Fourier Transform Properties

9 Key excerpts on "Fourier Transform Properties"

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  Practical Signals Theory with MATLAB Applications

  • Richard J. Tervo(Author)
  • 2013(Publication Date)
  • Wiley

    (Publisher)

  Many properties of the Fourier transform and its inverse can be revealed by manipulating the defining equations. This procedure establishes a firm mathematical basis for each of the properties and all the rules relating the behavior of signals expressed in one domain or the other. To illustrate this approach, some useful properties will be introduced immediately, after which the Fourier transform of the real nonperiodic rectangle signal will be computed. 5.2.1 Linearity of the Fourier Transform The Fourier transform is called a linear transformation since the Fourier transform of a linear combination of signals equals the linear combination of their respective Fourier transforms. This observation may be written as: THEOREM 5.1 (Linearity) If s 1 ðt Þ! F S 1 ð f Þ and s 2 ðt Þ! F S 2 ð f Þ then for constants k 1 and k 2 ½k 1 s 1 ðt Þþ k 2 s 2 ðt Þ! F ½k 1 S 1 ð f Þþ k 2 S 2 ð f Þ 178 Chapter 5 The Fourier Transform Proof: This result follows directly from the definition of the Fourier transform and the inherent linearity of integration: LHS ¼ Z þ N  N ½k 1 s 1 ðt Þþ k 2 s 2 ðt Þe j 2πft dt ¼ k 1 Z þ N  N s 1 ðt Þe j 2πft dt þ k 2 Z þ N  N s 2 ðt Þe j 2πft dt ¼ k 1 S 1 ð f Þþ k 2 S 2 ð f Þ¼ RHS The same approach can be applied to the inverse Fourier transform. The linearity of the Fourier transform is fundamental to its use in signals analysis. It implies that if a signal g ðt Þ can be represented as some linear combination of known signals fs n ðt Þg, then the Fourier transform G ð f Þ can be determined directly from the fS n ð f Þg without further computation. This technique can be applied either graphically or directly by computation.

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  An Introduction to Analog and Digital Communications

  • Simon Haykin, Michael Moher(Authors)
  • 2012(Publication Date)
  • Wiley

    (Publisher)

  2.5.  2.2 Properties of the Fourier Transform It is useful to have insight into the relationship between a time function and its Fourier transform and also into the effects that various operations on the function have on the transform This may be achieved by examining certain properties of the Fourier transform. In this section, we describe fourteen properties, which we will prove, one by one. These properties are summarized in Table A8.1 of Appendix 8 at the end of the book. PROPERTY 1 Linearity (Superposition) Let and Then for all constants and we have (2.14) The proof of this property follows simply from the linearity of the integrals defining and Property 1 permits us to find the Fourier transform of a function that is a linear combination of two other functions and whose Fourier transforms and are known, as illustrated in the following example. G 2 1f 2 G 1 1f 2 g 2 1t2 g 1 1t2 g1t2 G1f 2 g1t2. G1f 2 c 1 g 1 1t2  c 2 g 2 1t2 Δ c 1 G 1 1f 2  c 2 G 2 1f 2 c 2 , c 1 g 2 1t2 Δ G 2 1f 2. g 1 1t2 Δ G 1 1f 2 G1f 2. g1t2 G1f 2, g1t2 G1f 2 u1t2 g1t2  exp1t2 sin12pf c t2u1t2, exp1at2u1t2 Δ 1 a  j2pf FIGURE 2.5 Frequency function for Problem 2.2. G1f 2 26 CHAPTER 2  FOURIER REPRESENTATION OF SIGNALS AND SYSTEMS 0.366 t g( t) (a) 1.0 –1.0 0 t g( t) (b) 1.0 –— 1 a — 1 a FIGURE 2.6 (a) Double-exponential pulse (symmetric). (b) Another double-exponential pulse (odd-symmetric). EXAMPLE 2.3 Combinations of Exponential Pulses Consider a double exponential pulse (defined by (see Fig. 2.6(a)) (2.15) This pulse may be viewed as the sum of a truncated decaying exponential pulse and a truncated rising exponential pulse. Therefore, using the linearity property and the Fourier-transform pairs of Eqs. (2.12) and (2.13), we find that the Fourier transform of the double exponential pulse of Fig. 2.6(a) is We thus have the following Fourier-transform pair for the double exponential pulse of Fig.

