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Optimum and Sub-Optimum Detection of Physics-Based Ultra-Wideband Signals In Presence of Inter-Symbol Interference Robert C. Qiu Department of Electrical and Computer Engineering Center for Manufacturing Research Tennessee Tech University Cookeville, Tennessee 38505 Tel: 931-372-3847, email: [email protected] Abstract— This paper represents a major step toward receiver structures under the framework of the per-path pulse distortion using more realistic channel models. We extend our previous framework to include the important phenomenon of inter-symbol interference (ISI). ISI ultimately limits the maximum achievable data rate. In this paper we find that the per-path pulse distortion has impact on this maximum achievable data rate in the presence of ISI. As examples of this discovery, we will investigate the optimum receiver structure and two sub-optimum receivers with zeroforcing and minimum mean square error (MMSE) equalizers. We will use the high-rise building channel model as the underlying pulse propagation model. This model captures a lot of properties that are not available in the IEEE 802.15.4a model. Index Terms—Ultra-wideband, Physics-based signals, detection.
I. INTRODUCTION odern communication theory originated from the
of communication engineers to Mattempts understand what they were doing in the most general terms. The limit of digital wireless communication networks depends primarily on four basic laws and their underlying theories, which attributed respectively to: Maxwell and Hertz, Shannon, Moore, and Metcalfe [1]. We followed this philosophy in the previous work [2-20] 1. This paper attempts to seek optimum solutions predicted by both Maxwell/Hertz and Shannon. In a recent three-paper series [7-9], we have established the optimum receiver structure paradigm based on the so-called physics-based multipath channel model [2-20]. This frequency dependence model has been very recently 1
All our conference and journal papers and Ph. D. dissertation are also available in our website http://iweb.tntech.edu/rqiu/index.htm.
accepted in the IEEE 802.15.4a [21]. As pointed by its channel model subcommittee in [21], the first suggestion of using such a model was made for UWB wireless systems in indoor environments one decade ago [13]. This model has found many applications, e.g. [22]. This paper represents a major step of our research [2,6]. We extend the previous paradigm to include the important phenomenon of inter-symbol interference (ISI) [27]. ISI ultimately limits the maximum achievable data rate. In this paper we find that the per-path pulse distortion has impact on this maximum achievable data rate in the presence of ISI. In the absence of ISI, the per-path pulse distortion only affects the front-end matched filter design of the optimum receiver, but the performance in terms of bit error rate (BER) is not affected if the symbol or pulse energy is identical with or without pulse distortion. This is the property of the matched filter. Once matched, the performance only depends on the energy of the pulses, independent of the pulse shape. However this property is invalid when ISI is present. As examples of this discovery, in this paper we investigate the optimum receiver structure and two sub-optimum receivers with zero-forcing and minimum mean square error (MMSE) equalizers. This model captures some properties that are not available in the IEEE 802.15.4a model. It may be informative if the physics-based model is in closed form and the performance loss in signal to noise ratio (SNR), β n defined in (14), will be explicitly connected with physical time-domain propagation problems. For narrowband systems, this connection may be unfeasibly complicated since most propagation mechanisms are overlapping in time domain. However, for UWB communications, most propagation mechanisms are separable in time domain. Explicitly connecting these mechanisms with system performance metrics helps us to understand the limits of UWB communications deterministically. One day the system can be designed to adapt itself to specific environments for optimum network
1
where Ψ( x, y , z ) determines the wavefront at the (x,y,z) in question, i.e., Ψ ( x, y , z ) − ct = 0 is the discontinuity
performance. Consequently understanding the physic-based multipath model is of a fundamental significance.
hypersurface on which
II. Physics-Based Channel and Signals One big challenge of UWB is the per-path pulse distortion. Mathematically the generalized channel model is expressed by [2-5] L
h(τ ) = ∑ An hn (τ ) ∗ δ (τ − τ n )
hn (τ ) = where
(1)
n =1
n
α n assumes an arbitrary real value (possibly random
function and the unit function. A special case of α n
= α0
of (4) was recently accepted in the IEEE 802.15.4a. Therein α 0 is a random variable varying between 0.8 and
hn (τ ) represents an arbitrary function that has finite
energy. Symbol “ ∗ ” denotes convolution and δ ( x ) is the Dirac Delta function. The statistical parameterization of hn (τ ) is a challenging task. Turin’s model expressed as L
An τ − (1+αn )U (τ ) , H n (ω ) = ( jω )α (4) Γ( −α n )
variable) and U (.) and Γ(.) are, respectively, the Gamma
where L generalized paths are associated with amplitude An , delay τ n , and per-path impulse response hn (τ ) . The
h(τ ) = ∑ Anδ (τ − τ n )
hn (τ ) has a finite jump. For an
important subclass of (3) and (4), we have
1.4 [21]. III. Optimum Detection of Physics-based Signals
We will incorporate the per-path impulse response into the optimal receiver design when ISI is present. We will pay special attention to the hn (τ ) . For PAM signals in a single-
(2)
n =1
is a special case of Eq. (1) when hn (τ ) = δ (τ ) . As an example, let us take a look at the diffracted rays. The geometric optics rays that are reflected by and pass though walls, furniture, trees, and etc, can be treated as a special form of a diffracted ray [4,8]. Our theory presuppose that we know the behavior of the corresponding impulse response E0 and H 0 (electrical field and magnetic field)
user system, the transmitted signal is expressed as
s(t ) =
∞
∑ b x(t − nT ) n
s
n =−∞
bn represents the n-th discrete information symbol with duration of Ts .
where
which are by definition solutions of Maxwell’s equations generated by the source function g(x, y,z )δ (t) . The wave propagation along a generalized ray can be modeled by the propagation of a singularity in spatial time along such a ray. In the neighborhood of the spatial time singularity at τ = τ α (corresponding to a spatial position along a ray or
Diffracted Signal d(t)and Template Signal v(t) 1.5
1
d(t) and v(t)
0.5
multipath), the impulse response of a generalized ray (any component of E0 ) has the behavior
0
-0.5
alpha= -1: 0.25: 0 (bottom to top)
∞ Cn − ξ ( τ τ ) (τ α − τ ) n , τ < τ α ∑ α n ! n =0 hn (τ ) = ∞ η (τ − τ ) Dn (τ − τ ) n ,τ > τ ∑ α α α n=0 n !
