Undersampling

Plot of sample rates (y axis) versus the upper edge frequency (x axis) for a band of width 1; grays areas are combinations that are "allowed" in the sense that no two frequencies in the band alias to same frequency. The darker gray areas correspond to undersampling with the maximum value of n in the equations of this section.

In signal processing, undersampling or bandpass sampling is a technique where one samples a bandpass filtered signal at a sample rate below the usual Nyquist rate (twice the baseband bandwidth, i.e. twice the upper cut-off frequency), but is still able to reconstruct the signal.

When one samples a bandpass signal, the samples are equal to samples of a low-frequency alias of the high-frequency signal. Such undersampling is also known as bandpass sampling, harmonic sampling, IF sampling, and direct IF-to-digital conversion.

Description

Real-valued signals have Fourier spectra with symmetry about zero. That is, they have a negative-frequency spectrum that is a mirror image of the positive-frequency spectrum. Sampling effectively shifts both sides of the spectrum by multiples of the sampling frequency. The criterion to avoid aliasing is that none of these shifted copies of the spectrum overlap.

In the case of the bandpass (non-baseband) signals, with low and high band limits f_L and f_H respectively, the condition for an acceptable sample rate is that shifts of the bands from f_L to f_H and from –f_H to –f_L must not overlap when shifted by all integer multiples of sampling rate f_s. This condition reduces to the constraint:

$\frac{2 f_H}{n} \le f_s \le \frac{2 f_L}{n - 1}$, for some n satisfying: $1 \le n \le \left\lfloor \frac{f_H}{f_H-f_L} \right\rfloor$

The highest n for which the condition is satisfied leads to the lowest possible sampling rates.

Important signals of this sort include a radio's intermediate-frequency (IF) or radio-frequency (RF) signal.

If n > 1, then the conditions result in what is sometimes referred to as undersampling, bandpass sampling, or using a sampling rate less than the Nyquist rate 2f_H obtained from the upper bound of the spectrum. See aliasing for a simpler formulation of this Nyquist criterion that specifies the lower bound on sampling rate (but is incomplete because it does not specify the gaps above that bound, in which aliasing will occur). Alternatively, for the case of a given sampling frequency, simpler formulae for the constraints on the signal's spectral band are given below.

Spectrum of the FM radio band (88–108 MHz) and its baseband alias under 44 MHz (n = 5) sampling. An anti-alias filter quite tight to the FM radio band is required, and there's not room for stations at nearby expansion channels such as 87.9 without aliasing.

Spectrum of the FM radio band (88–108 MHz) and its baseband alias under 56 MHz (n = 4) sampling, showing plenty of room for bandpass anti-aliasing filter transition bands. The baseband image is frequency-reversed in this case (even n).

Example: Consider FM radio to illustrate the idea of undersampling.

In the US, FM radio operates on the frequency band from f_L = 88 MHz to f_H = 108 MHz. The bandwidth is given by

$W = f_H - f_L = 108 \ \mathrm{MHz} - 88 \ \mathrm{MHz} = 20 \ \mathrm{MHz}$

The sampling conditions are satisfied for

$1 \le n \le \lfloor 5.4 \rfloor = \left\lfloor { 108 \ \mathrm{MHz} \over 20 \ \mathrm{MHz} } \right\rfloor$

Therefore, n can be 1, 2, 3, 4, or 5.

The value n = 5 gives the lowest sampling frequencies interval $43.2\ \mathrm{MHz}<f_\mathrm{s}<44\ \mathrm{MHz}$ and this is a scenario of undersampling. In this case, the signal spectrum fits between 2 and 2.5 times the sampling rate (higher than 86.4–88 MHz but lower than 108–110 MHz).

A lower value of n will also lead to a useful sampling rate. For example, using n = 4, the FM band spectrum fits easily between 1.5 and 2.0 times the sampling rate, for a sampling rate near 56 MHz (multiples of the Nyquist frequency being 28, 56, 84, 112, etc.). See the illustrations at the right.

When undersampling a real-world signal, the sampling circuit must be fast enough to capture the highest signal frequency of interest. Theoretically, each sample should be taken during an infinitesimally short interval, but this is not practically feasible. Instead, the sampling of the signal should be made in a short enough interval that it can represent the instantaneous value of the signal with the highest frequency. This means that in the FM radio example above, the sampling circuit must be able to capture a signal with a frequency of 108 MHz, not 43.2 MHz. Thus, the sampling frequency may be only a little bit greater than 43.2 MHz, but the input bandwidth of the system must be at least 108 MHz. Similarly, the accuracy of the sampling timing, or aperture uncertainty of the sampler, frequently the analog-to-digital converter, must be appropriate for the frequencies being sampled 108MHz, not the lower sample rate.

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 rate 216 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 open interval: $\left(-\frac12f_\mathrm{s},\frac12f_\mathrm{s}\right)$,

and the reconstructive interpolation function, or lowpass filter impulse response, is $\operatorname{sinc} \left( t/T \right)$.

To accommodate undersampling, the bandpass condition is that X(f) = 0 outside the union of open positive and negative frequency bands

$\left(-\frac{n}2f_\mathrm{s},-\frac{n-1}2f_\mathrm{s}\right) \cup\left(\frac{n-1}2f_\mathrm{s},\frac{n}2f_\mathrm{s}\right)$ for some positive integer $n\,$.

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:

$n\operatorname{sinc} \left(\frac{nt}T\right) - (n-1)\operatorname{sinc} \left( \frac{(n-1)t}T \right)$.

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.

See also

• Drizzle (image processing)