Abstract
Functions of a discrete variable, more commonly known as sequences, have associated transform theories. There are discrete variable analogues of Laplace Transforms, Fourier Transforms, and Fourier Series. The function domain in the latter case is discrete and finite: a divide-and-conquer algorithm in this results in the ubiquitous Fast Fourier Transform.
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- 1.
These conditions are not independent, but define classes of sequences analogous to the continuous-time case.
- 2.
We have written the formulas as though an intermediate ω j is under consideration, although it is easy to see that the result also holds for the “extreme” roots ω 0 and ω N−1.
- 3.
Among the possible excuses for this guess, we cite the observation that points on the unit circle are associated with “sinusoidal” discrete-time functions, coupled with an incipient desire to make a connection with the z-transform of the previous section.
- 4.
The systematic theory of such transforms is based on the theory of groups and is known as abstract harmonic analysis. A further example of this transform theory is discussed below as the finite Fourier transform.
- 5.
This choice is made for the sake of the convenience of the connection with the z-transform.
- 6.
See the problems below.
- 7.
The condition that this be the case is the same as the condition required to guarantee that (8.23) has a transformable solution for arbitrary forcing functions f. This is simply that p(e i θ) ≠ 0, 0 ≤ θ < 2π.
- 8.
The “bare formula” is of course already familiar from encounters with Fourier series. In effect, the new interpretation is the only point at issue.
- 9.
We note that the frequency range is | θ | < π, so that the range of frequencies is limited. Intuitive support for this idea may be found in digital systems, in which frequencies are derived by dividing down the clock frequency. The clock (highest frequency) corresponds to θ = π, since e in π = (−1)n.
- 10.
The reader may ponder the fact that the labels “time” and “frequency” are in the present situation exactly reversed from the interpretation in Chapter 2. This is another aspect of the duality evident throughout transform theory.
- 11.
This argument is an analogue of the λ 2 = 0 cases encountered in boundary value problems.
- 12.
This result is closely connected to the continuous-time sampling theorem. It is even possible to derive (8.44) by use of the continuous-time theorem by associating continuous-time functions (via the sampling expansion) to the sequences under consideration. The approach taken here avoids this diversion and tends to emphasize the “frequency domain” interpretations of discrete Fourier transforms.
- 13.
One consequence of this periodicity is that both the original function and the transform may be regarded as functions whose domain is the group of the integers mod N. This set is isomorphic to the group of powers of the N th roots of unity, and is conventionally identified with a set of equally spaced points on the unit circle of the complex plane. This identification corresponds to the practice of regarding continuous Fourier series as being defined on the unit circle.
- 14.
It might be noted that these results can also be considered as exercises in the multiplication of polynomials in the N th root of unity.
- 15.
For the purposes of this order of magnitude accounting, we do not distinguish between real- and complex-valued operations.
- 16.
DSP performance changes rapidly. The web sites of manufacturers such as Texas Instruments, Motorola and Analog Devices should be consulted for current data sheets, user guides, and application notes.
- 17.
This is true except for the origin of the subscript count. The FORTRAN heritage of Octave shows through in the convention of numbering subscripts starting from 1.
References
A.V. Oppenheim, R.W. Schafer, Digital Signal Processing (Prentice-Hal1, Englewood Cliffs, 1975)
L.R. Rabiner, C.M. Rader (eds.), Digital Signal Processing (IEEE Press, New York, 1972)
Further Reading
P. Cootner (ed.) The Random Character of Stock Market Prices (MIT Press, Cambridge, 1964)
J.R. Johnson, Introduction to Digital Signal Processing (Prentice-Hal1, Englewood Cliffs, 1989)
E.I. Jury, Theory and Application of the Z-Transform Method (Wiley, New York, 1964)
Mathworks Incorporated, Signal Processing Toolbox User’s Guide (Mathworks Incorporated, Natick, 2000)
W. Rudin, Fourier Analysis on Groups (Interscience, New York, 1962)
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Davis, J.H. (2016). Discrete Variable Transforms. In: Methods of Applied Mathematics with a Software Overview. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-43370-7_8
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