Frequency Modulation

Frequency Modulation (FM)
Contents
Slide 1 Frequency Modulation (FM)
Slide 2 FM Signal Definition (cont.)
Slide 3 Discrete-Time FM Modulator
Slide 4 Single Tone FM Modulation
Slide 5 Single Tone FM (cont.)
Slide 6 Narrow Band FM
Slide 7 Bandwidth of an FM Signal
Slide 8 Demod. by a Frequency Discriminator
Slide 9 FM Discriminator (cont.)
Slide 10 Discriminator Using Pre-Envelope
Slide 11 Discriminator Using Pre-Envelope (cont.)
Slide 12 Discriminator Using Complex Envelope
Slide 13 Phase-Locked Loop Demodulator
Slide 14 PLL Analysis

PLL Analysis (cont. 1)
Slide 16 PLL Analysis (cont. 2)
Slide 17 Linearized Model for PLL
Slide 18 Proof PLL is a Demod for FM
Slide 19 Comments on PLL Performance
Slide 20 FM PLL vs. Costas Loop Bandwidth
Slide 21 Laboratory Experiments for FM
Slide 21 Experiment 8.1 Spectrum of an FM
Signal
Slide 22 Experiment 8.1 FM Spectrum (cont. 1)
Slide 24 Experiment 8.2 Demodulation by a
Discriminator
Slide 25 Experiment 8.2 Discriminator (cont. 1)
Slide 26 Experiment 8.2 Discriminator (cont. 2)
Slide 27 Experiment 8.3 Demodulation by a
PLL
Slide 28 Experiment 8.3 PLL (cont.)
8-ii

Presented by : -
SSR

✬ ✩
✫8-1 ✪
FM was invented and commercialized after AM.
Its main advantage is that it is more resistant to
additive noise than AM.
Instantaneous Frequency
The instantaneous frequency of cosθ(t) is
Motivational Example Let θ(t) =
ωct. The instantaneous frequency of
.
FM Signal for Message m(t)
The instantaneous frequency of an FM wave with
carrier frequency ωc for a baseband message m(t) is
ω(t) = ωc + kωm(t)

✬ ✩
✫8-2 ✪
FM Signal Definition (cont.)
where kω is a positive constant called the
frequency sensitivity.
An oscillator whose frequency is controlled by
its input m(t) in this manner is called a voltage
controlled oscillator.
The angle of the FM signal, assuming the value
is 0 at t = 0, is
where
is the carrier phase deviation caused by m(t).
The FM signal generated by m(t) is
s(t) = Ac cos[ωct + θm(t)]

✬ ✩
✫8-3 ✪
Discrete-Time FM Modulator
A discrete-time approximation to the FM wave can
be obtained by replacing the integral by a sum. The
approximate phase angle is
where
The total carrier angle can be computed recursively
by the formula
θ(nT) = θ((n − 1)T) + ωcT + kωTm((n − 1)T)
The resulting FM signal sample is
s(nT) = Ac cosθ(nT)
Single Tone FM Modulation

✬ ✩
✫8-4 ✪
Let m(t) = Am cosωmt. Then
The modulation index is defined as β = kωAm =
peak frequency deviation

✬ ✩
✫8-5 ✪
ωm modulating frequency
Example: fc = 1 kHz, fm = 100 Hz, fs = 80
kHz, β = 5
Single Tone FM (cont.) It can be
shown that s(t) has the series exansion
-1
-0.5
0
0.5
1
0 500 1000 1500 2000
Time in Samples
-1
-0.5
0
0.5
1
0 500 1000 1500 2000
Time in Samples

✬ ✩
✫8-6 ✪
where Jn(β) is the n-th order Bessel function of the
first kind. These functions can be computed by the
series
Clearly, the spectrum of the FM signal is much more
complex than that of the AM signal.
• There are components at the infinite set of
frequencies {ωc + nωm; n = −∞,···,∞}
• The sinusoidal component at the carrier
frequency has amplitude J0(β) and can actually
become zero for some β.
Narrow Band FM Modulation
The case where |θm(t)| ≪ 1 for all t is called narrow
band FM. Using the approximations cosx ≃ 1 and

