The raised-cosine filter is an implementation of a low-pass
Nyquist filter, i.e., one that has the property of vestigial symmetry. This means that its spectrum exhibits odd
symmetry about \frac{1}{2T}, where T is the symbol-period of the communications system. Its frequency-domain description is a
piecewise-defined
function, given by: :H(f) = \begin{cases} 1, & |f| \leq \frac{1 - \beta}{2T} \\ \frac{1}{2}\left[1 + \cos\left(\frac{\pi T}{\beta}\left[|f| - \frac{1 - \beta}{2T}\right]\right)\right], & \frac{1 - \beta}{2T} or in terms of
havercosines: :H(f) = \begin{cases} 1, & |f| \leq \frac{1 - \beta}{2T} \\ \operatorname{hvc}\left(\frac{\pi T}{\beta}\left[|f| - \frac{1 - \beta}{2T}\right]\right), & \frac{1 - \beta}{2T} for :0 \leq \beta \leq 1 and characterised by two values; \beta, the
roll-off factor, and T, the reciprocal of the symbol-rate. The
impulse response of such a filter is given by: :h(t) = \begin{cases} \frac{\pi}{4T} \operatorname{sinc}\left(\frac{1}{2\beta}\right), & t = \pm\frac{T}{2\beta} \\ \frac{1}{T}\operatorname{sinc}\left(\frac{t}{T}\right)\frac{\cos\left(\frac{\pi\beta t}{T}\right)}{1 - \left(\frac{2\beta t}{T}\right)^2}, & \text{otherwise} \end{cases} in terms of the normalised
sinc function. Here, this is the "communications sinc" \sin(\pi x)/(\pi x ) rather than the mathematical one.
Roll-off factor The
roll-off factor, \beta, is a measure of the
excess bandwidth of the filter, i.e. the bandwidth occupied beyond the Nyquist bandwidth of \frac{1}{2T}. Some authors use \alpha=\beta. If we denote the excess bandwidth as \Delta f, then: :\beta = \frac{\Delta f}{\left(\frac{1}{2T}\right)} = \frac{\Delta f}{R_S/2} = 2T\,\Delta f where R_S = \frac{1}{T} is the symbol-rate. The graph shows the amplitude response as \beta is varied between 0 and 1, and the corresponding effect on the
impulse response. As can be seen, the time-domain ripple level increases as \beta decreases. This shows that the excess bandwidth of the filter can be reduced, but only at the expense of an elongated impulse response. ====
β = 0==== As \beta approaches 0, the roll-off zone becomes infinitesimally narrow, hence: :\lim_{\beta \rightarrow 0}H(f) = \operatorname{rect}(fT) where \operatorname{rect}(\cdot) is the
rectangular function, so the impulse response approaches h(t)=\frac{1}{T}\operatorname{sinc}\left(\frac{t}{T}\right). Hence, it converges to an ideal or
brick-wall filter in this case. ====
β = 1==== When \beta = 1, the non-zero portion of the spectrum is a pure raised cosine, leading to the simplification: :H(f)|_{\beta=1} = \left \{ \begin{matrix} \frac{1}{2}\left[1 + \cos\left(\pi fT\right)\right], & |f| \leq \frac{1}{T} \\ 0, & \text{otherwise} \end{matrix} \right. or :H(f)|_{\beta=1} = \left \{ \begin{matrix} \operatorname{hvc}\left(\pi fT\right), & |f| \leq \frac{1}{T} \\ 0, & \text{otherwise} \end{matrix} \right.
Bandwidth The bandwidth of a raised cosine filter is most commonly defined as the width of the non-zero frequency-positive portion of its spectrum, i.e.: :BW = \frac{R_S}{2}(\beta+1),\quad(0 As measured using a
spectrum analyzer, the radio bandwidth B in Hz of the modulated signal is twice the baseband bandwidth BW (as explained in [1]), i.e.: :B = 2 BW = R_S (\beta+1),\quad(0
Auto-correlation function The
auto-correlation function of raised cosine function is as follows: : R\left(\tau\right) = T \left[\operatorname{sinc}\left( \frac{\tau}{T} \right) \frac{\cos\left( \beta \frac{\pi \tau}{T} \right)}{1 - \left( \frac{2 \beta \tau}{T} \right)^2} - \frac{\beta}{4} \operatorname{sinc}\left(\beta \frac{\tau}{T} \right) \frac{\cos\left( \frac{\pi \tau}{T} \right)}{1 - \left( \frac{\beta \tau}{T} \right)^2} \right] The auto-correlation result can be used to analyze various sampling offset results when analyzed with auto-correlation. ==Application==