Pulse-Shaping Basics

Pulse shaping (i.e., base-band filtering) can vastly improve the behavior of wired or wireless communication links in an electrical system. With that in mind, Circuit Cellar columnist Robert Lacoste explains the advantages of filtering and examines Fourier transforms; random non-return-to-zero NRZ signaling; and low-pass, Gaussian, Nyquist, and raised-cosine filters.

Lacoste’s article, which appears in Circuit Cellar’s April 2014 issue, includes an abundance of graphic simulations created with Scilab Enterprises’s open-source software. The simulations will help readers grasp the details of pulse shaping, even if they aren’t math experts. (Note: You can download the Scilab source files Lacoste developed for his article from Circuit Cellar’s FTP site.)

Excerpts from Lacoste’s article below explain the importance of filtering and provide a closer look at low-pass filters:

WHY FILTERING?
I’ll begin with an example. Imagine you have a 1-Mbps continuous digital signal you need to transmit between two points. You don’t want to specifically encode these bits; you just want to transfer them one by one as they are.

Before transmission, you will need to transform the 1 and 0s into an actual analog signal any way you like. You can use a straightforward method. Simply define a pair of voltages (e.g., 0 and 5 V) and put 0 V on the line for a 0-level bit and put 5 V on the line for a 1-level bit.

This method is pedantically called non-return-to-zero (NRZ). This is exactly what a TTL UART is doing; there is nothing new here. This analog signal (i.e., the base-band signal) can then be sent through the transmission channel and received at the other end (see top image in Figure 1).

Note: In this article I am not considering any specific transmission channel. It could range from a simple pair of copper wires to elaborate wireless links using amplitude, frequency and/or phase modulation, power line modems, or even optical links. Everything I will discuss will basically be applicable to any kind of transmission as it is linked to the base-band signal encoding prior to any modulation.

Figure 1: Directly transmitting a raw digital signal, such as this 1-Mbps non-return-to-zero (NRZ) stream (top), is a waste of bandwidth. Using a pulse-shaping filter (bottom) reduces the required bandwidth for the same bit rate, but with a risk of increased transmission errors.

Now, what is the issue when using simple 0/5-V NRZ encoding? Bandwidth efficiency. You will use more megahertz than needed for your 1-Mbps signal transmission. This may not be an issue if the channel has plenty of extra capacity (e.g., if you are using a Category 6 1-Gbps-compliant shielded twisted pair cable to transmit these 1 Mbps over a couple of meters).

Unfortunately, in real life you will often need to optimize the bandwidth. This could be for cost reasons, for environmental concerns (e.g., EMC perturbations), for regulatory issues (e.g., RF channelization), or simply to increase the effective bit rate as much as possible for a given channel.

Therefore, a good engineering practice is to use just the required bandwidth through a pulse-shaping filter. This filter is fitted between your data source and the transmitter (see bottom of Figure 1).

The filter’s goal is to reduce as much as possible the occupied bandwidth of your base-band signal without affecting the system performance in terms of bit error rate. These may seem like contradictory requirements. How can you design such a filter? That’s what I will try to explain in this article….

LOW-PASS FILTERS

A base-band filter is needed between the binary signal source and the transmission media or modulator. But what characteristics should this filter include? It must attenuate as quickly as possible the unnecessary high frequencies. But it must also enable the receiver to decode the signal without errors, or more exactly without more errors than specified. You will need a low-pass filter to limit the high frequencies. As a first example, I used a classic Butterworth second-order filter with varying cut-off frequencies to make the simulation. Figure 2 shows the results. Let me explain the graphs.

Figure 2: This random non-return-to-zero (NRZ) signal (top row) was passed through a second-order Butterworth low-pass filter. When the cut-off frequency is low (310 kHz), the filtered signal (middle row) is distorted and the eye diagram is closed. With a higher cutoff (410 kHz, bottom row), the intersymbol interference (ISI) is lower but the frequency content is visible up to 2 MHz.

The leftmost column shows the signal frequency spectrum after filtering with the filter frequency response in red as a reference. The middle column shows a couple of bits of the filtered signal (i.e., in the time domain), as if you were using an oscilloscope. Last, the rightmost column shows the received signal’s so-called “eye pattern.” This may seem impressive, but the concept is very simple.

Imagine you have an oscilloscope. Trigger it on any rising or falling front of the signal, scale the display to show one bit time in the middle of the screen, and accumulate plenty of random bits on the screen. You’ve got the eye diagram. It provides a visual representation of the difficulty the receiver will have to recover the bits. The more “open” the eye, the easier it is. Moreover, if the successive bits’ trajectories don’t superpose to each other, there is a kind of memory effect. The voltage for a given bit varies depending on the previously transmitted bits. This phenomenon is called intersymbol interference (ISI) and it makes life significantly more difficult for decoding.

Take another look at the Butterworth filter simulations. The first line is the unfiltered signal as a reference (see Figure 2, top row). The second line with a 3-dB, 310-kHz cut-off frequency shows a frequency spectrum significantly reduced after 1 MHz but with a high level of ISI. The eye diagram is nearly closed (see Figure 2, middle row). The third line shows the result with a 410-kHz Butterworth low-pass filter (see Figure 2, bottom row). Its ISI is significantly lower, even if it is still visible. (The successive spot trajectories don’t pass through the same single point.) Anyway, the frequency spectrum is far cleaner than the raw signal, at least from 2 MHz.

Lacoste’s article serves as solid introduction to the broad subject of pulse-shaping. And it concludes by re-emphasizing a few important points and additional resources for readers:

Transmitting a raw digital signal on any medium is a waste of bandwidth. A filter can drastically improve the performance. However, this filter must be well designed to minimize intersymbol interference.

The ideal solution, namely the Nyquist filter, enables you to restrict the used spectrum to half the transmitted bit rate. However, this filter is just a mathematician’s dream. Raised cosine filters and Gaussian filters are two classes of real-life filters that can provide an adequate complexity vs performance ratio.

At least you will no longer be surprised if you see references to such filters in electronic parts’ datasheets. As an example, see Figure 3, which is a block diagram of Analog Devices’s ADF7021 high-performance RF transceiver.

Figure 3: This is a block diagram of Analog Devices’s ADF7021 high-performance transceiver. On the bottom right there is a “Gaussian/raised cosine filter” block, which is a key factor in efficient RF bandwidth usage.

The subject is not easy and can be easily misunderstood. I hope this article will encourage you to learn more about the subject. Bernard Sklar’s book Digital Communications: Fundamentals and Applications is a good reference. Playing with simulations is also a good way to understand, so don’t hesitate to read and modify the Scilab examples I provided for you on Circuit Cellar’s FTP site.

Lacoste’s full article is in the April issue, now available for membership download or single issue purchase. And for more information about improving the efficiency of wireless communication links, check out Lacoste’s 2011 article “Line-Coding Techniques,” Circuit Cellar 255, which tells you how you can encode your bits before transmission.