I was at the Consumer Electronics Show (CES) last week and spent a lot of time talking to various silicon vendors about their wireless offerings. During these discussions, the topic of beamforming came up numerous times. Beamforming maximizes the transmit energy and receive sensitivity of an antenna in a specific direction. Beamforming is becoming a critical technology for improving the data transfer rate of wireless systems — rates that are critical to making wireless technology a credible option for delivering reliable Internet Protocol (IP) video around a home. The reliable delivery of IP video over wireless will simplify the deployment of Fiber-to-the-Home systems (my focus) by eliminating the need to install Ethernet cables to every room, which is expensive.

These discussions brought back many memories. Years ago I spent a lot of time working on beamforming for military sonar and radar systems. Military and aerospace technology often finds a home in commercial applications once it becomes cost effective, and beamforming is now becoming inexpensive enough to be in every home. Because I was familiar with the technology, I decided that it would be worthwhile to put together some training material for my staff on how beamforming works. I began by writing a Mathcad worksheet for simulating a simple linear antenna array. This simulation seemed to be a good way to illustrate how beamforming works and I thought it would be worthwhile to cover here.

Background

Let’s begin by defining beamforming. As usual, let’s turn to the Wikipedia.

Beamforming is a signal processing technique used in sensor arrays for directional signal transmission or reception. This is achieved by combining elements in the array in a way where signals at particular angles experience constructive interference and while others experience destructive interference. Beamforming can be used at both the transmitting and receiving ends in order to achieve spatial selectivity. The improvement compared with an omnidirectional reception/transmission is known as the receive/transmit gain (or loss).

I really like this definition because it focuses on the critical role of interference in allowing us to either direct energy (acoustic or electromagnetic) in a desired direction or to receive energy from a desired direction. Using advanced methods, we can also reject sending or receiving energy from a specific direction, which is called null steering. Beamforming will be my focus here.

Beamforming is useful for wireless communication because these systems have limited transmit power and it is best to point the energy you transmit toward an actual receiver. When you are receiving, it is best to listen carefully in the direction of a transmitter and to reject noise coming from other directions. Null steering is useful when you have source of interference that you wish to reject, which could be something as common as a microwave oven making popcorn or a neighbor’s wireless system.

Since beamforming is a such a good thing to do, let’s now take a closer look at how it works.

Analysis

Reciprocity

As mentioned above, beamforming can be applied to both transmitting and receiving. In fact, transmit and receive beamforming are identical because of the principle of reciprocity, which states that the receive sensitivity of an antenna as a function of direction is the same as the transmit radiation pattern from the same antenna when transmitting. See the Wikipedia for a discussion of this topic.

Linear Antenna Array

For the purposes of this post, I will use the simple model of a linear array shown in Figure 1.

Figure 1: Linear Antenna Model.

To keep this discussion simple, I will make the following assumptions:

• The antenna is composed of a series of identical elements separated by λ/2, where λ= c/f, c is the speed of light, and f is the frequency of transmission.

  Since the dawn of radio, engineers have been directing radio power along specific directions using shaped reflectors (e.g. parabolas). You can analyze antenna arrays using the viewpoint that we are sampling these apertures using small antennas that I will refer to here as elements. Since we are sampling an aperture, λ/2 makes sense because it corresponds to the Nyquist sampling rate for a receiver with a wavelength of λ. I will not be taking a sampling viewpoint for the remainder of this post, but I may in future posts.

• Every element is a receiver — I will ignore transmitting.

  By the reciprocity theorem, everything I say for a receiving antenna will also be true for a transmitting antenna. I am making this assumption just to reduce the amount of redundancy in this discussion. You should think of every element as a small antenna. We are going to be working with arrays of small antennas.

• The receiver elements generate an output voltage or current that is proportional to the level of the signal impinging upon it.

  This means that that output of an antenna element is an accurate reproduction of the signal strength impinging upon it.

