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Chapter 7:
The Sampling Theorem
Abstract
The Sampling Theorem is cited often in engineering circles, and a description of the theorem is a necessary part of a book on sampling. However, the Sampling Theorem represents an ideal limit. The Sampling Theorem substitutes ideal pre- and post-filters for the sensor and display MTF, respectively. As a result, the real system behavior is not characterized. The Sampling Theorem does not provide useful guidance for the design of EO imagers. In this chapter, the Sampling Theorem is described and an example given of a near-ideal reconstruction of a sampled waveform. Also, some of the common misconceptions about the dictates of the Sampling Theorem will be discussed. Our primary purpose in describing the Sampling Theorem, however, is to demonstrate its limited value in evaluating real systems. 7.1 THEORY The Sampling Theorem states that, for a signal f(x) for which the Fourier transform has no components at or above frequency f samp â2 , the function can be entirely reconstructed by the series: f(x)=â n=ââ â f(nâf samp )sin(Ïxf samp ânÏ) (Ïxf samp ânÏ) . The function f(x) is sampled with sample frequency f samp . As discussed in Sections 1.5 and 3.1, the Fourier transform F(Ï) of f(x) is replicated at multiples of the sample frequency as shown in Figure 7.1. If the replicas of F(Ï) do not overlap the baseband, meaning that F(Ï) is band-limited to half the sample frequency, then F(Ï) can be exactly reconstructed by using an ideal filter. This is also shown in Figure 7.1. An ideal filter is a rect function, with MTF of one (1.0) out to half the sample frequency and zero beyond. The Fourier transform of a rect function is a sinc wave. Convolving the sampled data with a sinc wave in the spatial domain provides an ideal reconstruction filter in the frequency domain. Equation 7.1 represents the convolution in space. Each sample at location nâf samp is convolved with the function sin(Ïxf samp )â(Ïxf samp ) . Since the samples are delta functions at the sample locations, Equation 7.1 represents the convolution.