noneMost of the low-cost software-defined radios (SDRs) being developed and used by hams and other experimenters currently rely on the host PC ‘s soundcard to digitize the signal, and on DSP software running on the host to demodulate the digitized signal. The SDR hardware, whether it’s an LD-1, Softrock-40, Elektor, etc. is actually a downconverter that converts a slice of RF spectrum to “audio” so it can be digitized by the soundcard. The first part of this series will explain exactly how this conversion is accomplished. After reading this part, I hope the conversion process will have become intuitive for you, and will seem more real to you.
The conversion in this type of SDR is usually done using a “Tayloe detector” or “Quadrature Sampling Detector” (QSD). As far as I can tell, these two terms mean the same thing, so I’ll use “QSD” because it’s shorter.  A Quadrature Sampling Detector is actually two sampling detectors in quadrature, to produce an in-phase (I)output and a quadrature (Q) output. What’s a sampling detector? It can be as simple as a sample-and-hold.image005

A sample-and-hold, shown here, is nothing more than a  MOSFET switch and capacitor, as shown here. The output voltage is the voltage on the capacitor.  When the switch closes, the capacitor immediately charges up to the input voltage, and tracks the input voltage while the switch remains closed.  When the switch opens, the input voltage is stored on the capacitor until the next time the switch closes.  The switch is usually controlled by a pulse train, so the input signal is sampledat a constant rate, which is called the sampling rate or sampling frequency.

 

 

This is a Simulink modimage004el, a sort of executable block diagram, depicting a 10 Hz sine wave being sampled and held at 1000 samples per second:

 

 

  

  

  

 

The output of the sample-and-hold is displayed on an oscilloscope, and looks like this: image008

 

  

  

  

  

  

 The output waveform looks like a 10 Hz. Sine wave, but it isn’t perfectley smooth.  Let’s zoom in on a portion of it:

 image010

 The sample-and-hold action is clearly visible, and gives a kind of “staircase effect to the waveform.  The sample rate (1000 samples/second) is 100
times the frequency of the sine wave, so there are 100 “stairsteps” in each cycle.

Now, let’s increase the frequency to 100 Hz:

 

 image014image012

 Now the sample rate is only 10 times the signal frequency, so there are 10 samples per cycle.  The output is still recognizable as a sine wave, sort of.  Let’s increase the frequency to 495 Hz, so there are just over  two samples in each cycle:

 image018image016

 This is interesting!  It looks like a DSB-SC or SSB waveform.  Normally, the input signal applied to a sample-and-hold (or an analog to digital converter) ”must” be limited to no more than ½ the sample rate. That means we shouldn’t go above 500 Hz., but let’s try 505 Hz: 

 

image022 image020

It looks just like the 495 Hz. Waveform. The world didn’t implode when we went above ½ the sample rate.

By the way, ½ the sample rate is called the “Nyquist rate”, because the Nyquist theorem says that if you sample a waveform with a sample rate at least twice the highest signal frequency, you can perfectly reproduce the signal.  If the sample rate is less than twice the highest signal frequency, that is called “undersampling”.

Now let’s go further above the Nyquist rate, making the signal frequency 990 Hz. That’s only 10 Hz. Less than the sample rate,
so we’re taking 1.01 samples per cycle:

 image026image024

The output is a 10 Hz. Sine wave! The simple sample and hold circuit has converted the input waveform to the same waveform, but at a different frequency! The output frequency is equal to the difference between the sample rate and the input frequency. It’s as if the sample and hold were a mixer, and the pulse train controlling the sample and hold were the LO.

In fact, that’s exactly what is happening. A mixer multiplies the RF signal by the LO sine wave. A sample and hold multiplies its input by a sequence of ones and zeros. It’s the same thing.

 Lets’s go beyond the sample rate, and increase the signal frequency to 1010 Hz:

 image028image030

The output frequency is still 10 Hz, but the waveform has been inverted.  Now let’s try 9990 Hz, 10 Hz less than the tenth harmonic of the sample rate:

 image034image032

Again, the output frequency is 10 Hz, equal to the difference between the signal frequency and the nearest integer multiple of the sample rate.

In Part II, I’ll show how and why two sample and hold circuits may be combined to form a QSD.

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