• Every element has an omnidirectional sensitivity response.

  This means that every element is equally sensitive in all directions.

• Assume that the wavelength λ = 1.

  Expressing all lengths in units of λ does not limit this discussion in any way and is simpler to deal with analytically. This means that the elements are separated by 1/2 in units of λ.

Given these assumptions, we can now create a mathematical model for the receive output of a linear array as a function of beam angle. While the linear array example it simple to analyze, it illustrates the basic approach to analyzing larger and more complex antenna arrays.

Simple Beamforming Algorithm

What is Beamforming Computationally?

Beamforming computationally is simply the linear combination of the outputs of the elements, which a beam can be computed using Equation 1.

Eq. 1

where

• N is number of elements.
• k is an index variable.
• a_k is coefficient of the k^th element.
• x_k is the voltage response from the k^th element.
• ζ is the beam response.

Equation 1 is not difficult to compute, but we need to determine what coefficients we should use to enhance the antenna’s receive gain in a specified direction.

Intuitive View of Array Beamforming

Consider Figure 2, where we have a transmitter that is far away from our linear antenna. “Far away” in this case means that the transmitter is many wavelengths distant from the receiving antenna.

Figure 2: Illustration of the Phase Shift At Each Element.

For a transmitter that is far enough away, the wavefronts can be approximated as plane waves when they arrive at the receive antenna. When we evaluate Equation 1 for the situation shown in Figure 2 assuming that a_k=1 , the maximum response is generated when all wavefronts hit each element at the same time (perfect constructive interference). This will occur when the transmitter is located on a line that is perpendicular to the orientation of the linear array. If the transmitter moves away from the perpendicular, destructive interference begins to occur and the magnitude of the response reduces.

In order to steer the beam in different directions, we need to look closely at Figure 2 and observe that, for transmitters that are not along the perpendicular, there is a linearly increasing phase shift introduced along the array elements. Can we use the coefficients to cancel out the phase differences and rotate the direction of maximum antenna response? It turns out we can.

We can compute the phase shift in the signal received between each element using Equation 2. The key to deriving this equation is to note that each element receives the same signal, but at a slightly different time. This time can be modeled as a phase shift.

Eq. 2

where

• θ is the angle of target relative to a vector normal to the center of the element array.
• φ is the phase shift between each element.
• δt is the time delay of the signal between the elements.
• f is the transmit frequency.
• c is the speed of light.
• t is time.

If we can compensate for the phase shift, we can maximize our receiver’s response in the direction of the transmitter. That is exactly what we are going to do.

Beamforming Simulation

Here is the approach I am going to use to determine the output of a simple beamformer.

• Determine the distance from each element to the source using the Pythagorean theorem.
• Determine the amplitude and phase of the signal at each element using the distance.
• Evaluate Equation 1.
• Plot the output of Equation 1 as a function of transmitter angle.

Distance to the Transmitter

Equation 3 shows my Mathcad program for generating the distance between an element and the transmitter.

Eq. 3

where

• N is number of elements.
• r is the radial distance to the transmitter.
• θ is the angle of target relative to a vector normal to the center of the element array.
• k is the element index (labeled from left to right in Figure 1).

Amplitude and Phase of the Received Signal

Equation 4 gives the phase of the signal at element as a function of distance. This formula uses the fact that each wavelength of distance equals 2·π of phase.

Eq. 4

Evaluate Equation 1

My approach to evaluating Equation 1 is to break it into three parts.

• Generate a vector of steering coefficients.
• Compute a matrix of the element responses over a range of transmitter angles.
• Form the matrix product of the steering coefficients and the element responses, which is equivalent to evaluating Equation 1.

Figure 3 illustrates how I compute the compensating phase shifts for two different beams. It turns out that you can generate multiple beams in parallel and I will illustrate below.

Figure 3: Generate the Coefficients For Generating a Beam in A Specific Direction.

Figure 4 illustrates how to compute a matrix of element responses for a transmitter positioned along a range of angles from 0° to 180°.

