Lab-Report Digital Signal Processing

Transcription

1 Lab-Report Digital Signal Proessing Aplitude Modulation using DSP ethods Ν ω Nae: Dirk Beker Course: BEng Group: A Student No.: 9835 Date: 8//999

2 . Contents. CONTENTS. INTRODUCTION 3 3. AMPLITUDE MODULATION SYSTEMS 3 a) Matheatial justifiation for the frequeny shifting 4 b) Standard aplitude ulation 4 ) Balaned Aplitude Modulation 5 4. AMPLITUDE MODULATION USING DSP-METHODS 6 a) The Digital Osillator 7 b) Modulation and deulation 8 ) Low Pass Filter 8 5. REALISATION USING C a) The ulation signal b) The digital Osillator ) Modulation and Deulation d) Low Pass Filtering e) Output Plots 3 6. AMPLITUDES 4 7. CONCLUSION 5 8. APPENDIX 6 a) Coplete soure ode of the ulation software 6

3 . Introdution Over very any years Aplitude Modulation was the standard ulation shee beause of it s easy and heap realisation. For Aplitude ulation and deulation ultipliers are used, whih are realised by seiondutors or in earlier ties by vauu tubes. A ore ern way of aplitude ulation is the use of Digital Signal Proessing ethods. The arrier generation and also the ultipliation are done by a DSP Software. Aplitude Modulation is today still used beause of the sall onsuption of bandwidth in the broadast bands. 3. Aplitude Modulation Systes ω +ω ω ω ω figure - Standard AM - Syste ω ω Figure shows the blok diagra of an AM ulation and deulation syste. The ajor bloks are the two ultipliers and the low pass filter to reove the high frequeny parts of the down-ixed signal. AM ulation siply eans the shifting of a signal frequeny to another (usually higher) frequeny. The inforation, or better the ontent of the original (ulating) signal is transferred to another frequeny, the arrier. Frequeny shifting is done by ultipliation of two signal in the tie doain. Multipliation in the tie-doain orrespondents with frequeny shifting in the frequeny (ω) doain. ω" ##$ #+ω! ω" ##$ # ω! ω! ω" ##$ # figure - Standard Aplitude ulation ω 3

4 a) Matheatial justifiation for the frequeny shifting The Frequeny shifting an be proofed by applying the Fourier Transfor to a funtion f(t) ultiplied with an osine funtion. f (t) osω f (t) osω t = t jωt jωt ( f (t)e + f (t)e ) ( F( ω ω ) + F( ω ω )) where f(t) = ulating signal and ω = arrier signal This shows that ultipliation of a signal f(t) by a sinusoid frequeny ω shifts the spetru F(ω) by ±ω. Multipliation of a sinusoid osω t by f(t) aounts to ulating the sinusoid aplitude. This kind of ulation is alled (balaned) aplitude ulation. b) Standard aplitude ulation Balane aplitude ulation, like shown above results in loss of the arrier signal, whih arries only redundant inforation. But for different reasons the arrier is transitted at standard aplitude ulation. Therefore an offset is added to the arrier and the arrier is transitted as well. Matheatial justifiation for standard aplitude ulation: = Asin = A = A A = j = A j = Asin = Asin ( ω t) ( + os( ω t) ) j j jω t jω t jωt jωt ( e e ) + ( e + e ) jω t jω t jωt jωt ( e e ) + ( e + e ) jω A t jωt jωt jωt jωt jωt ( e + e ) ( e + e )( e e ) jω A t jωt j( ω +ω ) t j( ω +ω ) t j( ω ω ) t j( ω ω ) t ( e + e ) ( e e + e e ) A 4j ( ω t) [ jsin( ω + ω ) t) jsin( ( ω ω ) t) ] A 4j 4j ( ω t) + [ sin( ( ω + ω ) t) + sin( ( ω ω ) t) ] where is alled the ulation index (ratio of peak ulating signal to peak arrier signal), and A is the aplitude of the arrier signal. Standard aplitude ulated signals an be deulated by eans of siple. diodes. 4

