2. Continuous-wave modulation

Transcription

1 . Continuous-wave odulation 1. Appliation goal We study representations in tie and frequeny doain for two types of ontinuous wave odulation: aplitude odulation (AM) and frequeny odulation (FM).. Continuous-wave odulation: basi notions The purpose of a ouniation syste is to transport a signal representing a essage over a hannel and deliver a reliable estiate of that signal to a user. This essage ay be a speeh signal, the hannel aybe a ellular phone hannel or a satellite hannel. The essage signal (alled the odulating wave) that ontains inforation, generated by soures of inforation, is a baseband signal and it is transitted using a high frequeny signal, also alled arrier signal. A paraeter of the arrier wave is odified aording the odulating wave. The two ost oon used fors of arriers are the sine wave and the periodi pulse wave. Correspondingly, we identify two ain lasses of odulation: ontinuous-wave odulation (CW) and pulse width odulation. A ontinuous wave has three paraeters: aplitude, frequeny and initial phase; the orresponding odulation is aplitude odulation (AM), frequeny odulation (FM) and phase odulation (PM). FM and PM are also known as angle odulation. Modulation provides the eans for: (1) shifting the range of frequenies ontained in the essage signal into another frequeny range suitable for transission over the hannel, entered onto the arrier frequeny () perforing a orresponding shift bak to the original frequeny range after reeption of the signal Exaple: A radio syste operates effiiently into a frequeny range that is higher than 3 khz; on the other hand baseband signals are audio signals below khz (-khz). Frequeny shifting is ade using odulation; in order to listen to an FM radio hannel, we set the frequeny to a very high value (for instane 1MHz). 3. Aplitude odulation Consider a arrier wave: and the odulating wave os t A t (1) xt. The aplitude odulated signal is: K sam t A Kax t t A x t t A kax t t A os 1 a os 1 os () The onstant K a, easured in V 1, is alled the aplitude sensitivity of the odulator. The aplitude odulation works by varying the strength of the transitted signal in relation to the 1

2 inforation being sent. The aplitude of the arrier signal is varied aording to the odulating wave through the ter. The aplitude of the sine wave is by definition a positive value a k x t whih eans we ipose the ondition: k x( t) 1 a (3) If this ondition is not satisfied then an overodulation is obtained, whose effet is distorting the envelope of the AM signal and the orret reovery (deodulation) of the odulating wave is no longer possible. The axiu value of is the odulation degree, usually expressed as a perentage: k x() t a k 1 % ax t (4) ax Fig. 1 shows the sine arrier wave, a odulating wave (also sine wave), and two ases of aplitude odulation. Fig. 1 ) presents a orret odulation while in the next ase we have overodulation (fig. 1 d), when the odulation degree is higher than 1 (>1). For the nd ase, we notie distortions of the envelope as a diret result of phase inversion in the arrier signal The ondition to reover orretly the essage signal fro the envelope of the odulated signal is that the arrier frequeny to be high enough opared to the axiu variation speed of the odulating signal: f f (5) a) 1 Carrier signal, (t) Modulating wave, x(t) x 1-4 b) AM signal, =5% x 1-4 ) d) AM signal, =1% (overodulation) x tip x 1-4 Fig. 1: Aplitude odulation for odulating sine wave x(t) and overodulation.

3 If this ondition is satisfied, the odulation degree an be deterined using the tie representation of the signals, on the osillosope. The signal fro fig. 1 is given below, its envelope with a dotted line (fig ). 1.5 Aax 1 Ain.5 aplitudine The approxiation is: tip Fig. : Measuring the odulation degree using the osillosope. x 1-4 A A ax in A ax A in 1 [%] (6) In the exaple above, we notie that Aax 1.5V and Ain.5V, so we have =5%. The spetru for the AM signal results fro the Fourier transfor applied on the signal in relation (): ka a SAM ( ) A ( ) ( ) X ( ) X ( ) (7) An exaple is given in fig. 3. 3