Figure 4: Element Responses for a Range of Angles from 0° to 180°.

Plot the Output of Equation 1

Figure 5 shows a plot of sensitivity for two beams, at 45° and -30° off the perpendicular.

Figure 5: Two Beam Patterns for a 7 Element Antenna and a Transmitter at Range = 1000 λ.

Conclusion

There is a lot more to talk about here, but I was able to put together a simple beamforming example for my team. You can see from this example that one can “point” the response curve of the antenna in specific direction with just a bit of matrix math. Note that the antenna beam patterns have a main “lobe” and side “lobes.” In a later post, I will discuss how to reduce the amplitude of the sidelobes.

Back in 2003, I used an approximation for the logarithm function in a hardware application. When originally implemented, the function only had to work for a limited range of input. Recently, a customer has requested that we expand the range of operation for this function. This post examines how I went about expanding this approximation’s range of operation.

Engineers frequently have to approximate common mathematical functions. You might wonder why we still need to approximate these functions when there are excellent mathematical libraries available for all commonly used processors. There are two reasons:

• Speed
  Library functions are coded for accuracy and they frequently take a long time to execute. There are applications where accuracy is less important than speed and approximations may accurate enough and fast enough to solve your problem. For example, I have had to use approximations to the square root function when computing the magnitude of vectors in real-time navigation applications. Library functions simply were too slow.
• Cost
  Library functions require lots of memory and may force you to buy a faster processor in order to execute them. My cost limitations are usually so tough that I have to use cheap processors like AVRs, which have limited memory and throughput. I need to find inexpensive ways to implement math functions on these brain-dead computers.

We will examine my original implementation and how I went about expanding its range of operation, which engineers refer to as the function’s “dynamic range.”

Background

Decibel Basics

Technically, I view decibels as a scaling rather than a unit. Decibels are always expressed relative to a unit, in this case milliwatts. Equation 1 defines the dBm, which means decibels relative to one mW.

Eq. 1

where

• P_mW is the measured power in milliwatts [mW],
• P_dBm is the measured power expressed in decibels milliwatt (dBm)

I need to approximate Equation 1 over a range from – 6 dBm to 0 dBm with an accuracy of better than 0.5 dB. This post will use a 4^th-order polynomial to approximate the logarithm function. Equation 2 defines my polynomial model. I also include an equivalent matrix version.

Eq. 2

Now that we know a bit about decibels, let’s discuss how cable TV works.

Some Cable TV Basics

Figure 1 illustrates the scenario I find myself in. In general, one laser will drive multiple stages of amplification. Thus, one laser and a “tree” of EDFAs can serve thousands of homes.

Figure 1: General Optical Deployment Model using Lasers and EDFAs.

Analysis

My Original Polynomial Approximation

Figure 2 shows my Mathcad implementation of a minimum-maximum error curve fit routine. I chose to use a dynamic range from -6 dBm to 0 dBm (to be truthful, the actual range I used was slightly different but for reasons that are unimportant here).

Figure 2: My Original Determination of Logarithmic Approximation Coefficients.

Figure 3: Goodness of Fit for the Original Approximation.

A Wider Dynamic Range Version

Approach

A service provider wants me to expand the range of operation of my optical power measurement to -12 dB to 6 dBm from -6 dBm to 0 dBm. As I thought about it, I made the following observations.

• I can break this range into three parts related by a factor of 4:
• -12 dBm to -6 dBm
• -6 dBm to 0 dBm
• 0 dm to 6 dBm
• You can see these ranges are related by a factor of 4 by noting that
  

Given these observations, I can state Equation 3.

Eq. 3

So I can use my dB approximation over a wider dynamic range by using Equation 3 as shown in Equation 4.

Eq. 4

Results

Figure 4 shows the effectiveness of my approximation. This is not too bad and I can reuse software that has already been tested.

