# Performance Comparison of UWB Pulse Modulation Schemes under White Gaussian Noise Channels

Authors: O. Abedi; M. C. E. Yagoub
Journal: International Journal of Microwave Science and Technology
 (2012)

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## Abstract

In this paper, and for the first time, the performance of various ultra wideband pulse modulation schemes are discussed and compared in terms of narrowband interference robustness, symbol error rate, system complexity, data rate, and maximum transmit power with respect to transceiver distance and channel capacity. The channel model is the UWB IEEE 802.15.3a multipath indoor channel model under additive white Gaussian noise. The transmit power was evaluated by integrating the fifth derivative of the power spectrum density of the Gaussian pulse over the whole bandwidth.


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## Body

## 1. Introduction

In ultra wideband (UWB) system design, there are always tradeoffs between maximum transmit distance, data rate, transmit power, and system complexity. Different data modulation schemes such as pulse-position modulation (PPM), pulse-amplitude modulation (PAM), phase-shift keying (PSK), or on-off keying (OOK) have been used in UWB communications with relative success depending on the targeted application [1, 2]. In fact, their performance can significantly vary according to which system parameters are considered such as narrowband interference (NBI) robustness, symbol error rate (SER), system complexity, data rate, or maximum transmit power with respect to transceiver distance and channel capacity. For instance, if minimum complexity is important, then OOK modulation would be the best choice. However, it is very sensitive to noise. On the other side, if interference robustness and power efficiency are the parameters to consider, binary PSK (BPSK) and M-ary PPM (M-PPM) can be the best candidates. Nevertheless, M-PPM deals with multidimensional signals, whereas M-PAM has only one dimension [3, 4].This topic has been widely discussed in the last decade but most of published works [4] did not efficiently include all tradeoffs, such as comparison of bit error rate (BER), modulation level, channel capacity, and complexity. In fact, even if several technical papers have already discussed this issue, they mainly investigated the modulation performance for specific applications or system requirements and thus, to the best of our knowledge, did not cover the whole comparison spectrum [5].The objective of this paper is indeed to evaluate all these parameters and their impact on each scheme performance. For this aim, we considered the PPM in time-hopping mode (TH-mode), the OOK in direct-sequence mode (DS-mode), and the binary PAM (BPAM) in both TH- and DS-modes. The channel model is the UWB IEEE 802.15.3a indoor channel model under additive white Gaussian noise (AWGN) [6].  Note that the white noise channel model is largely used in the literature due to its relative simplicity. Because the target here is to compare the performance of different UWB pulse modulation schemes under the same channel model, a relatively simple model can be the best candidate. Note also that this channel model can be specified for different propagation scenarios, such as extremely indoor non-line-of-sight (ENLOS) or outdoor line-of-sight (LOS). In this work, we focused on the indoor propagation channel model to take advantage of UWB frequency selective fading.Also, the transmit power was evaluated by integrating the fifth derivative of the power spectrum density (PSD) of the Gaussian pulse over the whole bandwidth.


