Matched filters & pulse shaping

Key takeaways Square symbol pulses spread forever in frequency, so transmitters shape them. A raised-cosine pulse satisfies the Nyquist no-ISI criterion — it is zero at every other symbol’s sampling instant, so neighbours don’t smear together. Split it as a root-raised-cosine (RRC) filter at both transmitter and receiver: their cascade is a full raised cosine, and the receiver half is also a matched filter that maximizes SNR exactly where each symbol is decided.

The FIR lesson built filters that select frequency bands. This lesson uses a filter for a different job: shaping the symbols themselves so they can be read cleanly. It sits right before symbol decisions in the demodulation chain.

The problem with square pulses

The obvious way to send a symbol is a rectangular pulse — hold a level for one symbol period. But a sharp-edged pulse occupies an enormous bandwidth (its spectrum is a sinc that trails off slowly), spilling into adjacent channels. Band-limit it crudely and its tails ring into neighbouring symbols — inter-symbol interference (ISI), where each symbol contaminates the ones around it.

The Nyquist no-ISI criterion

Harry Nyquist’s insight: a pulse can be band-limited and ISI-free if its shape is zero at every symbol instant except its own. Then, sampled at the symbol centres, each pulse contributes only to its sample and nothing to the others — even though the pulses overlap in between.

A raised-cosine pulse peaks at its own symbol instant and is exactly zero at every neighbour's — so overlapping pulses cause no ISI when sampled at the centres.

The raised-cosine pulse is the classic shape that meets this criterion. A roll-off factor sets how gently its spectrum tapers — a small roll-off is bandwidth-tight but rings more; a larger one is gentler but wider.

Splitting the filter: root-raised-cosine

Here is the elegant part. Rather than shaping the whole pulse at the transmitter, the system splits the raised cosine into two root-raised-cosine (RRC) halves — one at the transmitter, one at the receiver — so that:

RRC(transmit)  *  RRC(receive)  =  full raised cosine  (no ISI)

That single choice buys two things at once:

The cascade is a full raised cosine, so the Nyquist no-ISI condition holds at the decision instants.
The receiver’s RRC is a matched filter for the transmitted pulse — its impulse response is the pulse shape reversed — which maximizes SNR at the symbol centre.

Why the matched filter wins

A matched filter correlates the incoming signal against the exact pulse it is looking for. All of a symbol’s energy piles up into one peak at the sampling instant, while noise — uncorrelated with the pulse — does not accumulate the same way. No other linear filter gives a better signal-to-noise ratio at that instant, which is why the RX RRC sits right before the symbol decision and timing loop. Cleaner peaks mean cleaner symbols and a lower bit error rate.

Quick check: why is pulse shaping split into a root-raised-cosine at both ends?

Recap

Square pulses waste bandwidth and cause inter-symbol interference (ISI).
A raised-cosine pulse meets the Nyquist no-ISI criterion — zero at every other symbol instant.
Splitting it into root-raised-cosine halves at TX and RX yields no ISI and a matched filter.
The matched filter maximizes SNR at the decision instant, lowering the error rate.

Next up: multiplying by an oscillator to shift a channel to baseband — mixing and downconversion.

Frequently asked questions

What is a matched filter and why does it maximize SNR?

A matched filter is a filter whose impulse response is a time-reversed copy of the transmitted symbol pulse. Correlating the incoming signal against the exact pulse shape gathers all of that symbol’s energy into a single peak at the decision instant, while noise, being uncorrelated with the pulse, does not add up the same way. The result is the highest possible signal-to-noise ratio at the moment the symbol is decided.

Why split the pulse shaping into a root-raised-cosine at each end?

The transmitter and receiver each apply a root-raised-cosine filter so that their cascade equals a full raised-cosine response. The full raised cosine satisfies the Nyquist no-ISI criterion, so symbols do not smear into their neighbours at the sampling instants, while splitting it in half also makes the receiver’s filter a matched filter for the transmitted pulse — one design serves both goals at once.