The spectrum of a signal refers to the frequency distribution of the signal's amplitude or power. It provides information about the different frequency components that make up the signal. The [[discrete Fourier transform (DFT)]] can be used to map a [[discrete-time signal]], $x(n)$, into a [[frequency domain representation]], $X(k)$ of N coefficients: $ X(k)=\sum_{n=0}^{N-1} x(n) e^{-j\frac{2\pi}{N}kn},\;0 \leq k \leq N-1 $ If the discrete-time signal is the result of [[sampling]] a continuous-time signal at the [[sampling frequency]] $f_{s}$ in Hz, the frequency samples of $X(k)$ are separated by: $ \Delta_{f} = \frac{f_{s}}{N} $ In the general case $X(k)$ is a complex number. The magnitude: $ |X(k)| = \sqrt{ (\mathrm{Re}(X(k)) )^{2} + ( \mathrm{Im}(X(k)) )^{2}} $ provides a better indication of the signal's energy distribution in frequency. This is called the **magnitude spectrum** of the signal. Given that the intensity of a sound is perceived on a logarithmic scale (see [[loudness]]) it is frequently measured in [[deciBel]] (dB). If the magnitude of $X(k)$ is less than $1$, the magnitude spectrum is commonly represented in [[deciBel Full Scale (dBFS)]]: $ |X(k)|_{dBFS} = 20 \log_{10}(|X(k)|) $