Linear systems theory provides a powerful set of tools for analyzing optical and electro-optical systems. The spatial impulse response of the system *h* is Fourier transformed to yield the optical transfer function, for which the magnitude is the modulation transfer function (MTF). This viewpoint adds insight into the behavior of an imaging system, particularly when several subsystems are combined. In such a case, we multiply the MTF of each subsystem to give the overall transfer function. This procedure is easier than the repeated convolutions that a spatial-domain analysis would require and expresses the system performance in terms of the performance of each subsystem.

**shift invariance** One component of the MTF for sampled-data systems is the sampling MTF, which accounts for the fact that image data is acquired at discrete spatial intervals Δ*x*. For a staring-array image receiver, the sampling interval is just the center-to-center pixel spacing of the array. A typical focal-plane-array imager, for example, is not shift invariantthe position of the image irradiance with respect to the sampling sites affects the final image data (see figure). Thus the measured MTF depends on the alignment of the target and the sampling sites.

*Changing the image irradiation during acquisition of a sample MTF can alter the result, which implies that the system is not shift-invariant.*

This shift variance violates one of the main assumptions required for an MTF analysis, which is that the system under test is shift invariant. To preserve the convenience of a transfer-function approach, we follow Park^{1} and define a shift-invariant, spatially averaged MTF by assuming that the scene being imaged is randomly positioned with respect to the sampling sites. This corresponds to the situation in which a natural scene is imaged, with an ensemble of individual alignments. For a one-dimensional rectangular sampling grid, the sampling impulse response is a rectangle function whose width is equal to the sampling interval:

*h*_{sampling}(*x*) = rect(*x*/Δ*x*)

Wider spaced sampling produces an image with poorer image quality. An average MTF can be defined as the magnitude of the Fourier transform of *h*_{sampling}(*x*), which yields a sinc-function sampling MTF

MTF_{sampling} = | F { rect(*x*/Δ*x*) } | = | sin(πξΔ*x*)/(πξΔ*x*) |

where ξ is the spatial frequency in cy/mm. This sampling MTF is equivalent to the average of the MTFs that would be realized for a uniformly distributed ensemble of source locations with respect to the sampling sitesessentially, it describes what happens when a natural outdoor scene, rather than a specially aligned test target, is imaged. When the alignment of the source to the detector is optimum, the MTF is broad; for other source positions, the MTF is narrower. The sampling MTF is the average over all possible MTFs. Thus defined, the average is a shift-invariant quantity, so that we can proceed with a usual transfer-function-based analysis, in which the sampling MTF multiplies the other MTF components for the system.

**sampling MTF in practice** In MTF testing, the user aligns the image irradiance with respect to the imager to produce the best-looking image. In this case, the sampling MTF does not apply. By tweaking the alignment, the user has selected the best of many possible MTFs for the system. The sampling MTF is a means to calculate the image quality degradation that occurs on average when there is no alignment.

Because typical test procedures exclude the sampling MTF, it is often forgotten in a systems analysis. However, because a natural scene has no net alignment with respect to the sampling sites, the sampling MTF helps accurately represent the practical performance of the system and should be included in the performance modeling. **oe**

*For further reading:*

*1. S. Park et al., Applied Optics 23 2572 (1984).*

2. M. Sensiper et al., Optical Engineering 32, 395 (1993).

3. A. Daniels et al., Optical Engineering 34, 860 (1995).

**Glenn Boreman**

*Glenn Boreman is professor of optics and electrical engineering at CREOL/School of Optics, University of Central Florida, Orlando, FL. **He is the author of the new tutorial text* "Modulation Transfer Function in Optical and Electro-Optical Systems," published by SPIE Press (2001).