Digital Signal Processing

By Steven W. Smith, Ph.D.

- 1: The Breadth and Depth of DSP
- 2: Statistics, Probability and Noise
- 3: ADC and DAC
- 4: DSP Software
- 5: Linear Systems
- 6: Convolution
- 7: Properties of Convolution
- 8: The Discrete Fourier Transform
- 9: Applications of the DFT
- 10: Fourier Transform Properties
- 11: Fourier Transform Pairs
- 12: The Fast Fourier Transform
- 13: Continuous Signal Processing
- 14: Introduction to Digital Filters
- 15: Moving Average Filters
- 16: Windowed-Sinc Filters
- 17: Custom Filters
- 18: FFT Convolution
- 19: Recursive Filters
- 20: Chebyshev Filters
- 21: Filter Comparison
- 22: Audio Processing
- 23: Image Formation & Display
- 24: Linear Image Processing
- 25: Special Imaging Techniques
- 26: Neural Networks (and more!)
- 27: Data Compression
- 28: Digital Signal Processors
- 29: Getting Started with DSPs
- 30: Complex Numbers
- 31: The Complex Fourier Transform
- 32: The Laplace Transform
- 33: The z-Transform
- 34: Explaining Benford's Law

Your laser printer will thank you!

Sample Spacing and Sampling Aperture

Figure 25-6 shows two extreme examples of sampling, which we will call a
perfect detector and a blurry detector. Imagine (a) being the surface of an
imaging detector, such as a CCD. Light striking anywhere inside one of the
square pixels will contribute *only* to that pixel value, and no others. This is
shown in the figure by the black sampling aperture exactly filling one of the
square pixels. This is an optimal situation for an image detector, because *all* of
the light is detected, and there is *no* overlap or crosstalk between adjacent
pixels. In other words, the sampling aperture is exactly equal to the sample
spacing.

The alternative example is portrayed in (e). The sampling aperture is
considerably larger than the sample spacing, and it follows a Gaussian
distribution. In other words, each pixel in the detector receives a contribution
from light striking the detector in a region *around* the pixel. This should sound
familiar, because it is the output side viewpoint of convolution. From the
corresponding input side viewpoint, a narrow beam of light striking the detector
would contribute to the value of several neighboring pixels, also according to
the Gaussian distribution.

Now turn your attention to the edge responses of the two examples. The
markers in each graph indicate the actual pixel values you would find in an
image, while the connecting lines show the *underlying curve* that is being
sampled. An important concept is that the shape of this underlying curve is
determined *only* by the *sampling aperture*. This means that the resolution in the
final image can be limited in two ways. First, the underlying curve may have
poor resolution, resulting from the sampling aperture being too large. Second,
the sample spacing may be too large, resulting in small details being lost
between the samples. Two edge response curves are presented for each
example, illustrating that the actual samples can fall anywhere along the
underlying curve. In other words, the edge being imaged may be sitting exactly
upon a pixel, or be straddling two pixels. Notice that the perfect detector has
*zero* or *one* sample on the rising part of the edge. Likewise, the blurry detector
has *three* to *four* samples on the rising part of the edge.

What is limiting the resolution in these two systems? The answer is provided
by the *sampling theorem*. As discussed in Chapter 3, sampling captures all
frequency components below one-half of the sampling rate, while higher
frequencies are lost due to aliasing. Now look at the MTF curve in (h). The
sampling aperture of the blurry detector has removed all frequencies greater
than one-half the sampling rate; therefore, *nothing* is lost during sampling. This
means that the resolution of this system is

completely limited by the sampling aperture, and not the sample spacing. Put another way, the sampling aperture has acted as an antialias filter, allowing lossless sampling to take place.

In comparison, the MTF curve in (d) shows that *both* processes are limiting the
resolution of this system. The high-frequency fall-off of the MTF curve
represents information lost due to the *sampling aperture*. Since the MTF curve
has not dropped to zero before a frequency of 0.5, there is also information lost
during sampling, a result of the finite *sample spacing*. Which is limiting the
resolution more? It is difficult to answer this question with a number, since they
degrade the image in different ways. Suffice it to say that the resolution in the
perfect detector (example 1) is mostly limited by the sample spacing.

While these concepts may seem difficult, they reduce to a very simple rule for
practical usage. Consider a system with some 10%-90% edge response
distance, for example 1 mm. If the sample spacing is greater than 1 mm (there
is less than one sample along the edge), the system will be limited by the *sample
spacing*. If the sample spacing is less than 0.33 mm (there are more than 3
samples along the edge), the resolution will be limited by the *sampling aperture*.
When a system has 1-3 samples per edge, it will be limited by both factors.