TRANSACTIONS 1440
IEEE ON COMMUNICATIONS, VOL. 50, NO. 9, SEPTEMBER 2002
All-Digital Impulse Radio With Multiuser Detection for Wireless Cellular Systems
Christophe J. Le Martret and Georgios B. Giannakis, Fellow, IEEE
AbstractImpulse radio is an ultrawideband system with attractive features for baseband asynchronous multiple-access, multimedia services, and tactical wireless communications. Implemented with analog components, the continuous-time impulse radio multiple-access model utilizes pulse-position modulation and random time-hopping codes to alleviate multipath effects and suppress multiuser interference. We introduce a novel continuous-time impulse radio transmitter model and deduce from it an approximate one with lower complexity. We also develop a time-division duplex access protocol along with orthogonal user codes to enable impulse radio as a radio link for wireless cellular systems. Relying on this protocol, we then derive a multiple-input/multiple-output equivalent model for full continuous-time model and a single-input/single-output model, for the approximate one. Based on these models, we finally develop design composite linear/nonlinear receivers for the downlink. The linear step eliminates multiuser interference deterministically and accounts for frequency-selective multipath while a maximum-likelihood receiver performs symbol detection. Simulations are provided to compare performance of the different receivers. Index TermsImpulse radio, multipath fading channels, timedivision duplex, ultrawideband systems, wireless cellular systems.
I. INTRODUCTION HE IDEA of transmitting digital information using ultrashort impulses was first presented in [1] and called Impulse Radio (IR). It relies on pulse-position modulation (PPM) and time diversity that is gained by repeating the same symbol many ( 1000) times, according to a random code, which embodies IR with a very high processing gain. The attractive features of IR can be summarized as follows: it transmits at baseband and thus no intermediate frequency nor carrier synchronization processing is needed; it consumes minimal power; and, it is robust against jamming and multipath. IR has also been extended to multiuser communications in [2], where it is known as impulse radio multiple-access (IRMA). Its principle is based on asynchronous user transmissions and statistical multiuser interference (MUI) suppression that relies on power control.
T
Paper approved by M. Z. Win, the Editor for Equalization and Diversity of the IEEE Communications Society. Manuscript received May 24, 2000; revised September 18, 2001 and February 20, 2002. This paper was presented in part at the 1st Sensor Array and Multichannel Signal Processing Workshop, Boston, MA, March 2000, and at the MILCOM00 Conference, Los Angeles, CA, October 2000. C. J. Le Martret is with THALES Communications France, TRS/TSI, 92231 Genneviliers Cedex, France (e-mail: [e-mail]). G. B. Giannakis is with the Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455 USA (e-mail: georgios @ ece.umn.edu). Publisher Item Identifier 10.1109/TCOMM.2002.802559.
