标签: Transform

In my seminars, I often tackle why we should understand at least a little electromagnetics theory, even for purely firmware people. But the subject is hard to understand and sometimes harder to believe, which is why the best book on the subject, " High Speed Digital Design, " is subtitled "A Handbook of Black Magic." Why is it important? You'll surely be probing your design with various tools like scopes and logic analysers, and every such probe has some impedance. As speeds get higher, that impedance is ever more likely to corrupt the operation of the device. But "speed" is poorly understood today. We equate clock rate with speed, which is only part of the story. Almost two hundred years ago polymath Jean-Baptiste Joseph Fourier showed that any periodic function can be expressed as the sum of sine waves of different amplitudes and frequencies. A square wave, like a microprocessor clock, is periodic and its Fourier series is:   In other words, a square wave is composed of the sum of the sine of the wave's frequency and each of its odd harmonics. (A harmonic is an integer multiple of the base frequency.) As you can see, no matter what the square wave's frequency is, it has harmonics that go to infinity, though at progressively lower amplitudes. A useful rule of thumb is that for most digital design we can ignore harmonics that exceed: f=0.5/T r Where T r is the square wave's rise time in nanoseconds. Every real-world signal takes time to transition from a zero to a one. Above f the Fourier components are down about 40 dB. Since a picture is worth a million bits, look at the following scope trace:   Figure 1—A square wave with 20 ns rise time. The top trace is a 1MHz square wave, just like a CPU clock. The bottom is an expanded view of the highlighted portion of the same trace. Note that what looks like a nice, quick zero to one transition actually takes quite a bit of time. In this case the rise time is 20 ns. Running 20 ns through the previous formula and it's clear that anything above 25MHz will be so far down we don't have to worry about them. But it does mean that 1MHz signal has important frequency components that far exceed the fundamental. That square wave comes from my scope's waveform generator. To show the Fourier effect more dramatically I built a circuit to improve the signal's rise time by feeding the signal through a fast gate.   Figure 2—Square wave fixer-upper circuit. It is possible to see the individual frequency components predicted by the Fourier series. Most modern scopes can compute the Fourier transform of a signal, as in the following screen capture.   Figure 3—10MHz sine wave The bottom trace is a simple 10MHz sine wave. Above it is the Fourier transform. Unlike a normal scope display where the horizontal axis is time and the vertical volts, the upper one is displayed in units of dB and frequency. In this case the screen's scale is set to 0MHz at the left and 20MHz all the way to the right. Note there's a very strong peak exactly at 10MHz, because 10MHz is the only frequency component in a 10MHz sine wave. Sure, there are some other things running around on that trace due to an imperfect waveform generator and some artifacts of the Fourier transform process. But the ugly stuff is 57 dB lower than the peak. That's 1/500,000 less than the 10MHz peak. Switching from dB to voltage makes this more obvious:   Figure 4—10MHz sine wave in volts. Here's the Fourier transform of the 20 ns rise time square wave:   Figure 5—Spectrum of a square wave with 20 ns rise time. The bottom trace is the 1MHz square wave from the scope's waveform generator, before going to the fixer-upper circuit. Above it is the Fourier transform. In this case the screen's scale is set to 0MHz at the left and 100MHz at the right. Each peak is one of the square wave's odd harmonics from the Fourier series. Unsurprisingly, the highest peak is at the wave's fundamental frequency of 1MHz. At 33MHz the signal is down 38 dB, and quickly rolls off from there. That's close enough to the rule of thumb for practical work. With the fixer-upper circuit the rise time is now 1 ns:   Figure 6—Spectrum of a square wave with 1 ns rise time. This looks different from the previous picture because now the Fourier transform window goes from 0MHz to 500MHz. There are a lot more harmonics displayed. But notice that the 40 dB point is now at 375MHz (the formula predicts 500MHz, again, close enough for a rule of thumb). It's important to remember that this is the spectrum of the same 1MHz signal; all that has changed is the rise time. If your little 8 bit system is clocking at just a few MHz, or even hundreds ofkHz, with sharp edges it may have all of the same high speed issues we usually attribute to much faster circuits. Sounds hard to believe. Some time ago a company asked for help with their 30 year old, 4MHz, Z80 design. A new batch of boards didn't work, though the design was unchanged. It seems the semiconductor vendor had increased the speed of some of the logic. Those gates that used to switch in 15 ns now did it in 5. The company had to redesign this board as a high-speed digital system simply because of the faster rise time. The takeaway is that we can't think of digital signals as we do simple sine waves. They are composed of a lot of harmonics, and, depending on rise time, those harmonics can have a huge effect on the circuit's operation. A 1MHz clock does not necessarily imply a slow circuit.

