What is Phase Modulation?
Phase modulation is described by the following equation:
Phase
modulated signal: Xpm(t) = Ac *cos[2pi fc +m(t)]
Where
m(t) is
the modulating signal. Kp is the modulating index. When digital information is
presented as fixed discrete phase offsets of the modulating signal such as 0,
pi/4, 3pi/4, … the phase modulation is referred to as phase shift keying and is
a form of digital modulation. When the modulating signal is continuous signal it is a Xpm(t) is a form of linear modulation. For example if the modulating signal is :
Modulating
signal: m(t) = kp sin[2 pi Fpm t + x(t)]
The modulating signal is referred to as a side band where
m(t) is itself a modulated signal, where Fpm
is
the sideband.
In the figure the carrier Fc is phase
modulated with m(t) with x(t) set to zero. The modulating signal is a sine
function. Since kp is equal to pi/2 when the sine() is equal to ‘1’ the phase
is advanced by 90 degrees, when sine() is equal to ‘-1’ the phase is retarded
by 90 degrees.
Figure 2 Time Domain Plot of Phase Modulated
Signal
Figure 2
shows a phase modulated signal where
- kp is pi/2 radians
-
m(t) = kp
* sin(2pi * Fpm * t), x(t) = 0
- Fc = 1MHz
- Fpm = 100KHz
Note how the modulated signal advances by 90deg when high
and retards by 90 when low.
Taking a 2^16 bin FFT of the real valued equation gives us
positive and negative frequency components. The spectrum of the modulated
signal shows frequency carrier, Fc, peaks at +/-1MHz and side band peaks at
multiple 100KHz away from Fc. Now that we have an understanding of the Linear
Phase Modulated signal, how do we get our information out of it?
Figure 3 Frequency
Domain Plot (65K point FFT) of real valued Phase Modulated Signal
Demodulating Linear PM signals
There are multiple methods of extracting the modulating
signal from a PM source. We will show one method using Hilbert transforms and
Arctangent calculation with complex math demodulation algorithm. To get the negative
frequency components of the complex valued waveform we remove the negative
image by preforming a Hilbert transform on the modulated waveform. Once the
complex valued waveform is obtained a frequency rotation or down conversion is
performed. A 3-dimensional plot shows the variation in time (z-axis) of the PM
signal with respect to the carrier. Both signals rotate around the IQ diagram
at a rate of 1MHz.
The carrier signal is used as our reference, the modulated
signal when compared to carrier we notice a difference in time, this is our phase
difference. The modulated signal is either advanced or retarded for positive or
negative phase modulating values.
Figure 4 Time Domain
3d Plot of PM signal and Carrier
We can remove the carrier by translating the modulated
signal to zero IF frequency. Doing this stops the carrier and modulation
waveforms from rotating on the IQ diagram. Fc is a DC component and equal to
zero. Since Fc is now zero after translation to zero frequency the 3d plot
would look like this:
Figure 5 Modulating
Signal m(t) translation to zero frequency (DC)
At this point the signal can be recovered by performing an
arctangent the inphase and quadrature components.
m(t) = arctan( (I/Q) * t )
Note: the signal recovered below contains
a slight phase offset with respect to the original signal.
Figure 6
Arctan of I/Q signal (Kp=pi/2)
