Dual Views of the Short Time Fourier Transform (STFT)

In the overlap-add formulation of Chapter 7, we used a hopping window to extract time-limited signals to which we applied the DFT. Assuming for the moment that the hop size $R=1$ (the ``sliding DFT''), we have

$$\zbox {X_m(\omega_k) = \sum_{n=-\infty}^\infty [w(n-m) x(n)] e^{-j\omega_k n}.} \protect$$ (9.1)

This is the usual definition of the Short-Time Fourier Transform (STFT) (§6.1). In this chapter, we will look at the STFT from two different points of view: the OverLap-Add (OLA) and Filter-Bank Summation (FBS) points of view. We will show that one is the Fourier dual of the other [10]. Next we will explore some implications of the filter-bank point of view and obtain some useful insights. Finally, some applications are considered.