**1. Introduction**

14 Will-be-set-by-IN-TECH

[5] Hao, M. & Wicker, S. [1995]. Performance evaluation of FSK and CPFSK optical communication systems: a stable and accurate method, *Journal of lightwave technology*

[6] Kunisch, J. & Pamp, J. [2002]. Measurement Results and Modeling Aspects for the UWB

[7] Kunisch, J. & Pamp, J. [n.d.]. *IMST-UWBW: 1-11 GHz UWB Indoor Radio Channel Measurement Data*, IMST GmbH. available at

[9] Molisch, A. F. et al. [2005]. IEEE 802.15.4a channel model - final report, *Tech. Rep.*

[10] Song, N., Wolf, M. & Haardt, M. [2007]. Low-complexity and Energy Efficient Non-coherent Receivers for UWB Communications, *Proc. 18-th Annual IEEE International Symposium on Personal Indoor and Mobile Radio Communications (PIMRC07)*, Greece. [11] Song, N., Wolf, M. & Haardt, M. [2009a]. A Digital Code Matched Filter-based Non-Coherent Receiver for Low Data Rate TH-PPM-UWB Systems in the Presence of MUI, *IEEE International Conference on Ultra Wideband (ICUWB 2009)*, Vancouver, Canada. [12] Song, N., Wolf, M. & Haardt, M. [2009c]. Performance of PPM-Based Non-Coherent Impulse Radio UWB Systems using Sparse Codes in the Presence of Multi-User Interference, *Proc. of IEEE Wireless Communications and Networking Conference (WCNC*

[13] Song, N., Wolf, M. & Haardt, M. [2010a]. A Digital Non-Coherent Ultra-Wideband Receiver using a Soft-Limiter for Narrowband Interference Suppression, *Proc. 7-th International Symposium on Wireless Communications Systems (ISWCS 2010)*, York, United

[14] Song, N., Wolf, M. & Haardt, M. [2010b]. A *b*-bit Non-Coherent Receiver based on a Digital Code Matched Filter for Low Data Rate TH-PPM-UWB Systems in the Presence of MUI, *Proc. of IEEE International Conference on Communications (ICC 2010)*, Cape Town,

[15] Song, N., Wolf, M. & Haardt, M. [2010d]. Time-Hopping M-Walsh UWB Transmission Scheme for a One-Bit Non-Coherent Receiver, *Proc. 7-th International Symposium on*

[16] Song, N., Wolf, M. & Haardt, M. [Feb., 2009b]. Non-coherent Receivers for Energy Efficient UWB Transmission, *UKoLoS Annual Colloquium*, Erlangen, Germany. [17] Song, N., Wolf, M. & Haardt, M. [Mar., 2010c]. Digital non-coherent UWB receiver based on TH-Walsh transmission schemes, *UKoLoS Annual Colloquium*, Günzburg , Germany. [18] TG4a, I. . [2007]. Part 15.4: Wireless Medium Access Control (MAC) and Physical Layer (PHY) Specifications for Low-Rate Wireless Personal Area Networks (WPANs), *IEEE*

[19] Wolf, M., Song, N. & Haardt, M. [2009]. Non-Coherent UWB Communications, *FREQUENZ Journal of RF-Engineering and Telecommunications* . special issue on Ultra-Wideband Radio Technologies for Communications, Localisation and Sensor

*Wireless Communications Systems (ISWCS 2010)*, York, United Kingdom.

http://www.imst.com/imst/en/research/radio-networks/funk\_wel\_dow.php. [8] Le, B., Rondeau, T. W., Reed, J. H. & Bostian, C. W. [2005]. Analog-to-digital converters,

Radio Channel, *IEEE Conference on Ultra Wide-Band Systems and Technologies* .

*IEEE Signal Processing Magazine* 22(6): 69–77.

*Document IEEE 802.15-04-0662- 02-004a*.

*2009)*, Budapest, Hungary.

*Standard for Information Technology* .

Kingdom.

South Africa.

applications.

(13): 1613–1623.

Impulse-radio ultra-wideband (IR-UWB) is a promising transmission scheme, especially for short-range low-data-rate communications, as, e.g., in wireless-sensor networks. One of the main reasons for this is its potential to employ noncoherent, hence low-complexity, receivers even in dense multipath propagation scenarios, where channel estimation required for coherent detection would be overly complex due to the large signal bandwidth and hence rich multipath propagation [23].

