Finite-Resolution Digital Receiver Design for Impulse ... - IEEE Xplore
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Finite-Resolution Digital Receiver Design for Impulse Radio Ultra-Wideband Communication Lei Ke, Student Member, IEEE, Huarui Yin, Member, IEEE, Weilin Gong, Member, IEEE, and Zhengdao Wang, Senior Member, IEEE
Abstract—Receiver design for impulse radio based ultrawideband (UWB) communication is a challenge. High sampling rate high resolution digital receiver is usually difficult to implement. Some tradeoffs can be made on the digital receiver, such as limiting amplitude resolution to only one bit, which results in a previously considered monobit receiver. In this paper, we consider the design of finite-resolution digital UWB receivers. We derive the optimal post-quantization processing, and analyze the achievable bit-error rate performance using an approximation of the log-likelihood ratio. Optimal thresholds for 4- and 3-level quantization are obtained. Training-based receiver template estimation is presented. Our work discloses the incremental gain that additional quantization levels offer and the results provide useful guidelines for designing impulse radio UWB digital receivers. Index Terms—Ultra-wideband, finite resolution, impulse radio, digital receiver, bit error rate, quantization threshold.
I. I NTRODUCTION
U
LTRA-WIDEBAND (UWB) communication has drawn attention recently from both academia and industry [1]– [5]. Receiver design for UWB communications is a challenge due to the wide bandwidth nature of the received signal. One popular method of generating UWB signals is impulse radio (IR) [1], [2]. IR-UWB receivers can be divided into two categories: coherent and noncoherent receivers. The receiver discussed in [2] is a standard coherent receiver. To maximize the signal to noise ratio (SNR) at receiver, an matched filter is used, which is the optimal receiver under additive white Gaussian noise (AWGN) channel. Matched filter can have problems with distortion, timing mismatch [6], and when implemented using analog delay lines, difficulty of adjusting the delays. The performance can also be sensitive to timing jitter [7]. Another way of implementing the receiver is to perform frequency domain processing [6], [8], whose performance depends on the number of projection functions or filter bank branches.
Manuscript received November 2, 2007; revised March 15, 2008 and May 31, 2008; accepted September 16, 2008. The associate editor coordinating the review of this paper and approving it for publication was N. Arumugam. The work in this paper was supported in part by the National Science Foundation under Grant No. 0431092. L. Ke and Z. Wang are with the Department of Electrical and Computer Engineering, Iowa State University, Ames, IA 50011 USA (e-mail: {kelei, zhengdao}@iastate.edu). H. Yin and W. Gong are with the Department of Electronic Engineering and Information Science, University of Science and Technology of China, Hefei, Anhui 230027, P.R. China (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/T-WC.2008.071223
Coherent UWB receivers require channel estimation, which is usually difficult to perform due to the wideband nature of the channel. To alleviate the problem, transmitted-reference (TR) UWB receiver was proposed in [9]. The TR technique dates back to 1960’s [10]. Other similar receivers such as autocorrelation receiver [11], differential receiver [12] have also been proposed. Detailed performance analysis and comparison can be found in [13] and [14]. To alleviate the 3-dB non-coherent performance loss, a hybrid matched filter correlation receiver has been proposed in [15], which performs a despreading operation first before multiplication operation. The issue of tradeoff between performance and complexity is present in both coherent and noncoherent receivers. Such tradeoff was considered in [16] for rake and autocorrelation receivers, and TR based systems. Both coherent and noncoherent receivers can be implemented digitally. To implement the coherent matched filter digitally, high sampling rate is usually required [17]. For example, assuming a pulse duration of one nano-second (ns), an oversampling factor of 10 corresponds to a sampling rate of 10 giga samples per second (Gsps), which can be difficult to implement at high resolution of, say, 8 bits per sample. One-bit analog-to-digital converter (ADC) was used to sample the received signal in [18], and the resulting receiver was termed monobit receiver. Such a one-bit ADC can be realized by fast comparator. It is expected that tens of Gsps sampling rate can be achieved. In addition, single-bit ADCs are also attractive because of their lower power consumption and lower cost compared to multi-bit ADCs. Bit error rate (BER) expressions for matched filter and TR receivers are derived in [18]. And it is shown that using a one-bit analog sigma-delta modulator (SDM) with significant oversampling, BER can be made close to that of the matched filter in AWGN channel. A b-bit receiver for TR-UWB system is discussed in [19]. The relationship between ADC resolution and receiver performance with inter-pulse interference is examined based on the analysis of the statistical distribution of quantization noise. Similar to [6], it was demonstrated that a 3-bit receiver can already achieve good performance. The proposed receiver therein has an analog matched-filter front end, which can be difficult to implement. The quantization range is optimized by numerical method. Similar work can be found in [20], where an ad hoc channel estimator is proposed. In this paper, we focus on the issue of finite-resolution ADC design and associated receiver performance in terms
c 2008 IEEE 1536-1276/08$25.00
KE et al.: FINITE-RESOLUTION DIGITAL RECEIVER DESIGN FOR IMPULSE RADIO ULTRA-WIDEBAND COMMUNICATION
of quantization resolution and threshold for time-domain IRUWB coherent receivers. Our contributions are summarized as follows: 1) Optimal finite-resolution digital detector based on the maximum-likelihood (ML) rule is derived. 2) Accurate BER expressions through a Gaussian approximation are obtained. 3) Optimal quantization thresholds based on the analytical BER expressions are obtained through numerical search. 4) A training based template estimation method using the ML criterion is proposed. Compared to the quantization-noise based performance analysis in [19], our approach in this paper is to first derive the optimal detector architecture, and then quantify the performance by approximating the optimal combining coefficients. The optimal architecture turns out to be a weighted sum of the quantized bits, rather than processing all the bits in the samples jointly as performed in [19]. The rest of the paper is organized as follows: The system model is presented in Section II. A two-bit quantization based receiver is designed and its performance is analyzed in Section III. The results are generalized in Section IV, where the receiver performance under fading channel is also briefly discussed. ML estimation of the detection templates is proposed in Section V. Simulation results are presented in Section VI, and Section VII concludes the paper. II. S YSTEM M ODEL Consider a single-user IR-UWB system using binary pulse amplitude modulation. Let w(t) denote the pulse shape at the receiver, which incorporates the transmitter pulse the antenna differentiation effects, and the dispersive channel. The pulse w(t) also incorporates the possible time-hopping (TH) sequence or direct-sequence (DS) spreading code at the transmitter if they are used. We model the channel as dispersive and its impulse response is included in w(t). When the channel is time varying, we assume it to be quasi-static so that our results apply for each channel coherent interval. We can write the received signal as r(t) =
∞
dk w(t − kT ) + n(t)
(1)
k=0
where dk is the kth transmitted symbol, which is equal to ±1 with equal probability 1/2; T is the symbol duration; n(t) is AWGN with power spectral density N0 /2. The received signal is filtered by a receiver pre-sampling ideal low-pass filter (LPF), and then sampled with sampling period Ts ; see Fig. 1. We assume that the filter has bandwidth B = 1/(2Ts ) and unit response inside the passband frequency. The filtering bandwidth usually needs to be large enough to contain most of the signal energy, and yet not too large so that unnecessary noise can be excluded. The number of digital samples that can be obtained within one symbol is Ns = T /Ts , the smallest integer larger than or equal to T /Ts . The filtered pulse shape is denoted as wref (t), which is the convolution of w(t) and the filter’s impulse response. We first assume the availability of wref (t) in deriving our main results. Template estimation will be discussed in Section V. We call
r(t)
LPF
iTs
Quantize
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DSP
dˆ
Fig. 1. Sampling Structure: The received signal is low-pass filtered, sampled, quantized, and processed by the digital signal processor.
wref (t) the reference signal, which is the signal to be matched with if an analog matched filter is used. We assume that the maximum channel delay is much smaller than the symbol duration so that the inter-symbol interference (ISI) is negligible. Considering single symbol transmission and reception, the filtered signal can be written as r (t) = d · wref (t) + n (t)
(2)
where d is the binary symbol, and n (t) is the filtered noise. The ith noise sample n (iTs ) is Gaussian distributed with zero mean and variance N0 B. After sampling and normalization with respect to the sampling frequency, we have (3) ri = d · si + ni , i = 0, 1, . . . , Ns − 1 √ √ Ts r (iTs ), si = Ts wref (iTs ) and ni = where ri = √ Ts n (iTs ); ni has zero mean and variance N0 /2. Then ri is quantized to a certain number of discrete values. Denote the number of quantization levels as L. A b-bit digital receiver has 2b quantization levels. We point out that in our paper the number of quantization levels is not necessarily a power of 2. In addition, we do not exclude cases where L is odd, say L = 3, which may offer a desired tradeoff between complexity and performance. Let ai denote the ith sample after quantization. Collect the samples as a vector a = [a0 , a1 , . . . , aNs −1 ].
(4)
The goal of the digital signal processing (DSP) part of the receiver is to make optimized decision for d based on a. Without the constraints on sampling frequency, quantization level and with perfect timing, we know that the optimal analog receiver is the well-known matched filter. However, a digital implementation of the matched filter requires a direct high sampling rate ADC which is difficult. Thus some tradeoffs must be made. In our paper, the tradeoff is the constraint on the number of quantization levels. Our goal is to study the design and performance of finite-resolution digital receivers. We will first examine the design issues in a 4-level receiver. The analysis will give us the insight into the general cases. III. T WO -B IT R ECEIVER A. Maximum-Likelihood digital processing As we can foresee, the 2-bit receiver is actually the extension of monobit receiver [18]. One difference between them is that the 2-bit receiver performance depends on the quantization threshold. We assume the same quantization threshold for the positive and negative regions. Let θ denote the quantization threshold. The quantizer output ai in a 2-bit, or 4-level,
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1.2
written as if if if if
ri > θ, 0 < ri ≤ θ, − θ < ri ≤ 0, ri ≤ −θ,
Empirical Gaussian 1
i = 0, 1, ...Ns − 1.
