Constant envelope binary OFDM phase modulation - Military ...
CONSTANT ENVELOPE BINARYOFDM PHASEMODULATION Steve C. Thompson John G. Proakis James R. Zeidler Center for Wireless Communications University of California, San Diego La Jolla, CA 92093-0407 sct @ucsd.edu
Angle Modularor
_.
ABSTRACT
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1. INTRODUCTION
The high peak-to-average power ratio (PAPR) inherent to orthogonal frequency division multiplexing (OF'DM) is one of its primary drawbacks. The superposition of the orthogonal subcarriers yields a waveform with a large dynamic range that is not well suited for nonlinear power amplifiers (PA) used in the wireless environment [I], [2]. To reduce the undesirable effect of nonlinear intermodulation distortion output power backoff (OBO) is typically required. This O B 0 reduces the operational efficiency of the PA and also the signal-to-noise ratio (SNR) of the system. For example, a class-A power amplifier operating with a 10 dB O B 0 has an efficiency of less than 5% and a 10 dB loss in signal power [I]. Still, with this large backoff in power, intermodulation distortion is not eliminated completely which impacts bit-error-rates (BER). The nonlinear distortion imposes out-of-band spectral growth thus reducing the spectral efficiencies of OFDM. Furthermore, the low power efficiency can be detrimental to mobile devices which operate on battery power.
~
t
In this paper a new approach is presented to alleviate the
undesirable efects due to the high peak-to-average power ratio of orthogonal frequency division multiplexing. A class of~. siRnalinR called OFDM angle modulation is presented. These constant envelope OFDM based sinnals guarantee a OdB PAPR thus are well suited for eficient nonlinear power amplification. This paper focuses on binary OFDM phase modulation, which is a special case of OFDM angle modulation.
- -,,
OFDM
Data
1
*%~@koi,
Power Amplifier
.
t
High PAPR
Fig. 1.
.$
0 dB PAPR
Standard OFDM with transformation (shaded)
[IO]. Pre-distortion techniques such as clipping and filtering, peak windowing, and peak cancellation have been studied [I 1]-[13]. Researchers are actively pursuing amplifier designs with ereater linearity and efficiency to accommodate signals with large PAPR's [I] and signal processing combined with amplifier design techniques have been reported [141. The effectiveness of these methods vary and each comes with its own inherent trade-off in terms of bit-error-rate performance, implementation complexity, a n d o r spectral efficiency. '
This work investigates a new approach to PAPR reduction in which the OFDM waveform is transformed into a conslantenvelope OdB peak-to-average power ratio waveform. A simple block diagram is shown in Fig. 1. The nature of the transformation is to use the OFDM waveform to angle modulate the carrier signal using either phase modulation (OFDM-PM) or frequency modulation (OFDM-FM). The constant envelope property of the resultant signal achieves the goal of PAPR reduction and is well suited for highly efficient amplification. The out-of-band spectral growth associated with nonlinear distortion is eliminated. This paper presents the general form of OFDM angle modulation including OFDM-FM and OFDM-PM. Bandwidth considerations are discussed and performance of binary OFDMPM is studied in the additive white Gaussian noise channel with a proposed receiver structure. System performance comparisons are made between OFDM and OFDM-PM.
Much attention has been given to the PAPR problem [31-[6]. Coding techniques whereby data symbols are mapped to a subset of OFDM waveforms with relatively low PAPR have 11. THEBASEBAND OFDM WAVEFORM been extensively investigated [7], [8].Techniques based on scrambling subcarrier phase andlor amplitude in a pseudorandom way known at the receiver have been presented [9], The OFDM baseband waveform can be represented by N-1
dkeJ2*'',
~ ( t= )A ,
This work was supported by the UCSD Center for Wireless Communications and by the CoRe research gant 00-10071
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where h, is the modulation index for OFDM-PM. Since $ ( t ) is required to be real-valued the real part of the OFDM message signal is used in (3) and (4).
