View source for Third harmonic sampling and QSD - HW9

== Max Woesner ==
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=== Homework #9 - Third harmonic sampling and QSD ===

This page describes how third harmonic sampling and a QSD (quadrature sampling detector) work.<br><br>
<b>References:</b><br>
[http://groups.yahoo.com/group/softrock40 SoftRock-40 SDR interest group]<br>
[http://www.flex-radio.com/News.aspx?topic=publications FlexRadio SDR-1000 QEX articles]<br>

====Third Harmonic Sampling====
Third harmonic sampling is used in some applications, such as the SoftRock-40 software defined radio, to divide the local oscillator frequency by three. <br>  

Consider the simple representation of part of a software defined radio below.<br><br>
[[Image:simpleradio2.jpg]]<br>

The local oscillator can be defined as <math> sgn(v_{LO}(t)) \!</math>, giving us a square wave.  Note that <math> sgn(v_{LO}(t)) = \begin{cases} 1,  & \mbox{ } v_{LO}(t)>0 \\ 0, & \mbox{ } v_{LO}(t)=0 \\ -1, & \mbox{ } v_{LO}(t)<0 \end{cases} \!</math><br>

The output voltage of the radio, or <math> v_O \!</math> is <math> v_O = v_{RF}(t) \cdot sgn(v_{LO}(t)) \!</math><br><br>
Now <math> sgn(v_{LO}(t)) \!</math> can be written as the series <math> cos(\omega _{LO}t) + \frac{1}{2}cos(2 \omega _{LO}t) + \frac{1}{3}cos(3 \omega _{LO}t) + ... \!</math><br><br>
We are interested in the third harmonic term, or <math> \frac{1}{3}cos(3 \omega _{LO}t) \!</math><br><br>
This frequency is three times that of <math> cos(\omega _{LO}t) \!</math>.  Note that the period decreases by a factor of three.<br><br>
[[Image:thirdharmoniccosine.jpg]]<br><br>
We can use the third harmonic to drive the mixer in the radio at a frequency three time that of the local oscillator.<br>  

The advantage of using the third harmonic rather than using a higher speed oscillator is that it costs less to buy both a lower frequency oscillator as well as the parts that are driven by the oscillator. <br>  

The disadvantage is that there is some loss in the signal when the third harmonic is used, specifically <math> 20 log_{10}(1/3) \ db = -9.54 \ db \!</math><br>

We can now apply the same concept to a radio with parallel doubly balanced mixers.  The radio could be represented such as in the diagram below.<br><br>
[[Image:Simpleradioouble2.jpg]]<br><br>
We could use a mixer like this one.<br><br>
[[Image:dbmixer.jpg]]<br><br>

====Quadrature Sampling Detector====
If you read the above section about third harmonic sampling, then you might be wondering how we input both <math> sine \!</math> and <math> cosine \!</math> into the parallel mixers.  The answer is that we can use a quadrature sampling detector, or QSD.<br>

The local oscillator <math> (sine) \!</math> is directly fed to the lower-channel mixer and is delayed 90<math>^\circ (cosine) \!</math> to feed the upper-channel mixer.  The upper channel provides the in-phase signal <math> I(t) \!</math>, and the lower channel provides the quadrature signal <math> Q(t) \!</math>.<br><br>
[[Image:Simpleradioouble3.jpg]]<br><br>
A common QSD is the Tayloe detector, named after designer Dan Tayloe.  The Tayloe detector can be thought of as a four-position rotary switch.  The switch revolves at the same rate as the carrier frequency.  Each of the four switch positions is connected to a sampling capacitor, and the 50 ohm antenna impedance is connected to the rotor.  Each capacitor will track the carrier's amplitude for exactly one-quarter of the cycle since the switch rotor is turning at exactly the RF carrier frequency, causing the switch to sample the signal at 0°, 90°, 180° and 270°.  If the switch is off, each capacitor will hold its value for the remainder of the cycle.  The 180° and 270° outputs carry redundant information with the 0° and 90° outputs respectively.  Therefore the 0° and 180° outputs can be summed differentially to produce the in-phase signal (I).  Similarly, the 90° and 270° can be summed to form the quadrature signal (Q). <br><br>
[[Image:Tayloe max.jpg]]<br><br>