Chapter 2. Signal Processing and Modulation

Transcription

1 Chaper 2. Signal Proessing and Modulaion 2.1. The Naure of Eleroni Signals Sai and Quasi-Sai Signals Sai signals are by definiion unhanging over a long period of ime. Suh signals are essenially DC levels, while quasi-sai signals are hose ha hange very slowly suh as he drif on a sensor Periodi and Repeiive Signals Periodi signals are hose ha repea hemselves on a regular basis. These inlude sine, square and sawooh waves. Their naure is suh ha eah waveform is idenial. Repeiive signals are periodi in naure, bu he exa shape may hange slighly wih ime. ECG signals are an example of his ype Transien and Quasi Transien Signals Transien signals are eiher one ime only signals while quasi-ransien signals are hose whih are periodi bu wih a duraion whih is very shor ompared o he period of he waveform. Pulsed radar signals are good examples of hese Sinusoidal Signals Mos aousi and eleromagnei sensors exploi he properies of sinusoidal signals In he ime domain, suh signals are onsrued of sinusoidally varying volages or urrens onsrained wihin wires, where v () Signal, A Signal ampliude (V), ω Frequeny (rad/s), f Frequeny (Hz), Time (s). v ( ) = A osω = A os2πf, (2.1)

2 Sinusoidal elerial signals an be generaed by he appropriae frequeny seleive feedbak (shown below) or by feedbak aross an induive-apaiive ank irui. Figure 2.1: Sinusoidal volage signal generaed by an osillaor In he frequeny domain, a oninuous sinusoidal signal of infinie duraion an be represened in erms of is posiion on he frequeny oninuum and is ampliude only. Mos praial signals are no of infinie duraion and so here is some unerainy in he measured frequeny whih is represened in he frequeny domain by a finie speral widh. From a mahemaial perspeive, his is equivalen o windowing he oninuous sinusoidal signal using a reangular pulse of duraion τ. Beause windowing, or mulipliaion, in he ime domain beomes onvoluion in he frequeny domain, he oninuous signal sperum mus be onvolved by he sperum, or Fourier ransform, of a reangular pulse o obain he sperum of he windowed signal. As his Fourier ransform is he Syn funion sin( ωτ / 2) F ( ω ) = τ (2.2) ωτ / 2 and he sperum of a oninuous sinusoidal signal is an impulse δ(ω), he resulan onvoluion is jus he Syn funion. I an be seen from he equaion for he Syn funion ha as he duraion of he signal dereases, τ 0, is speral widh inreases unil, in he limi, when he signal an be represened by an impulse δ(), he speral widh is infinie. This relaionship is shown graphially in he following figure.

3 Time Frequeny τ τ/2 τ/2 τ/2 τ/2 1/τ 1/τ τ 0 Figure 2.2: Mapping he relaionship beween he duraion of a pulse and is sperum 2.3. The Fourier Series All oninuous periodi signals an be represened by a fundamenal frequeny sine wave and a olleion of sine and/or osine harmonis of ha fundamenal sine wave. The Fourier series for any waveform an be expressed as a0 v( ) = + an os( nω) bn sin( nω) 2 +, (2.3) n= 1 n= 1 where: a n b n - Ampliudes of he harmonis (an be zero), n - Ineger. The ampliude oeffiiens an be alulaed as follows, a b n n 2 = T 2 = T 0 0 v( )os( nω) d v( )sin( nω) d (2.4) The ampliude erms are non-zero a speifi frequenies deermined by he Fourier series. Beause only erain frequenies, deermined by ineger n, are allowed, he sperum is disree. The erm a 0 /2 is he average value of v() over a omplee yle. In general, hough he harmoni series is infinie, he oeffiiens beome so small ha heir onribuion is onsidered o be negligible.