In these definitions the complex-valued OFDM baseband waveform (1) is used. Typically this signal is computed in discrete-time by applying the discrete Fourier tsansform (DFT) to the complex-valued data symbols. Alternately, since a real-valued signal is needed for @(I), the discrete cosine transform (DCT) could be used to calculate an OFDM waveform with 2N PAM data symbols modulating cosines separated by 1/2T Hz [IS]. Either approach has the same bandwidth requirement.
N-1
%[dk]COS2nkt/T -3[dk] sin2nkrlT k=O
and N-1
3[v(t)] =A,
~
n O M l i k J time: f I T
where A, is a gain constant, N is the number of subcarriers, T is the signaling interval, and data symbol dk modulates the k" subcarrier eJ2*/'. The data symbols are chosen from a complex set defined by an M-point signal constellation such as PSK or QAM. The real and imaginary parts of v(f) are %[v(t)] = A Y
0.1
9[dk]cos2nkf/T+R[dk]sin2nkt/T k=O
Figure 2 illustrates the large dynamic range of the OFDM signal. Random phase alignment of the subcarriers results in large signal peaks. Figure 3 plots the squared envelope, % [ v ( r ) ] * + 3 [ ~ ( r ) ]with ~ , the peak and average signal powers. For this example the peak-to-average power ratio is more than 6dB.
IV. BINARYOFDM PHASE MODULATION A special case of OFDM angle modulation uses phase modulation with binary data symbols resulting in the binary OFDM-PM bandpass signal
111. OFDM ANGLEMODULATION
The OFDM angle modulation handpass waveform is represented as s ( t ) =A,cos(2nfcf +@(f)) (2) where A, and fc are the carrier amplitude and frequency. For OFDM-FM the instantaneous frequency of the carrier is proportional to the OFDM message thus the phase signal in (2) is .
@(r) -2nh,[%[v(r)]dr
(3)
for 0 5 f 5 T where dk E {kl} for all k. It is convenient to define the message signal as m ( f )= @(r)/hp.The constant A, is chosen to normalize the average power of the message signal, that is P,, = Jlm(r)df/T = 1 with A, = ,/?@?. The ! + = J l @ ( f ) d r / T= h:. The power of the phase signal is then P average power of (5) is Aj/2 and the signal energy is E, = A:T/2. Since there are N bits of information per transmission the average bit energy is !Ea =AfT/2N.
where h, is the modulation index. In OFDM-PM the carrier phase is proportional to the OFDM message signal, hence
@(f)= hpRlV(f)l
Figure 4 shows the binary OFDM message signal from Fig. 2 and the resulting constant envelope OFDM-PM bandpass signal.
(4)
622
Ha PAPR
WB PAPR
Fig. 4. (a) Binary OFDM message signal and (b) the resulting OFDM-PM signal. N = 16. h, = 0.8. / c = 20/T
Fig. 5 . Block diagram of OFDM-PM receiver
A. Bandwidth Considerations
HI(f)centered at fc. This front-end filter is assumed to pass s(r) with negligible distortion while rejecting out-of-band noise. Its output is
In general, evaluating the power spectral density of an angle modulated process if quite involved due to the nonlinear nature of the modulation. However some simple observations can give a rough idea about the bandwidth requirements of OFDM-PM.