✬ ✩
✫8-7 ✪
sinx ≃ x for |x| ≪ 1, the FM signal can be
approximated as:
s(t) = Ac cos[ωct + θm(t)]
= Ac cosωctcosθm(t) − Ac sinωctsinθm(t)
≃ Ac cosωct − Acθm(t)sinωct
or in complex notation
This is similar to the AM signal except that the
discrete carrier component Ac cosωct is 90◦ out of
phase with the sinusoid Ac sinωct multiplying the
phase angle θm(t). The spectrum of narrow band
FM is similar to that of AM.
The Bandwidth of an FM Signal
The following formula, known as Carson’s rule is
often used as an estimate of the FM signal
bandwidth:

✬ ✩
✫8-8 ✪
BT = 2(∆f + fm) Hz
where ∆f is the peak frequency deviation and
fm is the maximum baseband message
frequency component.
Example
Commercial FM signals use a peak frequency
deviation of ∆f = 75 kHz and a maximum baseband
message frequency of fm = 15 kHz. Carson’s rule
estimates the FM signal bandwidth as BT = 2(75 +
15) = 180 kHz which is six times the 30 kHz
bandwidth that would be required for AM
modulation.
FM Demodulation by a Frequency
Discriminator
A frequency discriminator is a device that converts
a received FM signal into a voltage that is

✬ ✩
✫8-9 ✪
proportional to the instantaneous frequency of its
input without using a local oscillator and,
consequently, in a noncoherent manner.
An Elementary Discriminator
Elementary FM Discriminator

✬ ✩
✫8-10 ✪
(cont.)
• When the instantaneous frequency changes
slowly relative to the time-constants of the
filter, a quasi-static analysis can be used.
• In quasi-static operation the filter output has
the same instantaneous frequency as the input
but with an envelope that varies according to
the amplitude response of the filter at the
instantaneous frequency.
• The amplitude variations are then detected
with an envelope detector like the ones used
for AM demodulation.

8-11 ✪
✬An FM Discriminator Using the✩
Pre-Envelope
When θm(t) is small and band-limited so that
cosθm(t) and sinθm(t) are essentially band-limited
signals with cutoff frequencies less than ωc, the
pre-envelope of the FM signal is s+(t) = s(t) + jsˆ(t)
= Acej(ωct+θm(t))
The angle of the pre-envelope is ϕ(t) =
arctan[ˆs(t)/s(t)] = ωct + θm(t) The derivative
of the phase is
which is exactly the instantaneous frequency. This
can be approximated in discrete-time by using FIR
filters to form the derivatives and Hilbert transform.
Notice that the denominator is ✫the squared
envelope of the FM signal.

✫8- 12
✬Discriminator Using the Pre-Envelope✩
(cont.)
This formula can also be derived by observing
so
The bandwidth of an FM discriminator must be at
least as great as that of the received FM signal which
is usually much greater than that of the baseband
message. This limits the degree of noise reduction
that can be achieved by preceding the ✫discriminator
by a bandpass receive filter.

8-13 ✪
✬A Discriminator Using the Complex Envelope ✩
The complex envelope is s˜(t) = s+(t)e−jωct = sI(t) +
jsQ(t) = Acejθm(t)
The angle of the complex envelope is
ϕ˜(t) = arctan[sQ(t)/sI(t)] = θm(t)
The derivative of the phase is
Discrete-Time Discriminator Realization ✪

✬ ✩
✫8-15 ✪
PLL Analysis
The PLL input shown in the figure is the noisless
FM signal s(nT) = Ac cos[ωcnT + θm(nT)]
This input is passed through a Hilbert transform
filter to form the pre-envelope s+(nT) = s(nT) +
jsˆ(nT) = Acej[ωcnT+θm(nT)]
The pre-envelope is multiplied by the output of the
voltage controlled oscillator (VCO) block.
The input to the z−1 block is the phase of the
VCO one sample into the future which is φ((n +
1)T) = φ(nT) + ωcT + kvTy(nT)
Starting at n = 0 and iterating the equation, it follows
that

✬ ✩
✫8-16 ✪
φ(nT) = ωcnT + θ1(nT)
where
The VCO output is
v(nT) = e−jφ(nT) = e−j[ωcnT+θ1(nT)]

✬ ✩
✫8-17 ✪
The multiplier output is
p(nT) = Acej[θm(nT)−θ1(nT)]
The phase error can be computed as
This is shown in the figure as being computed by
the C library function atan2(y,x) which is a four
quadrant arctangent giving angles between −π and
π. The block consisting of the multiplier and arctan
function is called a phase detector.