4 An ECG rae, for example wih a fundamenal frequeny of abou 1.2Hz an be reprodued wih 70 o 80 harmonis (a bandwidh of abou 100Hz) Figure 2.3: Typial ECG rae A square wave on he oher hand may require up o 1000 harmonis o reprodue he sharp ransiions as shown in he figure below, depending on he appliaion. Figure 2.4: Ampliudes of Fourier oeffiiens o produe a square wave If insuffiien Fourier oeffiiens are used in he reonsruion of he signal i will be disored as shown in he following example. Figure 2.5: Effe on he reonsrued signal of limiing he number of Fourier oeffiiens

5 2.4. Sampled Signals To proess signals wihin a ompuer (Digial Signal Proessing) requires ha hey be sampled periodially and hen onvered o a digial represenaion using an Analog o Digial Converer (ADC). Figure 2.6: Digiising a signal To ensure aurae represenaion he signal mus be sampled a a rae whih is a leas double he highes signifian frequeny omponen of he signal. This is known as he Nyquis rae. In addiion, he number of disree levels o whih he signal is quanised mus also be suffiien o represen variaions in he ampliude o he required auray. Mos ADCs quanise o 12 or 16 bis whih represen 2 12 = 4096 or 2 16 = disree levels. Afer he signal has been proessed, i is ofen neessary o generae an analog oupu. This funion is performed by a Digial o Analog Converer (DAC) Figure 2.7: Typial Configuraion for a digial signal proessing appliaion The reonsruion proess generally involves holding he signal onsan (zero order hold) during he period beween samples as shown in he following figure. This signal is hen leaned up by passing i hrough low-pass filer o remove high frequeny omponens generaed by he sampling proess.

6 Figure 2.8: Analog Reonsruion of a sampled signal using a zero order hold Generaing Signals in MATLAB MATLAB inludes a number of buil-in funions ha make i easy o generae boh periodi and aperiodi signals. Generaion of a square wave wih ampliude A, fundamenal frequeny w0 (rad/s) and duy yle rho as a perenage. % generae a square wave A = 1; w0 = 10*pi; rho = 50; = 0:0.001:1; sq = A*square(w0*, rho); plo(,sq) axis([0,1,-1.1,1.1]); Generaion of a riangular wave wih ampliude A, fundamenal frequeny w0 (rad/s) and widh W. % generae a riangular wave A = 1; w0 = 10*pi; W = 0.5; = 0:0.001:1; ri = A*sawooh(w0*, W); plo(,ri) grid axis([0,1,-1.1,1.1]); Generaion of a sine wave wih ampliude A, fundamenal frequeny w0 (rad/s) and sar phase angle phi (rad).

7 % generae a sine wave A = 1; w0 = 10*pi; phi = pi/4; = 0:0.001:1; sine = A*sin(w0* + phi); plo(,sine) grid axis([0,1,-1.1,1.1]); An exponenially damped sine wave is easily generaed by aking he produ of an exponenial funion and a sine wave. x( ) = Asin( φ) a ω o + e (2.5) % generae a exponenially % deaying sine wave A = 1; w0 = 10*pi; phi = pi/4; a = 6; = 0:0.001:1; expsine = A*sin(w0* + phi).*exp(-a*); plo(,expsine) grid axis([0,1,-1.1,1.1]); Oher useful MATLAB funions inlude he following:, COS, CHIRP, DIRAC, GAUSPULS, PULSTRAN, RECTPULS, SINC and TRIPULS Aliasing If he analog signal is no sampled a a leas wie he frequeny of he highes frequeny omponen, hen hese high frequeny signals are aliased down o a lower frequeny as shown in he following figure. Figure 2.9: Inerpreaion of aliasing effes in he ime domain

8 In he frequeny domain, a generi analog signal may be represened in erms of is ampliude and oal bandwidh as shown in he figure below. A sampled version of he same signal an be represened by a repeaed sequene spaed a he sample frequeny (generally denoed f s ). If he sample rae is no suffiienly high, hen he sequenes will overlap, and high frequeny omponens will appear a a lower frequeny (albei wih redued ampliude) as shown in he figure. Analog Signal Sperum Sampled Signal Sperum Aliased Signal -fs fs Sampled Signal Sperum -fs Signal no Aliased fs Figure 2.10: Inerpreaion of aliasing effes in he frequeny domain In mos appliaions, an ani aliasing (low pass) filer ensures ha he high frequeny signals are suffiienly aenuaed prior o sampling. A ypial filer will aenuae hese unwaned signals by beween 40 and 60dB (1/100 o 1/1000) in volage Filering A filer is a frequeny seleive nework ha passes erain frequenies of an inpu signal and aenuaes ohers The hree ommon ypes of filer are: High Pass Low Pass Band Pass High pass filer bloks signals below is uoff frequeny and passes hose above. Low pass filer passes signals below is uoff frequeny and aenuaes hose above.

