Walsh codes. CDMA: signals and their properties

Lecture 17. Walsh functions and their applications

Walsh functions. Basic definitions. Ways to Order Walsh Functions

The Walsh functions are a natural extension of the Rademacher function system, obtained by Walsh in 1923 and representing a complete system of orthonormal rectangular functions.

The set of Walsh functions, ordered by frequency, is usually denoted as follows:

Walsh functions ordered by frequency are similar to trigonometric functions can be divided into even cal(i,t) and odd sal(i,t)

Figure 17.1 shows the first eight wal functions w(i,t).

Figure 17.1

It is clear that the frequency of each subsequent Walsh function is greater than or equal to the frequency of the previous Walsh function and has one more zero crossing in the open interval t. This is where the name “frequency ordering” comes from.

Discretizing the Walsh functions shown in Figure 17.1a at eight equally spaced points results in the (8x8) matrix shown in Figure 17.1b. This matrix is ​​denoted by H w(n) where n=log 2 N and the matrix will have size NxN.

Walsh functions, when ordered by frequency, in the general case can be obtained from the Rademacher functions r k (x) using the formula:

where w is the number of the Walsh function; k – Rademacher function number;
exponent of the Rademacher function, which takes the value 0 or 1 as a result of summation modulo two, i.e. according to the rule: 11=00=0; 10=01=1 bits of a binary number w. For example, for the sixth Walsh function ( w=6), included in a system of size N=2 3 =8, the product (17.4) consists of three factors of the form: for k=1
at k=2
at k=3
. A number in the binary system is written as a combination of zeros and ones. In our case, the value w and its categories are shown in table 17.1

Table 17.1

w 0 – the most significant digit of the number, w 3 – least significant digit of the number w.

The exponents of the Rademacher functions are equal to:
;
;
and therefore,

wal(6,x)=r 1 1 (x)r 2 0 (x)r 3 1 (x)=r 1 (x)r 3 (x)

The rule for obtaining exponents for the Rademacher function is shown schematically in Table 17.1, where the arrows indicate the summable digits of the number w and Rademacher functions, to which the resulting exponent relates. From Figure 17.1 it can be seen that even numbers of Walsh functions refer to even functions, and odd numbers refer to odd functions. Another way of ordering is Paley ordering. When ordered by Paley, the analytical notation of the Walsh function has the form:

p 1 is the least significant digit of a binary number, p n is the most significant digit of a binary number. When ordering according to Paley, to form Walsh functions, it is necessary to take the product of Rademacher functions raised to a power, the numbers of which coincide with the numbers of the corresponding digits of the double representation of the number p, and the exponent of each function is equal to the contents of the corresponding digit, i.e. 0 or 1. Moreover, the least significant Rademacher function corresponds to the least significant digit of the binary combination of the number p. In accordance with this rule, table 17.2 shows the values ​​of Paley-ordered Walsh functions.

Table 17.2

The Rademacher functions in the table are shown in the form:
. A comparison of the products and powers of the Rademacher functions written in tables 17.1 and 17.2 shows that there is a correspondence between the Walsh functions ordered by Paley and Walsh, which is reflected in the last column of table 17.2. In accordance with the Paley-ordered Walsh functions, a matrix of samples H p (n) can also be constructed, similar to that shown in Figure 17.1b.

The next common ordering method is Hadamard ordering. The Hadamard functions har(h,x) are formed using the Hadamard matrices. A Hadamard matrix H N of order N=2 n is a square matrix with dimensions NxN and elements 1, which has the property

For example, starting from H 1 = 1 we find:

Comparing the resulting matrix H 8 with the matrix of samples for the Walsh function, Ordered by Walsh (Figure 17.1b), we see that the following correspondence exists between the first eight functions ordered by Walsh and Hadamard:

and can serve as the basis for the spectral representation of signals. Any function that is integrable over the interval 0x1 and is a mathematical model of an electrical signal can be represented by a Fourier series using the system of Walsh functions

Where
- dimensionless time normalized to an arbitrary interval T.

The Walsh functions, like the Rademacher functions, take only two values: -1 and 1. For any m – wal 2 (m,x)=wal(0,x)=1.

Walsh functions are periodic functions with period equal to 1.