-1
-- alpha=0
(3)
-1.5
0
0.1
0.2
0.3
0.4
0.5 time t (ns)
0.6
0.7
Figure 1 Pulse waveform as a function of in Equation (4).
where ξ (τ ) and η (τ ) are rather general. Then the corresponding asymptotic frequency response has the form The received signals are represented as ∞ [4,8] ∞ t Dm 1 η ( )t m e − t dt − m ∫0 ∞ m ! ( j ) j ω ω H n ( jω ) ≈ e − jk Ψ ∑ ∞ 1 t m −t m = 0 Cm )t e dt ξ( m ∫0 m ! ( − jω ) − jω
Incident Waveform
__Red dashed Template Pulse v(t)
r (t ) = ∑ bn y (t − nTs ) + n (t )
0.8
α
0.9
1
defined as
(5)
k =0
where n (t ) is AWGN. It has been shown for a narrowband system [23] that the optimum receiver structure is illustrated in Fig. 2.
2
The sampled sequence is processed by a maximum likelihood sequential estimator (MLSE) detector that was first derived by Forney (1972). This receiver structure is optimal in terms of minimizing the probability of transmission error in detecting the information sequence. The assumption of Forney (1972) was that the receive signal has finite energy. If in the UWB receiver design we assume that y(t) is of finite energy, the structure of Fig. 2 can immediately be applied to our UWB problem [2,6]. As a result the output of the matched filter is expressed as ∞
q(t ) = ∑ bn R yy (t − nTs ) + v (t )
(6)
qk = bk +
∞
∑
n =0,n ≠ k
where the
bn Rk −n + vk
(8)
bk term represents the expected information
symbol of the k-th sampling period, the 2nd term is the ISI, and vk is the additive Gaussian variable at the k-th sampling point. The sequence of
qk will be processed at the symbol
level by MLSE (Viterbi algorithm) before being sent to decision circuits.
where R yy ( t ) = y ( t ) ⊗ y * ( − t ) = Rxx ( t ) ⊗ h ( t ) ⊗ h * ( − t )
If the duration of each symbol is small such that R yy (t ) = 0 at instants t = (n − k )Ts for n ≠ k , (or,
is the autocorrelation of y (t ) and v (t ) is the response of
Rk −n = 0 for n ≠ k ), no ISI occurs. As a result (8)
n =0
the matched filter to AWGN noise n (t ) .
Rxx (t ) is the
autocorrelation of x ( t ) . See Appendix A for explicit expressions for R yy ( t ) and the cross-correlation Rxy ( t ) .
= kTs , we denote = R yy (t )δ [t − (n − k )Ts ]
Sampling at instants t
Rk − n
After the sampler in Fig. 2, the discrete signals at times t = kTs of (6) are given by As given in (6), the output of the matched
is Ryy (t ) = Rxx (t ) ⊗ Rhh (t ) .
filter
Rxx (t ) is
typically of a smooth shape. Finally, a time-reversal scheme is used to shift the design complexity from the receiver to the transmitter [26]. In time-reversal a signal is precoded such that it focuses both in time and in space at a particular receiver. Due to temporal focusing, the received power is concentrated within a few taps and the task of equalizer design becomes much simpler than without focusing. If we use the pre-filter (at the transmitter) which
h (t )
, the output of the matched has an impulse response of filter (at the receiver) is Ryy (t ) = Rxx (t ) ⊗ Rhh (t ) . If the first tap is used (shown in Fig.3), the collected energy is above 50% of the total energy. Pulse distortion is ignored in Fig. 3 since IEEE 802.15.3a model models the per-path impulse response as a Dirac function. It is natural to study the impact of pulse distortion on the Rhh (t ) . The next section provides such a model. ∞
qk = R yy (t ) ∑ bnδ [t − ( n − k )Ts ] + v(t )δ ( t − kTs ) n =0
∞
= ∑ bn Rk −n + vk ;
(7)
k=0,1,2,"
n =0
where
R0 = [ Rk −n ] |k =n (the energy of the n-th symbol) can
be regarded as an arbitrary scale factor and conveniently set to 1. From (7) we obtain
reduces to
qk = bk + vk It is, therefore, sufficient to detect the symbols independently one-by-one for each given instant t = kTs . This is the renowned matched filter for isolated symbol detection. As expected, once matched, the detection depends on the energy E y = R0 = R yy (0) of the symbol only, not the received composite pulse waveform y (t ) . As a result, per-path pulse distortion has no impact on the single symbol matched filter detector for isolated symbol. However, according to (8), we will find that this is not the case when ISI is present. We will evaluate this effect using some closed-form expressions for system performance. If the number of the overlapping symbols is less than ten, the MLSE (in Fig.2) may be even feasible for the state-ofthe-art signal processing capability. Received signal
Matched filter
y * (−t ) r (t ) = y (t ) + n(t )
Sampler clock
t = kT
MLSE (Viterbi algorithm) Output data
Figure 2. Optimal receiver structure in presence of ISI in an AWGN channel.
IV. Sub-Optimum Detection of Physics-based Signals Let us simplify optimum receiver structures in two directions: front-end blocks and sub-optimum decisions. First the matched filter can be simplified in a sub-optimum manner. Since the per-path pulse waveform has been included in the channel impulse, the resultant matched filter is sometimes called “Generalized RAKE” receiver structure [5], of which Altes’ receiver structure is a special case [5]. In absence of per-path pulse distortion, the matched filter
3
reduces to the conventional RAKE receiver of Price & Green (1958). Secondly the optimal decisions can be simplified. Let us follow Kailith and Poor [23] and restrict ourselves to a linear detector (zero-forcing and MMSE) with a matched filter front end [6]. The detected bits are
bˆn = sign( zn ) (9) where z = My , the matrix M is arbitrary, and the vector y has elements of y (t ) |t =nTs . Further we assume bn to take on the values of ±1 with equal probabilities. For zeroforcing, the probability that bn is in error is simply
σ
2
defined in (6), and an is the energy of
(10)
R = [Rk -n ] is
y (t − nTs ) .
Under some general conditions, the error probability of the linear MMSE detector
an M n,n Pe (n) ≈ Q σ ( MRM ) n ,n 2 −2 −1 where M = ( R + σ Da ) and Da = diag ( an ) .