Subsequent works have focused on optimizing the efficiency of IRMA by characterizing the channel [3][5], improving the modulation format [6], [7] and addressing networking aspects [8], [9]. Recently, an application of IRMA has been considered in [10] for the radio link in a multimedia PCS communication scenario. The aim of this work is to explore usage of the IR concept as the radio link in a wireless cellular setting that consists of (micro-)cells with a few users (say less than 32). In all IRMA schemes proposed so far, the interference due to other users is randomized and only statistically suppressed, provided that (strict) power control is successfully applied. This solution may be well motivated for ad hoc architectures, but prevents one from taking advantage of multiuser detection (MUD). Indeed, the latter brings benefits over the statistical MUI cancellation when the number of users is small and thus the independent Gaussian approximation of the interference is no longer valid. MUD alleviates the need for power control and facilitates channel equalization to mitigate multipath effects. Equalization has not been explicitly addressed for conventional IR systems. On the other hand, based on a pragmatic propagation model, it has been verified recently that IR performance degrades severely if multipath effects are not accounted for [11]. RAKE reception offers an option, but its complexity increases when more than 50 fingers are required for reliable performance [3], [5]. Zero-forcing (ZF) or minimum mean-square error (MMSE) equalizers outperform RAKE receivers and their digital implementations are well motivated for IR. To apply MUD to IRMA, we first present a new continuous-time model for the PPM-IRMA scheme (Section II). This model is realizable by parallel linear modulators whose inputs can be expressed as the output of a spreading operator, that is fed by a nonlinear transformation of the symbols. Based on this model, we then derive an approximate model that affords a simpler transmitter design but requires sampling faster than the chip rate. When the period of the pseudorandom hopping sequence is an integer multiple of the number of frames per symbol, the spreading operator can be implemented using filterbanks. Next, we develop a time-division duplex (TDD) protocol to enable IRMA operation in a wireless cellular environment. Our protocol relies on orthogonal code designs and it is based on alternate slotted transmissions between users and the base station (BS) (Section III). We address the problem of time synchronization with the BS through guard times that ensure nonoverlapping transmissions. Relying on the TDD protocol, we pursue then an all-digital IRMA transmission model that is equivalent to the
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Fig. 1. Time scale representation of the different parameters in the IRMA (1) for the qth symbol of user m. The zoom on the second frame shows the pulse placed c in the second chip (~ (k) = 2) and shifted by (s (q) = 1).
continuous-time one and can be represented in discrete-time as a multiple-input/multiple-output (MIMO) system (Section IV-A). We present also the discrete-time model derived from the approximate one which has a simpler receiver structure and turns out to be single-input/single-output (SISO) (Section IV-B). Based on the discrete-time modeling, we propose three digital receivers for the downlink of an IRMA cellular system for both MIMO and SISO models (Section V). They are composed of a linear filter for channel mitigation and user separation, followed by a maximum-likelihood (ML) detector for symbol recovery. Finally, we derive a symbol error rate (SER) bound for the zero forcing receiver and provide simulations of the different receivers in a multiuser, frequency selective multipath propagation environment (Section VI). II. CONTINUOUS-TIME PPM-IRMA To introduce notation and facilitate the transition from the original continuous-time PPM to our novel model, we first review the conventional PPM-IRMA briefly. A. Conventional PPM-IRMA Modeling In PPM-IRMA, each user (say the th) transmits each infordrawn from the alphabet mation symbol repeatedly over frames each of duration . Specifically, let, and ting the frame index to be denoting integer-floor, the th symbol can be written with . The same signal is as: times using a position-hopping sequence transmitted having possible hops (chips) per frame. With denoting , where is a chip duration, we thus have guard time introduced to account for processing delay at the receiver between two successive received frames (see, e.g., [10]). of For the th frame and depending on the value , the chip-pulse (also known as the monothe code of duration , is positioned at the th chip cycle)
interval. Within this chip interval, the monocycle is shifted by to implement the PPM with . With these notational conventions, the th users transmitted waveform is given by (see, e.g., [6])
(1) where is the amplitude which controls the transmitted power. is a periodic pseudorandom sequence [2] with The code period . Fig. 1 illustrates the time scale representation of the parameters in IR. B. Novel PPM-IRMA Modeling We present here a novel model for the IR that relies on the fact that the PPM signal can be expressed as the sum of linear modulators, fed by a nonlinear transformation of the information symbols. In each segment of duration corresponding to repetitions of a single symbol, the pulse stream is shifted in time according to the symbol value; e.g., . One way to model this is to it is shifted by , if have parallel branches, each realizing a shifted version of the pulse stream. In order to generate the signal, we then only need to select one branch out of depending on the symbol value. , Adopting this viewpoint and defining we can re-express (1) as (2) with
(3)
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where captures the branch selection process according to the following definition: for Hence, by defining the time-shifted pulses and recalling that for rewritten as if otherwise. (4)
, (3) can be
Fig. 2. Continuous-time PPM-IRMA model (mth user).
