My recent column about undersampling included a table that showed upper and lower sampling frequencies for a carrier signal at 55MHz with an 8MHz bandwidth. You also can look at sample rates graphically; an example appears below. This information relates the highest-frequency component in a signal to a minimum sample rate.
A plot of minimum-sample rate/bandwidth (y) vs. highest-frequency component/bandwidth (x) ratios.
The chart shows normalized data as a ratio of frequency to bandwidth, BW. The y axis represents values of minimum sample frequency divided by bandwidth: y = (fs/BW). The x axis represents values of the highest-frequency component divided by bandwidth, or center frequency (fc), plus half the bandwidth, divided by bandwidth: x = [fc + (BW/2)]/BW.
By using the definitions of minimum sample frequency, and rearranging the equation:
fs = (2∙fc + BW)/(n + 1) or 2∙[fc + (BW/2)]/(n + 1)
Thus, y = fs/BW or 2∙[fc + (BW/2)]/[BW∙(n + 1)]
Now substitute x for the portion of the equation shown above in bold to yield: y = (2∙x)/(n + 1). Remember that x and y values represent unit-less ratios.
If you look at the graph above and use the 55MHz signal (fc) with an 8MHz bandwidth (BW), you can calculate [fc + (BW/2)]/BW and find that value on the x axis [(55 MHz + 5 MHz)/8 MHz] 7.4. From that point, go upward until you reach the line for n = 6, and then go horizontally to the y axis and find the fs/BW value. I found the y-axis value 2.11. From that ratio, calculate the minimum sampling frequency, fs. In this example, 2.11 = fs/BW, so fs = 2.11∙BW = 2.11∙8 MHz = 16.9 MHz, which comes close to the value shown in a table in the November column. That table included upper and lower sampling frequencies.
You also can graph the ratios of highest sample frequency to bandwidth against values of the highest-frequency component divided by bandwidth.
This plot includes the maximum sample rate data and shows areas
of restricted values you should not use for undersampling.
This image shows that plot with extrapolated lines. The shaded areas represent restricted zones in which you cannot choose a corresponding fs/BW value from the y axis. The vertical green line represents the [fc + (BW/2)]/BW ratio for the 55MHz signal. The green horizontal line picks off one sample point on the y axis that corresponds to 26.2 Msamples/sec.
Algebra steps.
In practice, you should stay away from restricted-area boundaries. The DSP practitioner Richard Lyons recommends using an intermediate value in the white zones between restricted zones. For anyone interested in more information, I highly recommend Rick's book.
Do you have a measurement question you'd like answered here? Contact me at tituslabs@comcast.net.
Reference:
- Lyons, Richard G., "Understanding Digital Signal Processing," 2 ed., Pearson Education, Inc., 2004.
ISBN: 978-0-13-108989-7.
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