(11)
β n = 1/ ( R -1 )n ,n
(14)
For MMSE equalizers we can similarly define the performance loss in SNR using (11). From (14) we reach
β n = 1 − ρ 2 for two symbols case. As a result β n depends on per-path pulse distortion hn (τ ) through R yy (τ ) in (13) and (6). See also Appendix A. The β n for the n-th sample depends on the whole physics-based channel model (Eq. (1)) and symbol sampling rate. Consequently understanding the physic-based multipath model is of a fundamental significance. The formulas of (5)-(14) are of course valid in absence of per-path pulse distortion, i.e. hn (τ ) = δ (τ ) . The non-ISI case was studied in [7-9]. With suitable manipulation as in [23], the formulation of (5)-(14) will be also valid for multi-user detection. The formulations form a unified framework that connects physics-based time domain channel model with detection theory [6]. Autocorrelation of UW B Channel 1.2 CM3, Channel 1, Energy Ratio = 50% 1
It is observed from (10) and (11) that per-path pulse distortion affects the system performance through the output of the matched filter, i.e., R yy ( t ) , sampled at instants t
2
= (n − k )Ts . When R is diagonal for ISI-free
0.8 Normalized Amplitude
where
an Pe ( n ) = Q σ ( R -1 )n ,n is the noise density, the matrix
to 1 − ρ , from (12). Compared with the isolated single symbol detector, the performance loss in SNR for zeroforcing detector can be generally defined as
0.6
0.4
0.2
situations, the per-path pulse distortion affects each symbol energy separately. As a result
(
R -1 = [ R ]n ,n n ,n
)
−1
0
for -0.2 -100
symbols n=0,1,2,…. The energy for each symbol after the equalizer depends on its own energy of that symbol only. However in general the time-domain overlapping of distorted pulses make the matrix R non-diagonal (ISI occurs). Inter-pusle overlapping combined with per-path pulse distortion makes receiver design very challenging. Example 1 Two-Symbol Case To gain some insights, let us consider a two-symbol case. Denote
1 R = ρ -1
where
ρ
−1
1 1 = 1 1 − ρ 2 − ρ
−ρ 1
(12)
ρ = R1 = R−1 = R yy (τ ) |τ =± T ≤ 1 (13). From (10) and (11) the performance only depends on the -1 diagonal elements of R , which are inversely proportional s
-80
-60
-40
-20
0
20
40
60
80
100
Time (ns)
Figure 3 Auto-correlation of a channel impulse response
Rhh (t ) = h(t ) ∗ h( −t ) for CM3 channel. V. Multi-Path Capture and Time-Reversal
Here the optimum receiver structure of Fig. 2 is used as a heuristic approach to derive some new structures. In Fig. 3, we show the auto-correlation of the channel impulse response, Rhh (t ) = h (t ) ⊗ h( −t ) , using a typical IEEE 802.15.3a model. The
Rhh (t ) is normalized by Rhh (0) , the
energy of the channel impulse response. Figure 2 shows Rhh (t ) / Rhh (0) . It is interesting to notice that the first peak has an energy that is around 50% [26]. We can view each pulse as a chip in a PN code and the channel impulse response can be regarded as a pseudo-noise code where the width of multipath delay spread corresponds to the length of
4
the PN code. This observation first made in [28] intuitively suggests [26] that it is simpler to base the detection on Rhh (t ) , rather than the noisy y (t ) .
x
x=b
VI. Numerical Results
y = −a
φ0
(ρ ,φ ) x
2a
Tx
D1
D2
Rx y=a
y
Fig. 5 An impulse plane wave pulse is incident on the conducting rectangular obstacle. Pulse diffracted by two parallel planes 0.08
0.06
The uppper edge The lower edge
Amplitude
0.04
0.02
0
-0.02
a=5 m The high-order terms Incident Angle =pi/5
-0.04 Observation Angle=2pi/3 -0.06
0
20
40
60 Time(ns)
80
100
120
Pulse diffrac ted by two parallel planes 1.5 The sum of two autocorrelation terms
1
Auto-Correlation
In the simulations, the second order Gaussian pulse is used for incident pulse for all cases. The per path pulse distortion occurs for all paths. We use two simple structures as examples as shown in Fig. 4 and 5. Only diffracted pulses exist for these structures. The frequency domain mathematical formulations of these structures valid for frequencies above 100MHz are well adopted in the literature. Since these formulations are valid for the spectrum of our interest (2-10.6 GHz) in UWB, we convert these formations into the time domain using inverse Laplace transform to give us the impulse response of the channel. The pulse response of a specific pulse is obtained by convolution of the so obtained results with the UWB pulse waveform, as shown in Fig. 6(a) and 7(a). The autocorrelation can be easily calculated as shown in Fig. 6(b) and 7(b). These time domain and frequency domain results are compared and match well. See Appendix B. Fig. 7(a) gives results using four different techniques. The frequency domain techniques are given in [30]. We give time domain results for Kirchhoff technique. GTD results are not given here to save space. All parameters and notations follow the widely adopted frequency domain results in the literature [29,30]. Compared with indoors, the multipath paths are not so rich but pulse distortion seems be another crucial issue. For high-rise buildings (Fig.5) (such as apartments) in urban and suburban environments, pulse distortion caused by diffraction phenomenon needs be stressed although this is not an issue for narrowband system.
Cross-correlation term Cross-correlation term
0.5
0
-0.5
-1 -100
Fig.4 An impulse plane wave pulse is incident on the two parallel conducting planes. The incident and observation angles are, resp.,
φ0 and φ . The width of the two planes is
2a.
To gain insights, we choose two simple environments that have well-known frequency domain harmonic solutions. Fig. 4 can be used to model the UWB pulse propagation around the corner of two streets and two apartments.
-50
0
50
100
150
Time(ns)
Fig. 6 (a), (b) The channel response of two parallel planes (Fig. 4) to the UWB pulse with the incident pulse shape of the second order Gaussian.
Fig. 5 can model the shadowing phenomenon. We ignore antenna effects for simplicity. Antennas can be modeled as linear systems that has effective effects included in equation (1). Another reason for choosing Fig. 4 and Fig. 5 for examples is the ISI consideration. In Fig. 5 although pulse distortion is more severe than
5
z
Pulse througn rec tangular screen 0.2 FD-GTD TD-GTD FD-Kirchhoff TD-Kirchhoff
0.15
0.1
Amplitude
0.05
0
-0.05
-0.1
-0.15
-0.2
0
0.5
D1=500m
D2 =450m
a=15m
b=10 m
1
1.5
2 Time(ns)
2.5
3
3.5
10
System Performance in Two Parallel Plane Model
0
W ithout MMSE Equalizer 10
10
-1
-2
BER
Fig.4 (comparing Fig. 6(a) and 7(a)), the ISI is less severe since the delay spread in Fig. 4 is much bigger than Fig.5. Fig. 8 shows the impact of the diffracted pulse on the ISI in the two parallel planes model (streets, high-rise buildings). We performed Monte Carlo simulations to obtain the BER using the MMSE equalizer (25 taps) for data rate of 200 Mbps. The matched filter is assumed as suggested in the optimum structure of Fig. 2. This gives the performance bound for the MMSE equalizer since channel estimation would introduce some loss in performance. The analytical channel model in Appendix B gives us the capability to study some system issues of which are otherwise incapable, such as the dependence of path loss on pulse delay and distortion and the explicit dependence of these parameters on environmental parameters (street width, building heights, antenna positions and heights, etc.).
10
10
10
MMSE Equalizer
AWGN
-3
-4
-5
0
2
4
6
8
10
12
14
16
18
Eb/No
Fig. 8 The BER of a UWB system in two parallel planes (Fig.4) in presence of the ISI.