Walsh functions have the multiplicative property; multiplication of any two Walsh functions is also a Walsh function:

The average value of the Walsh function wal(i,x), for i0 is equal to zero.

The Walsh function system is a composite system and consists of even and odd functions, denoted respectively:

The relative error of approximation of the signal f(x) by a finite number of Walsh functions is determined by the formula

Where
- signal energy at a unit normalized interval.

Questions for self-study

Find expressions for the Walsh functions in terms of the Rademacher functions wal(7,x), wal(9,x), wal(13,x) in Walsh, Paley and Hadamard ordering.

List and explain the basic properties of Walsh functions.

Expand in a Walsh series, restricting yourself to the first eight Walsh functions of the functions sin x,cos x and build them.

Describe the advantages and disadvantages of each of the considered ways of ordering Walsh functions.

Calculate the values ​​of the first 8 coefficients of the Fourier–Walsh series expansion of the following signals:

The IS95c system (CDMA-2000-1x) uses code division multiple access technology (see PSP and characteristics), thanks to the use of this technology, the method of addressing channels, mobile and base stations in the system is also implemented using codes in a special way. To explain the principles implemented in this system, this section will first explain some technical concepts, and then address addressing issues in detail.

Radio channel configuration

Radio configuration (RC) defines the configuration of physical channels based on the specific data rate. Each RC defines a set of data rates, based on 9.6 or 14.4 kbit/s. These are the two existing data rates supported by the IS95c. Each RC also defines the spectral width (spreading rate SR1) and encoding type. There are currently five radio link configurations defined in cdma2000-1x for the forward link, and three for the return link.

Spreading Rate: Chip speed of the forward or reverse channel. IS95c uses SR1 (Spreading Rate 1): Same as “1XRTT.” The CDMA forward and reverse channel uses forward spread spectrum with a pseudo-random sequence at a chip speed of 1.2288 MHz.

RC2 configuration based on a speed of 14.4 kbit/s also supports speeds of 9.6, 4.8, 2.4 and 1.5 kbit/s for voice running in SR1 n=9 R=1/2.

The RC3 configuration, based on 9.6 kbps, also supports 4.8, 2.7, and 1.5 kbps for voice, while for data streams with channel code configurations are used - supporting speeds of 19.2, 38.4, 76.8, and 153.6 kbps /s and works in SR1 and uses channel coding with parameters n=9 R=1/2.

RC4 configuration for data transmission uses streams with a change in the channel code - supporting speeds of 9.6, 19.2, 38.4, 76.8, 153.6 and 307.2 kbit/s and operates in SR1 and uses turbo codes.

RC5 - used only for data transmission, streams with channeling code configurations are used - supporting speeds 14.4, 28.8, 57.6, 115.2 and 230.4 works in SR1 uses special. encoding and, thanks to a standardized range of speeds, is the most preferred configuration for data transmission.

The RC configuration also determines the operating mode of the radio transmission path, for example, the RC3 mode uses new method modulation, see Fig. 1, and RC1 mode is fully compatible with CCC IS95a, see. pic 1.

In this section we will consider the system in RC1 mode.

Codes used in the IS-95c system.

SSMS uses three types of codes: short and long m-sequences and Walsh codes.

Short PSP

The short PSP consists of two pseudo-random scrambling sequences PSP - I and PSP - Q (the symbols I and Q correspond to the physical purpose and indicate the in-phase and quadrature components in the modulator). The period of each of the named PSP contains 215 chips, the repetition rate of which, according to the standard, is 1.2288 Mchip / s. Direct calculation shows that exactly 75 periods of short PSP fit into one two-second segment. Structurally short PSPs are M - sequences of length

N=2-1 with characteristic polynomials

f i = x 15 + x 13 + x 9 + x 8 + x 7 + x 5 +1 and

f Q = X 15 + X 12 + X 11 + X 10 + X 6 + X 5 + X 4 + X 3 +1,

extended by adding a zero symbol to a chain of 14 consecutive zeros in each period.

Long PSP

Long PSP symbols have a repetition rate of 1.2288 Mchip/s. The formation of a long PSP is carried out using a polynomial

f( x) = x 42 + x 35 + x 33 + x 31 + x 27 + x 26 + x 25 + x 22 + x 21 + x 19 + + X 18 + X 17 + X 16 + X 10 + X 7 + X 6 + X 5 + X 3 + X 2 + X + 1.