0.8
With equal prior probability P (d = ±1) = 1/2, the optimal decision rule is ML: P (a|d = +1) ml ˆ d = sgn log = sgn[Λ] (6) P (a|d = −1) where Λ is the log-likelihood ratio (LLR), and sgn denotes the sign function. Since the noise samples are i.i.d., Λ can be expressed as Λ=
N s −1 i=0
P (ai |d = +1) . log P (ai |d = −1)
Define A(1) = {i : ai = ±1} and A(2) = {i : ai = ±2}. The LLR can be expressed as (4-level,1) (4-level,2) a i ci + sgn[ai ]ci (9) Λ= i∈A(1)
i∈A(2)
where (4-level,1)
= log
˜ ˜ Q(−s i ) − Q(θ − si ) ˜ ˜ Q(−θ − si ) − Q(−s i)
(10)
(4-level,2)
= log
˜ − si ) Q(θ . ˜ + si ) Q(θ
(11)
ci and ci
(4-level,1)
(4-level,2)
and ci are two templates We remark that ci of 4-level receiver that are used to combine the quantized samples. Unlike the monobit receiver, the 4-level receiver needs two templates for the quantizer outputs. The LLR depends on the shape of the reference signal wref (t), quantization threshold θ, noise variance N0 /2, and the sampling period Ts . When Ns is small, an exact BER analysis may be feasible. For multipath channel and when Ns is large, direct analysis of LLR is too complicated and may not be able to disclose the inside relationship of these parameters. Thus we will develop an approximate BER analysis in the following. B. Gaussian approximation of the LLR We can rewrite LLR in (9) as Λ=
N s −1 i=0
f (ai )
0.6 E b/N0 =10dB
(12)
E b/N0 =15dB
0.4
0.2
0 −10
0
10
20
30
40
50
60
LLR
(7)
√ 2 Let Q(x) := x (1/ 2π)e−x /2 dx
be the Gaussian Q ˜ := Q(x/ N0 /2). We can write function and also define Q(x) P (ai |d) as: ⎧ ˜ − dsi ), Q(θ ai = +2, ⎪ ⎪ ⎪ ⎨Q(−ds ˜ ˜ ai = +1, i ) − Q(θ − dsi ), P (ai |d) = (8) ˜ ˜ ⎪ Q(−θ − dsi ) − Q(−dsi ), ai = −1, ⎪ ⎪ ⎩˜ Q(θ + dsi ), ai = −2. ∞
E b/N0 =0dB
(5)
CDF
receiver can be ⎧ +2, ⎪ ⎪ ⎪ ⎨ +1, ai = ⎪ −1, ⎪ ⎪ ⎩ −2,
Fig. 2. Comparison between CDF of LLR and Gaussian distribution of 4level receiver under
different Eb /N0 in multipath channel. The quantization threshold is θ = 0.5 N0 /2.
where f (ai ) =
(4-level,1)
, if ai ∈ A(1) a i ci . (4-level,2) sgn[ai ]ci , if ai ∈ A(2)
(13)
The LLR is a function of f (ai ), and f (ai )’s are discrete random variables. Because LLR is the summation of f (ai )’s, its probability mass function (PMF) is the convolution of f (ai )’s PMF. If f (ai )’s were identically distributed, then we would be able to resort to the central limit theorem (CLT) and claim that Λ is asymptotically Gaussian. However, we can make the following intuitive justification for the Gaussian assumption. In practice, TH sequence or DS code is usually employed for multiple access and/or for spreading the symbol energy into multiple pulses. Also, UWB channels usually have large delay spread. For these reasons, the number Ns of samples per symbol is usually large. When Ns is large, there will be relatively large number (say more than 10) of the samples sharing a similar signal value si (cf. (3)). The CLT can be applied to these samples first to yield an approximately Gaussian distributed random variable, and Λ in (12) is then a summation of such Gaussian like random variables, and hence approximately Gaussian distributed as well. In Fig. 2, we show the empirical cumulative distribution function (CDF) of Λ together with the Gaussian approximation when a UWB symbol is transmitted through a multipath channel under several SNRs, where Eb is the bit energy of the transmitted symbol. The ideal received pulse in a nondispersive channel is chosen to be the second derivative of a Gaussian pulse
2
2 t t · exp −2π (14) wrec (t) = 1 − 4π Tps Tps where Tps = 0.29 ns is a pulse shaping parameter such that the pulse duration Tp is about 0.7 ns [2]. We assume that DS
KE et al.: FINITE-RESOLUTION DIGITAL RECEIVER DESIGN FOR IMPULSE RADIO ULTRA-WIDEBAND COMMUNICATION
code is used, the received signal for one symbol is N c r(t) = δi wrec (t − iTc ) ∗ h(t) + n(t)
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(15)
i=1
where h(t) is the multipath channel, and ∗ stands for convolution. The channel h(t) is chosen to be the first realization of standard CM1 channel model of IEEE P802.15 group [21]. The time delay of this particular channel is around 96 ns. Nc is the length of DS code and Tc is the chip duration. δi ∈ {±1} is the ith entry of the DS code. Let T = Nc Tc + Td, where Td is the guarding period to avoid ISI. We choose the DS code to be an m-sequence with length Nc = 127. We choose Tc = 1 ns, Td = 73 ns and Ts = 0.1 ns such that Ns = 2000. We can see from Fig. 2 that despite the fact that f (ai )’s are not identically distributed, a Gaussian approximation is relatively accurate. Treating Λ as a Gaussian random variable, we can estimate 2 ), where d the BER performance as Q(dmin / 4σeq min = 2E(Λ|d = 1) denotes the equivalent Euclidean distance 2 = V ar(Λ|d = between the two constellation symbols, and σeq 1) denotes the equivalent noise variance. Let g(x) denote the probability density function (pdf) of a Gaussian random variable with zero mean and variance N0 /2. By introducing linear approximation and assuming that the number of samples is large enough, we can evaluate E(Λ|d = 1) and V ar(Λ|d = 1) (see Appendix) as (4)
E(Λ|d = 1) = (W1
(4) 2(W1
where Ebref
(4)
+ W2 )Ebref (4) W2 )Ebref
V ar(Λ|d = 1) = + Ns −1 2 ∞ 2 = i=0 si = −∞ wref (t)dt, and (4)
W1
(4)
W2
= =
(16) (17)
4(g(0) − g(θ))2 , ˜ ˜ Q(0) − Q(θ)
(18)
4g(θ) . ˜ Q(θ)
(19)
2
We can see that with the Gaussian assumption on Λ, the BER is only a function of the quantization threshold, the noise variance and the bit energy of reference signal; i.e., it depends on the pulse waveform only through Ebref . However, Ebref may not equal Eb , depending on the channel and the lowpass filter. As the LPF filters out the signal energy outside the cut-off frequency band, making Ebref and Eb different. However, as long as the sampling frequency is twice the cutoff frequency of LPF, our analysis does not change. Increasing sampling rate will not increase Ebref if sampling rate is larger than the Nyquist rate. Thus, although oversampling could be used in our proposed system, it does not bring performance improvement. Under the Gaussian assumption, Nyquist rate sampling is sufficient for the finite-resolution digital receiver. C. Optimization of the quantization threshold Since the BER performance depends on the quantization (4) (4) threshold θ through W1 and W2 , which are functions of θ only, we can optimize the BER performance by choosing θ
Fig. 3.
BER sensitivity with respect to threshold of 4-level receiver.
to be
⎛
(4-level) θopt
= arg min Q ⎝ θ
⎞ (4) (4) W1 (θ) + W2 (θ) ref ⎠ Eb (20) 2 (4)
(4)
where the dependence on θ of W1 and W2 has been (2) (4) included explicitly. Using the definitions of W1 and W2 in (18) and (19), we have performed a numerical search (4-level) for optimal threshold θopt , which is found to be θopt ≈
0.98 N0 /2. Another interesting question is that whether the BER performance is sensitive to the threshold or not. To answer this question, we first define a parameter η(θ) as η(θ) =
Pe (θ) − minθ (Pe (θ)) maxθ (Pe (θ)) − minθ (Pe (θ))
(21)
where Pe (θ) is the BER as a function of θ when Ebref /N0 is fixed and both the maximization and minimization are over the whole range of θ ∈ [0, ∞). Thus η(θ) satisfies 0 ≤ η(θ) ≤ 1 and it indicates how close Pe (θ) is to the minimum BER. The relationship among η(θ), Ebref /N0 and the quantization threshold θ is shown in Fig. 3. We can see that η(θ) does not change much when threshold is varying near the optimal value. Thus the receiver is robust to the choice of quantization threshold. IV. G ENERAL F INITE R ESOLUTION R ECEIVER A. Even number of quantization levels We can extend the previous discussion to any case when the number of quantization levels L is even. Assuming Llevel quantization thresholds θ : {θm−1 , m = 1, ..., L/2}, the analytical minimum BER for L-level receiver when L is even can be derived as ⎞ ⎛ L/2 (L) ref W E m b ⎠ m=1 Pe(L-level,min) (Ebref ) = min Q ⎝ (22) θ 2
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is given as
(L) Wm =
10
4[g(θm−1 ) − g(θm )]2 , ˜ m) ˜ m−1 ) − Q(θ Q(θ
m = 1, . . . , L/2
where we set θ0 = 0 and θL/2 = ∞ to make this equation self(L) contained. The factors Wm ’s indicate how much contribution each corresponding template provides to the BER performance. Define A(m) = {i : ai = ±m}, m = 1, . . . , L/2. The decision variable, LLR, can be written as Λ=
L/2
(L-level,m) sgn[ai ]ci
−1
(23) 10
BER
(L)
and Wm
10
(24)
10
2[g(θm−1 ) − g(θm )] s, = ˜ m−1 ) − Q(θ ˜ m) i Q(θ m = 1, . . . , L/2, i = 0, 1, . . . , Ns − 1. (25)
10
−2
−3
2−level minimum BER 3−level minimum BER 4−level minimum BER 6−level minimum BER 8−level minimum BER Matched filter BER
−4
m=1 i∈A(m)
where the optimal receiver templates are (L-level,m) ci
B. Odd number of quantization levels Now we consider the cases where the number L of quantization levels is odd. We take 3-level quantization as an example. It will be a good tradeoff method between the BER performance and implementation issue. The 2-level receiver can be viewed as a special case of 4-level receiver where θ (4) (4) is chosen to be zero such that W1 = 0. However, W2 is not maximized when θ = 0. The intuition is that at some samples the sampled pulse amplitude of si is relatively small and hence these samples are easily corrupted by noise. Thus, if we can ignore these bad samples, we can achieve a better BER performance comparing to 2-level receiver while we still use only one template. The quantization scheme of a 3-level receiver is as follows ⎧ ⎪ ⎨+1, if ri > θ, ai = 0, if − θ < ri ≤ θ, i = 0, 1, ...Ns − 1. (26) ⎪ ⎩ −1, if ri ≤ −θ, The 3-level quantizer can be viewed as a 4-level quantizer followed by an “eraser” that sets the ±1 samples to 0. We can see that the LLR for 3-level receiver is related to 4-level case. Because P (ai = 0|d = +1) = P (ai = 0|d = −1), the LLR does not depend on the ai = 0 samples. The performance can be derived from 4-level receiver by dropping the contribution of the first template. We have (4)
E(Λ|d = 1) = W2 Ebref V ar(Λ|d = 1) = and
(4) 2W2 Ebref
⎛ (3-level) θopt (4)
= arg min Q ⎝ θ
⎞ (4) W2 (θ) ref ⎠ Eb . 2
−5
4
5
6
7
8
9
E ref /N b
Fig. 4.