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  Sinusoids

  Theory and Technological Applications

  • Prem K. Kythe(Author)
  • 2014(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  3 FourierTransforms Fourier transforms (FTs) constitute the central theme of this book. They are used in every technological application which starts from Chapter 4 onward. The purpose of the Fourier transform is to represent an image in terms of sine and cosine functions (sinusoids). Using these two basic functions, it is easy to perform processing operations, especially frequency domain filtering, which after inversion using inverse Fourier transform (IFT) is converted back to the spatial domain. Using Fourier transform, any digital image can be represented as a weighted sum of sine and cosine functions. These weights are used to reconstruct the image using the inverse Fourier transform. Linear image processing operations can be implemented using convolution in the spatial domain or by filtering in the frequency domain. Some image processing operations perform better if frequency domain solutions are used. 3.1 Definitions Let f ( x ), -∞

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  Fourier Transforms in Spectroscopy

  • Jyrki Kauppinen, Jari Partanen(Authors)
  • 2011(Publication Date)
  • Wiley-VCH

    (Publisher)

  24 2 General properties of Fourier transforms and { H ( f )} = cos { H ( f )} + i sin { H ( f )}. (2.7) Often, i sin { H ( f )} equals zero. Figure 2.1: The amplitude spectrum | H ( f )| and the phase spectrum θ( f ). In the following, a collection of theorems of Fourier analysis is presented. They contain the most important characteristic features of Fourier transforms. 2.1 Shift theorem Let us consider how a shift ± f 0 of the frequency of a spectrum H ( f ) affects the corresponding signal h(t ), which is the Fourier transform of the spectrum. We can find this by making a change of variables in the Fourier integral: { H ( f ± f 0 )} = ∞ −∞ H ( f ± f 0 )e i 2π f t d f g= f ± f 0 dg=d f = ∞ −∞ H (g)e i 2π(g∓ f 0 )t dg = e ∓i 2π f 0 t ∞ −∞ H (g)e i 2π gt dg = h(t )e ±i 2π f 0 t . Likewise, we can obtain the effect of a shift ±t 0 of the position of the signal h(t ) on the spectrum H ( f ), which is the inverse Fourier transform of the signal. These results are called the shift theorem. The theorem states how the shift in the position of a function affects its transform. If H ( f ) = −1 {h(t )}, and both t 0 and f 0 are constants, then { H ( f ± f 0 )} = { H ( f )}e ∓i 2π f 0 t = h(t )e ∓i 2π f 0 t , −1 {h(t ± t 0 )} = −1 {h(t )}e ±i 2π f t 0 = H ( f )e ±i 2π f t 0 . (2.8) 2.2 Similarity theorem 25 This means that the Fourier transform of a shifted function is the Fourier transform of the unshifted function multiplied by an exponential wave or phase factor. The same holds true for the inverse Fourier transform. If a function is shifted away from the origin, its transform begins to oscillate at the frequency given by the shift. A shift in the location of a function in one domain corresponds to multiplication by a wave in the other domain. 2.2 Similarity theorem Let us next examine multiplication of the frequency of a spectrum by a positive real constant a.

