标签: ratio

All data streams experience jitter, some of which is statistically random, so that means we must characterize it with statistics. In jitter analysis, random jitter (RJ) is often described as having a Gaussian distribution.   RJ is typically quantified as an rms value equal to σ of the Gaussian formula. Since Gaussian distributions are unbounded, a peak-to-peak value of RJ requires a measure of probability of bit errors, or BER (bit error ratio). For a typical specification of BER = 10 -12 , (1 error for every 10 12 bits) the peak-to-peak value of RJ is closely approximated by 14σ, where σ is one standard deviation.   Figure 1 shows a plot of the familiar Gaussian distribution. We are often concerned with probability in any given area of this bell curve. Table 1 shows the probability of events occurring in the peak and tail regions for different spans of σ.   Figure 1. Gaussian distribution.   Table 1. Probability of peak and tail areas vs. number of σ.   At ±6σ, we see that the probability of the tails is about two in a billion. For ±7σ, the probability of the tails is about 2.6 in a trillion, which is roughly how we came to use 14σ to bound RJ at BER = 10 -12 . Even though we frequently talk about probabilities of less than one in a trillion, few people actually have an intuitive grasp of such large/small numbers. I came across a very interesting example to illustrate Gaussian distributions using the distribution of height in American men.   The US has approximately 300 million people, out of whom approximately 100 million are non-acromegalic adult men (i.e., no abnormal pituitary problems affecting their height). The mean height of men in the US is about 5'9" with a standard deviation of 3". Knowing how Gaussian distributions work, we can estimate how many men are in each height bracket, shown below in Table 2.   Table 2. The probability of the height of men.   We see that about 68 million men are within ±1σ of the mean, which is between 5'6" and 6' tall. We refer to this range as average height. We think of men between 6' and 6'3" as being relatively tall, but there is still a large population of them, about 13.6 million or 13.6 percent. The numbers drop off very quickly for increasing heights, as expected.   What is interesting is that Gaussian statistics tell us that there should be no men in the US who are above 7'3" tall. But, many sports fans can immediately point out that Yao Ming is 7'6". Someone of Yao Ming's height is statistically unlikely in the normal height distribution of American men.   So, why is Yao Ming Chinese? Because the US does not have enough adult men to realize someone who is greater than 6σ in height! Instead, Yao Ming was born in China, where the current population of adult men is about 500 million, which is five times more likely to grow a 6σ person. Furthermore, the mean height of men in China is approximately 5'7", which makes Yao Ming's stature even more statistically rare!   Yao Ming, basketball player for the Houston Rockets (retired).   Many readers will quickly point out that India also has a population comparable to China's. Is there at least one Indian man who is 6σ above the mean in height? The answer is a resounding yes! Sunil Chaudhary currently claims to be the tallest living man in India at a height of 7’6", a number that neatly fits into our understanding of 6σ in Gaussian distributions.   So, in jitter analysis, when we are looking for that less-than-one-in-a-trillion event that determines our BER compliance, or when we test for violations with a PRBS31 (2 31 −1) pattern -- 2.4 billion bits long -- I like to tell people I'm searching for the "Yao Ming of jitter."   Daniel Chow

Most likely, you won't be surprised to hear that this blog relates to my ongoing Bodacious Acoustic Diagnostic Astoundingly Superior Spectromatic (BADASS) Display project. As you may recall, this little beauty is going to boast a 16 x 16 array of tricolored LEDs. In my previous blog on this project, in which we discussed the physical implementation and presentation, I noted that I had spaced out the LEDs more on the horizontal axis than on the vertical axis so as to increase the visual appeal of the display.   My vertical spacing between pixels is 33.33mm. This spacing is fixed because I'm using NeoPixel Strips from Adafruit -- the type with 30 NeoPixels per meter -- with the LEDs controlled by an Arduino Mega boasting an Atmel microcontroller. For my display, I decided to use a multiplication factor of 1.5, thereby giving me a horizontal spacing of 33.33mm * 1.5 ≈ 50mm.   A reader then suggested this:   You might want to consider employing the Golden Ratio (1.618:1 -- close enough for government work) in your rectangular layout. Doing so several times would be even cooler.   Hmm, that's certainly something to think about. In mathematics, two quantities are said to be in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. This is much easier to visualize graphically (well, it would be, wouldn’t it?) as illustrated below:   On the left we see a "golden line," while a "golden rectangle" is shown on the right. If we do the math, as illustrated below, we discover that the golden ratio works out at ≈1.618.   The golden ratio pops up all over the place in nature. For example, it is expressed in the arrangement of branches along the stems of plants, the branching on veins and nerves in animals, the proportions of chemicals in compounds, the geometry of crystals, and… the list goes on.   Many people seem to find things in proportion to the golden radio to be aesthetically pleasing. Based on this, some artists and architects have created their works so as to reflect this proportion (many of the proportions of the Parthenon on Greece are said to exhibit the golden ratio, for example).   Now, I was aware of the golden ratio (What engineer isn’t?), but I've never actually heard about it being employed in industrial design, so I decided to perform a simple experiment. Remember that each of my LEDs is going to be accompanied by a 20mm-diameter brass washer. First, I created an array in which the horizontal and vertical separations between elements were both 33.33mm as illustrated below:   Horizontal separation = 1.0x vertical separation.   Next, I made the horizontal spacing 1.5X the vertical spacing (the way I currently have things set up on the BIGASS Display), that is, 33.33mm * 1.5 ≈ 50mm as illustrated below:   Horizontal separation = 1.5x vertical separation.   Finally, I related the horizontal spacing to the vertical spacing using the golden ratio, that is, 33.33mm * 1.618 ≈ 54mm as illustrated below:   Horizontal separation = 1.6x vertical separation. I then went for a wander around the building to see what people thought. Well, wouldn’t you know it? Almost everyone had gone out for lunch! Happily, I did manage to track down two engineers. I talked to them independently and I didn’t mention anything about golden radios. Just to give them some reference to work from, however, I did explain that this was to be an array of LEDs presenting the spectral information from an audio stream, with the amplitude of the various frequency elements being presented on the vertical axis.   One of the engineers said he preferred the first pattern -- the one with equal horizontal and vertical spacing -- on the basis that one could present more information in a smaller area. Well, that certainly makes sense in an engineering-over-aesthetics sort of way. Interestingly enough, the other engineer said he preferred to have a wider horizontal spacing for visual interest. Furthermore, after glancing back and forth for a while, he said that -- although he couldn’t explain why -- he was more drawn to the last example (the one employing the golden ratio).   The difference between my 1.5 ratio and the golden ratio, which resulted in horizontal separations of 50mm and 54mm, respectively, is only 4mm, but that was enough to persuade this engineer to prefer the golden ratio.   Of course, a sample of two people is statistically meaningless, but it does make you think. I'm not ready to change the layout of my BIGASS Display quite yet (since the thought of re-laying out all 256 LEDs brings tears to my eyes), but I'm not ruling it out either.   As I mentioned earlier, I've never actually heard about the golden ratio being employed in industrial design -- have you? Also, if you have the time, it would be great if you could click the links above, print out the larger versions of my images, perform your own survey amongst your coworkers, and then report your findings back here. Based on these results, I will decide if a change in direction is on the cards for my BIGASS display.

