Enhance Your Signal Processing Toolbox with Complex Notation

Transcription

1 Enhane Your Signal Proessing Toolbox with Complex Notation Barry L. Dorr, P.E. Dorr Engineering, In. At first glane this notation seems lie an unneessary ompliation rather than an aid, and has aused many a good engineer to wal away from valuable signal proessing information. As the speed of DSP's and digital hardwa eeps inasing, softwa and digital hardwa a plaing traditional analog hardwa, maing today's gizmos and gadgets smarter, mo liable, less expensive, and mo power effiient than ever befo. As an embedded systems engineer, you may have found yourself doing mo signal proessing than you originally bargained for! This artile about omplex notation is a survival guide to some of the most portant basis of signal proessing. Begin ading nearly any artile on signal proessing and you will quily enounter omplex notation. Signals that ould be easily psented as sine s or osines beome Re(e j( ωt-φ and we often minded that j is the squa root of -1. It gets worse a sple sinusoid atually ontains both positive and negative fquenies, but fquently only positive fquenies a used. We also see modulated signals ferd to as "omplex envelopes." At first glane this notation seems lie an unneessary ompliation rather than an aid, and has aused many a good engineer to wal away from valuable signal proessing information. Most signal proessing textboos provide an exellent view of omplex notation, but these desriptions a sometes mo general and involved than the woring engineer wants. This artile psents the basis of omplex numbers as used in signal proessing, and will provide you with the tools for using them. Through the examples used in this artile you will gain a lear understanding of AC iruit analysis using phasors, and also get introdued to some of the ey omponents of a QPSK eiver. THE BASICS OF COMPLE NUMBERS Figu 1 shows a omplex number. The al part of the number, Re(C, is psented by the horizontal omponent and the aginary part of the number, Im(C, is psented by the vertial omponent. The omplex number appears mediately useful beause, lie a sine wave, it has both a magnitude and an angle. If we hange the sign of the aginary part we obtain the omplex onjugate of the number, and we designate the omplex onjugate with the overbar symbol.

2 C a +b C b Φ a Figu 1- Complex number The omplex number an be psented various ways: C a + j b Equation 1 [ os( φ + sin( φ ] C C j whe : C a + b and φ tan 1 b a Equation The Euler identity shown below forms the basis of omplex notation. C jφ C e Equation 3 Note that the tangular forms of equations 1 and allow onvenient addition and subtration of omplex numbers. The polar form of equation 3 allows onvenient multipliation and division. If Φ is a phase angle whih inases linearly with te then the omplex number in Figu 1 rotates in a ounter-lowise irle. Letting the magnitude of the omplex number be unity, equation beomes C os( ωt + j sin( ωt Equation 4 and it s evident that the al part of this expssion is a sple osine wave. Furthermo, equations 1 through 4 an be easily manipulated to yield other useful forms Page of 11

3 e j ω t os( ωt + j sin( ωt Equation 5 e os( ωt j sin( ωt Equation 6 e os( ωt e sin( ωt e j + e Equation 7 Equation 8 Several things a adily appant from equations 5 through 8. First they a lated - equation 7 is the sult of adding equations 5 and 6, and equation 8 is the sult of subtrating equation 6 from equation 5. Equations 5 and 6 an also be viewed as two ounter-rotating vetors in whih the omplex parts always anel theby aounting for the al sult. Equation 6 an be viewed as two ounter-rotating vetors whe the al parts anel. Equations 1 through 8 an also be used to derive useful identities suh as: x + x Re( x Equation 9 x y x y Equation 10 But the most signifiant observation is that if we wish to tae advantage of omplex exponentials, we must either use the abstration of omplex numbers (equations 5 and 6 or the abstration of negative fquenies (equations 7 and 8. LINEAR FILTERING AND PHASORS The problem of determining the steady state sponse of a linear networ to a sinusoidal input is an exellent appliation of omplex notation. However, woring the problem one will show that steady-state analysis an be done virtually by inspetion using the phasor tehnique. This setion will infore what we ve oved so far by showing why the phasor tehnique wors. Digital and analog filters an be desribed by omplex fqueny domain transfer funtions. For example the lowpass RC filter shown in Figu has the transfer funtion: V H ( ω V Out in ( ω 1 ( ω ( 1+ jωrc Equation 11 This transfer funtion an be evaluated at diffent fquenies. At DC (ω0 the sult is 1 + j0. At ω1/rc the sult is j0.707, at ω the sult is 0. Page 3 of 11

4 Figu - RC filter A property of this filter is that its transfer funtion has onjugate odd symmetry. This means that H ω H ω Equations 7 and 8 show that the sinusoidal exitation also has odd symmetry. ( ( We will use the superposition property of linear networs to determine the output of the filter. Stated mathematially: f ( a + b f ( a + f ( b Equation 1 In other words, the output due to the sum of two input signals is equal to the sum of the outputs due to the individual inputs. A qui glane at equations 5 through 8 suggests that this property should ome in handy. The input to the networ is the osine from equation 7. Using superposition, the output signal is sply the sum of the outputs due to the two omponents in equation 7. e e Equation 13 Vo ( ω, t H ( ω + H ( ω Applying the identity in equation 10 yields: e e V ( ω, t H ( ω + H ( ω o Equation 14 Then apply the identity from equation 9: { e H ( } V ( ω, t Re ω Equation 15 o o jφ { ( ( ω e H ω e } V ( ω, t Re Equation 16 j { ( ωt + φ ( ( ω H ω e } V ( ω, t Re Equation 17 o V o ( ω, t H( ω os( ωt + φ( ω Equation 18 Page 4 of 11