Differential pulse-amplitude-modulated IR-UWB in combination with autocorrelation-based detection constitutes an attractive variant of noncoherent detection schemes [11]. The inherent loss in performance of traditional noncoherent autocorrelation-based differential detection (DD), as compared to coherent detection based on explicit channel estimation, can be alleviated by advanced autocorrelation-based detection schemes operating on the output of an extended autocorrelation receiver (ACR). This ACR delivers correlation coefficients of symbols separated by several symbol durations. In this context, block-based detection schemes, which partition the receive symbol stream into (possibly overlapping) blocks and thus process multiple symbols jointly, have proven to enable power-efficient, yet low-complexity detection in both uncoded and coded IR-UWB transmission systems [3, 6, 11, 12, 15–18].

In this chapter, a comprehensive review of block-based detection schemes is presented. Starting with an exposition of the operation in uncoded schemes, we discuss the generation of soft output, required in coded IR-UWB systems employing autocorrelation-based detection. For the design of such systems, an information theoretic performance analysis of IR-UWB transmission with autocorrelation-based detection delivers design rules for coded IR-UWB systems. In particular, optimum rates for the applied channel code are derived, which improve the overall power efficiency (i.e., required signal-to-noise ratio to guarantee a desired error rate) of the system. The chapter concludes with a brief summary.

©2013 Schenk and Fischer, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0),which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ©2013 Schenk and Fischer, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### **2. IR-UWB system model with autocorrelation-based detection**

### **2.1. IR-UWB system model**

Throughout this chapter binary pulse-amplitude-modulated IR-UWB transmission in combination with bit-interleaved coded modulation (BICM), as shown in Fig. 1, is considered. Avoiding up-/downconversion due to operation at a carrier frequency, transmission takes place in the baseband; hence, all signals are real valued. The sequence of information bits (assumed to be equiprobable and independent, i.e., with maximum entropy) is encoded with a channel code of rate *R*c. After symbolwise mapping from (interleaved) codeword bits *ck* to binary information symbols *ak* ∈ {±1}, differential encoding is performed, yielding the transmit symbols *bk* ∈ {±1}, where *bk* = *bk*−<sup>1</sup>*ak* and *<sup>b</sup>*<sup>0</sup> = 1. The IR-UWB receive signal, after propagation through an UWB multipath channel, is given by [23]

$$r(t) = \sum\_{k=0}^{+\infty} b\_k p(t - kT) + n(t) \tag{1}$$

definitions for the end-to-end impulse and the noise, assuming no inter-symbol interference,

The noise components in ¯*n<sup>k</sup>* are modeled as uncorrelated Gaussian random variables with

Autocorrelation-based noncoherent detection of IR-UWB, cf., e.g., [2, 3, 6, 9, 15], requires to compute the correlation of the current symbol with up to *L* preceeding symbols, as shown in Fig. 1. Significant gains are achieved by adopting the integration interval to the channel characteristics at hand [23], i.e., choosing *T*<sup>i</sup> < *T* in the order of the expected channel delay spread. Simplified, larger *T*<sup>i</sup> lead to decreased performance, but become inevitable in case of only coarse synchronization or insufficient knowledge of the channel characteristics2. Defining the time-bandwidth product *N* = *f*s*T*<sup>i</sup> and *r<sup>k</sup>* as the part of ¯*r<sup>k</sup>* relevant for the ACR integration, i.e., (typically the first) *N* successive components out of *N*s, in an digital

*zk*−*l*,*<sup>k</sup>* <sup>=</sup> *<sup>r</sup>*<sup>T</sup>

The correlation coefficients serve as input for various detection schemes, cf., Sec. 3 and [3, 6, 9, 10, 12, 15, 23]. E.g., symbolwise differential detection (DD) utilizes only the correlation coefficient of the current symbol and its predecessor, i.e., *<sup>L</sup>* = 1, and, since *bk*−<sup>1</sup>*bk* = *ak*, the

We explicitly point out the major drawback of an autocorrelation-based receiver, namely the required accurate analog delay lines in an analog implementation, or the large sampling rate<sup>3</sup> in an all-digital implementation. Especially approaches based on the principle of compressed sensing (CS) promise to circumvent these problems [14, 24]. These approaches avoid sampling the receive signal at the (possibly prohibitively) large Nyquist rate by taking fewer measurements in a different domain (e.g., frequency or some transform domain). In [14] it has been shown, that a CS-front-end can readily be applied prior to an ACR, i.e., via direct correlation of the measurements, thus also avoiding the need for computationally complex CS-reconstruction algorithms. In combination with autocorrelation-based DD the inherent loss in performance of CS/ACR-based detection is proportional only to the square root of the

Based on the all-digital implementation, we introduce an equivalent discrete-time system

<sup>2</sup> A typical setting for realistic IR-UWB scenarios, e.g., modelled by the IEEE channel models [7, 8] is *T*<sup>i</sup> = 33 ns, whereas *<sup>T</sup>* <sup>=</sup> 75 ns to avoid inter-symbol interference. With *<sup>f</sup>*<sup>s</sup> <sup>=</sup> 12 GHz, we have *<sup>N</sup>*<sup>s</sup> <sup>=</sup> 900 and *<sup>N</sup>* <sup>≈</sup> 400 [23]. <sup>3</sup> With the advance in micro electronics, one can expect that an all-digital implementation becomes realistic within no

*k*−*l*

*<sup>k</sup>* = sign(*zk*−1,*k*).

