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Chapter 3: Frequency Modulation (FM)
EE456 – Digital CommunicationsProfessor Ha Nguyen
September 2016
EE456 – Digital Communications 1
Chapter 3: Frequency Modulation (FM)
Angle Modulation
In AM signals the information content of message m(t) is embedded as amplitudevariation of the carrier.
Two other parameters of the carrier are frequency and phase. They can also bevaried in proportion to the message signal, which results in frequency-modulated
and phase-modulated signals.
Frequency modulation (FM) and phase modulation (PM) are closely related andcollectively known as angle modulation. In our study, we will mainly focus on FM.
EE456 – Digital Communications 2
Chapter 3: Frequency Modulation (FM)
Instantaneous Frequency
Consider a generalized sinusoidal signal c(t) = A cos θ(t), where θ(t) is thegeneralized angle and is a function of t.
Over the infinitesimal duration of ∆t between [t1, t2], draw a tangential line ofθ(t), which can be described by equation ωct+ θ0.
It is clear from the figure that, over the interval t1 < t < t2 one has:
c(t) = A cos θ(t) = A cos(ωct+ θ0), t1 < t < t2.
This means that, over the small interval ∆t, the angular frequency of c(t) is ωc,which is the slope of the tangential line of θ(t) over this small interval.
EE456 – Digital Communications 3
Chapter 3: Frequency Modulation (FM)
For a conventional sinusoid A cos(ωct+ θ0), the generalized angle is a straightline ωct+ θ0 and the angular frequency is fixed.
For a generalizes sinusoid, the angular frequency is not fixed but varies with time.At every time instant t, the instantaneous frequency is the slope of angle θ(t) attime t:
ωi(t) =dθ(t)
dt
The equivalent relationship between angle θ(t) and the instantaneous frequencyωi(t) is:
θ(t) =
∫ t
−∞
ωi(α)dα
EE456 – Digital Communications 4
Chapter 3: Frequency Modulation (FM)
Phase Modulation (PM) and Frequency Modulation (FM)
In PM, the angle θ(t) is varied linearly with the message signal m(t):
θ(t) = ωct+ kpm(t), (assuming θ0 = 0)
sPM(t) = A cos[ωct+ kpm(t)], (where kp is a constant)
The instantaneous angular frequency ωi(t) of the PM signal is
ωi(t) =dθ(t)
dt= ωc + kp
dm(t)
dt,
which varies linearly with the derivative of the message.
If the instantaneous angular frequency ωi(t) varies linearly with the message,then we have frequency-modulated (FM) signal:
ωi(t) = ωc + kfm(t), (where kf is a constant)
θ(t) =
∫ t
−∞
ωi(α)dα =
∫ t
−∞
[ωc + kfm(α)]dα = ωct+ kf
∫ t
−∞
m(α)dα
sFM(t) = A cos
[
ωct + kf
∫ t
−∞
m(α)dα
]
EE456 – Digital Communications 5
Chapter 3: Frequency Modulation (FM)
Relationship Between FM and PM
FM ( )s t
PM ( )s t
PM and FM are very much related. It is not possible to tell from the time waveform
whether a signal is FM or PM. This is because either m(t),dm(t)
dt, or
∫m(α)dα can
be treated as a message signal.
EE456 – Digital Communications 6
Chapter 3: Frequency Modulation (FM)
PM and FM Circuits (Analog)
Note: RFC stands for radio-frequency choke
EE456 – Digital Communications 7
Chapter 3: Frequency Modulation (FM)
Example 3
The figure below shows a message signal m(t) and its derivative. Suppose that theconstants kf and kp are 2π × 105 and 10π, respectively, and the carrier frequency fcis 100 MHz.
(a) Write an expression of the instantaneous frequency of the FM signal. Determinethe minimum and maximum values of the instantaneous frequency.
(b) Write an expression of the instantaneous frequency of the PM signal. Determinethe minimum and maximum values of the instantaneous frequency.
(c) Sketch the FM and PM signals and offer your comments.
