5 **5.2. Unified complex Hadamard transform sequences**

6 The considered DS-CDMA downlink wireless communication system uses the orthogonal 7 unified complex Hadamard transform spreading sequences. These so-called orthogonal 8 unified complex Hadamard transform sequences are easy to generate. Larger sets of 9 complex sequences are also available. They are categorized into two groups: the half-10 spectrum property orthogonal unified complex Hadamard transform spreading sequences 11 and the non-half-spectrum property orthogonal unified complex Hadamard transform 12 spreading sequences. Consider briefly how these sequences can be generated and note the 13 main definitions and remarks discussed in [96, 97].

A unified complex Hadamard transform matrix **A** of order *<sup>n</sup>* 14 *N* 2 is a square matrix with 15 elements 1 and *j* , and is constructed by [96,97]

$$A\_n = \bigotimes\_{i=1}^n A\_1 = A\_1 \otimes A\_{n-1} = \underbrace{A\_1 \otimes \cdots \otimes A\_1}\_{n \text{ times}} \tag{196}$$

where denotes the Kronecker product, and *A*<sup>1</sup> 17 is defined as

$$A\_1 = \begin{bmatrix} a\_1 & a\_1 a\_3 \\ a\_2 & -a\_2 a\_3 \end{bmatrix} \tag{197}$$

19 with

54 BookTitle

3 the Rake receiver?

132 Contemporary Issues in Wireless Communications

10 autocorrelation properties.

1 What are benefits under GR implementation in DS-CDMA downlink wireless 2 communication system in comparison with other conventional receivers, for example,

4 To give answers on these questions we carry out our analysis using, for instance, the 5 orthogonal 4-phase complex sequences in the DS-CDMA downlink wireless communication 6 system. These sequences are generated by the unified complex Hadamard transform matrix 7 discussed in [96], the correlation properties of which are studied in [97], where it is shown 8 that the unified complex Hadamard transform sequences possess the better autocorrelation 9 properties in comparison with the WH sequences, which are characterrized by very poor

11 The use of orthogonal unified complex Hadamard transform sequences by the transmitter as 12 channelization spreading codes scrambled by long PN sequences and further processing 13 these sequences by the GR allows us to maintain the orthogonality between the users, and at 14 the same time, to reduce the effect of multipath fading and interference from other users. A 15 coherent GR [6], for example, can be used to combat the adverse effects of short-term 16 multipath fading in mobile radio propagation environments. Owing to computational 17 simplicity of the signal-to-interference-plus-noise ratio (SINR) in comparison with the 18 probability of error, SINR is mostly used for evaluating and selecting code sequences among 19 several candidates. Therefore, in this section, we investigate the SINR at the GR output 20 when the unified complex Hadamard transform spreading sequences are generated by 21 transmitter in the DS-CDMA downlink wireless communication system and compare this 22 with the SINR at the GR output under transmission of WH real sequences. It is shown that 23 the SINR at the GR output is independent of the phase offsets between different paths when 24 the unified complex Hadamard transform spreading sequences are generated by the 25 transmitter in the DS-CDMA downlink wireless communication system. The SINR at the GR 26 output of the same system is a function of the squared cosine of path phase offsets under 27 generation of WH real sequences by the transmitter. Because of this, as a direct result, the bit 28 error rate (BER) performance of GR employing by the DS-CDMA downlink wireless 29 communication system when the unified complex Hadamard transform spreading 30 sequences are generated by the transmitter is better that that of the system with the WH 31 sequences under Gaussian approximation. Also, we carry out a BER performance 32 comparison of the DS-CDMA system employing the GR with the same system using then 33 conventional receiver, for example, the Rake receiver [95]. Comparative analysis shows us a 34 great superiority in the BER performance under GR employment in the DS-CDMA

35 downlink wireless communication system over the use of the Rake receiver.

36 This section is organized as follows. At first, we present some basic definitions of the unified 37 complex Hadamard transform sequences. Additionally, we study the DS-CDMA downlink 38 wireless communication system model under the GR employment when the unified 39 complex Hadamard transform spreading sequences are generated and propagated in a 40 multipath fading channel. Further, we investigate the SINR performance at the GR output

$$a\_1, a\_2, a\_3 \in \{1, -1, j, -j\} \quad \text{and} \quad j = \sqrt{-1}. \tag{198}$$

There are 64 different matrices for *A*<sup>1</sup> 21 satisfying (197) with elements 1 and *j* ,

$$A\_{\!\!\!}A\_{\!\!\!\! }^\* = A\_{\!\!\!\! }^\* A\_{\!\!\! } = 2I\_2 \tag{199}$$

23 and

$$\left| \det \begin{pmatrix} A\_1 \\ \end{pmatrix} \right|^2 = 2^2,\tag{200}$$

1 where indicates the complex conjugate. Hence, the unified complex Hadamard transform 2 matrix is orthogonal. Furthermore, the unified complex Hadamard transform matrices 3 contain a WH transform matrix as a special case, with

$$a\_1 = a\_2 = a\_3 = 1 \tag{201}$$

in the matrix *A*<sup>1</sup> 5 .

56 BookTitle

6 The unified complex Hadamard transform matrices have two categories of 32 basic matrices, depending on whether <sup>3</sup> *a* in (197) is imaginary or not [97]. If <sup>3</sup> 7 *a* is imaginary, the matrix 8 group is called the half-spectrum property unified complex Hadamard transform. 9 Otherwise, the group is called the non-half-spectrum property unified complex Hadamard 10 transform. The unified complex Hadamard transform spreading sequence *ck* , *k* 1,, *N* is 11 defined by the *k*-th row of the unified complex Hadamard transform matrix. It has been 12 shown in [97] that the non-half-spectrum property unified complex Hadamard transform 13 sequences have very similarly poor autocorrelation properties as WH sequences, and some 14 of the half-spectrum property unified complex Hadamard transform sequences exhibit a 15 reasonable compromise between the autocorrelation and cross-correlation functions. In this 16 section, we just consider the half-spectrum property unified complex Hadamard transform sequences, i.e., *a j* or *j* <sup>3</sup> 17 .