(5) and is an integer in [0, ], we infer Because in (5) is shifted by an integer multiple of . It is thus that as a linearly modulated waveform with possible to view , and express it as symbol rate (6) is a sequence that depends on and . where linear Thus, (2) can be interpreted as the superposition of . modulators each with a different pulse function Note that the index in (5) denotes the frame number while in (6) corresponds to the chip index across the frame. Because with these two indices are related by , we deduce that , where and denotes Kroneckers , the latter delta. Observing that can be written as if otherwise. (7)
model of PPM-IRMA by moving the time-shifted form of the to the spreading code itself. The outPPM pulse-shaper for will then be delayed versions puts and only one transmit filter will be sufficient. The of price paid for such a simplification is increase in the transmitted . Parsing symbol rate in order to realize subchip delays segments, the new symbol rate becomes the chip length into , with . The approximation stems from the are approximated by integer delays fact that the real delays . Certainly, we can render the rounding sufficiently large. error as small as we wish by choosing The spreading implementation can accommodate the approxzeros between succesimate model by simply inserting . sive chips of the sequences denotes the new code sequence, we have If (9) The approximate modeling is depicted in Fig. 4, where the delayed filterbank precoder outputs are defined as (10) Based on (10), we can then define (11) to obtain the transmitted signal for the chip-oversampled PPM-IRMA transmissions as (12)
in terms of the chip To express the generic th symbol and , which index , we recall that . Substituting the latter in our eximply that , we arrive at pression for (8) Hence, the continuous-time PPM-IRMA transmission can be stands for the depicted as in Fig. 2, where the notation spreading operation defined by (8). has period , then it can be If the hopping code readily verified by direct substitution that the period of in (7) is . To illustrate the link between and with an example, let us consider , , and . Using (7), we then find [see also Fig. 3(a)]. Unlike the conventional model, we have assumed here for . In fact, our novel model can encompass simplicity that as well, by setting , with integer the case to take its values in and restricting the sequence , where . C. Approximate Chip-Oversampled PPM-IRMA Model With reference to the continuous-time model of Fig. 2, we develop here an approximate chip-oversampled discrete-time
D. Filterbank Implementation of IRMA With Block-Periodic Codes If we restrict the time hopping sequence period to be an integer , we will see multiple of the number of frames, i.e., that the spreading can be implemented using filterbanks. Such a block-periodic code that will be adopted henceforth, implies a information block spreading operation where each block of bearing symbols is spread by the same hopping sequence over frames. The block-periodic code structure will prove useful in developing can be easily adjusted our digital receivers. The parameter and will turn out to be the number of transmitted symbols per burst in the TDD protocol of Section III-A.
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(a)
(b)
N
Code sequences for the novel PPM-IRMA model. (a) (n) = f010010001100100001010100g obtained from ~ (n) = f11200210g for = 4, = 2 and = 3. (b) Codes (n) deduced from (n) for filterbank implementation. (c) Filterbank implementation of the PPM-IRMA spreading for ). branch (N = Fig. 3.
a
N
KN N
c
c
c
(c)
c
K
Fig. 4. Approximate continuous-time PPM-IRMA model (mth user).
Because in (7) has period and is spread over symbols, it is convenient to express in block form with denoted as after the th transmitted block of size with , , and setting . We can then use (8) to obtain . Recall now that when and remains constant over chips. Hence, the th transmitted block is given by [cf. (8)] (13) where is defined as for otherwise. (14) , Using the same example as in Fig. 3(a) for the code with . From Fig. 3(b) depicts the code this perspective, our PPM-IRMA model can be viewed as a multicode CDMA system (see, e.g., [12] and [13]).