0
Analytical BER of different receivers.
L-level receiver when L is odd can be derived in a similar way, given as ⎞ ⎛ L/2 (L) ref W E m b ⎠ m=1 Pe(L-level,min) (Ebref ) = min Q ⎝ (30) θ 2 where θ : {θm−1 , m = 1, ..., L/2} and (L+1) Wm , 1 < m ≤ L/2 (L) Wm = . 0, m=1
(31)
Similarly, define A(m) = {i : ai = ±(m − 1)}, m = 1, . . . , L/2. The decision variable, LLR, can be written as L/2
Λ=
(L-level,m)
sgn[ai ]ci
.
(32)
m=1 i∈A(m)
The corresponding templates are ((L+1)-level,m) c , (L-level,m) ci = i 0,
1 < m ≤ L/2 , m=1
i = 0, . . . , Ns − 1.
(33)
The optimal thresholds are different from the even number quantization cases though.
(27)
C. Incremental gain of additional quantization levels
(28)
The analytical BER curves of different receivers are shown in Fig. 4. We can see that the performance of 3-level receiver is between the 2-level and 4-level receivers. By dropping the bad samples, the receiver can acquire 1 dB gain from the monobit receiver. The cost is merely a high rate comparator and the calculation requirements are the same as monobit receiver, because the 3-level receiver only needs one optimal template. BER curves of other odd level receivers are not shown because the improvements are not as significant as that of 3-level receiver. In order to have a better view of how the quantization levels affect the BER performance, we list the optimal quantization
(29)
Although W2 and the associated template of 3-level receiver have the same expressions as those of the 4-level receiver, the optimal quantization threshold is different.
The optimal (3-level) ≈ 0.61 N0 /2 through threshold is found to be θopt numerical search and BER is also insensitive to the threshold around the optimal value. The analytical minimum BER for
KE et al.: FINITE-RESOLUTION DIGITAL RECEIVER DESIGN FOR IMPULSE RADIO ULTRA-WIDEBAND COMMUNICATION
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TABLE I P ERFORMANCE GAIN OF FINITE - RESOLUTION RECEIVER . L
Δ/( N0 /2) L/2 (L) m=1 Wm incremental level gain (dB) number of bits incremental bit gain (dB)
2 0 1.2732 – 1 –
3 0.61 1.6196 1.0451 – –
4 0.98 1.7650 0.3735 2 1.4186
L/2 (L) step Δ and the factor m=1 Wm , which affects the BER performance in (22) and (30), as well as the dB gain that can be obtained by additional quantization levels or bits in Table. I. We use Δ = θm − θm−1 , m = 1, . . . , L2 − 1; i.e., uniform quantization level spacing. We define the incremental level gain as the gain in SNR in dB offered by increasing the number of quantization levels from the previous listed value in Table I to the current value. Similarly, incremental bit gain is defined as the gain in SNR obtainable by increasing the number of quantization bits from the previous integer value to the current one. We can see that the incremental gain is very small (less than 0.1 dB) when L ≥ 8, which means that the high-resolution ADC (more than 3 bits) is not necessary. We can use the 8-level receiver to achieve a desired performance, while the BER is slightly larger than the theoretical matchedfilter performance. Similar observations were made in [19], based on quantization noise analysis. D. BER Analysis in Fading Channel Previous discussion only focuses on one transmission block, where the channel is quasi-static. But fading exists so that in some time intervals the fading is severe and the BER becomes worse. The BER under fading channel can be averaged over the pdf of Ebref as ∞ Pe = Pe(L-level,min) (x)p(x)dx (34) 0
(L-level,min) Pe (x)
where is the analytical BER given in (22) or (30), and p(x) is the pdf of Ebref . The pdf can be obtained by Monte Carlo simulations over the standard channel model [21]. V. T RAINING BASED T EMPLATE E STIMATION We have analyzed the BER performance of our proposed finite-resolution receiver, assuming that the perfect knowledge of reference signal is available. In practice, the reference signal needs to be estimated, from the finite-resolution samples. In this section, we propose a training based reference signal estimation method using the ML criterion. The optimal templates of our proposed receiver rely on the knowledge of the reference signal si . Thus, in order to obtain the values of the templates, we first need to estimate si . In the following, we present the ML template estimation solution using the 4-level receiver as an example. In the training period, we transmit an all-one training (m) sequence of M symbols. Let ai denote the ith sample of (k) mth training symbol and Ni denote the cardinality of the
6 0.71 1.8824 0.2796 – –
8 0.56 1.9284 0.1047 3 0.3843
10 0.47 1.9513 0.0513 – –
12 0.41 1.9645 0.0292 – –
14 0.36 1.9728 0.0184 – –
16 0.32 1.9784 0.0124 4 0.0925
(m)
set {m|ai = k, 1 ≤ m ≤ M }, where k ∈ {±1, ±2} and 0 ≤ i ≤ Ns − 1. Recall that Ns is the length of the template. For each sample we have an M × 1 training-based data vector (1) (2) (M) αi = [ai , ai , . . . , ai ]. The log-likelihood function of si is V (αi |si ) := log P (αi |si ) (k) Ni log P (ai = k|si , d = 1) =
(35) (36)
k∈{±1,±2}
i = 0, . . . , Ns − 1. (−2)
Based on the collected sample counts N i = [Ni (+1) (+2) Ni , Ni ], the ML estimate of si is sˆi = arg max V (αi |si ), si
i = 0, 1, . . . , Ns − 1.
(−1)
, Ni
,
(37)
In practice, the ML estimate of si can be obtained using a lookup table. The table is indexed by si , and each row of the table stores 4 entries: {log P (ai = k|si , d = 1), k = ±1, ±2}. For each N i vector, we can perform the inner product of it with each row. The row that gives the maximum inner product corresponds to the ML estimate of si . Assuming the function V (α|si ) is concave, a bisection search of the table will be sufficient. Corresponding to each row, it is not (4-level,1) necessary to store the value si . Instead, we can store ci (4-level,2) and ci which are the needed weights. To summarize, the weighting template can be estimated as follows: 1) The transmitter sends an all-one training sequence to the receiver. (k) 2) The receiver counts Ni for all Ns samples. (k) 3) The receiver uses the Ni to find the ML estimate of si according to (37), and retrieve the corresponding (4-level,1) (4-level,2) weights (such as ci and ci ). This step is performed for i = 0, 1, . . . , Ns − 1. The ML solution of si is asymptotically optimal when the length of training sequence M is large in the sense that it can achieve the Cramer-Rao lower bound asymptotically [22]. VI. S IMULATIONS AND D ISCUSSION In this section, we provide simulation results of proposed finite-resolution receiver design and BER analysis. In our simulations, we omitted TH sequence or DS code for simplicity. The conclusions will not change if they are included. The received pulse is the same as (14) with Tp = 0.7 ns. The sampling frequency is chosen to be 8/Tp in the following simulation. We will use the first realization of standard CM1 channel model [21] to test our analytical optimal quantization
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Fig. 5. Comparison between the analytical BER curve and numerical results of 4-level (2-bit) receiver under multipath channel.
Fig. 6. Comparison between the analytical BER curve and numerical results of 3-level receiver under multipath channel.
thresholds, where the channel is normalized such that its Euclidean norm is 1. Fig. 5 and Fig. 6 show the comparison of our analytical BER curves and numerical results for 4-level and 3-level receivers, respectively. We can see that the analytical curves are very good estimations of the minimum BER that the receivers can offer by choosing appropriate quantization thresholds. Fig. 7 and Fig. 8 illustrate the BER sensitivity with respect to quantization threshold of 4-level and 3-level receivers, respectively. We can see clearly that the optimal thresholds obtained by simulations are close to the analytical optimal thresholds. We can also see that η does not change much around the optimal thresholds we derived, suggesting that the receivers are robust to the choice of quantization thresholds. We also evaluated the BER performance under fading channel; see Fig. 9 and Fig. 10. The channel is the standard CM1 channel model [21] with 100 realizations. The analytical and numerical simulation results of 4-level and 3-
Fig. 7. Simulation results of BER sensitivity with respect to threshold of 4-level receiver under multipath channel.
Fig. 8. Simulation results of BER sensitivity with respect to threshold of 3-level receiver under multipath channel.
level receivers are shown as functions of transmitted Eb /N0 . As the optimal quantization thresholds have been verified in the previous simulation, we used the optimal values in these simulations. We can see that the analytical performance is in good agreement with simulated performance when full channel state information (CSI) is available. The effect of the training sequence length was also investigated using the lookup table technique we mentioned in Section V. Our lookup table quantizes si in the range [−5, 5] with a step size 0.005. It can be seen from Fig. 9 and Fig. 10 that the performance improves as the length of training sequence increases. The performance of 200-bit and 500-bit training is about 1 dB and 0.5 dB away from the full CSI performance, respectively, when Eb /N0 is around 10 dB. Using a 1000-bit training sequence, the performance degradation is reduced to about 0.3 dB. When the number of quantization levels is two, our proposed optimal receiver performs better but only slightly better than full-resolution template filtering method in [18]. One
KE et al.: FINITE-RESOLUTION DIGITAL RECEIVER DESIGN FOR IMPULSE RADIO ULTRA-WIDEBAND COMMUNICATION
10
BER
10
10
10
10
−1
−2
−3
Analytical BER Full CSI simulation 1000bit training 500bit training 200bit training Matched filter BER
−4
Fig. 11.
The illustration of linear approximation.
