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QPSK Modulation Demystified__
This article sets out to explain how phase shift
modulation works using high school mathematics
(trigonometric identities).
Ever since the turn of the century, people have
realised the importance of the need to communicate.
Since the early days of electronics, as advances in
technology took place, the boundaries of both local
and global communication have been eroded, resulting
in a world that is smaller and hence more easily
accessible for the sharing of knowledge and
information. The pioneering work done by Bell and
Marconi have formed the cornerstone of the
information age that exist today and have paved the
way for the future of telecommunications.
Traditionally local communication was done over
wires as this presented a cost effective way of
ensuring reliable information transfer. For long
distance communications, transmission of information
over radio waves was needed and although it was
convenient from a hardware viewpoint, it raised
doubts over the corruption of the information and
was often dependent on high power transmitters to
overcome weather conditions, large buildings and
interference from other sources of electromagnetics.
The various modulation techniques offered different
solutions in terms of cost effectiveness and quality
of received signal but, until recently were still
largely analogue. Frequency modulation and phase
modulation presented a certain immunity to noise,
whilst amplitude modulation was simpler to
demodulate. However more recently since the advent
of low cost microcontrollers and the introduction of
domestic mobile telephones and satellite
communications, digital modulation has gained
popularity. With digital modulation techniques come
all the advantages that traditional microprocessor
circuits have over their analogue counterparts. Any
shortfalls in the communications link can be
eradicated using software. Information can now be
encrypted, error correction can ensure more
confidence in received data and the use of DSP can
reduce the limited bandwidth allocated to each
service.
As
with traditional analogue systems, digital
modulation can use amplitude, frequency or phase
modulation with different advantages. As frequency
and phase modulation techniques offer more immunity
to noise, they are the preferred scheme for the
majority of services in use today and will be
discussed in detail below.
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Digital Frequency Modulation__
A
simple variation from traditional analogue frequency
modulation (FM) can be implemented by applying a
digital signal to the modulation input. Thus the
output takes the form of a sinewave at two distinct
frequencies. To demodulate this waveform, it is a
simple matter of passing the signal through two
filters and translating the resultant back into
logic levels. Traditionally, this form of modulation
has been called Frequency Shift Keying (FSK)
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Digital Phase Modulation__
Spectrally, Digital Phase Modulation (or Phase Shift
Keying - PSK) is very similar to Frequency
Modulation. It involves changing the phase of the
transmitted waveform instead of the frequency, these
finite phase changes representing digital data. In
its simplest form, a phase modulated waveform can be
generated by using the digital data to switch
between two signals of equal frequency but opposing
phase. If the resultant waveform is multiplied by a
sinewave of equal frequency, two components are
generated: one cosine waveform of double the
received frequency and one frequency independent
term, whose amplitude is proportional to the cosine
of the phase shift. Thus filtering out the higher
frequency term yields the original modulating data
prior to transmission. Although, conceptually, this
is difficult to picture, the mathematical proof of
the above will be shown later.
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Quadrature Phase Shift Modulation__
Taking the above concept of PSK one stage further,
it can be supposed that the number of phase shifts
is not limited to only two states. The transmitted
“carrier” can undergo any number of phase changes
and by multiplying the received signal by a sinewave
of equal frequency will demodulate the phase shifts
into frequency independent voltage levels.
This is indeed the case in Quadrature Phase Shift
Keying (QPSK). With QPSK, the carrier undergoes four
changes in phase and can thus represent four binary
bits of data. While this may seem insignificant at
first glance, a modulation scheme has now been
supposed that enables a carrier to transmit four
bits of information instead of two, thus effectively
doubling the bandwidth of carrier.
The proof of how phase modulation, and hence QPSK,
is demodulated is shown below.
