Wireless Communication

Chapter: Analog and Digital Transmission
Section: Multi-Carrier Modulation, OFDM, Effect of Doppler

Effect of Doppler on BER for MC-CDMA

Contributed by Jean-Paul Linnartz

This section addresses the local-mean BER B1. For MC-CDMA (but not for OFDM), the BER B0 for one specific user signal converges to the local-mean BER if the number of subcarriers is sufficiently large and the transmit bandwidth largely exceeds the coherence bandwidth. In section 6, simulations are used to verify the accuracy this approximation and to investigate the behaviour for systems with fewer subcarriers. The decision variable for user bit 0, after combining all subcarrier signals consists of

(21)

where x0 is the wanted signal, xMUI is the multi-user interference (due to imperfect restoration of the subcarrier amplitudes), xICI is the intercarrier interference (due to crosstalk b m,n between an and ym), and xnoise is the noise.

This not only has a direct contribution resulting from all subcarrier components, but also indirect components which leaked into other subcarriers but still contribute to x because of the non-zero autocorrelation of the Walsh-Hadamard code during despreading. Nonetheless, the expected value is

(23)

The variance of x0 vanishes for large N, i.e., the system sees a non-fading channel.

 The value of xMUI highly depends on the choice of C and on the channel. For orthogonal spreading codes (S n c0[n]ck[n] = 0) and a non-dispersive channel (b n,n is constant with subcarrier frequency n), xMUI can be made zero by taking wn,m = d n,m. For a dispersive channel, the orthogonality of spreading codes is eroded but the MUI level can remain low if the weight factors wn,m are appropriately chosen, as shown in a previous section. In such case, the variance of the MUI can be evaluated by observing that for any two orthogonal codes cj[n] and ck[n] with j ¹ k, one can partition the set of subcarrier indixes n with n = 0, 1, N-1 into two sets, both with exactly N/2 elements, such that A- = {n: cj[n]ck[n] = -1/N} and A+ = {n: cj[n]ck[n] = +1/N} [24]. Here A+ È A- = A ensures that S A+ È A- cj[n]ck[n] = 0. Hence

(25)

Because of independence of user symbols and channel properties, and mutual independence of user signals,

(26)

If we may assume that fading of the subcarriers is independent, we can write

(27)

and since A+ Ç A- = Æ ,

(28)

Thus,

(29)

The square of the triple sum simplifies because of Ebk bj*= d kj and Ech b i,j b k,l = d ijd kl and similar properties. We take, for reasons soon to be discussed,

E[c0[n] ck[n - m]]2 = N-2. So,

(33)

Thus

s ICI2 = S D ¹ 0pD M02 Ts2. (34)

Here the question arises whether a system designer can chose the spreading matrix C such that the ICI is mitigated. This poses requirements on the cross-correlation S c0[n]ck[n-m]. It is the time-frequency dual of the well known problem of finding good codes for asynchronous DS-CDMA with good cross and autocorrelation properties to combat delay spread. Walsh-Hadamard codes have no particular properties to achieve good autocorrelation properties, and their auto-correlation behaviour can be approximated by the behavior of randomly chosen codes. Here the situation here more involved, because each term c0[n]ck[n-m] is multiplied by b m,n and wn,n, which are complex valued with random arguments. Thus, even if the code had good auto-correlation properties, these erode the corresponding attenuation of ICI.

s noise2 = N M02 N0Ts. (35)

Since we consider ensembles of many different channels, xMUI, xICI and xnoise are zero-mean complex Gaussian. So, the local-mean BER becomes, with

 

(36)

We introduce the figure of merit z , and rewrite (36) as EN/N0 = z P0 Ts/N0. Thus z is a system parameter, which gives the improvement of MC-CDMA in a Rayleigh fading channel, over narrowband transmission in a non-fading channel. For very poor local-mean signal to noise ratios (large P0 Ts/N0), the noise largely dominates over the MUI, and the MC-CDMA MMSE receiver acts mainly as a maximum ratio combiner Since wn,n @ b n,n/ N0,

M11 ® E b n,n2/N0 = P0 / N0

M02 ® E b n,n2/N02 = P0 / N02

M22 ® E b n,n4/N02 = 2P02 / N02

and z tends to unity (0 dB).

 

 



Wireless Communication © Jean-Paul M.G. Linnartz, 2001.