frac{mk}{n}\right).$$

The forward DFT is analagous to

$$A(k) = \sum_{m=0}^{n-1} a(m) \exp (-i \omega n).$$

• $a(m)$ corresponds to the input sequence
• $n$ is the number of samples in the time domain
• $k$ is the number of cycles
• $m$ is the current index/sample
• $\frac{k}{n}$ corresponds to the frequency of the signal ($k$ cycles per $n$ samples)
• Relationship between angular frequency and temporal frequency: $\omega = 2 \pi f$