Code-multiplexed UWB transmitted-reference radio

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 56, NO. 12, DECEMBER 2008

2125

Code-Multiplexed UWB Transmitted-Reference Radio Antonio A. D’Amico and Umberto Mengali, Life Fellow, IEEE Abstract—In traditional transmitted reference (TR) ultrawideband systems the reference component is time-shifted and orthogonal relative to the data-bearing signal. This paves the way to a correlation receiver in which the local template is derived from the incoming waveform using a delay line. As analog delay lines are difficult to implement with current technology, an alternative TR system has recently been proposed in which reference and data components are made orthogonal by a frequency shift rather than a time shift. The resulting receiver has no delay lines and has better performance compared to the traditional scheme. In the present paper we discuss a third way to achieve orthogonality, i.e., by modulating reference and data components with two distinct code sequences. Even in this case the receiver has no delay lines. However, it is simpler to implement and has better performance than the frequency-shift based receiver. Index Terms—Low complexity receivers, transmitted-reference schemes, ultra-wideband communications.

I. I NTRODUCTION

U

LTRA-WIDEBAND (UWB) impulse radio conveys information by transmitting sequences of sub-nanosecond pulses. In an indoor environment each pulse generates hundreds of echoes that, in principle, can be combined in a Rake receiver to exploit the rich channel diversity. Unfortunately the implementation of a Rake is complex because of: (a) the large number of fingers needed to capture a significant fraction of the signal energy [1], [2]; (b) the high sampling rates required by the extremely large transmission bandwidth; (c) the intensive computation involved in estimating gains and delays of the channel paths [3]. Instead of estimating gains and delays, the overall channel response might be measured as a single entity and used as a local template in a correlation receiver. With an ideal template the detection performance would be the same as that of a Rake with infinite fingers and perfect knowledge of the channel paths. However, correlation receivers and Rakes have comparable complexity [4]. Transmitted reference (TR) systems have a simpler structure and can capture the entire signal energy without requiring channel estimation [5], [6], [7]. They operate by transmitting a reference pulse before each data pulse and correlating the channel response to the former with that to the latter. Letting D be the time separation between reference and data pulses and calling r(t) the received waveform, the decision variable

Paper approved by X. Wang, the Editor for Multiuser Detection and Equalization of the IEEE Communications Society. Manuscript received February 27, 2007; revised June 14, 2007. The authors are with the Department of Information Engineering, Via Caruso, 56100 Pisa, Italy (e-mail: {antonio.damico, umberto.mengali}@iet.unipi.it). Digital Object Identifier 10.1109/TCOMM.2008.070105

is formed by computing r(t − D) in an analog delay line and integrating the product r(t)r(t − D) over a symbol period. Notwithstanding this apparent simplicity the implementation of TR receivers is conditioned by the presence of the delay element, which is difficult to build in a integrated fashion [8], [9] and with the required accuracy (a fraction of nanosecond) [10]. For example, in [11] a delay of 20 ns is realized with a 20 foot coaxial cable. Clearly, this is not a viable solution in an integrated receiver. Other TR schemes have been proposed wherein the orthogonality between data and reference pulses is based on a frequency shift rather than a time shift [12], [13]. In this way the delay element is avoided. In particular, in [12] the data component (i.e., the collection of the information bearing pulses) and the reference component (the collection of the reference pulses) have each a comblike spectrum, with teeth of width W and at a distance 2W from each other. The two spectra are shifted by W with respect to each other, which makes them separable. The separation is difficult, however, because W must be small to guarantee that data and reference undergo the same distortions in travelling through the channel. A simpler implementation is achieved in [13] and is referred to as slightly frequency-shifted (SFS) TR scheme. Here, data and reference have standard bell-shaped spectra, shifted from each other by the symbol rate 1/Ts . With low rate transmissions the shift is much smaller than the channel coherence bandwidth so that data and reference are distorted in the same way. Their separation at the receiver relies on their orthogonality over a period of Ts seconds. The decision statistics is computed by squaring the incoming waveform, multiplying it by a sinusoidal wave at frequency 1/Ts , and integrating the result over Ts seconds. Finally, reference [14] describes a multi-carrier differential signaling scheme that avoids the analog delay element and has better performance than SFS-TR at higher data rates. The receiver implementation, however, looks much more complex than in [13]. In the present paper we parallel the approach in [13] but we base our development on code multiplexing rather than frequency multiplexing. In our scheme, data and reference are derived from amplitude modulating the same pulse train with two orthogonal code sequences. This feature is exploited to extract the reference at the receiver. As in the SFS-TR scheme, no delay lines are needed. Compared with SFS-TR, the code multiplexed TR scheme (henceforth, CM-TR) has advantages in terms of implementation and performance. As we shall see, at low data rates its bit-error-rate (BER) is virtually identical to that of SFS-TR. When the data rate increases, however, CM-TR takes the lead

