[hpsdr] sampling

[Nyall Davies]

Here is my simple explanation of sampling and dynamic range. I trust that it 
will help.

1) Sampling:

Nyquist states that we can reconstruct to the waveform if we sample at twice 
the waveform frequency.

Thus an AD sampling at say 100 MHz is OK for up to 50 MHz input ( but not 
quite as at precisely 50 MHz we get all identical samples out ie it looks 
like dc  )

To get to 100 MHz with a 100 MHz AD we could use two of them and we would 
set their timing to give phase and quadrature. ie staggered by 2 1/2 nS. We 
would then have two channels of information. Alternatively we can produce 
the same information by mixing with local oscillators in phase and quad. 
This is what Mercury does.

The easy way to think of it is "For one channel Nyquist says "half the 
sampling frequency;" for two channels he says "the sampling frequency."

2) Dynamic range - number of bits:

The dynamic range of a perfect AD is approx 6 dB per bit. (The quantization 
noise is 11 dB below one bit and the max RMS is 2 root 2 (9 dB) below the 
top bit for a sin wave)
Real ADs have a less dynamic range because of internal noise and jitter so 
we get an ENOB effective number of bits figure typically 13 bits for a fast 
16 bit AD. 6dB * 13 = a dynamic range of 78 dB. not good enough but we can 
now trade bits for speed.

Think of simply adding two samples together - noise will grow 3 dB per bit 
and signals will add coherently 6dB. There is a gain of 3dB in signal noise 
ratio which is what matters. ie we get one bit (6db) increase for each 4 
times reduction in speed. However Nyquist still rules meaning we are trading 
dynamic range for bandwidth and we get word growth ie more useful bits.

If we start with two channels at 100 MHz we have a 100 MHz bandwidth and say 
13 bits or ~78dB dynamic range.
The signal processing produces a final bandwidth of say 192 kHz so we have a 
increase in dynamic range of 10 log (100 Mhz / 0.192 MHz) = 27 dB.
(less any processing losses) - Total dynamic range 78 + 27 = 105 dB. We have 
to allow for the word growth in the signal processing

Put the sampling rate up to say 150MHz and we can have another 1.76 dB 
dynamic range at which point the increase in cost per dB becomes 
significant.

If we reduce the final bandwidth to 3 KHz we can have 18 dB more.

3) Undersampling.

We may undersample as long as the bandwidth of interest (192 kHx) meets the 
Nyquist criterion however there are some perfomance issues to keep an eye 
on. The AD aperture jitter is obvoiusly a greater proportion of the cycle at 
144 MHz than at 30 MHz say. This will give an increase in noise.

SN due to jitter = 20 Log[1/(2 pi f t)]  where f is the frequncy being 
sampled and t is the rms jitter ( due to aperture of AD and clock generator)


Nyall G8IBR







 1177320330.0