Expression (13) is identical to the one given in [14, eq. (1)], where it is shown that is the output of the filterbank shown in Fig. 3(c). We thus infer that our PPM-IRMA transmitted sequence can also be implemented with a discrete-time filterbank. The filterbank implementation can accommodate the approximate model of Section II-C by modifying the upsampling times that of the linear factor. Because the symbol rate is . model (6), the upsampling factor becomes We have presented a novel IRMA model and an approximate version of it. When the hopping sequence is block-periodic, we have shown that these models can be implemented using filterbanks which turns out to be very flexible when the models have to be adjusted to the transmission protocol of the next section. The block-periodic assumption does not modify the power spectral density of the IR signal since, as discussed in [15], the spectral lines are only affected by the value of the . Moreover, although the block-periodic assumption ratio facilitates the implementation of the spreading operation by using filterbanks, the derivations in the rest of the paper can be generalized to any value of the time hopping period. In this case, the discrete-time equivalent model in Section IV and the digital receivers derived in Section V would have to be slightly modified accordingly (see discussion by the end of Section IV). III. TDD IRMA Unlike existing IRMA schemes that consider asynchronous transmissions through frequency-flat channels and suppress MUI statistically, we propose here a novel IRMA approach using orthogonal codes in a synchronous or quasi-synchronous
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(a)
(b) Fig. 5. TDD-IRMA slots. (a) Successive slots for TDD in uplink/downlink pairs. (b) Timing parameters for TDD-IRMA slots. (a)
context, coupled with TDD for transmissions through frequency-selective channels. This is achieved by assigning to each user different orthogonal time-hopping sequences and designating two time slots for transmission: one for the uplink and one for the downlink. A. TDD-IRMA Protocol Because IR transmits at baseband, there is no carrier frequency and hence, we cannot use frequency-division duplexing (FDD) as in most multiuser systems. Here, we have to resort to TDD in order to provide a full duplex link between the users and the BS. Successive time slots in TDD are designated for the downlink and the uplink as shown in Fig. 5(a). We assume that the users have a common time reference so that transmissions can essentially be considered as quasi-synchronous as in IS-95, where even in the uplink users attempt to synchronize with the pilot waveform broadcasted by the BS. The term quasi-synchronous means that although a time reference is present, small offsets arising due to the time jitter of each users clock and relative propagation delays between the users and the base station, are allowed but must be accounted for. Because of the block structure we have introduced in information symbols Section II, each IRMA user transmits during one time slot, for both downlink and uplink sessions. Thus, the downlink transmitted signal is composed of a burst symbols, where is the maximum number of conveying active users, followed by a silent interval (no signal) of approximately the same duration. Conversely for the uplink, each user sends information symbols within the same time slot during the silent period of the downlink session and follows it up with a silent interval to enable the downlink burst transmission. The is timing of these slots is represented in Fig. 5(b), where the burst duration. As depicted in Fig. 5(b), the slot has duration with and and is thus longer than the burst. determines the time between the end of The timing offset the BS burst and the beginning of the users bursts, while denotes the time between the end of the users bursts and the beginning of the BS burst. These offsets are set to account for the asynchronism and also for the propagation channel. Let us to be the asynchronism between the BS and user ; define the channel length in the uplink between user and the BS; the channel length in the downlink between the BS and and user . Clearly, the transmitted burst from the BS should not
(b)
l < 0.
Fig. 6.
