Frequency Modulation Simulation Using Matlab

EXPERIMENT #4

FREQUENCY MODULATION

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Purpose: The objectives of this laboratory are: 1. To investigate frequency modulation characteristics in the frequency domain. 2. To implement a classical double-tuned FM demodulator and measure its characteristics. 3. To implement a modern PLL FM demodulator and measure its characteristics. 4. To investigate the effect of FM signal bandwidth on the detected signal-to-noise ratio.

Equipment List 1. PC with Matlab and Simulink

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Frequency Modulation FM results when the time derivative of the phase of the carrier is varied linearly with the message signal m(t). The frequency deviation is proportional to the derivative of the phase deviation. Thus, the instantaneous frequency of the output of the FM modulator is maximum when the message signal m(t) is maximum and minimum when m(t) is minimum.

Carson’s Rule: Carson’s Rule is used to determine the bandwidth of the FM wave. According to Carson’s Rule, the bandwidth is given by: BW = 2(β+1)fm

Hertz.

Laboratory Procedure Determining Constants: Before proceeding to perform the experiment, the following steps were performed: 1.

Calibrate the multiplier and determine the multiplier constant.

2.

Determine the VCO conversion constant Ko

3.

Set the VCO’s frequency for 5 kHz.

4.

Verify the outputs of the 1st order Low Pass Filter.

1. Multiplier Constant, Km With a 1V p-p sinusoidal voltage at both inputs of the multiplier, the output was observed and the multiplier constant was calculated to be 0.206. 2. VCO Conversion Constant, Ko An external voltage may control the output frequency of the VCO. The change in the output frequency per change in the dc input voltage was measured. Ko = ∆f / ∆v Ko = 1 / 0.5 = 2 kHz/sec/volt ∴ Ko = 4π103 = 12566 rad/sec/volt

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FM Transmission: The following schematic was implemented.

Figure 4 A (a) FM detector

Figure 4 A (b) FM Input signal

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Figure 4 A (c) PSD of message signal

Figure 4 A (d) Limter - Unmasked

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The output of the function generator is set to 1 kHz (modulation frequency fm = 1kHz) and no output level.

Figure 4 A (e)Band pass block parameters

Figure 4 A (f)Output of Limiter A 5kHz carrier “delta” function was observed on the signal analyzer. We increased the output level of the function generator by pressing the Delta Level key on the generator and selecting

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delta of 0.1 volt. Setting the Vpp to 0 volts and incrementing the Vpp by 0.1 V increments, we increased the level until a of β = 0.5 was achieved.

Figure 4 A (g)Output of 4k Band pass filter

Figure 4 A (h)Output of 8k Band pass filter

Figure 4 A (i)Block parameters of envelope detector

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Figure 4 A (j)Output of difference block

Figure 4 A (k)Output of Cheby filter β = Ko A./ (2πfm);

A = Vpp /2

The above procedure is repeated for β = 1 and β = 2. We observed that the carrier disappears at A = 1.05 V. Setting the frequency axis on the spectrum analyzer to a linear scale, the approximate bandwidth for different values of β was observed. The results are tabulated below:

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Figure 4 A (l) Received signal

Figure 4 A (m) Recovered signal

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Figure 4 A (n) Recovered signal

Figure 4 A (o) Input signal

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The output is read from the frequency counter. By varying the frequency, and observing the output, we see that the discriminator output follows the following characteristic. The signal is also monitored on the oscilloscope. PLL Detection In this part of the experiment, we detect the Fm signal using a PLL. Using VCO #1, we made an FM signal by setting the center frequency of the VCO in open loop to 5 kHz and putting the function generator’s signal (1kHz 0 Vpp) and putting the function generator’s signal (1kHz 0Vpp) into the input of the VCO #1.

figure 4 B (a) FM PLL The PLL circuit is built according to the schematic shown below. The LPF with 1kHz cutoff frequency is to remove high frequency components from the detected signal. It is not a part of the PLL. In the open loop, the VCO#2 is set for 5kHz.

