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 (BTs):
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 σh = α/√2π
Its integral from −∞ to ∞ is, of course, 1.
Let us now express the Gaussian filter in the discrete-time domain. Let t0 = Ts/OSR be an integer oversample of the symbol duration and t = kt0, k being the sample index. The discrete-time impulse response becomes
Substituting Eq. B.2 and dropping explicit ...