r(r) = s ( t ) + n ( t ) = s ( t ) +n,(t)cos2nfcr -n,(t)sin2nfct
where n,(t) and ns(t) are the quadrature components of the zero-mean filtered Gaussian noise n(t). Equation 7 can be represented as
As shown above, the modulation index controls the phase signal power Tq= h$ For very small h, the phase excursions are low and the OFDM-PM signal approximates to s ( f )% A,cosZnf,t -A,hpm(r)sin2nfCt
(7)
r(t) = s ( t )
+ J=-cos(2nft+
tan-' [ns(f ) h( t ) ] ) = s ( t ) V.(t)cos (2nf,t +e&))
(6)
(8)
+
This can be seen by expanding s(r) into a Taylor series and taking only the first two terms [16]. This narrowband case shows that the bandwidth of the signal is at least 2W where W is the bandwidth of the OFDM message m(t). As h, becomes larger (6) is no longer valid and the bandwidth of the signal broadens. As discussed in 1171, a bandwidth approximation know as the Carson's rule is B = 2(max[$(t)l+ l)W =2(h,&+ 1)W. With a large number of subcarriers this Carson bandwidth is overly conservative since rnax[$(t)] has a low occurrence rate of approximately 2-N. However, for large N the phase signal is accurately modeled as a Gaussian distributed process thus the root-mean-square bandwidth B = 2&W = 2h,W is a more appropriate measure j171. For the purposes of this paper the bandwidth of OFDM-PM is B = max[2W,2hPW].
where V,,(t) and e,(() represent the envelope and the phase of the noise process. Equation (8) simplifies to r(t) =R(f)cos(2nfcr+$(t) +$&))
(9)
where R ( r ) is the envelope of r(t) and
is the phase error term. Treatment of these trigonometric identities are standard in modem textbooks 1161, 1171. The phase deviation of the angle demodulator input is
v(t)= @ ( t -)+ @ e @ )
(11)
With the assumption of a high carrier-to-noise ratio (CNR) >> V,(t), (IO) becomes
A,
B. AWGN Analysis In this section the performance of a binary OFDM-PM receiver is evaluated. As shown in Fig. 5 the OFDM-PM receiver consists of an angle demodution front-end used to detect the phase of the received signal followed by a standard OFDM receiver to make decisions on the transmitted data symbols. Each block will be analyzed below.
The output of the angle demodulator is Y ( t )= W r )
(13)
where K is a gain. With the large CNR assumption and choosing K = lfh,, ~ ( t=)m ( r ) . + W )
I) Angle Demodulation Front-end: The zero-mean additive white Gaussian noise (AWGN), nw(r), with power-spectral density S.,(f) = N0/2 is added to s ( f ) and the result is passed through the bandpass filter with transfer function
(14)
where Vdt) Y,(t) = -sin(@,(t) hdc 623
- h,m(t))
(15)
2 ) OFDM Receiver: From (20) the zero mean additive Gaussian noise is flat over the OFDM message frequency band thus appears white. The OFDM receiver block correlates the noisy input with the N subcarriers in parallel. The AID converter samples z ( t ) at N equally spaced time instances over 0 2 r IOdB) is made to linearize the analysis. Ideal filters are assumed in the receiver frontend. Estimates are made on the transmitted data symbols with the use of the discrete Fourier transform. The modulation index controls the spectral spreading and detection performance of OF'DM-PM. Improved hit-errorrate performance is seen with increased modulation index at 625
fixed transmission power. Performance comparisons to ideal binary OFDM are made with a 6dB output power backoff differential. Comparable BER performance i s observed with improved utilization of the power amplifier at the cost of spectral efficiency.
1151 . . [I61 [I71
VI.
