# DFT: Basis functions and Significance of dividing frequency by Sample length

A time domain signal can be decomposed into sinusoids which are based on basis functions.

For a N sampled input, a cosine basis function is defined as:

$$C_k[i] = \cos\left[\dfrac{2\pi k i}{N}\right]$$

Range of Frequency $k = 0 \ldots \frac{N}{2}$, Range of $i = 0 \ldots N-1$

Why is the sinusoid angle divided by $N$?

• Hi Raj. So the sampling period is implicit in a sampled signal. Simply, if one of the cosines has a period of 1 sample, its frequency is equal to fs; this would occur when $k = N$. In your example, bin $k = N/2$ will always equal nyquist: $fs/2$, because the period of the sinusoid is 2 samples. This is the definition of the Nyquist frequency is a baseband signal. The normalisation of the frequency removes dependence on the sampling frequency. Hope that helps. As someone who needs to understand things intuitively as well as mathematically, it took me a while to get my head around this – kippertoffee Jul 29 '15 at 8:12