The Gaussian low-pass filter has a transfer function given by

The parameter *α* is related to *B*, the 3-dB bandwidth of the baseband Gaussian shaping filter. It is commonly expressed in terms of a normalized 3-dB bandwidth-symbol time product (*BT _{s}*):

As *α* increases, the spectral occupancy of the Gaussian filter decreases and the impulse response spreads over adjacent symbols, leading to increased ISI at the receiver. The impulse response of the Gaussian filter in the continuous-time domain is given by

which could easily be rearranged (Eq. B.4) to reveal its fit with the canonical form of a zero-mean Gaussian random variable with standard deviation *σ _{h}* =

Its integral from −∞ to ∞ is, of course, 1.

Let us now express the Gaussian filter in the discrete-time domain. Let *t*_{0} = *T _{s}*/OSR be an integer oversample of the symbol duration and

Substituting Eq. B.2 and dropping explicit ...

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