Abstract
Sampling is one of the main processes in an analog-to-digital converter. The sampling theory is examined and the crucial elements are extensively discussed. The relation with other techniques such as modulation and sampling of noise is described. The second section discusses the design of time-discrete filters. These filters form an important building block in several conversion structures especially in sigma-delta conversion. Down-sample filters transform the bit-stream format into a more usable pulse code format. The essential properties of finite impulse response and infinite impulse response filters are reviewed.
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Notes
- 1.
“No value is defined” does not imply that the value is zero! There is simply no value.
- 2.
Strange in the sense that many normal mathematical operations cannot be performed, e.g., δ 2(t) does not exist.
- 3.
For clarity this chapter uses for time-domain signals normal print, while their spectral equivalents are in bold face. The suffix s refers to sampled sequences.
- 4.
The Fourier transform definition and its inverse require that a factor 1∕2π is added somewhere. Physicists love symmetry and add \(1/\sqrt{2\pi }\) in front of the transform and its inverse. Engineers mostly shift everything to the inverse transform, see Eq. 2.15. More attention is needed when a single-sided engineering type Fourier transform is used instead of a mathematically more accurate double sided.
- 5.
Other originators for this criterion are named as E. Whittaker and V.A. Kotelnikov. Landau proved in 1967 for non-baseband and non-uniform sampling that the average sample rate must be twice the occupied bandwidth.
- 6.
Precise mathematicians will now argue that a signal with frequency f = f s, ny ∕2 cannot be reconstructed, so they should read here: \(f = f_{s,ny}/2 - \Delta f\), where \(\Delta f\) goes to zero.
- 7.
The only storage in the early days of CDs were video recorders. The 44.1 ks/s sample rate was chosen such that the audio signal exactly fits to a video recorder format (25 fields of 588 lines with 3 samples per line) of 44.1 ks/s.
- 8.
Keep track of: dsp.rice.edu/cs for an overview of all compressive sampling developments.
- 9.
In this book the term down-sampling is used for sampling an analog signal with the purpose to perform a frequency shift of the band of interest. Subsampling in Sect. 2.3.3 removes samples in a predetermined manner from an existing sample stream, but does not change the signal band.
- 10.
Subsampling by a rational factor (a division of two integers) or an irrational factor requires to calculate the signal at each new sample moment by interpolation of the existing samples. This technique is often applied in image processing and is used to combine data from sources with asynchronous clocks. Some fast-running hardware is needed to carry out the interpolation.
- 11.
Formally the result of a Fourier transform reflects the intensity of a process or signal at a frequency. Therefore the result has the dimension “events per Hz” or “Volt per Hertz.”
- 12.
Thermal noise in electronics is modeled as a noise source connected to a real impedance. For circuit calculations this works, but impedances are not noisy, electrons with random energy are.
- 13.
As thermal noise is an atomic phenomenon, its frequency span ends where sub-atomic mechanisms, as described by quantum physics, start. A rule of thumb limits the standard noise spectrum at 1 THz.
- 14.
Very annoying “T” for absolute temperature as well as for fixed time periods.
- 15.
A peak–peak value is often used for jitter, but peak–peak values for stochastic processes have no significance if the process and the corresponding number of observations are not identified.
- 16.
For some reason deciBel is spelled with single “l” although it was named after A.G. Bell. Similarly the letter “a” was lost in “Volta.”
- 17.
Here a strictly formal derivation requires some 10 pages, please check out specialized literature, e.g., [31].
- 18.
In financing a monthly moving average is a very simple FIR filter with 12 taps and simply “1” as multiplication factor.
- 19.
The human ear is sensitive to delay variations down to the microsecond range.
- 20.
The definition for Euler’s relation is: e j π + 1 = 0. According to Feynman this is the most beautiful mathematical formula as it is relates the most important mathematical constants to each other.
- 21.
An equivalent analog filter would require 10–12 poles.
- 22.
In the period 1970–1980 the charge-coupled device was seen as a promising candidate for storage, image sensing, and signal processing. Analog charge packets are in this multi-gate structure shifted, split and joint along the surface of the semiconductor. Elegant, but not robust enough to survive the digital era.
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Pelgrom, M. (2017). Sampling. In: Analog-to-Digital Conversion. Springer, Cham. https://doi.org/10.1007/978-3-319-44971-5_2
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DOI: https://doi.org/10.1007/978-3-319-44971-5_2
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