Timing frames for the TDD-IRMA slots. (a) Case
l 0. (b) Case
overlap in time with the received bursts sent by the users and vice versa. In order to select values for and which satisfy this conand are dition, we will assume that the channel lengths larger (in absolute value) than the asynchronism . This assumption, made for simplicity, is reasonable since every user can have either poshas a time reference. The asynchronism itive or negative values. Moreover, in an absolute time reference, if the BS sees user with delay , then user will see the . Accordingly, we depict in Fig. 6 the base station with delay timing structure of the different slots in the uplink and the downlink, for both positive and negative values of . The same picleads to the same expressions for ture for negative values of guard-intervals and . These values must be selected in order , to handle the worst case scenario. Defining the worst case can be accommodated by choosing (15) For both the uplink and the downlink, the transmitted burst seconds while the subsequent silent has duration seconds. In order interval has duration to match the all-digital modeling, the different time intervals must be multiples of the sample duration. Thus, defining and , where is the integer ceiling of , the timing will be set to and which seconds with leads to a new silent duration of . symbols per user are transmitted every For this protocol, ) seconds; thus, we deduce that the bit rate per user ( ) is given by (with alphabet size b/s (16)
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We will now see how to design orthogonal codes for the different users to ensure deterministic MUI suppression at the receivers for the downlink. B. Designing Orthogonal IRMA Codes The digital models described in Section II can accommodate the silent signal portion by simply padding the code sequences chips, the number with zeros. Since the silent period lasts and therefore the length of the of trailing zeros will be . codes becomes An orthogonal IRMA scheme capable of eliminating MUI in the downlink is possible by assigning to each user, one of the chip positions in each of the frames, with none of the chips belonging to more than one user. This is equivalent to having , where orthogonality is orthogonal spreading sequences defined as follows. and deDefinition 1: Two code sequences , where fined as in (7) are orthogonal if and only if for , and stands for transpose. To build such orthogonal codes, we recall (7) and establish the following equivalence. and , have Proposition 1: If according to (7), and , then corresponding sequences
Fig. spreading 7. MIMO PPM-IRMA model for the downlink (mth user receiver).
It follows from (2) and (6) that the chip-sampled matched filter output of the th branch at the receiver is Because there are chips per frame, we deduce that the . Using maximum number of users we can accommodate is orthogonal codes Proposition 1, we can define the set of for the TDD-IRMA transmission scheme as with (17) With no other constraint than orthogonality (one can also use correlation constraints for synchronization purposes), a simple means of constructing sequences satisfying (17) is to generate them randomly, user after user, while checking for orthogonality. (18) and where stands for convolution and , where is the length of the -sampled . channel Casting (8) into a matrix form, we can express the th transof length by the mitted block of the th branch vector (19) where is the of length
IV. DISCRETE-TIME EQUIVALENT MODEL We derive here the discrete-time equivalent model of the PPM-IRMA in the single-user case for simplicity. We first present in Section IV-A the MIMO model deduced from the model of Section II-B. Then we develop an approximate SISO model in Section IV-B which leads to a simplified receiver. A. Chip-Sampled MIMO Model and we sample The transmitted symbols are sent at a rate the received signal at the same rate. Adhering to PPM, this can only be achieved by passing the received signal through parprior to sampling. We allel filters matched to the pulses thus arrive at the MIMO continuous-time PPM-IRMA transmission model shown in Fig. 7.
vector representing the symbol block with the following notational convention:
denoting the with th column . At the th receiver, according to the TDD-IRMA protocol in Section III, it suffices to collect the first samples per transmitted burst to enable symbol recovery. The th received block corresponding to the th branch, defined as , can be expressed as (20)
and code matrix of user
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where and given by
are
Toeplitz convolution matrices
. . . . . . We can reexpress (20) for vectors of size which leads to
.. ..
. .
. . . (21) . . . by defining the
(22) where , and matrix given by
Fig. 8. SISO PPM-IRMA model for the downlink (mth user receiver).
, is a block
. . .
. . .
..
.
. . .
