Derivation of the Initial Synchronization Proce- Proce-dure [1, 2]Proce-dure [1, 2]

There are three possible PA-Preamble series, as shown in Fig. 2.15. Because the PA-Pramble series are known, we utilize this knowledge to derive the initial DL synchronization algorithm.

Although there are three different PA-Preambles with different bandwidth, 5, 10, and 20 MHz, but the commonality is that all three PA-Preambles, whose length is all 216 points, locate in the middle part of the bandwidth. Therefore, when the MS receives the signal, it only need to observe a 5-MHz bandwidth because there is no PA-Preamble signal outside this bandwidth, whatever the system bandwidth. In other words, we can do downsampling for the 10-MHz and the 20-MHz signal to the 5MHz bandwidth without losing information on PA-Preamble.

The received PA-Preamble (including CP) can be represented as

y576 = Γ(δ) · T576· h576+ η576 (3.1)

where y576 = [y448, y449, ..., y511, y0, y1, ..., y511]⁰, the received PA-Preamble symbol, δ is the normalized carrier frequency offset (what the normalization is whit respect to subcarrier spacing), T₅₇₆ is the 576 × 576 Toeplitz matrix of the transmitted PA-Preamble symbol as

T₅₇₆ =

h₅₇₆ is the channel response vector, Γ(δ) is the 576 × 576 diagonal matrix summarizing the effect of the CFO as

Γ(δ) =

and η₅₇₆ is the additive white Gaussian noise (AWGN) vector.

3.2.1 Coarse Timing Synchronization

Fig. 3.1 depicts a model about the 576-points power sum with the window sliding. We know the information of TTG + RTG =165 µs in [5], so it is reasonable to suppose RTG is 45 µs, about 256 sampling periods, and CP factor is 1/8 in our study. We can also know the power of PA-Preamble is larger than the common data symbol because the amplitude of PA-Preamble is boosted before transmitting [5].

0 200 400 600 800 1000 1200

576 points moving power sum under AWGN channel with 0 dB

Power

timing index

Figure 3.2: 576 points power sum under AWGN in 0 dB [1].

Refer to Fig. 3.1. We consider summing the signal power in a 576-point window. With the window sliding, we can decide the coarse timing as the point with the maximum power sum.

This technique can actually be interpreted as quasi-maximum likelihood (ML) noncoherent detection of the preamble timing.

According to [1], Figs. 3.2 and 3.3 show the results of power sum with the window sliding in 0 dB of signal-to-noise ratio (SNR), under the AWGN channel and the SUI-5 channel with mobility 350 km/h. The rayleighchan, a Matlab function, leads to an initial delay of the generated channel, even if we set the delay of the direct path zero. Figs. 3.5 depict this phenomenon and we must compensate it in [1].

Note that the PA-Preamble timing we get by the above method has an offset to the real PA-Preamble timing due to multipath and noise effects. We will handle these problems in fine timing synchronization.

0 200 400 600 800 1000 1200 600

800 1000 1200 1400 1600 1800 2000

X: 581 Y: 1881

576 points moving power sum under SUI−5 channel with mobility 350km/h in 0 dB

Power

timing index

Figure 3.3: 576 points power sum under SUI-5 at mobility 350 km/h in 0 dB [1].

3.2.2 Estimation of Fractional Carrier Frequency Offset

Eq. (3.1) gives the received PA-Preamble signal. We attempt an ML estimation of δ from it. It turns out that a truly ML estimation is quite complex because T₅₇₆ is not circulant.

However, if the coarse timing lands us in the CP and if we sacrifice the available signal power in the CP, then we can obtain a reduced-complexity solution. Let y512 denote the received PA-Preamble symbol after removal of the CP. It is given by

y₅₁₂ = Γ(δ) · T_xn· h + η, (3.4)

where x_n = [x₀, x₁, ..., x₅₁₁]⁰ (the transmitted PA-Preamble symbol), T_xn is a 512 × 512 circulant matrix given by

Figure 3.4: Channel impulse response of PB channel [1].

h is the channel impulse response vector,

Γ(δ) =

and η is an AWGN vector. Due to possibly incorrect identification of the PA-Preamble starting time from the coarse timing synchronization, there may be a circular shift of the elements in the h vector from their original positions.

Figure 3.5: Channel impulse response of SUI-5 channel [1].

Eq. (3.4) can then be rewritten as:

y₅₁₂ = Γ(δ) · F^H · F · T_xn· F^H · F · h + η (3.7)

= Γ(δ) · F^H · (F · T_xn · F^H) · (F · h) + η (3.8)

= Γ(δ) · F^H · Dk· H + η, (3.9)

where F is the normalized 512 × 512 FFT matrix, F^H is the corresponding normalized IFFT matrix, H is the channel frequency response vector, and D_k is a diagonal matrix of the PA-Preamble sequence in the frequency domain, with k being the PA-Preamble index.

