Dear Utkarsh,

All the formulation of the gaussian distribution function are based on the original formulation of the normalized gaussian distribution function f(t) as

f(t)= (1/sqr (2 pi) *sigma) exp - t^2/ sigma^2,

where sigma is the standard deviation. The square of sigma is called the variance and it is physically equal to the power contained in a random signal such as the white noise.

In case of pulse shaping gaussian filters, one may reformulate the above equation to suit the function of the shaping filter where one has one has a symbol with a duration Ts, the symbol time and one shapes it using gaussian shaping filter with a three dB cut off frequency = B,

In this case one uses the formula which you brought in part 2). This formulation is given in detail in link:https://www.mathworks.com/help/signal/examples/fir-gaussian-pulse-shaping-filter-design.html

g(t) = (sqr.pi/a) e^(−π^2t^2 /a^2).

Comparing this formula to the above standard formula, one gets

sigma= a/( sqr 2) pi,

For an interpretation of a , one transforms g(t) to the frequency domain such that G(f)= exp - a^2 f^2,

The three dB bandwidth of this filter B is given by:

B= (1/ aTs) (sqr. log2/ 2),

Given B, and Ts one can determine a which can be made to vary from 0.5 to 2. In fact, one can make trade off between the extension of the pulse and the filter bandwidth.

Best wishes

Dear Dr. Zekry, thanks a lot for the detailed explanation.

You have mentioned that in case of pulse shaping gaussians, the expression contains 'pi^2'. Whereas the expression that i mentioned in part 2, uses 'pi' instead of 'pi^2'. Is that formula correct ?

Dear Utkarsh Singh,

As per Gaussian PDF is concerned it is asymmetric (about the mean) bell shape smooth curve.It is also known as normal(0, sigma)Shape of the curve solely depends upon mean Variance. Variance gives you the spread of the curve i.e. large sigma means more flat and wide spread. less sigma means narrow and more concentrated about the mean. but area is constant. If is distributed about the origin mean is zero. Forms depends on mean=0(origin) or other value.

Dear Utkarsh,

Not at all. You are welcome.

The formula which i brought is taken from the reference cited in my first comment. This formula is correct since it conforms with its Fourier transform given also in my first post. I forgot to comment on the formula you brought in your question i think there is a term pi missing.

Best wishes