EE (082) Chapter IV: Angle Modulation Lecture 21 Dr. Wajih Abu-Al-Saud

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1 EE 70- (08) Chapter IV: Angle Modulation Leture Dr. Wajih Abu-Al-Saud Effet of Non Linearity on AM and FM signals Sometimes, the modulated signal after transmission gets distorted due to non linearities in the hannel, for exple. In general, when the transmitted modulated signal is affeted by hannel non linearity in the hannel, the demodulated signal beomes a distorted version of the message signal. We an easily show that the effet non linearities on different types of plitude modulated signals is devastating, while frequeny modulated signals are immune to non linearities. In fat, the effet of non linearities on FM signals an be used for generating wideband FM signals from narrowband FM signals, whih is an important feature. Non Linearity in AM: Consider the hannel shown below with a DSBSC input signal. The hannel is a non linear hannel in whih the output signal of the hannel is the sum of the input signal and other powers of the input signal. Let g DSBSC (t) be g = m()os( t ω t). DSBSC The output signal of the hannel f(t) beomes ( ω ) ( ω ) ( ω ) ( ω ) ( ω ) ( ω ) f = t ()os t + a mt ()os t + a mt ()os t = ()os t t + ()os t t + ()os t t = ( t) os( ωt) + os( ) os( ) os( ) + ω t + ω t t + ω = + + os ωt + os ωt + os os 4 ωt + ωt ( t) ( t) = + ( t) + os( ωt) + os( ωt) + os( ωt) 4 4 ( ) ( ) ( ) ( ) Around 0 Around ω Around Around ω ω So, unless a is zero, it is lear that the original modulated signal g DSBSC (t) given above annot be extrated from the reeived signal f(t) beause the terms with frequeny around ω does not ontain only m(t) but also m (t). So, DSBSC (and AM modulation in general) is vulnerable to non linearities. Non Linearity in FM: Again, onsider the se hannel shown given above with an FM input signal.

2 EE 70- (08) Chapter IV: Angle Modulation Leture Dr. Wajih Abu-Al-Saud g FM (t) a (.) + a (.) + a (.) q(t) Let g FM (t) be gfm = A os ωt + kf. The output signal of the hannel q(t) beomes q = aa os ωt + kf + a Aos ωt + kf + a A os ωt + k f aa = aa os ωt + kf + + os ωt + kf aa os t kf m( α) os ωt kf ω + aa aa aa = + aa os ωt kf os ωt kf DC + + Around ω with kf = kf Around ω with kf = kf aa os ω t k f 4 Around ω with kf = kf In this ase, it is lear that all the oeffiients of the different terms are simply onstants. The terms are in fat different FM signals with different frequenies and different values of the pareter k f. But, the important onlusion is that the original FM signal an easily be extrated from the output signal of the non linear hannel using a filter entered at the arrier frequeny and bandwidth equal to the bandwidth of the FM signal. So, FM signals are IMMUNE (does not get daged) to non linearities. Use of Non linearity for Manipulating FM Signals The fat that passing an FM signal through a non linear devie results in a set of FM signals with different arrier frequenies and different pareter k f (and therefore different frequeny variation pareter ω), we an use this proess for manipulating the arrier frequeny ω and/or the frequeny variation ω. In general, passing the FM

3 EE 70- (08) Chapter IV: Angle Modulation Leture Dr. Wajih Abu-Al-Saud signal through a non linear devie with a maximum non linearity power of P will give P different FM signals as shown in the blok diagr below. g FM = Aos ωt + kf qt ( ) ontains the following os ωt + k f os ωt + k f os Pωt + Pkf g () FM ( output ) t B os [ Pω] t + [ Pk f ] = Passing the output of the P power non linear devie through a BPF with enter frequeny Pω and bandwidth equal to the bandwidth of the last FM signal will extrat that signal and rejet the other FM signal. This proess an be used to obtain wideband FM signals from narrowband FM signals as will be desribed through a set of exples next. We will assume that the non linear devies that we use in the following exples have built-in BPFs to eliminate the undesired FM omponents. Generation of Wideband FM Signals Indiret Method for Wideband FM Generation: Consider the following blok diagr A narrowband FM signal an be generated easily using the blok diagr of the narrowband FM modulator that was desribed in a previous leture. The narrowband FM modulator generates a narrowband FM signal using simple omponents suh as an integrator (an OpAmp), osillators, multipliers, and adders. The generated narrowband FM signal an be onverted to a wideband FM signal by simply passing it through a non linear devie with power P. Both the arrier frequeny and the frequeny deviation f of the narrowband signal are inreased by a fator P. Sometimes, the desired inrease in the arrier frequeny and the desired inrease in f are different. In this ase, we inrease f to the desired value and use a frequeny shifter (multipliation by a sinusoid followed by a BPF) to hange the arrier frequeny to the desired value. Exple : A narrowband FM modulator is modulating a message signal m(t) with bandwidth 5 khz and is produing an FM signal with the following speifiations f = 00 khz,