## 2. Modulation Schemes

### 2.1. M-PAM

In pulse-amplitude modulation, the same waveform is sent with different amplitudes corresponding to different data being transmitted. The BPAM can be presented using antipodal, monocyclic, or Gaussian pulses. The transmitted signal for M-PAM can be expressed as [7]
(1)SPAM(t)=∑j=-∞∞∑m=1Mδm(j)dmP(t-jTs),
where δ(j)=[δ1(j),δ2(j)…,δM(j)] is an indicator transmitter pulse vector, that is, for each known unit impulsej, only one element of the vector is equal to unity otherwise zero; Ts is the pulse symbol period, M is the modulation level, and P is the pulse shape impulse response. For multiuser case, dm is the amplitude of independent and identically distributed (IID) random signals according to an equiprobable distribution. The related PSD can be formulated as [4]
(2)SPAM(f)=SCPAM(f)+SdPAM(f),
where SCPAM(f) and SdPAM(f) are, respectively, the desired continuous part (nonperiodic) and the undesired discrete part (periodic), which induces more spectral lines and lower mutual information [4]. For real data with equal probability of occurrence, SPAM(f) becomes [6, 7]
(3)SPAM(f)=σd2Ts|P(f)|2+μd2Ts2∑j=-∞∞|P(jTs)|2δ(f-jTs),
where σd2 and μd are the variance and mean of the random data sequence, respectively. In case of repetition pulse coding [6, 8], each random data contain number of pulses with various amplitude (PAM case) that indicate just one bit of data. Let us consider the symbol period limited by trade of data rate and pulse numbers. For instance, for wider Ts, the data rate is indeed lower but does have better performance with higher pulse repetition coding (increasing pulse numbers), and inversely. On the other hand, if a time-hopping PAM (TH-PAM) is considered, they are dependent on frequency, which is not the case in this study.In narrowband, using a filter bank receiver or a mixer bank receiver can mitigate the narrowband interference while splitting the UWB bandwidth to the narrowband. Thus, the average power or variance of the UWB signals can be calculated by integrating the PSD in frequency domain [9, 10]
(4)Var(SPAM(t))=∫-∞+∞SPAM(f)df=∫-∞+∞[∑j=-∞∞|P(jTS)|2σd2Ts|P(f)|2μd2TS2μd2TS2+μd2Ts2∑j=-∞∞|P(jTs)|2δ(f-jTs)]df=σd2Ts∫-∞+∞|P(f)|2df+μd2TS2μd2TS2+μd2Ts2∑j=-∞∞|P(jTs)|2.Assuming the carrier frequency of the receiver isfc0 and the bandwidth of the bandpass filter is B, the UWB interference power can be expressed as [10]
(5)PI=σd2Ts∫-∞+∞|P(f)HT(f)|2df+μd2Ts2∑(j/T)±fc0∈[-B/2B/2]|P(jTs)HT(jTs)|2,
with HT(f)=Hch(f)HB(f), where HB(f) and Hch(f) are the Fourier transform for the filter and the channel impulse response, respectively. For an ideal bandpass filter, the above equation can be formulated as [10]
(6)PI=2σd2Ts∫fc0-(B/2)fc0+(B/2)|P(f)|2df+2μd2Ts2∑(j/T)+fc0∈[-B/2B/2]|P(jTs)|2.Therefore, interference power can be affected by the following factors.(i)
The symbol period (1/Ts); in fact, increasing Ts will both increase the pulse repetition frequency and decrease the energy spectral density, while the corresponding continuous component of the power spectral density remains unchanged. So the pulse repetition leads to a discrete line spectrum. The line spectral density is composed of the delta function and separated by the pulse repetition frequency.(ii)
The energy spectral density (ESD) of the pulse waveform,|P(f)|2.(iii)
The power spectral density (PSD) of the UWB chip sequenceSPAM(f) determined by σd2 and μd for real and mutually uncorrelated sequences. Therefore, a proper pseudorandom (PR) code, directly related to the pulse repetition frequency and corresponding PSD, could be selected to reduce the discrete spectral components. If the mean μd is not zero, discrete components (spectral lines) will appear in SPAM(f). However, for a sequence of ±1, these components will disappear. On the other hand, if the UWB pulse train is modulated by an all “one” sequence (σd2=0,μd=1), the PSD of SPAM(f) will be composed only of discrete components. Thus, if these components, which generally have a stronger PSD than continuous ones [2, 5], are in the band of narrowband systems, they will have a bigger effect on the performance of the victim receiver.An alternative is to send alternating positive and negative pulses [2, 9]. This line coding scheme is often called alternate mark inversion (AMI). Following [10], it has been demonstrated that with equal probable binary elements, the PSD is given by
(7)SAMI(f)=12Ts|P(f)|2[1-cos(2πfTs)],
which shows there are no spectral lines, and thus, the bipolar PAM signal has a lower bit error rate compared to the nonbipolar PAM signal, which suffers from the presence of more spectral lines (Figure 1).Figure 1
BER AWGN channel comparison with various polarizationk.


