**3. Advanced detection schemes for IR-UWB**

### **3.1. Multiple-symbol differential detection**

One of the most powerful detection schemes is based on the principle of multiple-symbol differential detection (MSDD), cf., [1] and its modifications for IR-UWB detection [3, 6, 15]. In MSDD the stream of receive symbols is decomposed into blocks of *L* + 1 symbols (note that the blocks have to overlap by at least one symbol), and for each block the blockwise-optimal sequence of *L* information symbols is decided jointly based on the correlation coefficients corresponding to this block. The decision metric given a hypothesis of information symbols grouped into a vector ˜*a* and the corresponding hypothesis of the ACR output ˜*x*, reads

$$\Lambda(\tilde{\mathfrak{a}}) = \sum\_{k=1}^{L} \left( \sum\_{l=0}^{k-1} \left( |z\_{l,k}| - \tilde{x}\_{l,k} z\_{l,k} \right) \right) . \tag{6}$$

The blockwise-optimal sequence *a*MSDD of hard-output MSDD is given as the sequence with minimum decision metric.

To fully exploit the benefits of channel coding, reliability information on the estimated codeword bits should be delivered to the subsequent channel decoder, i.e., so-called soft-output MSDD (SO-MSDD) should be performed. Sticking to the so-called max-log approximation, in terms of log-likelihood ratios (LLRs) reliability information corresponds to the (scaled) difference of the decision metric of the optimum sequence [12], i.e.,

$$\Lambda^{\mathsf{MSDD}} = \Lambda(\mathfrak{a}^{\mathsf{MSDD}}) = \min\_{\mathfrak{A} \in \{\pm 1\}^{\mathsf{L}}} \Lambda(\mathfrak{a}) \tag{7}$$

and the decision metric of the corresponding counter hypothesis, i.e., the minimum metric with the restriction *<sup>a</sup>*˜*<sup>k</sup>* <sup>=</sup> <sup>−</sup>*a*MSDD *<sup>k</sup>* , i.e., for *k* = 1, . . . , *L*,

$$\Lambda\_k^{\overline{\rm MSDD}} = \min\_{\mathfrak{A} \in \{\pm 1\}^L, \mathfrak{A}\_k = -a\_k^{\rm MSDD}} \Lambda(\mathfrak{a}) \,. \tag{8}$$

Finally, the reliablitiy of the *k*th symbol/codeword bit is proportional to

$$\text{LLR}\_{k} \sim a\_{k}^{\text{MSDD}} \left( \Lambda\_{k}^{\overline{\text{MSDD}}} - \Lambda^{\text{MSDD}} \right) \,. \tag{9}$$

In the case of SO-DD (*L* = 1), the LLRs are directly given as the (scaled) ACR output, i.e., LLRDD *<sup>k</sup>* ∼ *zk*−1,*<sup>k</sup>* [12].

Utilizing the triangular structure of the decision metric, an efficient solution to the MSDD search problem (7) is obtained by employing the sphere decoder algorithm [6, 18, 19]. In the case of SO-MSDD, incorporating modifications in the sphere decoder algorithm proposed for efficient soft-output detection in multi-antenna systems [21], the *L* + 1 search problems per block, (7) and (8), can be solved in a single sphere decoder run per block using the single-tree-search soft-output sphere decoder [12, 21]. Thus, SO-MSDD can be realized at only moderate complexity increase compared to hard-output MSDD.

### **3.2. Decision-feedback differential detection**

4 Will-be-set-by-IN-TECH

where *E*<sup>i</sup> = *p*<sup>T</sup> *p* denotes the captured pulse energy. It is composed of the phase transition from *bk*−*<sup>l</sup>* to *bk*, i.e., *xk*−*l*,*<sup>k</sup>* = *bk*−*lbk*, and "information × noise" and "noise × noise" terms,

A detailled analysis of the components of the equivalent noise term in (5) shows that already for moderate time-bandwidth products *N* it is reasonable to approximate the respective terms as uncorrelated Gaussian random variables [9, 10, 14]. In particular, the "information ×

*N* products of independent Gaussian random variables, is zero-mean with variance *N*(*σ*<sup>2</sup>

Consequently, *<sup>η</sup>k*−*l*,*<sup>k</sup>* may be modeled as a zero-mean Gaussian random variable with variance

and symbols, the equivalent noise samples at different time instances and ACR branches are