4 EE 70- (08) Chapter IV: Angle Modulation Leture Dr. Wajih Abu-Al-Saud f = 5 Hz. We would like to use this signal to generate a wideband FM signal with the following speifiations f = 5 MHz, f = 77 khz. Show the blok diagr of several systems that will perform this funtion and speify the haratestis of eah system Solution: We see that the ratio of the arrier frequenies is f f 6 5*0 = = 450, 00*0 and the ratio of the frequeny variations is f 77*0 = = f Therefore, we should feed the narrowband FM signal into a single (or multiple) non linear devie with a non linearity order of f / f = 00. If we do this, the arrier frequeny of narrowband FM signal will also inrease by a fator of 00, whih is higher than what is required. This an easily be orreted by frequeny shifting. If we feed the narrowband FM signal into a non devie of order f /f, we will get the orret arrier frequeny but the wrong value for f. There is not way of orreting the value of f for this signal without affeting the arrier frequeny. System : System : In this system, we are using a single non linear devie with an order of 00 or multiple devies with a ombined order of 00. It is lear that the output of the non linear devie has the orret f but an inorret arrier frequeny whih is orreted using a the frequeny shifter with an osillator that has a frequeny equal to the differene between the frequeny of its input signal and the desired arrier frequeny. We ould also have used an osillator with a frequeny that is the sum of the frequenies of the input signal and the desired arrier frequeny. This system is haraterized by having a frequeny shifter with an osillator frequeny that is relatively large.

5 EE 70- (08) Chapter IV: Angle Modulation Leture Dr. Wajih Abu-Al-Saud System : In this system, we are using two non linear devies (or two sets of non linear devies) with orders 44 and 50 (44*50 = 00). There are other possibilities for the fatorizing 00 suh as *00, 4*550, 8*75, 0*0,. Depending on the available omponents, one of these fatorizations may be better than the others. In fat, in this ase, we ould have used the se fatorization but put 50 first and 44 seond. We want the output signal of the overall system to be as shown in the blok diagr above, so we have to insure that the input to the non devie with order 50 has the orret arrier frequeny suh that its output has a arrier frequeny of 5 MHz. This is done by dividing the desired output arrier frequeny by the non linearity order of 50, whih gives.7 Mhz. This allows us to figure out the frequeny of the required osillator whih will be in this ase either..7 = 0.5 MHz or.+.7 = 5.9 MHz. We are generally free to hoose whih ever we like unless the available omponents ditate the use of one of them and not the other. Comparing this system with System shows that the frequeny of the osillator that is required here is signifiantly lower (0.5 MHz ompared to 55 MHz), whih is generally an advantage. We also an bring the frequeny shifter before all the non linear devies and therefore redue the frequeny of the required osillator to the minimum value by finding the required arrier frequeny at the input of eah non linear devie to insure that the arrier frequeny of the final output of non linear devies is the desired final arrier frequeny. For more exerises on this WB FM generation method, refer to the problems at the end of hapter 5 in your textbook.

6 EE 70- (08) Chapter IV: Angle Modulation Leture Dr. Wajih Abu-Al-Saud Diret Method of Generating WB FM Signals This method is simple in the sense that it uses a single omponent: the voltage-ontrolled osillator (VCO). As desribed in the setion of Carrier Aquisition for DSBSC systems, VCOs are devies that produe a sinusoid with a frequeny that is proportional to the input signal. So, if the input signal to a VCO is the message signal, the output of the VCO will be an FM modulated signal of the message signal sine the frequeny of this FM signal hanges aording to the input message signal. m(t) Voltage Controlled Osillator (VCO) FM Modulator g FM (t)