V. CONCLUSION This paper extends our previous framework to the case when the ISI is present. In the UWB system design the connection of the ISI with the per-path pulse distortion adds a unique new dimension to the existing issues of a narrowband system. This is the first step towards addressing this problem. Analytically and numerically we illustrate that the diffracted pulses indeed complicate the system analysis. In the future, we will do more statistical analysis once the experimental statistical channel modeling that is able to capture the pulse distortion is made available. The future work involves the estimation of channel parameters related to pulse distortion. ACKNOLEDGMENT
4
The author would like to thank Chenming (Jim) Zhou and Qiang Zhang for their help.
7(a) A UW B Pulse Diffracted by A Rectangular Screen 1
Appendix A Cross-Correlation and AutoCorrelation
FD-GTD TD-GTD FD-Kirchhoff TD-Kirchhoff
0.8
L −1
Rxy (τ ) = ∑ al Rxx (τ ) ∗ hl (τ ) * δ (τ − τ l )
0.6
(A.1)
l =0
R y ( t ) y ( t ) (τ ) = Rx ( t ) x ( t ) (τ ) ∗ [h (τ ) ∗ h ( −τ ) ] ( A.2)
AutoCorrelation
0.4
L −1
R yy (τ ) = ∑ al2 {Rxx (τ ) * [ hl (τ ) ∗ hl ( −τ ) ]} +
0.2
l =0
0
L −1
L −1
i =0 i ≠k
k =0
∑ ∑ a a {R i k
-0.2
-0.4
xx
(A.3)
(τ ) ∗ [ hi (τ ) * hk ( −τ ) ]} * δ [τ − (τ k − τ i )]
For the commonly used second derivative Gaussian 2 pulse x ( t ) = d 2 / dt 2 e − t , we have Rxx (τ ) = d 4 / dτ 4 e −τ / 2 . 2
-0.6 -10
-8
-6
-4
-2
0 Time Shift(ns)
2
4
6
8
10
7(b) Fig. 7 The channel response of a rectangular obstacle (Fig. 5) to the UWB pulse with the incident pulse shape of the second order Gaussian.
Appendix B Closed Form Expression for Impulse Response
The impulse response of the channel of two parallel conducting planes (Fig.4) is given by
6
U (τ ) δ (τ − a / c ( sin φ + sin φ0 ) − ρ / c ) − A0 + ∗ τ δ (τ + a / c ( sin φ + sin φ0 ) − ρ / c ) ∞ − + − + − − δ τ / sin φ sin φ 2(2 1) / ρ / a c m a c c ( ) ( ) 0 1 h(τ ) = A1 ∑ U (τ ) ∗ 3/ 2 + (2 m 1) + + − + − δ τ a / c sin φ sin φ 2(2 m 1) a / c ρ / c 0 = m ( ) ( ) 0 δ (τ − a / c ( sin φ + sin φ0 ) − 2(2m)a / c − ρ / c ) − ∞ 1 ∗ U ( τ ) − A1 ∑ 3/ 2 δ (τ + a / c ( sin φ + sin φ0 ) − 2(2m )a / c − ρ / c ) m =1 (2m )
(B.1) A0 =
2
π
sin 12 φ sin 12 φ0 ρ / c cos φ + cos φ0 1
1 sin 12 φ sin 12 φ0 1 1 1 + A1 = 2π cos φ + cos φ0 cos φ cos φ0 ρ a / c
Note that the term of pulse shape with U (τ ) / τ is the first term diffraction, and other terms of pulse shape with U (τ ) are the multiply diffracted rays but are relatively weak as shown in Fig. 6(a). Using Kirchhoff technique, the impulse response of the channel of a single conducting rectangular obstacle Fig.5 is given by t 1 U (τ ) δ (τ ) − δ (τ ) − a ∗ (B.2) π τ + ta τ 1 h (τ ) = 2 δ (τ − d / c ) ∗ tb 1 U (τ ) δ τ ( ) − π τ t + τ b t a = π a 2 ( D 1 + D 2 ) /( 2 cD 1 D 2 )
c = 0.3 m/ns. 1. 2. 3. 4. 5. 6. 7.
8.
9.
tb = π b2 ( D1 + D2 ) /(2cD1D2 )
Reference
A. J. Viterbi, “Four laws of nature and society,” Epilogue, Wireless communications (Signal processing prospective), Edited by H. V. Poor and G. W. Wornell, Prentice-Hall, 1998 R. C. Qiu, H. Liu, S. Shen, M. Guizani, ”Ultra-wideband for Multiple Access Communications,” IEEE Communications Magazine, To appear, Feb. 2005. R. C. Qiu, “UWB Pulse Propagation and Detection,” Book Chapter, Ultra-Wideband Wireless Communications, Editors: S. Shen, M. Guizani, R. C. Qiu, T. Le-Ngoc, John Wiley and Sons, 2005. R. C. Qiu, Book Chapter “UWB Pulse Propagation Processes,” UWB Wireless Communications - A Comprehensive Overview, Eurasip, 2005. R. C. Qiu, “UWB Wireless Communications,” Book Chapter, Design and Analysis of Wireless Networks, Nova Science Publishers, 2004. R. C. Qiu, “Ultra-wideband Wireless Communications,” Seminar slides, Army Research Lab, Adephi, MD, June 4, 2004. R.C. Qiu, “A generalized time domain multipath channel and its application in ultra-wideband (UWB) wireless optimal receiver design: part III system performance analysis,” IEEE Trans. Wireless Communications, to appear. R.C. Qiu, “A generalized time domain multipath channel and its application in ultra-wideband (UWB) wireless optimal receiver design: part II wave-based system analysis,” IEEE Trans. Wireless Communications, Vol. 3, No.6, pp. 1-14, Nov. 2004. R.C. Qiu, “A Study of the Ultra-wideband wireless propagation channel and optimum UWB receiver design,Part I” IEEE J. Selected Areas in Commun. (JSAC), the First Special Issue on UWB Radio in Multiple Access Wireless Comm., Vol. 20, No. 9, pp. 1628-1637, December 2002.