5 ) Balaned Aplitude Modulation The opportunity of balaned aplitude ulation is the suppression of the arrier signal, whih ontains no useful inforation and onsues a lot of energy, when transitting. Hene balaned Aplitude Modulation an be desribed as: =Af(t)os(ω arrier t) where f(t) is the ulating signal and ω arrier the arrier frequeny. The ain differene in generating balaned aplitude ulation to standard aplitude ulation is the issing offset of the () offset in the ulating signal. Matheatial justifiation for balaned aplitude ulation: (see also a) Matheatial justifiation for the frequeny shifting) = Asin = A A = 4j A = ( ω t) os( ω t) j A = 4j A = j jw t jw t jwt jwt ( e e ) ( e + e ) j( w+ω ) t j( w+ω ) t j( w ω ) t j( w ω ) t ( e e + e e ) sin j ( ω + ω ) sin( ω ω ) ( jsin( ω + ω ) jsin( ω ω ) ( sin( ω + ω ) + sin( ω ω ) j where is the ulation index and A the aplitude of the arrier signal. 5

6 4. Aplitude Modulation using DSP-ethods DAC ADC ulating signal LPF (Low ADC Multiply Multiply DAC Pas Filter) Deulated Signal arrier Software-Siulation Aplitude ulated arrier (Transission) figure 3 - Aplitude ulation using DSP ethods with analogue transission In Aplitude Modulation Systes using Digital Signal Proessing ethods the generation of the arrier and the aritheti (ultipliation and arrier generation) are done via a Digital Signal Proessor. The ulating signal is fed via an ADC (Analogue to Digital Converter) to the syste. The transission (the hannel) an be either digital or analogue. For an analogue transission the digital aplitude ulated arrier ust be onverted to an analogue one and bak to a digital at the reeiver. Objetives of the Lab were to design an Aplitude Modulation and Deulation Syste siulated in C. The ulating signal was to ipleent via a sinus funtion and the arrier signal was to ipleent via a nd order IIR filter. The low pass filter to reove the reaining was to realise by eans of a FIR filter, a so alled onvolver. 6

7 a) The Digital Osillator The osillator for the arrier signal was to ipleent by eans of a nd order reursive Funtion. The nd Order differential equation for an analogue osillator was given by: d y(t) = ω y(t) + ω dt x(t) Applying bakwards differene approxiation leads to: d y(t) + ω dt [ y(nt) y(nt T) ]/ T [ y(nt T) y(nt T) ] T y(n) y(n ) + y(n ) + ω T y(n) y(n ) + y(n ) + ω y(n ) y(n) + ω T o y(n) = + ω y(t) = ω o T... where x(n) is x(t) y(n ) + + ω T o x(n) + + ω T y(n) ω ω T x(n) = + ω T o T y(n) ω o x(n) = y(n ) + ω / T + ω T x(n) = o y(n ) usually an ipulse to start the syste T y(nt) = ω x(nt) The proble of the bakward differene approxiation is, that it hanges the frequeny of the osillator and the apability to osillate without signal loss beause of the hange of the oeffiients (Transfors in Signals and Systes, Peter Kraniauskas, Addison-Wesley, Page 3). A better alulation of the oeffiients results in the following equation (C Algoriths for Real-Tie DSP, Paul M. Ebree, Prentie Hall, :995, Page 78) y(n) = y(n ) y(n ) + x(n) T = f saple where, = e dt d = daping( = ), os( ωt) and ω = πf = e Osillate dt For the apability of osillating the poles of the Z-Transfer funtion have to be on the unit irle, whih is not given by the bakward differene approxiation. The frequeny of the arrier signal an be easily heked by oparing with the ulation frequeny. The ratio should be /. 7

8 b) Modulation and deulation Hene the ulation and the deulation is only based on siple ultipliation, they an be realised very siple only using standard ultipliation provided by the DSP iruit. ) Low Pass Filter The transfer funtion of a FIR filter H(z) is given by: H(z) = The ultipliation by z - eans a delay of one unit of tie in the tie doain. The lowpass filter an be desribed by the equation: y(n) = N n= N k= h(k)x(n k) where h(k) are the filter oeffients and x(n - k) the sapled input values Coing fro the global definition of a low pass filter realised via a onvolver we an say: N h(n)z n y(n) = h(k)x(n k) = k= y(n) = h()x(n) + h()x(n -) + h(k) * x(n) where * eans onvolution h()x(n - ) + + h(n -)n(n - N + ) The output is just a linear weighted su of present and past inputs. So the FIR filter is alled a Running average filter. The filter funtion an be desribed by the following blok diagra, where z - represents the unit delay: x(n) x(n-) x(n-) x(n-n+) Z - Z - Z - h() h() h() h(n-) + y(n) figure 4 8

