The MLT Sine Window

The modulated lapped transform (MLT) [160] uses the sine window, defined by

$$w(n) = \sin\left[\left(n+\frac{1}{2}\right)\frac{\pi}{2M}\right], \quad n=0,1,2,\ldots,2M-1\,.$$ (4.24)

The sine window is used in MPEG-1, Layer 3 (MP3 format), MPEG-2 AAC, and MPEG-4 [200].

Properties:

• Side lobes 24 dB down
• Asymptotically optimal coding gain [159]
• Zero-phase-window transform (``truncated cosine window'') has smallest moment of inertia over all windows [202]:

  $$\int_{-\pi}^{\pi} \omega^2 W(\omega)d\omega =$$ (4.25)

  
  

Note that in perceptual audio coding systems, there is both an analysis window and a synthesis window. That is, the sine window is deployed twice, first when encoding the signal, and second when decoding. As a result, the sine window is squared in practical usage, rendering it equivalent to a Hann window ($\cos^2$ ) in the final output signal (when there are no spectral modifications).

It is of great practical value that the second window application occurs after spectral modifications (such as spectral quantization); any distortions due to spectral modifications are tapered gracefully to zero by the synthesis window. Synthesis windows were introduced at least as early as 1980 [213,49], and they became practical for audio coding with the advent of time-domain aliasing cancellation (TDAC) [214]. The TDAC technique made it possible to use windows with 50% overlap without suffering a doubling of the number of samples in the short-time Fourier transform. TDAC was generalized to ``lapped orthogonal transforms'' (LOT) by Malvar [160]. The modulated lapped transform (MLT) is a variant of LOT used in conjunction with the modulated discrete cosine transform (MDCT) [160]. See also [287] and [291].