5 Beause of the onjugate symmetry of both the input signal and the transfer funtion, the operations in equations 13 through 18 a idential for any networ. As a sult, to find the sponse of a networ due to a sine wave, you sply need to now the magnitude and phase sponse of the networ at the desid fqueny. In iruit analysis omplex numbers a psented using shorthand alled phasor notation as shown below. C C e C φ j ( ωt + φ Equation 19 Setting the angular fqueny to 1/RC, the steady state problem above is solved quily using phasors as follows: Equation 0 V o ( t os( ω t 45 Equation 1 NARROWBAND SIGNALS AND COMPLE NOTATION In signal proessing we fquently wor with narrowband signals as shown in Figu 3. A narrowband signal has its energy onentrated around a fqueny usually near the enter of the signal's bandwidth alled the arrier. Channelized signals an usually be tated as narrowband signals. Complex notation allows us to fous only on the bandwidth ontaining the signal rather than the bandwidth ontaining both the signal and the arrier. It also suggests hardwa and softwa strutus that aid proessing of lowpass signals. Figu 3 - Narrowband signal To illustrate, onsider the signal proessing used for QPSK data transmission shown in Figu 4. Page 5 of 11

6 si Lowpass Filter si os(ωt + m(t Bandpass Filter os(ωt G Complex Equalizer sq -sin(ωt -sin(ωt Lowpass Filter G e -j(ωtφ From arrier overy From adaptive equalizer sq Loal Osillator (T Loal Osillator (R Figu 4 - QPSK transmission system QPSK, or Quadratu Phase Shift Keying is a data transmission tehnique whe groups of two bits of data a enoded into one of four phase shifts of a sinusoidal arrier. The phase shift is held for the symbol te, and then it is updated based on the next pair of bits in the input bit stam. Sine the symbol rate is usually muh lower than the arrier fqueny, the sult is a narrowband signal. It is natural to use omplex notation to psent the two bits omprising eah symbol. The index is used to indiate that the signal psents a pair of bits in a serial bit stam. Signals s i and s q a the th in phase and quadratu signal omponents that tae values of ±1, and s denotes the omplex signal s i + j s q. The omplex baseband signal ould also be psented as e j φ whe φ is 45, 135, 5, or 315 deges. Mapping these signals on the omplex plane sults in the onstellation shown in Figu 5. q i-1 q1 i1 q1 i i-1 q-1 i1 q-1 Figu 5 - QPSK onstellation The two multipliers at the left of Figu 4 up-onvert the omplex baseband signal to the arrier fqueny. This is shown graphially in Figu 6 whe multipliation by e j ω t shifts the omplex baseband spetrum to the right, and taing the al part adds the negative fqueny omponent. Page 6 of 11

7 Step 1 - Multiplying the omplex baseband signal by e shifts the omplex baseband spetrum to the arrier fqueny ω -ω DC Step - Retaining only the al part adds the negative fqueny omponent. The sult is a al signal. ω Figu 6 - Up-onverter operations Equations to 5 desribe the operation of the up-onverter. The output is the al modulated signal m(t 1. m t Re jω {( s + j s e t } ( Equation i q {( s + j s ( os( ω t + j sin( ω } m( t Re t Equation 3 i q { s os( ω t s sin( ω t + j ( s os( ω t + s sin( ω } m( t Re t i q Eq. 4 m( t s os( ω t s sin( ω t Equation 5 i q q i At the eiver the signal enounters a bandpass filter used for jeting out-of-band signals. The bandpass filter sponse is not symmetri about the arrier fqueny and it distorts the signal slightly. The two multipliers and lowpass filters after the bandpass filter a used to down onvert the signal to its lowpass psentation. We'll elaborate on the down onverter sine it is a ommonly enounted signal proessing operation. One way to explain the operation of the down onverter is to begin with the signal as desribed by equation 5, multiply the sines and osines and note that the lowpass filter moves the double fqueny terms sulting from the multipliations. But instead of using sines and osines, we'll use our newfound sills with omplex notation. For the top arm we have: jω { s e t } os( ω t + φ Re Equation 6 The identities in equations 9 and 10 and the Euler identities allow us to write this as: 1 This expssion for m(t is valid for the th symbol only. Refer to the fenes for an aurate desription of a modulated data signal. Page 7 of 11

