Modeling AWGN channel

Section 2: Modeling an Additive

White Gaussian Noise Channel

 This section models noise at the output y(t) of the channel in Section 1. h(t) in figure 1.1 is used to represent the combined effect of modulator, demodulator and channel. In this course, only discrete-input modulators are of interest. Hence, input symbols X take their values in a finite sample while the channel output value Y can take on any values along the real line. Such a model is called a Discrete-Input Continuous-Output channel and is characterized by the conditional probability density function

 Equation 2.1:

                                                  k = 0,1,……,q-1 

 relating the real output value to all possible inputs. The most representative of such a model is the Additive White Gaussian Noise Channel for which

 Equation 2.2:

                                                                

 where N is the zero mean Gaussian random variable with variance and X = for k = 0,1,….,q-1. For a given X, it follows that Y is Gaussian with mean  and variance . Using this new information, the previous conditional probability density function becomes

 Equation 2.3:

                                               

 

For every input sequence , i=1,2,…,n, there is a corresponding output sequence that can be expressed as

 Equation 2.4:

                                                         i=1,2,….,n

 

For a memoryless channel, we can express the conditional probability density function for each given input sequence as

 Equation 2.5:

                     

 

 

The typical model of such a channel is shown below in Figure 2.1.

 Figure 2.1: