Root Raised Cosine (RRC) Filters and Pulse Shaping in Communication Systems
Erkin Cubukcu
Abstract
This presentation briefly discusses application of the Root Raised Cosine (RRC) pulse shaping in the space telecommunication. Use of the RRC filtering (i.e., pulse shaping) is adopted in commercial communications, such as cellular technology, and used extensively. However, its use in space communication is still relatively new. This will possibly change as the crowding of the frequency spectrum used in the space communication becomes a problem. The two conflicting requirements in telecommunication are the demand for high data rates per channel (or user) and need for more channels, i.e., more users. Theoretically as the channel bandwidth is increased to provide higher data rates the number of channels allocated in a fixed spectrum must be reduced. Tackling these two conflicting requirements at the same time led to the development of the RRC filters. More channels with wider bandwidth might be tightly packed in the frequency spectrum achieving the desired goals. A link model with the RRC filters has been developed and simulated. Using 90% power Bandwidth (BW) measurement definition showed that the RRC filtering might improve spectrum efficiency by more than 75%. Furthermore using the matching RRC filters both in the transmitter and receiver provides the improved Bit Error Rate (BER) performance.
In this presentation the theory of three related concepts, namely pulse shaping, Inter Symbol Interference (ISI), and Bandwidth (BW) will be touched upon. Additionally the concept of the RRC filtering and some facts about the RRC filters will be presented. Avionic Systems Analysis Pulse ShapinginCommunication Systems Root RaisedCosine Filters Friday, May 18, 2012 Erkin Cubukcu & Avionic Systems Analysis 5/7/2012 ● ● ● ● ● ● ● ● ● Conclusions BER Plots Signal Spectra Filters Link Modelingwith the RRC Facts aboutRRC Root RaisedCosine(RRC) (RC) Raised Cosine Filter (LPF) Pass Ideal Low and Bandwidth Intersymbol Interference (ISI), Pulse Shaping, Outline 2 Avionic Systems Analysis 5/7/2012 Digital Modulation Pulse Shaping,IntersymbolInterference (ISI), andBandwidth 3 Avionic Systems Analysis 5/7/2012 ● ● ● ● Theory in frequency). One possible solution is touseanIdealLowPassFilter (ILPF)(rectangular This iscalledInter-symbol Interference(ISI) points (intime) therewillbesignal(ta If channels are toonarrowthesymbolswill betoowide,hence.Atsampling Two conflictingrequirements!: ● ● Need forNarrowBandwidth(more users,morechannels,lessnoise). Demand forHighdatarates(more information). Pulse Shaping,IntersymbolInterference (ISI), andBandwidth(con’t) ils) oftheprevious andnextsymbols. 4 Avionic Systems Analysis 5/7/2012 ● ● ● Narrow bandwidthchannel No ISI Issues is aproblem(jitter inthesystemmightbedetrimental). attainable, requireextremeprecise synchronization,synchronization If Physically unrealizable and difficult toapproximate.
sin
t / t T / T Ideal LowPassFilter(ILPF) 5 Avionic Systems Analysis 5/7/2012 ● ● ● H h ( ( Effects of jittermaybeminimized to attain Much simpler that Nyquist by shown It is t f ) ) the impulse response will have zeros at uniformly spacedintervals. the impulseresponsewillhave zerosat If thefrequencycharacteristic hasoddsymmetryatthecutofffrequency, sin
T 0 T
2
t / t T 1 / Solution : RaisedCosine(RC)Filters T cos 1 cos (
4
T 2 t t 2 f / / T T 2 1 ) 2
T 0 f 1 2
T f 1 2
T 1 f 2
T 1 2
T (Eq. 2) (Eq. 1) 6 Avionic Systems Analysis 5/7/2012 ● ● Frequency Response Impulse Response Plot of Raised Cosine (RC)filter Plot ofRaised 7 Avionic Systems Analysis 5/7/2012 ● ● ● In addition, it hasanotherterm In addition, pass filter. low ideal crossings aslike has asincterm Impulse responsenow jitter. That decaysin timehencereduces Raised CosineFilter (Cont) 1 sin
cos (
4
t
/ t T / 2 T t t 2 / / T T the tails reducing the impactof reducing the tails 2 ) that ensures that it has zero that ithas thatensures 8 Avionic Systems Analysis
● 5/7/2012 ● ● c t fielLFwr sdtebsbn bandwidth would be If idealLPFwereusedthebaseband B Since
Bandwidth for a realizable RC filter
B B sin So thebaseband transmission bandwidth
t t f / ( ( / t c 1 1 T T / T
1 Bandwidth ofRaisedCosine(RC)Filter / ) ) 2
/ Where thesampling time is T f T c 2 H c T T ( f
Nyquist Bandwidth ) T 0 / T Nyquist Bandwidth times f
1 c 2 T
T
T
/ c
/( 1 2
f c ) 1 / 2 f c 9 Avionic Systems Analysis ● ● 5/7/2012 ● ● called RootRaisedCosinefilter. frequency domainandusethis newfilterintheTxandRx.Thisisso One wayofachieving it istotakes discussed above. be RaisedCosine(RC)as The overallchanneltransferfunction must H cosine filtertransfer characteristic When thetransmitterandreceiver filtersarecascaded one getsraised Or rrc
( H ) H rc rc
(
( H ) ) Root RaisedCosine(RRC)Filter
(
H ) rc rrc H , tx rc
(
(
1 2 ) ) H ( 1 rrc H , cos rc rx
(
(
) ) / quare rootoftheraisedcosine filterin
2 c
)
2 c 10 Avionic Systems Analysis 5/7/2012 ● ● H h ● ( By takingsquarerootofRCfilter frequencyresponse,onegets. t ( ) Finding itsimpulseResponse is alittlebittricky. f Impulse response canalsobegeneratednumerically usingIFFT. )
2
0 T Root RaisedCosineFilter (RRC) (con’t) T T 2 cos[( 1 1 cos
)
1 T t / T ( 4
f
] t sin[( 1 / T 2
T
) 2 1 4
t /
)
T t / T f 0 ] 1 2
1 T f 2
T 1 f 2
T 1 2
T 11 Avionic Systems Analysis ● ● ● ● ● ● 5/7/2012 narrow (high Q) filters in the RF bands isdifficult. in the filters narrow (highQ) base band in the RRC filtersareimplemented in stop bandisreduced. attenuation Smaller rolloffgives narrower bandwidth.Ho should besmaller.) bandwidth efficiencyrolloff (However, for no bandwidth restrictions itwould be easier increases As rolloff eye inthediagram Nyquist bandwidth of occupied beyondthe The rolloff factoris a measureoftheexce Decreasing the number ofsamples (filter band. However,inimplementation itslengthshould be reducedtoafinitevalue. taps RRC theoretically of has infinitenumber Facts aboutRRC delay) reduces the stop bandattenuation. delay) reducesthe ss bandwidth i.e., thebandwidth of thefilter, 1/2T, where 1/T issymbolrate. 1/2T, opens up.This if therewere means that on thereceiver if oneusedalargerolloff. wever, itssidelobes increasesso soithasinfiniteattenuationinthe stop as a digital filter. Since implementing as adigitalfilter. 12 Avionic Systems Analysis ● ● ● ● 5/7/2012 PAPR ofanRRC willincrease with linearity) more and/or (higher PAPRrequireshigherbackoff off High PAPRreduces power amplifierefficien PAPR is by combinationof determined (PAPR). pulse s consider in issue to Another increased filter length increased filter reduced excess bandwidth Pulse shaping signal the Constellation of Modulation Facts aboutRRC(cont.) 5 5 haping is the Peak-to-Average PowerRatio haping isthe cy since it mustoperatewith large back 13 Avionic Systems Analysis 5/7/2012 is a high levelblockdiagram ofthismodel. a is perf efficiencyError Rate(BER) andBit Adeveloped andsimulatedtostudythe spectral Model was SIMULINK Link ModelingwiththeRRCFilters ormance of the RRC filters. Below ormance oftheRRC 14 Avionic Systems Analysis 5/7/2012 Link ModelingwiththeRRCFilters(Cont.) pulse shaping is the last stage of transmitter before (DAC and) PA.transmitter before (DAC the laststageof pulse shapingis the channel (filtered) tocomply with up sampledandpulseshaped the modulator is symbols