Understanding Root-Raised Cosine (RRC) Pulse Shaping Filters
In digital transmission systems, Root-Raised Cosine (RRC) pulse shaping filters are deployed at both the transmitter and receiver ends to limit occupied transmission bandwidth and eliminate Inter-Symbol Interference (ISI) in accordance with the Nyquist ISI Criterion.
Nyquist Criterion & RRC Cascade
To eliminate ISI at optimum sampling instants, the combined transmitter and receiver filter cascade must yield a full Raised Cosine (RC) magnitude response:
- $$H_{\text{total}}(f) = H_{\text{RRC, Tx}}(f) \times H_{\text{RRC, Rx}}(f) = H_{\text{RC}}(f)$$
Splitting the filter equally across both ends ensures matched filtering performance, maximizing output Signal-to-Noise Ratio (SNR) in AWGN channels.
Occupied & Excess Bandwidth Formulas
The total bandwidth ($B_{\text{total}}$) occupied by an RRC-filtered signal depends on symbol rate ($R_s$) and the roll-off factor ($\alpha \in [0, 1]$):
- $$B_{\text{total}} = \frac{R_s}{2}(1 + \alpha)$$
- $$B_{\text{excess}} = \alpha \times \frac{R_s}{2}$$
Where $\alpha = 0$ represents the theoretical minimum Nyquist brick-wall filter, and $\alpha = 1$ expands the occupied bandwidth by 100%.