Abstract
Discrete-time periodic signals, i.e., signals defined on ℤ(T)/ℤ(T p ), have become important because they are the only class of one-dimensional signals that can be processed directly by a computer, for the reason that they are fully specified by a finite number of values N=T p /T. This assertion holds also in the frequency domain, where the dual group is
and the Fourier transform is a discrete-frequency periodic function that is specified by the same number of values N=F p /F. It is worth insisting on the fact that all other 1D signals (continuous- time, discrete-time) when simulated on a digital computer must be related to and approximated by signals on ℤ(T)/ℤ(T p ).
In this chapter signals on ℤ(T)/ℤ(T p ) are developed with the purpose of formulating the background for a study of the signals with a digital computer. Implementation techniques will be seen in the next chapter. In particular, the discrete Fourier transform (DFT) and the discrete cosine transform (DCT) will be studied in great detail.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
“Symmetry” is intended in the generalized sense introduced at the end of Chap. 4.
- 2.
This choice yields the property that the DCT and IDCT matrices are each others transpose.
References
N. Ahmed, T. Natarajan, K.R. Rao, Discrete cosine transform. IEEE Trans. Comput. C–23, 90–93 (1974)
G. Cariolaro, T. Erseghe, The fractional discrete cosine transform. IEEE Trans. Signal Process. 50, 902–911 (2002)
G. Cariolaro, T. Erseghe, P. Kraniauskas, N. Laurenti, A unified framework for the fractional Fourier transform. IEEE Trans. Signal Process. 46, 3206–3219 (1998)
G. Cariolaro, T. Erseghe, P. Kraniauskas, N. Laurenti, Multiplicity of fractional Fourier transforms and their relationships. IEEE Trans. Signal Process. 48, 227–241 (2000)
T. Erseghe, G. Cariolaro, An orthonormal class of exact and simple DFT eigenfunctions with a high degree of symmetry. IEEE Trans. Signal Process. 51(10), 2527–2539 (2003)
B. Haskell, A. Puri, A.N. Netravali, Digital Video: An Introduction to MPEG-2 (Chapman & Hall, London, 1997)
A.K. Jain, Fundamentals of Digital Image Processing (Prentice Hall, Englewood Cliffs, 1999)
J.L. Mitchell, MPEG Video Compression Standard (Chapman & Hall, London, 1997)
W.B. Pennebaker, J.L. Mitchell, JPEG Still Image Data Compression Standard (Van Nostrand/Reinhold, London, 1993)
P.P. Vaidyanathan, Multirate Systems and Filter Banks (Prentice Hall, Englewood Cliffs, 1993)
M. Vetterli, J. Kovac̆ević, Wavelets and Subband Coding (Prentice Hall, Englewood Cliffs, 1995)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2011 Springer-Verlag London Limited
About this chapter
Cite this chapter
Cariolaro, G. (2011). Signals on ℤ(T)/ℤ(T p ). In: Unified Signal Theory. Springer, London. https://doi.org/10.1007/978-0-85729-464-7_12
Download citation
DOI: https://doi.org/10.1007/978-0-85729-464-7_12
Publisher Name: Springer, London
Print ISBN: 978-0-85729-463-0
Online ISBN: 978-0-85729-464-7
eBook Packages: EngineeringEngineering (R0)