# Third harmonic sampling and QSD - HW9

## Max Woesner

### Homework #9 - Third harmonic sampling and QSD

References:
SoftRock-40 SDR interest group

#### 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.

Consider the simple representation of part of a software defined radio below.

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

The output voltage of the radio, or $v_O \!$ is $v_O = v_{RF}(t) \cdot sgn(v_{LO}(t)) \!$

Now $sgn(v_{LO}(t)) \!$ can be written as the series $cos(\omega _{LO}t) + \frac{1}{2}cos(2 \omega _{LO}t) + \frac{1}{3}cos(3 \omega _{LO}t) + ... \!$

We are interested in the third harmonic term, or $\frac{1}{3}cos(3 \omega _{LO}t) \!$

This frequency is three times that of $cos(\omega _{LO}t) \!$. Note that the period decreases by a factor of three.

We can use the third harmonic to drive the mixer in the radio at a frequency three time that of the local oscillator.

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.

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

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.

We could use a mixer like this one.

If you read the above section about third harmonic sampling, then you might be wondering how we input both $sine \!$ and $cosine \!$ into the parallel mixers. The answer is that we can use a quadrature sampling detector, or QSD.
The local oscillator $(sine) \!$ is directly fed to the lower-channel mixer and is delayed 90$^\circ (cosine) \!$ to feed the upper-channel mixer. The upper channel provides the in-phase signal $I(t) \!$, and the lower channel provides the quadrature signal $Q(t) \!$.