The Gaussian or normal distribution is very important, largely because of the Central Limit Theorem which we shall prove below. Because of this (and as part of the proof of this theorem) we shall show here that a Gaussian pdf has a Gaussian characteristic function too.

A Gaussian distribution with mean μμ and variance σ2 σ 2 has pdf:

Thus the characteristic function of a Gaussian pdf is also Gaussian in magnitude, e−u2⁢σ22 u 2 σ 2 2 , with standard deviation 1σ 1 σ , and with a linear phase rotation term, ei⁢u⁢μ u μ , whose rate of rotation equals the mean μμ of the pdf. This coincides with standard results from Fourier analysis of Gaussian waveforms and their spectra (e.g. Fourier transform of a Gaussian waveform with time shift).

If two variables, X 1 X 1 and X 2 X 2 , with Gaussian pdfs are summed to produce XX, their characteristic functions will be multiplied together (equivalent to convolving their pdfs) to give

Further Gaussian variables can be added and the pdf will remain Gaussian with further terms added to the above expressions for the combined mean and variance.

The central limit theorem states broadly that if a large number NN of independent random variables of arbitrary pdf, but with equal variance σ2 σ 2 and zero mean, are summed together and scaled by 1N 1 N to keep the total energy independent of NN, then the pdf of the resulting variable will tend to a zero-mean Gaussian with variance σ2 σ 2 as NN tends to infinity.

This result is obvious from the previous result if the input pdfs are also Gaussian, but it is the fact that it applies for arbitrary input pdfs that is remarkable, and is the reason for the importance of the Gaussian (or normal) pdf. Noise generated in nature is nearly always the result of summing many tiny random processes (e.g. noise from electron energy transitions in a resistor or transistor, or from distant worldwide thunder storms at a radio antenna) and hence tends to a Gaussian pdf.

Although for simplicity, we shall prove the result only for the case when all the summed processes have the same variance and pdfs, the central limit result is more general than this and applies in many cases even when the variance and pdfs are not all the same.

Figure 1(a) shows an example of convergence when the input pdfs are uniform, and NN is gradually increased from 11 to 5050. By N=12 N 12 , convergence is good, and this is how some 'Gaussian' random generator functions operate - by summing typically 1212 uncorrelated random numbers with uniform pdfs.

For some less smooth or more skewed pdfs, convergence can be slower, as shown for a highly skewed triangular pdf in Figure 1(b); and pdfs of discrete processes are particularly problematic in this respect, as illustrated in Figure 1(c).

Figure 1: Convergence toward a Gaussian pdf (Central Limit Theorem) for 3 different input pdfs for N=1 N 1 to 5050. Note that the uniform pdf (a) with smallest higher-order moments converges fastest. Curves are shown for N=123468101215203050 N 1 2 3 4 6 8 10 12 15 20 30 50 .

(a) (b) (c)