EE456 – Digital Communications 8
Chapter 3: Frequency Modulation (FM)
Solution:
(a) For FM, we have:
fi(t) =ωi(t)
2π= fc +
kf
2πm(t) = 108 + 105m(t)
[fi(t)]min = 108 + 105[m(t)]min = 99.9 MHz
[fi(t)]max = 108 + 105[m(t)]max = 100.1 MHz
(b) For PM, we have:
fi(t) =ωi(t)
2π= fc +
kp
2πm(t) = 108 + 5m(t)
[fi(t)]min = 108 + 5[m(t)]min = 99.9 MHz
[fi(t)]max = 108 + 5[m(t)]max = 100.1 MHz
EE456 – Digital Communications 9
Chapter 3: Frequency Modulation (FM)
(c) Sketches of the FM and PM signals are shown below.
FM ( )s t
PM ( )s t
Observations:
Because m(t) increases and decreases linearly with time, the instantaneousfrequency of the FM signal increases linearly from 99.9 to 100.1 MHz over ahalf-cycle, and then decreases linearly from 100.1 MHz to 99.9 MHz over theremaining half-cycle.
Because m(t) switches back and forth from a value of −20, 000 to 20, 000, thecarrier frequency switches back and forth from 99.9 to 100.1 MHz everyhalf-cycle of m(t).
EE456 – Digital Communications 10
Chapter 3: Frequency Modulation (FM)
Comparison of AM, FM and PM Signals with the same massage m(t)
Can you tell which signals on the right are AM, FM and PM, respectively?
0 5 10−0.5
0
0.5
t
Messagem(t)
0 5 10−1
0
1
t
dm(t)
dt
0 5 10−0.5
0
0.5
1
t
∫t−∞m(α
)dα
0 5 10−2
0
2
t
0 5 10−2
0
2
t
0 5 10−2
0
2
tEE456 – Digital Communications 11
Chapter 3: Frequency Modulation (FM)
Comparison of AM, FM and PM Signals with the same massage m(t)
0 5 10−0.5
0
0.5
t
Messagem(t)
0 5 10−1
0
1
t
dm(t)
dt
0 5 10−0.5
0
0.5
1
t
∫t−∞m(α
)dα
0 5 10−2
0
2
t
s AM(t)
0 5 10−2
0
2
t
s PM(t)
0 5 10−2
0
2
t
s FM(t)
EE456 – Digital Communications 12
Chapter 3: Frequency Modulation (FM)
Comparison of AM, FM and PM Signals under the same amount of noiseCompared to AM, FM and PM signals are much less susceptible to additive noise andinterference. This is because of two reasons: (i) Additive noise/interference acts onamplitude, and (ii) the message is embedded in amplitude in AM, while is isembedded in frequency/phase in FM/PM.
0 1 2 3 4 5−0.5
0
0.5
t
Messagem(t)
0 1 2 3 4 5−1
0
1
t
dm(t)
dt
0 1 2 3 4 5−0.5
0
0.5
1
t
∫t−∞m(α
)dα
0 1 2 3 4 5−2
0
2
t
s AM(t)
0 1 2 3 4 5−2
0
2
ts P
M(t)
0 1 2 3 4 5−2
0
2
t
s FM(t)
EE456 – Digital Communications 13
Chapter 3: Frequency Modulation (FM)
Power and Bandwidth of Angle-Modulated Signals
Since the amplitude of either PM or FM signal is a constant A, the power of anangle-modulated (i.e., PM or FM) signal is always A2/2, regardless of the valueof kp, kf , and power of m(t).
Unlike AM, angle modulation is nonlinear and hence its spectrum/bandwidthanalysis is not as simple as for AM signals.
To determine the bandwidth of an FM signal, define
a(t) =
∫ t
−∞
m(α)dα
sFM(t) = Aej[ωct+kfa(t)] = Aejkfa(t)ejωct ⇒ sFM(t) = ℜ{sFM(t)}
Expanding the exponential ejkfa(t) in power series gives:
sFM(t) = A
[
1 + jkfa(t) −k2f
2!a2(t) + · · ·+ jn
knf
n!an(t) + · · ·
]
ejωct
sFM(t) = ℜ{sFM(t)}
= A
[
cos(ωct) − kfa(t) sin(ωct)−k2f
2!a2(t) cos(ωct) +
k3f
3!a3(t) sin(ωct) + · · ·
]
EE456 – Digital Communications 14
Chapter 3: Frequency Modulation (FM)
Observations:
The FM signal consists of an unmodulated carrier and variousamplitude-modulated terms, such as a(t) sin(ωct), a2(t) cos(ωct), a3(t) sin(ωct),etc.
