Understanding Angle Modulation: FM and PM Signals, Lab Reports of Computer Science

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Part-I

Experiment 6:-Angle Modulation

1. Introduction

1.1 Objective

This experiment deals with the basic performance of Angle Modulation - Phase Modulation (P M) and Frequency Modulation (F M). The student will learn the basic differences between the linear modulation methods (AM DSB SSB). Upon completion of the experiment, the student will:

• Understand ANGLE modulation concept.
• Learn how to generate F M signal.
• Learn how to generate P M signal.
• Learn how to build F M demodulator.
• Get acquainted with Bessel Function.
• Understand the difference between the linear and nonlinear modulation.

1.1.1 Prelab Exercise

1.Find the maximum frequency deviation of the following signal; and verify your results in the laboratory. Carrier sinewave frequency 10.7 MHz, amplitude 1 V p-p with frequency deviation constant 10.7 kHz/V ,modulated by sinewave frequency 10 kHz amplitude 1 V p-p.

2. Explain what is Carson’s rule.
3. What is the difference between NBFM and wideband FM refer to the Spectral component of the two.
4. Print a graph with Matlab or other software of the following FM signal: wc = 15MHz, A=5, Am=1, wm=1KHz, Kf=7.5, t=0 to 12 seconds. Show a. Modulation frequency versus time. b. FM signal. c. Differentiated FM signal. d. Differentiated FM signal followed by LP F.

1.1.2 Necessary Background

To understand the properties of angle-modulated waveforms (F M and P M), you need a working knowledge of Fourier transform theory. We will also use the Bessel function, but will present the basic theory as it is needed. Finally, the actual systems for modulating and demodulating angle modulated waveforms require a knowledge of linear sys- tems,oscillators and phase-locked loops.

1.1.3 Background Theory

An angle modulated signal, also referred to as an exponentially modulated signal, has the form

Sm(t) = A cos[wt + φ(t)] = Re{A exp[jwt + jφ(t)]} (1) The instantaneous phase of Sm(t) is defined as φi(t) = wt + φ(t) and the instantaneous frequency of the modulated signal is defined as

wi(t) =

d dt

[wt + φ(t)] = w +

d(φ) dt

The functions φ(t) and d( dtφ )are referred to as the instantaneous phase and frequency deviations, respectively. The phase deviation of the carrier φ(t) is related to the baseband message signal s(t). Depending on the nature of the relationship between φ(t) and s(t) we have different forms of angle modulation. In phase modulation, the instantaneous phase deviation of the carrier is linearly proportional to the input message signal, that is,

φ(t) = kps(t) (3) where kp is the phase deviation constant (expressed in radians/volt or degrees/volt). For frequency modulated signals, the frequency deviation of the carrier is proportional to the message signal, that is,

d(φ) dt

= kf s(t) (4)

-0.

-0.

-0.

0

1

Fig - 2 - : Bessel Function

n\β 0 0.2 0.5 1 2 5 8 10 0 1.00 0.99 0.938 0.765 0.224 -0.178 0.172 -0. 1 0 0.1 0.242 0.440 0.577 -0.328 0.235 0. 2 0.005 0.031 0.115 0.353 0.047 -0.113 0. 3 0.02 0.129 0.365 -0.291 0. 4 0.002 0.034 0.391 -0.105 -0. 5 0.007 0.261 0.186 -0. 6 0.131 0.338 -0. 7 0.053 0.321 0. 8 0.018 0.223 0. 9 0.126 0. 10 0.061 0. 11 0.026 0. Table-1 Bessel function jn(β) Order 0 1 2 3 4 5 6 β for 1st zero 2.40 3.83 5.14 6.38 7.59 8.77 9. β for 2nd zero 5.52 7.02 8.42 9.76 11.06 12.34 13. β for 3rd zero 8.65 10.17 11.62 13.02 14.37 15.70 17. β for 4th zero 11.79 13.32 14.80 16.22 17.62 18.98 20. β for 5th zero 14.93 16.47 17.96 19.41 20.83 22.21 23. β for 6th zero 18.07 19.61 21.12 22.58 24.02 25.43 26. Table-2 Zeroes of Bessel function: Values for β when jn(β) = 0

1. Equation -7 indicates that the phase relationship between the sideband components is such that the odd-order lower sidebands are reversed in phase.
2. The number of significant spectral components is a function of β (see Table-1). When β ¿ 1 , only J 0 , and J 1 , are significant so that the spectrum will consist of carrier plus two sideband components, just like an AM spectrum with the exception of the phase reversal of the lower sideband component.
3. A large value of β implies a large bandwidth since there will be many significant sideband components.
4. Transmission bandwidth of 98% of power always occur after n = β + 1, we note it in table-1 with underline. 5.Carrier and sidebands null many times at special values of β see table-.

