# Mysterious pCell, or DIDO under the microscope

So, today after reading the translation article about pCell , I hit my face in the face a couple of times. Because it’s easy for anyone who has a physical level in mobile networks to understand what “mysterious motimatics (! Sic)” DIDO is all about.

Let's start from afar. Where do the legs grow from.

#### Orthogonal sets

An orthogonal set in mathematics is the set or subset of elements, where for any x and y from this set the following conditions are satisfied:

1) f (x, y) = 0 if x! = Y
2) f (x, x) = 1

Where the operation f is a scalar product that fulfills all three properties of the scalar product.

Moreover, the operation f and the elements of the set can be anything. So f can be both a banal operation of the scalar product of vectors and an integral. And the elements of the set can be either vectors or even functions. For example, when approximating functions, systems of orthogonal polynomials are often used. But that is another topic.

Back to our rams. Suppose we have two numbers (scalar) a and b and an orthogonal subset B from the set A. Take 2 elements x and y from B and make up such an element (a * x + b * y) that will belong to A, but not belong B. We get the following chains of operations:

1) f ((a * x + b * y), x) = a * f (x, x) + b * f (y, x) = a * 1 + b * 0 = a ,
2) f ((a * x + b * y), y) = a * f (x, y) + b * f (x, x) = a * 0 + b * 1 = b

Thus, in order to obtain an initial scalar from a composite element, it is enough to take the scalar product from this element and the original element of the orthogonal set.

If it is not yet clear how this relates to the topic, I will rephrase the previous sentence.

Thus, in order to obtain from the composite signal the original signal , it is sufficient to take the dot product of the received signal and source element orthogonal set.

Is starting to clear up, isn't it?

#### Orthogonal Codes

Orthogonal codes are the usual set of orthogonal vectors. In telecommunication systems they are used universally. For example, their application can be found in CDMA and W-CDMA technologies. The idea is that each bit that is transmitted through the physical medium must be encoded with a specific orthogonal code. Here, by “coded” is meant the banal operation of multiplying a number by a vector. And so, after encoding through the physical medium, not a bit is transmitted, but a whole vector multiplied by the value of the bit. And each element of such a vector is called chip . The multiplication operation itself is called channelization, and the orthogonal code is called channelization code.

What this gives is shown in the 2 properties of orthogonal sets in the previous section. Using the encoding of the signal with orthogonal codes, it becomes possible to transmit several different signals simultaneously on the same physical frequency. Of course, the maximum number of signals is still limited by the length of the code. For a code with a length of 2 elements - a maximum of 2 signals can be transmitted, for 3 elements - 3 signals, etc. I want to emphasize that this is being used now . The base station equipment encodes the signal and the receivers in mobile phones receive a signal intended for them, using the orthogonal code assigned to them.

In reality, there are a lot of nuances, such as the generation of orthogonal codes for transmission on the fly, but these are details.

So what can be innovative in receiving a signal from several radio points at the same time, what is declared in pCell? That's right - nothing. The only difference is that by creating several separate transmitting points, the system developers received an additional crap associated with synchronizing the time of the signals (the transmitters should send the beginning of the frame synchronously, otherwise there will be an offset between the orthogonal codes and the magic will stop working).

The rest of the filling and theory has been used for a long time, and I can’t name all this except marketing blurring of the eyes.