Since a(t) is an integral of m(t), if M(f) is band-limited to [−B,B], then A(f)is also band-limited to [−B,B].
The spectrum of a2(t) is the spectrum of A(f) ∗A(f) (where ∗ is the integralconvolution operation) and is band-limited to [−2B, 2B]. Similarly, the spectrumof an(t) is band-limited to [−nB, nB].
The spectrum of sFM(t) consists of an unmodulated carrier, plus spectra of a(t),a2(t), . . . , an(t), . . . , centered at ωc.
Clearly, the bandwidth of sFM(t) is theoretically infinite!
For practical message signals, because n! increases much faster than |kfa(t)|n,
we haveknf an(t)
n!≈ 0 for large n. Hence most of the modulated-signal power
resides in a finite bandwidth.
Carson’s rule for Bandwidth Approximation of an FM Signal (captures 98% of total power):
BFM = 2(∆f +B) = 2B(β + 1)
where ∆f = kfmmax −mmin
2 · 2πis defined as the peak frequency deviation
β =∆f
Bis the deviation ratio
EE456 – Digital Communications 15
Chapter 3: Frequency Modulation (FM)
Spectral Analysis of Tone FM
When the message m(t) is a sinusoid, namely m(t) = Am cos(ωmt), and withthe initial condition a(−∞) = 0, one has
a(t) =Am
ωmsin(ωmt)
β =∆f
B=
Amkf
ωm
sFM(t) = Ae(jωct+jkfAm/ωm sin(ωmt))
= Ae(jωct+jβ sin(ωmt)) = Aejωct(
ejβ sin(ωmt))
Since ejβ sin(ωmt) is a periodic signal with period T = 2π/ωm, it can beexpanded by the exponential Fourier series:
ejβ sin(ωmt) =∞∑
n=−∞
Dnejnωmt
where Dn =ωm
2π
∫ π/wm
−π/ωm
ejβ sin(ωmt)e−jnωmtdt
=1
2π
∫ π
−πej(β sinx−nx)dx = Jn(β)
︸ ︷︷ ︸
nth-order Bessel function of the first kind
EE456 – Digital Communications 16
Chapter 3: Frequency Modulation (FM)
It then follows that
sFM(t) = A∞∑
n=−∞
Jn(β)ej(ωct+nωmt)
sFM(t) = A∞∑
n=−∞
Jn(β) cos((ωc + nωm)t)
Observations:
The tone-modulated FM signal has a carrier component and an infinite numberof sidebands of frequencies ωc ± ωm, ωc ± 2ωm,. . . ,ωc ± nωm. This is verydifferent from DSB-SC spectrum of tone-modulated AM signal!
The strength of the nth sideband at ωc + nωm is A2Jn(β), which quickly
decreases with n. In fact, there are only a finite number of significant sidebandspectral lines.
In general, Jn(β) is negligible for n > β + 1, hence the bandwidth oftone-modulated FM signal is approximated as:
BFM = 2(β + 1)fm = 2(∆f + B)
EE456 – Digital Communications 17
Chapter 3: Frequency Modulation (FM)
Plot of Jn(β)
0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2 3.6 4 4.4 4.8 5.2 5.6 6 6.4 6.8 7.2 7.6 8 8.4 8.8 9.2 9.6 10−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
β
Jn(β
)
J0(β)
J1(β)
J2(β)J3(β) J4(β) J5(β) J6(β)
Two important properties: J−n(β) = (−1)nJn(β)∞∑
n=−∞
J2n(β) = 1
EE456 – Digital Communications 18
Chapter 3: Frequency Modulation (FM)
Table of Jn(β)
EE456 – Digital Communications 19
Chapter 3: Frequency Modulation (FM)
Illustration of Tone FM Spectrum
FM()
/2
Sf
AFM()
/2
Sf
A FM()
/2
Sf
A FM()
/2
Sf
AFM()
/2
Sf
A
EE456 – Digital Communications 20
Chapter 3: Frequency Modulation (FM)
Example 4
The figure below shows a message signal m(t) and its derivative. Suppose that theconstant kf = 2π × 105.
