Modulation exercises. Chapter 3

Transcription

1 Chaper 3 Modulaion exercises Each problem is annoaed wih he leer E, T, C which sands for exercise, requires some hough, requires some concepualizaion. Problems labeled E are usually mechanical, hose labeled T require a plan of aack, hose labeled C usually have more han one defensible answer. Recall he represenaion of bandlimied signals. Definiion x(); < <, is a real bandlimied signal wih carrier f c (Hz) if jx(f )j = 0, jf f c j > 2 B x and B x << f c as shown in Figure 3.. X(f) B x -f c f c f Figure 3.: FT of bandlimied signal Theorem Le x be a bandlimied signal. and ^x is Hilber ransform. Then x; ^x can be represened as 8; x() = A() cos[2ßf c + ()] ^x() = A() sin[2ßf c + ()] 9

2 0 CHAPTER 3. MODULATION EXERCISES Moreover, he ampliude A() and phase () can be obained from x; ^x and he carrier signal as follows. Firs obain he complex baseband signal z: z() =[x() +j ^x()]e j2ßfc ; hen A() =jz()j; () = arg z(): Moreover, jz(f )j =0; jfj > 2 B x. In he conex of modulaion, A and are he modulaing signals, and x is he modulaed (ransmied) signal wih carrier f c. Thus he demodulaor, afer receiving x, mus firs obain ^x, hen z, and hen recover A and.. E Wha is z in he following wo cases where we use f c =00Hz. (a) x() = cos(2ß 99) +sin(2ß 0). (b) x is given in Figure 3.2. X(f) f Figure 3.2: Bandlimied signal in Problem 2. T Repea Problem bu now use f c =05Hz. 3. T Given a modulaing signal f ha is bandlimied o W Hz, a carrier cos(! c ) wih! c >> 2ßW, a nonlineariy g, and a band-pass filer ha only passes frequencies wihing! c ±2ßW or! c ± 2ßW, we wan o build a modulaor ha has oupu x() =A[ + fif()] cos! c ] by combining he componens as in Figure 3.3. Assume jf ()j <<. Compue fi for he following nonlineariies. (a) g(x) = ( x; x<0 0; x 0

3 f() + g(.) BPF x() cos ω c Figure 3.3: Modulaion scheme in Problem 2 (b) g(x) = ( x; jxj < =2 0; jxj =2 4. E Consider a narrow band signal of he form 8; x() =cos(! c + ()) Assume X(!) is zero excep where jj!j! c j < 2ßWj. (Hin. Below you may use j!j = (j!)( jsgn!).) (a) Find y, z in he upper arrangemen of Figure 3.4. (b) Find z in he lower arrangmen. Here he bandpass filer only passes he frequencies jj!j! c j < 2ßWj. x() y() z() H(ω) = a + b ω envelope deecor x() y() hard sgn(y()) z() H(ω) = a + b ω limier BPF Figure 3.4: Deodulaion schemes in Problem 4 5. T The arrangmen in Figure 3.5 is a scrambler (i-iv) followed by a descrambler (v-vii). The single sideband modulaors generae, for an inpu signal f, he oupu signal ffi() = f ()cosw ^f ()sinw. The LPF has frequency response H(!) =, j!j» W, and H(!) = 0, oherwise. The signal m a inpu i is given by is Fourier ransform M in he lower par of he figure. (a) Give graphical represenaions in he frequency domain of he signals a ii-vii.

