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:<math>y_k[n] \ \triangleq \ x_k[n]*h[n] = \sum_{m=1}^{M} h[m] \cdot x_k[n-m].</math>
Then, for ''kL'' + ''M'' ≤ ''n'' ≤ ''kL'' + ''L'' + ''M'' − 1, and equivalently '''''M''''' ≤ ''n'' − ''kL'' ≤ '''''L'' + ''M'' − 1''', we can write''':'''
:<math>y[n] = \sum_{m=1}^{M} h[m] \cdot x_k[n-kL-m] \ \ \triangleq \ \ y_k[n-kL].</math>
The task is thereby reduced to computing ''y''<sub>''k''</sub>[''n''], for '''''M''''' ≤ ''n'' ≤ '''''L'' +'' M'' − 1'''. The process described above is illustrated in the accompanying figure.
Now note that if we periodically extend ''x''<sub>''k''</sub>[''n''] with period ''N'' ≥ ''L'' + ''M'' − 1, according to''':'''
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