(a) Since m(t) is periodic with a fundamental frequency f0 = 12×10−4
, it can be
represented as m(t) =∑
∞
k=−∞ake
j2πkf0t. Show that a0=0 and
ak =
{4
π2k2, k odd
0, k even
(b) Assume that the essential bandwidth of m(t) to be the frequency of its thirdharmonic, estimate the bandwidth of the FM signal when the modulating signal ism(t).
(c) Repeat Part (b) if the amplitude of m(t) is doubled (i.e., if m(t) is multiplied by2).
(d) Repeat Part (b) if m(t) is time-expanded by a factor of 2 (i.e., if the period ofm(t) is 4× 10−4).
EE456 – Digital Communications 21
Chapter 3: Frequency Modulation (FM)
Narrow-Band FM (NBFM)
sFM(t) = A
[
cos(ωct)− kfa(t) sin(ωct)−k2f
2!a2(t) cos(ωct) +
k3f
3!a3(t) sin(ωct) + · · ·
]
When kf is very small such that |kfa(t)| ≪ 1, then all higher order terms in theabove expression are negligible, except for the first two terms. We then have agood approximation of an FM signal:
sFM(t) ≈ A[cos(ωct) − kfa(t) sin(ωct)
](1)
The above approximation is a linear modulation similar to that of an AM signalwith the message signal being a(t).
Because the bandwidth of a(t) is the same as the bandwidth of m(t), which is BHz, the bandwidth of the narrowband FM signal in (1) is 2B Hz.
It is pointed out that the sideband spectrum for a NBFM signal has a phase shiftof π/2 with respect to the carrier, whereas the sideband spectrum of an AMsignal is in phase with the carrier.
The expression of the NBFM signal in (1) suggests a method of generating aNBFM signal by using a DSB-SC modulator (see Fig. 1-(a) on the next slide).
The output of the NBFM modulator in Fig. 1-(a) has some amplitude variations(distortion). Such distortion can be removed by using a hard-limiter and abandpass filter as shown in Fig. 1-(b).
The analysis of Fig. 1-(b) shall be explored in Assignment 2.
EE456 – Digital Communications 22
Chapter 3: Frequency Modulation (FM)
( )m t∫ ∑
2π
( )a t
cos( )c
A tω
sin( )c
A tω−
( )sin( )f cAk a t tω−
NBFM signal
( )cos[ ( )]c
A t t tω ϕ+
( )cos[ ( )]c
A t t tω ϕ+4
cos[ ( )]ct tω ϕπ
+
Figure 1: Generating a NBFM signal.
EE456 – Digital Communications 23
Chapter 3: Frequency Modulation (FM)
Demodulation of FM Signals
Signal at point b : sFM(t) = A cos[
ωct+ kf∫ t−∞
m(α)dα]
Signal at point c :
sFM(t) =d
dt
{
A cos
[
ωct+ kf
∫ t
−∞
m(α)dα
]}
= A[ωc + kfm(t)] sin
[
ωct+ kf
∫ t
−∞
m(α)dα − π
]
Signal at point d : A[ωc + kfm(t)]
Signal at point e : kfm(t)
EE456 – Digital Communications 24
Chapter 3: Frequency Modulation (FM)
A Practical (Continuous-Time) Differentiator
Recall that the frequency response of an ideal differentiator is H(f) = j2πf .
A differentiator can be approximated by a linear system whose frequency responsecontains a linear segment of a positive slope.
One simple device would be an RC high-pass filter. The RC frequency responseis simply
H(f) =j2πfRC
1 + j2πfRC≈ j2πfRC, if 2πfRC ≪ 1.
Thus, if the parameter RC is very small such that its product with the carrierfrequency ωcRC ≪ 1, the RC filter approximates a differentiator.