8 s e jω t + s e jω t ( j ( ωt+ φ j ω + φ e t + e Equation 8 1 s e ( ω φ ( ω φ ( ω + ω + φ ( ω + ω + φ + 1 t t j t j t j j t t s e s e + s e + Equation 9 Whe: ω ω ω Re 1 1 Equation 30 { } j ( ω t φ j (( ω + ω t + φ { s e } + Re s e The lowpass filter moves the double fqueny term on the right. Multiplying by yields the final sult: Re j ( ω φ { t s e } Equation 31 Silarly the sult for the bottom arm is: j ( ω φ { t } Im s e Equation 3 The I and Q branhes at the output of the down-onverter a lowpass signals and an be sampled at a muh lower rate than the modulated signal, whih dues both power onsumption and hardwa ost. Figu 4 shows the I and Q branhes ombined into a single heavy line indiating that the signal is omplex. A lear advantage of omplex notation is that the fqueny and phase diffenes between the transmit and eive osillators a maintained by equations 31 and 3. The next step for the QPSK eiver to over the transmitted data bits s i and s q is to ort for the fqueny offset ( ω and phase offset ( φ between transmitter and eiver. The proess of moving these offsets is alled arrier overy, and the eiver has the means to detet them and internally generate the signal e j( ωt-φ The proess of applying the ortion is often ferd to as de-rotation beause the fqueny offset auses the baseband phasor to rotate at the diffene fqueny. However this is sply omplex multipliation as shown by equation 33 and plemented in Figu 7. Though the strutu appears formidable, it an be done in just a few lines of DSP ode. ( x x ( y + y x y x y + j( x y + x y + Equation 33 Page 8 of 11

9 - + xy y + xy y Figu 7 - Complex multiplier The final step for overing the transmitted data bits is to move the distorting effets of the bandpass filter at the eiver input. The modem has the means for identifying the distortion based on statistial harateristis of the distorted input or orted output. This proess is alled adaptive equalization, and the sult of the equalization proess is the omplex filter shown in Figu 4 at the right of the de-rotator. Befo desribing the omplex filter, observe the signal spetrum shown in Figu 8. The operation of the down onverter an be viewed as shifting the positive portion of the spetrum left by ωˆ and disarding the negative fqueny term. The de-rotator further shifts the signal so that the overall shift pisely mathes the fqueny (and phase of the up-onverter. However the shifted signal is not symmetrial about DC. The i and q signals from the de-rotator a al signals. But when viewed as i + j*q, the sult is omplex and the asymmetrial spetrum is perfetly aeptable. The negative fqueny omponent is disarded. The positive fqueny omponent is shifted left by ω -ω DC ω Figu 8 - Bandpass and lowpass spetrums The oversplifiation is for the sae of larity. In this example, the output from the arrier overy would have a slight phase error beause its input has been distorted by the bandpass filter. Modern demodulators fquently perform arrier overy and equalization jointly. Page 9 of 11

10 Equalization of the asymmetrial signal spetrum quis an asymmetrial filter for the equalizer. To see how this wors note that when a signal is passed through a linear filter, the sulting signal spetrum is equal to the spetrum of the input signal multiplied by the fqueny sponse of the filter. If both the spetrum and the filter a omplex we have: ( S ( f + j S ( f ( H ( f + j H ( S( f H ( f f Equation 34 S [ S ( f H ( f + S ( f H ( ] ( f H ( f S ( f H ( f + j f Equation 35 Figu 9 shows an plementation of the omplex equalizer. The i and q signals from the equalizer pisely math those at the transmitter. Figu 9 - Complex filter ANOTHER TOOL IN THE TOOLBO! As engineers, we strive to avoid unneessary omplexity, and abstrations suh as aginary numbers and negative fquenies sometes fall into this ategory and end up on the hopping blo. In ality, omplex notation is a valuable tool for any engineer woring with signal proessing beause: Woring with exponentials is muh less tedious than woring with sines and osines. Signal proessing artiles and tehnial papers use omplex notation extensively. Complex notation suggests hardwa and softwa strutus for plementing signal proessing funtions. The intent of this artile has been to provide a set of tools for understanding and woring with omplex notation. Most signal proessing textboos (see fenes below inlude a hapter or setion dediated to narrowband signals. These a exellent fenes, and most inlude muh of the mathematial rigor not inluded he. The information in this artile should allow you to beze through the basis and get the most out of that next signal proessing artile. Page 10 of 11

11 REFERENCES [1] J. Proais, Digital Communiations, MGraw Hill, 1983, Chapter 3 [] H. Meyr, M. Moenelaey, A. Fehtel, Digital Communiation Reeivers, Wiley Intersiene, Chapter 1. [3] M. Shwartz, W. Bennett, S. Stein, Communiation Systems and Tehniques, MGraw Hill, 1966, Chapter 1. About the author: Barry Dorr is the owner/psident of Dorr Engineering Servies, In., a onsulting firm loated in San Maros, CA. DESI speializes in signal proessing, modem development, and embedded servos. He holds a BSEE from California Polytehni State University and an MSEE dege from San Diego State University, and is a gisted Professional Engineer in the state of California. Outside of wor he enjoys his hildn and playing the trombone. Mr. Dorr has also teahes a ourse in Embedded Servo Systems: He an be ahed at bdorr@dorngineering.om. Page 11 of 11