1.1.5 Spectrum of Frequency Modulated Signal

Since angle modulation is a nonlinear process, an exact description of the spectrum of an angle-modulated signal for an arbitrary message signal is more complicated than linear process. However if s(t) is sinusoidal, then the instantaneous phase deviation of the angle-modulated signal is sinusoidal and the spectrum can be relatively easy to obtained. If we

assume s(t) to be sinusoidal then s(t) = Am cos wmt (10) then the instantaneous phase deviation of the modulated signal is

φ(t) =

kf Am wm

sin wmt For F M (11)

φ(t) = kpAm cos wmt For P M (12) The modulated signal, for the FM case, is given by

Sm(t) = A cos(wt + β sin wt) (13) where the parameter β is called the modulation index defined as

β = kf Am wm

For FM (14)

β = kpAm For PM The parameter β is defined only for sinewave modulation and it represents the maximum phase deviation produced by the modulating signal. If we want to compute the spectrum of Sm(t) given in Equation 11, we can express Sm(t) as Sm(t) = Re{A exp(jwt) exp(jβ sin wmt)} (15) In the preceding expression, exp(jβ sin wmt) is periodic with a period Tm = (^) w^2 πm. Thus, we can represent it in a Fourier series of the form

exp(jβ sin wmt) =

X^ ∞

−∞

Cx(nfm) exp(j 2 πnf ) (16)

Where

Cx(nfm) =

wm 2 π

Z (^) wπ M − (^) wπM

exp(jβ sin wmt) exp(−jwmt)dt (17)

2 π

Z (^) π

−π

exp[j(β sin θ − nθ)]dθ = jn(β)

Where jn(β) known as Bessel functions. Combining Equations 14, 15 and 13, we can obtain the following ex- pression for the F M signal with tone modulation:

Sm(t) = A

X^ ∞

−∞

jn(β) cos[(w + nwm)t] (18)

The spectrum of Sm(t) is obtained from the preceding equation. An example is shown in Figure- 3 The spectrum of an F M signal has several important properties:

Amplitude

f c

β=0.

f c+ f m Amplitude

f c

β=2 f^ c+^ f^ m

f c+2 f m f c-3 f m

FM spectrum

Fig - 3 - : FM spectrum

1. The F M spectrum consists of a carrier component plus an infinite number of sideband components at fre- quencies f ± nfm (n = 1, 2 , 3 ...). But the number of significant sidebands depend primarily on the value of β. In comparison, the spectrum of an AM signal with tone modulation has only three spectral components (at frequencies f , f + fm, and f − fm ).
2. The relative amplitude of the spectral components of an F M signal depend on the values of jn(β). The relative amplitude of the carrier depends on j 0 (β) and its value depends on the modulating signal (unlike AM modulation where the amplitude of the carrier does not depend on the value of the modulating signal).

Where φ(t) = kf

Z (^) t

−∞

s(τ )dτ (25) For NBF M, the maximum value of |φ(t)| is much less than one (another definition for N BF M ) and hence we can write s(t) as

Sm(t) = A[cos φ(t) cos wt − sin φ(t) sin wt] (26) ≈ A cos wt − Aφ(t) sin wt Using the approximations cosφ = 1 and sinφ ≈ φ, when φ is very small. Equation-26 shows that a N BF M signal contains a carrier component and a quadrature carrier linearly modulated by (a function of) the baseband signal. Since s(t) is assumed to be bandlimited to fm therefore φ(t) is also bandlimited to fm,. Hence, the bandwidth of N BF M is 2 fm, and the N BF M signal has the same bandwidth as an AM signal.

1.1.7 Narrow Band F M Modulator

According to Equation-22, it is possible to generate N BF M using a system such as the one shown in Fig-4. The signal is integrated prior to modulation and a DSB modulator is used to generate the quadrature component of the N BF M signal. The carrier is added to the quadrature component to generate an approximation to a true N BF M signal. The output of the modulator can be approximated by

Sm(t) ≈ A cos[wt + φ(t)] (27)

Integrator DSB Modulator

90 Shift

NBFM

S(t) (^) φ (t) A^ φ(^ t)sinwt

A coswt

NBFM Modulator

Fig - 4 - : NBFM modulator

Voltage

Control

Oscillator

S(t) Acos[wt+ φ (t)]

Wideband FM modulator

Fig - 5 - : FM modulator

The approximation is good as long as the deviation ratio D = ∆fmf , is very small.