Figure 4: Wider Dynamic Range Version of the dB Approximation.

Conclusion

This was a nice example of using a property of logarithms to solve a common engineering problem.

This material, along with some cheap prototyping hardware from Radio Shack, helped me train my kids. Some of Mims’s books can be picked up at Radio Shack as well.

By the way, I also made my kids build their own PCs. This proved to be educational and fun.

My sons always tease me about my interest in space. In order to understand my interest in space, you need to understand what it was like being a boy during the 1960s. I had my own “October Sky” boyhood. I built rockets, read everything I could about space, built electronic circuits, and dreamt of someday being an engineer and space traveler. While the space traveler part didn’t work out, I was fortunate in that I did everything else that I dreamt about it. I have met numerous engineers who are now in the 50s who had the same experiences.

I still read everything I can find about space. Recently, I was reading about Mars Science Laboratory (MSL) and noticed that it has a Radioisotope Thermoelectric Generator (RTG) for a power source. Figure 1 shows an artist’s rendering of the MSL with two RTGs, which NASA refers to as the Radioisotope Power System (RPS).

Figure 1: Mars Science Laboratory Rover with Radioisotope Power System (aka RTG-based Power System).

Background

Referencing the Wikipedia, lets start with a definition of what an RTG is:

A radioisotope thermoelectric generator (RTG, RITEG) is an electrical generator that obtains its power from radioactive decay. In such a device, the heat released by the decay of a suitable radioactive material is converted into electricity by the Seebeck effect using an array of thermocouples.

Figure 1 (Source) shows a photograph of the RTG used on MSL, which is called the Multi-Mission RTG (MMRTG).

Figure 1: Photograph of MSL's MMRTG.

I saw the following statement in The Atlantic Magazine about the MMRTG:

The 43kg MMRTG is designed to produce 125 watts of electrical power at the start of the mission, falling to about 100W after 14 years. (NASA/Kim Shiflett)

Let’s see if we can use this one statement to derive some information about the MSL’s RTG.

Analysis

Amount of Power Required

A thermoelectric generator requires heat to produce electricity. Unfortunately, thermoelectric generators are notoriously inefficient. NASA has reported that their efficiency level is about 6.2%. This means that for every 1000 W put in, only ~62 W of electricity comes out. Since this generator is specified to put out 125 W, we need a heat source that produces . The Wikipedia entry for the MMRTG states that it dissipates 2 kW of power, so we appear to have the correct total power figure and efficiency.

Heat Generated by Radioactive Decay

When a plutonium-238 atom decays, it emits an alpha particle with a decay energy of 5.593 Million Electron Volts (MEV). It is this decay energy is the source of heat (i.e. energy) that will be converted to electricity by thermocouples. This heat is enough to cause the Pu-238 to glow (see Figure 2, Source).

Figure 2: A Glowing Pu-238 RTG Element.

Radioactive decay versus time is usually modeled mathematical using the concept of half-life. Equation 1 shows how half-life is used to model decay.

Eq. 1

where

• t is time in years.
• t_HL is the half life of Pu-238 (87.74 years).
• N is the number of atoms in a sample of radioactive material.
• N₀ is the number of atoms in a sample of radioactive material at t=0.

To compute the power radiated by a sample of Pu-238, we need to determine the number of decays that occur per second. We can compute the number of decays per second by differentiating Equation 1, which I show in Equation 2.

Eq. 2

Let’s say we want to compute the amount of heat generated by a gram of Pu-238. We need to determine the number of Pu-238 atoms in a one gram sample and apply Equation 2, which tells us that Pu-238 generates 0.568 W/gm of heat. Figure 4 illustrates this calculation. We can use this number to estimate the amount of Pu-238 that we will need to generate 2 kW of thermal power.

Figure 4: Calculation of Heat Generated Per gm of Pu-238.