### 2.2. OOK

Compared to others, OOK can be seen as the simplest scheme. In this technique, sending a pulse corresponds to bit “1” and no pulse means “0”. Therefore, because of the equal probability of either symbols,σd2=0.25,μd=0.5, and the PSD becomes [6, 10]
(8)SOOK(f)=14Ts|P(f)|2+14Ts2∑j=-∞∞|P(jTs)|2δ(f-jTs).In a variant (antipodal signals), the transmitter sends a pulse of durationTp much higher than the bit time Tb to indicate bit “1” and the negative of this pulse for “0”. Therefore, with equiprobable elements, we have σd2=1,μd=0 and thus, a PSD of [10]
(9)Sant(f)=1Ts|P(f)|2,
showing that the PSD of antipodal does not have any spectral lines or discrete components. Note that OOK and binary-PPM are similar. Therefore, only 2-PPM performance was evaluated to avoid duplication.


### 2.3. M-PPM

In PPM, the chosen bit to be transmitted influences the position of the UWB pulses. The signal at the output of the transmitter cascade,s(t), can be expressed as [1, 2, 6]:
(10)SPPM(t)=∑j=-∞∞∑m=1MP(t-jTs-cmTC-amε),
where j is the number of transmitted bits, Ts is the symbol period, cm is the time-hopping code, cmTc is the  pseudorandom code in chip duration TC, am depends on j, and ε is the time shift associated with TH-PPM.The chip time intervalTC represents only one pulse in the time frame period. Note that amε+Tp<Tc with Tp the pulse period.The Fourier transform of the PSD of M-PAM can be expressed as [11]
(11)SPPM(f)=∫-∞+∞SPPM(t)e-j2πftdt.
Leading to the following PSD, expressed in terms of continuous and discrete parts
(12)PSPPM(f)=PSCPPM(f)+PSdPPM(f)
with
(13)PSCPPM(f)=|P(f)|2Ts[1-(1Msin(πfT)sin(πfT/M))2]PSdPPM(f)=|P(f)|2Ts2(1Msin(πfT)sin(πfT/M))2×[∑j=-∞∞δ(f-jTs)],
which can be simplified to [6, 11]
(14)PSPPM(t)=1Ts2∑j=-∞∞∑m=1M|P(jMTs)|2δ(f-jMTs),
showing a smother PSD than of PAM, leading to less narrowband interference. The single pulse of the UWB signal is placed into one of N=Tsym/Th slots, where N is the number of slots per symbol, Th is the pulse width and Tsym is the symbol time. The number of bits per one PPM symbol is equal to log2N.Note that by increasingM (modulation level) the data rate will increase as well. Furthermore, the average transmitted power will be increased by a factor of M.In this case, theN slots are divided into L groups. Each group includes N/L slots with a single pulse. With this, the average bit number will be [11]
(15)Nb′=log2(N/L)L=L(log2N-log2L)N>>L,Nb′=Llog2N=Nb.Thus, modifying the bit rate can improve the throughput of number of pulses perN slots, but it will raise other issues.(i)
A larger bandwidth requirement, as the number of pulses per symbol interval increases.(ii)
The pulse width relaxation will be limited.(iii)
All symbols can be orthogonal. To solve this for large N, one can use predefined pulse arrangement such that no pulse is put in the same position. In other words, to avoid pulse interpath interference (IPI) or pulse jamming, we should decrease the pulse period or increase the chip period.Note that more information on PSK modulation can be found in [4, 6]. Because of its complexity, it has been introduced here just for comparison.