−5
4
6
8
10
12
14
E /N b
0
Fig. 9. Comparison between the analytical and numerical results in fading channel as well as the effect of training sequence length on BER performances for 3-level receiver. 10
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mance should be better than that of the receiver in [19], with a price of higher receiver complexity when the quantization resolution is high. When the number of quantization levels is large, the two should be close, modulo a 3 dB difference in SNR due to non-coherent TR modulation. VII. C ONCLUSIONS
BER
10
10
10
10
−2
−3
Analytical BER Full CSI simulation 1000bit training 500bit training 200bit training Matched filter BER
−4
−5
4
6
8
10
12
14
E /N b
0
Fig. 10. Comparison between the analytical and numerical results in fading channel as well as the effect of training sequence length on BER performances for 4-level receiver.
main contribution of this paper is however, to quantify the theoretically achievable performance of finite-resolution digital receivers. Another contribution compared to [18] is that we considered multiple quantization levels in this paper, and derived the optimal detector structure. We did not consider the possibility of sigma-delta modulation in this paper. In general the benefit of sigma-delta modulation is not obvious when Nyquist-rate sampling is used. We note that the performance analysis in [19] based on modeling the quantization noise is not as accurate as our performance result when the number of quantization levels is small. For 3-level receiver, comparing the curve in Fig. 9 with 200 training symbols and the corresponding curve in [19, Fig. 5, 1 bit], the receiver in this paper has about 4 dB gain at BER of 10−2 , whereas the complexity is comparable. Note that due to the non-coherent TR modulation, an inherent 3dB performance loss is incurred. In general, our receiver perfor-
We investigated the design and performance of finiteresolution digital receiver for impulse-radio ultra-wideband communication system over multipath dispersive channels. We evaluated the effect of quantization threshold and presented analytical BER expressions based on a Gaussian assumption. The optimal quantization thresholds for 3- and 4-level receivers have been obtained. Our results show that 3-level receiver can acquire 1 dB gain over monobit receiver and we can use 8-level receiver to achieve a performance 0.2 dB within that of the matched filter receiver. Higher resolution ADCs are not recommended as the additional BER gains are too small to justify the increased implementation complexity. We proposed a template estimation scheme based on the maximum-likelihood criterion. We only considered single user transmission with no interference in this paper. Possible future work includes evaluating receiver performance in the multiuser as well as narrow-band interference cases. VIII. A PPENDIX In this appendix, we derive the closed-form expression of the BER for 4-level receiver assuming Gaussian distribution of the LLR. Divide the Gaussian pdf into several regions {S (k) : k = 1, ..., 5}, as shown in Fig. 11. Denote P (S (k) ) as the probability that ni = ri − si falls in to the region of S (k) . We can rewrite (8) as P (ai |d = 1) = ⎧ P (S (3) ) + P (S (4) ) + P (S (5) ), ⎪ ⎪ ⎪ ⎨2P (S (1) ) + P (S (2) ), ⎪ P (S (2) ) + P (S (3) ) + P (S (4) ), ⎪ ⎪ ⎩ P (S (5) ),
if if if if
ai ai ai ai
= +2, = +1, = −1, = −2.
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By definition, the expectation and variance of f (ai ) when “1” is transmitted are E(f (ai )|d = 1) = (4-level,1) [P (ai ci (4-level,2) +ci [P (ai
= 1|d = 1) − P (ai = −1|d = 1)] = 2|d = 1) − P (ai = −2|d = 1)]
and
and V ar(Λ|d = 1) =
N s −1
V ar(f (ai )|d = 1)
i=0 (4)
(4)
N s −1
(4) (4) s2i −(W1 +W2 )2
= 2(W1 + W2 )
N s −1
s4i
i=0 (4) (4) (4) (4) 4 = 2(W1 + W2 )Ebref − (W1 +W2 )2 wref (t)dt Ts i=0
V ar(f (ai )|d = 1) (4-level,1) 2
=[ci
] [P (ai = 1|d = 1)+P (ai = −1|d = 1)]
+[ci
] [P (ai = 2|d = 1)+P (ai = −2|d = 1)]
(4-level,2) 2
−E 2 [f (ai )|d = 1] We next use a linear approximation on the weighting signals and the probability terms. First, we look at the probability terms. When si is small, P (S (3) ) ≈ P (S (4) ). We have P (ai = 1|d = 1) − P (ai = −1|d = 1) ≈ 2[P (S
(1)
) − P (S
(3)
)] = 2[g(0) − g(θ)]si
P (ai = 2|d = 1) − P (ai = −2|d = 1) ≈ 2P (s(4) ) = 2si g(θ) Using the approximation loge (1 + x) ≈ x and the approximation discussed previously, we have (4-level,1)
P (S (2) ) + 2P (S (1) ) P (S (2) ) + 2P (S (3) ) 2(P (S (1) ) − P (S (3) )) ≈ P (S (2) ) + 2P (S (3) ) 2(g(0) − g(θ)) si ≈ ˜ ˜ Q(0) − Q(θ)
≈ log
and (4-level,2)
ci
P (S (3) ) + P (S (4) ) + P (S (5) ) P (S (5) ) ˜ ˜ Q(θ + si ) − Q(θ − si ) ≈ ˜ + si ) Q(θ 2g(θ) si . ≈ ˜ Q(θ)
= log
Thus, we have E(Λ|d = 1) = =
N s −1 i=0 N s −1 i=0
E(f (ai )|d = 1) N s −1 4(f (0) − f (θ))2 2 4f (θ)2 2 si + si ˜ ˜ ˜ Q(0) − Q(θ) Q(θ) i=0 (4)
(4)
W1
=
(4) (W1 (4)
= (W1
N s −1
+
(4) W2 )
+
i=0 (4) W2 )Ebref
W2
si
(4)
(as Ts 1).
Finally, we want to comment that if si is large, then our linear approximation is not accurate. However, the effect of such inaccuracy should be small in fading channels because the overall performance is dominated by “deep fade” events and hence small si cases. R EFERENCES
and
ci
(4)
= 2(W1 + W2 )Ebref
[1] R. Scholtz, “Multiple access with time-hopping impulse modulation,” in Proc. IEEE MILCOM, vol. 2, pp. 447–450, 1993. [2] M. Z. Win and R. A. Scholtz, “Ultra-wide bandwidth time-hopping spread-spectrum impulse radio for wireless multiple-access communications,” IEEE Trans. Commun., vol. 48, pp. 679–689, Apr. 2000. [3] L. Yang and G. B. Giannakis, “Ultra-wideband communications: an idea whose time has come,” IEEE Signal Processing Mag., vol. 21, no. 6, pp. 26–54, June 2004. [4] C. J. Le Martret and G. B. Giannakis, “All-digital impulse radio with multiuser detection for wireless cellular systems,” IEEE Trans. Commun., vol. 50, no. 9, pp. 1440–1450, Sept. 2002. [5] Z. Xu, “Trends in ultrawideband transceiver design,” in Ultra-Wideband Wireless Communications and Networks, S. Shen, M. Guizani, R. C. Qiu, and T. Le-Ngoc, Eds. Chapter 7, John Wiley & Sons, 2006. [6] L. Feng and W. Namgoong, “An oversampled channelized UWB receiver with transmitted reference modulation,” IEEE Trans. Wireless Commun., vol. 5, no. 6, pp. 1497–1505, June 2006. [7] W. M. Lovelace and J. K. Townsend, “The effects of timing jitter and tracking on the performance of impulse radio,” IEEE J. Select. Areas Commun., vol. 20, no. 9, pp. 1646–1651, Sept. 2002. [8] S. Hoyos and B. Sadler, “Ultra-wideband analog-to-digital conversion via signal expansion,” IEEE Trans. Veh. Technol., vol. 54, no. 5, pp. 1609–1622, May 2005. [9] R. Hoctor and H. Tomlinson, “Delay-hopped transmitted-reference RF communications,” in Proc. IEEE Conference on Ultra Wideband Systems and Technologies, 2002, pp. 265–269, 2002. [10] C. Rushforth, “Transmitted-reference techniques for random or unknown channels,” IEEE Trans. Inform. Theory, vol. 10, no. 1, pp. 39–42, Jan. 1964. [11] J. Choi and W. Stark, “Performance of autocorrelation receivers for ultra-wideband communications with PPM in multipath channels,” in Proc. IEEE Conference on Ultra Wideband Systems and Technologies, 2002, pp. 213–217, 2002. [12] M. Ho, V. Somayazulu, J. Foerster, and S. Roy, “A differential detector for an ultra-wideband communications system,” in Proc. IEEE Vehicular Tech. Conf., vol. 4, pp. 1896–1900, 2002. [13] Y.-L. Chao and R. Scholtz, “Optimal and suboptimal receivers for ultrawideband transmitted reference systems,” in Proc. IEEE GLOBECOM, vol. 2, pp. 759–763, 2003. [14] Y.-L. Chao and R. Scholtz, “Ultra-wideband transmitted reference systems,” IEEE Trans. Veh. Technol., vol. 54, no. 5, pp. 1556–1569, May 2005. [15] F. Tufvesson, S. Gezici, and A. Molisch, “Ultra-wideband communications using hybrid matched filter correlation receivers,” IEEE Trans. Wireless Commun., vol. 5, no. 11, pp. 3119–3129, Nov. 2006. [16] J. Choi and W. Stark, “Performance of ultra-wideband communications with suboptimal receivers in multipath channels,” IEEE J. Select. Areas Commun., vol. 20, no. 9, pp. 1754–1766, Sept. 2002. [17] V. Lottici, A. D’Andrea, and U. Mengali, “Channel estimation for ultrawideband communications,” IEEE J. Select. Areas Commun., vol. 20, no. 9, pp. 1638–1645, Sept. 2002.
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[18] S. Hoyos, B. M. Sadler, and G. R. Arce, “Monobit digital receivers for ultrawideband communications,” IEEE Trans. Wireless Commun., vol. 4, no. 4, pp. 1337–1344, Apr. 2005. [19] J. Tang, Z. Xu, and B. M. Sadler, “Performance analysis of b-bit digital receivers for TR-UWB systems with inter-pulse interference,” IEEE Trans. Wireless Commun., vol. 6, no. 2, pp. 494–505, Feb. 2007. [20] S. Franz and U. Mitra, “Quantized UWB transmitted reference systems,” IEEE Trans. Wireless Commun., vol. 6, no. 7, pp. 2540–2550, July 2007. [21] J. Foerster, “Channel modeling sub-committee report final,” IEEE Working Group for Wireless Personal Area Networks (WPANs) P802.1502/490r1-SG3a, Feb. 2003. [22] S. M. Kay, Fundamentals of Statistical Signal Processing: Detection Theory. Prentice-Hall, Inc., 1998.
