Pulse shaping

Pulse shaping is used to turn discrete-time signals into analog signals with well-defined spectral properties.

The frequency domain representation of the raised-cosine filter writes as $$G\left(f\right)=\left\{ \begin{array}{c} 1,\quad\left|f\right|\leq\frac{1-\alpha}{2T}\\ \frac{1}{2}\left[1+\cos\left(\frac{\pi T}{\alpha}\left[\left|f\right|-\frac{1-\alpha}{2T}\right]\right)\right]=\cos^{2}\left(\frac{\pi T}{2\alpha}\cdot\left[\left|f\right|-\frac{1-\alpha}{2T}\right]\right),\:\frac{1-\alpha}{2T}\leq\left|f\right|\leq\frac{1+\alpha}{2T}\\ 0,\quad\mathrm{otherwise} \end{array}\right.$$ with time-domain impulse response $$g\left(t\right)\cdot T=\left\{ \begin{array}{c} 1,\quad t=0\\ \frac{\sin\left(\pi/\left(2\alpha\right)\right)}{\pi/\left(2\alpha\right)}\frac{\pi}{4},\quad\left|t\right|=\frac{T}{2\alpha}\\ \frac{\sin\left(\pi t/T\right)}{\pi t/T}\frac{\cos\left(\alpha\pi t/T\right)}{1-\left(2\alpha t/T\right)^{2}},\quad\mathrm{otherwise} \end{array}\right.$$ The parameter $\alpha$ is referred to as roll-off factor and allows to trade off spectral efficiency (best case $\alpha=0$) and filter order (i.e., required length of impulse response $g\left(t\right)$, least demanding for $\alpha=1$). Typically, $\alpha$ is chosen to be well below $0.5$.

Specifically, for the spectral most compact case $\alpha=0$ we obtain the ideal "brickstone"-spectrum $$G\left(f\right)=\left\{ \begin{array}{c} 1,\quad\left|f\right|\leq\frac{1}{2T}\\ 0,\quad\mathrm{otherwise} \end{array}\right.$$ with time-domain impulse response $$g\left(t\right)\cdot T=\left\{ \begin{array}{c} 1,\quad t=0\\ \frac{\sin\left(\pi t/T\right)}{\pi t/T},\quad\mathrm{otherwise} \end{array}\right.$$

For a symbol rate $R_{s}=1/T$ the required baseband bandwidth is $$B=\frac{1}{2}\left(1+\alpha\right)\cdot R_{s}$$ In the passband (i.e. at a carrier frequency greater or equal half the information bandwidth of the signal), the required bandwidth is twice $$B=\left(1+\alpha\right)\cdot R_{s}$$ but, as a return of investment, we can also convey twice the information with complex baseband signals (turning into real signals in the passband).

The raised cosine impulse fulfills the First Nyquist Condition, $$g\left(k\cdot T\right)=\left\{ \begin{array}{c} 1,\quad k=0\\ 0,\quad k\neq0 \end{array}\right.$$ which means that there is no intersymbol interference at the adjacent discrete-time sampling instants $t=k \cdot T, \, k\in\mathbb{Z}$. The raised cosine filter is regularly used for pulse shaping in digital communication systems.