[hpsdr] A/D

[Glenn Thomas]

At 06:44 AM 4/21/2007, Ben Hall wrote:

<snip>

>However, Nyquist theory gives me some heartburn:
>
>The 2X rule is true - you can use a 60 MHz ADC to capture a 30 MHz
>signal and reconstruct the signal.  But, you've got to assume a wave
>form shape, as sampling at 2X gives you no information as to the shape
>of the waveform, as you're only looking at two points per waveform
>period and don't have enough information to ascertain the waveform's
>shape.  At a 2X sample rate, you can't tell if the signal recorded is a
>sine wave, a square wave, a triangle wave, or what.

<snip>

Yup. Here's why. Many years ago a guy name Fourier proved 
mathematically that any repetitive waveform, no matter how complex, 
can be expressed as the arithmetic sum of pure sine functions of 
varying phase, amplitude and frequency. For example, a square wave is 
the arithmetic sum of the fundamental and the odd harmonics: sin(x) + 
sin(3x)/3 + sin(5x)/5 and so forth. Get yourself a piece of graph 
paper and demonstrate this to your self. The more odd harmonics of 
the form sin(n*x)/n you add, the more it looks like a square wave.

So, if you sample a 1 MHz square wave at just over the Nyquist rate 
(2 MHz plus a bit) and eliminate everything above the Nyquist 
frequency with an antialiasing low pass filter, the only frequency 
component you will sample is the fundamental, sin(x). This looks 
exactly like a sine wave because that's exactly what it is. The 
higher harmonics that made it a square wave have been filtered out 
because they exceed the Nyquist frequency. A similar analysis applies 
to any other waveform you might like to sample, sawtooth, triangle, etc.

Note that many waveforms are not the sum of harmonics. They are the 
sum of components that are apparently unrelated in frequency. An 
example is a .9 MHz carrier amplitude modulated by a 1 KHz tone, 
sin(c - t) + sin(c) + sin(c + t), a carrier with upper and lower 
sidebands. In this case, our "2 MHz plus a little" sampling rate will 
capture all three frequency components.

73 de Glenn Thomas wb6w




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