generatedby The modulatorgeneratesonesymbol for eachpairofdatabits. The Tx ModelwithRRCPulseShaping bandwidthrestrictions. Typically, the 15 Avionic Systems Analysis 5/7/2012 Link ModelingwiththeRRCFilters(Cont.) a pair of bits. received sample falls quadrant the one sampleperconstellation symbol.Demodulatorfinds out which give optimum points(i.e.,down-sampled) to the sampled at output is The is if Tx Intersymbol Interference,ISI, nearly zero give a filter filtersoutthesignal(i.e.,equalizesto RRC Rx Modelwith RRC Filter and basedonthatdecision generates also using RRC filter pulseshaping). also usingRRC 16 Avionic Systems Analysis 5/7/2012 RC Filter Impulse Response, rolloff 0.4, 0.7, and 1
Magnitude 10 12 14 -4 -2 0 2 4 6 8 -4
x 10 -3 -3 Symbol rate 1 Msps -2 Raised Cosine Filter Response Filter Cosine Raised -1 time ( 0 s) 1 2 RC, rolloff 1 rolloff RC, 0.7 rolloff RC, 0.4 rolloff RC, 3 4
17 Avionic Systems Analysis 5/7/2012 Just theplotofequation
Mag. (dB) -100 -50 -40 -30 -20 -10 -90 -80 -70 -60 0
0.5 Ideal RRC FrequencyResponse 1 with rolloff 0.4, 0.7, and 1.0 with rolloff0.4,0.7,and 1.5 Root Raised Filter Response Filter Raised Root Freq. (Hz) 2 2.5 3 Ideal rolloff RRC, 1 Ideal rolloff RRC, 0.7 Ideal rolloff RRC, 0.4 3.5 Symbol rate 1 Mbps x 10 6 4
18 Avionic Systems Analysis 5/7/2012 Symbol rate 1 Mbps Frequency Response of Realizable Root Raised of RealizableRoot Response Frequency Mag. (dB) -100 -90 -80 -70 -60 -50 -40 -30 -20 -10 0 Cosine Filter with Rolloff 0.4, 0.7, and 1 Filter with Rolloff0.4,0.7, and Cosine
0.5 1 1.5 Root Raised Filter Response Filter Raised Root Freq. (Hz) 2 2.5 3 3.5 rolloff 1 rolloff 0.7 rolloff 0.4 rolloff x 10 6 4
19 Avionic Systems Analysis 5/7/2012 Delay is half of the filter length in ti the filterlengthin half of Delay is
Mag. (dB) Effect ofthelengthFilter (Realizable) -100 -90 -80 -70 -60 -50 -40 -30 -20 -10 0
0.5 delay 2, 4, and 8 sym (rolloff 8 sym 0.4) 2,4,and delay 1 me. Hereisgivenasthesymbollength 1.5 Root Raised Filter Response Filter Raised Root Freq. (Hz) 2 2.5 3 3.5 del 8 sym del 4 sym del 2 sym del x 10 6 4
20 Avionic Systems Analysis 5/7/2012 OQPSK Tx RRCOutput Spectrum foraSymbolRateof6Msps ● ● ● Measured 90% 2-sided BW = 2.78 MHz = 90% 2-sidedBW Measured 3.48 MHz = 99% 2-sidedBW Measured Rolloff 0.35,Groupdelay 4,
power (dB) -70 -60 -50 -40 -30 -20 -10 0
OQPSK Spectrum at the RRC filter o/p, Sym Rate 6 Msps, OSR 16 OSR Msps, 6 Rate Sym o/p, RRC filter the at Spectrum OQPSK -2 Group Delay is 4sym is Group Delay 0.35 is rolloff -1.5 w/ Category A Masks -1 2-sided BW =3*1.35 = 4MHz. -0.5 freq (Hz) freq 0 0.5 1 Data Rate <2 Msps Data Rate >2 Msps o/p Tx 1.5 2 x 10 7
21 Avionic Systems Analysis 5/7/2012 90% measured 2-sided BW = 1.87 MHz. 1.87 = 90% measured 2-sidedBW MHz. 2.32 = 99% measured 2-sidedBW 2-sided =2*1.35 = 2.7 MHz. BW rolloffGroup delay4sym, 0.35 BPSK Tx RRC Output Spectrum foraSymRateof2Msps BPSK TxRRCOutput
power (dB) -70 -60 -50 -40 -30 -20 -10 0
-1.5 BPSK Spectrum at the RRC filter o/p, Sym Rate 2 Msps, OSR 16 OSR Msps, 2 Rate Sym o/p, RRC filter the at Spectrum BPSK Group Delay is 4sym is Delay Group 0.35 rolloff is -1 w/ Category A Masks -0.5 freq (Hz) freq 0 0.5 Data Rate <2 Msps Data Rate >2 Msps o/p Tx 1 x 10 1.5 7
22 Avionic Systems Analysis 5/7/2012
power (dB) -100 -90 -80 -70 -60 -50 -40 -30 -20 -10 0
-2 Group del is 4symb. is del Group 0.3. is Rolloff -1.5 Power Amplifier (PA) OutputSpectrum -1 Mod and Tx o/p Spectra, OSR 16 OSR Spectra, and Tx o/p Mod -0.5 for 6 MspsOQPSK freq (Hz) freq 0 0.5 1 NTIA Spurious Emissions Data Rate <2 Msps Data Rate >2 Msps o/p Tx Mod o/p 1.5 2 x 10 7