Lei Ke received the B.E. degree and the M.Sc. degree in Electronic Engineering and Information Science from the University of Science and Technology of China (USTC), Hefei, China, in 2003 and 2006, respectively. He is currently pursuing his Ph.D. with the Department of Electrical and Computer Engineering at Iowa State University. His general interests include wireless communication and signal processing.
Huarui Yin received his Bachelor’s Degree in 1996, and Ph.D. degree in 2006, both in Electronic Engineering and Information Science from University of Science and Technology of China, at Hefei, Anhui. Since 1999, he has been with the department as a lecturer. His research interests include digital signal processing, ultrawide bandwidth communication and software defined radio.
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Weilin Gong received his B.S. degree in applied physics, and Ph.D. degree in Electronic Engineering and Information Science from University of Science and Technology of China, Hefei, China, in 2003 and 2008 respectively. Currently he is a post-doc with the University of Science and Technology of China. His research focuses on communication theory and signal processing for ultra wideband communication.
Zhengdao Wang received his B.S. degree in Electronic Engineering and Information Science from the University of Science and Technology of China (USTC), 1996, the M.Sc. degree in Electrical and Computer Engineering from the University of Virginia, 1999, and Ph.D. in Electrical and Computer Engineering from the University of Minnesota, 2002. He is now with the Department of Electrical and Computer Engineering at the Iowa State University. His interests are in the areas of signal processing, communications, and information theory. He served as an associate editor for IEEE T RANS ACTIONS ON V EHICULAR T ECHNOLOGY from April 2004 to April 2006, and has been an Associate Editor for IEEE S IGNAL P ROCESSING L ETTERS since August 2005. He was a co-recipient of the IEEE S IGNAL P ROCESSING M AGAZINE Best Paper Award in 2003 and the IEEE Communications Society Marconi Paper Prize Award in 2004.
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 12, DECEMBER 2008
Finite-Resolution Digital Receiver Design for Impulse Radio Ultra-Wideband Communication Lei Ke, Student Member, IEEE, Huarui Yin, Member, IEEE, Weilin Gong, Member, IEEE, and Zhengdao Wang, Senior Member, IEEE
Abstract—Receiver design for impulse radio based ultrawideband (UWB) communication is a challenge. High sampling rate high resolution digital receiver is usually difficult to implement. Some tradeoffs can be made on the digital receiver, such as limiting amplitude resolution to only one bit, which results in a previously considered monobit receiver. In this paper, we consider the design of finite-resolution digital UWB receivers. We derive the optimal post-quantization processing, and analyze the achievable bit-error rate performance using an approximation of the log-likelihood ratio. Optimal thresholds for 4- and 3-level quantization are obtained. Training-based receiver template estimation is presented. Our work discloses the incremental gain that additional quantization levels offer and the results provide useful guidelines for designing impulse radio UWB digital receivers. Index Terms—Ultra-wideband, finite resolution, impulse radio, digital receiver, bit error rate, quantization threshold.
I. I NTRODUCTION
U
LTRA-WIDEBAND (UWB) communication has drawn attention recently from both academia and industry [1]– [5]. Receiver design for UWB communications is a challenge due to the wide bandwidth nature of the received signal. One popular method of generating UWB signals is impulse radio (IR) [1], [2]. IR-UWB receivers can be divided into two categories: coherent and noncoherent receivers. The receiver discussed in [2] is a standard coherent receiver. To maximize the signal to noise ratio (SNR) at receiver, an matched filter is used, which is the optimal receiver under additive white Gaussian noise (AWGN) channel. Matched filter can have problems with distortion, timing mismatch [6], and when implemented using analog delay lines, difficulty of adjusting the delays. The performance can also be sensitive to timing jitter [7]. Another way of implementing the receiver is to perform frequency domain processing [6], [8], whose performance depends on the number of projection functions or filter bank branches.
Manuscript received November 2, 2007; revised March 15, 2008 and May 31, 2008; accepted September 16, 2008. The associate editor coordinating the review of this paper and approving it for publication was N. Arumugam. The work in this paper was supported in part by the National Science Foundation under Grant No. 0431092. L. Ke and Z. Wang are with the Department of Electrical and Computer Engineering, Iowa State University, Ames, IA 50011 USA (e-mail: {kelei, zhengdao}@iastate.edu). H. Yin and W. Gong are with the Department of Electronic Engineering and Information Science, University of Science and Technology of China, Hefei, Anhui 230027, P.R. China (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/T-WC.2008.071223
Coherent UWB receivers require channel estimation, which is usually difficult to perform due to the wideband nature of the channel. To alleviate the problem, transmitted-reference (TR) UWB receiver was proposed in [9]. The TR technique dates back to 1960’s [10]. Other similar receivers such as autocorrelation receiver [11], differential receiver [12] have also been proposed. Detailed performance analysis and comparison can be found in [13] and [14]. To alleviate the 3-dB non-coherent performance loss, a hybrid matched filter correlation receiver has been proposed in [15], which performs a despreading operation first before multiplication operation. The issue of tradeoff between performance and complexity is present in both coherent and noncoherent receivers. Such tradeoff was considered in [16] for rake and autocorrelation receivers, and TR based systems. Both coherent and noncoherent receivers can be implemented digitally. To implement the coherent matched filter digitally, high sampling rate is usually required [17]. For example, assuming a pulse duration of one nano-second (ns), an oversampling factor of 10 corresponds to a sampling rate of 10 giga samples per second (Gsps), which can be difficult to implement at high resolution of, say, 8 bits per sample. One-bit analog-to-digital converter (ADC) was used to sample the received signal in [18], and the resulting receiver was termed monobit receiver. Such a one-bit ADC can be realized by fast comparator. It is expected that tens of Gsps sampling rate can be achieved. In addition, single-bit ADCs are also attractive because of their lower power consumption and lower cost compared to multi-bit ADCs. Bit error rate (BER) expressions for matched filter and TR receivers are derived in [18]. And it is shown that using a one-bit analog sigma-delta modulator (SDM) with significant oversampling, BER can be made close to that of the matched filter in AWGN channel. A b-bit receiver for TR-UWB system is discussed in [19]. The relationship between ADC resolution and receiver performance with inter-pulse interference is examined based on the analysis of the statistical distribution of quantization noise. Similar to [6], it was demonstrated that a 3-bit receiver can already achieve good performance. The proposed receiver therein has an analog matched-filter front end, which can be difficult to implement. The quantization range is optimized by numerical method. Similar work can be found in [20], where an ad hoc channel estimator is proposed. In this paper, we focus on the issue of finite-resolution ADC design and associated receiver performance in terms
c 2008 IEEE 1536-1276/08$25.00
KE et al.: FINITE-RESOLUTION DIGITAL RECEIVER DESIGN FOR IMPULSE RADIO ULTRA-WIDEBAND COMMUNICATION
of quantization resolution and threshold for time-domain IRUWB coherent receivers. Our contributions are summarized as follows: 1) Optimal finite-resolution digital detector based on the maximum-likelihood (ML) rule is derived. 2) Accurate BER expressions through a Gaussian approximation are obtained. 3) Optimal quantization thresholds based on the analytical BER expressions are obtained through numerical search. 4) A training based template estimation method using the ML criterion is proposed. Compared to the quantization-noise based performance analysis in [19], our approach in this paper is to first derive the optimal detector architecture, and then quantify the performance by approximating the optimal combining coefficients. The optimal architecture turns out to be a weighted sum of the quantized bits, rather than processing all the bits in the samples jointly as performed in [19]. The rest of the paper is organized as follows: The system model is presented in Section II. A two-bit quantization based receiver is designed and its performance is analyzed in Section III. The results are generalized in Section IV, where the receiver performance under fading channel is also briefly discussed. ML estimation of the detection templates is proposed in Section V. Simulation results are presented in Section VI, and Section VII concludes the paper. II. S YSTEM M ODEL Consider a single-user IR-UWB system using binary pulse amplitude modulation. Let w(t) denote the pulse shape at the receiver, which incorporates the transmitter pulse the antenna differentiation effects, and the dispersive channel. The pulse w(t) also incorporates the possible time-hopping (TH) sequence or direct-sequence (DS) spreading code at the transmitter if they are used. We model the channel as dispersive and its impulse response is included in w(t). When the channel is time varying, we assume it to be quasi-static so that our results apply for each channel coherent interval. We can write the received signal as r(t) =
∞
dk w(t − kT ) + n(t)
(1)
k=0
where dk is the kth transmitted symbol, which is equal to ±1 with equal probability 1/2; T is the symbol duration; n(t) is AWGN with power spectral density N0 /2. The received signal is filtered by a receiver pre-sampling ideal low-pass filter (LPF), and then sampled with sampling period Ts ; see Fig. 1. We assume that the filter has bandwidth B = 1/(2Ts ) and unit response inside the passband frequency. The filtering bandwidth usually needs to be large enough to contain most of the signal energy, and yet not too large so that unnecessary noise can be excluded. The number of digital samples that can be obtained within one symbol is Ns = T /Ts , the smallest integer larger than or equal to T /Ts . The filtered pulse shape is denoted as wref (t), which is the convolution of w(t) and the filter’s impulse response. We first assume the availability of wref (t) in deriving our main results. Template estimation will be discussed in Section V. We call
r(t)
LPF
iTs
Quantize
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DSP
dˆ
Fig. 1. Sampling Structure: The received signal is low-pass filtered, sampled, quantized, and processed by the digital signal processor.
wref (t) the reference signal, which is the signal to be matched with if an analog matched filter is used. We assume that the maximum channel delay is much smaller than the symbol duration so that the inter-symbol interference (ISI) is negligible. Considering single symbol transmission and reception, the filtered signal can be written as r (t) = d · wref (t) + n (t)
(2)
where d is the binary symbol, and n (t) is the filtered noise. The ith noise sample n (iTs ) is Gaussian distributed with zero mean and variance N0 B. After sampling and normalization with respect to the sampling frequency, we have (3) ri = d · si + ni , i = 0, 1, . . . , Ns − 1 √ √ Ts r (iTs ), si = Ts wref (iTs ) and ni = where ri = √ Ts n (iTs ); ni has zero mean and variance N0 /2. Then ri is quantized to a certain number of discrete values. Denote the number of quantization levels as L. A b-bit digital receiver has 2b quantization levels. We point out that in our paper the number of quantization levels is not necessarily a power of 2. In addition, we do not exclude cases where L is odd, say L = 3, which may offer a desired tradeoff between complexity and performance. Let ai denote the ith sample after quantization. Collect the samples as a vector a = [a0 , a1 , . . . , aNs −1 ].