c 2008 IEEE 0090-6778/08$25.00 

2126

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 56, NO. 12, DECEMBER 2008

and exhibits significant gains. Also, transmitter and receiver are simpler to implement as they do not involve frequency conversion. The rest of paper is organized as follows. The next section describes the signal model and provides transmit/receive structures for CM-TR. To underline similarities/differences with SFS-TR, an overview of the latter scheme is provided. The error probability of CM-TR is computed in Section III assuming a data rate sufficiently low as to exclude any inter-frame interference (IFI). In these conditions the BER performance is independent of the sequences being used. In Section IV we allow a moderate amount of IFI and we look for the code sequences that minimize its effects. Performance comparisons between CM-TR and SFS-TR are made in Section V with analytical and simulation tools. Conclusions are drawn in Section VI. II. CM-TR S CHEME We start with an overview of SFS-TR. The information is transmitted at 1/Ts bit/s and, on the kth signaling interval Ik := {kTs ≤ t < (k + 1)Ts }, the signal √ s(t) = p(t− kTs )+ ak p(t− kTs ) 2 cos[2π(t− kTs )/Ts ] (1) is input to the channel, where ak is the kth data symbol (equal to ±1) and Nf −1  cm g(t − mTf ) (2) p(t) = m=0

is a regular train of g(t)-shaped pulses (monocycles), each with support 0 ≤ t < Tg . The monocycles are very short (on the order of the nanosecond) and are confined in frames of length Tf >> Tg . Also, their polarity is switched according to a Nf −1 code sequence {cm }m=0 , with cm = ±1. The purpose of this sequence is to smooth the signal spectrum1. Further smoothing may be obtained by dithering the monocycles according to some time-hopping code. This option is not considered in the sequel as it does not affect the system performance while it formally complicates the discussion. Note that, as a symbol interval has Nf frames, we have Ts = Nf Tf . From (1) it is seen that the signal is made of two parts: a reference component, which does not bear information, and a data component, which is proportional to the data symbol. The latter is frequency shifted by 1/Ts , which makes it orthogonal to the former for Nf ≥ 2. In [13] it is shown that the orthogonality still holds at the receiver, provided that Tf is made longer than the channel delay spread so that IFI is avoided. In these conditions a heuristic detection strategy is proposed in [13] wherein the variable  (k+1)Ts +τ √ zk = r2 (t) 2 cos[2π(t − kTs − τ )/Ts ]dt (3) kTs +τ

is computed and the decision a ˆk = ±1 is made, depending on the sign of zk . In equation (3) r(t) represents the received waveform and τ is the channel propagation delay (defined later). Figure 1 illustrates the block diagram of the SFS-TR receiver. 1 Actually, no code is used in [13]. We adopt a code sequence here as it simplifies the introduction of the CM-TR scheme later.

r(t)

( • )2

aˆ k

zk




2 cos [ 2 (t
kTs
 ) / Ts ] Fig. 1.

Block diagram of the SFS-TR receiver.

Turning our attention to CM-TR we observe that (1) has the following structure s(t) = p(t − kTs ) + ak p(t − kTs )q(t − kTs ) with q(t) =

√ 2 cos(2πt/Ts )

(4) (5)

and we wonder whether other choices of q(t) exist that guarantee orthogonality between reference and data components, i.e., that satisfy the relationship  (k+1)Ts p2 (t − kTs )q(t − kTs )dt = 0 (6) kTs

We maintain that a possible answer is Nf −1

q(t) =



c˜m cm rectTf (t − mTf )

(7)

m=0 N −1

N −1

f f is a code sequence orthogonal to {cm }m=0 , where { cm }m=0 i.e. Nf −1  cm cm = 0  (8)

m=0

and rectTf (t) is a rectangular pulse of unity height and support 0 ≤ t < Tf . In fact, substituting (2) and (7) into (6) yields, after some straightforward passages,  Tg  (k+1)Ts Nf −1  p2 (t − kTs )q(t − kTs )dt = g 2 (t)dt cm cm  kTs

0

m=0

(9) from which our claim follows in view of (8). Signal s(t) in (4) may be rewritten in a form that suggests a simple transmitter implementation. Bearing in mind (2) and (7) and exploiting the fact that c˜m and cm take values ±1, it is easily checked that Nf −1

p(t)q(t) =



c˜m g(t − mTf )