The proof begins by defining Euler’s relation, from
whence *all the trigonometric identities can be
derived* (if only I had known this in high
school)
Euler’s Relation states:
Now consider multiplying two sinewaves together,
thus
=
=
**(Equation
1)**
From Equation 1, it
can be seen that multiplying two sine waves together
(one sine being the incoming signal, one being the
local oscillator at the receiver mixer), results in
an output frequency
double that of the
input (at half the amplitude) superimposed on a dc
offset of half the input amplitude.
Similarly, multiplying
by
gives
**
**
x=
=
which gives an output
frequency ( )
double that of the input, with no dc offset.
It is now fair to make
the assumption that multiplying
by
any phase shifted sinewave (+),
yields a “demodulated” waveform with an output
frequency double that of the input frequency, whose
dc offset varies according to the phase shift,
.
To
prove this,
x
=
=
=
=
=
Thus the above proves the supposition that the phase
shift on a carrier can be demodulated into a varying
output voltage by multiplying the carrier with a
sinewave local oscillator and filtering out the high
frequency term. Unfortunately, the phase shift is
limited to two quadrants - a phase shift of 45
degrees cannot be distinguished from a phase shift
of 315 degrees (360 degrees – 45 degrees) Therefore,
to accurately decode phase shifts present in all
four quadrants, the input signal needs to be
multiplied by both sinusoidal and cosinusoidal
waveforms, the high frequency filtered out and the
data reconstructed. The proof of this, expanding on
the above mathematics, is shown below.
Thus,
x
=
=
=
=
An
LTspice^{®} simulation verifies the above theory.
The circuit can be downloaded here:
LTspice simulation of QPSK
Modulation
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FIG 1__
FIG 1 shows an LT Spice simulation of a simple
demodulator circuit. V2 is the incoming signal (that
is going to undergo a phase shift) and V1 and V3 are
the sine and cosine local oscillators respectively.
B1
and B2 are the LTspice way of doing a voltage
multiplication. B1 is an arbitrary voltage source
whose output is the value V(SIN) x V(IN), so the
voltage across R4 is the resultant output when these
two voltages have been multiplied together. B2 is
the same, but the resultant output is the
multiplication of V(COS) and V(IN).
All voltages are 1V amplitude at 1kHz, centred about
0V.
With no phase shift on VIN, the I and Q outputs are
shown below:
We
can see that the I_Out waveform is as expected when
two sinewaves are multiplied together – we get a
cosine output of half the amplitude but twice the
frequency (2kHz) on a dc offset equal to
For a phase shift of 0 degrees on the input
oscillator, this equates to an offset of
Looking at Q_Out, this is the result of multiplying
a sine (with 0 phase shift) by a cosine. We have a
waveform with twice the frequency, half the
amplitude and a dc offset equal to
If
we apply a phase shift to the input of 90 degrees,
we get the following output:
Here we see that I_Out has undergone a dc shift
equal to
And Q_Out has undergone a dc shift equal to
The above steps can be repeated for different phase
shifts on VIN to show that the offset voltage of
both the I and Q outputs shifts in proportion to the
phase shift. It can be shown that if the phase shift
is a multiple of 90 degrees, this results in a dc
offset voltage of 0.5V between the I and Q channels.
As we can have four 90 degree phase shifts in 360
degrees, this allows us to transmit 4 bits over a
single carrier, just by phase shifting the carrier.
Thus if the output voltages are low pass filtered
and fed into a dual ADC, the receiver can pick out
the dc offset and demodulate the data.
The above theory is perfectly acceptable and it
would appear that removing the data from the carrier
is a simple process of low pass filtering the output
of the mixer and reconstructing the four voltages
back into logic levels. In practice, getting a
receiver local oscillator exactly synchronised with
the incoming signal is not easy. If the local
oscillator varies in phase with respect to the
incoming signal, the dc offset will change, its
magnitude equal to the phase difference.
Therefore, it is not a simple matter of detecting
the dc offsets with comparators. A bit more
intelligence is needed.
Linear Technology's range of RF products and high
speed low power data converters are ideally suited
to perform such demodulation tasks.
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