10. R.C. Qiu and I.T. Lu, “Multipath resolving with frequency dependence for broadband wireless channel modeling,” IEEE Trans. Veh. Tech., Vol. 48, No. 1, pp. 273-285, January 1999. 11. M. McClure, R.C. Qiu, and L. Carin, “On the Superresolution of Observables from Swept-Frequency Scattering Data,” IEEE Trans. Antenna Propagation, Vol. 45, No. 4, pp. 631-641, April 1997. 12. I-Tai Lu and R.C. Qiu, “A Novel High-Resolution Algorithm for Complex-Direction Search,” J. Acoustics Society of America, Vol. 104, No. 1, pp. 288-299, 1998. 13. R.C. Qiu, Ph.D. Dissertation, Digital Transmission Media: UWB Wireless Channel, MMIC, and Chiral Fiber, Polytechnic University, Brooklyn, New York, January 1996. Also available at http://iweb.tntech.edu/rqiu/publications3.htm 14. R.C. Qiu and L. Carin, “Early-To-Late-Time Superresolution Signal Processing of Time-Domain Scattering Data,” USNC/URSI, National Radio Science Meeting, Boulder, Colorado, USA, January 1995. 15. R.C. Qiu and I-Tai Lu, “A Novel High-Resolution Algorithm for Ray Path Resolving and Wireless Channel Modeling,” IEEE Princeton/Central Jersey Sarnoff Symposium, Princeton, NJ, April 28, 1995. 16. R.C. Qiu and I-Tai Lu, “Modeling the Indoor Radio Propagation Channel using the High Resolution DSP Algorithms,” 6th IEEE International Symposium on Personal, Indoor, and Mobile Radio Communications (PIMRC 95), Toronto, Canada, September 27-29, 1995. 17. R. C. Qiu and I-Tai Lu, A Novel Approach with Superresolution and Ray Frequency Dependence Trace for Channel Modeling and Applications in CDMA Cellular Systems, IEEE Wireless Communications Systems Symposium "Wireless Trends in 21st Century," Long Island, New York, Nov. 27-29, 1995. 18. R.C. Qiu and I.T. Lu, “A Novel Model of Time-Dispersion Multipath Channel for Wideband CDMA Systems,” IEEE ICUPC’96, pp. 350354, September 1996. 19. R.C. Qiu and I-Tai Lu, “Wideband Wireless Multipath Channel Modeling with Path Frequency Dependence,” IEEE International Conference on Communications (ICC'96), Dallas, Texas, June 23-27, 1996. 20. R.C. Qiu, “A Theoretical Study of the Ultra-Wideband Wireless Propagation Channel Based on the Scattering Centers,” Proc. IEEE VTC’98, Ottawa, Canada, May 1998. 21. Andy Molisch et al, “IEEE 802.15.4a channel models-final report,” IEEE 802.15.4a Standard Committee, Nov. 2004. 22. B. Sadler and A. Swami, “On the performance of episodic UWB and direct-sequence communication system, ” IEEE Trans. Wireless Communications, To appear. 23. T. Kailath, H. V. Poor, “Detection of stochastic processes,” IEEE Trans. Inform. Theory, vol. 44, No. 6, pp. 2230-2259, Oct. 1998. 24. IEEE 802.15.3a Task Group, http://ieee802.org/15/pub/TG3a.html 25. IEEE 802.15.4a Task Group, http://www.ieee802.org/15/pub/TG4a.html 26. M. Emami, et al “Application of time-reversal with MMSE equalizer to UWB communications,” IEEE Comm. Lett., To appear. 27. Y. Li, A. Molisch, J. Zhang,“Channel estimation and signal detection for UWB,” Proc. Of WPMC 2003. 28. M. Weisenhorn, W. Hirt, “Performance of Binary Antipodal Signaling over the Indoor UWB MIMO Channel,” ICC’03, Anchorage, AK, pp. 2872-2878, May 11-15, 2003. 29. J. J. Bowman, T. B. A. Senior, P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering By Simple Shapes, Hemisphere Publishing, 1987. 30. H. Mokhtari and P. Lazaridis, “Comparative study of lateral profile knife edge diffraction and ray tracing technique using GTD in urban environment,” IEEE Trans. Veh. Tech., vol. 48, No.1, pp. 255-261, Jan 1999.
7
I. INTRODUCTION odern communication theory originated from the
of communication engineers to Mattempts understand what they were doing in the most general terms. The limit of digital wireless communication networks depends primarily on four basic laws and their underlying theories, which attributed respectively to: Maxwell and Hertz, Shannon, Moore, and Metcalfe [1]. We followed this philosophy in the previous work [2-20] 1. This paper attempts to seek optimum solutions predicted by both Maxwell/Hertz and Shannon. In a recent three-paper series [7-9], we have established the optimum receiver structure paradigm based on the so-called physics-based multipath channel model [2-20]. This frequency dependence model has been very recently 1
All our conference and journal papers and Ph. D. dissertation are also available in our website http://iweb.tntech.edu/rqiu/index.htm.
accepted in the IEEE 802.15.4a [21]. As pointed by its channel model subcommittee in [21], the first suggestion of using such a model was made for UWB wireless systems in indoor environments one decade ago [13]. This model has found many applications, e.g. [22]. This paper represents a major step of our research [2,6]. We extend the previous paradigm to include the important phenomenon of inter-symbol interference (ISI) [27]. ISI ultimately limits the maximum achievable data rate. In this paper we find that the per-path pulse distortion has impact on this maximum achievable data rate in the presence of ISI. In the absence of ISI, the per-path pulse distortion only affects the front-end matched filter design of the optimum receiver, but the performance in terms of bit error rate (BER) is not affected if the symbol or pulse energy is identical with or without pulse distortion. This is the property of the matched filter. Once matched, the performance only depends on the energy of the pulses, independent of the pulse shape. However this property is invalid when ISI is present. As examples of this discovery, in this paper we investigate the optimum receiver structure and two sub-optimum receivers with zero-forcing and minimum mean square error (MMSE) equalizers. This model captures some properties that are not available in the IEEE 802.15.4a model. It may be informative if the physics-based model is in closed form and the performance loss in signal to noise ratio (SNR), β n defined in (14), will be explicitly connected with physical time-domain propagation problems. For narrowband systems, this connection may be unfeasibly complicated since most propagation mechanisms are overlapping in time domain. However, for UWB communications, most propagation mechanisms are separable in time domain. Explicitly connecting these mechanisms with system performance metrics helps us to understand the limits of UWB communications deterministically. One day the system can be designed to adapt itself to specific environments for optimum network
1
where Ψ( x, y , z ) determines the wavefront at the (x,y,z) in question, i.e., Ψ ( x, y , z ) − ct = 0 is the discontinuity
performance. Consequently understanding the physic-based multipath model is of a fundamental significance.
