ELEC Counications Engineering Laboratory ---- Frequency Modulation (FM) 1. Objectives On copletion of this laboratory you will be failiar with: Frequency odulators (FM), Modulation index, Bandwidth, FM signals in the tie and frequency doain, FM deodulation using quadrature detector. ** Make sure the Frequency Modulation workboard is properly connected to your PC before beginning this laboratory.. Theory Frequency Modulation (FM) Basics The equation of a sinusoidal signal can be represented by: where: v : the instantaneous voltage, V ax. : the axiu voltage aplitude, ω : the angular frequency, φ : the phase. v= V sin( ωt+ φ) (1) ax. A steady signal corresponding to the above equation conveys little inforation. To convey inforation, the signal ust be varied and the variations are used to carry inforation. This process is called odulation. Any of the paraeters, aplitude, frequency and phase, in the signal ay be varied to convey inforation. In ffrequency odulation (FM), variations in frequency is used to convey inforation. We shall think in ters of the angular frequency ω. The signal whose frequency is being varied is called the carrier wave. The signal doing the variation is called the odulating signal. For siplicity, suppose both carrier wave and odulating signal are sinusoidals, i.e.,. v = V sinω t (where the subscript c denotes carrier) () c c c and v cos = V ωt ( denotes odulating signal) (3) What is frequency? If the frequency is varying, how do we define it? We can no longer count the nuber of cycles over a longish interval per second. Instead we define frequency as the rate of change of phase. This is consistent with the siple definition, because at a constant (angular) frequency ω radians/second the phase is changing at ω radians per second, which is ω/π cycles per second. Since we can only define what the instantaneous frequency is by reference to the phase, we ust look at the phase in order to derive an expression for the frequency-odulated (FM) signal. 1
Phase of the FM signal For an unodulated carrier v = V sinω t, the phase is: c c c s = ω t (4) c We want the odulating signal to vary the carrier frequency, ω c, so that its frequency takes the for: ω = ω + Dcosω t (5) c where D denotes the peak frequency deviation and is related to the aplitude of the odulating signal v by the 'frequency slope' of the frequency odulator (VCO) say k radians/s per V. The peak value of v produces the deviation D, so: D = kv (6) The total phase change undergone at tie t is found by integrating the angular frequency, i.e., ( ) s = ωc + Dcos ωt dt = ωct+ ( D ω)sinωt (7) (If you are not failiar with integration, you will have to take this result on trust). So the FM signal can be expressed as: Vcsin ωct+ ( D ω) sinωt (8) where inforation is carried in the ter ( D ω )sinω t. Modulation Index In the expression of (8), the coefficient (D/ω ) turns out to be quite iportant and is given the nae odulation index. It is often represented by the Greek letter beta, β. So we ay write the FM signal as: [ ω βsinω ] v = V sin t+ t (9) v c c where β is the odulation index (D/ω ). In this expression, the factor sin[ ωct βsinωt] is of the for sin ( a+ b) which can be expanded to sin acosb cos asin b we obtain: ( ) ( ) + (let us call it F) +. Applying this expansion to F, F = sinωctcos β sinωt + cosωctsin β sinωt (1) FM Sidebands The coplicated function of (1) can be expanded, using atheatics, into a series of ters like this: F = ( β )sinω t + ( β)[sin ( ω + ω )t sin ( ω ω )t] c 1 c c + ( β)[sin ( ω + ω )t sin ( ω ω )t] c c + ( β)[sin ( ω + 3 ω )t sin ( ω 3 ω )t] 3 c c + ( β)[sin ( ω + 4 ω )t sin ( ω 4 ω )t] +. 4 c c (11) where (β), 1 (β), (β) etc, are constants whose values depend only on β. They are called Bessel Functions. There is an infinite series of these functions and so we have an infinite nuber of FM sidebands. But in practice the values of the Bessel functions becoe very sall as the series goes on. For exaple, when β =