Amount of Pu-238 Required

Since we know that we require 2 kW of thermal power and that Pu-238 generates 0.568 W/gm, we see that we need . This agrees with the value 3.5 kg stated in this presentation from the Department of Energy, so I think we understand how much Pu-238 is required.

Degradation of Performance Over

The quote from The Atlantic Magazine states that the electrical power available from the MMRTG drops to 100 W after 14 years. Let’s assume that we have a fixed percentage of degradation year over year, which is similar to a compound interest problem. We can determine the percentage decline per year as shown in Equation 3.

Eq. 3

So the output power is declining by 1.58% every year.

The amount of heat available from a radioactive source degrades every year because the amount of radioactive material reduces every year because of decay. We can compute this decline in thermal power as shown in Equation 4.

Eq. 4

So thermal power declines by 0.79% every year, which is about half of the electrical power drop reported by NASA. However, the thermocouples also degrade in performance every year. So the 1.58% degradation rate is composed of two parts: (1) the reduction in available thermal power due to radioactive decay, and (2) the reduction in thermocouple conversion efficiency over time.

It turns out that there is data from Voyager on RTG power reduction. Over a 33 year period, Voyager has seen its available electrical power decline to 67% of its initial value, though its available thermal power has only declined by 83.4%. This means that Voyager has seen a decline in electrical power generation of 1.7% per year while its thermal power has degraded by 0.79% per year. So that 1.6% degradation expected for the MMRTG by NASA seems reasonable.

Conclusion

I was able to calculate some important characteristics of NASA’s power system for the Mars Science Laboratory. I could see problems like this being good ones for calculus students to sharpen their skills on.

One of my sons asked me if I could work through a PageRank calculation example a couple of different ways (algebraic and iterative). It was an interesting exercise and I thought it would be worth documenting here. I used Mathcad for both my algebraic and iterative solutions.

The Wikipedia has an excellent definition of the PageRank algorithm, which I will quote here.

PageRank is a link analysis algorithm, named after Larry Page[1] and used by the Google Internet search engine, that assigns a numerical weighting to each element of a hyperlinked set of documents, such as the World Wide Web, with the purpose of “measuring” its relative importance within the set.

PageRank is computed using a relatively simple function (see Equation 1), but a number of web-based examples treat the weighting of inbound links from sites external to a particular group of pages as a special case. I did not see any explicit calculation examples, so I thought I would include this calculation here.

Background

PageRank views the web as graph, with inbound links being viewed as measure of the significance of a web page. Equation 1 shows the PageRank equation. Note that this equation can be solved several different ways. One approach involves eigenvalues, which my son does not know about yet. The equation can be solved algebraically or iteratively. I will use both approaches for this example.

Eq. 1

where

• N is the number of pages.
• d is called the damping factor and it is an arbitary weighting factor.
• PR(p_i is the PageRank of page p_i.
• L(p_i) is the number of outbound links from page p_i.
• M(p_i) is the set of links to page p_i.

Equation 2 shows a matrix form of Equation 1. The matrix form is most likely the form used “out in the wild.”

Eq. 2

where

• R is the PageRank vector.
• 1 is a column vector with all elements equal to 1.
• t is a discrete time variable (really a sequence number).

Note that many examples of PageRank are computed using a variant of Equations 1 and 2 that multiplies the PageRank value by the number of pages (N·PageRank). Equation 3 illustrates Equation 2 modified with the substitution . Equation 1 can be modified similarly. This is the equation that Ian used for his examples. Since I am going to duplicate his results, I will multiply my results by N.

Eq. 3

There has been quite a bit written about the nuances of this equation because of its importance in determining a web page’s position in a list of search results. I am not concerned about those details here. I am focused here on the calculation of the PageRank for a specific set of pages.

Analysis

Example

My son was using Ian Roger’s excellent site for learning about the details of PageRank. The question he had is on Example 10, which assigns a PageRank of 1 to an external page. Figure 1 shows the Example 10′s web page configuration. Ian’s PageRank results are shown in the boxes, which represent web pages. I want to show the details on obtaining Ian’s results as an illustration of how to handle an external link from a page with a defined PageRank.