(10)

m=0

Thus, substituting this result into (4) along with (2) produces Nf −1

s(t) =



Nf −1

cm g(t−mTf −kTs )+ak

m=0



c˜m g(t−mTf −kTs )

m=0

(11)

or, alternatively, Nf −1

s(t) =



bk,m g(t − mTf − kTs )

(12)

m=0

where bk,m = cm + ak c˜m

(13)

D’AMICO and MENGALI: CODE-MULTIPLEXED UWB TRANSMITTED-REFERENCE RADIO

are ternary symbols belonging to the alphabet {−2, 0, +2}. Relation (11) shows that reference and data components are obtained by amplitude modulating the same pulse train with two orthogonal code sequences. Also, relation (12) indicates that s(t) may be obtained from a single pulse generator with a ternary output. Contrary to SFS-TR, no frequency conversion is needed. To proceed further we concentrate on the CM-TR receiver. The propagation occurs on a L-path channel with impulse response L−1  α δ(t − τ ) (14) c(t) = =0

where δ(t) is the Dirac delta function and α and τ are the gain and delay of the th path. Without loss of generality  and we refer to τ := τ0 as we take τ0 < τ1 ... < τL−1 the channel propagation delay. Such a delay is estimated by the synchronizer at the receiver. For now we assume perfect estimation. The receive filter has a rectangular transfer function with a bandwidth B sufficiently large to pass the signal pulses undistorted. Thus, letting τ = τ − τ , the overall system response to g(t) may be expressed as h(t − τ ), with h(t) =

L−1 

α g(t − τ )

(15)

=0

and from (12)-(13) the received waveform becomes

r(t)

2127

( • )2




r(t) =

m=0

where n(t) represents thermal noise. In this paper we consider low to average data-rate transmissions and, for the time being, we take Tf longer than the duration Th of h(t) so that no IFI occurs. Later this constraint is removed and situations involving IFI are considered. In the absence of IFI it is easily found that reference and data components in (16) are orthogonal. As the same occurs in SFS-TR, we adopt for CM-TR a detection scheme similar to that in [13]. In particular, the√decision variable is computed as in (3), except that function 2 cos(2πt/Ts ) in the integral is replaced by the q(t) in (7) zk =

⎡

r (t) ⎣ 2

Nf −1



⎤

c˜m cm rectTf (t − mTf − kTs − τ )⎦ dt

m=0

kTs +τ

(17)

The right hand side may be suitably rearranged carrying the summation out of the integral and splitting the integration into Nf frame intervals 

Nf −1

zk =



m=0

c˜m cm

kTs +(m+1)Tf +τ

kTs +mTf +τ



yk,m

r2 (t)dt

zk

Fig. 2.

(18)



The resulting receiver scheme is shown in Fig. 2. Comparisons with Fig. 1 are interesting as they show the simpler implementation of CM-TR. In fact, while the multiplier in Fig. 1 is an ultra wideband mixer, that in Fig. 2 is only a sign changer (recall that c˜m cm = ±1). Its purpose

aˆ k

Block diagram of the CM-TR receiver.

is to switch the sign of the integrator output according to the value of c˜m cm . A physical interpretation of (18) is worthwhile. Assume for simplicity that the noise is negligible. Then, substituting (16) into (18) and bearing in mind that the h(t)-pulses do not overlap, the quantity yk,m is found to be yk,m = (cm + ak c˜m )2 Eh

(19)

where Eh is the energy of h(t). From (15) we have Eh = Eg

L−1  L−1 

α αm ρ(τ − τm )

(20)

=0 m=0

where Eg is the energy of g(t) and  ∞ 1 g(t − τ )g(t − τm )dt ρ(τ − τm ) = Eg −∞

(21)

Thus, substituting (19) into (18) and exploiting the orthogonality of the code sequences, yields zk = 2ak Nf Eh

(cm + ak c˜m )h(t − mTf − kTs − τ ) + n(t) (16)

