Fm Exercises

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1) An FM signal is given as:

( )

100 cos 2 100

( )

t c s t π f t m τ τd −∞ ⎡ ⎤ = _⎢ + _⎥ ⎣

∫

⎦ where m(t) is a

rectangular periodic pulse:

a) Sketch the instantaneous frequency of this signal as a function of time. b) Determine the peak frequency deviation.

2) An angle modulated waveform is given as s t

( )

=20 cos 10⎡_⎣ 6t+10 cos 500

(

t

)

⎤_{⎦ .} a) What is the instantaneous frequency of s t

( )

?

b) What is the approximate bandwidth of s t

( )

? c) If s t

( )

is FM, what is m t

( )

?

d) If s t

( )

is PM, what is m t

( )

? 3) A signal m t

( )

is as shown below:

In an experiment, we use FM and PM using this signal and the same carrier frequency.

a) Find a relation between k_p and k_f such that the “maximum phase” of the modulated signals in both cases are equal.

b) If k_p =k_f = , then what is the maximum instantaneous frequency in each 1 case? m(t) Tp t

( )

m t 1 -1 1 2 3 t

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4) A signal m t₁

( )

is as shown below:

and another signal m t₂

( )

is given as m t₂

( )

=sinc 2 10

(

× 4t

)

. The value of signals are in terms of Volts, and t is in terms of sec.’s.

a) If m t₁

( )

is frequency modulated on a carrier with frequency 106 Hz and a frequency deviation constant (kf) equal to 5 Hz/V, what is the maximum

instantaneous frequency of the modulated signal?

b) If m t1

( )

is phase modulated on a carrier with frequency 10

6

Hz and a phase deviation constant (kp) equal to 3 rad/V, what is the maximum instantaneous

frequency of the modulated signal? What is the minimum instantaneous frequency of the modulated signal?

c) If m t2

( )

is frequency modulated on a carrier with frequency 10

6

Hz and a frequency deviation constant (kf) equal to 103 Hz/V, what is the maximum

instantaneous frequency of the modulated signal? What is the bandwidth of the modulated signal?

(Note that m t2

( )

is not narrowband, therefore the bandwidth must be approximated as 2Δ +2W, where kf max

{

m t

( )

}

W

Δ = is the frequency deviation, and W is the bandwidth of the message signal.)

5) An angle modulated signal is formed as s t

( )

= Accos⎡⎣ωct+θ

( )

t ⎤⎦ . The signal is distorted through the channel, so the observation becomes

( ) ( )

dcos

(

c d

)

r t =s t +A ⎡_⎣ ω ω+ t⎤_{⎦ .}

a) Express the envelope of r t

( )

using phasor vector diagrams and corresponding magnitudes.

b) On the same vector diagram, obtain phase of r t

( )

.

c) What is the approximate instantaneous frequency of r t

( )

if Ac Ad? d) What is the approximate instantaneous frequency of r t

( )

if Ad Ac? 105

1 3 4 t

( )

1

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1. We have periodic rectangular message signal

( )

0 0 1, 0 2 1, 2 0 t T x t T t < < ⎧ = ⎨₋ ₋ _{< <} ⎩ where T is 0

the period.The signal is FM modulated A_ccos 2

(

π f t_c +φ

( )

t

)

, where

( )

t kp t x

( )

d

φ λ λ

−∞

=

∫

a. Take φ

( )

0 =0 and plot φ

( )

t between −T0 2< <t T0 2.

b. If the input is periodic, so is the FM wave, and the F.S. coefficients of the FM wave can be obtained by the equation:

( ) 0 0 0

1

j t n t n T

c

e

dt

T

φ − ω ⎡ ⎤ ⎣ ⎦

=

∫

_{, with the} F.S. representation:

( )

Re

( c 0) j n t c c n n

x t

A

c e

ω ω ∞ + =−∞

⎧

⎫

=

_⎨

_⎬

⎩

∑

⎭

. Use this relation to

show that the F.S. coefficients of the FM wave for the given signal is

2 2

1 sinc

sinc

2

j j n j n n

n

c

=

e

πβ

_⎢

⎡

⎛

_⎜

+

β

_⎟

⎞

e

π

+

_⎜

⎛

−

β

⎞

_⎟

e

− π

⎤

_⎥

⎝

⎠

⎝

⎠

⎣

⎦

, where 0 fT β = .

c. Roughly sketch the magnitude of the FM spectrum when β is a very large integer.

