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FBS and Perfect Reconstruction

An important property of the STFT established in Chapter 8 is that it is exactly invertible when the analysis window satisfies the constant-overlap-add constraint. That is, neglecting numerical round-off error, the inverse STFT reproduces the original input signal exactly. This is called the perfect reconstruction property of the STFT, and modern filter banks are usually designed with this property in mind [287].

In the OLA processors of Chapter 8, perfect reconstruction was assured by using FFT analysis windows $ w$ having the Constant-Overlap-Add (COLA) property at the particular hop-size $ R$ used (see §8.2.1).

In the Filter Bank Summation (FBS) interpretation of the STFT (Eq.$ \,$ (9.1)), it is the analysis filter-bank frequency responses $ W(\omega-\omega_k)$ that are constrained to be COLA. We will take a look at this more closely below.

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