## 2.1. M-PAM

In pulse-amplitude modulation, the same waveform is sent with different amplitudes corresponding to different data being transmitted. The BPAM can be presented using antipodal, monocyclic, or Gaussian pulses. The transmitted signal for M-PAM can be expressed as [7]
(1)SPAM(t)=∑j=-∞∞∑m=1Mδm(j)dmP(t-jTs),
where δ(j)=[δ1(j),δ2(j)…,δM(j)] is an indicator transmitter pulse vector, that is, for each known unit impulsej, only one element of the vector is equal to unity otherwise zero; Ts is the pulse symbol period, M is the modulation level, and P is the pulse shape impulse response. For multiuser case, dm is the amplitude of independent and identically distributed (IID) random signals according to an equiprobable distribution. The related PSD can be formulated as [4]
(2)SPAM(f)=SCPAM(f)+SdPAM(f),
where SCPAM(f) and SdPAM(f) are, respectively, the desired continuous part (nonperiodic) and the undesired discrete part (periodic), which induces more spectral lines and lower mutual information [4]. For real data with equal probability of occurrence, SPAM(f) becomes [6, 7]
(3)SPAM(f)=σd2Ts|P(f)|2+μd2Ts2∑j=-∞∞|P(jTs)|2δ(f-jTs),
where σd2 and μd are the variance and mean of the random data sequence, respectively. In case of repetition pulse coding [6, 8], each random data contain number of pulses with various amplitude (PAM case) that indicate just one bit of data. Let us consider the symbol period limited by trade of data rate and pulse numbers. For instance, for wider Ts, the data rate is indeed lower but does have better performance with higher pulse repetition coding (increasing pulse numbers), and inversely. On the other hand, if a time-hopping PAM (TH-PAM) is considered, they are dependent on frequency, which is not the case in this study.In narrowband, using a filter bank receiver or a mixer bank receiver can mitigate the narrowband interference while splitting the UWB bandwidth to the narrowband. Thus, the average power or variance of the UWB signals can be calculated by integrating the PSD in frequency domain [9, 10]
(4)Var(SPAM(t))=∫-∞+∞SPAM(f)df=∫-∞+∞[∑j=-∞∞|P(jTS)|2σd2Ts|P(f)|2μd2TS2μd2TS2+μd2Ts2∑j=-∞∞|P(jTs)|2δ(f-jTs)]df=σd2Ts∫-∞+∞|P(f)|2df+μd2TS2μd2TS2+μd2Ts2∑j=-∞∞|P(jTs)|2.Assuming the carrier frequency of the receiver isfc0 and the bandwidth of the bandpass filter is B, the UWB interference power can be expressed as [10]
(5)PI=σd2Ts∫-∞+∞|P(f)HT(f)|2df+μd2Ts2∑(j/T)±fc0∈[-B/2B/2]|P(jTs)HT(jTs)|2,
with HT(f)=Hch(f)HB(f), where HB(f) and Hch(f) are the Fourier transform for the filter and the channel impulse response, respectively. For an ideal bandpass filter, the above equation can be formulated as [10]
(6)PI=2σd2Ts∫fc0-(B/2)fc0+(B/2)|P(f)|2df+2μd2Ts2∑(j/T)+fc0∈[-B/2B/2]|P(jTs)|2.Therefore, interference power can be affected by the following factors.(i)
The symbol period (1/Ts); in fact, increasing Ts will both increase the pulse repetition frequency and decrease the energy spectral density, while the corresponding continuous component of the power spectral density remains unchanged. So the pulse repetition leads to a discrete line spectrum. The line spectral density is composed of the delta function and separated by the pulse repetition frequency.(ii)
The energy spectral density (ESD) of the pulse waveform,|P(f)|2.(iii)
The power spectral density (PSD) of the UWB chip sequenceSPAM(f) determined by σd2 and μd for real and mutually uncorrelated sequences. Therefore, a proper pseudorandom (PR) code, directly related to the pulse repetition frequency and corresponding PSD, could be selected to reduce the discrete spectral components. If the mean μd is not zero, discrete components (spectral lines) will appear in SPAM(f). However, for a sequence of ±1, these components will disappear. On the other hand, if the UWB pulse train is modulated by an all “one” sequence (σd2=0,μd=1), the PSD of SPAM(f) will be composed only of discrete components. Thus, if these components, which generally have a stronger PSD than continuous ones [2, 5], are in the band of narrowband systems, they will have a bigger effect on the performance of the victim receiver.An alternative is to send alternating positive and negative pulses [2, 9]. This line coding scheme is often called alternate mark inversion (AMI). Following [10], it has been demonstrated that with equal probable binary elements, the PSD is given by
(7)SAMI(f)=12Ts|P(f)|2[1-cos(2πfTs)],
which shows there are no spectral lines, and thus, the bipolar PAM signal has a lower bit error rate compared to the nonbipolar PAM signal, which suffers from the presence of more spectral lines (Figure 1).Figure 1
BER AWGN channel comparison with various polarizationk.


