# How to Use a Pulse Shaping Filter for Communications

### Key Takeaways

• A pulse shaping filter is used in communication channels to manipulate a waveform or pulses to have the desired shape in the time domain.

• A pulse shaping filter can be a physical circuit, but normally it’s a mathematical function that is used as a signal processing algorithm.

• The goal of using a pulse shaping filter is to transfer digital data through a bandwidth limited channel by converting it to an equivalent modulated analog signal.

These instruments are used to analyze pulse shaping filter responses for arbitrary signals.

LTI systems form the basis of the majority of electronics analysis, including design and analysis of communication channels. Even if there are nonlinear components in your system, much of your circuitry can be treated as an LTI subsystem. Understanding signal propagation is an important part of LTI systems design, and each portion of a system can affect signal shape, frequency content, and phase with broadband response.

A fundamental tool for calculating, analyzing, and designing time-frequency domain relationships between signals is a pulse shaping filter. Although it sounds like this might be a type of circuit, this is normally just a concept used as a mathematical function in designing communication channels. These techniques are normally implemented in an MCU, FPGA, or other PLD with sufficient computational power, but newer analog pulse shaping filter designs are possible as well. The next time you need to design a high speed channel to accommodate limited bandwidth and prevent ISI, you’ll use a digital pulse shaping filter or an analog pulse shaping filter circuit.

## Pulse Shaping Filter Types

There are many types of pulse shaping filters that can be implemented at a hardware, software, or firmware level. The three principle types of pulse shaping filters widely used in telecom are:

• Sinc filter—this filter has a rectangular transfer function, and the time-domain impulse response is a sinc function.

• Raised cosine filter—this pulse shaping filter uses a cosine function with a parameter to control the curvature of the filter’s transfer function in the frequency domain.

• Gaussian filter—as its name implies, this filter has a Gaussian transfer function.

These filters and other filtration functions are centered at zero frequency and are symmetric. Other common pulse shaping filters are the square root raised cosine filter, Nyquist, and square-root Nyquist filters. Other filters can be custom-designed by engineering the desired impulse response or transfer function, but the filter design depends on the signal integrity goals in a communication channel.

## Why Are Pulse Shaping Filters Used?

When we want to transmit a digital signal between two points, it’s important to note that the bandwidth of any digital signal is infinite. Unfortunately, all physical communication channels (including transmission lines on a PCB) have finite bandwidth. For short links between two components, finite bandwidth does not matter as long as a significant fraction of the signal’s power spectral density is transmitted through the channel.

However, for long links, total insertion loss and ISI can significantly distort a transmitted signal to the point where equalization in the receiver cannot recover it. The finite bandwidth of a real transmission line, coaxial cable, twisted pair cable, or other transmission medium means there is another form of interference seen at the receiver when a stream of digital pulses is sent down a transmission line: intersymbol interference, or ISI. The interaction between an input digital signal, the finite bandwidth of the channel, and the formation of ISI at the receiver is shown below.

Transmission of infinite bandwidth signals over bandlimited media leads to ISI at the receiver.

The formation of ISI results in part because of the limited bandwidth of real transmission media. In the above graphic, the bandwidth is limited to produce ISI for two reasons:

1. Bandlimited transmission medium—all transmission media are bandlimited. For example, on a PCB, a transmission line acts as a low pass filter at ~GHz frequencies due to the parasitic capacitance with respect to the PCB substrate.

2. Bandlimited receiver—the receiver is also bandlimited, particularly when recovering digital signals due to the finite sampling rate in the receiver. For this reason, many designers simply take the bandwidth limit of a channel to be the Nyquist frequency corresponding to the signal’s data rate.

Although you can’t increase transmission media bandwidth, you can modify the frequency content of the transmitted signal using pulse shaping filters.

## How Pulse Shaping Filters are Used

A pulse shaping filter is placed at the transmit end in order to prevent ISI. These filters all operate as you would expect a typical filter to work in the frequency domain: they modify an input signal based on the transfer function of the filter. A pulse shaping filter’s job is to turn a discrete-time sequence of digital data into a continuous analog signal. Furthermore, they reduce the bandwidth of the analog signal so that it does not exceed the channel’s bandwidth.

The relationship between the input bitstream and the modulated signal sent through the transmission medium are related using a Fourier transform/inverse transform. In fact, this produces the symmetry about zero frequency when applied to the filter’s impulse response and transfer function. This relationship, or the convolution theorem, determine the signal content sent down the transmission medium. The example below shows where convolution is used to determine the sinc response when a pulse shaping filter is used in a bandlimited channel.

Example application of a sinc pulse shaping filter over bandlimited media.

## Getting to Analog Pulse Shaping Filters

The algorithm shown above is normally implemented with a DSP core in an ASIC, or as an algorithm in an FPGA or MCU. However, analog pulse shaping filters are desirable and are an active research topic. The goal in designing these analog filter algorithms is to offload computational tasks from an FPGA/MCU/DSP core and implement them directly as hardware. CMOS circuits that can perform standard pulse shaping filter algorithms are of great interest to IC designers. By implementing these algorithms at the hardware level, the size of the circuit needed to perform these is much smaller. The algorithm can then be implemented in an IC without needing a DSP core.

If you’re a circuit designer and you want to create an analog pulse shaping filter, you’ll need a complete set of circuit modeling and simulation tools to engineer the filter response you need for your system. You can then simulate formation of modulated signals and demodulation at the receiver to recover your data. The goal here is to inspect ISI and distortion to ensure signals can be recovered accurately at the receiver end of your communication channel.

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