Walsh codes

The Walsh codes used in the system are designated as: W n N , where N is the code length, n is the row in the Walsh-Hadamard matrix. This matrix is ​​constructed by an iterative algorithm (see Fig. 2). At each iteration, any codeword obtained at the previous step is converted into two new ones by doubling the length by repeating twice in one word and repeating with a change in sign in the other. So if C k , a certain word obtained at the k-th step, its “descendants” at the k+1-th step will be words of the form (C k ,C k),(C k ,-C k), thus starting from the trivial words of length 1 equal to 1, in k iterations you can obtain 2 k code vectors of length N=2 k whose orthogonality is obvious (see Fig. 2.).

Using this method, you can create a Walsh code whose dimension is equal to 2 k X 2 k(k- positive integer). The Walsh code set is characterized by a 64 x 64 (RC1) or 128 x 128 (RC3) matrix, where each row corresponds to a separate code. Since the elements of the Walsh code set are mutually orthogonal, their use makes it possible to divide the forward communication channel into 64 (RC1) or 128 (RC3) orthogonal signals.

Direct Channel Addressing

Channel addressing.

The cdma2000-1x Forward Channel, while maintaining IS95a compatibility, uses the same structure for the forward channel pilot (F-Pilot), synchronization channel (F-Sync) and signaling (F-Paging).

Also in CDMA2000-1x, each user is assigned their own direct traffic channel (F-Traffic), which may include:

Eight additional channels (F-SCCHs) for RC1 and RC2;

Three additional channels (F-SCHs) for RC3 to RC9;

Two dedicated control channels (F-DCCHs);

F-FCHs are used for voice transmission, F-SCCHs, and F-SCHs are used for data transmission. The base transceiver station may also send the zero or first F-DCCHs. The F-DCCH is associated with traffic channels (either FCH and SCH, or SCCH) and may contain signaling data and transmit power control data.

In this manual, we will take a closer look at the main channels:

pilot channel (f-pilot channel);

synchronization channel (f-synchronization channel);

personal call channel (f-paging channel);

direct traffic channel (forward traffic channel).

In RC1 mode, the mapping of logical channels to physical channels in the forward direction is carried out using a system of orthogonal Walsh functions of length 64: w i , i= 0,1,..., 63, where i is the number of the Walsh function. The CDMA-2000 standard provides for the organization of one pilot channel, one synchronization channel, from one to seven call channels (depending on the subscriber load on the BS) and from 55 to 62 direct traffic channels, since some call channels can be used as traffic channels. The correspondence between logical and physical channels is shown in Fig. 4.

In RC3 mode, the mapping of logical channels to physical channels is carried out in the same way as in RC1, with the only difference being that, thanks to the use of quadrature phase modulation, the number of Walsh codes used is increased from 64 to 128 - accordingly, the number of possible addressable channels is doubled compared to the RC1 mode.

1. Pilot channel

According to Fig. Pilot channel 5 is assigned a zero Walsh function w0 , i.e. a sequence of only zeros.

2. Channel synchronization

After the block interleaver, the data stream is directly spread spectrum by adding modulo 2 with the Walsh function assigned to the synchronization channel w 32.

3. Channelpersonal call

After scrambling the decimated long PSP of period 2 42-1, the data stream is subjected to spectrum expansion in the same way as was done for the channels already considered: it is summed modulo two with the Walsh function assigned to the channel from the set W 1 -W 7 . This is followed by combination with the remaining channels (inputs P 1 - P 7 in Fig. 2), and then (in the modulator) multiplication with a complex short bandwidth and transfer to the carrier.

4. Direct traffic channel

One of the Walsh sequences w 8 + w 31 and w 33 + w 63 with a chip speed of 1.2288 Mchip/s, with the Walsh sequence number uniquely identifying the forward traffic channel number.

Addressing base stations.

