Pulse-shaping FIR filter

Question

I understand that pulse-shaping is performed to limit the bandwidth of the transmitted signal, but I would like to know how pulse-shaping can be implemented using an FIR filter.

Answer 1

You literally use the desired sampled pulse shape as impulse response, i.e. as taps, of the FIR.

Answer 2

The important thing to remember is that when you're in transmit, you don't have to actually do the operations that are demanded by theory, you just have to emit a signal that is exactly the same as if you did*.

You want to emit a signal that is, sample-for-sample, the same as if you took your starting data, generated a train of impulses from it, then filtered that resulting train of impulses.

So to build a really simple transmitter that just does textbook NRZ, at baseband, with a rectangular "pulse shaping filter", you you could just feed a trains of '1's and '0's through a shift register to a digital output, subtract half VCC, and voila! you have a baseband, NRZ signal that operates between +VCC/2 and -VCC/2. At no point did you generate either a Dirac impulse or a Kronecker delta and run it through a FIR filter.

I can't think of a good way to give a general case for this, but say you have a pulse shape that's 3 pulse durations long, and you're sending a baseband NRZ signal but with pulse shaping. Then at time $t$, the output of your transmitter should just be sum of the effect of the three previous bits.

So let $T$ be a bit time, let $x_n$ be the value of the "impulse" for the $n^{th}$ bit, let your pulse shape be $p(\tau)$, where $p(\tau) = 0\ \forall\ {0 < \tau < 3T}$, and let $t_p$ be an "inter-pulse" time in the range $t_p \in [0, T)$. Then at time $t = nT + t_p$, your output needs to be $$x_n p(\tau) + x_{n-1}p(\tau + T) + x_{n-2} p(\tau + 2T).$$

You make this happen any way you can. I'm not sure what the most popular way to do this in industry would be, but I'd probably oversample the bit time by enough to keep my analog filtering simple (probably something between 4x and 16x). Up to around 1990, the prevalent way to do this would have been to use a return-to-zero pulse running through an analog filter; today digital electronics are far cheaper than analog electronics except at very high speeds, so it would almost certainly be oversampled such that any analog filtering would be no more than second-order.

* This is true of processing received signals, as well, but the constraint of starting with an actual messy noisy corrupted physical signal means that you're much more closely bound to following the theoretical treatment.