Editing Modified discrete cosine transform (section)

=== Smoothness and discontinuities ===

We have seen above that the MDCT of 2''N'' inputs (''a'', ''b'', ''c'', ''d'') is equivalent to a DCT-IV of the ''N'' inputs (−''c''<sub>''R''</sub> − ''d'', ''a'' − ''b''<sub>''R''</sub>).
The DCT-IV is designed for the case where the function at the right boundary is odd, and therefore the values near the right boundary are close to 0. If the input signal is smooth, this is the case: the rightmost components of ''a'' and ''b''<sub>''R''</sub> are consecutive in the input sequence (''a'', ''b'', ''c'', ''d''), and therefore their difference is small.
Let us look at the middle of the interval: if we rewrite the above expression as (−''c''<sub>''R''</sub> − ''d'', ''a'' − ''b''<sub>''R''</sub>) = (−''d'', ''a'') − (''b'', ''c'')<sub>''R''</sub>, the second term, (''b'', ''c'')<sub>''R''</sub>, gives a smooth transition in the middle.
However, in the first term, (−''d'', ''a''), there is a potential discontinuity where the right end of −''d'' meets the left end of ''a''.
This is the reason for using a window function that reduces the components near the boundaries of the input sequence (''a'', ''b'', ''c'', ''d'') towards 0.