## 2.2. OOK

Compared to others, OOK can be seen as the simplest scheme. In this technique, sending a pulse corresponds to bit “1” and no pulse means “0”. Therefore, because of the equal probability of either symbols,σd2=0.25,μd=0.5, and the PSD becomes [6, 10]
(8)SOOK(f)=14Ts|P(f)|2+14Ts2∑j=-∞∞|P(jTs)|2δ(f-jTs).In a variant (antipodal signals), the transmitter sends a pulse of durationTp much higher than the bit time Tb to indicate bit “1” and the negative of this pulse for “0”. Therefore, with equiprobable elements, we have σd2=1,μd=0 and thus, a PSD of [10]
(9)Sant(f)=1Ts|P(f)|2,
showing that the PSD of antipodal does not have any spectral lines or discrete components. Note that OOK and binary-PPM are similar. Therefore, only 2-PPM performance was evaluated to avoid duplication.


## 2.3. M-PPM

In PPM, the chosen bit to be transmitted influences the position of the UWB pulses. The signal at the output of the transmitter cascade,s(t), can be expressed as [1, 2, 6]:
(10)SPPM(t)=∑j=-∞∞∑m=1MP(t-jTs-cmTC-amε),
where j is the number of transmitted bits, Ts is the symbol period, cm is the time-hopping code, cmTc is the  pseudorandom code in chip duration TC, am depends on j, and ε is the time shift associated with TH-PPM.The chip time intervalTC represents only one pulse in the time frame period. Note that amε+Tp<Tc with Tp the pulse period.The Fourier transform of the PSD of M-PAM can be expressed as [11]
(11)SPPM(f)=∫-∞+∞SPPM(t)e-j2πftdt.
Leading to the following PSD, expressed in terms of continuous and discrete parts
(12)PSPPM(f)=PSCPPM(f)+PSdPPM(f)
with
(13)PSCPPM(f)=|P(f)|2Ts[1-(1Msin(πfT)sin(πfT/M))2]PSdPPM(f)=|P(f)|2Ts2(1Msin(πfT)sin(πfT/M))2×[∑j=-∞∞δ(f-jTs)],
which can be simplified to [6, 11]
(14)PSPPM(t)=1Ts2∑j=-∞∞∑m=1M|P(jMTs)|2δ(f-jMTs),
showing a smother PSD than of PAM, leading to less narrowband interference. The single pulse of the UWB signal is placed into one of N=Tsym/Th slots, where N is the number of slots per symbol, Th is the pulse width and Tsym is the symbol time. The number of bits per one PPM symbol is equal to log2N.Note that by increasingM (modulation level) the data rate will increase as well. Furthermore, the average transmitted power will be increased by a factor of M.In this case, theN slots are divided into L groups. Each group includes N/L slots with a single pulse. With this, the average bit number will be [11]
(15)Nb′=log2(N/L)L=L(log2N-log2L)N>>L,Nb′=Llog2N=Nb.Thus, modifying the bit rate can improve the throughput of number of pulses perN slots, but it will raise other issues.(i)
A larger bandwidth requirement, as the number of pulses per symbol interval increases.(ii)
The pulse width relaxation will be limited.(iii)
All symbols can be orthogonal. To solve this for large N, one can use predefined pulse arrangement such that no pulse is put in the same position. In other words, to avoid pulse interpath interference (IPI) or pulse jamming, we should decrease the pulse period or increase the chip period.Note that more information on PSK modulation can be found in [4, 6]. Because of its complexity, it has been introduced here just for comparison.


