Lx=d: SundayResearch

Input actual data Let's input some signal to the filter that we designed last time. Table 1 shows the input of constant signal. Constant signal will be boring, but, We start with a simple one. Table 1 Constant input To make sure, I will show you how to compute the Table 1's red output. Here y_n and n=1, In this case, the filter gets the first three inputs. The inputs are all the same (= 1), therefore all the outputs are also the same. Let's compute the transfer function. We compute all the time the transfer function in digital filter. What, Jojo, You! Be surprised. (it's a bit old and does anybody know Jojo's strange adventure by Araki Hirohiko?) The ratio of input and output is equal to the transfer function! Here is a small details. In the Table, y_n has a value at n=0,8 since we can compute the cos value. But there usually is no n=-1 value in the real data acquisition (If we start with n=0, then no data at n=-1). Therefore, it is also possible to say there i...

This is about filter design of Hamming's digital filter. The chapter 3.9 is outstanding, so I would like to make a note. While ago there was a soccer game. How to suppress the Vuvuzela sound (but other sounds should pass) was a popular topic at that time. In this case, you analyze the frequency of Vuvuzela sound and design a filter which doesn't pass that frequency. The design of digital filter is: which frequency of the signal will be passed and which frequency of the signal will not be passed. See the following article for instance, http://blogs.mathworks.com/loren/2010/06/30/vuvuzela-denoising-with-parametric-equalizers/ In chapter 3.9, a simple non recursive filter design is described as an example. The filter looks like the following. This filter uses three samples. The design of filter is to determine the a,b to fit to your desire. First, we will calculate the transfer function of this filter. As I explained in a former blog post, a transfer function is an eige...

Last time I use sin and cos, but this relationship becomes simpler if we use Euler's formula. Let's apply the same operator T. Wow again. This is also eigenfunction of operator T. This function is based on trigonometric functions. Therefore, we use these trigonometric function as the basis of the frequency domain analysis. Eigenvalues show us a brief overview of the operation and its function. Assume we have an input x and an output y, operator T is applied to the input x, then the result is the y. If eigenvalue exists, we could write it as the following. This means, the input is transfered to the output and how much transfered is λ. Therefore, signal processing people call this λ as a transfer function. Why it is called function? λ looks like a constant. Usually, λ is not a function of input x, but it usually has some parameter, means this is a function. For example, in the former equation, λ is not a function of x, but a function of ω.  In signal processing, x is usual...

Eigenvalue and transfer function In the Hamming's book, he repeats to mention about the merit of using trigonometric function as a basis in signal processing. Unfortunately, that is not the main point of this blog, therefore, I could not compare it with the other bases. I will just stick to this basis with assuming this is a good one. Let's see the eigenvalue of trigonometric function according to the example of Hamming's book. The first example in his book is  A sin x + B cos x. We apply a transformation operation. Then, let's see something is changed or not. If something doesn't change, it will be a eigenvector and we will also see its eigenvalue. Transform T is a shift operation of the origin of coordinate like T: x → x' + h . Why someone wants to shift the coordinate? For example, signal processing usually doesn't matter when you start to measure the signal sequence. When you started to measure the signal, then, that point is the origin. Usually the...

Eigenvalue and Eigenvector Function case Interestingly, the same story is repeated again in function. (Well, ``interesting'' is just my personal feeling. So, many might not agree with this. I found this --- the same story repeated again, but in the different level --- interesting in mathematics. Like Hitchhiker's Guide to the galaxy's jokes have some mathematical structure.) So far, we apply an ``operation'' to a scalar or a vector. Then, we again apply an operation to a function. We want to know what is the substance of the ``operation'' instead of the each result of operation. We could not know the substance at once, but we could know the response of the function with an operation. Usually, a function is an operation to a scalar or a vector, therefore, it is a bit confusing to think about an operation on an operation. Let's see an example. Let's assume a function f and a scalar or a vector x , this function f can be an operation on x , ...