4 X(ω) 1 πaδ(ω+ω) upper sideband -ω- ω -ω kaa/ lower sideband -ω+ ω SAM(ω) lower sideband ω- ω ω πaδ(ω-ω) upper sideband ω+ ω Fig. 3: The spetru of the AM signal. The spetru has two Dira ipulses orresponding to the arrier; the effet of the odulation is the frequeny shifting of the spetru fro the baseband on the arrier frequeny. The bandwidth oupied by the AM signal above the arrier is the upper sideband and the bandwidth oupied by the AM spetru below the arrier is the lower sideband. We an see fro the figure that the bandwidth of the odulated signal, is double of the bandwidth of the essage (odulating) signal, B: B B (8) B AM AM M The ain advantage of aplitude odulation is the siple ipleentation. As a onsequene it was used fro the beginning in the field of radio transission. Transission of the radio signal using AM was heaper. However, aplitude odulation has low energy effiieny. Although the arrier does not have itself inforation, it is present in the AM spetru, so there is a waste of power. For the transission of the inforation, only one sideband is neessary; if both the arrier and the other sidebands are suppressed at the transitter, no inforation is lost. This way, the hannel needs to provide only the sae bandwidth as the essage signal. The types of AM odulation that save bandwidth are: suppressed-arrier Aplitude Modulation single sideband Aplitude Modulation, SSB-AM vestigial sideband Aplitude Modulation,VSB-AM For the AM spetru, onsider the ase of a sine odulating wave. The wavefor of the AM signal is shown in fig. 1 and the odulation degree is k A / A. The signal s expression is a obtained fro () using the partiular for for the odulating wave x( t) A os( t) : or: s ( t) A 1 os t os( t) (9) AM A A sam ( t) A os( t) os( ) t os( ) t (1) 4

5 SAM(ω) πa πa/ -ω- ω -ω -ω+ ω ω- ω ω ω+ ω Fig. 4: The spetru of the AM signal for odulating sine wave. ω This spetru orresponds in tie to a su of three sine waves with different frequenies, and : a SAM A ka X X (11) The power of the AM signal is A A A A Ps (1) 8 8 4, 4. Frequeny odulation Another way to odulate a arrier is to alter its angle phase, aording to the essage; whereas the aplitude reains onstant. The advantage is the signal is ore robust against noise and interferene. The disadvantage is the inrease in bandwidth. Again, onsider a arrier wave: t A os t The odulated signal is a rotating vetor with aplitude A and angle t sfm ( t) A os i t A os t k f xd (13) The angular veloity of this vetor is the instantaneous frequeny of the odulated signal t For the FM ase, the instantaneous frequeny is kf d t i i (14) t dt k x t (15) i f Hz / V is the frequeny sensitivity. For oparison, we onsider AM signals: the phase is (even onsidering the no odulation ase) i () t t and the aplitude isn t onstant, it depends on tie; while for FM signals, the aplitude is onstant and the inforation is sent through the phase or the instantaneous frequeny. i : 5

6 t If we denote y( t) x( ) d, relation (1) beoes: s ( t) A os( t)os( k y( t)) sin( t)sin( k y( t)) (16) FM F F There are two types of frequeny odulation: - if ax { k y( t )} <<, the effet of the odulation on the arrier is sall and the FM t f / bandwidth isn t uh larger than the spetru of the arrier. This is narrow band frequeny odulation. - if ax { k y( t )} >>, the FM bandwidth is uh larger than the spetru of the arrier. t f / This is wide band frequeny odulation. For a sine odulating wave x t A os t the instantaneous frequeny is: os t k A t (17) i f The frequeny deviation is the axiu instantaneous differene between an FM odulated arrier frequeny and the noinal arrier frequeny: kf A; and the odulation index is The FM signal beoes: kf A os sin FM (18) s t A t t (19) or: os( )os( sin( )) sin( )sin( sin( )) s t A t t A t t () FM An essential harateristi for FM signals is that the frequeny deviation is proportional with the odulating signal s aplitude; but it does not depend on its frequeny. Again we have narrow band odulation or wide band odulation for 1 radian or 1 radian respetively. The funtions os( sin( t )) and sin( os( t )) are periodi with period / ; hene they are deoposed into Fourier series: os( sin( t)) J( ) Jk ( )os( kt) (1) and: k1 sin( sin( t)) Jk 1( )os(( k 1) t) () where J ( ) are Bessel funtions of the first kind, and order p. We have: P k sfm ( t) A Jn( )os( n ) t (3) n f 6

7 Fig. 5: Bessel funtions, J ( ) Zero rossings for the Bessel funtions: the values of β for whih Jk(β) =. n Although the odulating wave is band liited, we an see that s () t is nonband liited, oposed of an infinite series of sine waves, separate on the frequeny axis by. For 1 (narrow band odulation), only J( ) and J1( ) have signifiant values, the other Bessel funtions an be approxiated with zero. The FM signal is: FM or: s ( t) A J ( )os( t) A J ( )os( ) t A J ( )os( ) t (4) FM 1 1 7