It is no longer surprising to hear that this column relates to my ongoing Bodacious Acoustic Diagnostic Astoundingly Superior Spectromatic (BADASS) Display project. As you may recall, this little beauty is going to boast a 16 x 16 array of tricolored LEDs. In my previous blog on this project, in which we discussed the physical implementation and presentation, I noted that I had spaced out the LEDs more on the horizontal axis than on the vertical axis so as to increase the visual appeal of the display.   My vertical spacing between pixels is 33.33mm. This spacing is fixed because I'm using NeoPixel Strips from Adafruit -- the type with 30 NeoPixels per meter -- with the LEDs controlled by an Arduino Mega boasting an Atmel microcontroller. For my display, I decided to use a multiplication factor of 1.5, thereby giving me a horizontal spacing of 33.33mm * 1.5 ≈ 50mm.   A reader then suggested this:   You might want to consider employing the Golden Ratio (1.618:1 -- close enough for government work) in your rectangular layout. Doing so several times would be even cooler.   Hmm, that's certainly something to think about. In mathematics, two quantities are said to be in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. This is much easier to visualize graphically (well, it would be, wouldn’t it?) as illustrated below:   On the left we see a "golden line," while a "golden rectangle" is shown on the right. If we do the math, as illustrated below, we discover that the golden ratio works out at ≈1.618.   The golden ratio pops up all over the place in nature. For example, it is expressed in the arrangement of branches along the stems of plants, the branching on veins and nerves in animals, the proportions of chemicals in compounds, the geometry of crystals, and… the list goes on.   Many people seem to find things in proportion to the golden radio to be aesthetically pleasing. Based on this, some artists and architects have created their works so as to reflect this proportion (many of the proportions of the Parthenon on Greece are said to exhibit the golden ratio, for example).   Now, I was aware of the golden ratio (What engineer isn’t?), but I've never actually heard about it being employed in industrial design, so I decided to perform a simple experiment. Remember that each of my LEDs is going to be accompanied by a 20mm-diameter brass washer. First, I created an array in which the horizontal and vertical separations between elements were both 33.33mm as illustrated below:   Horizontal separation = 1.0x vertical separation.   Next, I made the horizontal spacing 1.5X the vertical spacing (the way I currently have things set up on the BIGASS Display), that is, 33.33mm * 1.5 ≈ 50mm as illustrated below:   Horizontal separation = 1.5x vertical separation.   Finally, I related the horizontal spacing to the vertical spacing using the golden ratio, that is, 33.33mm * 1.618 ≈ 54mm as illustrated below:   Horizontal separation = 1.6x vertical separation. I then went for a wander around the building to see what people thought. Well, wouldn’t you know it? Almost everyone had gone out for lunch! Happily, I did manage to track down two engineers. I talked to them independently and I didn’t mention anything about golden radios. Just to give them some reference to work from, however, I did explain that this was to be an array of LEDs presenting the spectral information from an audio stream, with the amplitude of the various frequency elements being presented on the vertical axis.   One of the engineers said he preferred the first pattern -- the one with equal horizontal and vertical spacing -- on the basis that one could present more information in a smaller area. Well, that certainly makes sense in an engineering-over-aesthetics sort of way. Interestingly enough, the other engineer said he preferred to have a wider horizontal spacing for visual interest. Furthermore, after glancing back and forth for a while, he said that -- although he couldn’t explain why -- he was more drawn to the last example (the one employing the golden ratio).   The difference between my 1.5 ratio and the golden ratio, which resulted in horizontal separations of 50mm and 54mm, respectively, is only 4mm, but that was enough to persuade this engineer to prefer the golden ratio.   Of course, a sample of two people is statistically meaningless, but it does make you think. I'm not ready to change the layout of my BIGASS Display quite yet (since the thought of re-laying out all 256 LEDs brings tears to my eyes), but I'm not ruling it out either.   As I mentioned earlier, I've never actually heard about the golden ratio being employed in industrial design -- have you? Also, if you have the time, it would be great if you could click the links above, print out the larger versions of my images, perform your own survey amongst your coworkers, and then report your findings back here. Based on these results, I will decide if a change in direction is on the cards for my BIGASS display.