EE456 – Digital Communications 25
Chapter 3: Frequency Modulation (FM)
FCC FM Standards
EE456 – Digital Communications 26
Chapter 3: Frequency Modulation (FM)
FM Stations in Saskatoon
EE456 – Digital Communications 27
Chapter 3: Frequency Modulation (FM)
Stereo FM
EE456 – Digital Communications 28
Chapter 3: Frequency Modulation (FM)
Review of Discrete-Time Processing of Continuous-Time Signals
( )cH jω
ˆ( )jdH e ω
The frequency response of the discrete-time LTI system, Hd(ejw) is periodic with
period 2π. Over −π ≤ w ≤ π, it is simply a frequency-scaled version of Hc(ω):
Hd(ejw) = Hc (ωfs) , −π ≤ w ≤ π
where fs = 1T
is the sampling frequency.
EE456 – Digital Communications 29
Chapter 3: Frequency Modulation (FM)
Example: Discrete-Time Low-Pass Filter
( )cH jω
( )cH jω
ˆ( )jdH e ω
ˆ( )jdH e ω
ω
EE456 – Digital Communications 30
Chapter 3: Frequency Modulation (FM)
Discrete-Time Integrator
EE456 – Digital Communications 31
Chapter 3: Frequency Modulation (FM)
An illustration of the backward difference, forward difference, and trapezoid rule forapproximating the integral of a continuous-time signal using discrete-time processing.
EE456 – Digital Communications 32
Chapter 3: Frequency Modulation (FM)
Realization of discrete-time integrators: (a) A realization of the discrete-time integrator based on
the trapezoid rule. (b) A realization of the discrete-time integrator based on the backward
difference. (c) A rearrangement of (b) to produce the more traditional system block diagram of an
accumulator.
EE456 – Digital Communications 33
Chapter 3: Frequency Modulation (FM)
Freq. Responses: Ideal Integrator, Accumulator, Trapezoid-Rule Integrator
Ideal Integrator: Hideal(ejw) = 1
jw.
Accumulator: Hacc(z) =1
1−z−1, Hacc(ejw) = 1
1−e−jw .
Trapezoid-Rule Integrator: Htrap(z) = 0.5 1+z−1
1−z−1, Htrap(ejw) = 0.5 1+e−jw
1−e−jw .
−3 −2 −1 0 1 2 30
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
ω (radians/sample)
Magnituderespon
se
−π π−π/2 π/2
|Hideal(ejω)|
|Hacc(ejω)|
|Htrap(ejω)|
The accumulator works very well as a DT integrator, especially forsmall-bandwidth signals.
EE456 – Digital Communications 34
Chapter 3: Frequency Modulation (FM)
Discrete-Time Differentiator
ˆ( )jdH e ω
ˆ( )jdH e ω
ω
EE456 – Digital Communications 35
Chapter 3: Frequency Modulation (FM)
H(ω) =
{jω, |ω| ≤ Wc
0, otherwise⇒ Hd(e
jw) =
{j ωT, |ω| ≤ WcT
0, WcT < |ω| ≤ π
hd[n] =1
2π
∫ WcT
−WcTjω
Tejwndω =
WcT
πT
cos(WcTn)
n−
1
πT
sin(WcTn)
n2
The impulse response has infinite support ⇒ The discrete-time system is an IIR filter.For the special case of full-bandwidth, i.e., when WcT = π, the impulse response is
hd[n] =
{1T
(−1)n
n, n 6= 0
0, n = 0
The first few samples of the impulse response for the full-bandwidth differentiator areshown below.
EE456 – Digital Communications 36
Chapter 3: Frequency Modulation (FM)
An Approximate Discrete-Time Differentiator
By truncating the impulse response to n = −1, 0, 1, the differentiator consists ofthe three center coefficients. The output of such a differentiator is
y[n] =1
T(x[n+ 1]− x[n− 1])
The above system is non-causal. It can be made causal by introducing a delay of1 sample:
y[n] =1
T(x[n]− x[n− 2])
[ ]x n1z− 1
z−
+
−∑ [ ] [ ] [ 2]y n x n x n= − −
approximate a differentiator
with a delay of 1 sample
Ignoring the scaling factor 1T, the impulse response of the above approximate
differentiator is h[n] = δ[n]− δ[n− 2] .