A pair of PSP - I and PSP - Q or, equivalently, a complex PSP. This complex short bandwidth is the same for all 64 CDMA channels and is used by all BSs of the system, but with different cyclic shifts. The difference in cyclic shifts allows the MS to separate signals emitted by BSs of different cells or sectors, i.e., it allows you to identify the number of the BS or sector. For different BSs, the shift changes with a constant step equal to 64 chip x PILOT_INC, where the system parameter PILOT_INC takes values ​​from 1 to 4. Thus, with a minimum step, 2 15 /2 6 =2 9 =512 short bandwidth shifts are available, i.e., conflict-free existence of a network consisting of 512 BS is possible. If it is necessary for the network to consist of a larger number of BSs, then with territorial planning of the network it can be easily achieved that BSs with the same cyclic shifts of short PSPs cannot simultaneously be in the radio visibility zone of the MS.

On the other hand, the PRP shift step uniquely determines the cell (or sector) size at which the MS can reliably distinguish PRPs having a minimum time shift. It is easy to see that with a minimum shift of 64 chips, the cell radius will be about 15.5 km.

Back channel addressing

In the reverse channel (uplinks)

Access channel (access channel);

Reverse traffic channel traffic channel).

The asynchrony of code division makes it irrational to use Walsh functions as channel-forming sequences (signatures) of physical channels, since with relative time shifts they cannot maintain orthogonality and have very unattractive cross-correlation properties. Therefore, various cyclic shifts of the long PSP of period 2 42 -1 are responsible for the separation of channels in the uplink. Walsh functions in the reverse channel are also used, but in a different capacity: to organize another stage of noise-resistant coding of data transmitted by the MS.

The general structure of the reverse communication channel of the IS-95c system is illustrated in Fig. 6. The access and return traffic channels that are used by the MS are associated with certain paging channels. As a result, one personal call channel can have up to n = 32 access channels and up to t = 64 return traffic channels.

Rice. 6. Structure of the reverse channel SSMS standard IS-95c

1. Channel access

Channel access provides connection between the MS and the BS until the MS has tuned in to the reverse traffic channel assigned to it. The access channel selection process is random - the MS randomly selects a channel number from the range O...ACC_CHAN, where ACC_CHAN is a parameter transmitted by the BS in the access parameters message. An orthogonal modulator maps (encodes) groups of 6 binary symbols into a Walsh function of length 64. This operation is the encoding of 6-bit blocks (64,6) with an orthogonal code. With optimal (“soft”) decoding, the energy gain from using such a code asymptotically tends to 4.8 dB (45]. At the same time, in many sources the procedure under consideration is called orthogonal modulation or Walsh modulation. The 6 symbol group is replaced by the Walsh function according to the following rule: the decimal value of a 6-bit binary number corresponding to a group of 6 bits uniquely determines the number of the Walsh function. For example, if a group of 6 symbols of the form (010110) is supplied to the input of an orthogonal modulator, then it corresponds to the decimal value 22, which means this group is replaced by the modulator with the Walsh function w 22, consisting of 64 characters. As a result of orthogonal modulation, the data rate increases to

The stream of orthogonally modulated data is subjected to direct spectrum spreading using a long PSP with a certain cyclic shift that uniquely identifies a given MS, which makes it possible to identify it at the BS, and therefore implement code separation of subscribers. The cyclic shift of a long memory bandwidth is determined by a generator mask 42 bits long, which is constructed from the BS identifier, call and access channel numbers. After spectrum expansion (modulo 2 summation with a long memory bandwidth and converting Boolean symbols to bipolar ones), the flow follows at the speed of the chips, i.e. e. 1.2288 Mchip/s, enters the quadrature channels of the phase modulator, where it is scrambled by two short PSPs (PSP-I and PSP-Q) of period 2 15. All MSs in a given cell use the same short PRP offset. Since the return channel uses offset quadrature PSK (OQPSK), a delay element is introduced in the Q arm of the modulator for half the duration of the chip. The use of OQPSK reduces the depth of unwanted dips in the signal envelope, and therefore reduces the required linear dynamic range of the MS transmitter power amplifier.

Engineers selected signals, the use of which should improve the basic characteristics of systems (communication quality, noise immunity), relying only on their intuition. The turning point was the creation of the theory of signal formation, processing and transmission. It allows you to determine the effectiveness of using a specific ensemble (set) of signals, based only on knowledge of their auto- and intercorrelation characteristics.