( ) =.4 1 ( ) =.577 ( ) =.353 3 ( ) =.19 4 ( ) =.34 5 ( ) =.7 A Practical Approxiation Rule Because the higher-order sidebands becoe very sall, in practice the bandwidth of the FM signal ay be restricted to a finite bandwidth. The practical rule, often called Carson s Rule, is used to take the bandwidth required as: ( ) B= F + F (1) d where B is the bandwidth, F d the frequency deviation and F is the bandwidth of the odulating signal, all in the sae units. 3. Task A --- Concept of FM This task introduces the idea of FM. Before you start it is necessary to appreciate soe fundaental concepts described above. In AM, it is the carrier aplitude that is odulated by the odulating signal which contains inforation. In FM it is the frequency that is odulated by the odulating signal. The aplitude is changed. When no odulation applied, the carrier is at its noinal frequency, i.e. the carrier frequency. The odulating signal causes the carrier frequency to deviate, i.e. to ove above and below the noinal value. With the axiu possible deviations, the carrier frequency is oved up and down by the aount of the frequency deviation, thus the bandwidth is about twice the frequency deviation. However, this would take up a very large aount of frequency spectru. A set liit is norally ade on the aount that the carrier can deviate fro its noinal frequency and this is called the axiu deviation. Different systes use different values of axiu deviation, depending on a nuber of factors. Soe of which are very coplex Bandwidth of FM signal It is iportant that we can understand and estiate the bandwidth of the transitted signal so that the transission paraeters can be chosen to fit into the available spectru. Clearly the bandwidth ust be at least equal to twice the deviation, as the carrier actually oves above and below its noinal frequency by that aount, but it also depends on how fast the frequency is being changed, i.e. on the bandwidth of the odulating signal. The atheatical analysis of FM described above shows that an FM signal has sidebands far above and below the axiu deviation. However the power in these sidebands decreases quickly as they becoe further away fro the carrier and it can be shown that, for practical purposes, a good approxiation to the bandwidth is given by: ( ) B= F + F (13) d where F d the frequency deviation and F is the bandwidth of the odulating signal. This is called Carson s Rule, and the bandwidth B can be viewed as containing the ajority of the transitted power, certainly sufficient for successful deodulation. 3
Modulation Index As we have seen, the bandwidth of an FM signal depends on both the frequency deviation and the bandwidth of the odulating signal. It ight be thought that, in order to keep the bandwidth as narrow as possible, all FM systes should be operated with very sall deviations. However, there are advantages to use a large frequency deviation. The ain one is an apparent iproveent in noise perforance. A specific bandwidth can be the result of a large F d and a narrow F or a sall F d and a large F. The ratio of the frequency deviation to the bandwidth of the odulating signal is called the odulation index and is an iportant paraeter in describing a FM syste. Thus the odulation index is given by: MI = F F (14) d Modulation index is soeties represented by the Greek letter beta (β). In this task, the frequency odulator is realized using a voltage controlled oscillator (VCO). A control signal fro a hardware board is applied to the VCO and the VCO output is exained using an oscilloscope and spectru analyser. Using this configuration, we can see how an external signal can change the oscillator frequency. 4. Task B --- Generation of FM using a VCO In this task, a sine wave signal is used to frequency-odulate a carrier so that you can investigate the signal in both the tie and frequency doains. You can adjust the aount of deviation and hence change the odulation index and see the effects. Notice that, when the odulation index is sall, the FM signal in the frequency doain shown on the spectru analyzer is siilar to that of an AM signal. You should try to relate the explanation of the FM bandwidth given in the previous background pages with the observations you ake in this task. 5. Task C --- Deodulation of FM using quadrature detectors The quadrature detector as shown in Fig. splits the incoing FM signal into two paths. One path is connected directly to one input of a phase detector. The other path contains a siple network which shifts the phase of the signal in proportion to its frequency deviation. Figure 1 Figure Phase Shifter Let us consider the nature of a typical phase shifter shown in Fig. first. The circuit diagra can be regarded as a siple potential divider, with input at port 1 and output at port. The upper ar has ipedance and the lower ar jω L+ R (15) 1 jω C (16) where ω is the angular frequency. 4