EE456 – Digital Communications 37
Chapter 3: Frequency Modulation (FM)
The system function H(z) = 1− z2 has 2 zeros at 0 and π.
The frequency response is
H(ejw) = 1− z−2
∣∣∣∣z=ejw
= 1− e−j2w
= e−jw(ejw − e−jw) = e−jw︸ ︷︷ ︸
delay of 1 sample
(2j sin ω︸ ︷︷ ︸
≈ω for ω small
) ≈ 2jωe−jw
−3 −2 −1 0 1 2 30
0.5
1
1.5
2
2.5
3
3.5
ω (radians/sample)
Magnituderespon
se
−π π−π/2 π/2
|Happrox(ejω)|
|Hideal(ejω)|
The above length-3 FIR approximation to a differentiator works reasonably wellfor small-bandwidth signal, about |ω| ≤ 0.2π
EE456 – Digital Communications 38
Chapter 3: Frequency Modulation (FM)
Better Approximations of a Discrete-Time Differentiator
Use a Blackman window (Matlab command blackman) to approximate an idealdifferentiator as an FIR filter.
0 0.1 0.2 0.3 0.4 0.50
0.5
1
1.5
2
2.5
3
frequency (cycles/sample)
Frequen
cyrespon
se
N=3, 7, 11, 15, 19, 23, 27, 31
EE456 – Digital Communications 39
Chapter 3: Frequency Modulation (FM)
0 0.1 0.2 0.3 0.4 0.5−60
−50
−40
−30
−20
−10
0
10
frequency (cycles/sample)
Frequen
cyrespon
se(dB)
N=3, 7, 11, 15, 19, 23, 27, 31
EE456 – Digital Communications 40
Chapter 3: Frequency Modulation (FM)
Building FM Transmitter and Receiver in Lab # 3
Transmitter
+ FM[ ] cos( [ ])
cs n n nω θ= +
+∑
[ ] [ 1] 2 [ ]fn n k m nθ θ π= − + ⋅ ⋅ (cycles/sample)
cf
[ ]m n
fk
EE456 – Digital Communications 41
Chapter 3: Frequency Modulation (FM)
Receiver
cos[( ) ]c
nω ω+ ∆
sin[( ) ]c
nω ω+ ∆
cf f+ ∆
[ ]c
x n
[ ]s
x n
[ ]c
y n
[ ]s
y n
∑
1z
− 1z
−
1z
− 1z
−
' [ 1]c
y n −
' [ 1]s
y n −
'[ 1]nθ −
+
+
+
−
−
−
[ 1]s
y n −
[ 1]c
y n −
FM[ ]s n
EE456 – Digital Communications 42
Chapter 3: Frequency Modulation (FM)
Analysis of the FM Demodulator
sFM(t) = cos
[
ωct+ kf
∫ t
−∞
m(α)dα
]
= cos [ωct + θ(t)]
sFM[n] = cos [ωcnTs + θ(nTs)] = cos (ωcn+ θ[n])
yc[n] = cos(∆ωn− θ[n]); where ∆ω = 2π∆f
ys[n] = sin(∆ωn− θ[n]);
y′c[n− 1] ≈d
dtyc(t)
∣∣∣∣t=(n−1)Ts
, where yc(t) = cos(∆ωt− θ(t)), ∆ω =∆ω
Ts
= −(∆ω − θ′(t)) sin(∆ωt − θ(t))
∣∣∣∣t=(n−1)Ts
= −(∆ω − θ′[n− 1]) sin(∆ω(n− 1) − θ[n− 1])
Similarly,
y′s[n− 1] ≈ (∆ω − θ′[n− 1]) cos(∆ω(n− 1) − θ[n− 1])
Finally,
y′c[n− 1]ys[n− 1]− y′s[n− 1]yc[n− 1]
= (θ′[n− 1]−∆ω)[cos2(∆ω(n− 1)− θ[n− 1]) + sin2(∆ω(n− 1)− θ[n− 1])]
= (θ′[n− 1]−∆ω) = θ′[n− 1]−∆ω
In the above ∆ω is the DC offset due to error in the receiver’s local oscillator, whileθ′[n− 1] is proportional to m[n− 1].
EE456 – Digital Communications 43