In digital telecommunication, pulse shaping is the process of changing the waveform of transmitted pulses. Its purpose is to make the transmitted signal suit better to the communication channel by limiting the effective bandwidth of the transmission. By filtering the transmitted pulses this way, the intersymbol interference caused by the channel can be kept in control. In RF communication pulse shaping is essential for making the signal fit in its frequency band.
Need for pulse shaping[]
Transmitting a signal at high modulation rate through a band-limited channel can create intersymbol interference. As the modulation rate increases, the signal's bandwidth increases. When the signal's bandwidth becomes larger than the channel bandwidth, the channel starts to introduce distortion to the signal. This distortion is usually seen as intersymbol interference.
The signal's spectrum is determined by the pulse shaping filter used by the transmitter. Usually the transmitted symbols are represented as a time sequence of dirac delta pulses. This theoretical signal is then filtered with the pulse shaping filter, producing the transmitted signal. The spectrum of the transmission is thus determined by the filter.
In many baseband communication systems the pulse shaping filter is implicitly a boxcar filter. Its spectrum is of the form sin(x)/x, and has significant signal power at frequencies higher than symbol rate. This is not a big problem when optical fibre or even twisted pair cable is used as the communication channel. However, in RF communications this would waste bandwidth, and only tightly specified frequency bands are used for single transmissions. In other words, the channel for the signal is band-limited. Therefore better filters have been developed, which attempt to minimise the bandwidth needed for a certain symbol rate.
Pulse filters[]
A typical NRZ coded signal is implicitly filtered with a boxcar filter.
Not all filters can be used as a pulse shaping filter. The filter itself must not introduce intersymbol interference — it needs to satisfy certain criteria. Nyquist ISI criterion is commonly used criterion for evaluation of filters, because it relates the frequency spectrum of the transmitter signal to intersymbol interference.
Examples of pulse-shaping filters that are commonly found in communication systems are:
- The trivial boxcar filter
- Sinc shaped filter
- Raised-cosine filter
- Gaussian filter
Sender side pulse shaping is often combined with a receiver side matched filter to achieve optimum tolerance for noise in the system. In this case the pulse shaping is equally distributed to the sender and receiver filters. The filters' amplitude responses are thus pointwise square-roots of the system filters.
Other approaches that eliminate complex pulse shaping filters have been invented. In OFDM, the carriers are modulated so slowly that each carrier is virtually unaffected by the bandwidth limitation of the channel.
Boxcar filter[]
The boxcar filter results in infinitely wide bandwidth for the signal. Thus its usefulness is limited, but it is used widely in wired baseband communications, where the channel has some extra bandwidth and the distortion created by the channel can be tolerated. It is often a natural choice, because digital electronics create such signals naturally. In optical telecommunications, the transmitting and receiving equipment cause the limitation for data speeds. Therefore it makes sense to use simplest possible waveforms. Boxcar filter however, is usually not considered a real pulse shaping filter.
Sinc filter[]
Amplitude response of raised-cosine filter with various roll-off factors
Theoretically the best pulse shaping filter would be the sinc filter, but it cannot be implemented precisely. It is a non-causal filter with a relatively slowly decaying tails. It is also problematic from synchronisation point of view as any phase error results theoretically infinitely steeply increasing intersymbol interference.
Raised-cosine filter[]
Raised-cosine filter is practical to implement and it is in wide use. It has a parametrisable excess bandwidth, so communication systems can choose a trade-off between a more complex filter and spectral efficiency.
Gaussian filter[]
This gives an output pulse shaped like a Gaussian function.
See also[]
- Nyquist ISI criterion
- Raised-cosine filter
- Matched filter
- Femtosecond pulse shaping
References[]
- John G. Proakis, "Digital Communications, 3rd Edition" Chapter 9, McGraw-Hill Book Co., 1995. ISBN 0-07-113814-5
- National Instruments Signal Generator Tutorial, Pulse Shaping to Improve Spectral Efficiency
- National Instruments Measurement Fundamentals Tutorial, Pulse-Shape Filtering in Communications Systems