The transfer function between port 1 and port is [ ω ] ω ( ω ) e e1 = 1 j C R+ j L+ 1 j C (17) = 1 1 ω LC+ jωcr (18) It is a series tune circuit with the resonant frequency given by: ( ω C) ω L = 1, or ω = (19) LC 1 and the quality factor Q, given by: Q = ω L / R = 1/( ω CR) () Using (19) and (), the transfer function of (18) can re-written as: e e = 1/[1 ω / ω + j ω/ ω Q] (1) 1 which has a phase expression given by: φ = ω / ω Q arctan 1 ( ω / ω ) () Let y = ω/ω o, () can be re-written as: Replacing y by ω/ω o yields: ( y ) ( y ) y φ = arctan y/ Q 1 (3) φ = arc cot Q 1 (4) φ = π + arctanq y 1 y (6) ( ) [ ] ( ) arctan Q ( ) φ = π + ω ω ωω ( ) arctan Q ( )( ) ( ) ( ) (7) (8) φ = π + ω ω ω+ ω ωω φ = π + arctan{[ Qdω ω + dω /[( ω + dω) ω ]} (9) where dω = (ω - ω o ) has been used. Assue dω is very sall copared with ω o, then (9) becoes: ( ) Qd ( ) arctan[ Qd ] φ = π + arctan[ ω( ω )/( ω ω )] (3) φ = π + ω ω (31) If dω is sufficiently sall, the arguent of the arctan is sall and therefore close to the arctan value in radians. A good approxiation can be obtained: ( ) [ Qd ] φ = π + ω ω (3) It ust be noted that if Q has a large value (high Q), these approxiations ay becoe invalid for quite oderate values of deviation (dω). Thus the value of Q ust be kept low for good approxiation for (3). However, for good sensitivity, Q should be high. So a coproise is needed in the design is needed. 5
Quadrature Detector Consider the quadrature detector shown in Fig. 1. The FM signal fro the output of the phase shift network and the original FM signal are fed to the of the phase detector. The phase detector is siply a ixer with a bandpass filter. The phase detection can be drawn as shown in Fig.. Figure 3 As the frequency varies, both detector inputs vary together in frequency, but the one fro the output of the phase shifter shifts in phase relative to the other, and such relative phase shift is proportional to the frequency deviation of the FM signal, as explain as follows. Let input signal #1 be the FM signal and input signal # be the output signal fro the phase shifter, i.e., Input #1: sin + ( ) Vc ωct D ω sinωt (33) V 'sin c ωct + D ω sinωt + π + Qdω ω (34) Input #: ( ) where V c is the aplitude odified by the phase shifter. The signal at the output of the ixer is: + ( ) + + + ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) V ' sin c ωct D ω sinωt π Qdω ω Vcsin ωct D ω sinωt = V ' cvc cos π + Qdωω V ' cvc cos ωct + D ω sinωt + π+ Qdωω = V ' cvc sin Qdωω V ' cvc cos ωct + D ω sinωt + π+ Qdωω (36) where the nd ter is at a higher frequency and so will be reoved by the filter. If Qdω ω is sall, the 1 st ter of (36) can be re-written as: (( V ' c V c Q) dω ω ) (37) which will appear at the filter output. In (37), ω is the instantaneous frequency of the odulated signal given by (5), i.e., ω = ω + Dcosω t c Recalling dω = ω ω and so = ω + Dcosω t ω (38) c Substituting (38) into (37) and reoving the DC ter yield the signal at phase detector output. ( VV' QD) ω cos( ω t) (39) c c where has the sae for as the odulating signal in (3). A inor coplication is that ost phase detectors produce their outputs for 9 degrees phase difference between the input signals. This is the required condition when the FM signal is at its centre frequency, so an additional constant 9 degree phase shift is added to one of the paths. When unodulated, the two inputs to the phase detector are at 9 degrees apart, or in quadrature; hence the nae of the detector. 6