1.1.8 Wide Band F M Modulator

There are two basic methods for generating F M signals known as direct and indirect methods. The direct method makes use of a device called voltage controlled oscillator (V CO) whose oscillation frequency depends linearly on the modulation voltage. A system that can be used for generating a P M or F M signal is shown in Figure-5. The combination of message differentiation that drive a V CO produces a P M signal. The physical device that generates the F M signal is the V CO whose output frequency depends directly on the applied control voltage of the message signal. V CO 0 s are easily implemented up to microwave frequencies using the reflex klystron.. Integrated circuit V CO 0 s are also used at

lower frequencies. At low carrier frequencies it may be possible to generate an F M signal by varying the capacitance of a parallel resonant circuit. The main advantage of direct F M is that large frequency deviations are possible, for relatively wide range of modulating frequency. The main disadvantage of the method is the instability of the carrier frequency.

1.1.9 Demodulation of FM Signals

An F M demodulator is required to produce an output voltage that is linearly proportional to the input frequency. Circuits that produce such response are called discriminators. If the input to a discriminator is an F M signal, is

Sm(t) = A cos[wt + kf

Z (^) t

−∞

s(τ )dτ]

the discriminator output will be yd(t) = kdkf s(t) where kd is the discriminator constant. The characteristics of an ideal discriminator are shown in Figure-6.

Frequency

Differentiator

Envelope

Detector

S m (t) S' m (t) y^ d

Output Voltage

Linear range

Slope k d

Differentiator and BPF as FM Discriminator

Fig - 6 - : FM discriminator

An approximation to the ideal discriminator characteristics can be obtained by the use of a differentiator followed by an envelope detector (see Figure-6). If the input to the differentiator is Sm(t), then the output of the differentiator is S

0 m(t) =^ −A[w^ +^ kf^ s(t)] sin[wt^ +^ φ(t)]^ (28) With the exception of the phase deviation φ(t), The output of the differentiator is both amplitude and frequency modulated. Hence envelope detection can be used to recover the message signal. The baseband signal is recovered without any distortion if Max{kf s(t)} = 2π∆f < w, which is easily achieved in most practical systems.

1.2 Required Equipment

1.Spectrum Analyzer (SA) HP − 8590 L. 2.Oscilloscope HP − 54600 A. 3.Signal Generator (SG) HP − 8647 A.

4. Arbitrary Waveform Generator (AW G) HP − 33120 A. 5.Double Balanced Mixer Mini-Circuit ZP − 2.
5. Two Combiner/Splitter Mini-Circuit
6. 20 MHz low pass filter. 8.Envelope detector RC = 20μ sec

1.3 Experiment procedure

During this experiment you learn how to measure the F M modulation characteristics and Bessel function using spec- trum analyzer, in frequency domain.

1. Connect the Test and Measurement (T &M ) equipment as indicated in Fig.. 7 ).
2. Adjust the AW G as follow: DC volt, amplitude 530 mV.
3. Signal-generator HP − 8647 A ,frequency 10.7 MHz, amplitude 0 dBm. External DC F M modulation, frequency deviation 20 kHz.

Fig - 8 - : First Null

Amplitude

f c

Amplitude

Frequency

2∆ f fm

Measurement of fm and frequency deviation

Fig - 9 - :

5. In that state you tune the system to accurate frequency deviation ∆f = β ∗ fm = 2. 4 ∗ fm. now calculate ∆f print the results.
6. Repeat the above procedure for ∆f = 50, 100 kHz and compare your measurement to the specification of the signal generator HP − 8647 A. Another way to approximately measure ∆f with spectrum analyzer is to find carrier frequency, then measuring the sideband spacing using a sufficiently small IF filter and then the peak frequency deviation is measured by selecting an IF bandwidth wide enough to cover all major sidebands.
7. Set the IF bandwidth of the spectrum analyzer to 1 kHz and measure the modulation frequency.
8. Set the IF bandwidth of the spectrum analyzer to 100 kHz and measure the frequency deviation (see figure 9) print the results.

