We assume that the real and imaginary components of {x(n)} and {WNkn} are represented by ‘b’ bits. Consequently, the computation of product x(n). WNkn requires four real multiplications. Each real multiplication is rounded from 2b bits to b bits and hence there are four quantization errors for each complex valued multiplication.

We know that, the variance of the quantization errors is directly proportional to the size N of the DFT. So, every fourfold increase in the size N of the DFT requires an additional bit in computational precision to offset the additional quantization errors.

We know that the variance of the signal sequence is (2/N)2/12=\\(\\frac{1}{3N^2}\\)

Additive white noise model is the model that we use in the statistical analysis of round off errors in IIR and FIR filters.

Since DFT plays a very important role in many applications of DSP, it is very important for us to know the effect of quantization errors in its computation. In particular, we shall consider the effect of round off errors due to the multiplications performed in the DFT with fixed point arithmetic.

The Quantization errors due to rounding off are uniformly distributed random variables in the range (-Δ/2,Δ/2) if Δ=2-b. This is one of the assumption that is made about the statistical properties of the quantization error.

We know that each of the quantization has a variance of Δ2/12=2-2b/12.

The 4N quantization errors are mutually uncorrelated. This is one of the assumption that is made about the statistical properties of the quantization error.

Since the computation of single point DFT of a sequence of length N involves N number of complex multiplications, it contains 4N number of quantization errors.

According to one of the assumption that is made about the statistical properties of the quantization error, the 4N quantization errors are uncorrelated with the sequence {x(n)}.