Basic Concepts

The code sequences used in CDMA systems for signal transmission consist of N elementary symbols (chips). Each information symbol of the signal is added to one N-symbol sequence, which is called a “spreading sequence”, since the “resulting” signal is emitted into the air with a deliberately spread spectrum. The gain in communication quality depends both on the number of symbols (length) of the sequence and on the characteristics of the set of signals, primarily their cross-correlation properties and modulation method.

Sequence length. In the domestic literature, signals whose base is significantly greater than unity (B=TF>>1, where T is the duration of the signal element, F is the frequency band) are usually called complex. In relation to the original (information) signal, a complex signal is noise with almost the same spectral power density.

It is known that the more the spectrum of a signal is “stretched” on the air, the lower its spectral density. Thanks to this property, signals with a large base can be used in a “foreign” (already occupied) frequency band “on a secondary basis”, having an arbitrarily small impact on the system operating there.

Characteristics. The entire set of code sequences used in CDMA is divided into two main classes: orthogonal (quasi-orthogonal) and pseudo-random sequences (PSR) with a low level of cross-correlation (Fig. 1).

In an optimal CDMA receiver, the incoming signals, which are essentially additive white Gaussian noise, are always processed using correlation methods. Therefore, the search procedure is reduced to finding a signal that is maximally correlated with the individual subscriber code. The correlation between two sequences (x(t)) and (y(t)) is done by multiplying one sequence with a time-shifted copy of the other. Depending on the type of sequence, CDMA systems use different correlation methods:

• autocorrelation, if the multiplied pseudo-random sequences have the same form, but are shifted in time;
• mutual, if PSPs have different types;
• periodic if the shift between two PSPs is cyclic;
• aperiodic if the shift is not cyclic;
• on part of the period if the result of the multiplication includes only segments of two sequences of a certain length.

In order to obtain a gain in communication quality when using any of the correlation processing methods, it is necessary that the ensemble of signals have “good” autocorrelation properties. It is desirable that the signals have a single autocorrelation peak, otherwise false synchronization along the side lobe of the autocorrelation function (ACF) is possible. Note that the wider the spectrum of emitted signals, the narrower the central peak (main lobe) of the ACF.

Pairs of code sequences are selected so that mutual correlation function(VKF) had a minimum value for their pairwise correlation. This guarantees a minimum level of mutual interference.

Consequently, the choice of an optimal ensemble of signals in CDMA comes down to searching for a structure of code sequences in which the central peak of the ACF has the highest level, and the side lobes of the ACF and the maximum spikes of the ACF are as minimal as possible.

Orthogonal codes

Depending on the method of formation and statistical properties, orthogonal code sequences are divided into actually orthogonal and quasi-orthogonal. A distinctive feature of the sequence is the cross-correlation coefficient pij, which generally varies from -1 to +1.

In signal theory it has been proven that the maximum achievable value of the cross-correlation coefficient is determined from the condition

The minimum value of the VCF provides codes in which the correlation coefficients for any pairs of sequences are negative ( transorthogonal codes). Cross correlation coefficient orthogonal sequences, by definition, is equal to zero, i.e. O? ij =0. For large values ​​of N, the difference between the correlation coefficients of orthogonal and transorthogonal codes can practically be neglected.

There are several ways to generate orthogonal codes. The most common is using Walsh sequences of length 2n, which are formed based on the rows of the Hadamard matrix

Repeating the procedure multiple times allows you to form a matrix of any size, which is characterized by the mutual orthogonality of all rows and columns.

This method of generating signals is implemented in the IS-95 standard, where the length of the Walsh sequences is chosen to be 64. Note that the difference between the rows of the Hadamard matrix and the Walsh sequences is only that the latter use unipole signals of the form (1,0).

Using the Hadamard matrix as an example, it is easy to illustrate the principle of constructing transorthogonal codes. Thus, one can verify that if the first column consisting of only ones is deleted from the matrix, then orthogonal Walsh codes are transformed into transorthogonal ones, in which for any two sequences the number of symbol mismatches exceeds the number of matches by exactly one, i.e. O? ij = -1/(N-1).

Another important type of orthogonal codes is biorthogonal a code that is formed from an orthogonal code and its inversion. The main advantage of biorthogonal codes compared to orthogonal ones is the ability to transmit a signal in half the frequency band. For example, the biorthogonal block code (32,6) used in WCDMA allows the transmission of a TFI transport format signal.