4 2 CHAPTER 3. MODULATION EXERCISES i ii iii iv v vi vii SSB X LPF SSB X LPF m() cos 2W cos 2W M(ω) -W W ω Figure 3.5: The scrambler-descrambler of Problem 5 (b) Explain in one senence in which sense he signal a iv is a scrambled version of he inpu signal. 6. T Consider he PAM ransmission sysem (you don need o know wha PAM is) in which he frequency response of he channel and he LPF combined is given by ( H(!) = 2 ( + cos! ); j!j < 2ß 2 0; j!j 2ß (a) Find he (non-causal) impulse response of his sysem. (b) Le he inpu o he PAM sysem be x() =5ffi() +ffi( ) + 4ffi( 2): Use he resul of he previous par o obain a formula for he oupu y. (c) Wha are he values of y(0);y();y(2);y(3)? (d) If he clock a he receiving end is delayed by fi seconds he sampled oupu is y(fi );y(+ fi );y(2 + fi );y(3 + fi ). Le fi =0:2: Use your calculaor o compue one of hese values. 7. T A es channel for binary ransmission has he frequency response H(!) =. The inpu ff+i! o he channel is given in Figure 3.6. The inpu on he lef corresponds o he binary inpu 0 (case A), and on he righ corresponds o (case B). The second binary digi is sampled a he oupu a =:5. (a) The oupu a =:5 will depend on wheher he firs binary digi is 0 or, and denoe i as y A (:5) and y B (:5), respecively. Wha is he lowes possible value of ff if we have he requiremen y B (:5) y A (:5)» 0:0: y A (:5)

5 3 0 (Case A) (Case B) Figure 3.6: Tes signal for Problem 7 (b) Explain inuiively why he requiremen is easy o saisfy for large values of ff. In paricular, wha happens o he raio above as ff!. 8. T (PSK Specra) Consider phase shif keying using phase deviaion consan ffi = ß. Tha is a 0 is represened by cos(2ßf c ) and a is represened by cos(2ßf c + ß). Assume a carrier frequency of 0.0 MHz. Assume daa is sen a he rae of 000 bis per second. (a) Skech he magniude specrum of he PSK signal for he periodic bi sequence (Hin: he bi sequence can be represened by a square wave. Choose he square wave o be even in ime). (b) For he sequence above, esimae he bandwidh wihin which 90% of he signal power can be found. (Numerical mehods are appropriae). 9. E(Hilber Transform) Le Le m() = sin00ß ß sin200ß + : ß x() =m()cos(0 5 ß) +( ß Λ m())sin(05 ß): Skech he specra M (f ) and X(f ). Wha ype of modulaion does his sysem provide? 0. T (Asynchronous Demodulaion) Consider deecion of a one-modulaed AM double-sideband wih carrier (DSB-WC) signal p[x 4 () =(+cos2ßf m )cos2ßf c using an ideal diode followed by low pass filer, where he oupu of he deecor is x 5 (). (As shown in figure 3.7, ideal recificaion of an AM DSB-WC signal can be considered as a muliplicaion by a square wave a he carrier frequency f c ). For f c = Hz and f m = 2 0 3, skech x 4 ();x 5 ();x 6 () and heir specra. Explain using a ime domain skech of x 6 () why he carrier is necessary for his asynchronous demodulaion scheme.. T This Malab exercise les you simulae he asynchronous demodulaion described above. Download he hwk8.zip file from he class web page, and follow he insrucions o creae he vecor radioes, which represens.5 seconds of a 50kHZ cenered radio signal

6 4 CHAPTER 3. MODULATION EXERCISES Inpu Volage x4() T Ideal Diode x5() x4() x5() H(f) x6() s() s() f (low pass filer wih cuoff freq. 20KHz) T Figure 3.7: Asynchronous demodulaion (DSB-LC) sampled a 400 housand samples per second ( T = 2:5μs). The radio band is from 50kHz o 500kHz and conains several saions. (a) Plo radioes and a recified radioes (= x 5 ) from.0 o.05 seconds. (Hin: consider using abs.) (b) Wha is he impulse response h lp () for a windowed, causal, ideal low pass filer wih cuoff 20 khz? (Choose an appropriae window lengh.) Skech H lp (f ). (c) Use conv o filer x 5 () wih h lp (n T ). Plo he resul from.0 o.05 seconds. Downsample from 400 khz o 8 khz, and play wih sound. Describe wha you hear, and explain any arifacs you hear.