ACKNOWLEDGMENTS
[I81 [I91
The authors want to thank Professor Ray Pettit from California State University, Northridge and Dr. Richard North from SPAWAR Systems Center, San Diego. Also, thanks to Mike Geile from Nova Engineering (Cincinnati, OH) for his creative ideas at the genesis of this research. REFERENCES [ l ] F. H. Raab er. al., "Power Amplifiers and Transmitters for RF and Microwave:' IEEE Tmnrocrions on Micmwove Theory and Techniques, vol. 50, pp. 814-826. Mar. 2002. [2] E. Costa. M. Midrio, and S . hpolin. "Impact of Amplifier Nonlinearities on OFDM Transmission System Performance:' IEEE Coniniunicnrions Lerrers, vol. 3, pp. 37-39, Feb. 1999. [3] K. Patersan and V. Tarokh. "On the Existence and Construction of Good Codes with Low Peak-to-Average Power Ratios:' IEEE Trnnroctionr on Infomarion nteory, vol. 46. pp. 1974-1987. Sept. 2oM1. [4] G.Wunder and H. Boche. ''Upper Bounds on the Statistical Distribution of the Crest-Factor in OFDM Transmission:' IEEE Trrmraclionr on Information Theory, vol. 49. pp. 488494, Feb. 2003. 151 1. Tellado. L. Hw. and 1. Cioffi. "Maximum-Likelihwd Detection of Nonlinearly Distorted Multicanier Symbols by Iterative Decoding,'' IEEE Transactions on Cornnumications Theory, vol. 51. pp. 218-228, Feb. 2003. 161 X. Wang, T. Tjhung, and C. Ng. "Reduction of Pealr-to-Average Power Ratio of OFDM Systems using a Companding Technique:' IEEE 7ransarriom on Brondeosring, vol. 45, pp. 303-307. Sept. 1999. [7] R. V. Nee, "OFDM Codes for Peak-tc.Average Power Reduction and Ermr Correction:' in Proceedings of IEEE GLOBECOM, (London), pp. 74b744. Nov. 1996. 181 V. Tarokh and H. Jafarkhani. "On the Computation and Reduction of the Peak-to-Average Power Ratio in Multicarrier Communications:' IEEE Tramoerionr on Comnrunicotionr. vol. 48, pp. 3 7 4 , Jan. 2000. [9] S . Muller and J. Huber, "OFDM with Reduced Peak-to-Average Power Ratio by Optimum Combination of Pmial Transmit Sequences:' Elerrronic Leners, vol. 33. pp. 368-369. Feb. 1997. [IO] S. Muller, R. W. Bauml, R. F. H. Fisher, and 1. B. Huber, "OFDM with Reduced Peak-to-Averige Power Ratio by Multiple Signal Representation:' Annals of Telecomrnunicntionr. vol. 52, pp. 58-67. Feb. 1997. [ I l l M. Pauli and H. Kuchenbecker, "Minimization of the Intemodulation Distortion of a Nonlinearly Amplified OFDM Signal:' Wireless Perkonnl Conmenicoriom. vol. 4, pp. 93-101, Jan. 1997. [I21 T. May and H. Kohling, "Reducing the Peak-to-Average Power Ratio in OFDM Radio Transmission Systems:' in Proceedings of IEEE Veliiculor Technology Conference, vol. 3, (Ottawa, Canada), pp. 27742778, May 1998. [I31 X. Li and L. 1. Cimini. "Effects of Clipping and Filtering on the Performance of OFDM:' in Proceedings of IEEE Vehicular Technology Conference. vol. 3. pp. 1634-1638. May 1997. [I41 W. Liu. 1. La". and R. Cheng, "Considerations on Applying OFDM in a Highly Efficient Power Amplifier:' IEEE Tranracrionr on Cirruirs
626
ond Sysienis II: Analog ond Digilnl Signal Processing. vol. 46, no. I I , pp. 1329-1336, 1999. 1. Tan and G. L. Stuber, "Constant Envelope Multi-Carrier Modulation:' in Proceedings of IEEE Milirory Contrnunicarionr Conference. vol. I, (Anaheim), pp. 6 0 7 4 1 1, Oct. ZWZ. R. Zeirner and W. Tranter, Principles of Comnzunicorionr: Syslenu, Modulorion, and Noise. Wiley, 4* ed., 1995. I. G. Proakis and M. Salehi, Conintunicorion Systems Engineering. Prentice-Hall, 1994. 1. 1. Downing, Modulation Syrrenis and Noire. Prentice-Hall, 1964. 1. G. Proakis. Digirol Cornmunicotionr. McGraw-Hill, 4Ih ed., ZWI.
Angle Modularor
_.