(23) The model in (20)(23) is MIMO, but as we will see next, the discrete counterpart of the approximate model in Section II-C is SISO. B. Oversampled SISO Model Consider the modulated signal (12) propagating through a , corrupted by additive noise and linear channel . filtered at the receiver with the filter matched to the pulse The resulting SISO model is shown in Fig. 8. Then, in order to obtain the discrete-time equivalent model of the approximate continuous-time PPM-IRMA, we sample the received signal . The discrete-time equivalent channel is thus at a rate . Defining the noise , the discrete-time equivalent PPM-IRMA model is given by (24) is the length of the -sampled channel and where is given by (11). the sequence Casting (10) into a matrix form, we can express the th transby the vector mitted block of the th branch (25) where is the of length vector representing the symbol block with the following notational convention: and is the code matrix for as the downshift opthe spreading of branch . Defining erator of order for column vectors with zero padding for new can be expressed as entries, vectors
with . Thus, using (25) we can cast (11) in vector form and the th transmitted block . for user is given by Because the sample rate has increased, the quantities , , defined for the TDD-IRMA protocol in Section III have to be modified accordingly. Thus, we define the equivalent -samas , pled duration for , and and . Likewise, the number of trailing zeros for the code sequence has to be equal to and thus, the code length becomes . At the th receiver, it suffices to collect the first samples per transmitted burst to enable symbol rereceived vector is defined as covery. Then, the th . can be expressed as And, according to (24), (26) where by is an ( ) Toeplitz convolution matrix given
. . . . . .
.. ..
. .
. . . (27) . . .
. and We have seen that an approximation of the MIMO model leads to a SISO one which has lower complexity. It has one pulse-shaper for the transmitter and one receive-filter whereas the MIMO model needs pulse-shapers and receive filters. However, the price to pay for such a simplification is an increase of the sampling rate. If the pulses are equally spaced in time , ), the SISO model is equal to the ( MIMO and the sampling rate is multiplied by . Finally, we will see in the next section that the channel in the SISO model is always invertible, which is not true for the MIMO one.
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As mentioned in Section II-D, the proposed TDD-IRMA model can also accommodate nonblock periodic codes. In this case, the spreading code matrix will be different from one burst to the next. They will have to be indexed by , the burst symbols). Thus the matrix in (19) number (block of in (25) for the SISO case, will for the MIMO case and (resp. ) have to be replaced by matrices and . In the same with way, the receivers developed in the next section will have to be modified accordingly. V. DIGITAL RECEIVERS FOR THE DOWNLINK We describe here three linear receivers for the downlink of a cellular multiuser scheme (see [16] and reference therein), using the TDD and the orthogonal code design described in Section III for both MIMO and SISO models. We assume that the known channel is time invariant, but adaptive variants of these receivers or successive interference cancellers can be also derived to handle slowly-varying channels. Although not considered here, nonlinear receivers such as DFEs (see, e.g., [16]) are also applicable. Because PPM is a nonlinear modulation, the receivers will operate in two stages: 1) a linear filtering stage to eliminate channel effects and separate the users, and 2) a nonlinear processing stage to recover the symbols. A. MIMO Model The th transmitted symbol block for the downlink is given , while the received block at the th by receiver is (28) Based on the vector model (28), a multichannel finite-impulse response (FIR) receiver can be described of dimension as follows: by a matrix , where is the estimated vector of the symbols transformed by the nonlinear and is the filtered noise. function , we obtain different linear Depending on how we select receivers and possible choices include ZF, matched filter (MF), and MMSE. These receivers are given by the following: ZF (a.k.a. Decorrelating) Receiver (29)
where
is an . . .
block matrix . . . (32)
with where
blocks given by is . . . is . Matrix and block matrix is . . .
