Nyquist zero-ISI criteria; one pulse shaping method

Nyquist criterion for zero ISI

For an overall pulse $p(t)$ sampled every $T_b$ seconds, there is no ISI if

$$p(nT_b) = \begin{cases} 1, & n = 0 \\ 0, & n \ne 0 \end{cases} \quad\Longleftrightarrow\quad \sum_{n=-\infty}^{\infty} P\left(f - \frac{n}{T_b}\right) = T_b$$

The ideal solution is the rectangular spectrum of bandwidth $B_0 = R_b/2$, $p(t) = \text{sinc}(t/T_b)$, but it is not realizable and its slowly decaying tails make it very sensitive to timing errors.

Raised cosine pulse shaping

The raised cosine spectrum keeps the zero-ISI property but rolls off gradually:

$$P(f) = \begin{cases} \dfrac{1}{2B_0}, & 0 \le |f| < f_1 \\[2mm] \dfrac{1}{4B_0}\left[1 + \cos\dfrac{\pi(|f| - f_1)}{2B_0 - 2f_1}\right], & f_1 \le |f| < 2B_0 - f_1 \\[2mm] 0, & |f| \ge 2B_0 - f_1 \end{cases}$$

with $B_0 = \frac{R_b}{2}$, roll-off factor $\alpha = 1 - \frac{f_1}{B_0}$ ($0 \le \alpha \le 1$).

Transmission bandwidth:

$$B_T = B_0(1 + \alpha) = \frac{R_b}{2}(1+\alpha)$$

Pulse:

$$p(t) = \text{sinc}(2B_0t)\,\frac{\cos(2\pi\alpha B_0 t)}{1 - 16\alpha^2B_0^2t^2}$$

The sinc factor keeps zeros at $t = nT_b$ (zero ISI); the second factor makes the tails decay as $1/|t|^3$.

 P(f)
   |---------.                alpha = 0
   |----.     '.  .           alpha = 0.5
   |      '.    '.  '.        alpha = 1
   |        '.    '.  '.
   +----------+-----+----+--> f
   0       Rb/2  3Rb/4  Rb

Example: $R_b = 10$ kbps, $\alpha = 0.5$ gives $B_T = 5000 \times 1.5 = 7.5$ kHz.

Advantages: realizable filters, small ISI under timing jitter, wide eye opening. Cost: extra bandwidth $\alpha B_0$. The filter is often split equally as root raised cosine filters at transmitter and receiver.