Note that orthogonal codes have two fundamental disadvantages.

1. The maximum number of possible codes is limited by their length (in the IS-95 standard the number of codes is 64), and accordingly, they have a limited address space.

To expand the ensemble of signals, along with orthogonal ones, quasi-orthogonal sequences. Thus, the draft cdma2000 standard proposes a method for generating quasi-orthogonal codes by multiplying Walsh sequences by a special masking function. This method allows one to obtain a set of quasi-orthogonal sequences Quasi-Orthogonal Function Set (QOFS) using one such function. Using m masking functions and an ensemble of Walsh codes of length 2n, one can create (m+1) 2n QOF sequences.

2. Another disadvantage of orthogonal codes (and those used in the IS-95 standard are no exception) is that the cross-correlation function is equal to zero only “at a point,” i.e. in the absence of a time shift between codes. Therefore, such signals are used only in synchronous systems and mainly in direct channels (from the base station to the subscriber).

The ability to adapt the CDMA system to different transmission rates is achieved through the use of special orthogonal sequences with a variable spreading factor (OVSF, Orthogonal Variable Spreading Factor), called variable length codes. When transmitting a CDMA signal that was created using such a sequence, the chip speed remains constant, but the information speed changes by a factor of two. The 3rd generation standards propose to use orthogonal Gold codes with multiple transmission rates (multirate) as OVSF codes. The principle of their formation is quite simple; Fig. explains it. 3, which shows a code tree that allows you to build codes of different lengths.

Each level of the code tree determines the length of the codewords (spreading factor, SF), with each subsequent level doubling the possible number of codes. So, if at level 2 only two codes can be generated (SF=2), then at level 3 four code words (SF=4) are generated, etc. The complete code tree contains eight levels, which corresponds to SF=256 (only the lower three levels are shown in the figure).

Thus, the ensemble of OVSF codes is not fixed: it depends on the spreading factor SF, i.e. in fact, it depends on the channel speed.

It should be noted that not all code tree combinations can be simultaneously implemented in the same CDMA cell. The main condition for choosing a combination is the inadmissibility of violating their orthogonality.

Pseudo-random sequences

Along with orthogonal codes, a key role in CDMA systems is played by PRP, which, although generated in a deterministic manner, have all the properties of random signals. However, they differ favorably from orthogonal sequences by being invariant to time shifts. There are several types of PSP with different characteristics. Simply put, today they appeared technical means, capable of “deriving” any ensemble of sequences with given properties.

m-sequences

One of the simplest and extremely effective means generating binary deterministic sequences - using a shift register (RS). The sequence at the output of an n-bit PC with feedback is always periodic, and its period n (the number of cycles after which the circuit returns to its original state) does not exceed 2n.

Theoretically, using an n-bit register and appropriately selected feedback logic, it is possible to obtain a sequence of any length N in the range from 1 to 2 n inclusive. The maximum length sequence, or m-sequence, will have a period of 2 n -1.

The m-sequence autocorrelation function is periodic and two-valued:

The level of side maxima of the autocorrelation function (Fig. 4) does not exceed the value

Codes Golda are formed by character-by-character addition modulo 2 of two m-sequences (Fig. 5). The WCDMA draft specifies three types of Gold codes: primary and secondary orthogonal Gold codes (both 256 bits long) and long code.

Orthogonal Gold codes are created based on a 255-bit m-sequence with the addition of one redundant symbol. The primary synchronization code has an aperiodic autocorrelation function and is used for initial synchronization. The secondary sync code is an unmodulated orthogonal Gold code that is transmitted in parallel with the primary sync code. Each secondary sync code is selected from 17 different Gold codes (C1,...,C17).

The long code for the forward channel is 40,960 chip long Gold code fragments. The WCDMA-based communication system is asynchronous, and neighboring base stations use different Gold codes (512 in total), repeated every 10 ms. Asynchronous principle operation of base stations makes them independent of external synchronization sources. It is intended to use a long code in the reverse channel, but only in those cells where the multi-user detection mode is not enabled.

Code family Kasami contains 2 k sequences with a period of 2 n-1. They are considered optimal in the sense that for any “preferred” pair it is ensured maximum value autocorrelation function equal to (1+2 k).