2) A frequency-sweep generator is a device that produces a sinusoidal output whose instantaneous frequency linearly increases from f₁ at t=0 to f₂ at t =T . Write the FM wave angle expression if this signal is fed to a modulator with kf = . 1 3) The general instantaneous angle expression for an angle modulation scheme can

be described as θ_c

( )

t =2π f t_c +φ

( )

t , and the instantaneous frequency is

( )

c

( )

d

f t t

dtθ

= . In this expression, we know phase and frequency modulation counterparts. However, we can define two more modulation techniques: • Phase integral modulation:

( )

t K d x t

( )

dt

φ = , and

• Phase acceleration modulation: f t

( )

f_c K t x

( )

λ λd −∞

= +

_∫

.

Describe the modulated signal with these definitions. Construct a table which shows φ

( )

t , f t

( )

, max value of φ

( )

t , and max value of f t

( )

for PM, FM, phase integral, and phase acceleration modulation.

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4) An angle modulated (PM or FM) wave with fm=10kHz, β =2.0, 100Ac = , and 30

c

f = kHz. Write an expression for the instantaneous frequency: f t

( )

. 5) When an FM signal (x t_c

( )

) passes through a system (H f

( )

), applying phase

linearity approxiation similar to the phase and group delay analysis, the system output can be approximated as:

( )

_{( )} cos

{

( )

_{( )}

}

c c _f _{f t} c _f _{f t}

y t ≈A H f ₌ ω t+φ t + )H f ₌ which means that

[

]

( ) cos ( ) with ( ) ( ) ( )

c c c y y

y t = A _⎣⎡ωt+φ t _⎦⎤ φ t =φ t +)H f t . Define the output instantaneous frequency as ( ) 1 ( )

2

y c y

f t f φ t

π

= + to obtain the output signal and its instantaneous frequency when the system is given as: H f

( )

= and 1

( )

(

)

(

)

3

1 c 3 c

H f =α f − f +α f − f

) over the positive frequency band.

6) Consider a bandpass signal:

( )

cos _c 0.2 cos _m sin _c

v t = ωt+ ω t ω t

a) Show that this is a combination of an AM and an FM signal. b) Sketch the phasor diagram, and indicate FM – AM portions.

7) A sinusoidal signal at 2kHz is FM modulated using a carrier, and the resulting frequency deviation is observed to be 5 kHz. What is the bandwidth occupied by the FM wave? If the amplitude of the message sinusoid is multiplied by a factor of 3 and its frequency is lowered to 1kHz, what is the new bandwidth of the FM wave?

8) Consider a narrowband FM signal approximately defined by:

( )

ccos 2

(

c

)

csin 2

(

c

) (

sin 2 m

)

s t ≈ A π f t −βA π f t πf t

a. Determine the envelope of this modulated signal. What is the ratio of the maximum to minimum value of this envelope? Plot this ratio versus β between 0 and 0.3 (the narrowband range).

b. Determine the average power of the narrowband FM signal, expressed as a percentage of the average power of the carrier wave. Plot this result versus β between 0 and 0.3 (the narrowband range).

c. By expanding the instantaneous angle θ_i

( )

t of the narrowband FM signal

( )

s t in the form of power series, and restricting the modulation index β to be less than 0.3 for narrowband, show that

( )

(

)

3 3

(

)

2 sin 2 sin 2 3 i t f tc f tm f tm β θ ≈ π +β π − π

Accordingly, obtain the power ratio of the third harmonic to fundamental harmonic component for β =0.3 in s t

( )

.

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9) The sinusoidal message m t

( )

=A_mcos 2

(

π f t_m

)

is applied to a phase modulator with phase sensitivity=kp. The carrier wave has amplitude Ac, and frequency fc. a. Determine the spectrum of the resulting PM signal assuming that

0.3 m p A k

β = < (narrowband assumption).

b. Construct a phasor diagram for this modulated signal, and compare it to that of a single tone FM signal.