9 The input x(n) is ultiplied by the oeffiient h(n). Then x(n) is delayed by one step and ultiplied with the next oeffiient h(n+). After that x(n) is delayed again and ultiplied with the next filter oeffiient h(n+). All these single ultipliation are added together and one run of an x(n) value through all the filter oeffiients results in one output y(n). So it takes at least N steps, till the filter orretly works. It s the tie required for the st value to arrive at the filter-output. This filter ethod is realised by eans of a onvolver, whih stores and shifts the x-values and ultiplies the by the orret filter oeffiients. Figure 4 shows the standard blok diagra of a transversal filter. The oeffiients of the filter were alulated using the Fourier design ethod The requireents were: f utoff =khz,, f saple =5kHz, no given transition band, no given stop band attenuation Hene a retangular window funtion was used: Corner frequeny of ideal filter : ω = π =.4π 5 thus the ideal frequeny response : jαω e, ω.4π H D ( ω) = where α =,.4π < ω π IFT of the frequeny response : h h h h h D D D D D (n) = π (n) = π (n) = π H( ω)e.4π.4π e e e e jω(n α) e (n) = π j(n α) (n) = sin π(n α) π π jαω jαω jωn dω jωn jωn.4π dω + π dω = π = πj(n α) [.4π(n α) ].4π.4π e e N jωn jω(n α) dω dω j.4π(n α) ( e ) fro where the oeffiients an be diretly alulated sine no window funtion is used For the iddle filter-oeffiient L Hospital is used to alulate it (/)!: [ Aπ(n α) ] sin h(n) = π(n α) Aπ h(n) = π h(n) = A =.4 = [ os( naπ Aπα) ] when n = α differntiate Nuerator and Denoinator seperatly 9

10 5. Realisation using C a) The ulation signal In usual aplitude ulation signal, where speeh or equivalent signals are ulated, the ulation signal is fed to the DSP via an analogue to digital onverter. In the lab-siulation the ulation signal was to generate using a sinus funtion with a frequeny of khz. The following line shows the realisation using C: y[]=sin(n*wn), where ωn is the noralised signal frequeny of khz figure 4 - Modulating signal Figure 4 shows the ulating signal output of the progra. b) The digital Osillator The osillator was designed based on the following differene equation derived via the bakwards differene approxiation. Only the oeffiients were to deterine new, in order to full-fill the requireents, whih said the osillator frequeny to be khz at a sapling frequeny of 5kHz. y(n) = y(n ) y(n ) + x(n) T = f saple where, = e d = daping( = ), os( ωt) and ω = πf = e π khz Hene : = os( ) = kHz = dt Osillate dt The resulting ode is the following: x[]=.68339*x[]-x[]+i; x[]=x[]; // shifting osillator array x[]=x[]; figure 5 - Digital arrier signal

11 ) Modulation and Deulation The ulation and the deulation is based on siple ultipliation. Hene the ode for it was very siple to ipleent: // Modulating =(ap+(-ap*.5)*signal)*arrier; // Deulating de=-*arrier; The variable ap is used for hanging between balaned and standard aplitude ulation. It s value is previous set for balaned ulation and for standard ulation. The fator and.5 (=-.5) are hosen to get appropriate ulating results, beause the ratio of the arrier and the ulation signal should not be to high ( ), or to low ( ), to get best visible results for Standard aplitude ulation. The values for balaned ulation are A= and =. figure 6 - Modulated and deulated signal figure 7 - Balane Aplitude Modulated and deulated signal Figure 5 and figure 6 show the ulated and deulated but not low pass filtered outputs of the software.

