Fm Exercises

t ⎡ ⎤ 1) An FM signal is given as: s ( t ) = 100 cos ⎢ 2π f c t + 100 ∫ m (τ ) dτ ⎥ where m(t) is a −∞ ⎣ ⎦ rectangular periodic pulse:

m(t)

t Tp

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 ⎡⎣106 t + 10 cos ( 500t ) ⎤⎦ .

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: m (t )

1 3 1

2

t

-1

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 = 1 , then what is the maximum instantaneous frequency in each case?

4) A signal m1 ( t ) is as shown below: m1 ( t )

105 1

3

4

t

and another signal m2 ( t ) is given as m2 ( t ) = sinc ( 2 ×104 t ) . The value of signals are in terms of Volts, and t is in terms of sec.’s. a) If m1 ( t ) 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 m1 ( t ) is phase modulated on a carrier with frequency 106 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 m2 ( t ) is frequency modulated on a carrier with frequency 106 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 m2 ( t ) is not narrowband, therefore the bandwidth must be approximated as 2Δ + 2W , where Δ =

{

k f max m ( t )

}

W and W is the bandwidth of the message signal.)

is the frequency deviation,

5) An angle modulated signal is formed as s ( t ) = Ac cos ⎡⎣ωc t + θ ( t ) ⎤⎦ . The signal is

distorted through the channel, r ( t ) = s ( t ) + Ad cos ⎡⎣(ωc + ωd ) t ⎤⎦ .

so

the

observation

becomes

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 ?

0 < t < T0 2 ⎧ 1, 1. We have periodic rectangular message signal x ( t ) = ⎨ where T0 is ⎩−1, − T0 2 < t < 0 the period.The signal is FM modulated Ac cos ( 2π f c t + φ ( t ) ) , where t

φ (t ) = k p ∫ x (λ ) dλ −∞

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:

cn =

1 j ⎡⎣φ ( t ) − nω0t ⎤⎦ e dt , with the T0 T∫0

⎧ ∞ j (ωc + nω0 )t ⎫ ⎬ . Use this relation to F.S. representation: xc ( t ) = Ac Re ⎨ ∑ cn e ⎩ n =−∞ ⎭ show that the F.S. coefficients of the FM wave for the given signal is

1 cn = e jπβ 2

⎡ ⎛n+β sinc ⎜ ⎢ ⎝ 2 ⎣

⎞ jπ n 2 ⎛n−β + sinc ⎜ ⎟e ⎠ ⎝ 2

⎞ − jπ n 2 ⎤ ⎟e ⎥ , where ⎠ ⎦

β = fT0 .

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 f 2 at t = T . Write the FM wave angle expression if this signal is fed to a modulator with k f = 1 . 3) The general instantaneous angle expression for an angle modulation scheme can be described as θ c ( t ) = 2π f c t + φ ( t ) , and the instantaneous frequency is

d θc ( t ) . In this expression, we know phase and frequency modulation dt counterparts. However, we can define two more modulation techniques: d • Phase integral modulation: φ ( t ) = K x ( t ) , and dt f (t ) =

t

•

Phase acceleration modulation: f ( t ) = f c + K ∫ x ( λ ) d λ . −∞

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.

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

f c = 30kHz . Write an expression for the instantaneous frequency: f ( t ) .

5) When an FM signal ( xc ( t ) ) 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:

yc ( t ) ≈ Ac H ( f ) f = f

(t )

{

cos ωc t + φ ( t ) + ) H ( f ) f = f

(t )

} which means that

yc (t ) = Ac cos ⎡⎣ωc t + φ y (t ) ⎤⎦ with φ y (t ) = φ (t ) + )H [ f (t ) ] . Define the output 1  φ y (t ) to obtain the output signal and 2π its instantaneous frequency when the system is given as: H ( f ) = 1 and

instantaneous frequency as f y (t ) = f c +

)H ( f ) = α1 ( f − f c ) + α 3 ( f − f c ) over the positive frequency band. 3

6) Consider a bandpass signal: v ( t ) = cos ωc t + 0.2 cos ωmt sin ωc 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: s ( t ) ≈ Ac cos ( 2π f c t ) − β Ac sin ( 2π f c t ) sin ( 2π f m 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 θi ( 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

θi ( t ) ≈ 2π f ct + β sin ( 2π f mt ) −

β3

sin 3 ( 2π f m t )

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

9) The sinusoidal message m ( t ) = Am cos ( 2π f mt ) is applied to a phase modulator

with phase sensitivity= k p . The carrier wave has amplitude Ac , and frequency f c . a. Determine the spectrum of the resulting PM signal assuming that β = Am k p < 0.3 (narrowband assumption). b. Construct a phasor diagram for this modulated signal, and compare it to that of a single tone FM signal.