In my seminars, I usually discuss the importance of understanding at least a little electromagnetics theory, even for purely firmware people. But the subject is hard to understand and sometimes harder to believe, which is why the best book on the subject, " High Speed Digital Design, " is subtitled "A Handbook of Black Magic." Why is it important? You'll surely be probing your design with various tools like scopes and logic analysers, and every such probe has some impedance. As speeds get higher, that impedance is ever more likely to corrupt the operation of the device. But "speed" is poorly understood today. We equate clock rate with speed, which is only part of the story. Almost two hundred years ago polymath Jean-Baptiste Joseph Fourier showed that any periodic function can be expressed as the sum of sine waves of different amplitudes and frequencies. A square wave, like a microprocessor clock, is periodic and its Fourier series is:   In other words, a square wave is composed of the sum of the sine of the wave's frequency and each of its odd harmonics. (A harmonic is an integer multiple of the base frequency.) As you can see, no matter what the square wave's frequency is, it has harmonics that go to infinity, though at progressively lower amplitudes. A useful rule of thumb is that for most digital design we can ignore harmonics that exceed: f=0.5/T r Where T r is the square wave's rise time in nanoseconds. Every real-world signal takes time to transition from a zero to a one. Above f the Fourier components are down about 40 dB. Since a picture is worth a million bits, look at the following scope trace:   Figure 1—A square wave with 20 ns rise time. The top trace is a 1MHz square wave, just like a CPU clock. The bottom is an expanded view of the highlighted portion of the same trace. Note that what looks like a nice, quick zero to one transition actually takes quite a bit of time. In this case the rise time is 20 ns. Running 20 ns through the previous formula and it's clear that anything above 25MHz will be so far down we don't have to worry about them. But it does mean that 1MHz signal has important frequency components that far exceed the fundamental. That square wave comes from my scope's waveform generator. To show the Fourier effect more dramatically I built a circuit to improve the signal's rise time by feeding the signal through a fast gate.   Figure 2—Square wave fixer-upper circuit. It is possible to see the individual frequency components predicted by the Fourier series. Most modern scopes can compute the Fourier transform of a signal, as in the following screen capture.   Figure 3—10MHz sine wave The bottom trace is a simple 10MHz sine wave. Above it is the Fourier transform. Unlike a normal scope display where the horizontal axis is time and the vertical volts, the upper one is displayed in units of dB and frequency. In this case the screen's scale is set to 0MHz at the left and 20MHz all the way to the right. Note there's a very strong peak exactly at 10MHz, because 10MHz is the only frequency component in a 10MHz sine wave. Sure, there are some other things running around on that trace due to an imperfect waveform generator and some artifacts of the Fourier transform process. But the ugly stuff is 57 dB lower than the peak. That's 1/500,000 less than the 10MHz peak. Switching from dB to voltage makes this more obvious:   Figure 4—10MHz sine wave in volts. Here's the Fourier transform of the 20 ns rise time square wave:   Figure 5—Spectrum of a square wave with 20 ns rise time. The bottom trace is the 1MHz square wave from the scope's waveform generator, before going to the fixer-upper circuit. Above it is the Fourier transform. In this case the screen's scale is set to 0MHz at the left and 100MHz at the right. Each peak is one of the square wave's odd harmonics from the Fourier series. Unsurprisingly, the highest peak is at the wave's fundamental frequency of 1MHz. At 33MHz the signal is down 38 dB, and quickly rolls off from there. That's close enough to the rule of thumb for practical work. With the fixer-upper circuit the rise time is now 1 ns:   Figure 6—Spectrum of a square wave with 1 ns rise time. This looks different from the previous picture because now the Fourier transform window goes from 0MHz to 500MHz. There are a lot more harmonics displayed. But notice that the 40 dB point is now at 375MHz (the formula predicts 500MHz, again, close enough for a rule of thumb). It's important to remember that this is the spectrum of the same 1MHz signal; all that has changed is the rise time. If your little 8 bit system is clocking at just a few MHz, or even hundreds ofkHz, with sharp edges it may have all of the same high speed issues we usually attribute to much faster circuits. Sounds hard to believe. Some time ago a company asked for help with their 30 year old, 4MHz, Z80 design. A new batch of boards didn't work, though the design was unchanged. It seems the semiconductor vendor had increased the speed of some of the logic. Those gates that used to switch in 15 ns now did it in 5. The company had to redesign this board as a high-speed digital system simply because of the faster rise time. The takeaway is that we can't think of digital signals as we do simple sine waves. They are composed of a lot of harmonics, and, depending on rise time, those harmonics can have a huge effect on the circuit's operation. A 1MHz clock does not necessarily imply a slow circuit.  

