electronics The field of electronics is a branch of physics and electrical engineering that deals with the emission, behaviour and effects of electrons using electronic devices. Electronics uses active devices to control electron flow by amplification ...

and

telecommunications Telecommunication is the transmission of information by various types of technologies over wire, radio, optical, or other electromagnetic systems. It has its origin in the desire of humans for communication over a distance greater than that ...

, pulse shaping is the process of changing the waveform of transmitted pulses 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 informa ...

. This is often done by limiting the bandwidth of the transmission and filtering the pulses to control

intersymbol interference In telecommunication, intersymbol interference (ISI) 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 a similar effect as noise, thus making ...

. Pulse shaping is particularly important in RF communication for fitting the signal within a certain frequency band and is typically applied after line coding and

modulation In electronics and telecommunications, modulation is the process of varying one or more properties of a periodic waveform, called the '' carrier signal'', with a separate signal called the ''modulation signal'' that typically contains informat ...

Need for pulse shaping

Transmitting a signal at high

modulation rate In a digitally modulated signal or a line code, symbol rate, modulation rate or baud rate is the number of symbol changes, waveform changes, or signaling events across the transmission medium per unit of time. The symbol rate is measured in ''b ...

through a band-limited channel can create

. The reason for this are Fourier correspondences (see

Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...

). A bandlimited signal corresponds to an infinite time signal, that causes neighbouring pulses to overlap. As the modulation rate increases, the signal's bandwidth increases. As soon as the spectrum of the signal is a sharp rectangular, this leads to a sinc shape 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 neighbouring pulses are perfectly aligned, i.e. in the zero crossings of each other. But this requires a very good synchronization and precise/stable sampling without jitters. As a practical tool to determine ISI, one uses the

Eye pattern In telecommunication, an eye pattern, also known as an eye diagram, is an oscilloscope display in which a digital signal from a receiver is repetitively sampled and applied to the vertical input, while the data rate is used to trigger the hori ...

, 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 a rectangular in the time domain (like this is usually done when drawing it), this would lead to an unlimited spectrum. In many base band communication systems the pulse shaping filter is implicitly a

boxcar A boxcar is the North American (AAR) term for a railroad car that is enclosed and generally used to carry freight. The boxcar, while not the simplest freight car design, is considered one of the most versatile since it can carry most ...

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 fibre An optical fiber, or optical fibre in Commonwealth English, is a flexible, transparent fiber made by drawing glass (silica) or plastic to a diameter slightly thicker than that of a human hair. Optical fibers are used most often as a means t ...

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. An example in other areas of electronics is the generation of pulses where the rise time 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. These descriptions here provide a working knowledge, that cover most effects, but do not include causality, which would lead to analytical functions/signals. To understand this completely, one needs the Hilbert transform, that 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 In mathematics and signal processing, an analytic signal is a complex-valued function that has no negative frequency components. The real and imaginary parts of an analytic signal are real-valued functions related to each other by the Hil ...

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 con ...

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 In electronics and signal processing mainly in digital signal processing, a Gaussian filter is a filter whose impulse response is a Gaussian function (or an approximation to it, since a true Gaussian response would have infinite impulse res ...

Sender side pulse shaping is often combined with a receiver side

matched filter In signal processing, a matched filter is obtained by correlating a known delayed 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 w ...

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 In telecommunications, orthogonal frequency-division multiplexing (OFDM) is a type of digital transmission and a method of encoding digital data on multiple carrier frequencies. OFDM has developed into a popular scheme for wideband digital commu ...

, 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 of the base form f(x) = \exp (-x^2) and with parametric extension f(x) = a \exp\left( -\frac \right) for arbitrary real constants , and non-zero . It is ...

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