ABSTRACT
: ?.,.'~..-.: ~
1. INTRODUCTION
The high peak-to-average power ratio (PAPR) inherent to orthogonal frequency division multiplexing (OF'DM) is one of its primary drawbacks. The superposition of the orthogonal subcarriers yields a waveform with a large dynamic range that is not well suited for nonlinear power amplifiers (PA) used in the wireless environment [I], [2]. To reduce the undesirable effect of nonlinear intermodulation distortion output power backoff (OBO) is typically required. This O B 0 reduces the operational efficiency of the PA and also the signal-to-noise ratio (SNR) of the system. For example, a class-A power amplifier operating with a 10 dB O B 0 has an efficiency of less than 5% and a 10 dB loss in signal power [I]. Still, with this large backoff in power, intermodulation distortion is not eliminated completely which impacts bit-error-rates (BER). The nonlinear distortion imposes out-of-band spectral growth thus reducing the spectral efficiencies of OFDM. Furthermore, the low power efficiency can be detrimental to mobile devices which operate on battery power.
~
t
In this paper a new approach is presented to alleviate the
undesirable efects due to the high peak-to-average power ratio of orthogonal frequency division multiplexing. A class of~. siRnalinR called OFDM angle modulation is presented. These constant envelope OFDM based sinnals guarantee a OdB PAPR thus are well suited for eficient nonlinear power amplification. This paper focuses on binary OFDM phase modulation, which is a special case of OFDM angle modulation.
- -,,
OFDM
Data
1
*%~@koi,
Power Amplifier
.
t
High PAPR
Fig. 1.
.$
0 dB PAPR
Standard OFDM with transformation (shaded)
[IO]. Pre-distortion techniques such as clipping and filtering, peak windowing, and peak cancellation have been studied [I 1]-[13]. Researchers are actively pursuing amplifier designs with ereater linearity and efficiency to accommodate signals with large PAPR's [I] and signal processing combined with amplifier design techniques have been reported [141. The effectiveness of these methods vary and each comes with its own inherent trade-off in terms of bit-error-rate performance, implementation complexity, a n d o r spectral efficiency. '
This work investigates a new approach to PAPR reduction in which the OFDM waveform is transformed into a conslantenvelope OdB peak-to-average power ratio waveform. A simple block diagram is shown in Fig. 1. The nature of the transformation is to use the OFDM waveform to angle modulate the carrier signal using either phase modulation (OFDM-PM) or frequency modulation (OFDM-FM). The constant envelope property of the resultant signal achieves the goal of PAPR reduction and is well suited for highly efficient amplification. The out-of-band spectral growth associated with nonlinear distortion is eliminated. This paper presents the general form of OFDM angle modulation including OFDM-FM and OFDM-PM. Bandwidth considerations are discussed and performance of binary OFDMPM is studied in the additive white Gaussian noise channel with a proposed receiver structure. System performance comparisons are made between OFDM and OFDM-PM.
Much attention has been given to the PAPR problem [31-[6]. Coding techniques whereby data symbols are mapped to a subset of OFDM waveforms with relatively low PAPR have 11. THEBASEBAND OFDM WAVEFORM been extensively investigated [7], [8].Techniques based on scrambling subcarrier phase andlor amplitude in a pseudorandom way known at the receiver have been presented [9], The OFDM baseband waveform can be represented by N-1
dkeJ2*'',
~ ( t= )A ,
This work was supported by the UCSD Center for Wireless Communications and by the CoRe research gant 00-10071
k=O
621 0-7803-814O.Om3$17.M00 2M3 IliZE
05t 5T
(1)
I
70
6
Ea
2
-
3
.-
0
F$
-2
e 9" -
t,
4
20
10
-8
-'OO
0.1
0.2
03
0.4
0.5
0.8
normflied lime: I I T
0.7
08
0.9
0-
0
1
0.2
0.3
0.4
0.S
OS
0.1
08
0.9
1
where h, is the modulation index for OFDM-PM. Since $ ( t ) is required to be real-valued the real part of the OFDM message signal is used in (3) and (4).