Figure 1: PageRank Example from Ian Roger's Website.

Algebraic Solution

When Ian uses a link from an external site, he sets the PageRank value to 1. Here is his rationale.

We’ll assume there’s an external site that has lots of pages and links with the result that one of the pages has the average PR of 1.0.

Algebraically, this is easy to handle. Figure 2 shows my solution implemented in Mathcad.

Figure 2: Algebraic Solution for Example 10.

Iterative Solution

Figure 3 shows how I setup my iterative solution. To force page A to have a PageRank of 1, I needed to remove page A from the R vector and M matrix, but add it back in so that page A’s contribution can be included. Again, it is a slight modification of Equation 3 so that I can force page A to have a PageRank of 1.

Figure 3: Setup for My Iterative Solution of Equation 2.

Figure 4: Iterative Solution of Equation 2.

Conclusion

I obtained the same results as Ian using two different approaches — algebraic and iterative. This example was different than most in that a particular web page was forced to a particular PageRank. I hope that I answered my son’s question.

The word bolide comes from the Greek βολίς (bolis) [2] which can mean a missile or to flash. The IAU has no official definition of “bolide”, and generally considers the term synonymous with “fireball”. The term generally applies to fireballs reaching magnitude −14 or brighter.[12] Astronomers tend to use “bolide” to identify an exceptionally bright fireball, particularly one that explodes (sometimes called a detonating fireball). It may also be used to mean a fireball which creates audible sounds.

I have seen two bolide meteors in my life. I saw my first one while walking home from Catholic education one evening during 8th grade. It was very bright and lasted only a short period of time. About 30 seconds after it vanished, I heard a rumble. Next day, the local newspaper reported that a meteor had been spotted and fragments had been picked up on the ground.

Last week, I was driving late in the evening when I saw a bright white streak come across the sky that appeared to drop a yellow and then a green fragment. I have watched meteors my whole life and this was the first time that I saw colors. I did not know that others have seen colors until I saw this article. This was pretty cool!

I have an application where a potentiometer may be useful. In fact, it would be useful if the potentiometer had a logarithmic resistance characteristic, which is also called an audio taper for reasons that I will cover later. I have never used a potentiometer with a logarithmic characteristic before and I thought it would be worth documenting what I learned during this effort.

Background

Logarithmic Misnomer

What is normally referred to as a logarithmic taper is really an exponential characteristic. A typical logarithmic taper potentiometer characteristic is shown in Figure 1 (Source).

Figure 1: Example of A Logarithmic/Audio Taper Potentiometer Resistance Characteristic

Potentiometer Specifications

Not all vendors include a graph of the resistance characteristics of their logarithmic potentiometers. Many of the vendors include a specification that says something similar to the following quote. (Source)

The “W” taper attains 20% resistance value at 50% of clockwise rotation (left-hand).

This specification means that the potentiometer has

• 0% of it full scale resistance value with the wiper at 0% of its full scale position
• 20% of it full scale resistance value with the wiper at 50% of its full scale position
• 100% of the total resistance value at 100% of its full scale position.

This type of specification gives you sufficient information to create an exponential curve fit, which I illustrate in Figure 2.

Figure 2: Illustration of the Fitting of an Exponential Function to the Specified Points.

The math associated with this curve fitting is shown in Figure 3.

Figure 3: Curve Fitting Math.

Equation 1 illustrates the basic form of the logarithmic potentiometer’s resistance characteristic R(x), where x is the wiper position as a percentage of full scale.

Eq. 1

where

• R₀ is a curve fitting parameter
• R₁ is a curve fitting parameter
• x is the wiper position as a percentage of full scale range

Human Hearing

The logarithmic taper is commonly called an audio taper because it is often used audio applications for loudness control. Understanding why involves knowing a little bit about human hearing. Figure 4 (Source) illustrates a human’s perception of loudness relative to the Sound Pressure Level (SPL). Sound pressure level is proportional to an audio amplifier’s output power. However, the ear is sensitive to the log of the sound pressure level. This is why the “logarithmic” taper is useful.