## 3. AWGN Channels

Under AWGN UWB channel models expressed by the IEEE 802.15.3a standard, symbol, or bit error rates can be easily expressed, leading to an optimum receiver [6, 7], consisting on an optimum demodulator (coherent demodulator). First, let us assume a zero system margin (no power loss between equipment connections), that is, Ms=0.For any binary modulation, the BER can be calculated using the distance between two symbols as [6]
(16)Pb=Q(d2N0).This distance is different for each modulation scheme(17)dPPM=2γb;dPSK=2γb;dOOK=2γb;dPAM<2γb,γb=ERx/N0=Nsγp is the average bit energy in the receiver and γp is its average pulse energy. Table 1 summarizes the BER and SER performance for different modulation schemes.Table 1
BER and SER in AWGN channel for different UWB modulations.
TH-BPPM
Orthogonal single pulse (Ns=1)
P
r
b
=
Q
(
N
s
γ
b
)
,
Q
(
x
)
=
1
2
erfc
(
x
2
)
s
m
=
{
s
0
=
+
p
T
x
[
1
-
R
0
(
m
)
]
,
s
1
=
-
p
T
x
[
1
-
R
0
(
m
)
]
whereRj(ε) is autocorrelation of the pulse Sm.
orthogonal multipulse signal with TH coding:
P
r
b
=
Q
(
N
s
γ
b
(
1
-
R
0
(
m
)
)
)
BPAM/BPSK
P
b
=
Q
(
2
γ
b
)
M-PPM
P
M
=
1
2
π
∫
-
∞
∞
[
1
-
(
1
2
π
∫
-
∞
y
e
-
x
2
/
2
d
x
)
M
-
1
]
⋯
⋯
exp
[
-
1
2
(
y
-
2
γ
s
)
2
]
d
y
P
b
=
2
k
-
1
2
k
-
1
P
M
≈
1
2
P
M
M-PAM
P
M
=
2
(
M
-
1
)
M
Q
(
6
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g
2
(
M
)
γ
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)
γ
b
γ
b
=
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s
ξ
b
N
0For simplicity, let us assumeNs=1 with a normalized pulse energy. From theory, and as shown in Figure 2, binary PAM has 3 dB more power efficiency than PPM. As M is increasing, the performance of M-PPM also increases. This is due to space dimensionality that makes more noise orthogonal and causes the elements of the covariance noise matrix, Qc, to be almost zero [9, 12]. Since the signal-to-noise ratio (SNR) is given by SNRr=STQc-1S, it will also increase. Also, M-PPM requires lower transmission power in high modulation levels. On the other hand, the performance of M-PAM becomes worse due to decreased symbol distance and as a result, |Qc| will increase leading to more interference at narrowband. Thus, by increasing the modulation level, the M-PPM will exhibit better performance.Figure 2
Symbol error rate for differentM levels of M-PAM and M-PPM (Ns=1).Note that (17) confirms what was mentioned above, that is, simulating 2-PPM and OOK parameters leads to similar performance. Therefore, we just simulated and compared the M-PAM and the M-PPM in terms of BER versus SNR while a comparison between OOK and PPM is not necessary.Table2 summarizes the comparison between the different modulations schemes in terms of noise interference, spectral lines, transceiver complexity, pulse energy variance, and bandwidths.Table 2
Comparison of various pulse modulation techniques in terms of bandwidth, spectral line, noise sensitivity, transceiver complexity, and pulse energy variance.
Bandwidth requirement(normalized)
Power requirement(normalized)
Noisesensitively
Spectral linepower distortion
Transceiver complexity
Pulse energyvariance
PAM
1
1
Highest(1 dimension)
Yes
Low
(
M
2
-
1
6
)
1
γ
s
PPM
(
M
lo
g
2
M
)
(
τ
T
s
)
2
M
lo
g
2
M
Lowest(M dimensions)
Yes(lower)
Highest(by M)
1
γ
s
OOK
1
1
Very high
Yes(high)
Lowest
1
γ
s
BPSK
τ
T
s
1.12
Lower(M/2 dimensions)
No
High(by M/2)
1
γ
sAs shown, the candidate pulse modulation is strongly related to the application. In fact, if simplicity is considered, the repeat pulse coding or OOK modulation will be the good option but in case of NBI the 4-PPM should be chosen to satisfy all spectrum requirements.


