blinkdagger

This is the eighth post in the blinkdagger signal processing series.

In this post, we’re going to discuss the proper window to choose based on the application and the situation. For the most part (95% of the time) you’ll be best off using the Hanning (Hann) Window because of its versatility and overall effectiveness across the board. Of course, there are exceptions, so read on to learn more about windows and their characteristics.

Contents

• The Fantastic Four
• Spectral Resolution and Spectral Leakage
• Amplitude Accuracy
• Examples of Three Different Windows
• Other Considerations
• References

The Fantastic Four

When it comes to windowing, there are four main things to consider: signal type, spectral resolution, spectral leakage, amplitude accuracy. Below is a nice table that is lifted from this document that shows which window to use in a particular situation:

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Calculator Madness!

If you thought $10 gift certificates to Amazon was good, then you’ll be amped up to hear about the prizes for the next TWO Monday Math Madness. Our partner in math madness, Sol Lederman, has secured two very cool TI calculators (that can be shipped internationally!) to give away in the next two Monday Math Madness contests!

Since the bounty has increased exponentially in value, we decided to spruce up the contest a little bit.

Paradigm Shift

This week’s Monday Math Madness supplies a different twist. Instead of asking readers to submit a solution to a problem that we post, we are instead asking readers to submit their own favorite math/logic problem. The problem that we deem the “best” will be featured in the following Monday Math Madness, and the person who submitted the problem will be rewarded with one of the calculators. For more information on the contest, visit Sol’s website. Essentially, we’re looking for interesting/clever/intriguing problems that have elegant/unexpected answers. Good luck to all!

Don’t Miss Out! Subscribe to our Feed!!

Since we’ll be hosting the next MMM at blinkdagger, you should subscribe to our feed so that you won’t miss out on an oppurtunity to win a fantastic TI-calculator. Our last contest involving the Martian Problem only had 9 submissions (only 3 were correct), so your chances could be pretty good!

Nostalgia

TI-Calculators have come a long way ever since my high school days. Back then, I was still using the ancient TI-85, shown here. It was a nice calculator that got me through AP Calculus. But after that, I moved on to the much improved TI-89, which I still have in my closet to this day.

The TI-85(1992) was designed as a powerful engineering and calculus calculator. It was the first TI calculator to have a link port and assembly programming capability (through an unintentional loophole). It has since been eclipsed by the TI-86, which features TI-85 compatibility along with more advanced features.

The TI-89 (1998) sports all of the power of a TI-92 Plus, but in a traditional handheld design. Along with the TI-92 Plus and Voyage 200 it is one of TI’s most sophisticated calculators, with such features as 3D graphing, upgradable flash ROM, built-in assembly language programming, and a generous amount of user-available memory. The TI-89 Titanium will eventually replace this model.

MMM #12 Winner

The results are in, and the winner for MMM #12 is Anneleen Van Geenhoven! She will be receiving a $10 Amazon gift certificate. We only had 9 submissions this time, but a whopping 2500+ visits. This was a challenging problem so thanks to everyone who participated!

Correct Submissions
Lieven Marchand
Kinks

Before We Tell You The Answer

Perhaps you needed a little help to solve the problem. If you are still up for solving the Martian problem consider the following hints.

Hints:

1. The dialogue can be interpreted as a series of problem statements. For example: Dan: Quan, you don’t know what X and Y are. implies that the product is not a unique product. We are also told that the numbers are greater than 2. So products like 12, 15, etc… cannot be the product told to Quan because there exist only one pair of factors that are greater than 2 for 12: 3×4 and 15: 3×5, making them ‘unique’.
2. You can proceed to decipher the problem statements to eliminate pairs of numbers that do not meet the requirements.
3. Again we are not looking for unique sum or product. If it is unique Quan and Dan would know the two numbers!
4. The answer is not 4 and 13! 4 and 13 is the answer for the range of numbers 2 to 99. But we specifically said that the numbers 2 and 99 are not included. If you google for the answer you would probably find 4 and 13

The Answer

The two numbers are 13 and 16

Here is Anneleen’s explanation:

We can already conclude a lot of things from the first statement of Dan (and the fact that Quan admits he was right).