✬ ✩
8-18 ✪
A less accurate, but computationally simpler,
estimate of the phase error when the error is small
is
ℑm{p(nT)} = sˆ(nT)cos[ωcnT + θ1(nT)]
− s(nT)sin[ωcnT + θ1(nT)]
= Ac sin[θm(nT) − θ1(nT)]
≃ Ac[θm(nT) − θ1(nT)]

✬ ✩
✫8-19 ✪
The phase detector output is applied to the
loop filter which has the transfer function
The accumulator portion of the loop filter which
has the output σ(nT) enables the loop to track carrier
frequency offsets with zero error. It will be shown
shortly that the output y(nT) of the loop filter is an
estimate of the transmitted message ✫m(nT).
Linearized Model for PLL
The PLL is a nonlinear system because of the
characteristics of the phase detector. If the
discontinuities in the arctangent are ignored, the
PLL can be represented by the linearized model
shown in the following figure.

✬ ✩
✫8-20 ✪
The transfer function for the linearized PLL is
Proof that the PLL is an FM Demodulator
At low frequencies, which corresponds to z ≃ 1,
L(z) can be approximated by

✬ ✩
✫8-21 ✪
Thus
and in the time-domain
Using the formula on slide 8-3 for θm gives
This last equation demonstrates that the PLL is an
FM demodulator under the appropriate conditions.

✫ 8- 22 ✪
✬Comments on PLL Performance✩
• The frequency response of the linearized loop
has the characteristics of a band-limited
differentiator.
• The loop parameters must be chosen to provide
a loop bandwidth that passes the desired
baseband message signal but is as small as
possible to suppress out-of-band noise.
• The PLL performs better than a frequency
discriminator when the FM signal is corrupted
by additive noise. The reason is that the
bandwidth of the frequency discriminator must
be large enough to pass the modulated FM
signal while the PLL bandwidth only has to be
large enough to pass the baseband message.
With wideband FM, the bandwidth of the
modulated signal can be significantly larger
than that of the baseband message.

✩
✫ 8- 23 ✪
Bandwidth of FM PLL vs. Costas Loop
The PLL described in this experiment is very
similar to the Costas loop presented in Chapter 6
for coherent demodulation of DSBSC-AM. However,
the bandwidth of the PLL used for FM
demodulation must be large enough to pass the
baseband message signal, while the Costas loop is
used to generate a stable carrier reference signal so
its bandwidth should be very small and just wide
enough to track carrier drifts and allow a
reasonable acquisition time.
✬Laboratory Experiments for
Frequency Modulation
Initialize the DSK as before and use a 16 kHz
sampling rate for these experiments.
Chapter 8, Experiment 1
Spectrum of an FM Signal

✬ ✩
✫ 8- 24 ✪
1. Set the signal generator to FM modulate an fc = 4
kHz sinusoidal carrier with an fm = 100 Hz sine
wave by doing the following steps:
(a) Make sure the signal type is set to a sine
wave.
(b) Press the blue “SHIFT” button and then the
“AM/FM” button.
(c) Set the carrier frequency by pressing the
“FREQ” button and setting the frequency to 4
kHz.
(d) Set the modulation frequency by pressing the
“RATE” button and setting it to 100 Hz.
Experiment 8.1
FM Spectrum (cont. 1.)
(e) Adjust the modulation index by pressing the
“SPAN” button and setting a value. The
displayed value is related to, but not, the
modulation index β.

✩
✫ 8- 25 ✪
2. Connect the FM output signal to the oscilloscope
and observe the resulting waveforms as you
vary the frequency deviation.
3. Use the FFT function of the oscilloscope to
observe the spectrum of the FM signal by
performing the following steps:
(a) Turn off the input channels to disable the
display of the time signals.
(b) Press “Math.”
(c) Under the oscilloscope display screen,
i. Set “Operator” to FFT.
ii. Set “Source 1” to your input channel.
iii. Set “Span” to 2.00 kHz.