## 4. Data Rate and Maximum Distance

To further investigate the modulation scheme performance, let us assume that the signal propagation occurs over free space. Thus, the free-space attenuationAm can be expressed as [6, 13]
(18)Am=4πD2LGTSR,
with D is the distance of propagation, SR is the antenna effective area, L is the system hardware loss, and GT is the transmitter antenna gain. In this work, we assumed L=1 (no losses in the system hardware). The pulse waveform is the 5th derivative of a monocyclic Gaussian pulse and its PSD is given by [13]
(19)Ps(f)=Amax(2πfδ)2ne-(2πfδ)2nne-n;Am=10-13.125;δ=51ps,
leading to the following receiving signal power
(20)Pr=MsSNRspecN02,
where N0 is the thermal noise power, SNRspec is the system requirement SNR, and H(f) is the frequency response of the indoor channel, which can be expressed as [13]
(21)|H(f)|=1Am(f)R(f),
where R(f) is the shadowing parameter that depends on the geography of the propagating environment (usually assumed to be 6 dB). The system margin Ms is assumed to be 1 dB. From this, the distance can be reformulated as [6, 7]
(22)D2=(GTGRc2/(4π)2Rb)2∫fLfH(Ps(f)/f2)dfMsSNRspec(1/2)N0,N0=kTs(f)=k(TA+(F(f)-1)T0),
where Ts is the noise temperature at the receiver, TA is the receiver antenna temperature, T0 is the room temperature (usually T0=300°K), k is the Boltzmann constant, F(f) is the noise factor of the receiving device, and Rb is the data rate. Given a specific SNR (i.e., the SER of a specific modulation scheme), one can evaluate the maximum transmit distance. Figure 3 shows the maximum distance as function of data rate for PAM and PPM modulation schemes. The following assumptions have been made(i)
SER is lower than10-5(Ns=1,Ts=3ns).(ii)
No fading margin required and PSD of noise is constant (Eb/N0).Figure 3
M-PAM and M-PPM signals data rate for short distances and uncoded systems.It is observed that BPAM (BPSK) has the best data rate over short distances with respect to complexity. However, by increasing the modulation levelM, the M-PPM data rate can be improved (SER8PPM<SER4PPM<SER2PPM) while the opposite occurs for M-PAM.


## 5. Channel Capacity

A channel with M-ary PPM/PAM modulation has discrete-valued inputs and continuous-valued outputs, which imposes an additional capacity calculation. Letsm be the encoded M-ary PPM/PAM signal vector input to the channel and ri theith diversity channel output vector corrupted by the AWGN noise vector. Let wi be an M-dimensional Gaussian vector with zero mean and variance σw2=N0/2 along each real direction (multipath). This output vector is given by:
(23)ri=sm+wi1≤i≤L,
where L is the number of multipaths. The channel capacity (bits/channel), with input signals restricted to an equiprobable M-ary signal constellation and with no restrictions on the channel output, can be written as [14]
(24)C=log2(M)-1M∑m=1M∫P(rsm)log2(∑k=1MP(r/sk)P(r/sm))dy,P(rsm)=∏i=1L[(1πN0)e(-(ri-E)2/N0)],
with P is the probability to receive a signal from the transmitter and N0 is the PSD of white noise. For M-PAM over an AWGN channel, we obtain
(25)C=log2(M)-1M∑k=0M-1×〈{log2∑p=0M-1exp(∑i=1L|wi|2-|sk+wi-sp|2N0)}〉,
where sp is one of the M-ary PAM signals. For M-PPM, the probability of receiving a signal is [7, 9, 14]
(26)P(rsm)=∏i=1L[(1πN0)M/2(∏j=1Me-rij2/2σT2)e-(rij-Eg)2/N0]rij=Eg+wij,σT2(n)=σpre2(n)+σpost2(n)+σw2,σpre2(n)=1N∑k=-∞-1h2(n-k),σpost2(n)=1N∑k=1nh2(n-k)forn>0,
where σT2 is the total noise variance, Eg is the signal energy, M is the signal dimension, σpre2(n) is the variance of the noise pulses preceding the present signal pulse at time n, σpost2(n) is the variance of the noise of the pulses following the present signal, and h(n) is the discrete channel response.Giving the following channel capacity model for symmetric channel (Figure4), and [9, 12, 15]
(27)CM-PPM=1M∑i=0M-1∑j=0,j≠iM-1PeM-1log[(Pe/(M-1))(1/M)]+1M∑i=0M-1∑j=0,j≠0M-1Pcmlog[Pcm(1/M)],CM-PPM=log2(M)+Pcmlog2(Pcm)+(1-Pcm)log2(1-PcmM-1).Figure 4
Channel capacity model for M-ary.Figure5 shows the channel capacity of various modulation schemes. In this figure, the capacity can be improved by increasing the modulation level for M-PPM, M-PAM, and M-PSK. Note that the theory details about this comparison are expressed in Table 2.Channel capacity comparison of PAM, PPM, and PSK in various modulation levels.
(a)
(b)
(c)