figure 4 B (b) Block parameters of carrier VCO – 5kHz

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VCO#1 is part of the FM transmitter, VCO #2 is part of the PLL detector. With the function generator putting out no signal, the VCO #1’s output frequency is varied. We note the dc voltages for the corresponding input frequencies. We observe that the discriminator curve generated using the PLL is more linear than the Double Tuned Detector.

figure 4 B (c) PSD of message signal

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figure 4 B (d) Input signal

figure 4 B (e) vco output

figure 4 B (f) Limiter

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figure 4 B (g)VCO spectrum

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figure 4 B (h) Recovered Signal spectrum

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Appendix

Pre – Lab

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Prelab Questions Consider a carrier signal (cos ω ct) being frequency modulated by a sinusoidal signal (A cos ω mt). The result can be expressed as a series of Bessel functions:

S(t) =

∞

∑J

n = -∞

n

( β ) cos[ (ωc + nωm )t ]

where Jn(β) are Bessel functions of nth order β = 2πkoA / ω m = koA / fm = modulation index ko = frequency deviation constant 1. for β = .5, 2, and 2, sketch the positive frequency domain representation (magnitude only). For β = 0.5 J0(β) = 0.9385 J1(β) = 0.2423 J2(β) = 0.0306 J3(β) = 0.0026

M a g n it u d e s p e c t ru m

o f s ( t ) fo r b e t a = 0 . 5

1

0 .9

0 .8

0 .7

M a g n it u d e

0 .6

0 .5

0 .4

0 .3

0 .2

0 .1

0 -3

-2

-1

0

D e vi a t i o n o f fr e q u e n c y fr o m

1 f

2

in m u lt ip le o f f c

m

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3

For β = 1 J0(β) = 0.7652 J1(β) = 0.4401 J2(β) = 0.1149 J3(β) = 0.0196 J4(β) = 0.0025

M a g n it u d e s p e c t ru m

o f s ( t ) fo r b e t a = 1

0 .8

0 .7

0 .6

M a g n itu d e

0 .5

0 .4

0 .3

0 .2

0 .1

0 -4

-3

-2

-1

0

1

D e v i a t i o n o f fr e q u e n c y f r o m

f

2

c

For β = 2 J0(β) = 0.2239 J1(β) = 0.5767 J2(β) = 0.3528 J3(β) = 0.1289 J4(β) = 0.0340 J5(β) = 0.0070 J6(β) = 0.0012

M a g n it u d e s p e c t ru m

3

4

in m u lt ip le o f f m

o f s ( t ) fo r b e t a = 2

0 .7

0 .6

M a g n it u d e

0 .5

0 .4

0 .3

0 .2

0 .1

0 -6

-4

-2

0

D e v i a t i o n o f fr e q u e n c y f r o m

2 f

4

in m u lt ip le o f f c

m

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2. If fc = 5,000 Hz, fm = 1,000 Hz and ko = 2,000 Hz/V, find RMS value of the modulating signal for β = 0.5, 1, and 2

P=

β+1

J ∑ β

n =−(

2 n −1)

2 ( β) A 2

2πk Ak o = o = A ⋅ 2000 = 2 A = modulation index β = For β = 0.5 ω f 1000 m m Then A = β/2 For β = 0.5 A = β/2 = 0.25 P = (0.24232 + 0.93852 + 0.24232 )*A2 /2 = 0.0312 J RMS value = P1/2 = 0.02441/2 = 0.1766 V For β = 1 A = β/2 = 0.5 P = ( 0.76522 + 0.44012 + 0.44012 + 0.11492 + 0.11492 )*A2 /2 = 0.1249 J RMS value = P1/2 = 0.12491/2 = 0.3534 V For β = 2 A = β/2 = 1 P = ( 0.22392 + 0.57672 + 0.57672 + 0.35282 + 0.35282 + 0.12892 + 0.12892 )*A2 /2 = 0.4987 J RMS value = P1/2 = 0.49871/2 = 0.7062 3. Using Carson’s rule, what is the approximate bandwidth occupied by s(t) for β = 1 and 2 For β = 1 Bandwidth = 2(β + 1)fm = 2(1 + 1)*1000 = 4000 Hz For β = 2 Bandwidth = 2(β + 1)fm = 2(2 + 1)*1000 = 6000 Hz

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