1.3.3 FM Spectrum and Bessel Function

FM spectrum based on properties of Bessel function. We start to verify F M spectrum according to Equations-7,8 and

1.Set the signal generator HP − 8647 A to frequency 10.7 MHz , amplitude 0 dBm, external AC F M modulation, FM-2 kHz.

2. Set the AW G to sinewave frequency 10 kHz, amplitude As necessary to proper external F M modulation (about 530 mv).
3. Set the spectrum analyzer to 10.7 MHz, span 200 kHz, bandwidth 1 kHz.
4. The spectrum of the signal look like AM-modulation, set marker on the carrier and two sidebands, print the results. what is the value of β? And what is the bandwidth of the FM signal? Is it narrow or wideband FM?
5. Change FM deviation to 50 kHz , record the amplitude of every spectral line, press MOD − OF F on Signal generator and measure the amplitude of the carrier without modulation.
6. Calculate β,and verify Equations-7,8 and 9 with carrier and sidebands what is the bandwidth of the FM signal? Is it narrow or wideband FM? How many sidebands contains 98% of signal energy?
7. Set the AW G to triangle wave frequency 10 kHz, amplitude As necessary to proper external F M modulation (about 530 mv).
8. Measure and record with spectrum analyzer the highest spectral component of the triangle signal (fm).
9. Connect the AWG to signal generator as indicated in Figure-7- and measure the bandwidth of the modulated signal,print the results and compare them to Carson’s rule

1.3.4 Narrow Band FM Modulator

In this part of the experiment, we generate N BF M signal, without the first stage- integrator, since our input signal will be the integral of the modulating signal.

1. Connect the Test and Measurement (T &M ) equipment according to Fig.-10.
2. Adjust the T &M equipment as follow: AW G LO- Sinewave frequency 10.7 MHz amplitude 7dbm. AW G R- Sinewave frequency 10 kHz amplitude -10dbm.(integral of the cosine input wave).

15.000,000 MHz

LPF 10.7 MHz

NBFM MODULATOR

HP-33120A

R

L

I

15.000,000 MHz

HP-33120A

90 Shift

NBFM

Spectrum analyzer HP-

Communication Test Set

Sinad 1. (^0 )

Fig - 10 - :

3. Set the spectrum analyzer to 10.7 MHz span 50kHz , watch the FM signal at spectrum analyzer, change the amplitude and frequency of the modulating frequency generator, which component of the FM signal changed? n\β 0 0.1 0.15 0. 0 0.00(Ref.) 0.00(Ref.) 0.00(Ref) 0.00(Ref.) 1 - ∞ -26.0(dB) -22.5(dB) -19.95(dB) Table-3 Small β logarithmic values for NBFM
4. Change the amplitude and frequency of the local oscillator , which component of the FM signal changed?
5. According to table-3 set the system to NBF M β = 0. 1 , calculate the frequency deviation and print the results.
6. Calculate the proper DC voltage that cause the same frequency deviation.
7. Verify your results by setting the AWG to the calculated DC voltage and measure on spectrum analyzer the frequency difference of the signal, with and without DC signal.

1.3.5 Power and Bandwidth of FM Signal

Set the spectrum to single sweep and make the measurement of the above signal as follow

1. Measure the power of each component of the FM above signal (signal with null carrier), measure the total bandwidth of the FM signal (signal with all the sidebands).
2. According to the criterion of 98% power calculate the power of the FM signal and the bandwidth of the signal.
3. what is the difference in percent between the measured and calculated power and bandwidth?

1.3.6 IF Filter as FM Discriminator

In this part of the experiment you demodulate FM signal using the linear region of the IF filter of the spectrum analyzer as frequency discriminator. You have to choose If bandwidth and video bandwidth wide enough to pass all sidebands of the signal, but with proper slope so the amplitude of the demodulated signal will be measurable.

1. Connect the AWG directly to spectrum analyzer as shown in figure-11.

15.000,000 MHz

FM Discriminator

HP-33120A Spectrum analyzer HP-

Fig - 11 - :

2. Set the AWG to frequency 10.7 MHz, amplitude 0 dBm, FM modulation frequency 1 kHz, deviation 4 kHz.
3. Set the spectrum analyzer to Center frequency 10.7 MHz, Span 100 kHz, Bandwidth as necessary to pass all the signal B = 2 ∗ (∆f + fm), Amplitude linear (why)?
4. Place the signal near the top of the screen and in the center of the screen.
5. Set the span to 0 kHz, and sweeptime to 20 ms, you see the time domain of modulation frequency near the top of the screen.