1) The most interesting thing is that the sum of the numbers x and y needs to be odd, in other words, that one of x and y is odd, and that the other one is even. Here is why: suppose the sum of the 2 numbers is even. Then it can be written as the sum of 2 prime numbers in some way. (This is the Goldbach’s conjecture, which is not yet proved as far as I know, but we are only working with numbers smaller than 100, and for those numbers this property holds anyway.) So, if Dan has been told an even number as sum, then he thinks that it is possible that x and y are exactly those primes (or 1 of those combinations, because the ‘decomposition’ in 2 primes is not at all unique). But in this case, the product of these 2 numbers is easily factorisable, so that Quan knows the 2 numbers immediately. So, Dan could never be sure that Quan doesn’t know the 2 numbers.

2) Likewise, the sum of the numbers can not be 4 more than a prime number. Let’s suppose that the sum x+y equals 4+p, where p is a prime number. Then Dan thinks that it is possible that {x, y} = {4, p}, but then the product xy = 4p, and this can be composed in only 1 way (because 2 is not allowed). So in this case also, Dan couldn’t be sure that Quan doesn’t know the numbers x and y.

3) It’s also not possible to have a sum which is 57 or more: otherwise the sum x+y could also be written as the sum of 53 and another number larger than 2, but when 53 is a factor of the product, Quan should know the numbers x and y, because 53 is a prime, and it can’t be multiplied by another factor because then it would exceed 98.
(So the reason why this argument holds is that 53 is the smallest prime which is larger than 98/2 = 49).

4) For any prime p larger than 10, the sum can’t be 3p, because otherwise p and 2p could be the numbers x and y, but in this case Quan would know x and y immediately: 2p^2 can’t be factorised in another way in 2 numbers (larger than 2 and smaller than 99). So, the following sums are also impossible: 33, 39, 51, 57, …

So, the sum x+y has to be one of the following:
13, 19, 25, 29, 31, 37, 43, 49, 53, 55

(The other way around, with any of these sums, it is also possible to prove that Dan’s first statement (about Quan knowing nothing) is right.)

Now let’s define the property P as follows:
We say that a number N has property P if and only if there is only 1 way to write this number as a product of 2 factors, (larger than 2 and smaller than 99) where these factors have as sum one of the numbers in the list above.

Now we have the following conditions:
1) After the first statement of Dan, Quan knows the two numbers. This means that the product he got needs to have property P.
2) Moreover, also Dan knows the numbers after Quan says he knows. In other words, for the sum Dan got, there can only be 1 possible decomposition s = a+b, for which the product ab has property P.

Let’s consider all the foregoing sums:
13: this can be 4+9 (the product 36 has property P)
or 3+10 (30 has property P)
and there are still other possibilities, but all that’s important is that condition 2 is not satisfied here
19: 3+16 or 4+15 (48 and 60 both have property P) or …, so condition 2 doesn’t hold
25: 3+22 or 6+19 are both possible (66 and 114 have property P), so condition 2 doesn’t hold
29: I’ll let this one open for a moment
31: 4+27, 5+26, (108 and 130 satisfy P) so condition 2 doesn’t hold
37: 3+34, 4+33 (102 and 132 satisfy P) so condition 2 is not satisfied
43: 6+37, 8+35 (222 and 280 satisfy P) so 2 is not satisfied
49: 6+43, 8+41 (258 and 328 satisfy P) so 2 is not satisfied
53: 4+49, 6+47 (196 and 282 satisfy P) so 2 is not satisfied
55: 5+50, 8+47 (250 and 376 satisty P) so 2 is not satisfied

So, from condition 2 we get that the sum needs to be 29. Now let’s check condition 1 for all possible sums of 2 terms to get 29:
3+26, product 78 = 6.13 doesn’t have property P
5+24, product 120 = 3.40 doesn’t have P
7+22, product 154 = 11.14 doesn’t satisfy P
9+20, product 180 = 4.45 doesn’t satisfy P
11+18, product 198 = 9.22 doesn’t satisfy P
13+16, product 208 satisfies P
15+14, product 210 = 10.21 doesn’t satisfy P
17+12, product 204 = 4.51 doesn’t satisfy P
19+10, product 190 = 5.38 doesn’t satisfy P
21+8, product 168 = 7.24 doesn’t satisfy P
23+6, product 138 = 3.46 doesn’t satisfy P
25+4, product 100 = 5.20 doesn’t satisfy P

So, we see that the given scenario is only possible when x = 13, and y = 16 (or vice versa of course).