shown in the plot. shown inthe output is also the Modulator of The spectrum 23 Avionic Systems Analysis 5/7/2012 90% measured 2-sided BW = 4.96 MHz. = 90% measured 2-sidedBW Rectangular pulseshapingdoes OQPSK Tx Output Spectrum for OQPSK TxOutputSpectrum a RectPulseat a Sym Rateof6Mspsw/ A Masks Category
power (dB) -70 -60 -50 -40 -30 -20 -10 0
OQPSK Spectrum at the Tx o/p (Rect Pulse), Sym Rate 6 Msps, OSR 16 -2 not comply with Category A -1.5 -1 -0.5 freq (Hz) freq 0 spectral mask requirements. 0.5 1 Data Rate <2 Msps Data Rate >2 Msps o/p Tx 1.5 2 x 10 7
24 Avionic Systems Analysis 5/7/2012 90% measured 2-sided BW = 3.34 MHz. 3.34 = 90% measured 2-sidedBW MHz. 2.22 = 99% measured 2-sidedBW Rectangular pulseshapingdoes notcomplywi BPSK Tx Output Spectrum for a Rectangular Pulse, for a Rectangular Tx OutputSpectrum BPSK for a SymbolRateof2Mspsw/ A Masks Category
power (dB) -70 -60 -50 -40 -30 -20 -10 0
-1.5 BPSK Spectrum at the Tx o/p (Rect Pulse), Sym Rate 2 Msps, OSR 16 -1 -0.5 freq (Hz) freq 0 th Category Aspectral mask requirements 0.5 Data Rate <2 Msps Data Rate >2 Msps o/p Tx 1 x 10 1.5 7
6 . 25 Avionic Systems Analysis 5/7/2012
BER 10 10 10 10 10 10 10 10 10 -8 -7 -6 -5 -4 -3 -2 -1 0 -2
-1 0 1 2 OQPSK, 6 Msps, Extrapolated (Cubic Spline) (Cubic Extrapolated 6 Msps, OQPSK, 3 4 OQPSK, 6Msps 5 6 Eb/No 7 8 9 Tx Rect & Rx I&D, Extr I&D, Rx & Rect Tx Extr Filt, No Rx & Rect Tx Extr I&D, Rx RRC & Tx Extr filt, No Rx RRC & Tx Extr RRC, Rx RRC & Tx Theory Tx Gauss (BT=0.85) & Rx I&D, Extr I&D, Rx & (BT=0.85) Gauss Tx Extr I&D, Rx & (BT=0.35) Gauss Tx 10 11 12 13 14 15
stated otherwise. unless error bits ran for+1000 Simulations are and 10-11EbNo dB extrapolated after (Cubic Spline) the curvesare Please Notethat 26 Avionic Systems Analysis 5/7/2012
BER 10 10 10 10 10 10 10 10 10 -8 -7 -6 -5 -4 -3 -2 -1 0 -2
-1 0 1 2 BPSK, 2BPSK, Msps, Extrapolated (Cubic Spline) 3 4 5 BPSK, 2 Msps Eb/No 6 7 8 9 w/Tx Rect & Rx No Filt,Extr No Rx & Rect w/Tx I&D,Extr Rx RRC& w/Tx Filt,Extr No Rx RRC& w/Tx RRC,Extr Rx RRC& w/Tx Theory w/Tx Gauss & Rx I&D,Extr Rx & Gauss w/Tx I&D,Extr Rx & Rect w/Tx 10 11 12 13 14 15
stated otherwise. unless error bits ran for+1000 Simulations are 10-11EbNo dB extrapolated after (Cubic Spline) the curvesare *Please Notethat and 27 Avionic Systems Analysis ● ● ● 5/7/2012 spectral efficiency and Bit Error Rate (BER) performance. spectral efficiencyandBitError Rate(BER) This study in the provides insight andguidance system design for performance. filters Matching TXandRX achieves optimumBitError Rate (BER) rectangular waveform). (i.e., efficient than NRZ no pulseshaping, is morebandwidth RRC Conclusions 28 Avionic Systems Analysis 6. 5. 4. 2. 1. 3. 5/7/2012 Interoperability Standards Book Volu (C3I) Constellation Control, Communication, andInformation Program Command, Reduction Intersymbol Interference and Gagnon,François Châtelain, Benoît, 9 Band-Limited DS/SSMA Systems" Tavares G.N.(1998) Tavares, L.M.; Ltd. ISBN 0-13-089399-4. Proakis, J.(1995). distribution ofsingle 12-14 March2008. carrier signals", ISCCSP 2008,Malta, S. Daumont,R.Basel,Y.Lout,"Root-R Glover, I.; Grant, P. (2004). 113814-5. 3.4.2.2 Spectral Emissions Mask for Category AMissions,p43 Spectral Emissions Maskfor 3.4.2.2 Spectral EmissionsMask fo 3.4..2.1 Digital Communications Digital Communications References . IEICE Trans. Communications.,Vol. E81-B, No. me 2:Spectrum andChannel Plan by NyquistPulse Optimization” Comments on"PerformanceofAsynchronous aised Cosine filter influences on PAPR influences on aised Cosinefilter r Spurious Emissions (NTIA), p42 , “Peak-to-Average Power Ratio and (3rd ed.).McGraw-Hill Inc. ISBN 0-07- (2nd ed.). Pearson Education (2nd ed.). 29