(4)
The goal of the digital signal processing (DSP) part of the receiver is to make optimized decision for d based on a. Without the constraints on sampling frequency, quantization level and with perfect timing, we know that the optimal analog receiver is the well-known matched filter. However, a digital implementation of the matched filter requires a direct high sampling rate ADC which is difficult. Thus some tradeoffs must be made. In our paper, the tradeoff is the constraint on the number of quantization levels. Our goal is to study the design and performance of finite-resolution digital receivers. We will first examine the design issues in a 4-level receiver. The analysis will give us the insight into the general cases. III. T WO -B IT R ECEIVER A. Maximum-Likelihood digital processing As we can foresee, the 2-bit receiver is actually the extension of monobit receiver [18]. One difference between them is that the 2-bit receiver performance depends on the quantization threshold. We assume the same quantization threshold for the positive and negative regions. Let θ denote the quantization threshold. The quantizer output ai in a 2-bit, or 4-level,
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1.2
written as if if if if
ri > θ, 0 < ri ≤ θ, − θ < ri ≤ 0, ri ≤ −θ,
Empirical Gaussian 1
i = 0, 1, ...Ns − 1.
0.8
With equal prior probability P (d = ±1) = 1/2, the optimal decision rule is ML: P (a|d = +1) ml ˆ d = sgn log = sgn[Λ] (6) P (a|d = −1) where Λ is the log-likelihood ratio (LLR), and sgn denotes the sign function. Since the noise samples are i.i.d., Λ can be expressed as Λ=
N s −1 i=0
P (ai |d = +1) . log P (ai |d = −1)
Define A(1) = {i : ai = ±1} and A(2) = {i : ai = ±2}. The LLR can be expressed as (4-level,1) (4-level,2) a i ci + sgn[ai ]ci (9) Λ= i∈A(1)
i∈A(2)
where (4-level,1)
= log
˜ ˜ Q(−s i ) − Q(θ − si ) ˜ ˜ Q(−θ − si ) − Q(−s i)
(10)
(4-level,2)
= log
˜ − si ) Q(θ . ˜ + si ) Q(θ
(11)
ci and ci
(4-level,1)
(4-level,2)
and ci are two templates We remark that ci of 4-level receiver that are used to combine the quantized samples. Unlike the monobit receiver, the 4-level receiver needs two templates for the quantizer outputs. The LLR depends on the shape of the reference signal wref (t), quantization threshold θ, noise variance N0 /2, and the sampling period Ts . When Ns is small, an exact BER analysis may be feasible. For multipath channel and when Ns is large, direct analysis of LLR is too complicated and may not be able to disclose the inside relationship of these parameters. Thus we will develop an approximate BER analysis in the following. B. Gaussian approximation of the LLR We can rewrite LLR in (9) as Λ=
N s −1 i=0
f (ai )
0.6 E b/N0 =10dB
(12)
E b/N0 =15dB
0.4
0.2
0 −10
0
10
20
30
40
50
60
LLR
(7)
√ 2 Let Q(x) := x (1/ 2π)e−x /2 dx
be the Gaussian Q ˜ := Q(x/ N0 /2). We can write function and also define Q(x) P (ai |d) as: ⎧ ˜ − dsi ), Q(θ ai = +2, ⎪ ⎪ ⎪ ⎨Q(−ds ˜ ˜ ai = +1, i ) − Q(θ − dsi ), P (ai |d) = (8) ˜ ˜ ⎪ Q(−θ − dsi ) − Q(−dsi ), ai = −1, ⎪ ⎪ ⎩˜ Q(θ + dsi ), ai = −2. ∞
E b/N0 =0dB
(5)
CDF
receiver can be ⎧ +2, ⎪ ⎪ ⎪ ⎨ +1, ai = ⎪ −1, ⎪ ⎪ ⎩ −2,
Fig. 2. Comparison between CDF of LLR and Gaussian distribution of 4level receiver under
different Eb /N0 in multipath channel. The quantization threshold is θ = 0.5 N0 /2.
where f (ai ) =
(4-level,1)
, if ai ∈ A(1) a i ci . (4-level,2) sgn[ai ]ci , if ai ∈ A(2)
(13)
The LLR is a function of f (ai ), and f (ai )’s are discrete random variables. Because LLR is the summation of f (ai )’s, its probability mass function (PMF) is the convolution of f (ai )’s PMF. If f (ai )’s were identically distributed, then we would be able to resort to the central limit theorem (CLT) and claim that Λ is asymptotically Gaussian. However, we can make the following intuitive justification for the Gaussian assumption. In practice, TH sequence or DS code is usually employed for multiple access and/or for spreading the symbol energy into multiple pulses. Also, UWB channels usually have large delay spread. For these reasons, the number Ns of samples per symbol is usually large. When Ns is large, there will be relatively large number (say more than 10) of the samples sharing a similar signal value si (cf. (3)). The CLT can be applied to these samples first to yield an approximately Gaussian distributed random variable, and Λ in (12) is then a summation of such Gaussian like random variables, and hence approximately Gaussian distributed as well. In Fig. 2, we show the empirical cumulative distribution function (CDF) of Λ together with the Gaussian approximation when a UWB symbol is transmitted through a multipath channel under several SNRs, where Eb is the bit energy of the transmitted symbol. The ideal received pulse in a nondispersive channel is chosen to be the second derivative of a Gaussian pulse
2
2 t t · exp −2π (14) wrec (t) = 1 − 4π Tps Tps where Tps = 0.29 ns is a pulse shaping parameter such that the pulse duration Tp is about 0.7 ns [2]. We assume that DS
KE et al.: FINITE-RESOLUTION DIGITAL RECEIVER DESIGN FOR IMPULSE RADIO ULTRA-WIDEBAND COMMUNICATION
code is used, the received signal for one symbol is N c r(t) = δi wrec (t − iTc ) ∗ h(t) + n(t)
5111
(15)
i=1
where h(t) is the multipath channel, and ∗ stands for convolution. The channel h(t) is chosen to be the first realization of standard CM1 channel model of IEEE P802.15 group [21]. The time delay of this particular channel is around 96 ns. Nc is the length of DS code and Tc is the chip duration. δi ∈ {±1} is the ith entry of the DS code. Let T = Nc Tc + Td, where Td is the guarding period to avoid ISI. We choose the DS code to be an m-sequence with length Nc = 127. We choose Tc = 1 ns, Td = 73 ns and Ts = 0.1 ns such that Ns = 2000. We can see from Fig. 2 that despite the fact that f (ai )’s are not identically distributed, a Gaussian approximation is relatively accurate. Treating Λ as a Gaussian random variable, we can estimate 2 ), where d the BER performance as Q(dmin / 4σeq min = 2E(Λ|d = 1) denotes the equivalent Euclidean distance 2 = V ar(Λ|d = between the two constellation symbols, and σeq 1) denotes the equivalent noise variance. Let g(x) denote the probability density function (pdf) of a Gaussian random variable with zero mean and variance N0 /2. By introducing linear approximation and assuming that the number of samples is large enough, we can evaluate E(Λ|d = 1) and V ar(Λ|d = 1) (see Appendix) as (4)
E(Λ|d = 1) = (W1
(4) 2(W1
where Ebref
(4)
+ W2 )Ebref (4) W2 )Ebref
V ar(Λ|d = 1) = + Ns −1 2 ∞ 2 = i=0 si = −∞ wref (t)dt, and (4)
W1
(4)
W2
= =
(16) (17)
4(g(0) − g(θ))2 , ˜ ˜ Q(0) − Q(θ)
(18)
4g(θ) . ˜ Q(θ)
(19)
2
We can see that with the Gaussian assumption on Λ, the BER is only a function of the quantization threshold, the noise variance and the bit energy of reference signal; i.e., it depends on the pulse waveform only through Ebref . However, Ebref may not equal Eb , depending on the channel and the lowpass filter. As the LPF filters out the signal energy outside the cut-off frequency band, making Ebref and Eb different. However, as long as the sampling frequency is twice the cutoff frequency of LPF, our analysis does not change. Increasing sampling rate will not increase Ebref if sampling rate is larger than the Nyquist rate. Thus, although oversampling could be used in our proposed system, it does not bring performance improvement. Under the Gaussian assumption, Nyquist rate sampling is sufficient for the finite-resolution digital receiver. C. Optimization of the quantization threshold Since the BER performance depends on the quantization (4) (4) threshold θ through W1 and W2 , which are functions of θ only, we can optimize the BER performance by choosing θ
Fig. 3.
BER sensitivity with respect to threshold of 4-level receiver.