In these definitions the complex-valued OFDM baseband waveform (1) is used. Typically this signal is computed in discrete-time by applying the discrete Fourier tsansform (DFT) to the complex-valued data symbols. Alternately, since a real-valued signal is needed for @(I), the discrete cosine transform (DCT) could be used to calculate an OFDM waveform with 2N PAM data symbols modulating cosines separated by 1/2T Hz [IS]. Either approach has the same bandwidth requirement.
N-1
%[dk]COS2nkt/T -3[dk] sin2nkrlT k=O
and N-1
3[v(t)] =A,
~
n O M l i k J time: f I T
where A, is a gain constant, N is the number of subcarriers, T is the signaling interval, and data symbol dk modulates the k" subcarrier eJ2*/'. The data symbols are chosen from a complex set defined by an M-point signal constellation such as PSK or QAM. The real and imaginary parts of v(f) are %[v(t)] = A Y
0.1
9[dk]cos2nkf/T+R[dk]sin2nkt/T k=O
Figure 2 illustrates the large dynamic range of the OFDM signal. Random phase alignment of the subcarriers results in large signal peaks. Figure 3 plots the squared envelope, % [ v ( r ) ] * + 3 [ ~ ( r ) ]with ~ , the peak and average signal powers. For this example the peak-to-average power ratio is more than 6dB.
IV. BINARYOFDM PHASE MODULATION A special case of OFDM angle modulation uses phase modulation with binary data symbols resulting in the binary OFDM-PM bandpass signal
111. OFDM ANGLEMODULATION
The OFDM angle modulation handpass waveform is represented as s ( t ) =A,cos(2nfcf +@(f)) (2) where A, and fc are the carrier amplitude and frequency. For OFDM-FM the instantaneous frequency of the carrier is proportional to the OFDM message thus the phase signal in (2) is .
@(r) -2nh,[%[v(r)]dr
(3)
for 0 5 f 5 T where dk E {kl} for all k. It is convenient to define the message signal as m ( f )= @(r)/hp.The constant A, is chosen to normalize the average power of the message signal, that is P,, = Jlm(r)df/T = 1 with A, = ,/?@?. The ! + = J l @ ( f ) d r / T= h:. The power of the phase signal is then P average power of (5) is Aj/2 and the signal energy is E, = A:T/2. Since there are N bits of information per transmission the average bit energy is !Ea =AfT/2N.
where h, is the modulation index. In OFDM-PM the carrier phase is proportional to the OFDM message signal, hence
@(f)= hpRlV(f)l
Figure 4 shows the binary OFDM message signal from Fig. 2 and the resulting constant envelope OFDM-PM bandpass signal.
(4)
622
Ha PAPR
WB PAPR
Fig. 4. (a) Binary OFDM message signal and (b) the resulting OFDM-PM signal. N = 16. h, = 0.8. / c = 20/T
Fig. 5 . Block diagram of OFDM-PM receiver
A. Bandwidth Considerations
HI(f)centered at fc. This front-end filter is assumed to pass s(r) with negligible distortion while rejecting out-of-band noise. Its output is
In general, evaluating the power spectral density of an angle modulated process if quite involved due to the nonlinear nature of the modulation. However some simple observations can give a rough idea about the bandwidth requirements of OFDM-PM.