(k+1)T  s +τ


c
m cm

Nf −1



yk, m

(22)

which says that ak equals the sign of zk , as indicated in Fig. 2. A few words about the synchronization algorithm are in order. As mentioned earlier, the propagation delay τ in (18) is not known and must be estimated. This can be done in a way similar to that indicated in [13]. Let τ˜ be a trial value of τ and consider the random variable Nf −1  kTs +(m+1)Tf +˜τ  τ) = c˜m cm r2 (t)dt (23) Λk (˜ m=0

kTs +mTf +˜ τ

It can be shown that its expectation E{Λk (˜ τ )2 } varies with τ˜ and has two peaks on the interval −Ts /2 ≤ τ˜ < Ts /2, the highest at τ˜ = τ , the lowest at τ˜ = τ + Ts /2. The former corresponds to perfect synchronization and can be found through a search as follows. The synchronizer varies τ˜ with a small resolution over the uncertainty range −Ts /2 ≤ τ˜ < Ts /2 and, for each τ˜, it computes E{Λk (˜ τ )2 } as an arithmetic mean over several symbols. The delay τ is estimated as the argument of the maximum mean. III. E RROR P ROBABILITY In this Section we compute the conditional error probability PCM−T R (e|h(t)) of CM-TR for a given channel response h(t). The unconditional probability is obtained by averaging over several channel realizations {h(i) (t)} N −1 1  PCM−T R ∼ PCM−T R (e|h(i) (t)). = N i=0

(24)

2128

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 56, NO. 12, DECEMBER 2008

Without loss of generality we concentrate on the detection of a0 and we restrict our attention to the interval t ∈ [τ, Ts + τ ). We assume that the channel response duration is shorter than the frame period so that IFI is absent. Substituting (16) into (18) and exploiting (8), the decision variable becomes

zS×N = 2

(m+1)T  f +τ

n(t)h(t − mTf − τ )dt mTf +τ (m+1)T  f +τ

Nf −1



zN ×N =

c˜m cm

m=0

n2 (t)dt

mTf +τ

Alternatively, (25) may be written as z0 = a0 Es + ν

(29)

where Es = 2Nf Eh = 2Nf Eg

L−1  L−1 

α αm ρ(τ − τm )

(30)

=0 m=0

is the received energy per symbol and ν = zS×N + zN ×N is an (approximately) Gaussian random variable with zero-mean and variance σν2 . Thus, the error probability is given by

 Es PCM−T R (e|h(t)) = Q (31) σν where Q(x) is the standard Gaussian tail function. The calculation of σν2 is straightforward and is skipped for brevity. Assuming BTf  1, the final result is σν2 = 2N0 Es + N02 BTs so that (31) becomes PCM−T R (e|h(t)) = Q



Es

 2N0 Es + N02 BTs

(32)  .

(33)

(34) On the other hand, under the same assumptions from (30) we have L−1  α2 (35) Es = Es =0

=0

We

So far IFI has been excluded assuming Tf longer than the channel response duration Th . As is seen from (33), in these conditions the error probability is independent of the code sequences being used. However, as the transmission rate increases and Tf decreases, eventually IFI starts appearing. In this section we investigate the IFI effects arising for Tf < Th ≤ 2Tf . To proceed, we split the channel response in two parts h(t) = h0 (t) + h1 (t − Tf )

(37)

with h0 (t) and h1 (t) having both support [0, Tf ]. The former represents the portion of h(t) within a frame interval while the latter is the tail of h(t) beyond that interval. Next, paralleling the arguments in the previous section it is found that the decision variable has still the form (25) in which, however, zS×S is given by

 Nf −1  zS×S = 2a0 Eh,0 Nf + Eh,1 cm−1 c˜m−1 cm c˜m m=1

+ 2a−1 Eh,1 cNf −1 c˜Nf −1 c0 c˜0  + 2γ Eh,0 Eh,1 f (a0 , a−1 , c, ˜c)

It is interesting to compare (33) with the error probability PSF S−T R (e|h(t)) of SFS-TR. Assuming for simplicity that the multipath components do not overlap (i.e., ρ(τ − τm ) = 0 for  = m) and calling Es = 2Nf Eg the transmitted energy per symbol, it is found [13, eq. (16)] ⎞ ⎛ L−1  ⎜ Es α2 cos(2πτ /Ts ) ⎟ ⎟ ⎜ =0 ⎟ PSF S−T R (e|h(t)) = Q ⎜ ⎟ ⎜ L−1  ⎠ ⎝ 5 2 + N 2 BT  N E α s 0  2 0 s =0

⎜ ⎟ ⎜ ⎟ =0 ⎜ ⎟ PCM−T R (e|h(t)) = Q ⎜  ⎟ L−1 ⎝ ⎠  2N0 Es α2 + N02 BTs

IV. O PTIMAL C ODE S EQUENCES

(26)

(˜ cm + a 0 cm )

⎞ α2

(28)

zS×S = 2a0 Nf Eh

m=0

L−1 

(27)