hypersurface on which
II. Physics-Based Channel and Signals One big challenge of UWB is the per-path pulse distortion. Mathematically the generalized channel model is expressed by [2-5] L
h(τ ) = ∑ An hn (τ ) ∗ δ (τ − τ n )
hn (τ ) = where
(1)
n =1
n
α n assumes an arbitrary real value (possibly random
function and the unit function. A special case of α n
= α0
of (4) was recently accepted in the IEEE 802.15.4a. Therein α 0 is a random variable varying between 0.8 and
hn (τ ) represents an arbitrary function that has finite
energy. Symbol “ ∗ ” denotes convolution and δ ( x ) is the Dirac Delta function. The statistical parameterization of hn (τ ) is a challenging task. Turin’s model expressed as L
An τ − (1+αn )U (τ ) , H n (ω ) = ( jω )α (4) Γ( −α n )
variable) and U (.) and Γ(.) are, respectively, the Gamma
where L generalized paths are associated with amplitude An , delay τ n , and per-path impulse response hn (τ ) . The
h(τ ) = ∑ Anδ (τ − τ n )
hn (τ ) has a finite jump. For an
important subclass of (3) and (4), we have
1.4 [21]. III. Optimum Detection of Physics-based Signals
We will incorporate the per-path impulse response into the optimal receiver design when ISI is present. We will pay special attention to the hn (τ ) . For PAM signals in a single-
(2)
n =1
is a special case of Eq. (1) when hn (τ ) = δ (τ ) . As an example, let us take a look at the diffracted rays. The geometric optics rays that are reflected by and pass though walls, furniture, trees, and etc, can be treated as a special form of a diffracted ray [4,8]. Our theory presuppose that we know the behavior of the corresponding impulse response E0 and H 0 (electrical field and magnetic field)
user system, the transmitted signal is expressed as
s(t ) =
∞
∑ b x(t − nT ) n
s
n =−∞
bn represents the n-th discrete information symbol with duration of Ts .
where
which are by definition solutions of Maxwell’s equations generated by the source function g(x, y,z )δ (t) . The wave propagation along a generalized ray can be modeled by the propagation of a singularity in spatial time along such a ray. In the neighborhood of the spatial time singularity at τ = τ α (corresponding to a spatial position along a ray or
Diffracted Signal d(t)and Template Signal v(t) 1.5
1
d(t) and v(t)
0.5
multipath), the impulse response of a generalized ray (any component of E0 ) has the behavior
0
-0.5
alpha= -1: 0.25: 0 (bottom to top)
∞ Cn − ξ ( τ τ ) (τ α − τ ) n , τ < τ α ∑ α n ! n =0 hn (τ ) = ∞ η (τ − τ ) Dn (τ − τ ) n ,τ > τ ∑ α α α n=0 n !
-1
-- alpha=0
(3)
-1.5
0
0.1
0.2
0.3
0.4
0.5 time t (ns)
0.6
0.7
Figure 1 Pulse waveform as a function of in Equation (4).
where ξ (τ ) and η (τ ) are rather general. Then the corresponding asymptotic frequency response has the form The received signals are represented as ∞ [4,8] ∞ t Dm 1 η ( )t m e − t dt − m ∫0 ∞ m ! ( j ) j ω ω H n ( jω ) ≈ e − jk Ψ ∑ ∞ 1 t m −t m = 0 Cm )t e dt ξ( m ∫0 m ! ( − jω ) − jω
Incident Waveform
__Red dashed Template Pulse v(t)
r (t ) = ∑ bn y (t − nTs ) + n (t )
0.8
α
0.9
1
defined as
(5)
k =0
where n (t ) is AWGN. It has been shown for a narrowband system [23] that the optimum receiver structure is illustrated in Fig. 2.
2
The sampled sequence is processed by a maximum likelihood sequential estimator (MLSE) detector that was first derived by Forney (1972). This receiver structure is optimal in terms of minimizing the probability of transmission error in detecting the information sequence. The assumption of Forney (1972) was that the receive signal has finite energy. If in the UWB receiver design we assume that y(t) is of finite energy, the structure of Fig. 2 can immediately be applied to our UWB problem [2,6]. As a result the output of the matched filter is expressed as ∞
q(t ) = ∑ bn R yy (t − nTs ) + v (t )
(6)
qk = bk +
∞
∑
n =0,n ≠ k
where the
bn Rk −n + vk
(8)
bk term represents the expected information
symbol of the k-th sampling period, the 2nd term is the ISI, and vk is the additive Gaussian variable at the k-th sampling point. The sequence of
qk will be processed at the symbol
level by MLSE (Viterbi algorithm) before being sent to decision circuits.
where R yy ( t ) = y ( t ) ⊗ y * ( − t ) = Rxx ( t ) ⊗ h ( t ) ⊗ h * ( − t )
If the duration of each symbol is small such that R yy (t ) = 0 at instants t = (n − k )Ts for n ≠ k , (or,
is the autocorrelation of y (t ) and v (t ) is the response of
Rk −n = 0 for n ≠ k ), no ISI occurs. As a result (8)
n =0
the matched filter to AWGN noise n (t ) .
Rxx (t ) is the
autocorrelation of x ( t ) . See Appendix A for explicit expressions for R yy ( t ) and the cross-correlation Rxy ( t ) .
= kTs , we denote = R yy (t )δ [t − (n − k )Ts ]
Sampling at instants t
Rk − n
After the sampler in Fig. 2, the discrete signals at times t = kTs of (6) are given by As given in (6), the output of the matched
is Ryy (t ) = Rxx (t ) ⊗ Rhh (t ) .
filter
Rxx (t ) is
typically of a smooth shape. Finally, a time-reversal scheme is used to shift the design complexity from the receiver to the transmitter [26]. In time-reversal a signal is precoded such that it focuses both in time and in space at a particular receiver. Due to temporal focusing, the received power is concentrated within a few taps and the task of equalizer design becomes much simpler than without focusing. If we use the pre-filter (at the transmitter) which
h (t )
, the output of the matched has an impulse response of filter (at the receiver) is Ryy (t ) = Rxx (t ) ⊗ Rhh (t ) . If the first tap is used (shown in Fig.3), the collected energy is above 50% of the total energy. Pulse distortion is ignored in Fig. 3 since IEEE 802.15.3a model models the per-path impulse response as a Dirac function. It is natural to study the impact of pulse distortion on the Rhh (t ) . The next section provides such a model. ∞
qk = R yy (t ) ∑ bnδ [t − ( n − k )Ts ] + v(t )δ ( t − kTs ) n =0
∞
= ∑ bn Rk −n + vk ;
(7)
k=0,1,2,"
n =0
where
R0 = [ Rk −n ] |k =n (the energy of the n-th symbol) can
be regarded as an arbitrary scale factor and conveniently set to 1. From (7) we obtain
reduces to
qk = bk + vk It is, therefore, sufficient to detect the symbols independently one-by-one for each given instant t = kTs . This is the renowned matched filter for isolated symbol detection. As expected, once matched, the detection depends on the energy E y = R0 = R yy (0) of the symbol only, not the received composite pulse waveform y (t ) . As a result, per-path pulse distortion has no impact on the single symbol matched filter detector for isolated symbol. However, according to (8), we will find that this is not the case when ISI is present. We will evaluate this effect using some closed-form expressions for system performance. If the number of the overlapping symbols is less than ten, the MLSE (in Fig.2) may be even feasible for the state-ofthe-art signal processing capability. Received signal
Matched filter
y * (−t ) r (t ) = y (t ) + n(t )
Sampler clock
t = kT
MLSE (Viterbi algorithm) Output data
Figure 2. Optimal receiver structure in presence of ISI in an AWGN channel.
