Let $x(n)$ be a finite duration complex-valued sequence of length $N .$

By definition, $X(k)=D F T[x(n)]=\sum_N x(n) \cdot W_{N}^{n k},$ where $k=0,1,2, \ldots N-1$ and W is the twiddle factor.

Let the real and imaginary components of $\mathrm{x}(\mathrm{n})$ and $W_{N}^{n k}$ be represented by $\mathrm{b}$ bits with fixed-point arithmetic. …