*zk*−*l*,*<sup>k</sup>* = *<sup>E</sup>*i*xk*−*l*,*<sup>k</sup>* + *<sup>η</sup>k*−*l*,*<sup>l</sup>* (4)

*<sup>n</sup>* = *f*s*N*0/2, which is the case for a square-root Nyquist low-pass receiver front-end

*r*¯*<sup>k</sup>* = *bkp*¯ + *n*¯ *<sup>k</sup>* . (2)

in Impulse-Radio Ultra-Wideband Transmission Systems

Coding, Modulation, and Detection for Power-Effi cient Low-Complexity Receivers

125

*r<sup>k</sup>* . (3)

we compactly write

filter with two-sided bandwidth *f*s.

**2.2. Autocorrelation-based detection**

implementation we have, for *l* = 1, . . . , *L*,

decision rule for the information symbols reads *a*DD

compression ratio (number of measurements over *N*s) [14].

model of ACR-based detection. The ACR-output can be written as

**2.3. Equivalent discrete-time system model**

later than the next decade.

variance *σ*<sup>2</sup>

where *T* is the symbol duration and *p*(*t*) denotes the overall receive pulse shape, resulting from the convolution of transmit (TX) pulse, receive (RX) filter, and channel (CH) impulse response; its energy is normalized to one, thus, the energy per information symbol<sup>1</sup> is given as *E*<sup>s</sup> = 1. We assume the channel to remain constant within one codeword. *n*(*t*) results from white Gaussian noise of two-sided power-spectral density *N*0/2 passed through the RX filter. To preclude inter-symbol interference, the symbol duration *T* is chosen sufficiently large, such that each pulse has decayed before the next pulse is received. For clarity, we do not explicitly consider the typically applied frame structure used for time-hopping and code-division multiple access, as it can be averaged out prior to further receive signal processing [6].

**Figure 1.** System model of coded IR-UWB transmission with autocorrelation-based detection.

For convenient representation and in view of an all-digital implementation of the receiver, we define the sampled receive signal of the *k*th symbol interval as ¯*r<sup>k</sup>* = [*r*(*kT*), *r*(*kT* + *T*s), ..., *<sup>r</sup>*(*kT* + (*N*<sup>s</sup> <sup>−</sup> <sup>1</sup>)*T*s)]T, where *<sup>N</sup>*<sup>s</sup> <sup>=</sup> *<sup>f</sup>*s*<sup>T</sup>* is the number of samples per symbol interval, and *f*<sup>s</sup> = 1/*T*<sup>s</sup> is the sampling rate (greater than or equal to the Nyquist rate). With respective

<sup>1</sup> For long bursts, the energy for the reference symbol may be neglected.

definitions for the end-to-end impulse and the noise, assuming no inter-symbol interference, we compactly write

$$
\bar{r}\_k = \mathbb{b}\_k \bar{p} + \bar{n}\_k \,. \tag{2}
$$

The noise components in ¯*n<sup>k</sup>* are modeled as uncorrelated Gaussian random variables with variance *σ*<sup>2</sup> *<sup>n</sup>* = *f*s*N*0/2, which is the case for a square-root Nyquist low-pass receiver front-end filter with two-sided bandwidth *f*s.

### **2.2. Autocorrelation-based detection**

2 Will-be-set-by-IN-TECH

Throughout this chapter binary pulse-amplitude-modulated IR-UWB transmission in combination with bit-interleaved coded modulation (BICM), as shown in Fig. 1, is considered. Avoiding up-/downconversion due to operation at a carrier frequency, transmission takes place in the baseband; hence, all signals are real valued. The sequence of information bits (assumed to be equiprobable and independent, i.e., with maximum entropy) is encoded with a channel code of rate *R*c. After symbolwise mapping from (interleaved) codeword bits *ck* to binary information symbols *ak* ∈ {±1}, differential encoding is performed, yielding the transmit symbols *bk* ∈ {±1}, where *bk* = *bk*−<sup>1</sup>*ak* and *<sup>b</sup>*<sup>0</sup> = 1. The IR-UWB receive signal, after

