# Digital Demodulation

Investigate the minimum distance rule and how received samples are mapped back to bits.

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Until now, we haven’t included the impact of noise. The symbols sent as $-1$ and $+1$ have been received at the same amplitude. Let’s include the additive white Gaussian noise and see how bits can be obtained from the equalized samples.

Additive white Gaussian noise (AGWN) results from the random motion of electrons in the receiver frontend.

• White light consists of all seven colors with equal intensity. Similarly, white noise implies a power spectral density that is equally spread across all frequencies.
• The noise amplitude is Gaussian distributed. This implies that the values close to the mean (the received signal itself) are more probable than values away from the mean.

## Minimum distance rule

When AWGN is added to the received samples, the equalized samples do not exactly map to the symbols $-1$ and $+1$ as before. Instead, they can assume any value around the mean, and the spread depends on the power of the noise. The symbol decisions are taken according to the minimum distance rule.

The minimum distance rule says that each sample should be mapped to the closest symbol. In terms of binary modulation, this implies that values greater than $0$ map to the symbol $+1$ and the values less than $0$ map to the symbol $-1$.

$\hat X[k] = \left\{ \begin{array}{l} -1, \quad r[k] < 0 \\ +1, \quad r[k] > 0 \\ \end{array} \right.$

Let’s apply this rule and demodulate the transmitted bits in the presence of noise. As opposed to the previous cases, some symbol decisions can be wrong when, for instance, noise amplitude is strong enough to push a $-1$ symbol beyond the boundary of $0$ and vice versa.

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