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

Model for OFDM and MC-CDMA with Doppler

Practical implementations of an OFDM transmission system use Fast Fourier Transforms (FFTs) to create and decompose user data signals on multiple parallel subcarriers [5]. However, in our evaluation we adhere to a continuous-time representation, but we see no significant shortcoming in such analysis. We consider a transmit signal s(t) of the multi-carrier form

(1)

where w _c is the carrier frequency, w _s is the subcarrier spacing, n is the subcarrier number, N the number of subcarriers, and a_n is the modulation of the n-th subcarrier carrying the user data. As illustrated in Figure 1, we use the following vector notation: For OFDM, vector A of length N carries a 'frame' of user data, with A = [a₀, a₁, ... a_N-1]^T, where the elements a_n are user symbols. In MC-CDMA, A = CB, where C is an N by N code matrix and B = [b₀, b₁, ... b_N-1]^T represents a frame of user data. We will refer to B as N user signals, without explicitly identifying whether or not all symbols come from the same end-user. The k-th column of C represents the 'spreading code' of user data stream k, and will be denoted as (c_k[0], .. c_k[N - 1])^T. A commonly used special case [12-14], which we will also consider here, is C = N^-1/2 WH_N where WH_N is the Walsh-Hadamard matrix of size N by N. In that case, C = C^-1 = C^H, so C C = I_N with I_N the N by N unit matrix. In another special case, namely that of C = I_N, the MC-CDMA system reduces to OFDM. For ease of analysis, we normalize the modulation as E b_ib_j* = d _ij, or equivalently EBB^H = I_N. Then E(AA^H) = EC(BB^H)C^H = CC^H = I^H. This implies that (in contrast to many studies for DS-CDMA) we address a "fully loaded" system, with N - 1 interfering signals.

Figure 1 illustrates such a transmitter. Frames are created by a serial-to-parallel (S/P) conversion of an incoming stream of data, applying the code spreading, an I-FFT and a parallel-to-serial conversion with prefix insertion. We will address the transmission of a single frame, and assume that interframe interference is avoided by choosing appropriate guard intervals. Hence, the elements of vectors A and B are constant with time. The frame duration, excluding any guard interval is T_s, where w _sT_s = 2p .

The Wide Sense Stationary Uncorrelated Scattering (WSSUS) multipath channel is modeled as a collection of I_w reflected waves. Each wave has its particular Doppler frequency offset w _i, path delay T_i and amplitude D_i, each of which is assumed to be constant. That is, we make the common assumption that the time-varying nature of the channel arises from the accumulation of multiple components, each of which has a linearly increasing phase offset, though with different slopes. The Doppler offset w _i = 2p f_i lies within the Doppler spread -2p fD £ w _{i £} 2p fD , with fD = vf_c/c the maximum Doppler shift. Here v is the velocity of the mobile antenna, c is the speed of light. The carrier frequency is 2p f_c = w _c. The received signal r(t) consists of the composition of all reflected waves, namely

(2)

here n(t) represents additive white Gaussian noise (AWGN). Detection of the signal at subcarrier m occurs by multiplication with the m-th subcarrier frequency, thus with exp{-jw _ct-jmw _st +j _m} during an appropriately chosen interval T_s. A phase compensation j _m = 0 is used.

Vector Y describes the outputs of the FFT at the receiver, with Y = [y₀, y₁, .., y_N-1]^T. Assuming rectangular pulses of duration T_s, we get, after some resequencing of terms in the exponent,

(3)

Here n_m represents the noise sampled at the m-th subcarrier. It can be shown to have variance N₀T_s, with N₀ the spectral power density of the AWGN. We denote the subcarrier offset as D = n - m, so

(4)

We rewrite the above expression as y_m = S _na_{nb m,n}T_s where b _m,n can be interpreted as the 'leakage' for a signal transmitted at subcarrier n and received at subcarrier m. Using sinc(x) = sin(p x)/p x and w _sT_s = 2p , we get

(5)

This result can be interpreted as sampling in frequency domain: the I_w multipath channel contributions appear weighed according to their individual Doppler offset w _i. It confirms that due to the Doppler shifts, the detected signal y_m contains contributions from all n subcarrier signals, not only from m = n. All b _m,n's with m ¹ n lead to ICI, with amplitudes weighed by sinc(D + w _i /w _s ).

Clarke [18] and Aulin [19] studied a uniform probability density of the angle q _i at which multipath waves arrive at the mobile [17], thus fq (q ) = 1/(2p ). The Doppler shift per wave equals f_i = (v/c)f_c cos(q _i), so

and (6)

For an omni-directional antenna, this leads to the U-shaped spectrum [17-19] of the Doppler spread, as follows:

(7)

Here, p_T is the local mean received power, per subcarrier. We combine this with the results from the signal model of the previous section. The variance of the ICI signal leaking from transmit subcarrier n into received subcarrier m = n + D equals

(8)

This PD can be expressed in terms of the ratio of the Doppler spread over the subcarrier spacing, defined as l = fD /f_s, D and p_T, namely after some polishing [10-11],

MATLAB (9)

Figure 2 plots the received power P₀, and the ICI powers P₁, P₂, and P₃ versus the normalized Doppler spread l for p_T = 1.

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