📄 qpsk modulation demystified - maxim-dallas.htm
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<TD
style="FONT-WEIGHT: bold; COLOR: #a5a5a5">APPLICATION
NOTE 686</TD></TR>
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<H1>QPSK Modulation
Demystified</H1></TD></TR></TBODY></TABLE>
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<TD><I>Readers are presented with step by step
derivations showing the operation of <A
href="http://www.maxim-ic.com/glossary/index.cfm/Ac/V/ID/244/Tm/QPSK">QPSK</A>
modulation and demodulation. The move from
analog communication to digital has advanced the
use of QPSK. Euler's relation is used to assist
analysis of multiplication of sine and cosine
signals. A SPICE simulation is used to
illustrate QPSK modulation of a 1MHz sine wave.
A phasor diagram shows the impact of poor
synchronization with the local oscillator.
Digital processing is used to remove phase and
frequency errors.</I><BR><BR><!-- BEGIN: DB HTML -->
<P>Since the early days of electronics, as
advances in technology were taking place, the
boundaries of both local and global
communication began eroding, resulting in a
world that is smaller and hence more easily
accessible for the sharing of knowledge and
information. The pioneering work by Bell and
Marconi formed the cornerstone of the
information age that exists today and paved the
way for the future of telecommunications. </P>
<P>Traditionally, local communication was done
over wires, as this presented a cost-effective
way of ensuring a reliable transfer of
information. For long-distance communications,
transmission of information over radio waves was
needed. Although this was convenient from a
hardware standpoint, radio-waves transmission
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.</P>
<P>The various modulation techniques offered
different solutions in terms of
cost-effectiveness and quality of received
signals but until recently were still largely
analog. Frequency modulation and phase
modulation presented a certain immunity to
noise, whereas amplitude modulation was simpler
to demodulate. However, more recently with the
advent of low-cost microcontrollers and the
introduction of domestic mobile telephones and
satellite communications, digital modulation has
gained in popularity. With digital modulation
techniques come all the advantages that
traditional microprocessor circuits have over
their analog 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.
</P>
<P>As with traditional analog 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.</P>
<P><FONT
face="Arial, Helvetica, sans-serif"><B><FONT
size=+1>Digital Frequency
Modulation</FONT></B></FONT></P>
<P>A simple variation from traditional analog
frequency modulation (<A
href="http://www.maxim-ic.com/glossary/index.cfm/Ac/V/ID/130/Tm/FM">FM</A>)
can be implemented by applying a digital signal
to the modulation input. Thus, the output takes
the form of a sine wave 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).</P>
<P><FONT
face="Arial, Helvetica, sans-serif"><B><FONT
size=+1>Digital Phase
Modulation</FONT></B></FONT></P>
<P>Spectrally, digital phase modulation, or
phase-shift keying (<A
href="http://www.maxim-ic.com/glossary/index.cfm/Ac/V/ID/690/Tm/PSK">PSK</A>),
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 sine wave 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.This is difficult to picture
conceptually, but mathematical proof will be
shown later.</P>
<P><FONT
face="Arial, Helvetica, sans-serif"><B><FONT
size=+1>Quadraphase-Shift
Modulation</FONT></B></FONT></P>
<P>Taking the above concept of PSK a stage
further, it can be assumed 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 sine wave of equal
frequency, will demodulate the phase shifts into
frequency-independent voltage levels.</P>
<P>This is indeed the case in quadraphase-shift
keying (QPSK). With QPSK, the carrier undergoes
four changes in phase (four symbols) and can
thus represent 2 binary bits of data per symbol.
Although this may seem insignificant initially,
a modulation scheme has now been supposed that
enables a carrier to transmit 2 bits of
information instead of 1, thus effectively
doubling the bandwidth of the carrier. </P>
<P>The proof of how phase modulation, and hence
QPSK, is demodulated is shown below.</P>
<P>The proof begins by defining Euler's
relations, from which all the trigonometric
identities can be derived.</P>
<P>Euler's relations state the following:</P>
<P> <IMG height=63
src="QPSK Modulation Demystified - Maxim-Dallas.files/T126Eqn1.gif"
width=337></P>
<P>Now consider multiplying two sine waves
together, thus</P>
<P> <IMG height=67
src="QPSK Modulation Demystified - Maxim-Dallas.files/T126Eqn2.gif"
width=378></P>
<P> <IMG height=55
src="QPSK Modulation Demystified - Maxim-Dallas.files/T126Eqn3.gif"
width=284></P>
<P>From Equation 1, it can be seen that
multiplying two sine waves together (one sine
being the incoming signal, the other being the
local oscillator at the receiver mixer) results
in an output frequency <IMG height=37
src="QPSK Modulation Demystified - Maxim-Dallas.files/T126Eqn4.gif"
width=75 align=middle> double that of the input
(at half the amplitude) superimposed on a dc
offset of half the input amplitude. </P>
<P>Similarly, multiplying <IMG height=14
src="QPSK Modulation Demystified - Maxim-Dallas.files/T126Eqn5.gif"
width=37 align=absBottom> by <IMG height=13
src="QPSK Modulation Demystified - Maxim-Dallas.files/T126Eqn6.gif"
width=37 align=absBottom> gives </P>
<P> <IMG height=61
src="QPSK Modulation Demystified - Maxim-Dallas.files/T126Eqn7.gif"
width=191></P>
<P>which gives an output frequency <IMG
height=17
src="QPSK Modulation Demystified - Maxim-Dallas.files/T126Eqn8.gif"
width=63 align=absBottom> double that of the
input, with no dc offset. </P>
<P><FONT
face="Arial, Helvetica, sans-serif">I</FONT>t is
now fair to make the assumption that multiplying
<IMG height=14
src="QPSK Modulation Demystified - Maxim-Dallas.files/T126Eqn5.gif"
width=37 align=absBottom> by any phase-shifted
sine wave <IMG height=17
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