An efficient receiver scheme for downlink ZP-CDMA

Abstract

A signal transmitted through a wireless channel may be severely distorted by intersymbol interference (ISI) and multiple access interference (MAI). In this paper, we propose an efficient CDMA receiver based on frequency domain equalization (FDE) with a regularized zero forcing (RZF) equalizer and parallel interference cancellation with a unit clipper decision function (CPIC) to combat both the ISI and the MAI. We call this receiver the FDE-RZF-CPIC receiver. This receiver is suitable for downlink zero padding CDMA cellular systems. The effects of the decision function, the channel estimation, the number of cancelled users, and the user loading on the performance of the proposed receiver are discussed in the paper. The bit error rate (BER) of the data received by the proposed receiver is evaluated by computer simulations. The experimental results show that the proposed receiver provides a good performance, even with a large number of interfering users. At a BER of 10⁻³, the performance gain of the proposed receiver is about 2 dB over the RAKE receiver with a clipper decision function and PIC in the half-loaded case (eight users) and is much larger in the full-loaded case (16 users).

Appendix: Toeplitz to circulant approximation

Let Q be an S × S Toeplitz matrix of the following form:

$${\mathbf{Q}} = \left[ {\begin{array}{*{20}c} {q\left( 0 \right)} & \cdots & {q\left( { - l} \right)} & {} & {\mathbf{0}} \\ \vdots & \ddots & {} & \ddots & {} \\ {q\left( k \right)} & {} & \ddots & {} & {q\left( { - l} \right)} \\ {} & \ddots & {} & \ddots & \vdots \\ {\mathbf{0}} & {} & {q\left( k \right)} & \cdots & {q\left( 0 \right)} \\ \end{array} } \right]$$

(43)

It can be approximated by an S × S circulant matrix Q ^c defined as [17, 18]:

$${\mathbf{Q}}^{\mathbf{c}} = \left[ {\begin{array}{*{20}c} {q\left( 0 \right)} & \cdots & \cdots & {q\left( { - l} \right)} & 0 & \cdots & {q\left( k \right)} & \cdots & {q\left( 1 \right)} \\ \vdots & \ddots & {} & {} & \ddots & \ddots & \cdots & \ddots & {} \\ \vdots & {} & \ddots & {} & {} & \ddots & \ddots & \cdots & {q\left( k \right)} \\ {q\left( k \right)} & {} & {} & \ddots & {} & {} & \ddots & \ddots & \cdots \\ 0 & \ddots & {} & {} & \ddots & {} & {} & \ddots & 0 \\ \vdots & \ddots & \ddots & {} & {} & \ddots & {} & {} & {q\left( { - l} \right)} \\ {q\left( { - l} \right)} & \vdots & \ddots & \ddots & {} & {} & \ddots & {} & \vdots \\ \vdots & \ddots & \vdots & \ddots & \ddots & {} & {} & \ddots & \vdots \\ {q\left( { - 1} \right)} & \cdots & {q\left( { - l} \right)} & \vdots & 0 & {q\left( k \right)} & \cdots & \cdots & {q\left( 0 \right)} \\ \end{array} } \right]$$

(44)

where each row is a circular shift of the row above and the first row is a circular shift of the last row. The primary difference between the matrices Q and Q ^c is that they differ only by elements added in the upper right and lower left parts to produce the cyclic structure in the rows. If S is large and the number of nonzero elements on the main diagonals of the matrix compared to the number of zero elements is small (i.e., the matrix is sparse), the number of elements added to the upper right and lower left corners does not affect the matrix. It can be shown from the eigen value distribution of both matrices that Q and Q ^c are asymptotically equivalent.

It is known that an S × S circulant matrix Q ^c can be diagonalized as follows [1]:

$${\mathbf{\Lambda }} = {\mathbf{\Psi }}^{ - {\mathbf{1}}} {\mathbf{Q}}^{\mathbf{c}} {\mathbf{\Psi }}$$

(45)

where Λ is an S × S diagonal matrix whose elements λ(s, s) are the eigen values of Q ^c and Ψ is an S × S unitary matrix of the eigen vectors of Q ^c. Thus, we have:

$${\mathbf{\Psi \Psi }}^{ * {\mathbf{t}}} = {\mathbf{\Psi }}^{ * {\mathbf{t}}} {\mathbf{\Psi }} = {\mathbf{I}}$$

(46)

The elements ψ(s ₁, s ₂) of Ψ are given by [17, 18]:

$$\psi \left( {s_1 ,s_2 } \right) = \exp \left[ {\frac{{j2\pi s_1 s_2 }}{S}} \right]$$

(47)

for s ₁, s ₂ = 0,1,........., S − 1 and j ² = −1.

The eigen values λ(s, s) can be called λ(s). For these eigen values, the following relation holds [17, 18]:

$$\lambda \left( s \right) = q\left( 0 \right) + \sum\limits_{m = 1}^k {q\left( m \right)\exp \left[ {\frac{{ - j2\pi ms}}{S}} \right]} + \sum\limits_{m = - l}^{ - 1} {q\left( m \right)\exp \left[ {\frac{{ - j2\pi ms}}{S}} \right]} $$

(48)

s = 0,1,........., S − 1.

Because of the cyclic nature of Q ^c, we define:

$$q\left( {S - m} \right) = q\left( { - m} \right)$$

(49)

and thus Eq. 48 can be written in the form [17, 18]:

$$\lambda \left( s \right) = \sum\limits_{m = 0}^{S - 1} {q\left( m \right)\exp \left[ {\frac{{ - j2\pi ms}}{S}} \right]} $$

(50)

for s = 0,1,........., S − 1.

Thus, the circulant matrix can be simply diagonalized by computing the discrete Fourier transform of the cyclic sequence q(0), q(1),........., q(S − 1).