(25)

with 

Es

(36) see that two concurrent circumstances make PCM−T R (e|h(t)) smaller than PSF S−T R (e|h(t)). On one side, the first term under square root in (36) is smaller than in (34). On the other, the numerator in the argument of the Q-function in (36) is greater than in (34) since cos(2πτ /Ts ) ≤ 1. This second circumstance becomes more and more important as the data rate increases and the ratios τ /Ts depart from zero.

z0 = zS×S + zS×N + zN ×N

Nf −1

so that (33) may also be written as ⎛

(38)

In this formula Eh,0 and Eh,1 are the energies of h0 (t) and h1 (t) respectively, γ is the correlation coefficient  Tf 1 γ :=  h0 (t)h1 (t)dt (39) Eh,0 Eh,1 0 and f (a0 , a−1 , c, ˜c) is a function of the symbols a0 and a−1 as well as of the code sequences c = (c0 , c1 , ..., cNf −1 ) and ˜ = (˜ c c0 , c˜1 , ..., c˜Nf −1 ) f (a0 , a−1 ,c, ˜c) = a−1 a0 c0 c˜Nf −1 + a−1 c˜Nf −1 c˜0 Nf −1



+ a0 cNf −1 c0 + a0

(cm−1 cm + c˜m−1 c˜m )

m=1 Nf −1

+ cNf −1 c˜0 +



(cm−1 c˜m + c˜m−1 cm )

m=1

The variance of ν = zS×N + zN ×N is found to be

(40)

D’AMICO and MENGALI: CODE-MULTIPLEXED UWB TRANSMITTED-REFERENCE RADIO

 σν2 = 4N0



  Nf (Eh,0 + Eh,1 ) + γ Eh,0 Eh,1 cNf −1 c0 +  (cm−1 cm + c˜m−1 c˜m ) + N02 BTs

m=1

The following remarks are of interest. With a UWB channel the correlation coefficient γ is very small since the individual pulses in h(t) are sparsely distributed in time and the integral in (39) takes negligible values. Accordingly (41) may be approximated as σν2 4N0 Nf (Eh,0 + Eh,1 ) + N02 BTs

m=1

(43)

The last term represents inter-symbol interference and, as expected, it vanishes as Eh,1 goes to zero. The useful term (proportional to a0 ) depends on the code sequences and has a strength that increases with the quantity Nf −1

Q :=



cm−1 c˜m−1 cm c˜m

(44)

m=1

Thus, the best receiver performance is achieved choosing c c such that Q is as large as possible. and ˜ The selection of the optimal code sequences can be addressed assuming that their length is a power of 2, say Nf = 2kf +1 , and they are picked up from the rows of a Nf × Nf Hadamard matrix C kf +1 of order kf + 1. As is known, Hadamard matrices of any order can be generated through the recursive relation   Ck Ck C k+1 = k = 1, 2, . . . (45) C k −C k with

 C 1 :=

 1 1 1 −1

(46)

Within this framework the optimal code sequences are found (i) as follows. Denoting ckf the ith row of C kf , from (45) it (i)

is seen that each ckf generates two rows of C kf +1 , namely (i)

(i)

(i)

(i)





 

(42)

which coincides with (32), as is checked bearing in mind that Eh,0 + Eh,1 = Eh and Es = 2Nf Eh . In conclusion, the noise component in the decision variable has the same strength as in the absence of IFI. Turning our attention to the signal component, from (38) we have for γ 1

 Nf −1  zS×S 2a0 Eh,0 Nf + Eh,1 cm−1 c˜m−1 cm c˜m + 2a−1 Eh,1 cNf −1 c˜Nf −1 c0 c˜0



(41)

 



Nf −1



2129

(ckf , ckf ) and (ckf , −ckf ). We call them conjugate rows. Note that there are Nf /2 pairs of such rows. In the Appendix it is shown that Q achieves the maximum Nf − 3 if and only if the code sequences form a conjugate pair. For example, for Nf = 4 (i.e., kf = 1) the maximum is Q = 1 and is achieved with the pairs (1, 1, 1, 1) and (1, 1, −1, −1) or (1, −1, 1, −1) and (1, −1, −1, 1).









   





Fig. 3. Performance comparisons between CM-TR and CR with AWGN and CM4 channels. Marks represent simulations; lines are drawn to ease the reading.