Eigenvalue and Eigenvector Vector case Next, let's think about vector. I assume the readers know about matrix multiplication a bit. When a matrix applies to a vector, then this generates a new vector. For example, a matrix can rotate a vector, or enlarge/shrink a vector. We say we can apply a matrix A to a vector x . The matrix A could be a rotation operation, or any. The result of application creates a new vector b .  A x = b If you see this, it looks like scalar's multiplication. But, usually A is quite complex and hard to understand what it is. But if there is a scalar λ and a vector x' , such that  A x' = λ x' . This means: We can replace a complex matrix A with a scalar value λ.  I think this sentence has the whole idea of this topic. Usually it is not possible to find such λ for any vector. But, we have a chance to find a specific x' and its relating a scalar value λ. When I saw this, I said, ``Wow.'' This is a powerful idea. If I...

Eigenvalue and Eigenvector Scalar case Let's start with multiplication of scalars. 3 x 4 = 12 If I could use variables a,x,b: ax = b. The first example shows x=4 times a=3 equals b =12. Here, I multiply x a times. I didn't multiply a times x. In the scalar case, these have no difference since the following commutative law works on scalars. ax = xa Please note, this is not always correct. Even we can not exchange the meaning in the scalar numbers. For example, assume there is a chocolate box that costs five Euro. We can buy two packages. This is 2 times 5 = 10 Euro. This is not two's 5 Euro time. We can double the 5 Euro chocolate, but we can not see five Euro time doubles. Usually it doesn't make sense: five Euro times (five times works, five Euro times has a problem). So I remind the order of operator is also important since vector is more strict about the operations.

Function Here I want to put matrix according to the order of this story, however, I will skip the matrix story for simplicity. Vector is a sequence of scalars. The sequence order of scalars is very important. If we changed the order of scalars, they are totally different vectors. For example, we had a vector that represents position, it was [direction distance] that means 50 degree from north, distance 5 km. If we exchange the direction and distance, it becomes [distance direction] that means 5 degree from north, 50 km distance. That is the different position (at least in the Euclidean space). If we can put 4 scalars. Figure 1's the second from the top shows the vector that has 4 scalars. A scalar is one (real) number and a vector is a sequence of scalars. If we add more scalars in a vector, what happens? If we add scalars more and more, infinite number of scalars are added, then it becomes a function as shown in the Figure. OK, I cheated here a bit. We need some more prerequ...

Scalar, vector, and function Scalar When I want to mention a quantity, I use numbers. A number always represents ``something.'' For example, if I said 130, what is this number means? The number itself has not so much meaning. This could be someone is 130cm tall, or Autobahn's speed limit is 130km/hour.  One number represent ``something'' e.g., ``tall cm'', ``speed limit km/hour.''  Figure 1 shows this. These single numbers are Scalers. It is just a number, why it has a special name Scalar? I think this is just for distinguishing a scalar and a vector (or a (complex) number). Figure 1 Scalar has some meaning Vector There are many stuffs I can not represent with a scalar. For instance, a place. The distance from my apartment to Zoo station can be represented by a scalar value. But, if you need a direction, I can't tell it by a scalar value. I could say, 50 degree from the north in clockwise direction, the distance is 5 km. Or go north 3...

Introduction There was a famous mathematician and computer scientist, Richard Hamming. I am reading his book, Digital Filters. I would like to write  something I understand about this book. Let's talk about eigenvalue and transfer function. But this is too sudden. Most of the people (including me) would say What is eigen-blah stuff? Therefore, I would like to start why it matters, what is the motivation to think about that, as usual in my blog. After reading some math book, I often said, ``I don't understand'' or ``So what?'' I want to say, ``Wow, that's great.'' If I said, ``Wow, that's great,'' then I usually understand what the purpose is and it is achieved in the paper. I try to explain this in the high school math only, but, I found out one step is missing. That is the relationship among scaler, vector, and function. I would like to explain these are all the same in some abstraction sense. Maybe high school students know a vector...