12 d) Low Pass Filtering The low pass filter was realised by eans of a onvolver. The order of the onvolver was inreased, until a interferene free low pass signal available was. The 5 oeffiients of the onvolver are alulated with every new run of the ulator ulation setion of the progra. The filter oeffiients are alulated by the following lines using the algorith fro 4) Low Pass Filter. The resulting progra ode is as follows: // Calulating Convolver Filter oeffiients for(n=; n<=5 ; n++) { h[n]=sin((.4)*pi*(n-5))/(pi*(n-5)); h[(5)-n]=h[n]; // Reove oents for printing filter oeffiients // printf("%d=>%f %d=>%f \n",n,h[n],(5-n),h[5-n]); } h[5]=.4; // Set id filter oeffiient (fro L'Hospital) A J K B@C :6 ; < = >? ABC D7 BC 9 ABC D7 :F G ; % %%% & &&& '( )* +/ - +, %%% &&& '( )* + + / C >E 7 AB 9 L< = >? M? A H 7 E C? B C I A BC D7 figure 8 - Working el of a onvolver OO X O X O PQRSTQSU RSW XX O X O X O X O PQRSTQSU RSW X Y Z Z [\[] ^ \] Y ` a ` b \] d D K N C +/ 7 II@K 6 BA N N N +, N N N + + / Figure 8 shows how the onvolver ipleented in the software works. Every new deulated value is written into the last storage of the onvolver. Then all present stores are ultiplied with their appropriate oeffiient.

13 The following lines show the progra ode of the onvolver: // Filtering via onvolver filter_out=; for(k=; k<=5; k++) { filter_out+=xstore[k]*h[k]; // onvoling xstore[k]=xstore[k+]; // shifting array } xstore[5]=de; /* writing new xstore(n) value */ // END OF CONOLER figure 9 - Low pass filtered signal Figure 9 shows the output of the low pass filter realised via onvolver. There are no visible interferene in the sine signal. e) Output Plots figure - Output plot for standard aplitude ulation 3

14 figure - Balaned aplitude ulation Figure and figure show the output plots of the prograe in both ulation es. The in/ax funtion provides the peak levels of the different signals and is realised via siple oparing funtion. 6. Aplitudes The following peak aplitudes an be read out fro the output plots (figure and ): signal arrier ulated deulated Standard.5.5 Balaned Standard Aplitude Modulation = Asin = Asin ( ω t) ( + os( ω t) ) A ( ω t) + [ sin( ( ω + ω ) t) + sin( ω ω ) t) ] Fro the equation above it an bee seen, that the axiu peak aplitude is.5, when the arrier is A= and the ulation index =.5 (values fro prograe). 4

15 Balaned Aplitude Modulation = A sin A = ( ω t) os( ω t) ( sin( ω + ω ) + sin( ω ω ) The axiu peak value of the ulated signal is equal, when A= and =, whih are the used values in the prograe. Deulation de de = A sin = A A ( ω t) + ( sin( ω + ω ) t) + sin( ( ω + ω ) t) sin( ω t) ( os( ω t) ) A 4 os + os ( ω + ω ) t) os( ω t) + (( ω ω ) t) The deulated signal is negative signed. It s aplitude is at standard aplitude ulation.5 (=-.5A-.5A-*.5A) with A= and =.5. The aplitude of the filtered signal was not easured, beause of the attenuation of the low pass filter. 7. Conlusion Aplitude ulation is very easy realisable with digital systes, when the systes are apable to alulate at the speed of the desired frequenies. Analogue aplitude ulation systes an be diretly onverted fro the tie to the disrete tie doain. The onversion has to be done very arefully, in order to prevent frequeny shifting like the bakward differene approxiation does. Although aplitude ulation is used sine the first days of the th entury, it is still very popular. The advantages of AM are the easy and heap way of realisation and the little onsuption of bandwidth. The disadvantages are the poor signal to noise ratio and the proneness to aplitude distortions. These disadvantages an t be redued only with the hange fro analogue to digital, beause the transission hannel is the sae analogue one than before. The effort of the future is to replae today s analogue based aplitude ulation systes with digital systes, like the hange fro RTTY and Morse-ode to digital transission systes like the aateur s paket-radio. 5

16 8. Appendix a) Coplete soure ode of the ulation software #inlude <graphis.h> #inlude <stdio.h> #inlude <onio.h> #inlude <ath.h> #define pi #define end 64 // // Aplitude ulation - DSP siulation using C // Dirk Beker, BENG /3, 9835 // // Balaned and Standard Aplitude Modulation // float x(int n); void differene(int ypos, int sale, int ap); int ain(void) { /* request autodetetion */ int gdriver = DETECT, ge, errorode, sale=3; har key; int type=; /* initialize graphis and loal variables */ initgraph(&gdriver, &ge, ""); setbkolor(black); do { /* start */ differene(4,sale,type); setolor(lightblue); oveto(,447);lineto(639,447); outtextxy(5,45,"+- for saling or Es for exit, B for toggling ulating e"); key=geth(); if (key=='+') sale++; if (key=='-') sale--; if ((key=='b') (key=='b')) type=abs(type-); // toggle if (sale<=) {sale=;}; leardevie(); lrsr; } while (key!=7); } /* losegraph (lean up) */ losegraph(); return ; 6