to be
⎛
(4-level) θopt
= arg min Q ⎝ θ
⎞ (4) (4) W1 (θ) + W2 (θ) ref ⎠ Eb (20) 2 (4)
(4)
where the dependence on θ of W1 and W2 has been (2) (4) included explicitly. Using the definitions of W1 and W2 in (18) and (19), we have performed a numerical search (4-level) for optimal threshold θopt , which is found to be θopt ≈
0.98 N0 /2. Another interesting question is that whether the BER performance is sensitive to the threshold or not. To answer this question, we first define a parameter η(θ) as η(θ) =
Pe (θ) − minθ (Pe (θ)) maxθ (Pe (θ)) − minθ (Pe (θ))
(21)
where Pe (θ) is the BER as a function of θ when Ebref /N0 is fixed and both the maximization and minimization are over the whole range of θ ∈ [0, ∞). Thus η(θ) satisfies 0 ≤ η(θ) ≤ 1 and it indicates how close Pe (θ) is to the minimum BER. The relationship among η(θ), Ebref /N0 and the quantization threshold θ is shown in Fig. 3. We can see that η(θ) does not change much when threshold is varying near the optimal value. Thus the receiver is robust to the choice of quantization threshold. IV. G ENERAL F INITE R ESOLUTION R ECEIVER A. Even number of quantization levels We can extend the previous discussion to any case when the number of quantization levels L is even. Assuming Llevel quantization thresholds θ : {θm−1 , m = 1, ..., L/2}, the analytical minimum BER for L-level receiver when L is even can be derived as ⎞ ⎛ L/2 (L) ref W E m b ⎠ m=1 Pe(L-level,min) (Ebref ) = min Q ⎝ (22) θ 2
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is given as
(L) Wm =
10
4[g(θm−1 ) − g(θm )]2 , ˜ m) ˜ m−1 ) − Q(θ Q(θ
m = 1, . . . , L/2
where we set θ0 = 0 and θL/2 = ∞ to make this equation self(L) contained. The factors Wm ’s indicate how much contribution each corresponding template provides to the BER performance. Define A(m) = {i : ai = ±m}, m = 1, . . . , L/2. The decision variable, LLR, can be written as Λ=
L/2
(L-level,m) sgn[ai ]ci
−1
(23) 10
BER
(L)
and Wm
10
(24)
10
2[g(θm−1 ) − g(θm )] s, = ˜ m−1 ) − Q(θ ˜ m) i Q(θ m = 1, . . . , L/2, i = 0, 1, . . . , Ns − 1. (25)
10
−2
−3
2−level minimum BER 3−level minimum BER 4−level minimum BER 6−level minimum BER 8−level minimum BER Matched filter BER
−4
m=1 i∈A(m)
where the optimal receiver templates are (L-level,m) ci
B. Odd number of quantization levels Now we consider the cases where the number L of quantization levels is odd. We take 3-level quantization as an example. It will be a good tradeoff method between the BER performance and implementation issue. The 2-level receiver can be viewed as a special case of 4-level receiver where θ (4) (4) is chosen to be zero such that W1 = 0. However, W2 is not maximized when θ = 0. The intuition is that at some samples the sampled pulse amplitude of si is relatively small and hence these samples are easily corrupted by noise. Thus, if we can ignore these bad samples, we can achieve a better BER performance comparing to 2-level receiver while we still use only one template. The quantization scheme of a 3-level receiver is as follows ⎧ ⎪ ⎨+1, if ri > θ, ai = 0, if − θ < ri ≤ θ, i = 0, 1, ...Ns − 1. (26) ⎪ ⎩ −1, if ri ≤ −θ, The 3-level quantizer can be viewed as a 4-level quantizer followed by an “eraser” that sets the ±1 samples to 0. We can see that the LLR for 3-level receiver is related to 4-level case. Because P (ai = 0|d = +1) = P (ai = 0|d = −1), the LLR does not depend on the ai = 0 samples. The performance can be derived from 4-level receiver by dropping the contribution of the first template. We have (4)
E(Λ|d = 1) = W2 Ebref V ar(Λ|d = 1) = and
(4) 2W2 Ebref
⎛ (3-level) θopt (4)
= arg min Q ⎝ θ
⎞ (4) W2 (θ) ref ⎠ Eb . 2
−5
4
5
6
7
8
9
E ref /N b
Fig. 4.
0
Analytical BER of different receivers.
L-level receiver when L is odd can be derived in a similar way, given as ⎞ ⎛ L/2 (L) ref W E m b ⎠ m=1 Pe(L-level,min) (Ebref ) = min Q ⎝ (30) θ 2 where θ : {θm−1 , m = 1, ..., L/2} and (L+1) Wm , 1 < m ≤ L/2 (L) Wm = . 0, m=1
(31)
Similarly, define A(m) = {i : ai = ±(m − 1)}, m = 1, . . . , L/2. The decision variable, LLR, can be written as L/2
Λ=
(L-level,m)
sgn[ai ]ci
.
(32)
m=1 i∈A(m)
The corresponding templates are ((L+1)-level,m) c , (L-level,m) ci = i 0,
1 < m ≤ L/2 , m=1
i = 0, . . . , Ns − 1.
(33)
The optimal thresholds are different from the even number quantization cases though.
(27)
C. Incremental gain of additional quantization levels
(28)
The analytical BER curves of different receivers are shown in Fig. 4. We can see that the performance of 3-level receiver is between the 2-level and 4-level receivers. By dropping the bad samples, the receiver can acquire 1 dB gain from the monobit receiver. The cost is merely a high rate comparator and the calculation requirements are the same as monobit receiver, because the 3-level receiver only needs one optimal template. BER curves of other odd level receivers are not shown because the improvements are not as significant as that of 3-level receiver. In order to have a better view of how the quantization levels affect the BER performance, we list the optimal quantization
(29)
Although W2 and the associated template of 3-level receiver have the same expressions as those of the 4-level receiver, the optimal quantization threshold is different.
The optimal (3-level) ≈ 0.61 N0 /2 through threshold is found to be θopt numerical search and BER is also insensitive to the threshold around the optimal value. The analytical minimum BER for
KE et al.: FINITE-RESOLUTION DIGITAL RECEIVER DESIGN FOR IMPULSE RADIO ULTRA-WIDEBAND COMMUNICATION
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TABLE I P ERFORMANCE GAIN OF FINITE - RESOLUTION RECEIVER . L
Δ/( N0 /2) L/2 (L) m=1 Wm incremental level gain (dB) number of bits incremental bit gain (dB)
2 0 1.2732 – 1 –
3 0.61 1.6196 1.0451 – –
4 0.98 1.7650 0.3735 2 1.4186
L/2 (L) step Δ and the factor m=1 Wm , which affects the BER performance in (22) and (30), as well as the dB gain that can be obtained by additional quantization levels or bits in Table. I. We use Δ = θm − θm−1 , m = 1, . . . , L2 − 1; i.e., uniform quantization level spacing. We define the incremental level gain as the gain in SNR in dB offered by increasing the number of quantization levels from the previous listed value in Table I to the current value. Similarly, incremental bit gain is defined as the gain in SNR obtainable by increasing the number of quantization bits from the previous integer value to the current one. We can see that the incremental gain is very small (less than 0.1 dB) when L ≥ 8, which means that the high-resolution ADC (more than 3 bits) is not necessary. We can use the 8-level receiver to achieve a desired performance, while the BER is slightly larger than the theoretical matchedfilter performance. Similar observations were made in [19], based on quantization noise analysis. D. BER Analysis in Fading Channel Previous discussion only focuses on one transmission block, where the channel is quasi-static. But fading exists so that in some time intervals the fading is severe and the BER becomes worse. The BER under fading channel can be averaged over the pdf of Ebref as ∞ Pe = Pe(L-level,min) (x)p(x)dx (34) 0
(L-level,min) Pe (x)
where is the analytical BER given in (22) or (30), and p(x) is the pdf of Ebref . The pdf can be obtained by Monte Carlo simulations over the standard channel model [21]. V. T RAINING BASED T EMPLATE E STIMATION We have analyzed the BER performance of our proposed finite-resolution receiver, assuming that the perfect knowledge of reference signal is available. In practice, the reference signal needs to be estimated, from the finite-resolution samples. In this section, we propose a training based reference signal estimation method using the ML criterion. The optimal templates of our proposed receiver rely on the knowledge of the reference signal si . Thus, in order to obtain the values of the templates, we first need to estimate si . In the following, we present the ML template estimation solution using the 4-level receiver as an example. In the training period, we transmit an all-one training (m) sequence of M symbols. Let ai denote the ith sample of (k) mth training symbol and Ni denote the cardinality of the
6 0.71 1.8824 0.2796 – –
8 0.56 1.9284 0.1047 3 0.3843
10 0.47 1.9513 0.0513 – –
12 0.41 1.9645 0.0292 – –
14 0.36 1.9728 0.0184 – –
16 0.32 1.9784 0.0124 4 0.0925
(m)
set {m|ai = k, 1 ≤ m ≤ M }, where k ∈ {±1, ±2} and 0 ≤ i ≤ Ns − 1. Recall that Ns is the length of the template. For each sample we have an M × 1 training-based data vector (1) (2) (M) αi = [ai , ai , . . . , ai ]. The log-likelihood function of si is V (αi |si ) := log P (αi |si ) (k) Ni log P (ai = k|si , d = 1) =
(35) (36)
k∈{±1,±2}
i = 0, . . . , Ns − 1. (−2)
Based on the collected sample counts N i = [Ni (+1) (+2) Ni , Ni ], the ML estimate of si is sˆi = arg max V (αi |si ), si
i = 0, 1, . . . , Ns − 1.
(−1)
, Ni
,
(37)
In practice, the ML estimate of si can be obtained using a lookup table. The table is indexed by si , and each row of the table stores 4 entries: {log P (ai = k|si , d = 1), k = ±1, ±2}. For each N i vector, we can perform the inner product of it with each row. The row that gives the maximum inner product corresponds to the ML estimate of si . Assuming the function V (α|si ) is concave, a bisection search of the table will be sufficient. Corresponding to each row, it is not (4-level,1) necessary to store the value si . Instead, we can store ci (4-level,2) and ci which are the needed weights. To summarize, the weighting template can be estimated as follows: 1) The transmitter sends an all-one training sequence to the receiver. (k) 2) The receiver counts Ni for all Ns samples. (k) 3) The receiver uses the Ni to find the ML estimate of si according to (37), and retrieve the corresponding (4-level,1) (4-level,2) weights (such as ci and ci ). This step is performed for i = 0, 1, . . . , Ns − 1. The ML solution of si is asymptotically optimal when the length of training sequence M is large in the sense that it can achieve the Cramer-Rao lower bound asymptotically [22]. VI. S IMULATIONS AND D ISCUSSION In this section, we provide simulation results of proposed finite-resolution receiver design and BER analysis. In our simulations, we omitted TH sequence or DS code for simplicity. The conclusions will not change if they are included. The received pulse is the same as (14) with Tp = 0.7 ns. The sampling frequency is chosen to be 8/Tp in the following simulation. We will use the first realization of standard CM1 channel model [21] to test our analytical optimal quantization
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Fig. 5. Comparison between the analytical BER curve and numerical results of 4-level (2-bit) receiver under multipath channel.
Fig. 6. Comparison between the analytical BER curve and numerical results of 3-level receiver under multipath channel.
thresholds, where the channel is normalized such that its Euclidean norm is 1. Fig. 5 and Fig. 6 show the comparison of our analytical BER curves and numerical results for 4-level and 3-level receivers, respectively. We can see that the analytical curves are very good estimations of the minimum BER that the receivers can offer by choosing appropriate quantization thresholds. Fig. 7 and Fig. 8 illustrate the BER sensitivity with respect to quantization threshold of 4-level and 3-level receivers, respectively. We can see clearly that the optimal thresholds obtained by simulations are close to the analytical optimal thresholds. We can also see that η does not change much around the optimal thresholds we derived, suggesting that the receivers are robust to the choice of quantization thresholds. We also evaluated the BER performance under fading channel; see Fig. 9 and Fig. 10. The channel is the standard CM1 channel model [21] with 100 realizations. The analytical and numerical simulation results of 4-level and 3-
Fig. 7. Simulation results of BER sensitivity with respect to threshold of 4-level receiver under multipath channel.