V. S IMULATION R ESULTS Computer simulations have been run to assess the performance of CM-TR and make comparisons with SFS-TR in terms of BER. The following setting has been chosen. The monocycle g(t) is shaped as the second derivative of a Gaussian function with duration Tg = 1 ns. The receive filter bandwidth is B = 4 GHz. The channel statistics are taken from the IEEE 802.15.4a models CM4 and CM5 [15]. As is known, CM4 represents an indoor non-line-of-sight (NLOS) office environment with rms delay spread of about 13 ns, while CM5 reflects a line-of-sight (LOS) outdoor scenario with rms delay spread of about 29 ns. Parameters Tf and Nf are varied so as to assess the influence of data rate and IFI. The BER performance is expressed as a function of the signal-to-noise ratio Eh /N0 . This is equivalent to comparing CM-TR and SFS-TR for the same ratio Es /N0 . The reason is that in both CM-TR and SFS-TR we have Es = 2Nf Eg and from (30) we see that Eh and Es are proportional (for a given channel). In general, the code sequences are chosen as conjugate rows of C kf +1 . In particular, denoting 1n the n-dimensional vector ˜ = (1Nf /2 , −1Nf /2 ). of all 1’s, we take c = 1Nf and c An exception is made in Fig. 7 where the difference in performance between conjugate and non-conjugate sequences is illustrated. Ideal synchronization is assumed, meaning that, for each SNR and channel realization, the position of the integration window is experimentally sought so as to minimize the BER. Figure 3 compares CM-TR with a correlation receiver (CR) in which the transmitted pulse g(t) is used as local template. The frame duration is Tf = 50 ns and Nf = 4, which corresponds to a data rate of Rb =5 Mbit/s. Propagation occurs through either an AWGN channel or a multipath CM4 channel. In the former case g(t) coincides with the channel response

2130

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 56, NO. 12, DECEMBER 2008











   



  





   

  







   





Fig. 4. Comparison between theoretical and simulation results for CM-TR and SFS-TR in the absence of IFI with CM4 channel. Solid and dashed lines represent theory, circles and squares are simulations.

so that CR has the big advantage of knowing the shape of the received signal.2 To make the comparison fair we assume that CM-TR has knowledge of the duration of g(t) so that, in computing the decision variable in (18), the integrations can be limited to intervals of size Tg (rather than Tf ) to improve the SNR. In these conditions it is seen from the figure that CR has a gain of about 3 dB over CM-TR. The situation is more than reversed in passing to CM4. This is so because the channel response is now made of hundreds of pulses and CR can only capture a very small fraction of the available signal energy. Figure 4 illustrates theoretical and simulation results for CM-TR and SFS-TR when Tf is large enough to avoid IFI. The channel is CM4 and Tf is 50 ns, as before. Solid lines represent CM-TR and are computed from (33), letting Es = 2Nf Eh . Dashed lines correspond to SFS-TR and are derived3 as indicated in [13, Section III]. Circles and squares represent simulation results. The data rate Rb is either 2.5 or 5.0 Mbit/s. As is seen, there is good agreement between theory and experiments and, as predicted earlier, the gap between CM-TR and SFS-TR widens as the data rate increases. Analogous results are shown in Fig. 5 where Tf is now reduced to 16 ns and the data rate is either 3.9 or 7.8 Mbit/s. Although Tf is slightly greater than the rms channel delay spread (13 ns), IFI takes place with many channel realizations and the BER expression (33) does not hold any longer. So, as opposed to Fig. 4, the solid/dashed lines in Fig. 5 do not represent the theory: they only serve to ease the reading. 2 We disregard signal distortions due to frequency-dependent behavior of antennas and free space. We also ignore that different environments and different look angles of a given antenna give rise to a variety of the channel response, making it difficult to implement a correlation receiver with a template suited to all circumstances. 3 Equation (34) has not been used since the assumption of pulse separability does not hold with CM4. The method in [13, Section III] is more general and applies with any channel.



    

   



Fig. 5. Comparison of CM-TR with SFS-TR in the presence of IFI with CM4 channel. Solid and dashed lines serve to ease the reading.

