# Probability Of error for a joint PDF

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#### M_A

##### Newbie level 3 If we want to compute the probablity of error for a 4-PAM for example assuming AWGN and ISI having discrete values then it's straightforward ,
But what if ISI is also gaussian, how can i get the pdf of such a function?

thanks i

#### bulx

##### Full Member level 4 Can you clarify what you mean by discrete ISI?
-b

#### M_A

##### Newbie level 3 ISI takes certain values with given probabilties, which is stastical independent
with the Gaussian Noise .

#### Bhanumurthy

##### Full Member level 5 Hi,
A fundamental concept in probability theory is 'A Random variable X is called a normal or Gaussian Random Variable if its pdf is of the form

fX(x) = 1/√2Πσ exp[-(x-µ)]²/2σ².
Hence rewrite your ISI in terms of above equation and u'll be able to plot the pdf.And then u can proceed for calculation of probability of error in terms of the complementary error function Q(z).

Regards
Bhanumurthy.

#### bulx

##### Full Member level 4 Question would be: What is probablistic in ISI? The pulse shapes are deterministic. The channel transfer function could be taken to be determinstic (but may be unknown), the rx and tx filters are also deterministic. The only probablistic quantity is the data itself. So are you refering to the discrete nature of the data when you say ISI is discrete? If so, ISI cannot be gaussian (as it is a continous distribution) where as data is discrete. Of course, added noise is random.

secondly, does the result that the sum of 2 normal independent random variables is also normal provide some direction?
-b

#### Bhanumurthy

##### Full Member level 5 Hi Bulx,
Central Limit Theorem provides some insight to u'r second question.
Regards
Bhanumurthy.

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