Matched Filtering and the Square-Root Raised Cosine

4.6 Matched Filtering and the Square-Root Raised Cosine

In many situations the receiver should use a filter similar to the one with impulse response $p_{t} (t)$ used in the transmitter. When discussing receivers for AWGN in Section 4.3, it was concluded that the optimal filter is a matched filter $p_{t}^{∗} (−t)$ delayed by $T_{sym}$ . Hence, disregarding the delay by $T_{sym}$ and channel $h_{c} (t)$ , Eq. (4.21) can be written with $p_{t} (t) = p_{sr} (t)$ and $p_{r} (t) = p_{sr}^{∗} (−t)$ . In this case, the overall impulse response of Block (4.19) is

p(t) = p_{sr} (t)∗ p_{sr}^{∗} (−t).

(4.31)

Note that the goal is to have $p(t)$ as a Nyquist pulse, not $p_{sr} (t)$ . As indicated in Section A.15, $F{p_{sr}^{∗} (−t)} = P_{sr}^{∗} (f)$ , therefore $P(f) = | P_{sr} (f) |^{2}$ and assuming that $P(f)$ is a non-negative “normal” raised cosine given by Eq. (4.28), a sensible choice for $p_{sr} (t)$ is to have

P_{sr} (f) = P^{1∕2} (f).

(4.32)

Hence, $p_{sr} (t)$ is called a “square-root” raised cosine (RRC) and $p(t)$ is a Nyquist pulse (achieves zero ISI). There are distinct definitions for this pulse and the one used in the companion code ak_rcosine.m is assumed here, which is compatible with the expression adopted in Matlab’s rcosine function. A RRC can be designed with ak_rcosine(1,L,’fir/sqrt’,r,P) or using Matlab, with the same syntax: rcosine(1,L,’fir/sqrt’,r,P).

An expression for the RRC pulse in time domain is

p_{sr} (t) = \frac{sin ((1 − r) \frac{πt}{T_{sym}}) + \frac{4rt}{T_{sym}} cos ((1 + r) \frac{πt}{T_{sym}})}{\frac{πt}{T_{sym}} [1 − {(\frac{4rt}{T_{sym}})}^{2}]},

(4.33)

which has singularities at $t = 0$ and $t = ± T_{sym} ∕(4r)$ . Using L’Hospital’s rule leads to

p_{sr} (t) |_{t=0} = 1 − r + \frac{4r}{π}

and

p_{sr} (t) |_{t=± T_{sym} ∕(4r)} = \frac{r}{\sqrt{2}} [sin (\frac{π}{4r}) (1 + \frac{2}{π}) + cos (\frac{π}{4r}) (1 − \frac{2}{π})],

respectively. Similar to the conversion from Eq. (4.28) to Eq. (4.29), a discrete-time version of Eq. (4.33) can be obtained by substituting $t∕ T_{sym}$ by $n∕L$ .

As mentioned, the RRC in Eq. (4.33) does not lead to zero ISI. Only the cascade of two of them achieves this property. Consider what motivates a RRC: if eventually two Nyquist pulses given by Eq. (4.28) are cascaded (one used at the Tx and its matched version at the Rx, for example), the overall effect is not zero ISI! But having a RRC at the Tx and its matched filter at the Rx would lead to zero ISI in case the channel is ideal (or a perfect equalization is assumed).