Transcript
TEL312 Electronic Communications Fundamentals
Angle Modulation – Basic Concepts Reference: Tomasi, Chapters 7 - 8
TEL312 Electronic Communications Fundamentals General Angle-Modulated Signal
s (t ) = Vc cos(2πf c t + θ (t ) )
θ (t ) = Phase Deviation in radians
If the modulating signal is proportional to the phase deviation, then we have phase modulation (PM):
θ (t ) = k p m(t )
where k p is the phase deviation sensitivity of the modulator, in radians/volt.
If the modulating signal is proportional to the angular frequency deviation, then we have frequency modulation (FM):
f (t ) =
dθ (t ) = 2π k f m(t ) = instantaneous frequency dt
where k f is the frequency deviation sensitivity of the modulator, in Hz/volt.
1
TEL312 Electronic Communications Fundamentals
Frequency Modulation Frequency modulation implies that
dθ dt
is proportional to the modulating signal.
dθ = 2π k m(t ) f dt Thus, in FM the instantaneous frequency varies linearly with the message signal.
dθ (t ) f (t ) = f c + 1 2π dt = f c + 1 (2πk f m(t ) ) 2π = f c + k f m(t )
k f is the deviation sensitivity of the FM modulator and has units of Hz per volt
TEL312 Electronic Communications Fundamentals The phase deviation θ(t) of FM signal is given by t
t
t
0
0
θ (t ) = ∫ dθ = ∫ 2π k f m(τ )dτ =2π k f ∫ m(τ )dτ 0
dt
Therefore, an FM signal can be expressed as:
s(t ) = Vc cos(2πf c t + θ (t ) )
t ⎛ ⎞ = Vc cos⎜⎜ 2πf c t + 2πk f ∫ m(τ )dτ ⎟⎟ 0 ⎝ ⎠
where Ec is the amplitude in volts, f c is the carrier frequency in Hz k f is the deviation sensitivity of the FM modulator in Hz/volt m(t) is the message signal in volts
2
TEL312 Electronic Communications Fundamentals Frequency deviation Consider a sinusoidal modulating information signal given by
m(t ) = Am cos(2πf mt ) The instantaneous frequency of the resulting FM signal equals
f (t ) = f c + k f m(t ) = f c + k f Am cos(2πf mt ) The maximum change in instantaneous frequency f(t) from the carrier frequency fc, is known as frequency deviation Δf. In the case of m(t ) = Am cos( 2πf m t ) , the peak frequency deviation is
Δf = k f Am
The frequency deviation is a useful parameter for determining the bandwidth of the FM-signals
TEL312 Electronic Communications Fundamentals Phase deviation of FM signal In the case where the message signal is a sinusoid, the phase deviation is:
t
t
θ (t ) = 2π k f ∫ m(τ )dτ = 2π k f ∫ Am cos(2πf mτ )dτ 0
0 t
= 2π k f Am ∫ cos(2πf mτ )dτ = 2π k f Am 0
=
k f Am fm
sin (2πf mt ) =
sin (2πf mt ) 2πf m
Δf sin (2πf mt ) = β sin (2πf mt ) fm
The ratio of the frequency deviation ∆f to the message frequency fm is called the modulation index of the FM signal. We denote it by: Δf = peak frequency deviation in Hz f m = message frequency in Hz
β=
Δf f m
β is unitless. For FM, it represents the depth of modulation achieved for a given modulating signal frequency.
3
TEL312 Electronic Communications Fundamentals
TEL312 Electronic Communications Fundamentals The FM signal is given by
s (t ) = Ac cos(2πf c t + θ (t ) )
In the case where the message signal is a sinusoid, the phase deviation is:
θ (t ) = β sin (2πf mt )
The resulting FM signal is:
s (t ) = Ac cos(2πf c t + θ (t ) )
= Ac cos(2πf c t + β sin (2πf mt ))
Depending on the value of the modulation index β, we may distinguish two cases of frequency modulation: -Narrow-Band FM -Wide-Band FM.
4
TEL312 Electronic Communications Fundamentals
Narrow-band Frequency Modulation For small values of β, cos(β sin(2π fm t)) ~ 1 sin(β sin(2π fm t)) ~ β sin(2π fm t) Thus the expression for FM signal can be expanded as: x ( t ) = Ac cos( 2π f c t ) − Ac sin( 2π f c t ) β sin( 2π f m t ) because cos( A + B ) = cos A cos B − sin A sin B which may be written as follows
{
}
x ( t ) = Ac cos( 2π f c t ) + 1 β Ac cos[ 2π ( f c + f m ) t ] − cos[ 2π ( f c − f m ) t ] 2 because
sin A sin B =
1 [cos( A − B ) − cos( A + B )] 2
TEL312 Electronic Communications Fundamentals
Amplitude spectrum (single-sided plot) Ac
1 βAc 2
1 βAc 2 f
fc -fm
fc
fc +fm
Bandwidth=2fm
5
TEL312 Electronic Communications Fundamentals
Wide-band Frequency Modulation The general expression for FM signal can be analyzed to give the spectral components of wide-band FM signal. In order to compute the spectrum of an angle-modulated signal with a sinusoidal message signal, let
θ (t) =
Δf fm
s in (2π f m t )
The corresponding FM signal
x(t ) = Ac cos(2πf ct + β sin(2πf mt )) and may alternatively be written as
x(t ) = Ac Re(e
jω c t jβ sin 2πf mt e )
where Re(x) denotes the real part of x. The parameter β is known as the modulation index and is the maximum value of phase deviation of the FM signal.
6