✬ ✩
✫8-26 ✪
Experiment 8.1
FM Spectrum (cont. 2.) iv. Set
“Center” to 4.00 kHz.
v. Use the “Horizontal” knob at the top left of
the control knob section to set the “FFT
Resolution” to “763 mHz” (0.763 Hz) or
“381 mHz” (0.381 Hz).

✬ ✩
✫8-27 ✪
Note: You can turn off the FFT by pressing
“Math” again.
4. Watch the amplitude of the 4 kHz carrier
component on the scope as the modulation
index is increased from 0. Remember that this
component should be proportional to J0(β).
5. Increase the modulation index slowly from 0
until the carrier component becomes zero for
the first and second times and record the
displayed SPAN values. Compare these
displayed values with the theoretical values of
β for the first two zeros of J0(β).
Experiment 8.1 FM Spectrum (cont. 3)
You can generate values of the Bessel function
by using the series expansion given on Slide 8-
5 or with MATLAB.
6. Plot the theoretical power spectra for a
sinusoidally modulated FM signal with β = 2, 5,

✬ ✩
✫8-28 ✪
and 10. Compare them with the spectra
observed on the oscilloscope.
FM Demodulation Using a Frequency
Discriminator
• Write a C program that implements the
frequency discriminator described on Slide 8-
12. Assume that:
– the carrier frequency is 4 kHz,
– the baseband message is band limited with
a cutoff frequency of 500 Hz, – the sampling
rate is 16 kHz.
Experiment 8.2 Discriminator
Implementation (cont. 1)
Use REMEZ87.EXE, WINDOW.EXE, or MATLAB
to design the Hilbert transform and FIR

✬ ✩
✫8-29 ✪
differentiation filters. Use enough taps to
approximate the desired Hilbert transform
frequency response well from 1200 to 6800
Hz. Try a differentiator bandwidth extending
from 0 to 8000 Hz. WINDOW.EXE gives good
differentiator designs. (Be sure to match the
delays of your filters in your
implementation.)
• Synchronize the sample processing loop with
the transmit ready flag (XRDY) of McBSP1.
Read samples from the ADC, apply them to
your discriminator, and write the output
samples to the DAC.
Experiment 8.2 Discriminator
Implementation (cont. 2)
• Use the signal generator to create a
sinusoidally modulated FM signal as you did
for the FM spectrum measurement
experiments. Attach the signal generator to the

✬ ✩
✫8-30 ✪
DSK line input and observe your demodulated
signal on the oscilloscope to check that the
program is working.
• Modify your program to add Gaussian noise to
the input samples and observe the
discriminator output as you increase the noise
variance. Listen to the noisy output with the PC
speakers. Does the performance degrade
gracefully as the noise gets larger?
Using a Phase-Locked Loop for FM
Demodulation
Implement a PLL like the one shown on Slide 8-13
to demodulate a sinusoidally modulated FM signal
with the same parameters used previously in this
experiment. Let α = 1 and choose β to be a factor of
100 or more smaller than α.

✬ ✩
✫8-31 ✪
• Compute and plot the amplitude response of
the linearized loop using the equation on the
bottom of Slide 8-17 for different loop
parameters until you find a set that gives a
reasonable response.
• Theoretically compute and plot the time
response of the linearized loop to a unit step
input for your selected set of parameters by
iterating a difference equation corresponding
to the transfer function.

✬
✫8-32 ✪
Experiment 8.3 PLL Demodulator✩
(cont.)
• Write a C program to implement the PLL. Test
this demodulator by connecting an FM signal
from the signal generator to the DSK line input
and observing the DAC output on the
oscilloscope.
• See if your PLL will track carrier frequency
offsets by changing the carrier frequency on the
signal generator slowly and observing the
output. See how large an offset your loop will
track. Observe any differences in behavior when
you change the carrier frequency smoothly and
slowly or make step changes.
• Modify your program to add Gaussian noise to
the input samples and observe the demodulated
output as the noise variance increases. How
does the quality of the demodulated output
signal compare with that of the frequency
discriminator at the same SNR.

Frequency Modulation

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