9 Band pass filer passes a range of frequenies while aenuaing hose boh above and below ha range. A fourh, less ommon onfiguraion, is a band-sop or noh filer ha aenuaes signals a a speifi frequeny or over a narrow range of frequenies and passes all oher frequenies. Filers an be haraerised by heir impulse responses in he ime domain, bu are usually haraerised by heir frequeny domain ampliude response H(ω) or Gain whih presens he raio of he oupu o inpu volage over he frequeny range of ineres. In his ourse we will only use sampled daa filers ha an be synhesized in MATLAB as shown in he figure. % Banspass filer fs = 200e+03; s = 1/fs; fma = 40.0e+03; bma = 10.0e+03; wl=2*s*(fma-bma/2);% lower band wh=2*s*(fma+bma/2);% upper band wn=[wl,wh]; % 6h order Buerworh filer [B,A]=buer(3,wn); [h,w]=freqz(b,a,1024); freq=(0:1023)/(2000*s*1024); %semilogx(freq,20*log10(abs(h))); plo(freq,abs(h)); grid ile('bandpass Filer Transfer Fn') xlabel('frequeny (khz)'); ylabel('gain') Figure 2.11: Buerworh bandpass filer ransfer funion generaed by MATLAB Noe ha he upper and lower limis of he pass band represen he half power poins (0.707 of he peak volage gain) If he gain was ploed in db hen i would be alulaed using 20*log 10 (Gain) o onver o power.

10 MATLAB implemens filers as he Dire Form II Transposed of he sandard differene equaion a(1)*y(n) = b(1)*x(n) + b(2)*x(n-1) b(n b +1)*x(n-n b ) -a(2)*y(n-1) a(n a +1)*y(n-n a ) where he oeffiiens A =[ a(1), a(2) a(n a +1)] and B = [b(1), b(2) b(n b +1)] are generaed when he filer is synhesised Filer Caegories The major filer aegories are as follows: Buerworh (maximally fla). Chebyshev (equi ripple). Bessel (linear phase). Ellipial Buerworh This approximaion o an ideal low pass filer is based on he assumpion ha a fla response a zero-frequeny is mos imporan. The ransfer funion is an all-pole ype wih roos ha fall on he uni irle. I exhibis fairly good ampliude and ransien haraerisis. Chebyshev The ransfer funion is also all-pole, bu wih roos ha fall on an ellipse. This resuls in a series of equi ampliude ripples in he pass band and a sharper uoff han he Buerworh. I exhibis good seleiviy bu poor ransien behaviour. Bessel Opimised o obain a linear phase response whih resuls in a sep response wih no overshoo or ringing and an impulse response wih no osillaory behaviour. Poor frequeny seleiviy ompared o he oher response ypes. Ellipi These filers have zeros as well as poles whih reae equi-ripple behaviour in he pass band similar o Chebyshev filers. Zeros in he sop band redue he ransiion region so ha exremely sharp roll-off haraerisis an be ahieved, for ha reason hey are ommonly used in anialiasing filers.

11 Figure 2.12: Lowpass filer omparison Noe ha he uoff frequeny (200Hz in his ase) speified in MATLAB is equal o he 3dB poin for he Buerworh filer and o he passband ripple for he Chebyshev and Ellipi filers The rae a whih he signal is aenuaed as a funion of frequeny is proporional o he order of he filer. The following figure shows he roll-off for Buerworh filers. The heoreial slope is 6n db/oave. Figure 2.13: Roll off for Buerworh lowpass filers

12 The Ear as a Filer Bank In he ear, sound waves are ransmied ino he ohlea whih apers in size like a one. Through he middle srehes he basilar membrane whih ges wider as he ohlea ges narrower. The vibraory movemen is ransmied as a sanding wave in he basilar membrane (muh like a snapping rope) wih he ampliude reahing a peak a a loaion dependen on frequeny due o he varying resonan haraerisis of he membrane as shown shemaially in he following figure. High frequeny peaks our oward he base (where he membrane is siffes and narrowes) while he low frequeny peaks our owards he apex. Hair ells res on he basilar membrane and onver hese vibraions o elero-hemial signals whih are ransmied o he brain for inerpreaion. The base resonaes a abou 20kHz while he apex resonaes a abou 20Hz Figure 2.14: The Basilar Membrane of he Cohlea depied unoiled and flaened showing he resonane for ravelling waves of differen frequenies The ohlear implan onsiss of a sring of elerodes, eah exied by a narrow band of frequenies, whih simulae he hair ell nerves direly. This allows some hearing o be resored in he ase when eiher he ilia or basilar membrane has been damaged.

