The structure of the receiver design can be classified into two types: the open-loop structure and the closed-loop structure. For example in parameter estimator, the structures are illustrated in Fig. 1.1. In the left part, the open-loop estimator uses averaging filter to smooth the sampled observations and feeds the produced material to the parameter estimator to calculate the value in one-shot process. Whereas, the closed-loop estimator uses the closed-loop adaptation that
Figure 1.1: Comparison of the structures of open-loop estimator and closed-loop estimator.
iteratively adjusts the estimated parameter according to the difference of the reconstructed result(s) and the new income(s). The use of the structures depends on the property of the estimation/detection target and the realization cost.
We consider a simple example of the estimate of the sample mean to explain the difference between the open-loop estimator and the closed-loop estimator. Let x0, x1, · · · , xN −1 be the input samples. We want to calculate the mean value of these samples. From the open-loop structure, we have
X =¯ 1 N
N −1X
n=0
xn. (1.1)
If the closed-loop structure is considered, we iteratively update the estimation via X¯n+1= ¯Xn+ µ¡
xn− ¯Xn¢
= (1 − µ) ¯Xn+ µxn. (1.2) A effective factor of the closed-loop estimator is the selection of the stepsize µ. The selection of the stepsize depends on two factors: the convergence time constant and the steady-state accu-racy. Smaller one provides smaller estimation mismatch but requires longer time to converge.
To achieve the same accuracy as the open-loop structure, the stepsize should be set to µ = N +12 . Another factor affects the convergence property is the initial condition. Good initial points yield good convergence properties as well as the final estimate results. Need to notice that the closed-loop estimator may not guarantee the optimal estimate results with the arbitrary initial conditions.
We give the examples of the estimator/detector structures for the carrier frequency estima-tion, the data detection and the channel estimation as follows.
1.4.1 Carrier Frequency Estimation
Consider the problem in carrier frequency estimation that we find out the ω form the observed signal samples, which are given by
r(n) = ejωn+jθ+ w(n) (1.3)
for n = 0 · · · N − 1, where θ is the phase offset and w(n) is the Gaussian noise. The maximally likely solution of the ω satisfies that
ˆ
In the open-loop structure, we may directly calculate the matrics of the quantized solution candidates as
and find out the ω which has the maximal J(ω). The metric calculations corresponding to the ω can be accomplished via the spectrum analysis with Fourier transform [68, Sec. 8.2.1].
In the closed-loop structure, we adopt the carrier recovery loop to estimate both the phase and the frequency offsets. One feasible recursions of the carrier recovery loop are given by
ˆ
2π is the phase error function in the recursion and (·)2π denotes the module operation with respect to a 2π interval. With suitable selection of KI, KP and the initial condition, the recursion of ˆω(n) would achieve the ML result.
1.4.2 Data Detection
Consider the data detection of the binary shift-keying modulation over the two-path channel with path gains = [1 h1]. Let [x(0) x(1) · · · x(N − 1)] be the transmitted sequence, where x(n) ∈ {1, −1}. The received signal can be represented by
r(n) = x(n) + h1x(n − 1) + w(n) (1.8) for n > 1 and r(0) = x(0) + w(0).
We can adopt the maximal likelihood sequence estimation(MLSE) by finding out the se-quence which minimizes the metric:
J (ˆx(0) · · · ˆx(N − 1)) = |r(0) − ˆx(0)|2+
N −1X
n=1
|r(n) − ˆx(n) − h1x(n − 1)|ˆ 2. (1.9) One structuralized search scheme of the MLSE is the Viterbi detector/decoder, which is widely adopted in the decoder of the convolutional code (CC).
Beside the MLSE approach, another feasible way is the linear equalizer. In matrix view, the received vector is presented by
r = Hx + w (1.10)
is the toeplitz channel matrix. Assume W be the equalization matrix, the equalized output is given by
˜
x = Wr (1.12)
and the detected data is obtained by performing the decision:
ˆ
x(n) = sgn(˜x(n)). (1.13)
According to the minimal mean-square error criterion, the equalization matrix has the formula-tion:
W = (HHH + σn2I)−1HH (1.14)
where σ2n is the noise variance. Both the MLSE and the linear equalizer obviously have the open-loop structures.
For complexity reason, we may adopt the interference cancellation (IC) to remove the inter-symbol interference (ISI) caused by the second path of the channel. One of the IC successively deletes the ISI terms generated by the prior tentative decided symbols. This kind IC is called as the successive IC (SIC), or the decision-feedback equalization in single carrier system. The recursion of the SIC/DFE is given by
ˆ
x(n) = sgn (r(n) − h1x(n − 1))ˆ (1.15) where sgn(·) is the sign operation or the decision in general, and ˆx(n) is the detection of x(n).
The DFE would cause the error propagation effect when the prior tentative decision is wrong.
Thus, this is a sub-optimal detection scheme.
In addition to SIC, another IC scheme treats the sequence detection as the signal restoration problem and adopts the iterative process to recovery the sequence. This kind IC is named as the parallel IC (PIC) since it operates in a parallel manner. One familiar PIC is the (constrained) gradient descent algorithm [24] as well as the projected Landweber iteration [82]. Let ˆxk−1 be k − 1st tentatively detected vector. The constrained gradient descent algorithm in kth iteration is given by non-expansive (NE) property [24]. This recursion does still not guarantee the optimal solution since the solution constrain (or the sign operation) does not satisfy the convex property. This kind PIC is also named as the iterative block decision feedback equalization (IBDFE) [13] in the viewpoint of the channel equalizer. Moreover, if the tentative detections are regenerated by the forward error correcting (FEC) decoder, this kind PIC is specially called the turbo DFE (TDFE), which is a simple version of the turbo equalization [56]. Both the SIC and PIC have the closed-loop structures.
1.4.3 Channel Estimation
Consider the coefficient estimation in the single path channel. Let x = [x(0) x(1) · · · x(N −1)]T be the transmitted pilot signal, which is known at the receiver. Let α be the channel gain to be identified. The received signal vector can be expressed as
r = xα + w (1.18)
where w is the noise vector. If the noise has Gaussian distribution, the maximal likelihood estimation of the channel gain is to determine the ˆα such that kr − xˆαk2 is minimal. The well-known least square (LS) solution of the estimation is given by
ˆ α =¡
xHx¢−1
xHr. (1.19)
The LS estimation in this case has the open-loop structure, which collects all the received samples and performs one-shot process to determine the channel coefficient.
Besides the LS estimation, we can adopt the adaptive least-mean square algorithm to train the channel coefficient. The cost function of the LMS adaptor in this case is J = E |r(n) − x(n)α|2. By taking gradient of J with respect to α and replacing α as ˆαn, which is the tentative solution at nth iteration, we have the recursion of the LMS estimator as
ˆ
αn+1= ˆαn+ µ (r(n) − αx(n)) x(n)∗ (1.20) where µ is the stepsize in the recursion. Obviously, the LMS for the channel estimation is a kind of the closed-loop structure in this case.