Figure 4: Loudness Level Versus Sound Pressure Level (dB).

Compensating for Hearing

Figure 5 shows what how the loudness is perceived by a person as the potentiometer’s wiper is moved. You can see that the perceived loudness increases approximately linearly for wiper positions above ~20%.

Figure 5: Potentiometer Resistance Model and Linearized Loudness Characteristic.

Conclusion

After all this research, I ended up not using the logarithmic potentiometer because it was not logarithmic. I ended up using another approach which I will discuss in a later post.

I used to watch the television show “MASH” years ago. They would refer to quick procedures for patient stabilization as “meatball surgery.” I recently encountered some “meatball math” as part of my engineering job. I call it “meatball” because the “patient”, a recalcitrant laser, had to be fixed quickly and rough approximations were totally acceptable. This application was a nice application of Mathcad and the implementation of the approximation in C will soon be going out in one our software loads.

Background

We use AVR processors from Atmel to perform simple controller functions. I like these processors so much that I have an AVR development system at home (you never know when you might want to cookup some controller code). In one of our products, we need to estimate the threshold current for a laser at a given temperature. It turns out that Equation 1 [Keiser] does a fair job of modeling the temperature variation of the threshold current.

Eq. 1

Equation 1 requires the computation of an exponential function. Unfortunately, a library version of the exponential function requires lots of memory space and my available AVR memory space is very small. However, it turns out that I do not need tremendous accuracy in this application – ±2% would be fine. I started to think in terms of a piecewise linear approximation. Let’s take a look at what this approximation involves.

Analysis

Requirements

We begin with some simple algorithm requirements:

• We need to compute the exponential function over an input range from 0 to 1.5
• The exponential function must be accurate within 2%.
• The routine must occupy much less memory than the accurate exponential library routine.

As always, I began my work by seeing if anyone else has already solved my problem. My first search turned up an article on a bifurcated linear approximation to the exponential function over the input range 0 to 1.

Bifurcated Approach

Figure 1 shows a plot of the bifurcated approximation. This equation meets my accuracy requirement, but it is not designed for my input range from 0 to 1.5. Let’s use Mathcad to fix this.

Figure 1: Bifurcated Linear Approximation to the Exponential Function.

Trifurcated Approach

I needed to get this approximation out quickly, so I decided to augment the range setup of Equation 1 (0 to 0.5, and 0.5 to 1) with a third range (1.0 to 1.5).

Figure 2 shows the form of my linear function and my error function, which I will minimize. I am using the “minimax” optimality criterion for my approximation.

Figure 2: Linear Approximation Prototype and My Error Function.

We can use solver to find the optimal solution. Figure 3 shows my solution setup. Note that I randomly chose the starting values. I ended up trying various random starting values to come to my final choice of coefficients.

Figure 3: Trifurcated Mathcad Solver Setup.

Note how I am also enforcing a continuity requirement at the end points of my ranges. The bifurcated solution did not enforce this criterion and their error function was not smooth. Figure 4 shows the results of my Mathcad work.

Figure 4: Plot of My Trifurcated Linear Approximation to the Exponential Function.

Conclusion

The equation shown in Figure 4 solved my problem. This was a good example of a “quick and dirty” solution to a common engineering dilemma — an accurate solution is too big or expensive and you need to find an approach that is good enough. That is exactly what was done here.

I am doing some interesting work with lasers this week. I thought it would be useful to provide some background on how we build and control lasers. We deploy a lot of lasers in outdoor applications, which means that special attention must be paid to the temperature characteristics of lasers.  Like batteries, lasers have characteristics that vary with temperature. These temperature variations would cause the laser’s output power to drop if actions were not taken to compensate for the temperature variations. These variations also make a laser subject to thermal runaway, also just like a battery. This week’s work will focus on temperature compensation. But first, let’s discuss how our lasers operate and are assembled. Once we have established that base, we can move on to related topics.