where *T* is the symbol duration and *p*(*t*) denotes the overall receive pulse shape, resulting from the convolution of transmit (TX) pulse, receive (RX) filter, and channel (CH) impulse response; its energy is normalized to one, thus, the energy per information symbol<sup>1</sup> is given as *E*<sup>s</sup> = 1. We assume the channel to remain constant within one codeword. *n*(*t*) results from white Gaussian noise of two-sided power-spectral density *N*0/2 passed through the RX filter. To preclude inter-symbol interference, the symbol duration *T* is chosen sufficiently large, such that each pulse has decayed before the next pulse is received. For clarity, we do not explicitly consider the typically applied frame structure used for time-hopping and code-division multiple access, as it can be averaged out prior to further receive signal processing [6].

*bkp*(*t* − *kT*) + *n*(*t*) (1)

**2. IR-UWB system model with autocorrelation-based detection**

propagation through an UWB multipath channel, is given by [23]

*r*(*t*) =

+∞ ∑ *k*=0

**Figure 1.** System model of coded IR-UWB transmission with autocorrelation-based detection.

<sup>1</sup> For long bursts, the energy for the reference symbol may be neglected.

For convenient representation and in view of an all-digital implementation of the receiver, we define the sampled receive signal of the *k*th symbol interval as ¯*r<sup>k</sup>* = [*r*(*kT*), *r*(*kT* + *T*s), ..., *<sup>r</sup>*(*kT* + (*N*<sup>s</sup> <sup>−</sup> <sup>1</sup>)*T*s)]T, where *<sup>N</sup>*<sup>s</sup> <sup>=</sup> *<sup>f</sup>*s*<sup>T</sup>* is the number of samples per symbol interval, and *f*<sup>s</sup> = 1/*T*<sup>s</sup> is the sampling rate (greater than or equal to the Nyquist rate). With respective

**2.1. IR-UWB system model**

Autocorrelation-based noncoherent detection of IR-UWB, cf., e.g., [2, 3, 6, 9, 15], requires to compute the correlation of the current symbol with up to *L* preceeding symbols, as shown in Fig. 1. Significant gains are achieved by adopting the integration interval to the channel characteristics at hand [23], i.e., choosing *T*<sup>i</sup> < *T* in the order of the expected channel delay spread. Simplified, larger *T*<sup>i</sup> lead to decreased performance, but become inevitable in case of only coarse synchronization or insufficient knowledge of the channel characteristics2. Defining the time-bandwidth product *N* = *f*s*T*<sup>i</sup> and *r<sup>k</sup>* as the part of ¯*r<sup>k</sup>* relevant for the ACR integration, i.e., (typically the first) *N* successive components out of *N*s, in an digital implementation we have, for *l* = 1, . . . , *L*,

$$z\_{k-l,k} = r\_{k-l}^{\\\\\top} r\_k \,. \tag{3}$$

The correlation coefficients serve as input for various detection schemes, cf., Sec. 3 and [3, 6, 9, 10, 12, 15, 23]. E.g., symbolwise differential detection (DD) utilizes only the correlation coefficient of the current symbol and its predecessor, i.e., *<sup>L</sup>* = 1, and, since *bk*−<sup>1</sup>*bk* = *ak*, the decision rule for the information symbols reads *a*DD *<sup>k</sup>* = sign(*zk*−1,*k*).

We explicitly point out the major drawback of an autocorrelation-based receiver, namely the required accurate analog delay lines in an analog implementation, or the large sampling rate<sup>3</sup> in an all-digital implementation. Especially approaches based on the principle of compressed sensing (CS) promise to circumvent these problems [14, 24]. These approaches avoid sampling the receive signal at the (possibly prohibitively) large Nyquist rate by taking fewer measurements in a different domain (e.g., frequency or some transform domain). In [14] it has been shown, that a CS-front-end can readily be applied prior to an ACR, i.e., via direct correlation of the measurements, thus also avoiding the need for computationally complex CS-reconstruction algorithms. In combination with autocorrelation-based DD the inherent loss in performance of CS/ACR-based detection is proportional only to the square root of the compression ratio (number of measurements over *N*s) [14].