Figure 6 illustrates BER curves with CM5. The frame duration is twice that in Fig. 5 (Tf = 32 ns) while Nf is reduced so as to keep the same data rates. It is seen that the superiority of CM-TR increases in passing to an outdoor scenario. Physically, this is due to the larger ratios τ /Ts . For example, at BER=10−3 and Rb = 3.9 Mbit/s, CM-TR and SFS-TR have virtually the same performance with CM4 whereas they are 2 dB apart with CM5. Finally, Fig. 7 shows performance degradations incurred in passing from conjugate to non-conjugate code sequences. In the latter case c is made of all 1’s while ˜c is an alternate sequence of +1 and -1. There are Nf =8 frames per symbol and Tf is 16 ns, which is comparable with the delay spread of the CM4 channel. In these conditions IFI takes place with many channel realizations. As is seen, conjugate sequences offer a significant advantage. VI. C ONCLUSIONS Code sequences have been indicated as a third option to transmit reference signals in UWB communications. Thus, there are three alternative ways to achieve orthogonality between reference and data components: time-, frequency- and code-multiplexing. The first way has the drawback that it involves analog delay lines. The second avoids delay lines and, as argued in [13], has better performance than timemultiplexing. We have shown that delay lines are also avoided with code-multiplexing. The advantage of code-multiplexing compared to frequency-multiplexing is that it allows a simpler transmitter and receiver implementation and has equal or better detection performance, depending on the data rate. VII. A PPENDIX In this appendix we prove that Q achieves a maximum equal ˜ are conjugate rows of C kf +1 . to Nf − 3 if and only if c and c To begin, we call b := c ◦ ˜c the Hadamard product of c and

D’AMICO and MENGALI: CODE-MULTIPLEXED UWB TRANSMITTED-REFERENCE RADIO



to Nf − 1. In other words, nc represents the number of sign changes in the components of b. We wonder: how does Q vary with nc ? From (47) it is seen that Q = Nf − 1 for nc = 0. Similarly it is readily checked that Q = Nf − 3 for nc = 1. In general we have (48) Q = Nf − 1 − 2nc

    





    





 



  

2131



Fig. 6. Comparison between CM-TR and SFS-TR in the presence of IFI and ISI with CM5 channel. Solid and dashed lines serve to ease the reading.

Note that (48) holds true even if b is the Hadamard product of two rows of C kf +1 since such products are included in Nf . ˜ and we wonder Now we reimpose the constraint b := c ◦ c whether there are two rows of C kf +1 , c and ˜c, such that nc = 0. Bearing in mind that bm = cm c˜m , we conclude that the answer is negative because, in order to have no sign changes, ˜ = −˜ c, which is impossible it must be either c = ˜c or c ˜ are distinct rows of C kf +1 . Thus, Q must because c and c be smaller than Nf − 1 and, because of (48), it can be either Nf − 3 or less. ˜ are conjugate, We maintain that Q equals Nf − 3 if c and c whereas it is smaller if they are not. To prove the first part of our claim we recall from Section IV that conjugate rows of (i) (i) (i) (i) C kf +1 can be written as c = (ckf , ckf ) and ˜c = (ckf , −ckf ), (i)

where ckf is a generic row of C kf . Then, from the definition of Hadamard product we have





(i)

(i)

(i)

(i)

b = c ◦ ˜c = (ckf , ckf ) ◦ (ckf , −ckf ) = (1Nf /2 , −1Nf /2 ) (49) where 1Nf /2 is a vector of all 1’s and length Nf /2. As the last term in (49) exhibits a single sign change, we conclude that Q = Nf − 3. The second part of the claim is proved assuming that c ˜ are not conjugate and belong to the upper/lower half4 and c (i) (i) of C kf +1 . Correspondingly we can write c = (ckf , ckf ) and







(j)

(j)

˜ = (ckf , ckf ) with i = j. Then c





(i)

(i)

(i)



   



˜, meaning that the mth component of b is bm = cm c˜m . Note c that bm takes value ±1. Thus, (44) may be rewritten as Nf −1



bm−1 bm

(j)

(i)

(j)

(i)

(j)

(j)

components of ckf ◦ ckf exhibit as many 1’s as -1’s and, in consequence, there must be at least a sign change in their pattern. It follows from (50) that there must be at least two sign changes in b, which makes us conclude that Q ≤ Nf − 5.

Fig. 7. CM-TR improved performance with conjugate sequences in the ˜ = (1Nf /2 , −1Nf /2 ). presence of IFI. The conjugate pair is c = 1Nf and c In the non conjugate pair c = 1Nf and ˜ c has an alternate sequence of +1 and -1.

Q=

(j)

b = c ◦ ˜c = (ckf , ckf ) ◦ (ckf , ckf ) = (ckf ◦ ckf , ckf ◦ ckf ) (50) (i) (j) On the other hand, as ckf and ckf are orthogonal, the

  

(47)

m=1

Next we temporarily ignore the structure of b induced by the relation b = c ◦ ˜ c and we take the bm ’s arbitrarily from the alphabet ±1. Also, we call Nf the set of Nf -dimensional vectors with these characteristics and, for a given b, we denote nc the number of times bm−1 bm = −1 as m varies from 1