## 6. Maximum Transfer Power

From a communication theory perspective, the most important characteristic of UWB systems is the power-limited regime operation. Following standards and regulations, there is a maximum allowed transmit powerPt. This limitation has a direct impact on the maximum transmit distance and data rate [8, 16]
(28)PtPr(dB)=C0+10χlog10(λ4πd)+XR(dB),
where
(29)Pt(dB)=10log10(MsRbEb0GTGR(4πλ)d2).In the above equations,XR is the average shadowing system, χ is the path-loss component, and C0 is the maximum UWB pulse power, assumed to be −41.42 dBm [8]. Eb0 is the energy per bit (1.82. 10−19 W) for a specified BER of 5.10-5.In Figure6, we compared the transfer power for different modulation levels at a fixed data rate (200 MHz) and as function of maximum transmitter power and distance. It can be seen that BPSK soft decision has a higher maximum distance compared to other modulation schemes.Figure 6
Maximum transfer power for M-PAM-M-PPM, BPSK for a fixed data rate of 200 Mb/s. The horizontal line (close to −5 dBm) represents the maximum transmit power specified by FCC regulations.By increasing the modulation levelM, M-PPM will be improved but M-PAM and M-PSK will exhibit a lower performance, due to more noise interference (noise dimensionality). Then again M-PPM has better narrowband interference robustness in terms of required less transmit power with respect to system complexity. Furthermore, the 4-PPM has an optimum modulation level which covers all tradeoff as sorted by Table 2.In Figure7, we compared the 4-PPM in various data rates using the same transmitter and channel propagation scenario as in Figure 5 for different modulation levels. As expected, by increasing distance, the data rate will decrease, requiring higher transmitter power. For short distance applications (<10 m), the data rate cannot exceed 200 Mb/s even with an LOS scenario and AWGN channel assumption. This demonstrated that UWB technology cannot practically provide the expected high data rate (>400–600 Mb/s) unless a pulse coding is considered. In fact, with more complexity and higher modulation levels, the data rate can be significantly improved but with the cost of higher transmitter power.Figure 7
Maximum transfer power for 4-PPM with various data rate. The horizontal line (close to −5 dBm) represents the maximum transmit power specified by FCC regulations.


## 7. Conclusion

Various UWB pulse modulation schemes were evaluated in this paper and their performance was compared in terms of narrowband interference robustness, symbol error rate, system complexity, data rate, and maximum transmit power with respect to transceiver distance and channel capacity.It appears that BPAM has the highest data rate in terms of maximum distance and transmit power. However, M-PPM has better power efficiency and channel capacity, at the expense of increased complexity and bandwidth.If bandwidth is not an issue, the M-PPM is the best option, due to lower spectral lines (smoother PSD), higher dimensionality (orthogonally of noise), and narrowband interference robustness. However, if the UWB receiver is single band, the M-PPM is not the optimum option in terms of bandwidth.If the system complexity and bandwidth are the limiting factors, OOK is the best choice, but has more spectral lines and higher probability of error which leads to higher risk of interference. Finally, because of signal orthogonally and lower spectral lines (smoother PSD), the 4-PPM (optimum modulation level) can be the best candidate for NBI robustness and can be used for unique UWB rake filter-bank receiver design.



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*Source: 590153-2012-10-02.xml*