Fundamentals of Fourier Analysis

2009 ◽  
Author(s):  
Robert J Marks II
Keyword(s):  
Fourier Transform ◽  
Discrete Time ◽  
Continuous Time ◽  
Fourier Integral ◽  
Time Signal ◽  
Continuous Sampling ◽  
Common Reference ◽  
Multidimensional Signals ◽  
Time Signals ◽  
The Fourier Transform

This chapter contains foundational material for modelling of signals and systems. Section 2.2 introduces classes of functions useful in signal processing and analysis. The Fourier transform, in Section 2.3, begins with the Fourier integral and develops the Fourier series, the discrete time Fourier transform and the discrete Fourier transform as special cases. The following material in this chapter can be skipped on a first reading. † denotes material relevant to multidimensional signals in Chapters 8 and 11. ‡ denotes material relevant to probability and stochastic processes in Chapter 4. ¶ denotes material used in continuous sampling in Chapter 10. There are a number of signal classes to which we will make common reference. Continuous time signals are denoted with their arguments in parentheses, e.g., x(t). Discrete time signals will be bracketed, e.g., x[n]. A continuous time signal, x(t), is periodic if there exists a T such that x(t) = x(t − T) for all t. The function x(t) = constant is periodic. A discrete time signal, x[n], is periodic if there exists a positive integer N such that x[n] = x[n − N] for all n. The function x[n] = constant is periodic.

Digital Filters

2011 ◽  
pp. 180-196
Author(s):  
Gordana Jovanovic-Dolecek
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Digital Filters ◽  
Time Signal ◽  
Digital Signals ◽  
Video Signals ◽  
Time Signals ◽  
Time Distance ◽  
Independent Variable ◽  
Distance Position

A signal is defined as any physical quantity that varies with changes of one or more independent variables, and each can be any physical value, such as time, distance, position, temperature, or pressure (Oppenheim & Schafer, 1999; Elali, 2003; Smith, 2002). The independent variable is usually referred to as “time”. Examples of signals that we frequently encounter are speech, music, picture, and video signals. If the independent variable is continuous, the signal is called continuous-time signal or analog signal, and is mathematically denoted as x(t). For discrete-time signals the independent variable is a discrete variable and therefore a discrete-time signal is defined as a function of an independent variable n, where n is an integer. Consequently, x(n) represents a sequence of values, some of which can be zeros, for each value of integer n. The discrete–time signal is not defined at instants between integers and is incorrect to say that x(n) is zero at times between integers. The amplitude of both the continuous and discrete-time signals may be continuous or discrete. Digital signals are discrete-time signals for which the amplitude is discrete. Figure 1 illustrates the analog and the discrete-time signals.

Discrete Time, Discrete Frequency

2021 ◽  
pp. 106-155
Author(s):  
Victor Lazzarini
Keyword(s):  
Signal Processing ◽  
Fourier Transform ◽  
Discrete Time ◽  
Time Domain ◽  
Continuous Time ◽  
Time Varying ◽  
Discrete Frequency ◽  
The Fourier Transform ◽  
Definition Of ◽  
The Time Domain

This chapter is dedicated to exploring a form of the Fourier transform that can be applied to digital waveforms, the discrete Fourier transform (DFT). The theory is introduced and discussed as a modification to the continuous-time transform, alongside the concept of windowing in the time domain. The fast Fourier transform is explored as an efficient algorithm for the computation of the DFT. The operation of discrete-time convolution is presented as a straight application of the DFT in musical signal processing. The chapter closes with a detailed look at time-varying convolution, which extends the principles developed earlier. The conclusion expands the definition of spectrum once more.

Deterministic Discrete-Time Signal Transforms

2021 ◽  
pp. 714-796
Author(s):  
Stevan Berber
Keyword(s):  
Fourier Transform ◽  
Discrete Time ◽  
Linear Time ◽  
Power Spectra ◽  
Time Signal ◽  
Linear Time Invariant ◽  
Energy Spectral Density ◽  
Analysis And Synthesis ◽  
Spectral Density Functions ◽  
Linear Time Invariant Systems

Chapter 15 presents a detailed analysis of discrete-time signals and systems in the frequency domain, including the theory of the discrete Fourier series, the discrete-time Fourier transform, and the discrete Fourier transform, and key examples relevant for the analysis and synthesis of signals processed in the discrete transceiver blocks of a communication system. Amplitude spectra, magnitude spectra, phase spectra, and power spectra are defined and calculated for typical signals. Using a unique notation that distinguishes between energy signals and power signals, the correlation function and power or energy spectral density functions are inter-related by proving the Wiener–Khintchine theorem. A comprehensive analysis of linear-time-invariant systems, using the notions of impulse responses, correlation functions, and power spectral densities for both power and energy signals, is presented. The basic theory of the z-transform is also presented.