,
(33)
blocks given by . with in (23) is not guaranteed to Due to its structure, matrix be full rank and therefore the ZF receiver (29) might not always exist. As for symbol detection, assuming that the noise is Gaussian, the optimal detector in the ML sense is where given by is the correlation matrix of . the filtered noise B. SISO Model For the downlink, the th transmit block of symbols is given , while the corresponding received by block vector at the th receiver is (34) Based on the vector model (34), a multichannel FIR receiver of dimension as can be described by a matrix , where follows: is the filtered noise. As for the MIMO model, , we obtain similar to (29)(31) depending on how we select different linear receivers. ZF (a.k.a. Decorrelating) Receiver
(35) MF (a.k.a. Rake) Receiver
(36) MMSE Receiver (37) where is is , , and ,
MF (a.k.a. Rake) Receiver (30) MMSE Receiver (31)
is is . is Toeplitz and the impulse response Inasmuch as has at least one nonzero value, has full and is thus always invertible. As a consequence, the rank channel is always invertible. We infer that for the ZF receiver, MUI is canceled due to the orthogonality among spreading
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codes. Specifically, since ) we have (assuming
,
B. Simulations We present here simulations to illustrate the behavior of the different receivers. The selected configuration is a binary PPM modulation ( ), eight users ( ) and two symbols ). As per [10], we have chosen a frame duration per burst ( ns ns and a maximum delay spread equal to 100 ns. . We have assumed a time guard of duration ; hence, Thus, we deduce the chip duration ns. For the MIMO model, the sampling rate is equal to which leads to a channel of length (assuming ). and , the SISO model Moreover, assuming which leads then to a channel length is obtained for . The channel is modeled by the SalehValenzuela model [11], [17] for indoor IR systems with the same parameters. This model is based on clusters of rays. The received signal is composed of attenuated and delayed versions of the transmitted signal arriving in clusters. The times of arrival are modeled as a Poisson process and the amplitudes as Gaussian. For the simulations, the receivers will be assumed to be synchronized on the strongest path. In the PPM-IRMA system, the received signal is the second derivative of the Gaussian function (normalized to have ); hence, we have , where is the correlation function of and ns is adjusted to yield a pulsewidth the parameter equal to 0.7 ns (see, e.g., [15]). Fig. 9 depicts the BER corresponding to the three receivers . We can see that for one channel trial and different values for the RAKE receiver performs poorly and exhibits a BER floor at high SNR. The MMSE performs the best while the ZF remains close to the MMSE. Fig. 9(a) shows that the BER of the ZF receiver is very close to the bound given by (39). Fig. 9(b) shows that, compared to Fig. 9(a), increasing the number of frames improves slightly the performance for the ZF and MMSE receivers, but does not change their relative behavior. Moreover, , the performance does not improve it shows that for further does not provide any any more and thus, increasing benefit. Fig. 10 shows the average BER of the three receivers over 100 Monte Carlo channel realizations and for the same parameters as in Fig. 9(a). It shows that the behavior of the receivers remains the same and that the difference between the MMSE and ZF increases. Simulations of the conventional IRMA (not shown here) were performed and confirmed the conclusion of [11], where even single-user performance was seen to suffer severely in the presence of multipath. VII. CONCLUSION An all-digital IR scheme was developed for ultra wideband multiple-access wireless cellular systems. It included a novel modeling of the conventional time-continuous IR along with a TDD protocol and orthogonal user code design. A discrete-time MIMO model was derived from the continuous-time one, along with an SISO approximate of it that turns out to have lower
(38) achieves (almost) error-free symbol Thus, equalizer recovery in the noise-free (high SNR) case, regardless of the channel. One can remark that the MIMO model does not hold in (29) may not always be this property, since matrix guaranteed to have full rank although it is a rare event. is As for symbol detection, assuming that the noise Gaussian, the optimal detector in the ML sense is given by with . We have derived three linear receivers for both MIMO and SISO models. We have assumed that the channel was known, which that can be obtained for instance by using probe sequences within the information symbols. We will now give a SER bound for the ZF receiver and show some simulations of the different receivers. VI. PERFORMANCE We first derive an upper bound on the SER for the ZF receiver and then present a simulation example of the proposed downlink TDD-PPM-IRMA scheme in a multipath environment for both MIMO and SISO models. A. SER Bound for the ZF Receiver Because the linear stage of the ZF receiver cancels the effect of the channel and the MUI, we can derive an upper bound for the SER. We present here the derivation using the MIMO receiver for simplicity, but the resulting expression is identical for the SISO as well. Provided that the matrix is invertible in (28), the output of the linear filter (29) can be . Then, given the symbol expressed as , the error probability for the ML detector is given by is th...