In the past year or so, there have been many hosting programs introduced to the DoTA scene. The most popular of these are: ListChecker, DoTA Client, and Delay Reducer. Each has its pros and cons and I will discuss all three in this post.

You can download and install each of the programs by following these links:

1. ListChecker
2. DoTA Client
3. Delay Reducer

Ease of Installation

Delay Reducer is hands down the easiest to install, and DoTA Client is not far behind. ListChecker is by far the most difficult to configure and install, and there have been entire posts and forums dedicated to troubleshooting the ListChecker program.

Ease of Use

Delay Reducer once again takes the cake here. All you have to do is open up the program before the game begins and host games as you normally would via Battle.net. You ccan set your latency using an ingame command which is nifty to ensure that you’re running at the optimal latency.

DoTA Client isn’t too difficult to use either. Some initial configuration and you’re ready to host just like you would host a normal game on Battle.net. There have been reported problems that people can’t connect to the correct server and that DoTA Client simply won’t work. It worked fine for me but I didn’t use it much longer than 2 weeks before I started using ListChecker again.

ListChecker is the most cumbersome of the three programs. You have to start the program, go into LAN, and then create the game using in-game commands. By far the annoying thing as you have to exit Battle.net to host the game. Once the game is over, you have to sign back into Battle.net for any post game discussions.

Latency

Delay Reducer and DoTA Client both have the flexibility of configuring the latency in game.

ListChecker has a default latency of 100ms, which is a great improvement over the Battle.net standard of 250 ms.

Stability

Stability is key in DoTA to ensure that people do not drop and that games will be fun for all. The king of stability is ListChecker. I’ve used ListChecker for almost a year now, and its stability is unparalleled. When you have people from different countries (or cross country) playing a game, ListChecker pretty much handles it like a champ.

Delay Reducer and DoTA Client seem to work much better when the people playing the game are much closer to each other.

Overall

Overall, I would have to designate ListChecker as the champion of Hosting Programs. Sure, the installation and ease of use is no walk in the park, but once you set up the program, all of your games will be lag-free and your games will be stable as a rock!

My next choice would be Delay Reducer for its incredibly simply interface and ease of use. I’ve use this one with minimal problems. Sometimes when I’m too lazy to exit Battle.net to host with ListChecker, I just use Delay Reducer instead.

The forgotten program here is DoTA Client. I haven’t used it in a while, but I had some problems with the stability issue so I went back to ListChecker. In addition, many people have problems connecting to the correct server.

Future Potential

DoTA Client by far has the most potential of any of these three programs. They offer a range of goodies such as anti-hack.

ListChecker hasn’t changed much since its introduction to the scene, so I predict it won’t be changing much in the future.

Delay Reducer has some potential to do some cool things as well.

In Summary

This is the seventh post in the blinkdagger signal processing series.

In the previous windowing post, we discussed what spectral leakage is and how it comes about. In this post, we are going to provide an example of what windowing can do to mitigate spectral leakage.

Contents

• Quick Recap from Last Time
• Windowing
• Apply the Window to the Input Signal
• Comparison of Non-windowed and Windowed Signal
• Next Time - How to Choose a Window that’s Best for You

Quick Recap from Last Time

In the last post, we tried taking the Fourier transform of the following sine wave:

fo = 10;   %frequency of the sine wave
Fs = 200; %sampling rate
Ts = 1/Fs; %sampling period
t2 = 0:Ts:0.95 -Ts; %time vector(notice the difference here!)
n2 = length(t2); %number of samples
y2 = 2*sin(2*pi*fo*t2);%plot the curve in the time domain
sinePlot = figure;
plot(t2,y2)

Which yielded the following result:

Let’s do an example of windowing to show how it can mitigate the effects of spectral leakage.

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