The likelihood function of y512 can be written as:

p(y₅₁₂|δ, H, k) = 1

(2πσ²_η)⁵¹² · exp(− 1

2σ²_ηky₅₁₂− Γ(δ) · F^H · D_k· Hk²), (3.10) in the likelihood function, there are three unknowns, namely δ, H and k. The ML estimation

arg max Note that (3.14) arises because the inner minimization of (3.13) is achieved with H = D^H_k · F · Γ^H(δ) · y512as can be obtained via standard least-square estimation technique. Since Dk· D^H_k is the same whatever for add k, we cannot solve for the optimal k from (3.14), but must find it through above other means, In addition, the minimization target in (3.14) is a function of δ only. Thus it is equivalent to:

arg min diagonal matrix whose ith diagonal element is the ith element in y₅₁₂.

Since the quantity D_k· D^H_k is the same for all three PA-Preamble series, the bracketed term in (3.19) is a known quantity for a given received PA-Preamble signal. Let M = Y^H · F^H · Dk· D^H_k · F · Y. Then the quantity to be maximized can be expressed as

γ^H(δ) · M · γ(δ) the nth diagonal of M.

Note that since Dk· D^H_k is diagonal, W , F^H· Dk· D^H_k · F is a circulant matrix. Indeed, because D_k· D^H_k is nearly periodic (with mostly every other element equal to 1 while others equal to zero) along the diagonal, W is nearly tri-diagonal and so is M. The three diagonal sums are given by where y_i,i is the ith diagonal element of Y, and w_i,i is the ith diagonal of W. Note that M₋₂₅₆ = M₂₅₆^∗ . Substituting (3.21)–(3.23) into (3.20) with all other terms set to null in order to reduce the effect of noise. We utilize the mathematic format of FFT of these three

X[f ] = resolution of estimating δ and the resolution of this derivation is 0.0039. Moreover, we can conclude δ = −_π¹ arctan={M₂₅₆}

<{M₂₅₆}, and this final result is quite similar to that of the Moose algorithm [8].

3.2.3 Jointly Integral CFO, PA-Preamble Index, Channel Estima-tion and Fine Timing Offset Searching

CFO is separated into two parts, FCFO and integral carrier frequency offset (ICFO), and the former have been estimation in the previous subsection. We can expect that the power of channel impulse response (CIR), the inverse fourier transform of H as obtained in (3.14), will be more concentrated if we compensate with the accurate CFO and use the correct one of the three possible PA-Preamble symbols. For example, Figs. 3.6 and 3.7 depict two CIRs obtained from using a combination of correct CFO and correct PA-Preamble index and from

using a combination of incorrect values. The simulation environment we choose in Figs. 3.6 and 3.7 is PB channel, 120 km/h, 0 dB in SNR, the correct ICFO 8, the correct PID 1 (10-MHz), the wrong ICFO 6, and the wrong PID 0 (5-MHz). We consider there are 21 possible ICFO explained in the Eq. (3.31), 3 PA-Preamble symbols and 256 timing locations in CIR, therefore, 21 × 3 × 64 = 16128 candidates in total. The method we use here is to do 64 points sum of squared CIR for these candidates and find out one which has the maximum of power sum. The reason why we choose the searching range of ICFO from −20 to 20 is that we assume a maximum mismatch of the local oscillator frequency of ±80 ppm, so that a wireless system with carrier frequency 2.5 GHz ±18.28 subcarriers of offset at the 10.9375 kHz subcarrier spacing of IEEE 802.16m, as given by

2.5G · 80ppm

10.94K ≈ 18.28. (3.31)

For the fine timing, since it is reasonable to assume that the CIR is mostly concentrated over a length not exceeding the CP length, we decide the ICFO, the PA-Preamble index and the fine timing offset by finding which one of all candidates has the maximum power sum over the CP length.

3.2.4 Overall Block Diagram

In summary, Fig. 3.8 shows the resulting overall block diagram of the derived initial DL synchronization method.

0 100 200 300 400 500 600 0

1 2 3 4 5 6 7x 10⁵

Figure 3.6: The estimated CIR with accurate ICFO, 8, compensating and correct PA-Preamble index, 1, under PB channel with 120 km/h, 0dB in SNR.

0 100 200 300 400 500 600

Figure 3.7: The CIR with the inaccurate ICFO, 6, compensating and incorrect PA-Preamble index, 0, under PB channel with 120 km/h, 0dB in SNR.

Coarse timing

Mean square value of IFFT H

Joint ICFO ,PID and fine timing Searching

≈

y⁵¹²

Chapter 4

Introduction to the DSP Implementation Platform

In this chapter, we introduce the architecture of the DSP chip because we implement the ini-tial synchronization on DSP chip. We use the DSP chip on the module is the TMS320C6416T made by Texas Instrument (TI). We introduce and the DSP chip, and what is more, we present the software development tool, Code Composer Studio (CCS), the code development technique.