### **2.3. Equivalent discrete-time system model**

Based on the all-digital implementation, we introduce an equivalent discrete-time system model of ACR-based detection. The ACR-output can be written as

$$z\_{k-l,k} = E\_{\text{i}} \mathbf{x}\_{k-l,k} + \eta\_{k-l,l} \tag{4}$$

<sup>2</sup> A typical setting for realistic IR-UWB scenarios, e.g., modelled by the IEEE channel models [7, 8] is *T*<sup>i</sup> = 33 ns, whereas

*<sup>T</sup>* <sup>=</sup> 75 ns to avoid inter-symbol interference. With *<sup>f</sup>*<sup>s</sup> <sup>=</sup> 12 GHz, we have *<sup>N</sup>*<sup>s</sup> <sup>=</sup> 900 and *<sup>N</sup>* <sup>≈</sup> 400 [23]. <sup>3</sup> With the advance in micro electronics, one can expect that an all-digital implementation becomes realistic within no later than the next decade.

#### 4 Will-be-set-by-IN-TECH 126 Ultra-Wideband Radio Technologies for Communications, Localization and Sensor Applications Coding, Modulation, and Detection for Power-Efficient Low-Complexity Receivers in Impulse-Radio Ultra-Wideband Transmission Systems <sup>5</sup>

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, summarized in the equivalent noise term

$$
\boldsymbol{\eta}\_{k-l,l} = \boldsymbol{b}\_{k-l} \boldsymbol{p}^{\mathsf{T}} \boldsymbol{n}\_{k} + \boldsymbol{b}\_{k} \boldsymbol{n}\_{k-l}^{\mathsf{T}} \boldsymbol{p} + \boldsymbol{n}\_{k-l}^{\mathsf{T}} \boldsymbol{n}\_{k} \,. \tag{5}
$$

approximation, in terms of log-likelihood ratios (LLRs) reliability information corresponds

and the decision metric of the corresponding counter hypothesis, i.e., the minimum metric

*<sup>k</sup>* <sup>=</sup> min *<sup>a</sup>*˜∈{±1}*<sup>L</sup>*, *<sup>a</sup>*˜*k*=−*a*MSDD

 ΛMSDD

In the case of SO-DD (*L* = 1), the LLRs are directly given as the (scaled) ACR output, i.e.,

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

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.

> *k*1,...,ˆ *ki*−<sup>1</sup>}

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

 

*i*−1 ∑ *l*=0 *z*ˆ

*kl*,*<sup>k</sup> <sup>b</sup>*DF-DD ˆ *kl*

 

*a*˜∈{±1}*<sup>L</sup>*

Coding, Modulation, and Detection for Power-Effi cient Low-Complexity Receivers

*k*

*<sup>k</sup>* <sup>−</sup> <sup>Λ</sup>MSDD

Λ(*a*˜) (7)

in Impulse-Radio Ultra-Wideband Transmission Systems

127

Λ(*a*˜) . (8)

. (9)

*k*<sup>0</sup> = 0, *b*DF-DD

. (11)

<sup>0</sup> = 1, the

(10)

to the (scaled) difference of the decision metric of the optimum sequence [12], i.e.,

*<sup>k</sup>* , i.e., for *k* = 1, . . . , *L*,

ΛMSDD

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

LLR*<sup>k</sup>* <sup>∼</sup> *<sup>a</sup>*MSDD *k*

only moderate complexity increase compared to hard-output MSDD.

Briefly sketched, following [15] and focusing on the first block, with ˆ

= sign

*ki* = argmax *<sup>k</sup>*∈{1,...,*L*}/{<sup>ˆ</sup>

> *i*−1 ∑ *l*=0 *z*ˆ *kl*,ˆ *ki b*DF-DD ˆ *kl*

receive symbols and previous decisions, rather than on the channel realization.

optimized decision order and the estimates are given by

*b*DF-DD ˆ *ki*

ˆ

**3.2. Decision-feedback differential detection**

with the restriction *<sup>a</sup>*˜*<sup>k</sup>* <sup>=</sup> <sup>−</sup>*a*MSDD

LLRDD

*<sup>k</sup>* ∼ *zk*−1,*<sup>k</sup>* [12].

ΛMSDD = Λ(*a*MSDD) = min

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 × noise" terms are zero-mean with variance *σ*<sup>2</sup> *<sup>n</sup>*, and the "noise × noise" term, as the sum of *N* products of independent Gaussian random variables, is zero-mean with variance *N*(*σ*<sup>2</sup> *<sup>n</sup>*)2. Consequently, *<sup>η</sup>k*−*l*,*<sup>k</sup>* may be modeled as a zero-mean Gaussian random variable with variance *σ*2 *<sup>η</sup>* = 2*σ*<sup>2</sup> *<sup>n</sup>* + *N*(*σ*<sup>2</sup> *<sup>n</sup>*)2. Since each *<sup>η</sup>k*−*l*,*<sup>k</sup>* results from the multipication of different parts of noise and symbols, the equivalent noise samples at different time instances and ACR branches are uncorrelated.

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 transmission chain.
