electronics Electronics is a scientific and engineering discipline that studies and applies the principles of physics to design, create, and operate devices that manipulate electrons and other Electric charge, electrically charged particles. It is a subfield ...

and

telecommunications Telecommunication, often used in its plural form or abbreviated as telecom, is the transmission of information over a distance using electronic means, typically through cables, radio waves, or other communication technologies. These means of ...

, pulse shaping is the process of changing a transmitted pulses'

waveform In electronics, acoustics, and related fields, the waveform of a signal is the shape of its Graph of a function, graph as a function of time, independent of its time and Magnitude (mathematics), magnitude Scale (ratio), scales and of any dis ...

to optimize the signal for its intended purpose or the

communication channel A communication channel refers either to a physical transmission medium such as a wire, or to a logical connection over a multiplexed medium such as a radio channel in telecommunications and computer networking. A channel is used for infor ...

. This is often done by limiting the bandwidth of the transmission and filtering the pulses to control intersymbol interference. Pulse shaping is particularly important in RF communication for fitting the signal within a certain frequency band and is typically applied after

line coding In telecommunications, a line code is a pattern of voltage, current, or photons used to represent digital data transmitted down a communication channel or written to a storage medium. This repertoire of signals is usually called a constrained ...

and

modulation Signal modulation is the process of varying one or more properties of a periodic waveform in electronics and telecommunication for the purpose of transmitting information. The process encodes information in form of the modulation or message ...

Need for pulse shaping

Transmitting a signal at high modulation rate through a band-limited channel can create intersymbol interference. The reason for this is Fourier correspondences (see

Fourier transform In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...

). A bandlimited signal corresponds to an infinite time signal, that causes neighboring pulses to overlap. As the modulation rate increases, the signal's bandwidth increases. When the spectrum of the signal is uniformly rectangular, a sinc shape results in the time domain. This happens if the bandwidth of the signal is larger than the channel bandwidth, leading to a distortion. This distortion usually manifests itself as intersymbol interference (ISI). Theoretically for sinc shaped pulses, there is no ISI, if neighboring pulses are perfectly aligned, i.e. in the zero crossings of each other. But this requires very good synchronization and precise/stable sampling without jitter. As a practical tool to determine ISI, one uses the Eye pattern, that visualizes typical effects of the channel and the synchronization/frequency stability. The signal's spectrum is determined by the modulation scheme and data rate used by the transmitter, but can be modified with a pulse shaping filter. This pulse shaping will make the spectrum smooth, leading to a time limited signal again. Usually the transmitted symbols are represented as a time sequence of dirac delta pulses multiplied with the symbol. This is the formal transition from the digital to the analog domain. At this point, the bandwidth of the signal is unlimited. This theoretical signal is then filtered with the pulse shaping filter, producing the transmitted signal. If the pulse shaping filter is rectangular in the time domain, the result is an unlimited spectrum. In many base band communication systems the pulse shaping filter is implicitly a

boxcar A boxcar is the North American (Association of American Railroads, AAR) and South Australian Railways term for a Railroad car#Freight cars, railroad car that is enclosed and generally used to carry freight. The boxcar, while not the simpl ...

filter. Its Fourier transform 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 fiber An optical fiber, or optical fibre, is a flexible glass or plastic fiber that can transmit light from one end to the other. Such fibers find wide usage in fiber-optic communications, where they permit transmission over longer distances and at ...

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 minimize the bandwidth needed for a certain symbol rate. An example in other areas of electronics is the generation of pulses where the

rise time In electronics, when describing a voltage or current step function, rise time is the time taken by a signal to change from a specified low value to a specified high value. These values may be expressed as ratiosSee for example , and . or, equiva ...

need to be short; one way to do this is to start with a slower-rising pulse, and decrease the rise time, for example with a step recovery diode circuit. The descriptions here provide a working knowledge that covers most effects but does not include causality, which would lead to analytical functions/signals. To understand this completely, one needs the Hilbert transform, which induces a direction by the convolution with the Cauchy Kernel. This couples the real and imaginary part of the baseband description, thereby adding structure. This immediately implies that either the real or the imaginary part are enough to describe an analytical signal. By measuring both in a noisy setting, one has a redundancy that can be used to better reconstruct the original signal. A physical realization is always causal, since an analytic signal carries the information.

Pulse shaping filters

Not every filter can be used as a pulse shaping filter. The filter itself must not introduce intersymbol interference — it needs to satisfy certain criteria. The

Nyquist ISI criterion In communications, the Nyquist ISI criterion describes the conditions which, when satisfied by a communication channel (including responses of transmit and receive filters), result in no intersymbol interference or ISI. It provides a method for ...

is a commonly used criterion for evaluation, 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: * 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 between 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.

Sinc filter

It is also called as Boxcar filter as its frequency domain equivalent is a rectangular shape. 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.

Raised-cosine filter

Raised-cosine is similar to sinc, with the tradeoff of smaller sidelobes for a slightly larger spectral width. Raised-cosine filters are practical to implement and they are in wide use. They have a configurable excess bandwidth, so communication systems can choose a trade off between a simpler filter and spectral efficiency.

Gaussian filter

This gives an output pulse shaped like a

Gaussian function In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function (mathematics), function of the base form f(x) = \exp (-x^2) and with parametric extension f(x) = a \exp\left( -\frac \right) for arbitrary real number, rea ...

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

Root Raised Cosine Filters & Pulse Shaping in Communication Systems
by Erkin Cubukcu (ntrs.nasa.gov). Telecommunication theory Signal processing