13 2.6. Analog Modulaion and Demodulaion A oninuous unmodulaed signal anno be used o measure range as here is no way of deermining when he signal was ransmied. For mos sensor appliaions, he ransmied signal is marked in some way eiher by alering is ampliude or frequeny. The round rip ime from he momen ha he mark is ransmied o when i is reeived an hen be used o deermine he range o he arge if he speed of propagaion is known. Marking he ransmied signal is known as modulaion Ampliude Modulaion (AM) AM is defined as a modulaion ehnique in whih he ampliude of he arrier is varied in aordane wih some haraerisi of he baseband modulaing signal. I is he mos ommon form of modulaion beause of he ease wih whih he baseband signal an be reovered from he ransmied signal. Figure 2.15:Time domain represenaion of ampliude modulaion For an unmodulaed arrier desribed earlier and for a baseband signal v b (), he AM signal v am () is desribed by he following equaion v am [ 1 v ( ) ] os 2πf ( ) = A +. (2.6) b For he example shown above, where he modulaing signal is a sinusoid wih a frequeny f a and ampliude A am hen he revised formula, v am [ 1 A os 2πf ] os 2πf ( ) = A +. (2.7) am In general A am <1 oherwise a phase reversal ours and demodulaion beomes more diffiul. a

14 The exen o whih he arrier has been ampliude modulaed is expressed in erms of a perenage modulaion whih is jus alulaed by muliplying A am by 100. To deermine he haraerisis of he signal in he frequeny domain, i an be rewrien in he following form (using a rig ideniy), v A Aam ( ) = A os2π f + [ os2π ( f fa) + os2π ( f + fa ]. (2.8) 2 am ) I an be seen ha his is made up from hree independen frequenies: The original arrier a a frequeny of f. A frequeny a he differene beween he arrier and he baseband signal A frequeny a he sum of he arrier and he baseband signal The sperum is shown in he figure. Figure 2.16: Simulaed and measured frequeny domain represenaion of ampliude modulaion

15 The ank irui in he rysal radio shown in he figure below is uned o resonae a he arrier frequeny and so operaes o sele only a single radio program whih passes ino he deeion sage. Demodulaion is hen ahieved wih he use of a simple diode reifier (rysal) and lowpass filer (apaior) and a pair of high-impedane headphones onvers he resulan envelope urren o an audio signal ha auraely reprodues he original modulaion. From he figure below i an be seen how simple he demodulaion of an AM signal an be. Figure 2.17: Demodulaion of an AM signal by a rysal radio Noe ha here is suffiien energy in he RF signal (wih he appropriae long wire anenna) o drive a se of high-impedane headphones direly wihou any amplifiaion (hene no baery) Frequeny Modulaion (FM) FM is a modulaion ehnique in whih he frequeny of he arrier is varied in aordane wih some haraerisi of he baseband modulaing signal. v fm ( ) = A os ω + k vb ( ) d (2.9)

16 The reason ha he modulaing signal is inegraed is beause variaions in he modulaing erm equae o variaions in he arrier phase. The insananeous angular frequeny an be obained by differeniaing he insananeous phase, d ω = ω + k vb ( ) d = + kvb ( ) d ω. (2.10) The deviaion of he insananeous frequeny from he arrier frequeny, ω /2π, is k δ f = f f = vb ( ) (2.11) 2π This shows ha he deviaion of he insananeous frequeny is direly proporional o he ampliude of he modulaing signal. Hene he ombinaion of an inegraor and phase modulaor produes frequeny modulaion. Figure 2.18: Time domain represenaion of frequeny modulaion For sinusoidal modulaion, he formula for FM is v fm [ ω β sin ω ] ( ) = A os +, (2.12) where β, whih is he maximum phase deviaion, is usually referred o as he modulaion index. The insananeous frequeny in his ase is a f f ω βω a = + osω a 2π 2π = f + βf osω a a (2.13)