R EFERENCES [1] R. J.-M. Cramer, R. A. Scholtz, and M. Z. Win, “Evaluation of an ultra-wide-band propagation channel,” IEEE Trans. Antennas and Propagation, vol. 50, pp. 561–570, May 2002. [2] M. Z. Win and R. A. Scholtz, “On the energy capture of ultrawide banwidth signals in dense multipath environments,” IEEE Commun. Lett., vol. 2, pp. 245–247, Sept. 1998. [3] V. Lottici, A. N. D’Andrea, and U. Mengali, “Channel estimation for ultra-wideband communications,” IEEE J. Select. Areas Commun., vol. 20, pp. 1638–1645, Dec. 2002. [4] C. Carbonelli and U. Mengali, “Synchronization algorithms for UWB signals,” IEEE Trans. Commun., vol. 54, pp. 329–338, Feb. 2006. [5] R. Hoctor and H. Tomlinson, “Delay-hopped transmitted-reference RF communications,” in Proc. IEEE Conf. on Ultra-Wideband Systems and Technologies, Baltimore, US, pp. 265–269, 2002. 4 A similar argument holds true if c and ˜ c belong to different halves of Ckf +1 .

2132

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 56, NO. 12, DECEMBER 2008

[6] M. Choi and A. Abidi, “A 6b 1.3 gsample/s A/D converter in 0.35um CMOS,” in Proc. IEEE Int. Solid-State Circuits Conference, vol. 438, pp. 126–127, 2001. [7] Y.-L. Chao, “Optimal integration time for UWB transmitted reference correlation receivers,” in Records of the 38th Asilomar Conference on Signals, Systems and Computers, vol. 1, Nov. 2004. [8] M. Casu and G. Durisi, “Implementation aspects of a transmittedreference UWB receiver,” Wireless Commun. and Mobile Computing, vol. 5, pp. 537–549, May 2005. [9] L. Feng and W. Namgoong, “An oversampled channelized UWB receiver with transmitted reference modulation,” IEEE Trans. Wireless Commun., vol. 5, pp. 1497–1505, June 2006. [10] “Active delay lines,” tech. rep., RCD Components Inc. [online report]. URL: http://www.rcd-comp.com/rcd/rcdpdf/A08-A14-SA08SMA14.pdf. [11] N. van Stralen, A. Dentinger, K. Welles II, R. Gaus Jr., R. Hoctor, and H. Tomlison, “Delay hopped transmitted reference experimental results,” in Proc. IEEE Conference on U W BST , May 2002. [12] W. Gifford and M. Z. Win, “On transmitted reference UWB communications,” in Proc. Asilomar Conference on Signals, Systems and Computers, 2004. [13] D. Goeckel and Q. Zhang, “Slightly frequency-shifted reference ultrawideband (UWB) radio,” IEEE Trans. Commun., vol. 55, pp. 508–519, Mar. 2007. [14] H. Xu, L. Yang, and D. L. Goeckel, “Digital multi-carrier differential signaling for UWB radios,” in Proc. IEEE GlobeCom, San Francisco, US, Nov. 2006. [15] A. F. Molisch, K. Balakrishnan, C.-C. Chong, S. Emami, A. Fort, J. Karedal, H. Schantz, U. Schuster, and K. Siwiak, “IEEE 802.15.4a channel model-final report,” Tech. Rep., Feb. 2005.

Antonio A. D’Amico received the Dr. Ing. Degree in Electronic Engineering in 1992 and the Ph. D. degree in 1997, both from the University of Pisa, Italy. He is currently a Research Fellow at the Department of Information Engineering of the University of Pisa. His research interests are in digital communication theory, with emphasis on synchronization algorithms, channel estimation and detection techniques. Umberto Mengali (M’69-SM’85-F’90) received his education in Electrical Engineering from the University of Pisa. In 1971 he obtained the Libera Docenza in Telecommunications from the Italian Education Ministry. Since 1963 he has been with the Department of Information Engineering of the University of Pisa where he is a Professor of Telecommunications. In 1994 he was a Visiting Professor at the University of Canterbury, New Zealand as an Erskine Fellow. He has served in the technical program committee of several international conferences and has been technical Co-Chair of the 2004 International Symposium on Information Theory and Applications (ISITA). His research interests are in digital communications and communication theory, with emphasis on synchronization methods and modulation techniques. He has co-authored the book Synchronization Techniques for Digital Receivers (Plenum Press, 1997). Professor Mengali is a member of the Communication Theory Committee and has been an editor of the IEEE T RANSACTIONS ON C OMMUNICATIONS from 1985 to 1991 and of the E UROPEAN T RANSACTIONS ON T ELECOM MUNICATIONS from 1997 to 2000.