8 t yt x d t sfm t A ost k f x d A ost os k f y t A sint sin k f y t yt A For narrow band odulation, k f A sfm t A ost A k f y tsin t. (5) 36 The FM spetru for narrow band odulation resebles the AM spetru, obtained in the sae partiular ase (odulating wave is a sine wave). X X SFM A A (6) Reall that: a SAM A ka X X For higher values of, the Bessel funtions of superior order an no longer be negleted, and the FM spetru is of wide band: A SFM Jn n n (7) n The aplitude of the oponent with the frequeny depends on the fator J. Opposed to the AM odulation, the aplitude of the orresponding FM signal spetru isn t onstant but depends on. The power of the FM signal is 1 T Ps x () FM t dt T (8) Supposing the arrier frequeny is a ultiple of the axiu frequeny of the spetru of the odulating wave, we have : 1 1 Ps A Jn A (9) n We have used the property of the Bessel funtions of first kind: k K J 1 The power of the odulated signal is the sae with the power of the arrier signal. Nearly all of the power (98%) of the FM signal lies within a bandwidth of: 1 BT f f f 1 For wide band odulation ( 1), we an approxiate the transission bandwidth as BT (3) f (31) This eans the bandwidth of the FM signal is onstant and doesn t depend on the odulating wave s axiu frequeny. 8

9 5. Pratial part We ake easureents and graphial representations in tie and frequeny for the AM and FM signals. We use two signal generators, GEN1 and GEN, an osillosope and a spetru analyzer. One of the signal generators is the soure of the odulating wave whose wavefor an be sine wave, square wave or triangle wave. The fundaental frequeny of the odulating wave and the arrier are set to: AM: f=khz; f=5khz FM: f=1khz; f=1 MHz. The essage signal will odulate a arrier with a arrier frequeny f. The arrier is generated by GEN and the odulation is ade by bringing the odulating signal (output 5) at the AM or FM input, depending on the type of odulation. Both signals (odulating wave and odulated wave) are visualized in tie on the osillosope using both hannels. The spetral analysis is ade using the spetral analyzer. The following paraeters are set: Center frequeny:.5mhz or 1MHz (sae value as f); Referene level 1dB Span width: khz/div or 1 khz/div (sae value as f); RBW: 3kHz RBW 5.1 AM study In this ase we will use three odulating wavefors: sine, triangle and square wave, and we will visualize and represent the odulating wave and the odulated signal as seen on the osillosope. In the ase of sine wave odulating signal, - Graphial representation of the AM spetru as seen on the spetru analyzer. Attention is paid to the agnitude and to the positioning of the spetral oponents on the frequeny axis. - Measure the odulation degree,, using the experiental ethod in the tie doain fro relation (6). - Measure the odulation degree,, fro this spetru. For this, easure the agnitude fro the arrier frequeny and the agnitude fro the frequeny. Magnitude vs. frequeny U db U V A A/ (freq. ) A (freq. ) The referene level is U 6V. ref U V lg U db U ref - Change the odulating wave to a square wave. What happens in the spetru of the AM signal, and how an you explain this. 9

10 5.. FM study - Visualize the odulating wave and the odulated signal as seen on the osillosope for odulating sine wave. How do you explain the spetru on the spetru analyzer? - Visualize and ake a graphial representation of the FM spetru as seen on the spetru analyzer, for the ase of narrow band odulation. - Inrease the aplitude of the odulating wave (thus inreasing β, the odulation index). Repeat the graphial representation. Copare the spetru with the previous ase and with the AM spetru. - Plot the harateristi of the FM odulator. 1

11 Worksheet for odulation Part 1: Aplitude odulation Modulating wave: frequeny f =, aplitude (peak-to-peak) A =, Carrier: frequeny f =, aplitude (peak-to-peak) A = Plot the odulating wave (sine, triangle and square), Plot the orresponding aplitude odulated signal for the three odulating waves, 11

12 For sine wave odulating signal: plot the spetru of the AM odulated signal For sine wave odulating signal: easure the odulation degree, 1) in the tie doain, Aax = Ain = A A ax in A ax A in 1 [%] = ) in the frequeny doain fro the spetru. For this, easure the agnitude fro the arrier frequeny and the agnitude fro the frequeny : Magnitude vs. frequeny U db U V A A/ (freq. ) A (freq. ) U V lg U db, referene level Uref 6V. U Fro the U[V] easureents: = A[V] / A [V]. ref Change the odulating wave to a square wave. What happens in the spetru of the AM signal, and how an you explain this: 1

13 Part : Frequeny Modulation for sine odulating wave Modulating wave: frequeny f =, Carrier: frequeny f =, aplitude A = Narrow band FM: Set the aplitude of the odulating wave A, at a sall value, suh that the spetru of the odulated wave has the sae bandwidth as the spetru of AM signal. aplitude (peak-to-peak) A =, Plot the spetru of the narrow FM odulated signal 13

14 Wide band FM: inrease the aplitude of the odulating wave, A, suh that the bandwidth of the odulated signal inreases. aplitude (peak-to-peak) A =, Plot the wide band FM signal (tie doain, as seen on the osillosope) Plot the spetru of the wide band FM signal 14

15 Inrease slowly the aplitude of the odulating wave, A, fro iniu to axiu, suh that the aplitude in the arrier frequeny f, drops below -4 db. Eah tie this happens, easure the aplitude of the odulating wave A. A [V] Plot the harateristi of the FM odulator: 15