17 So he maximum frequeny deviaion defined as Δf, when osω a = 0, is Δ f = βf a. (2.14) Even hough he insananeous frequeny lies wihin he range f +/-Δf, he speral omponens of he signal don lie wihin his range. Some manipulaion of he formula for v fm () shows ha he sperum omprises a arrier wih ampliude J o (β) wih sidebands spaed symmerially on eiher side of he arrier a offses of ω a, 2ω a, 3ω a,.as shown in he figure below. Theoreially, he bandwidh is infinie, however, for any β, mos of he power is onfined wihin a finie bandwidh. As a rule of humb (Carson s Rule), he bandwidh is wie he sum of he maximum frequeny deviaion plus he modulaing frequeny. Figure 2.19: Simulaed and measured frequeny domain represenaion of frequeny modulaion

18 Demodulaion of an FM signal is ommonly ahieved by onvering i ino AM and hen envelope deeing i. The simples mehod o perform he onversion o AM is o pass he signal hrough a frequeny sensiive irui suh as a low-pass or bandpass filer. The iruiry o perform his funion is known as a disriminaor FM Signal Inpu Bandpass Filer f1 Bandpass Filer f2 Deeor Deeor Lowpass Filer f3 Lowpass Filer f3 + - Demodulaed Signal Oupu Figure 2.20: A disriminaor onvers he FM signal ino an ampliude variaion and envelope dees he resuling AM signal In his example he FM signal is spli and passed hrough wo bandpass filers wih enre frequenies jus below, and jus above he arrier frequeny. The wo signals are hen deeed and filered o remove he residual arrier before he differene is aken. The ransfer funion for his proess is shown in he figure below. Figure 2.21: The differene signal from a pair of offse bandpass filers produes a symmerial ransfer funion o onver variaions in frequeny o variaions in ampliude In he ime domain, he FM signal afer deeion and filering produes wo symmerial demodulaed signals wih DC offses. The differene beween hese reprodues he original baseband signal as shown in he figure below.

19 Figure 2.22: Sages of FM demodulaion showing (a) he deeed and filered oupus from he bandpass filers and (b) he final demodulaed oupu obained by aking he differene beween hese signals Alernaive ehniques used in mos modern radio reeivers use eiher phase-loked loop or quadraure deeion ehniques o perform his funion Linear Frequeny Modulaion In mos aive sensors ha operae using he Frequeny Modulaed Coninuous Wave (FMCW) priniple, he frequeny is no modulaed sinusoidally, bu in a linear manner wih ime. ω A b. (2.15) b = Subsiuing ino he sandard equaion for FM, we obain he following resul v v fm fm ( ) = A os ω + Ab d Ab 2 ( ) = A os ω + 2 (2.16) Noe ha he phase modulaion follows a quadrai funion. In general i is no possible o oninue o inrease he frequeny indefiniely, so he modulaion ofen follows a sawooh or riangular funion.

20 Figure 2.23: Simulaed and measured frequeny domain represenaion of linear FM In FMCW sysems, a porion of he ransmied signal is mixed wih (muliplied by) he reurned eho as disussed in Chaper 11. The ransmi signal will be shifed from ha of he reeived signal beause of he round rip ime, τ, + = 2 ) ( 2 ) ( os ) ( τ τ ω τ A A v b fm. (2.17) Calulaing he produ of he ransmied signal and he delayed eho + + = ) ( 2 ) ( os 2 os ) ( ) ( τ τ ω ω τ A A A v v b b fm fm (2.18)

21 Equaing using he rig ideniy osaosb = 0.5[os(A+B)+os(A-B)] 2 A b 2 os ( 2ω Abτ ) + Ab + τ ωτ 1 2 v ou ( ) = (2.19) 2 A b 2 + os A τ. + ω τ τ b 2 The firs osine erm desribes a linearly inreasing FM signal (hirp) a abou wie he arrier frequeny wih a phase shif ha is proporional o he delay ime τ. This erm is generally filered ou. The seond osine erm desribes a bea signal a a fixed frequeny, Ab f bea = τ. (2.20) 2π I an be seen ha he signal frequeny is direly proporional o he delay ime τ, and hene is direly proporional o he round rip ime o he arge. The sperum of his oupu signal is shown in he following figure. Figure 2.24: Frequeny domain represenaion of FMCW sensor oupu 2.9. Pulse Coded Modulaion Tehniques Pulse Ampliude Modulaion (PAM) Modulaion in whih he ampliude of individual, regularly spaed pulses in a pulse rain is varied in aordane wih some haraerisi of he modulaing signal. For ime-of-fligh sensors, he ampliude is generally onsan. The abiliy of a pulsed sensor o resolve wo losely spaed arges is deermined by he pulse widh as shown in more deail in Chaper 11.