Deterministic Continuous-Time Signals and Systems

2021 ◽  
pp. 562-598
Author(s):  
Stevan Berber
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Linear Time ◽  
Continuous Variable ◽  
Linear Time Invariant ◽  
Time Signals ◽  
Analysis And Synthesis ◽  
Linear Time Invariant Systems ◽  
Definition Of ◽  
Time Invariant

Due to the importance of the concept of independent variable modification, the definition of linear-time-invariant system, and their implications for discrete-time signal processing, Chapter 11 presents basic deterministic continuous-time signals and systems. These signals, expressed in the form of functions and functionals such as the Dirac delta function, are used throughout the book for deterministic and stochastic signal analysis, in both the continuous-time and the discrete-time domains. The definition of the autocorrelation function, and an explanation of the convolution procedure in linear-time-invariant systems, are presented in detail, due to their importance in communication systems analysis and synthesis. A linear modification of the independent continuous variable is presented for specific cases, like time shift, time reversal, and time and amplitude scaling.

The Spectra of Filters

2021 ◽  
pp. 204-268
Author(s):  
Victor Lazzarini
Keyword(s):  
Fourier Transform ◽  
Impulse Response ◽  
Discrete Time ◽  
Filter Design ◽  
Finite Impulse Response ◽  
Infinite Impulse Response ◽  
Main Body ◽  
Analysis Tool ◽  
The Fourier Transform ◽  
Definition Of

This chapter now turns to the discussion of filters, which extend the notion of spectrum beyond signals into the processes themselves. A gentle introduction to the concept of delaying signals, aided by yet another variant of the Fourier transform, the discrete-time Fourier transform, allows the operation of filters to be dissected. Another analysis tool, in the form of the z-transform, is brought to the fore as a complex-valued version of the discrete-time Fourier transform. A study of the characteristics of filters, introducing the notion of zeros and poles, as well as finite impulse response (FIR) and infinite impulse response (IIR) forms, composes the main body of the text. This is complemented by a discussion of filter design and applications, including ideas related to time-varying filters. The chapter conclusion expands once more the definition of spectrum.

Field of a Uniformly Moving Charge in an Anisotropic Dispersive Medium

1973 ◽  
Vol 28 (6) ◽  
pp. 907-910
Author(s):  
S. Datta Majumdar ◽  
G. P. Sastry
Keyword(s):  
Electromagnetic Field ◽  
Fourier Transform ◽  
Point Charge ◽  
Rest Frame ◽  
Dispersive Medium ◽  
Scalar Potential ◽  
Fourier Integral ◽  
The Third ◽  
Dispersion Formula ◽  
The Fourier Transform

The electromagnetic field of a point charge moving uniformly in a uniaxial dispersive medium is studied in the rest frame of the charge. It is shown that the Fourier integral for the scalar potential breaks up into three integrals, two of which are formally identical to the isotropic integral and yield the ordinary and extraordinary cones. Using the convolution theorem of the Fourier transform, the third integral is reduced to an integral over the isotropic field. Dispersion is explicitly introduced into the problem and the isotropic field is evaluated on the basis of a simplified dispersion formula. The effect of dispersion on the field cone is studied as a function of the cut-off frequency.

scholarly journals On the uncertainty inequality as applied to discrete signals

2006 ◽  
Vol 2006 ◽  
pp. 1-22 ◽  
Author(s):  
Y. V. Venkatesh ◽  
S. Kumar Raja ◽  
G. Vidyasagar
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Sampling Theorem ◽  
Discrete Signals ◽  
Bandlimited Signal ◽  
Time Signals ◽  
Shannon Sampling Theorem ◽  
Heisenberg Uncertainty ◽  
Comparison Of The Results ◽  
Uncertainty Inequality

Given a continuous-time bandlimited signal, the Shannon sampling theorem provides an interpolation scheme forexactly reconstructingit from its discrete samples. We analyze the relationship between concentration (orcompactness) in thetemporal/spectral domainsof the (i) continuous-time and (ii) discrete-time signals. The former is governed by the Heisenberg uncertainty inequality which prescribes a lower bound on the product ofeffectivetemporal and spectral spreads of the signal. On the other hand, the discrete-time counterpart seems to exhibit some strange properties, and this provides motivation for the present paper. We consider the following problem:for a bandlimited signal, can the uncertainty inequality be expressed in terms of the samples, using thestandard definitions of the temporal and spectral spreads of the signal?In contrast with the results of the literature, we present a new approach to solve this problem. We also present a comparison of the results obtained using the proposed definitions with those available in the literature.

On: “Gravity Interpretation Using the Fourier Integral”, by M. E. Odegard and J. W. Berg, Jr. (GEOPHYSICS, June 1965, p. 127–138)

Geophysics ◽  
1975 ◽  
Vol 40 (2) ◽  
pp. 356-357
Author(s):  
Jay Gopal Saha
Keyword(s):  
Fourier Transform ◽  
Gravity Anomaly ◽  
Fourier Integral ◽  
Two Dimensional ◽  
Transform Method ◽  
Reverse Problem ◽  
Vertical Fault ◽  
Vertical Step ◽  
The Fourier Transform ◽  
Gravity Interpretation

In their paper, Odegard and Berg claim that from the gravity anomaly due to a two‐dimensional vertical fault the density, the throw, and the depth can be determined uniquely by a Fourier transform method. It is true that the solution of the reverse problem for a two‐dimensional vertical step is theoretically unique. The derivation of the Fourier transform by the authors, however, is erroneous.

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