Homework Nine

1. How does Third Harmonic Sampling work? (The Soft Rock receiver uses it.)
2. How does the quadrature sampling detector (QSD) work? (Look at the SDR-1000 QEX articles.)

Nick Christman

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Third Harmonic Sampling

The Softrock 40 software defined radio (SDR) uses a special technique when sampling the data -- this technique is referred to as 3rd Harmonic Sampling. The purpose of 3rd Harmonic Sampling is to eliminate the need to use a high frequency oscillator which is useful economically and physically. (In other words, a high frequency oscillator is expensive and is often incompatible with "cheaper" switches.) However, nothing in this world is perfect and because of this imperfection, Third Harmonic Sampling have a signal loss of $\textstyle 20\log _{10}({\frac {1}{3}})=-9.54db$ (Max Woesner).

So how does Third Harmonic Sampling work? Well, let's take for example a very basic SDR (similar to the one designed in Electronics II). In the most basic SDR there is a local oscillator that outputs a wave in the form of a square wave. Now, this square is not the product of 21st century sorcery, but instead it is the result of an "infinite" sinusoidal series that is evaluated at the desired local oscillation frequency. This sinusoidal series is refereed to as a Fourier series and is defined as

$\sum _{n=1}^{\infty }{\frac {1}{n}}cos(n\omega t)=cos(\omega t)+{\frac {1}{2}}cos(2\omega t)+{\frac {1}{3}}cos(3\omega t)+{\frac {1}{4}}cos(4\omega t)+\cdots$

Of course, the segment of this series we are particulary interested in is the third harmonic, or $\textstyle {\frac {1}{3}}cos(3\omega t)$. Notice the constant $\textstyle {\frac {1}{3}}$ in front of the cosine -- this is where the 9.54 db loss originates from (mathematically). From here, we simply use the third harmonic cosine as our local oscillator frequency.

Quadrature Sampling Detector

A Quadrature Sampling Detector (QSD) is used to create a cosine (or sine) waveform from a established sine (or cosine) waveform. (I say "sine or cosine" because the output waveform is simply sinusoidal -- there is really no way to distinguish between sine and cosine, we can only distinguish between 0 and 90 degrees.) At any rate, the most common QSD is known as a Tayloe Detector.

The Tayloe Detector is essentially an array of (mechanical or digital) switches which are switched at a rate equal to the desired RF frequency. Each switch position is then connected to a capacitor which allows the switch to remain in each position for one-quarter of a cycle (when it completes all positions, it will have completed a full cycle). This method establishes different angles offset by 90-degress, that is $\textstyle 0^{\circ }{\mbox{, }}90^{\circ }{\mbox{, }}180^{\circ }{\mbox{, and }}270^{\circ }$. This is exatly what we were seeking, because now we have a cosine-wave ($\textstyle 0^{\circ }{\mbox{ and }}180^{\circ }$) and a sine-wave ($\textstyle 90^{\circ }{\mbox{ and }}270^{\circ }$).

This is essentially how QSD works, though there is much more detail and mathematics that I was unable to find and that could probably be added in the future.