22 The following figure shows he relaionship beween he widh of a single pulse and is speral onen. Figure 2.25: Relaionship beween pulsewidh and frequeny for a single pulse Figure 2.26: Relaionship beween he duraion of a pulse and is sperum for pulsed ampliude modulaion Noe ha he widh of he pulse sperum in he frequeny domain inreases as he pulse widh dereases wih he following relaionship, 1 β, (2.21) τ where β is he speral widh (Hz) beween he half power (3dB) poins and τ is he pulsewidh (se). In a ime-of-fligh sensor, his relaionship deermines he bandwidh required by a reeiver o reeive pulses of a speifi duraion. A filer ha onforms o his relaionship is known as a mahed filer. I is formally defined in Chaper 11. In a repeaed sequene of pulses, he fine sruure of he sperum is deermined by he oal lengh of he observed sequene as shown in he figure below.

23 Figure 2.27: Relaionship beween pulsewidh and frequeny for a finie lengh sequene of pulses The following figures show he measured spera of a number of oninuous and pulsed signals wih differen periods (a) (b) () (d) Figure 2.28: Spera of pulsed signals wih duraions of (a) 50ns, (b) 100ns, () 500ns, (d) oninuous

24 2.10. Frequeny Shif Keying (FSK) This modulaion ehnique is he digial equivalen of linear FM where only wo differen frequenies are uilised. A single bi an be represened by a single yle of he arrier, or if he daa rae is no riial, hen muliple yles an be used. Figure 2.29: Two ommon mehods of generaing FSK Demodulaion an be ahieved by deeing he oupus of a pair of filers enred a he wo modulaion frequenies, f 1 and f 2 as shown, or by using a phase loked loop. Figure 2.30: Band pass filer mehod of demodulaing FSK signals The following figure shows an example of FSK modulaion wih muliple yles per bi. Figure 2.31: Example of frequeny shif keying (FSK)

25 In oheren FSK, he bi ransiion is synhronised wih he arrier frequeny so ha here are no phase disoninuiies. In he non-oheren opion here is no synhronisaion and large phase disoninuiies an our As he number of yles per bi dereases, he sperum shifs from being wo disin peaks enred he wo frequenies o a flaer, broad almos uniform peak as shown in he figures below. Figure 2.32: FSK sperum, 5 yles per bi Figure 2.33: FSK sperum, 1 yle per Bi FSK is a very simple modulaion ehnique and is sill exremely popular. I was originally used by elepriners whih operaed a abou 45bps, and was inrodued in 1962 for a Bell modem whih operaed a up o 300bps. Early PC s used a Kansas Ciy Inerfae whih used FSK o sore sofware on audio assees a up o 1200bps. I is now used in ouh-one phones and a myriad of oher ommuniaions sysems, operaing a speeds in exess of 1Mbps.

26 2.11. Phase Shif Keying (PSK) The mos ommon form of PSK is binary phase oding. The arrier, f o is swihed beween +/-180 aording o a digial base band, f m sequene. This modulaion ehnique an be implemened quie easily using a balaned mixer as shown in he figure, or wih a dediaed BPSK modulaor. fo - Carrier A BPSK Ou C D B fm - Digial Modulaion +1-1 Figure 2.34: Implemening BPSK using a balaned mixer When he modulaion signal f m is high (+1), diodes A and B are forward biased and he arrier f o is oupled o he direly o he oupu ransformer. However, when f m is low (-1), hen diodes C and D are forward biased and f o is oupled o he o he opposie erminals of he oupu ransformer whih resuls in a reversal of he phase. The modulaion and oupu waveforms are shown in he following figure. Figure 2.35: Example of binary phase shif keying wih one yle per bi Demodulaion is ahieved by muliplying he modulaed signal by a oheren arrier (a arrier ha is idenial in frequeny and phase o he arrier ha originally modulaed he BPSK signal).

