Time Varying OLA Modifications
In the preceding sections, we assumed that the spectral modification did not vary over time. We will now examine the implications of time-varying spectral modifications. The derivation below follows [9], except that we'll keep our previous notation:
Using in our OLA formulation with a hop size results in
Define to get
(9.42) |
Let's examine the term in more detail:
- describes the time variation of the tap.
- is a filtered version of the tap . It is lowpass-filtered by w and delayed by samples.
- Denote the th time-varying, lowpass-filtered, delayed-by- filter tap by . This can be interpreted as the weighting in the output at time of an impulse entering the time-varying filter at time .
This is a superposition sum for an arbitrary linear, time-varying filter .
Block Diagram Interpretation of Time-Varying STFT Modifications
Assuming is causal gives
This is depicted in Fig.8.17.
The term can be interpreted as the FIR filter tap at time . Note how each tap is lowpass filtered by the FFT window . The window thus enforces bandlimiting each filter tap to the bandwidth of the window's main lobe. For an -term length- Blackman-Harris window, for example, the main-lobe reaches zero at frequency (see Table 5.2 in §5.5.2 for other examples). This bandlimiting places a limit on the bandwidth expansion caused by time-variation of the filter coefficients, which in turn places a limit on the maximum STFT hop-size that can be used without frequency-domain aliasing. See Allen and Rabiner 1977 [9] for further details on the bandlimiting property.
Length L FIR Frame Filters
To avoid time aliasing, we restrict the filter length to a maximum of samples. Since is an arbitrary multiplicative weighting of the th spectral frame, the frame filter need not be causal. For odd , the filter impulse response indices may run from to , where
(9.43) |
This gives
This is the general length time-varying FIR filter convolution sum for time , when is odd.
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