17 /* END OF MAIN FUNCTION */ /* */ /* Osillators, ulator, deulator and filter */ void differene(int ypos, int sale, int ap) { int n, k, y_old, x_old, y_old, xplot, yplot, yplot, z_old, zplot, f_old=5, fplot; // ars for line drawing // (storing old points) int _old=9, plot; // ( -"- ) float xvalue, y[]={}, x[5]={},, de, i=; float fsignal=, fsignal=; // signal and osillator float fsaple=5; // and saple frequenies float wn=*pi*fsignal/fsaple; // noralisation of the and float wn=*os(*pi*fsignal/fsaple); // oeffiients of differene // equation float arrier, signal; // outputs vars float filter_out, xstore[5]={}, h[5]={}; // filter vars setolor(red); int saley=5; // saling in y diretion float arrierax=, signalax=, ax=, deax= ; // ax-in als // Calulating Convolver Filter oeffiients for(n=; n<=5 ; n++) { h[n]=sin((.4)*pi*(n-5))/(pi*(n-5)); h[(5)-n]=h[n]; // Reove oents for printing filter oeffiients // printf("%d=>%f %d=>%f \n",n,h[n],(5-n),h[5-n]); } h[5]=.4; // Set id filter oeffiient (fro L'Hospital) oveto(,ypos); i=; y_old=ypos;y_old=ypos+7;z_old=ypos+; // preset storings // for line funtion x_old=; for(n=; n<=end; n++) // ain loop { // Calulating frequeny (signal) y[]=sin(n*wn); signal=y[]; // for noralisation to (already is) if (signal>=signalax) signalax=signal; // Calulating frequeny (arrier) x[]=wn*x[]-x[]+i; arrier=x[]; // for noralisation to ( - "" - ) if (arrier>=arrierax) arrierax=arrier; // Modulating Y=A*f*(+*f) and hange between // Balaned and Standard ulation // Standard Modulation: A=(ap), =.5 // Balaned Modulation: A=, = // =(ap+(-ap*.5)*signal)*arrier; if (>=ax) ax=; // Deulating de=-*arrier; if (de<=deax) deax=de; // Filtering via onvolver filter_out=; for(k=; k<=5; k++) { filter_out+=xstore[k]*h[k]; // onvoling xstore[k]=xstore[k+]; // shifting array } xstore[5]=de; /* writing new xstore(n) value */ // END OF CONOLER

18 x[]=x[]; x[]=x[]; // shifting osillator array 8

19 // Drawing the wavefors // the old points are always stored in the _old vars setolor(red); // Raw signal xplot=n*sale; yplot=-signal*saley+ypos; line(x_old, y_old, xplot, yplot); y_old=yplot; setolor(green); // Carrier frequeny yplot=arrier*saley+ypos+7; line(x_old,y_old, xplot, yplot); y_old=yplot; setolor(blue); // Modulated signal zplot=*saley+ypos+7; line(x_old,z_old, xplot, zplot); z_old=zplot; setolor(lightred); // Deulated (ultiplied) plot=de*saley+ypos+95; line(x_old,_old, xplot, plot); _old=plot; setolor(lightgreen); // Filtered via onvolver if ((filter_out>4) (filter_out<-4)) filter_out=; fplot=filter_out*(5+*ap)+ypos+(37+6*ap); line(x_old,f_old, xplot, fplot); f_old=fplot; x_old=xplot; } i=; /* End of ipulse */ // Adding text to urves // setolor(white); outtextxy(,5,"modulating signal:"); outtextxy(,65,"carrier signal:"); outtextxy(,55,"modulated signal:"); outtextxy(,65,"deulated signal:"); outtextxy(,34,"deulated and filtered signal:"); // Printing ax values gotoxy(6,); printf("sigax: %.f \n",signalax); gotoxy(6,5); printf("arax: %.f \n",arrierax); gotoxy(6,); printf("ax: %.f \n",ax); gotoxy(6,7); printf("deax: %.f \n",deax); } 9