27 This produes he original BPSK signal plus a signal a wie he arrier whih an be filered ou. The speral widh of he arrier is widened by he modulaion proess as shown in he figure below. This is one of he modulaion ehniques ha is used for broadband ommuniaions for obvious reasons. Noe in his example ha he ampliude of he ransmied signal has been redued by 30dB beause he power has been spread, even hough he oal power ransmied (and hene he range performane) remains unalered (see Chaper 11). Figure 2.36: PSK Sperum for a 500MHz arrier and 1 yle per bi modulaion Sepped Frequeny In his modulaion ehnique, a sequene of pulses are ransmied eah a a slighly differen frequeny. The pulse widh (in he radar ase) is made suffiienly wide o span he region of ineres, bu beause i is so wide, i anno resolve individual arges wihin ha region. However, if all of he pulses are proessed ogeher, he effeive resoluion is improved beause he oal bandwidh is widened by he oal frequeny deviaion of he sequene of pulses as disussed below. If a signal is ransmied a a frequeny ω and i refles off a arge a a range R, hen he round rip delay ime is 2R τ =. (2.22)

28 The phase differene beween he ransmied signal and he eho an be deermined by mixing a porion of he ransmied signal wih he eho as follows and filering ou he high frequeny omponens, v v ou ou ( ) = osω.osω ( τ ) 1 ( ) = 2 [ osω (2 τ ) + osω τ ] (2.23) Subsiuing for τ and filering he high frequeny omponens leaves he phase erm only, The phase shif, φ, an be obained by replaing ω by 2πf v ou 1 2ω R ( ) = os. (2.24) 2 4πf R φ =. (2.25) I an be seen ha he phase shif will hange wih eah new frequeny f This phase hange is sinusoidal in naure and an be onsidered o be a synhei Doppler frequeny and an be proessed in he frequeny domain o deermine he arge range more auraely. If a number of losely spaed refleors are presen hey will eah have heir own unique Doppler frequeny and so an eah be resolved as a separae arge Convoluion Linear Time Invarian Sysems I is onvenien o desribe he relaionship beween he inpu x() and oupu y() signals of Linear Time Invarian (LTI) sysems in erms of he impulse response h() of he sysem. I an be shown ha an LTI sysem is ompleely haraerised by is impulse response x() = δ() applied a ime = 0 (or n = 0). Applying an impulse o he inpu of an unknown LTI sysem is herefore a good mehod of deermining is haraerisis. This is easily ahieved in he disree ime ase where he inpu is se equal o an impulse δ(n). However, in he oninuous ime ase, i is no possible o produe a rue impulse wih zero widh and infinie ampliude and an approximaion is used. As an arbirary inpu signal an expressed as he weighed superposiion of ime shifed impulses x() = x(τ)δ(-τ), he sysem oupu is jus he weighed superposiion of ime-shifed impulse responses. τ = y( ) = h( τ ) x( τ ) dτ = h( τ ) x( τ ) dτ (2.26) 0

29 This weighed superposiion is ermed he onvoluion inegral for oninuous ime sysems and he onvoluion sum for disree ime sysems. Given a sysem defined by is impulse response h(), he oupu, y() is found by onvolving he inpu x() wih his funion. If he Laplae ransform is aken of his onvoluion inegral, i an be shown ha Y ( s) = H ( s) X ( s) (2.27) Tha is, onvoluion in he ime domain is equivalen o mulipliaion in he Laplae domain. Addiionally, if s is replaed by jω, hen he equaion an be replaed by Y ( ω) = H ( ω) X ( ω) (2.28) whih desribes he frequeny response of he sysem. I is he magniude of his ransfer funion, H(ω), ha is onsidered when he haraerisis of various filers are onsidered in he following seion. In his ase, he mapping beween he ime domain and he frequeny domain is hrough he Fourier Transform. In he following example, he impulse response of a 3 rd order Buerworh bandpass filer ( normalised passband 0.2 o 0.3) is generaed. As his haraerises he filer ompleely, is Fourier Transform should reprodue he frequeny response auraely. Figure: MATLAB example showing he relaionship beween he impulse response of a bandpass filer and is Fourier Transform

30 % relaionship beween filers in he ime and frequeny domain % % generae a Buerworh bandpass filer wn=[0.2,0.3]; [b,a]=buer(3,wn) % generae he impulse response of he filer x=zeros(1,128); x(1)=1; y=filer(b,a,x); subplo(211), plo(y),grid, xlabel('sample (n)'), ylabel('h(n)') ile('impulse Response of Bandpass Filer') % ake he Fourier ransform of he impulse response f=ff(y); freq=(0:63)*1/64; subplo(212), plo(freq, abs(f(1:64))), grid, xlabel('normalised Frequeny'),ylabel(' H(w) ') %ile('frequeny Response') The Convoluion Sum Mos modern sysems are digial whih requires ha he inpu and oupu daa be sampled. In his ase he onvoluion sum replaes he onvoluion inegral and is defined by he following equaion where y(n) is he oupu, x(n) he inpu and h(n) he sysem impulse response x k = y ( n) = ( k) h( n k) (2.29) However, o be raable x(n) and h(n) are nonzero only over a finie inerval. In he example below N x = 4 and N h = 3 whih makes he oal duraion of he onvolved sequene N x + N h 1 = 6 To perform his funion manually, he order of h(n) is firs reversed before being shifed aross x(n) one sample a a ime. The oupu y(n) is equal o he sum of he produs of eah of he aligned erms