IV. Sub-Optimum Detection of Physics-based Signals Let us simplify optimum receiver structures in two directions: front-end blocks and sub-optimum decisions. First the matched filter can be simplified in a sub-optimum manner. Since the per-path pulse waveform has been included in the channel impulse, the resultant matched filter is sometimes called “Generalized RAKE” receiver structure [5], of which Altes’ receiver structure is a special case [5]. In absence of per-path pulse distortion, the matched filter
3
reduces to the conventional RAKE receiver of Price & Green (1958). Secondly the optimal decisions can be simplified. Let us follow Kailith and Poor [23] and restrict ourselves to a linear detector (zero-forcing and MMSE) with a matched filter front end [6]. The detected bits are
bˆn = sign( zn ) (9) where z = My , the matrix M is arbitrary, and the vector y has elements of y (t ) |t =nTs . Further we assume bn to take on the values of ±1 with equal probabilities. For zeroforcing, the probability that bn is in error is simply
σ
2
defined in (6), and an is the energy of
(10)
R = [Rk -n ] is
y (t − nTs ) .
Under some general conditions, the error probability of the linear MMSE detector
an M n,n Pe (n) ≈ Q σ ( MRM ) n ,n 2 −2 −1 where M = ( R + σ Da ) and Da = diag ( an ) .
(11)
β n = 1/ ( R -1 )n ,n
(14)
For MMSE equalizers we can similarly define the performance loss in SNR using (11). From (14) we reach
β n = 1 − ρ 2 for two symbols case. As a result β n depends on per-path pulse distortion hn (τ ) through R yy (τ ) in (13) and (6). See also Appendix A. The β n for the n-th sample depends on the whole physics-based channel model (Eq. (1)) and symbol sampling rate. Consequently understanding the physic-based multipath model is of a fundamental significance. The formulas of (5)-(14) are of course valid in absence of per-path pulse distortion, i.e. hn (τ ) = δ (τ ) . The non-ISI case was studied in [7-9]. With suitable manipulation as in [23], the formulation of (5)-(14) will be also valid for multi-user detection. The formulations form a unified framework that connects physics-based time domain channel model with detection theory [6]. Autocorrelation of UW B Channel 1.2 CM3, Channel 1, Energy Ratio = 50% 1
It is observed from (10) and (11) that per-path pulse distortion affects the system performance through the output of the matched filter, i.e., R yy ( t ) , sampled at instants t
2
= (n − k )Ts . When R is diagonal for ISI-free
0.8 Normalized Amplitude
where
an Pe ( n ) = Q σ ( R -1 )n ,n is the noise density, the matrix
to 1 − ρ , from (12). Compared with the isolated single symbol detector, the performance loss in SNR for zeroforcing detector can be generally defined as
0.6
0.4
0.2
situations, the per-path pulse distortion affects each symbol energy separately. As a result
(
R -1 = [ R ]n ,n n ,n
)
−1
0
for -0.2 -100
symbols n=0,1,2,…. The energy for each symbol after the equalizer depends on its own energy of that symbol only. However in general the time-domain overlapping of distorted pulses make the matrix R non-diagonal (ISI occurs). Inter-pusle overlapping combined with per-path pulse distortion makes receiver design very challenging. Example 1 Two-Symbol Case To gain some insights, let us consider a two-symbol case. Denote
1 R = ρ -1
where
ρ
−1
1 1 = 1 1 − ρ 2 − ρ
−ρ 1
(12)
ρ = R1 = R−1 = R yy (τ ) |τ =± T ≤ 1 (13). From (10) and (11) the performance only depends on the -1 diagonal elements of R , which are inversely proportional s
-80
-60
-40
-20
0
20
40
60
80
100
Time (ns)
Figure 3 Auto-correlation of a channel impulse response
Rhh (t ) = h(t ) ∗ h( −t ) for CM3 channel. V. Multi-Path Capture and Time-Reversal
Here the optimum receiver structure of Fig. 2 is used as a heuristic approach to derive some new structures. In Fig. 3, we show the auto-correlation of the channel impulse response, Rhh (t ) = h (t ) ⊗ h( −t ) , using a typical IEEE 802.15.3a model. The
Rhh (t ) is normalized by Rhh (0) , the
energy of the channel impulse response. Figure 2 shows Rhh (t ) / Rhh (0) . It is interesting to notice that the first peak has an energy that is around 50% [26]. We can view each pulse as a chip in a PN code and the channel impulse response can be regarded as a pseudo-noise code where the width of multipath delay spread corresponds to the length of
4
the PN code. This observation first made in [28] intuitively suggests [26] that it is simpler to base the detection on Rhh (t ) , rather than the noisy y (t ) .
x
x=b
VI. Numerical Results
y = −a
φ0
(ρ ,φ ) x
2a
Tx
D1
D2
Rx y=a
y
Fig. 5 An impulse plane wave pulse is incident on the conducting rectangular obstacle. Pulse diffracted by two parallel planes 0.08
0.06
The uppper edge The lower edge
Amplitude
0.04
0.02
0
-0.02
a=5 m The high-order terms Incident Angle =pi/5
-0.04 Observation Angle=2pi/3 -0.06
0
20
40
60 Time(ns)
80
100
120
Pulse diffrac ted by two parallel planes 1.5 The sum of two autocorrelation terms
1
Auto-Correlation
In the simulations, the second order Gaussian pulse is used for incident pulse for all cases. The per path pulse distortion occurs for all paths. We use two simple structures as examples as shown in Fig. 4 and 5. Only diffracted pulses exist for these structures. The frequency domain mathematical formulations of these structures valid for frequencies above 100MHz are well adopted in the literature. Since these formulations are valid for the spectrum of our interest (2-10.6 GHz) in UWB, we convert these formations into the time domain using inverse Laplace transform to give us the impulse response of the channel. The pulse response of a specific pulse is obtained by convolution of the so obtained results with the UWB pulse waveform, as shown in Fig. 6(a) and 7(a). The autocorrelation can be easily calculated as shown in Fig. 6(b) and 7(b). These time domain and frequency domain results are compared and match well. See Appendix B. Fig. 7(a) gives results using four different techniques. The frequency domain techniques are given in [30]. We give time domain results for Kirchhoff technique. GTD results are not given here to save space. All parameters and notations follow the widely adopted frequency domain results in the literature [29,30]. Compared with indoors, the multipath paths are not so rich but pulse distortion seems be another crucial issue. For high-rise buildings (Fig.5) (such as apartments) in urban and suburban environments, pulse distortion caused by diffraction phenomenon needs be stressed although this is not an issue for narrowband system.