This constant phase shift is usually added by eans of a siple inductor, as shown in the circuit diagra of the phase shifter. The output of the phase detector still contains a large coponent at twice the carrier frequency and the detector is usually followed by a filter that passes the baseband but not the carrier. Quadrature detectors are used extensively in doestic FM radios and in a lot of counications equipent. In this task, the sae odulator is used as in the Task B. The odulation is a sine wave so that the signal can be followed through the circuit. 6. Procedures 6.1 Task A --- Concept of FM Open the Discovery II by double clicking the icon on the desktop. Go to Telecounications/Telecounications Principles/Analogue Counications/Workboard: FM/Assignent: Generation of Frequency Modulation/ Practical: Concepts of Frequency Modulation/Perfor Practical to begin this task In this task, the hardware is configured as shown in Fig. 4. You have available an oscilloscope and a spectru analyzer. Figure 4 Set carrier level to half of the total scale. Use the oscilloscope or spectru analyzer to observe the signal when the anual frequency control is used to change the frequency. Observe the signal at point 16 and the output signal at point 4 when a dc voltage is applied to the oscillator. Observe the signal at point 4, and use the oscilloscope to easure the total frequency range of the oscillator. Use the spectru analyzer to verify the frequency range easured using the oscilloscope. Questions Q1. Is it easier to easure the frequency range on the oscilloscope or on the spectru analyzer? Q. Choose two voltages levels at the control input to the oscillator and easure the corresponding output frequencies, hence calculate the 'frequency slope' of the oscillator in kilohertz per volt. Q3. Can you see any aplitude variation over the frequency range? Should there be any? 7
6. Task B --- Generation of FM with VCO Go to Telecounications/ Telecounications Principles/ Analogue Counications/ Workboard: FM/ Assignent: Generation of Frequency Modulation/ Practical: Generation of FM with a VCO/Perfor Practical to begin this task In this task, the variable dc voltage used to control the VCO frequency is replaced by a sine wave oscillator. The sine wave generated is used to frequency-odulate the carrier. The hardware configuration is shown in Fig. 5: Figure 5 Set carrier level to about half of the total scale. Observe the signal at point 4 using the oscilloscope. Turn the odulation level up and down and observe the effects. Notice that the frequency is changing and where the output at point 4 has the axiu frequency. Observe the signal at point 3 and observe how the instantaneous frequency depends on the instantaneous value of the odulating signal. Use the spectru analyser to exaine the sidebands of the signal. Adjust the odulation level and observe that, at low deviation, only the sidebands at F c -F and F c +F are present. At higher deviation, i.e., with a larger odulation index, the higher-order sidebands appear. Questions Q1. By looking at the spectru of the odulated signal, can you estiate the frequency of the odulating signal? (Explain carefully how). Q. Would it be equally easy to estiate the bandwidth of the odulating signal fro the spectru if the odulating signal has a coplicated wavefor and so has any frequencies? Q3. As the odulation level varies, how constant are: (a) the carrier-frequency coponent of the odulated signal and (b) the aplitude of the odulated signal? 8
6.3 Task C --- Deodulation of FM using quadrature detectors This task shows how a quadrature detector works. Go to Telecounications/ Telecounications Principles/ Analogue Counications/ Workboard: FM/Assignent: Deodulation of Frequency Modulation signals/ Practical: Quadrature detector/ Perfor Practical to begin this task The hardware configuration of this task is in Fig. 6. Figure 6 Observe the signal at point 9 and observe the FM signal at different settings of odulation level. Note that the two signals at the inputs of the phase detector, i.e., at points 9 and 11. Set the odulation level to about half scale. Observe the signal at the phase detector output at point 1 and then after the post-detection filter at point 14. Questions Q1. Use the large oscilloscope to easure the phase shift between the two phase detector inputs when there is no odulation. Q. What frequencies ust the output filter: (a) pass or (b) reject? Q3. Would your answer to question (b) be altered if the phase coparator were iperfect in soe way? 9