31 x h refleed h Sum = Sum = 1x1 = Sum = 1x2 +1x1 = Sum = 1x3 + 1x2 + 1x1 = Sum = 1x3 + 1x2 + 1x1 = Sum = 1x3 + 1x2 = Sum = 1x3 = % onvoluion example % onvsum.m x = ([1,1,1,1]); h = ([0,1,2,3]); y = onv(x,h); plo(y, o ) Convoluion Example An unusual appliaion for he onvoluion proess is o use i o model he propagaion of a pulsed ime-of-fligh sensor. In his appliaion he impulse response of he round-rip propagaion o he poin arge is a signal ha is delayed in ime and aenuaed in ampliude h() = aδ(t-τ) where a represens he aenuaion and τ he round rip ime o he arge and bak. Consider an airraf flying owards a radar sysem as shown in he following figure: Radar Figure 2.37: Radar illuminaing an airraf arge

32 Assume ha he main poins of refleion from he airraf are lised in he following able: Table 2.1: Refleion onribuions from various pars of he airraf Conribuion Disane from Daum Magniude (m) Nose 2 1 Wing 6 1 Engine 8 1 Tail 14 1 Deermine he magniude of he eho signal if a pulse wih a lengh of 3m is ransmied. The ransmi signal is reaed as a sequene of impulses separaed by he sample inerval of 10m overing a oal of 3m in range The arge is reaed as a sequene of four impulses wih separaions as lised in he able The refleed signal reeived bak a he radar is well desribed by he onvoluion of he ransmied pulse and he array of poin refleors as shown in he figure. % onvoluion demo % % generae he ransmi pulse as a sequene of impulses 10m apar a=([zeros(1,70),ones(1,30),zeros(1,70)]); % generae he poin arge refleors b=zeros(1,170); b(20)=1; % Nose b(60)=1; % Wings b(80)=1; % Engine b(140)=1; % Tail % ake he onvoluion o deermine he reurn from all of he refleors =onv(a,b); % plo he resuls x=(1:170)/10; subplo(211),plo(x,a,x,a,'o'), grid; ile('transmi Pulse'),ylabel('ampliude'); subplo(212), plo(x,b,x,b,'o'),grid; ile('targe Refleors'),xlabel('range(m)'),ylabel('ampliude'); figure(2) xx=(1:339)/10; plo(xx,,xx,,'+'); grid ile('eho Pulse') xlabel('range(m)') ylabel('ampliude')

33 Figure 2.38: Malab simulaion oupu showing ransmi signal, arge sruure and eho signal This example does no onsider ha he ransmied signal is in fa sinusoidal in naure and hene he onvoluion should inlude boh ampliude and phase effes. Nor does no onsider he effes of he round-rip disane whih doubles he effeive spaing beween he refleors as explained in Chaper Referenes [1] N.Taub & D.Shilling, Priniples of Communiaion Sysems. MGraw Hill [2] B.Mahafza, Radar Sysems Analysis and Design using MATLAB. Chapman & Hall/CRC [3] N.Currie, Radar Refleiviy Measuremen: Tehniques and Appliaions. Areh House, 1989 [4] M.Skolnik, Radar Handbook 2 nd Ed. MGraw Hill [5] N.Currie & C.Brown. Priniples and Appliaions of Millimeer-Wave Radar. Areh House [6] T.Dunan, Eleronis and Nulear Physis. John Murray [7] J.Carr, Eleroni Cirui Guidebook: Sensors. Promp [8] H.Jaoboviz, Eleronis Made Simple. W.H.Allen [9] The ARRL Handbook for Radio Amaeurs: 2000 [10] S.Haykin & B.van Veen, Signals and Sysems. John Wiley [11] H. Durran-Whye, Signal Analysis and Transform Mehods, Course Noes

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