r(r) = s ( t ) + n ( t ) = s ( t ) +n,(t)cos2nfcr -n,(t)sin2nfct
where n,(t) and ns(t) are the quadrature components of the zero-mean filtered Gaussian noise n(t). Equation 7 can be represented as
As shown above, the modulation index controls the phase signal power Tq= h$ For very small h, the phase excursions are low and the OFDM-PM signal approximates to s ( f )% A,cosZnf,t -A,hpm(r)sin2nfCt
(7)
r(t) = s ( t )
+ J=-cos(2nft+
tan-' [ns(f ) h( t ) ] ) = s ( t ) V.(t)cos (2nf,t +e&))
(6)
(8)
+
This can be seen by expanding s(r) into a Taylor series and taking only the first two terms [16]. This narrowband case shows that the bandwidth of the signal is at least 2W where W is the bandwidth of the OFDM message m(t). As h, becomes larger (6) is no longer valid and the bandwidth of the signal broadens. As discussed in 1171, a bandwidth approximation know as the Carson's rule is B = 2(max[$(t)l+ l)W =2(h,&+ 1)W. With a large number of subcarriers this Carson bandwidth is overly conservative since rnax[$(t)] has a low occurrence rate of approximately 2-N. However, for large N the phase signal is accurately modeled as a Gaussian distributed process thus the root-mean-square bandwidth B = 2&W = 2h,W is a more appropriate measure j171. For the purposes of this paper the bandwidth of OFDM-PM is B = max[2W,2hPW].
where V,,(t) and e,(() represent the envelope and the phase of the noise process. Equation (8) simplifies to r(t) =R(f)cos(2nfcr+$(t) +$&))
(9)
where R ( r ) is the envelope of r(t) and
is the phase error term. Treatment of these trigonometric identities are standard in modem textbooks 1161, 1171. The phase deviation of the angle demodulator input is
v(t)= @ ( t -)+ @ e @ )
(11)
With the assumption of a high carrier-to-noise ratio (CNR) >> V,(t), (IO) becomes
A,
B. AWGN Analysis In this section the performance of a binary OFDM-PM receiver is evaluated. As shown in Fig. 5 the OFDM-PM receiver consists of an angle demodution front-end used to detect the phase of the received signal followed by a standard OFDM receiver to make decisions on the transmitted data symbols. Each block will be analyzed below.
The output of the angle demodulator is Y ( t )= W r )
(13)
where K is a gain. With the large CNR assumption and choosing K = lfh,, ~ ( t=)m ( r ) . + W )
I) Angle Demodulation Front-end: The zero-mean additive white Gaussian noise (AWGN), nw(r), with power-spectral density S.,(f) = N0/2 is added to s ( f ) and the result is passed through the bandpass filter with transfer function
(14)
where Vdt) Y,(t) = -sin(@,(t) hdc 623
- h,m(t))
(15)
2 ) OFDM Receiver: From (20) the zero mean additive Gaussian noise is flat over the OFDM message frequency band thus appears white. The OFDM receiver block correlates the noisy input with the N subcarriers in parallel. The AID converter samples z ( t ) at N equally spaced time instances over 0 2 r IOdB) is made to linearize the analysis. Ideal filters are assumed in the receiver frontend. Estimates are made on the transmitted data symbols with the use of the discrete Fourier transform. The modulation index controls the spectral spreading and detection performance of OF'DM-PM. Improved hit-errorrate performance is seen with increased modulation index at 625
fixed transmission power. Performance comparisons to ideal binary OFDM are made with a 6dB output power backoff differential. Comparable BER performance i s observed with improved utilization of the power amplifier at the cost of spectral efficiency.
1151 . . [I61 [I71
VI.
