Pulse shaping
Encyclopedia
In digital telecommunication, pulse shaping is the process of changing the waveform of transmitted pulses. Its purpose is to make the transmitted signal better suited 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.
Typically pulse shaping occurs after line coding and before modulation
.
. 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 base band 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.
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:
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.
Theoretically the best pulse shaping filter would be the sinc filter, but it cannot be implemented precisely. It is a non-causal filter
with relatively slowly decaying tails. It is also problematic from a synchronisation point of view as any phase error results in steeply increasing intersymbol interference.
Intersymbol interference
In telecommunication, intersymbol interference is a form of distortion of a signal in which one symbol interferes with subsequent symbols. This is an unwanted phenomenon as the previous symbols have similar effect as noise, thus making the communication less reliable...
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.
Typically pulse shaping occurs after line coding and before modulation
Modulation
In electronics and telecommunications, modulation is the process of varying one or more properties of a high-frequency periodic waveform, called the carrier signal, with a modulating signal which typically contains information to be transmitted...
.
Need for pulse shaping
Transmitting a signal at high modulation rate through a band-limited channel can create intersymbol interferenceIntersymbol interference
In telecommunication, intersymbol interference is a form of distortion of a signal in which one symbol interferes with subsequent symbols. This is an unwanted phenomenon as the previous symbols have similar effect as noise, thus making the communication less reliable...
. 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 base band communication systems the pulse shaping filter is implicitly a boxcar
Boxcar function
In mathematics, a boxcar function is any function which is zero over the entirereal line except for a single interval where it is equal to a constant, A; it is a simple step function...
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
Nyquist ISI criterion
In communications, the Nyquist ISI criterion describes the conditions which, when satisfied by a communication channel, result in no intersymbol interference or ISI...
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 boxcarBoxcar functionIn mathematics, a boxcar function is any function which is zero over the entirereal line except for a single interval where it is equal to a constant, A; it is a simple step function...
filter - Sinc shaped filter
- Raised-cosine filterRaised-cosine filterThe raised-cosine filter is a filter frequently used for pulse-shaping in digital modulation due to its ability to minimise intersymbol interference...
- Gaussian filterGaussian filterIn electronics and signal processing, a Gaussian filter is a filter whose impulse response is a Gaussian function. Gaussian filters are designed to give no overshoot to a step function input while minimizing the rise and fall time. This behavior is closely connected to the fact that the Gaussian...
Sender side pulse shaping is often combined with a receiver side matched filter
Matched filter
In telecommunications, a matched filter is obtained by correlating a known signal, or template, with an unknown signal to detect the presence of the template in the unknown signal. This is equivalent to convolving the unknown signal with a conjugated time-reversed version of the template...
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.Sinc filter
Causal filter
In signal processing, a causal filter is a linear and time-invariant causal system. The word causal indicates that the filter output depends only on past and present inputs. A filter whose output also depends on future inputs is non-causal. A filter whose output depends only on future inputs is...
with relatively slowly decaying tails. It is also problematic from a synchronisation point of view as any phase error results in 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 criterionNyquist ISI criterionIn communications, the Nyquist ISI criterion describes the conditions which, when satisfied by a communication channel, result in no intersymbol interference or ISI...
- Raised-cosine filterRaised-cosine filterThe raised-cosine filter is a filter frequently used for pulse-shaping in digital modulation due to its ability to minimise intersymbol interference...
- Matched filterMatched filterIn telecommunications, a matched filter is obtained by correlating a known signal, or template, with an unknown signal to detect the presence of the template in the unknown signal. This is equivalent to convolving the unknown signal with a conjugated time-reversed version of the template...
- Femtosecond pulse shapingFemtosecond pulse shapingIn optics, femtosecond pulse shaping is a technique that modifies the temporal profile of an ultrashort pulse from a laser. Pulse shaping can be used to shorten/elongate the duration of optical pulse, or to generate more complex pulses.-Introduction:...
- Pulse (signal processing)Pulse (signal processing)In signal processing, the term pulse has the following meanings:#A rapid, transient change in the amplitude of a signal from a baseline value to a higher or lower value, followed by a rapid return to the baseline value....