Fig. 8. Simulation results of BER sensitivity with respect to threshold of 3-level receiver under multipath channel.
level receivers are shown as functions of transmitted Eb /N0 . As the optimal quantization thresholds have been verified in the previous simulation, we used the optimal values in these simulations. We can see that the analytical performance is in good agreement with simulated performance when full channel state information (CSI) is available. The effect of the training sequence length was also investigated using the lookup table technique we mentioned in Section V. Our lookup table quantizes si in the range [−5, 5] with a step size 0.005. It can be seen from Fig. 9 and Fig. 10 that the performance improves as the length of training sequence increases. The performance of 200-bit and 500-bit training is about 1 dB and 0.5 dB away from the full CSI performance, respectively, when Eb /N0 is around 10 dB. Using a 1000-bit training sequence, the performance degradation is reduced to about 0.3 dB. When the number of quantization levels is two, our proposed optimal receiver performs better but only slightly better than full-resolution template filtering method in [18]. One
KE et al.: FINITE-RESOLUTION DIGITAL RECEIVER DESIGN FOR IMPULSE RADIO ULTRA-WIDEBAND COMMUNICATION
10
BER
10
10
10
10
−1
−2
−3
Analytical BER Full CSI simulation 1000bit training 500bit training 200bit training Matched filter BER
−4
Fig. 11.
The illustration of linear approximation.
−5
4
6
8
10
12
14
E /N b
0
Fig. 9. Comparison between the analytical and numerical results in fading channel as well as the effect of training sequence length on BER performances for 3-level receiver. 10
5115
−1
mance should be better than that of the receiver in [19], with a price of higher receiver complexity when the quantization resolution is high. When the number of quantization levels is large, the two should be close, modulo a 3 dB difference in SNR due to non-coherent TR modulation. VII. C ONCLUSIONS
BER
10
10
10
10
−2
−3
Analytical BER Full CSI simulation 1000bit training 500bit training 200bit training Matched filter BER
−4
−5
4
6
8
10
12
14
E /N b
0
Fig. 10. Comparison between the analytical and numerical results in fading channel as well as the effect of training sequence length on BER performances for 4-level receiver.
main contribution of this paper is however, to quantify the theoretically achievable performance of finite-resolution digital receivers. Another contribution compared to [18] is that we considered multiple quantization levels in this paper, and derived the optimal detector structure. We did not consider the possibility of sigma-delta modulation in this paper. In general the benefit of sigma-delta modulation is not obvious when Nyquist-rate sampling is used. We note that the performance analysis in [19] based on modeling the quantization noise is not as accurate as our performance result when the number of quantization levels is small. For 3-level receiver, comparing the curve in Fig. 9 with 200 training symbols and the corresponding curve in [19, Fig. 5, 1 bit], the receiver in this paper has about 4 dB gain at BER of 10−2 , whereas the complexity is comparable. Note that due to the non-coherent TR modulation, an inherent 3dB performance loss is incurred. In general, our receiver perfor-
We investigated the design and performance of finiteresolution digital receiver for impulse-radio ultra-wideband communication system over multipath dispersive channels. We evaluated the effect of quantization threshold and presented analytical BER expressions based on a Gaussian assumption. The optimal quantization thresholds for 3- and 4-level receivers have been obtained. Our results show that 3-level receiver can acquire 1 dB gain over monobit receiver and we can use 8-level receiver to achieve a performance 0.2 dB within that of the matched filter receiver. Higher resolution ADCs are not recommended as the additional BER gains are too small to justify the increased implementation complexity. We proposed a template estimation scheme based on the maximum-likelihood criterion. We only considered single user transmission with no interference in this paper. Possible future work includes evaluating receiver performance in the multiuser as well as narrow-band interference cases. VIII. A PPENDIX In this appendix, we derive the closed-form expression of the BER for 4-level receiver assuming Gaussian distribution of the LLR. Divide the Gaussian pdf into several regions {S (k) : k = 1, ..., 5}, as shown in Fig. 11. Denote P (S (k) ) as the probability that ni = ri − si falls in to the region of S (k) . We can rewrite (8) as P (ai |d = 1) = ⎧ P (S (3) ) + P (S (4) ) + P (S (5) ), ⎪ ⎪ ⎪ ⎨2P (S (1) ) + P (S (2) ), ⎪ P (S (2) ) + P (S (3) ) + P (S (4) ), ⎪ ⎪ ⎩ P (S (5) ),
if if if if
ai ai ai ai
= +2, = +1, = −1, = −2.
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By definition, the expectation and variance of f (ai ) when “1” is transmitted are E(f (ai )|d = 1) = (4-level,1) [P (ai ci (4-level,2) +ci [P (ai
= 1|d = 1) − P (ai = −1|d = 1)] = 2|d = 1) − P (ai = −2|d = 1)]
and
and V ar(Λ|d = 1) =
N s −1
V ar(f (ai )|d = 1)
i=0 (4)
(4)
N s −1
(4) (4) s2i −(W1 +W2 )2
= 2(W1 + W2 )
N s −1
s4i
i=0 (4) (4) (4) (4) 4 = 2(W1 + W2 )Ebref − (W1 +W2 )2 wref (t)dt Ts i=0
V ar(f (ai )|d = 1) (4-level,1) 2
=[ci
] [P (ai = 1|d = 1)+P (ai = −1|d = 1)]
+[ci
] [P (ai = 2|d = 1)+P (ai = −2|d = 1)]
(4-level,2) 2
−E 2 [f (ai )|d = 1] We next use a linear approximation on the weighting signals and the probability terms. First, we look at the probability terms. When si is small, P (S (3) ) ≈ P (S (4) ). We have P (ai = 1|d = 1) − P (ai = −1|d = 1) ≈ 2[P (S
(1)
) − P (S
(3)
)] = 2[g(0) − g(θ)]si
P (ai = 2|d = 1) − P (ai = −2|d = 1) ≈ 2P (s(4) ) = 2si g(θ) Using the approximation loge (1 + x) ≈ x and the approximation discussed previously, we have (4-level,1)
P (S (2) ) + 2P (S (1) ) P (S (2) ) + 2P (S (3) ) 2(P (S (1) ) − P (S (3) )) ≈ P (S (2) ) + 2P (S (3) ) 2(g(0) − g(θ)) si ≈ ˜ ˜ Q(0) − Q(θ)
≈ log
and (4-level,2)
ci
P (S (3) ) + P (S (4) ) + P (S (5) ) P (S (5) ) ˜ ˜ Q(θ + si ) − Q(θ − si ) ≈ ˜ + si ) Q(θ 2g(θ) si . ≈ ˜ Q(θ)
= log
Thus, we have E(Λ|d = 1) = =
N s −1 i=0 N s −1 i=0
E(f (ai )|d = 1) N s −1 4(f (0) − f (θ))2 2 4f (θ)2 2 si + si ˜ ˜ ˜ Q(0) − Q(θ) Q(θ) i=0 (4)
(4)
W1
=
(4) (W1 (4)
= (W1
N s −1
+
(4) W2 )
+
i=0 (4) W2 )Ebref
W2
si
(4)
(as Ts 1).
Finally, we want to comment that if si is large, then our linear approximation is not accurate. However, the effect of such inaccuracy should be small in fading channels because the overall performance is dominated by “deep fade” events and hence small si cases. R EFERENCES
and
ci
(4)
= 2(W1 + W2 )Ebref
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KE et al.: FINITE-RESOLUTION DIGITAL RECEIVER DESIGN FOR IMPULSE RADIO ULTRA-WIDEBAND COMMUNICATION
[18] S. Hoyos, B. M. Sadler, and G. R. Arce, “Monobit digital receivers for ultrawideband communications,” IEEE Trans. Wireless Commun., vol. 4, no. 4, pp. 1337–1344, Apr. 2005. [19] J. Tang, Z. Xu, and B. M. Sadler, “Performance analysis of b-bit digital receivers for TR-UWB systems with inter-pulse interference,” IEEE Trans. Wireless Commun., vol. 6, no. 2, pp. 494–505, Feb. 2007. [20] S. Franz and U. Mitra, “Quantized UWB transmitted reference systems,” IEEE Trans. Wireless Commun., vol. 6, no. 7, pp. 2540–2550, July 2007. [21] J. Foerster, “Channel modeling sub-committee report final,” IEEE Working Group for Wireless Personal Area Networks (WPANs) P802.1502/490r1-SG3a, Feb. 2003. [22] S. M. Kay, Fundamentals of Statistical Signal Processing: Detection Theory. Prentice-Hall, Inc., 1998.
Lei Ke received the B.E. degree and the M.Sc. degree in Electronic Engineering and Information Science from the University of Science and Technology of China (USTC), Hefei, China, in 2003 and 2006, respectively. He is currently pursuing his Ph.D. with the Department of Electrical and Computer Engineering at Iowa State University. His general interests include wireless communication and signal processing.
Huarui Yin received his Bachelor’s Degree in 1996, and Ph.D. degree in 2006, both in Electronic Engineering and Information Science from University of Science and Technology of China, at Hefei, Anhui. Since 1999, he has been with the department as a lecturer. His research interests include digital signal processing, ultrawide bandwidth communication and software defined radio.
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Weilin Gong received his B.S. degree in applied physics, and Ph.D. degree in Electronic Engineering and Information Science from University of Science and Technology of China, Hefei, China, in 2003 and 2008 respectively. Currently he is a post-doc with the University of Science and Technology of China. His research focuses on communication theory and signal processing for ultra wideband communication.
Zhengdao Wang received his B.S. degree in Electronic Engineering and Information Science from the University of Science and Technology of China (USTC), 1996, the M.Sc. degree in Electrical and Computer Engineering from the University of Virginia, 1999, and Ph.D. in Electrical and Computer Engineering from the University of Minnesota, 2002. He is now with the Department of Electrical and Computer Engineering at the Iowa State University. His interests are in the areas of signal processing, communications, and information theory. He served as an associate editor for IEEE T RANS ACTIONS ON V EHICULAR T ECHNOLOGY from April 2004 to April 2006, and has been an Associate Editor for IEEE S IGNAL P ROCESSING L ETTERS since August 2005. He was a co-recipient of the IEEE S IGNAL P ROCESSING M AGAZINE Best Paper Award in 2003 and the IEEE Communications Society Marconi Paper Prize Award in 2004.
