Explain the effects of under sampling with suitable examples.
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If the sampling theorem is interpreted as requiring twice the highest frequency, then the required sampling rate would be assumed to be greater than the Nyquist rate216 MHz. While this does satisfy the last condition on the sampling rate, it is grossly oversampled.Note that if a band is sampled with n > 1, then a band-pass filter is required for the anti-aliasing filter, instead of a lowpass filter.
As we have seen, the normal baseband condition for reversible sampling is that X(f) = 0 outside the interval: {\displaystyle \scriptstyle \left(-{\frac {1}{2}}f_{\mathrm {s} },{\frac {1}{2}}f_{\mathrm {s} }\right),}
and the reconstructive interpolation function, or lowpass filter impulse response, is {\displaystyle \scriptstyle \operatorname {sinc} \left(t/T\right).}
To accommodate undersampling, the bandpass condition is that X(f) = 0 outside the union of open positive and negative frequency bands
{\displaystyle \left(-{\frac {n}{2}}f_{\mathrm {s} },-{\frac {n-1}{2}}f_{\mathrm {s} }\right)\cup \left({\frac {n-1}{2}}f_{\mathrm {s} },{\frac {n}{2}}f_{\mathrm {s} }\right)}for some positive integer {\displaystyle n\,}.which includes the normal baseband condition as case n = 1 (except that where the intervals come together at 0 frequency, they can be closed).
The corresponding interpolation function is the bandpass filter given by this difference of lowpass impulse responses:
{\displaystyle n\operatorname {sinc} \left({\frac {nt}{T}}\right)-(n-1)\operatorname {sinc} \left({\frac {(n-1)t}{T}}\right)}.
On the other hand, reconstruction is not usually the goal with sampled IF or RF signals. Rather, the sample sequence can be treated as ordinary samples of the signal frequency-shifted to near baseband, and digital demodulation can proceed on that basis, recognizing the spectrum mirroring when n is even.
Further generalizations of undersampling for the case of signals with multiple bands are possible, and signals over multidimensional domains (space or space-time) and have been worked out in detail by Igor Kluvánek.
As we have seen, the normal baseband condition for reversible sampling is that X(f) = 0 outside the interval: {\displaystyle \scriptstyle \left(-{\frac {1}{2}}f_{\mathrm {s} },{\frac {1}{2}}f_{\mathrm {s} }\right),}
and the reconstructive interpolation function, or lowpass filter impulse response, is {\displaystyle \scriptstyle \operatorname {sinc} \left(t/T\right).}
To accommodate undersampling, the bandpass condition is that X(f) = 0 outside the union of open positive and negative frequency bands
{\displaystyle \left(-{\frac {n}{2}}f_{\mathrm {s} },-{\frac {n-1}{2}}f_{\mathrm {s} }\right)\cup \left({\frac {n-1}{2}}f_{\mathrm {s} },{\frac {n}{2}}f_{\mathrm {s} }\right)}for some positive integer {\displaystyle n\,}.which includes the normal baseband condition as case n = 1 (except that where the intervals come together at 0 frequency, they can be closed).
The corresponding interpolation function is the bandpass filter given by this difference of lowpass impulse responses:
{\displaystyle n\operatorname {sinc} \left({\frac {nt}{T}}\right)-(n-1)\operatorname {sinc} \left({\frac {(n-1)t}{T}}\right)}.
On the other hand, reconstruction is not usually the goal with sampled IF or RF signals. Rather, the sample sequence can be treated as ordinary samples of the signal frequency-shifted to near baseband, and digital demodulation can proceed on that basis, recognizing the spectrum mirroring when n is even.
Further generalizations of undersampling for the case of signals with multiple bands are possible, and signals over multidimensional domains (space or space-time) and have been worked out in detail by Igor Kluvánek.
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