## Review of STFT Filterbanks

Let's take a look at some of the STFT processors we've seen before,
now viewed as *polyphase filter banks*. Since they all use FFTs
to perform overlap-add decompositions of spectra, they are all
efficient, but most are *oversampled* in time and/or frequency as
``filter banks'' go. Oversampling is usually preferred outside of a
compression context, and normally required when spectral modifications
are to be performed. The STFT also computes a *uniform* filter
bank, but it can be used as the basis for a variety of non-uniform
filter banks, as discussed in §10.7, to give frequency
resolution more like that of hearing (§7.3).

For each selected STFT example below, a list of filter-bank properties is listed, followed by some discussion. Most of the properties are determined by the choice of FFT window and FFT hop size .

### STFT, Rectangular Window, No Overlap

- Perfect reconstruction
- Critically sampled (relies on aliasing cancellation)
- Poor channel isolation ( dB)
- Not robust to filter-bank modifications

*time-varying complex gains*applied to the filter-bank channel signals prior to remodulation and summing to reconstruct the signal (Chapter 9). In contrast to this, as discussed in Chapter 8, multiplicative spectral modifications in overlap-add systems having sufficient time-domain zero-padding yield perfect reconstruction of the filtered signal, even when their filter-bank interpretation obviously involves aliasing cancellation among channels in the frequency domain.

### STFT, Rectangular Window, 50% Overlap

- Perfect reconstruction
- Oversampled by 2 (less reliant on aliasing cancellation)
- Poor channel isolation ( dB)
- Not very robust to filter-bank modifications, but better

### STFT, Triangular Window, 50% Overlap

- Perfect reconstruction
- Oversampled by 2
- Better channel isolation ( dB)
- Moderately robust to filter-bank modifications

### STFT, Hamming Window, 75% Overlap

- Perfect reconstruction
- Oversampled by 4
- Aliasing from side lobes only
- Good channel isolation ( dB)
- Moderately robust to filter-bank modifications

### STFT, Kaiser Window, Beta=10, 90% Overlap

- Approximate perfect reconstruction (side lobes controlled by )
- Oversampled by
- Excellent channel isolation ( dB)
- Very robust to filter-bank modifications
- Aliasing from side lobes only

### Sliding FFT (Maximum Overlap), Any Window, Zero-Padded by 5

- Perfect reconstruction (always true when hop-size )
- Oversampled by
, where
- = window length (time-domain oversampling factor)
- 5 = zero-padding factor (frequency-domain oversampling factor)

- Excellent channel isolation (set by window side lobes)
- Extremely robust to filter-bank modifications
- No aliasing to cancel!

**Next Section:**

Wavelet Filter Banks

**Previous Section:**

MPEG Filter Banks