ACKNOWLEDGMENTS
[I81 [I91
The authors want to thank Professor Ray Pettit from California State University, Northridge and Dr. Richard North from SPAWAR Systems Center, San Diego. Also, thanks to Mike Geile from Nova Engineering (Cincinnati, OH) for his creative ideas at the genesis of this research. REFERENCES [ l ] F. H. Raab er. al., "Power Amplifiers and Transmitters for RF and Microwave:' IEEE Tmnrocrions on Micmwove Theory and Techniques, vol. 50, pp. 814-826. Mar. 2002. [2] E. Costa. M. Midrio, and S . hpolin. "Impact of Amplifier Nonlinearities on OFDM Transmission System Performance:' IEEE Coniniunicnrions Lerrers, vol. 3, pp. 37-39, Feb. 1999. [3] K. Patersan and V. Tarokh. "On the Existence and Construction of Good Codes with Low Peak-to-Average Power Ratios:' IEEE Trnnroctionr on Infomarion nteory, vol. 46. pp. 1974-1987. Sept. 2oM1. [4] G.Wunder and H. Boche. ''Upper Bounds on the Statistical Distribution of the Crest-Factor in OFDM Transmission:' IEEE Trrmraclionr on Information Theory, vol. 49. pp. 488494, Feb. 2003. 151 1. Tellado. L. Hw. and 1. Cioffi. "Maximum-Likelihwd Detection of Nonlinearly Distorted Multicanier Symbols by Iterative Decoding,'' IEEE Transactions on Cornnumications Theory, vol. 51. pp. 218-228, Feb. 2003. 161 X. Wang, T. Tjhung, and C. Ng. "Reduction of Pealr-to-Average Power Ratio of OFDM Systems using a Companding Technique:' IEEE 7ransarriom on Brondeosring, vol. 45, pp. 303-307. Sept. 1999. [7] R. V. Nee, "OFDM Codes for Peak-tc.Average Power Reduction and Ermr Correction:' in Proceedings of IEEE GLOBECOM, (London), pp. 74b744. Nov. 1996. 181 V. Tarokh and H. Jafarkhani. "On the Computation and Reduction of the Peak-to-Average Power Ratio in Multicarrier Communications:' IEEE Tramoerionr on Comnrunicotionr. vol. 48, pp. 3 7 4 , Jan. 2000. [9] S . Muller and J. Huber, "OFDM with Reduced Peak-to-Average Power Ratio by Optimum Combination of Pmial Transmit Sequences:' Elerrronic Leners, vol. 33. pp. 368-369. Feb. 1997. [IO] S. Muller, R. W. Bauml, R. F. H. Fisher, and 1. B. Huber, "OFDM with Reduced Peak-to-Averige Power Ratio by Multiple Signal Representation:' Annals of Telecomrnunicntionr. vol. 52, pp. 58-67. Feb. 1997. [ I l l M. Pauli and H. Kuchenbecker, "Minimization of the Intemodulation Distortion of a Nonlinearly Amplified OFDM Signal:' Wireless Perkonnl Conmenicoriom. vol. 4, pp. 93-101, Jan. 1997. [I21 T. May and H. Kohling, "Reducing the Peak-to-Average Power Ratio in OFDM Radio Transmission Systems:' in Proceedings of IEEE Veliiculor Technology Conference, vol. 3, (Ottawa, Canada), pp. 27742778, May 1998. [I31 X. Li and L. 1. Cimini. "Effects of Clipping and Filtering on the Performance of OFDM:' in Proceedings of IEEE Vehicular Technology Conference. vol. 3. pp. 1634-1638. May 1997. [I41 W. Liu. 1. La". and R. Cheng, "Considerations on Applying OFDM in a Highly Efficient Power Amplifier:' IEEE Tranracrionr on Cirruirs
626
ond Sysienis II: Analog ond Digilnl Signal Processing. vol. 46, no. I I , pp. 1329-1336, 1999. 1. Tan and G. L. Stuber, "Constant Envelope Multi-Carrier Modulation:' in Proceedings of IEEE Milirory Contrnunicarionr Conference. vol. I, (Anaheim), pp. 6 0 7 4 1 1, Oct. ZWZ. R. Zeirner and W. Tranter, Principles of Comnzunicorionr: Syslenu, Modulorion, and Noise. Wiley, 4* ed., 1995. I. G. Proakis and M. Salehi, Conintunicorion Systems Engineering. Prentice-Hall, 1994. 1. 1. Downing, Modulation Syrrenis and Noire. Prentice-Hall, 1964. 1. G. Proakis. Digirol Cornmunicotionr. McGraw-Hill, 4Ih ed., ZWI.
