how can sampling rates be lesser than the maximum frequency content of a bandpass signal ?

Pankaj,

There are many articles covering the process of undersampling or aliasing.  Here is a good introduction to ADC basics - undersampling is covered in one of the sections.

As long as the analog input bandwidth is wide enough for the sampling circuits to see the signal of interest, then you can use undersampling.

The common reference to the minimum input frequency of 2x input frequency is not entirely correct.  It implies that you want to maintain your sampled signal in the 1st nyquist (without undersampling).  However you could under-sample or subsample the input signal that it still has the same information, just shifted by the under sampling process to a different frequency.  In the frequency domain, a single tone does not take up much of the Nyquist BW, even if the tone is near the Fs/2, it still only requires 1 frequency bin, from this perspective it is easy to see that the nyquist BW can be reduced to a minimum of 2xBW without any loss of information.

It is better to consider the sampling rate required to maintain the input signal BW and not just the frequency, think 2xBW instead of 2xFrequency.  Theoretically for a single tone you could have a very low sample rate and it would be able to sample the signal without any loss of information other than a shift in the frequency of the signal.  If you have a signal over multiple frequencies such as tones between 88M and 108M, then you would need at least 2x20MHz to be able to sub-sample or undersample the signal without loss of information.

If you consider this in the frequency domain it will be easier to see the sampling needs vs the signal BW.

Ken.

Dear Ken,

Thanks for your reply and insightful clarifications on my queries.

With your explanation, I have come up with another doubt.

What are the parameters you take into account for the term "information". When you say "loss of information", what parameters get  lost.

When we sampling according to Nyquist criteria, we can retrieve the "information " about the frequency of the input signal. But if we undersample, we lose the info on the input frequency. So, what are the parameters/quantity that we can still extract from the undersampled signal???

Please help me with my doubts.

Typically if you are only looking at a sine wave the only information you have is frequency and amplitude. If the signal is not band limited to the 1st nyquist and the input BW of the ADC is wide enough then you will not be able to determine what frequency the signal came in at.

For example for 100Msps sampling rate an input of 10M and 90M would yield the same output samples.

In most systems the signal is at a specific carrier frequency and is modulated in some way: amplitude/phase/frequency. In this case you can use down sampling to move the carrier (with information) down to low frequency without losing the information on phase, frequency (assuming carrier frequency is known) or amplitude.

Ken

Hi Ken,

Thanks again for your reply. Your comments are truly insightful.

Hi Ken,
Your comment leads me to one more question.
As you said in your previous reply, " In this case you can use down sampling to move the carrier (with information) down to low frequency without losing the information on phase, frequency (assuming carrier frequency is known) or amplitude."

In what ways does the information about "phase" appear in the spectrum of the sampled signal ??? (because as far as i know the spectrum of a sine and cosine wave are same.)

The spectrum analyzer does not show phase. It is only used for amplitude and frequency. You will need to look at the complex output of your FFT to also be able to look at the phase information. A spectrum analyzer only displays the amplitude information of the signal at the associated frequency bin.

Ken.