Cross-correlation term Cross-correlation term
0.5
0
-0.5
-1 -100
Fig.4 An impulse plane wave pulse is incident on the two parallel conducting planes. The incident and observation angles are, resp.,
φ0 and φ . The width of the two planes is
2a.
To gain insights, we choose two simple environments that have well-known frequency domain harmonic solutions. Fig. 4 can be used to model the UWB pulse propagation around the corner of two streets and two apartments.
-50
0
50
100
150
Time(ns)
Fig. 6 (a), (b) The channel response of two parallel planes (Fig. 4) to the UWB pulse with the incident pulse shape of the second order Gaussian.
Fig. 5 can model the shadowing phenomenon. We ignore antenna effects for simplicity. Antennas can be modeled as linear systems that has effective effects included in equation (1). Another reason for choosing Fig. 4 and Fig. 5 for examples is the ISI consideration. In Fig. 5 although pulse distortion is more severe than
5
z
Pulse througn rec tangular screen 0.2 FD-GTD TD-GTD FD-Kirchhoff TD-Kirchhoff
0.15
0.1
Amplitude
0.05
0
-0.05
-0.1
-0.15
-0.2
0
0.5
D1=500m
D2 =450m
a=15m
b=10 m
1
1.5
2 Time(ns)
2.5
3
3.5
10
System Performance in Two Parallel Plane Model
0
W ithout MMSE Equalizer 10
10
-1
-2
BER
Fig.4 (comparing Fig. 6(a) and 7(a)), the ISI is less severe since the delay spread in Fig. 4 is much bigger than Fig.5. Fig. 8 shows the impact of the diffracted pulse on the ISI in the two parallel planes model (streets, high-rise buildings). We performed Monte Carlo simulations to obtain the BER using the MMSE equalizer (25 taps) for data rate of 200 Mbps. The matched filter is assumed as suggested in the optimum structure of Fig. 2. This gives the performance bound for the MMSE equalizer since channel estimation would introduce some loss in performance. The analytical channel model in Appendix B gives us the capability to study some system issues of which are otherwise incapable, such as the dependence of path loss on pulse delay and distortion and the explicit dependence of these parameters on environmental parameters (street width, building heights, antenna positions and heights, etc.).
10
10
10
MMSE Equalizer
AWGN
-3
-4
-5
0
2
4
6
8
10
12
14
16
18
Eb/No
Fig. 8 The BER of a UWB system in two parallel planes (Fig.4) in presence of the ISI.
V. CONCLUSION This paper extends our previous framework to the case when the ISI is present. In the UWB system design the connection of the ISI with the per-path pulse distortion adds a unique new dimension to the existing issues of a narrowband system. This is the first step towards addressing this problem. Analytically and numerically we illustrate that the diffracted pulses indeed complicate the system analysis. In the future, we will do more statistical analysis once the experimental statistical channel modeling that is able to capture the pulse distortion is made available. The future work involves the estimation of channel parameters related to pulse distortion. ACKNOLEDGMENT
4
The author would like to thank Chenming (Jim) Zhou and Qiang Zhang for their help.
7(a) A UW B Pulse Diffracted by A Rectangular Screen 1
Appendix A Cross-Correlation and AutoCorrelation
FD-GTD TD-GTD FD-Kirchhoff TD-Kirchhoff
0.8
L −1
Rxy (τ ) = ∑ al Rxx (τ ) ∗ hl (τ ) * δ (τ − τ l )
0.6
(A.1)
l =0
R y ( t ) y ( t ) (τ ) = Rx ( t ) x ( t ) (τ ) ∗ [h (τ ) ∗ h ( −τ ) ] ( A.2)
AutoCorrelation
0.4
L −1
R yy (τ ) = ∑ al2 {Rxx (τ ) * [ hl (τ ) ∗ hl ( −τ ) ]} +
0.2
l =0
0
L −1
L −1
i =0 i ≠k
k =0
∑ ∑ a a {R i k
-0.2
-0.4
xx
(A.3)
(τ ) ∗ [ hi (τ ) * hk ( −τ ) ]} * δ [τ − (τ k − τ i )]
For the commonly used second derivative Gaussian 2 pulse x ( t ) = d 2 / dt 2 e − t , we have Rxx (τ ) = d 4 / dτ 4 e −τ / 2 . 2
-0.6 -10
-8
-6
-4
-2
0 Time Shift(ns)
2
4
6
8
10
7(b) Fig. 7 The channel response of a rectangular obstacle (Fig. 5) to the UWB pulse with the incident pulse shape of the second order Gaussian.
Appendix B Closed Form Expression for Impulse Response
The impulse response of the channel of two parallel conducting planes (Fig.4) is given by
6
U (τ ) δ (τ − a / c ( sin φ + sin φ0 ) − ρ / c ) − A0 + ∗ τ δ (τ + a / c ( sin φ + sin φ0 ) − ρ / c ) ∞ − + − + − − δ τ / sin φ sin φ 2(2 1) / ρ / a c m a c c ( ) ( ) 0 1 h(τ ) = A1 ∑ U (τ ) ∗ 3/ 2 + (2 m 1) + + − + − δ τ a / c sin φ sin φ 2(2 m 1) a / c ρ / c 0 = m ( ) ( ) 0 δ (τ − a / c ( sin φ + sin φ0 ) − 2(2m)a / c − ρ / c ) − ∞ 1 ∗ U ( τ ) − A1 ∑ 3/ 2 δ (τ + a / c ( sin φ + sin φ0 ) − 2(2m )a / c − ρ / c ) m =1 (2m )
(B.1) A0 =
2
π
sin 12 φ sin 12 φ0 ρ / c cos φ + cos φ0 1
1 sin 12 φ sin 12 φ0 1 1 1 + A1 = 2π cos φ + cos φ0 cos φ cos φ0 ρ a / c
Note that the term of pulse shape with U (τ ) / τ is the first term diffraction, and other terms of pulse shape with U (τ ) are the multiply diffracted rays but are relatively weak as shown in Fig. 6(a). Using Kirchhoff technique, the impulse response of the channel of a single conducting rectangular obstacle Fig.5 is given by t 1 U (τ ) δ (τ ) − δ (τ ) − a ∗ (B.2) π τ + ta τ 1 h (τ ) = 2 δ (τ − d / c ) ∗ tb 1 U (τ ) δ τ ( ) − π τ t + τ b t a = π a 2 ( D 1 + D 2 ) /( 2 cD 1 D 2 )
c = 0.3 m/ns. 1. 2. 3. 4. 5. 6. 7.
8.
9.
tb = π b2 ( D1 + D2 ) /(2cD1D2 )
Reference
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