Additive White Gaussian Noise

Additive white Gaussian noise (AWGN) is a basic noise model used in Information theory to mimic the effect of many random processes that occur in nature. The modifiers denote specific characteristics:• 'Additive' because it is added to any noise that might be intrinsic to the information

system.• 'White' refers to idea that it has uniform power across the frequency band for the

information system. It is an analogy to the color white which has uniform emissions at all frequencies in the visible spectrum.

• 'Gaussian' because it has a normal distribution in the time domain with an average time domain value of zero. The noise samples have a Gaussian distribution

Optimal signal detection in AWGN LTI channelThe theory for signal transmission over AWGN LTI channels is very well developed and covered in many excellent text books. Many fundamental theorems in signal detection theory have been developed during World War II, to improve and automate the radar detection of enemy airplanes and ships. The theory of the matched filter receiver is of particular interest. The concept was introduced by D.O. North with the RCA labs in Princeton, in 1943.

Figure: possible implementation of a matched filter receiver. The signal is multiplied by a locally stored reference copy and integrated over time (correlation).

Matched Filter & it’s Operation:The matched filter correlates the incoming signal with a locally stored reference copy of the transmit waveform. The matched filter maximizes the signal-to-noise ratio for a known signal. It can be shown to be the optimal detector if • the channel produces Additive White Gaussian Noise (AWGN), • the channel is linear and time-invariant (LTI), and • an exact time reference is available, the signal amplitude as a function of time is

precisely known.

Huffman Encoding

This problem is that of finding the minimum length bit string which can be used to encode a string of symbols. One application is text compression:

What's the smallest number of bits (hence the minimum size of file) we can use to store an arbitrary piece of text?

Huffman's scheme uses a table of frequency of occurrence for each symbol (or character) in the input. This table may be derived from the input itself or from data which is representative of the input. For instance, the frequency of occurrence of letters in normal English might be derived from processing a large number of text documents and then used for encoding all text documents. We then need to assign a variable-length bit string to each character that unambiguously represents that character.

Encoding Huffman- Data

The encoding for each character must have a unique prefix. If the characters to be encoded are arranged in a binary tree:

An encoding for each character is found by following the tree from the route to the character in the leaf: the encoding is the string of symbols on each branch followed.

Decoding Huffman-encoded Data

Example

Line Coding and Decoding