This approximation is only valid under the following prerequisites, which typically are fulfilled in common IR-UWB systems: i) the symbol duration is chosen sufficiently large, such that no inter-symbol interference is present, ii) the integration interval of the ACR and the time-bandwidth product *N* are chosen sufficiently large, such that the Gaussian approximation holds, iii) the receiver front-end filter is a square-root Nyquist low-pass with two-sided bandwidth *f*s to avoid correlations of the noise samples, and iv) the channel remains constant over the block of at least *L* + 1 symbols. We emphasize that this model not only enables the subsequent information theoretic analysis of ACR-based detection of IR-UWB, but also serves as a tool for efficient numerical simulations of the IR-UWB

One of the most powerful detection schemes is based on the principle of multiple-symbol differential detection (MSDD), cf., [1] and its modifications for IR-UWB detection [3, 6, 15]. In MSDD the stream of receive symbols is decomposed into blocks of *L* + 1 symbols (note that the blocks have to overlap by at least one symbol), and for each block the blockwise-optimal sequence of *L* information symbols is decided jointly based on the correlation coefficients corresponding to this block. The decision metric given a hypothesis of information symbols grouped into a vector ˜*a* and the corresponding hypothesis of the ACR output ˜*x*, reads

*<sup>k</sup>*−*<sup>l</sup> <sup>p</sup>* <sup>+</sup> *<sup>n</sup>*<sup>T</sup>

*<sup>n</sup>*)2. Since each *<sup>η</sup>k*−*l*,*<sup>k</sup>* results from the multipication of different parts of noise

*k*−*l*

*<sup>n</sup>*, and the "noise × noise" term, as the sum of

*n<sup>k</sup>* . (5)

*<sup>n</sup>*)2.

*<sup>η</sup>k*−*l*,*<sup>l</sup>* <sup>=</sup> *bk*−*lp*T*n<sup>k</sup>* <sup>+</sup> *bkn*<sup>T</sup>

summarized in the equivalent noise term

noise" terms are zero-mean with variance *σ*<sup>2</sup>

**3. Advanced detection schemes for IR-UWB**

Λ(*a*˜) =

*L* ∑ *k*=1 *<sup>k</sup>*−<sup>1</sup> ∑ *l*=0 

The blockwise-optimal sequence *a*MSDD of hard-output MSDD is given as the sequence with

To fully exploit the benefits of channel coding, reliability information on the estimated codeword bits should be delivered to the subsequent channel decoder, i.e., so-called soft-output MSDD (SO-MSDD) should be performed. Sticking to the so-called max-log


 

. (6)

**3.1. Multiple-symbol differential detection**

*σ*2 *<sup>η</sup>* = 2*σ*<sup>2</sup>

uncorrelated.

transmission chain.

minimum decision metric.

*<sup>n</sup>* + *N*(*σ*<sup>2</sup>

A closely-related detection scheme is blockwise decision-feedback differential detection (DF-DD), cf., [5] and its modifications for IR-UWB detection [15], which decides the symbols within each block in a successive manner taking into account the feedback from already decided symbols within the block. The blockwise processing of the receive signal enables to optimize the decision order, such that in each step the most reliable symbol is decided next, resulting in almost the performance of MSDD at lower and in particular constant complexity.

Briefly sketched, following [15] and focusing on the first block, with ˆ *k*<sup>0</sup> = 0, *b*DF-DD <sup>0</sup> = 1, the optimized decision order and the estimates are given by

$$\hat{k}\_{l} = \underset{k \in \{1, \ldots, L\} / \{\hat{k}\_{l}, \ldots, \hat{k}\_{l-1}\}}{\text{argmax}} \left| \sum\_{l=0}^{i-1} z\_{\hat{k}\_{l},k} b\_{\hat{k}\_{l}}^{\textsf{DFT-DD}} \right| \tag{10}$$

$$b\_{\vec{k}\_l}^{\mathsf{DF-DD}} = \text{sign}\sum\_{l=0}^{i-1} z\_{\vec{k}\_l, \vec{k}\_l} b\_{\vec{k}\_l}^{\mathsf{DF-DD}}.\tag{11}$$

Basically, the optimized decision order forces reliable decisions for the first decided symbols, which then strongly influence the upcoming decisions. In contrast to the related detection scheme BLAST in multiple-antenna systems, sorting is done per block based on the actual receive symbols and previous decisions, rather than on the channel realization.