Kasami code sequences are implemented using three shift registers connected in series (u, v and w) with different feedback(Fig. 6), each of which forms its own m-sequence. To obtain Kasami code sequences with given properties, the sequences v and w must have different shifts.

Kasami codes with a length of 256 bits are used as short sequences in the return channel (WCDMA project) in those cells that use multi-user detection.

Barker sequences

Pseudorandom sequences with a small aperiodic ACF value are capable of ensuring synchronization of transmitted and received signals in a fairly short period of time, usually equal to the length of the sequence itself. The Barker sequences are the most famous (see table).

The efficiency of sequences with aperiodic ACF is usually assessed by the quality indicator F, which is defined as the ratio of the squares of the in-phase components of the signal to the sum of the squares of its out-of-phase components. Thus, a measure of the effectiveness of aperiodic correlation of a binary sequence is the quality indicator.

Paul Feyerabend (b. 1924).

Thomas Kuhn (b. 1922).

Imre Lakatos (1921–1974).

The Walsh functions are a natural extension of the Rademacher function system, obtained by Walsh in 1923 and representing a complete system of orthonormal rectangular functions.

The set of Walsh functions, ordered by frequency, is usually denoted as follows:

Walsh functions, ordered by frequency, similar to trigonometric functions, can be divided into even cal(i,t) and odd sal(i,t)

Figure 17.1 shows the first eight wal functions w(i,t).

Figure 17.1

It is clear that the frequency of each subsequent Walsh function is greater than or equal to the frequency of the previous Walsh function and has one more zero crossing in the open interval tÎ. This is where the name “frequency ordering” comes from.

Discretizing the Walsh functions shown in Figure 17.1a at eight equally spaced points results in the (8x8) matrix shown in Figure 17.1b. This matrix is ​​denoted by H w(n) where n=log 2 N and the matrix will have size NxN.

Walsh functions, when ordered by frequency, in the general case can be obtained from the Rademacher functions r k (x) using the formula:

where w is the number of the Walsh function; k – Rademacher function number; exponent of the Rademacher function, which takes the value 0 or 1 as a result of summation modulo two, i.e. according to the rule: 1Å1=0Å0=0; 1Å0=0Å1=1 bits of a binary number w. For example, for the sixth Walsh function ( w=6), included in a system of size N=2 3 =8, the product (17.4) consists of three factors of the form: for k=1 for k=2 for k=3 . A number in the binary system is written as a combination of zeros and ones. In our case, the value w and its categories are shown in table 17.1

Table 17.1

w 0 – the most significant digit of the number, w 3 – least significant digit of the number w.

The exponents of the Rademacher functions are equal to: ; ; and therefore,

wal(6,x)=r 1 1 (x)×r 2 0 (x)×r 3 1 (x)=r 1 (x)r 3 (x)

The rule for obtaining exponents for the Rademacher function is shown schematically in Table 17.1, where the arrows indicate the summable digits of the number w and Rademacher functions, to which the resulting exponent relates. From Figure 17.1 it can be seen that even numbers of Walsh functions refer to even functions, and odd numbers refer to odd functions. Another way of ordering is Paley ordering. When ordered by Paley, the analytical notation of the Walsh function has the form:

p 1 is the least significant digit of a binary number, p n is the most significant digit of a binary number. When ordering according to Paley, to form Walsh functions, it is necessary to take the product of Rademacher functions raised to a power, the numbers of which coincide with the numbers of the corresponding digits of the double representation of the number p, and the exponent of each function is equal to the contents of the corresponding digit, i.e. 0 or 1. Moreover, the least significant Rademacher function corresponds to the least significant digit of the binary combination of the number p. In accordance with this rule, table 17.2 shows the values ​​of Paley-ordered Walsh functions.

Table 17.2

The Rademacher functions in the table are shown in the form: . A comparison of the products and powers of the Rademacher functions written in tables 17.1 and 17.2 shows that there is a correspondence between the Walsh functions ordered by Paley and Walsh, which is reflected in the last column of table 17.2. In accordance with the Paley-ordered Walsh functions, a matrix of samples H p (n) can also be constructed, similar to that shown in Figure 17.1b.