ASN9 - 3rd Harmonic & QSD

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Problem Statement

Explain how Third Harmonic Sampling works and explain how a Quadrature Sampling Detector works too.

Solution

When considering the properties of an oscillator's wave it is important to include its Fourier series. From electronics last year we were able to construct a square wave by summing multiple sine waves with the same basic n * ω or rather the summation . As indicated in the problem statement we are interested in the third one, or rather the third harmonic:

Comparing this to our original function of cos(ωt) we have three times the frequency or rather a third of the period.

For the soft rock 40 radio this property is used to maintain a cost effective ham radio while achieving a high quality signal output. It is important for the mixer to have a very high frequency but oscillators that operate at the frequencies required are very expensive so a cheaper oscillator may be purchased while maintaining the function of the radio.

Unfortunately their is loss attributed to increasing the oscillator frequency. Because our amplitude in the series is reduced by 1/n for each harmonic we are only at 1/3 the amplitude for the third harmonic. Rather this is a loss.

In electronics we built a radio that required both the real and imaginary parts of the signal to go through the mixer. This requires us to mix with a sine and cosine wave simultaneously in different parts of our circuitry. To do this it is necessary to use a QSD (Quadrature Sampling Detector). This can be built with either diodes and transformers or transistors and transformers. I believe we attempted both designs and finally settled on the transistor schematic. The concept that allowed me to understand the QSD had to do with the unit circle.

(I apologize for the large size, I didnt bother resizing it.)

As the oscillator signal is sent through the QSD it in turn rotates around the unit circle. First it is at 0° then it rotates to 90° then to 180° and finally to 270° before returning to 0°. In my mind their is a switch opperating like a clock hand in the center of the circle and wires to hook up with it at each of the axis'. The signals received during these rotations then sent out to the two mixers. The 0° and 180° are sent out to the cosine wave and the 90° and 270° are sent to the sine wave. Although it is only a matter of phase as to which signal they are to mix with and can be adjusted as to the designers preference.

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 $\textstyle O\deg {\mbox{ and }}90\deg$.)

Describe how third harmonic sampling and QSD (Quadrature Sampling Detector works.

For this assignment I will look third harmonic sampling and QSD as used by SoftRock-40 software defined radio. For SoftRock they needed a high frequency oscillator to provide the necessary signal to mix with the signal coming from the antenna. The reason this oscillation needed to be so high was due to there design which used one oscillator to create both the sin and cos waveform for there mixer. To do this they needed a oscillator with a frequency four times higher than the frequency they wished to sample. Getting an oscillator at these speeds is both difficult and expensive so they came up with a clever way to use the third harmonic form their oscillator to achieve the frequency necessary.

It is known by using Fourier Series a square wave can be made up of a series of cos wave i.e. $\sum _{n=1}^{\infty }{\frac {1}{n}}cos(n\omega t)$.

SoftRock decided that they could use the 3rd harmonic of this i.e ${\frac {1}{3}}cos(3\omega t$ to get the required oscillation without the hassle and cost of using a high frequency oscillator. Stated another way they need an oscillator that has a frequency three time slower than the one originally needed. The disadvantage to this is that the signal strength is a third smaller or around a 9.5 dB loss.