It is truly a funny old world. When I was slogging my way through math lessons at high school, and cursing the folks who invented things like logarithms, I would never have dreamed that I would one day read a book on equations "Just for the fun of it!" But things change as you grow older. I just finished reading In Pursuit of the Unknown – 17 Equations That Changed the World by Ian Stewart, and I have to say that – on the whole – this was a very good read. I would dare to say that the majority of us have at least a passing familiarity with history's great equations, such as Pythagoras's theorem, Newton's Law of Gravity, or Einstein's E = mc 2 . I would also dare to suggest that most of us don't know as much as we think we know with regard to the ways in which these equations were derived and the things they lead to. The author has a very friendly and engaging style, and he provides detailed explanations of things I may once have known, but (if so) have long forgotten. We start with Pythagoras's Theorem, which describes how the three sides of a right-angled triangle are related. This isn't a long chapter, but I came away thinking "Wow ... I didn't know this ... or that ... or that!" One thing I really liked about this book is that it's packed with nuggets of knowledge and tidbits of trivia. For example, clay tablets show that the Babylonians knew about the 3-4-5 triangle at least a thousand years before Pythagoras. The interesting thing is how the fundamental equations introduced at the beginning of the book keep on reappearing in different guises as we stroll through more complex concepts later on. I cannot tell you how many times Pythagoras's theorem popped up throughout the book (a bit like the mathematical equivalent of Whac-A-Mole). Chapter 2 is on Logarithms. Before reading this chapter I knew how to use these little rascals, but I had forgotten exactly what they were or how they were derived. By the end of this chapter I was squirming in my seat with excitement. From here we plunge into Calculus, Newton's Law of Gravity, the Square Root of Minus One, Euler's Formula for Polyhedra, Normal Distribution, the Wave Equation, the Fourier Transform, the Navier-Stokes Equation, Maxwell's Equations, the Second Law of Thermodynamics, Relativity, Schrödinger's Equation, Information Theory, Chaos Theory, and the Black-Scholes Equation. It has to be said that some of the equations and topics are more challenging than others, but all-in-all I really enjoyed myself. I discovered all sorts of things, like how the Wave Equation (which was originally developed in the mid-1700s to explain how strings vibrate) lead to huge advances in our understanding of water waves, sound waves, and light waves. I was also captivated by the discussions on how harmonious ratios of the fourth (4:3) and the fifth (3:2) ended up providing a theoretical basis for a musical scale and led to the scale(s) now used in most Western music. The real interesting thing to me was the discussion on how each tone is derived as a ratio, and how each tone can be divided into two intervals, each close to a semitone, to obtain a 12-note scale (think of the white and black keys on a piano – there are 12 for each octave). The thing is that the way in which each tone is divided can be achieved in several different ways, yielding slightly different results. However it is done, there can be subtle but audible problems when changing the key of a piece of music. This completely blew me away – I used to play the piano and trumpet and trombone (not all at the same time, you understand), and I had always assumed that you could simply swap one key for another and continue on your merry way. Later we run into the Scottish physicist and mathematician James Clerk Maxwell (1831-1879). Maxwell's Equations provided the first major unification of physical forces, showing that electricity and magnetism are intimately related. (You can only imagine my surprise when the Wave Equation from Chapter 8 jumped back into the picture.) This lead to the prediction that electromagnetic waves exist and that they travel at the speed of light (finally, after reading this book I understand how this all came about); in turn, this lead to the realisation that light itself is an electromagnetic wave. All of this motivated the subsequent invention of radio, radar, television, and most modern communication systems. As always, I could waffle on and on about this until you begged me to stop. Suffice it to say, however, that I think this book has something for everyone, from younger and/or less-informed readers to engineers and scientists. Even folks who consider themselves to be knowledgeable about mathematics will learn something here, such as the historical background with regard to how their prized mathematical tools came into being.