Laser Basics

Lasers emit light when they are driven by current above a level called the threshold current (symbolically I_Threshold).  The more current you drive into the laser above the threshold level, the more optical power that the laser emits. Figure 1 illustrates this characteristic.

Figure 1: Idealized Laser Output Power Versus Input Current Characteristic.

Let’s summarize the key points of Figure 1:

• No optical power is emitted until the drive current exceeds a given value called I_Threshold.
• For drive currents above threshold, the laser emits an optical power proportional to the amount that the drive current exceeds the threshold level
• The constant of proportionality is called Slope Efficiency (SE)

We can model this characteristic using Equation 1.

Eq. 1

This power versus drive current characteristic by itself does not cause any problems. However, I_Threshold and SE both vary with temperature and this is the source of my woes. Figure 2 shows how the laser characteristics vary with temperature for a real device. Notice how the slope efficiency reduces with increasing temperature. That is the key problem, as I will discuss below.

Figure 2: Slope Efficiency Graph for an Actual Laser.

Controlling Laser Power

Communication Using Lasers

Digital communications using fiber optics is similar to many other forms of digital communications in that we encode binary information by sending light down the fiber at two different power levels. The high-level is called a “1″ and the low-level is called a “0″. Figure 3 illustrates what an idealized optical bit stream looks like.

Figure 3: Example of a Digital Bit Stream Using Optical Power Modulation.

Because SE decreases with temperature, maintaining constant “1″ and “0″ levels means increasing the drive current with increasing temperature. It turns out that this is a more difficult problem to solve than you might think. Figure 4 shows how we convert drive current into optical power. We try to control two parameters:

• I_Bias, which controls the average power level
• I_Modulation, which controls the difference between P₀ and P₁.

Figure 4: Conversion of Drive Current Into Optical Output Power.

Over the years, we have used various temperature compensation techniques. It turns out that controlling a laser’s output power near room temperature is simple. However, particularly at high temperature, problems develop because the drive current becomes so high that the laser self-heats. This self-heating results in lower SE, which means more drive current. The laser is now in thermal runaway, which means our control circuitry will need to shut it down so it does not damage itself.

We have used feedback to control the laser’s average output power . However, controlling the average power is not sufficient. Receivers actually detect the difference in power between a “1″ and a “0″. Reduced SE also reduces the difference between a “1″ and “0″. So we need an approach that will both increase the difference between the “1″ and “0″ drive levels and maintain an average power setting.

We have used a mathematical model for the variation of SE with temperature and have used this model to predict the required increase in the difference between the “1″ and “0″ drive levels with temperature. This has proven to be a mediocre solution. The temperature characteristics of a laser vary by vendor and lot. We need something better — we need a two feedback loops. One feedback loop to control average power and another to control the difference in levels between a “1″ and “0″. It turns out that a new “dual-loop” laser drivers are now becoming available. Future posts will discuss how these devices work.

Basic Laser Construction

Before I go too far into controlling a laser, let’s talk about how commercial laser systems are constructed. Figure 5 illustrates how lasers are typically configured for use in communications applications. The laser is mounted so that the vast majority of the optical power generated goes into the fiber, with a tiny fixed percentage tapped off for measurement by a photodiode (called the monitor photodiode). In Figure 2, the light going to the left is focused into fiber by some type of lens, which is commonly a ball lens but can be other types.

Figure 5: Standard Laser Configuration for Communications Applications.

Figure 6 shows how lasers are packaged into transistor-outline (TO) cans, which is how we often see them.

Figure 6: Illustration of Laser TO-Can Packaging.

Conclusion

This post was intended to provide some background for the posts to follow. These posts will look at applying lasers in communications applications.