Abstract
The concept of signal conditioning holds true both for synthesis and analysis domain. When we are considering the discrete signal processing concept, the signal must be discrete both in time/ space domain and in frequency domain. But by Discrete Time Fourier Transform (DTFT), the discrete time signal is transformed into continuous frequency signal in analysis domain. Therefore, our objective should be to condition/ process the continuous periodic spectrum such that the spectrum can be used by discrete signal processors like computers and hand-held devices like mobile phones, directly. In the present chapter, we have first introduced the classical algorithm of Discrete Fourier Transform (DFT). A complete flow from analog continuous time signal to discrete spectrum analysis is represented. Next, a clockwise rotated phasor (twiddle factor) is presented to simplify the DFT algorithm. Some interesting observations related to the DFT operation both in electrical signal (1D) and image (2D) are also presented at the rear end of the chapter.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
DTFT always generates frequency-repetitive signal with \( F_{repeat} \) interval \( 2\pi \).
- 2.
The direction of phasor is clockwise as it’s defined as a negative exponential.
- 3.
Dimension 1: X-direction of the image, Dimension 2: Y-direction of the image,
Dimension 3: Intensity of the image.
- 4.
Entropy is measured as non-homogeneity or randomness in a signal/ message [6].
- 5.
weight \( F\left( {u,v} \right) \)
References
Cooley, J.W., Tukey, J.W.: An algorithm for the machine calculation of complex Fourier series. Math. Comput. 19(90), 297–301 (1965)
Carl, F.G.: Nachlass: Theoria interpolationis methodo nova tractata, Werke band 3, pp. 265–327. Königliche Gesellschaft der Wissenschaften, Göttingen (1866)
Heideman, M.T., Johnson, D.H., Burrus, C.S.: Gauss and the history of the fast Fourier transform. IEEE ASSP Mag. 1(4), 14–21 (1984)
Heideman, M.T., Burrus, C.S.: On the number of multiplications necessary to compute a length-2n DFT. IEEE Trans. Acoust. Speech. Sig. Proc. 34(1), 91–95 (1986)
Gonzalez, R.C., Woods, R.E.: Digital Image Processing. Pearsen Education, Inc, Upper Saddle River (2002)
Das, A.: Digital Communication: Principles and System Modelling. Springer, Berlin (2010)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Das, A. (2012). Discrete Fourier Transform. In: Signal Conditioning. Signals and Communication Technology. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28818-0_7
Download citation
DOI: https://doi.org/10.1007/978-3-642-28818-0_7
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-28274-4
Online ISBN: 978-3-642-28818-0
eBook Packages: EngineeringEngineering (R0)