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  Partial Differential Equations

  A First Course

  • Rustum Choksi(Author)
  • 2022(Publication Date)
  • American Mathematical Society

    (Publisher)

  Before continuing, a few comments about definition (6.5) are in order. • The Fourier transform is related to the original function in an awfully convoluted and nonlocal way, meaning that to find ̂ ? at any point ? (say, ? = 3 ), we need to incorporate (integrate) the values of the original function ? everywhere. • For each ? , it follows that ̂ ?(?) is a complex number. So while ? , ?, and the values of ?(?) are real, in general, the values of ˆ ?(?) are complex. However, as we shall see, the Fourier transform of several important functions has no imaginary part; i.e., it is real valued. 2 The definition naturally extends to complex-valued functions (cf. Section 6.2.5). 6.2. • Definition of the Fourier Transform and Its Fundamental Properties 207 • By Euler’s formula ? −𝑖?? = cos ?? − 𝑖 sin ??, (6.6) meaning that ? −𝑖?? lies on the unit circle in the complex plane. However, note that in this chapter the integration is, and will always be, with respect to a real variable. This “real” integration of the complex-valued integrand can be viewed as the separate integration of its real and complex parts. • As a function of ? , ˆ ? is bounded. Indeed, since |? −𝑖?? | = 1 , we have for all ? ∈ ℝ , | ˆ ?(?)| ≤ ∫ ∞ −∞ |?(?)| |? −𝑖?? | ?? = ∫ ∞ −∞ |?| ?? < ∞. In fact, one can show that ˆ ? is also a continuous function of ? (see Section 6.5). • The Fourier transform is a linear operation on functions. That is, if ? 1 and ? 2 are integrable functions and ?, ? ∈ ℝ (or even ℂ ), then ˆ (?? 1 + ?? 2 ) (?) = ? ˆ ? 1 (?) + ? ˆ ? 2 (?). We give a simple yet important example. Example 6.2.1. Let ? > 0 be fixed and consider the characteristic function of the interval [−?, ?] : ?(?) = { 1 if |?| < ?, 0 if |?| ≥ ?. We find that ˆ ?(?) = ∫ ∞ −∞ ?(?) ? −𝑖?? ?? = ∫ ? −? ? −𝑖?? ?? = ? −𝑖?? − ? 𝑖?? −𝑖? = 2 sin ?? ? , where we used Euler’s formula (6.6) in the last equality.

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  Time-Frequency/Time-Scale Analysis

  • Patrick Flandrin(Author)
  • 1998(Publication Date)
  • Academic Press

    (Publisher)

  Finally, we mention as a third and more pragmatic reason, that the collection of these advantages has led to the development of a large number of algorithms, programs, processors, and machines for fre-quency analysis, all of which contribute to its good reputation for practical use. Limitations. Important though it is from a mathematical point of view, the Fourier analysis possesses several restrictions concerning its physical interpretation and its range of applicability. In order to explain this, let us look at the usual definition of the Fourier transform Evidently, the computation of one frequency value X(i/) necessitates the knowledge of the complete history of the signal ranging from — oo to +oo. Conversely, the inverse Fourier transform is given by xit)= / X(i/)e^2^'''di/. (1.2) J —'X, Hence, any value xif) of the signal at one instant t can be regarded as an in-inite superposition of complex exponentials, or everlasting and completely nonlocal waves. Even if this mathematical point of view may reveal the true properties of a signal in certain cases (ciuasi-monochromatic situations, steady state, etc.), it can also distort the physical reality. This happens, for example, with transient signals, which vanish outside a certain time Chapter 1 The Time-Fmqimncy PnAilem 11 interval (e.g., by switching a machine on and off). Although the nullity of its values is reflected by the Fourier analysis, this occurs only in an ar-tificial manner: it results from an infinite superposition of virtual waves that interfere such that they annihilate each other. Hence the situation on the domain, where the signal vanishes, can be described as a dynamic zero (there exist waves whose resulting contribution is zero by interference). This contradicts any proper understanding of the real pliyeical situation as a static zero (the signal does not exist). Citations. In this regard we wish to give prominence to three particu-larly appropriate citations.

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  Applied Digital Signal Processing

  Theory and Practice

  • Dimitris G. Manolakis, Vinay K. Ingle(Authors)
  • 2011(Publication Date)
  • Cambridge University Press

    (Publisher)

  Therefore, there are four types of signal and related Fourier transform and series representations which are summarized in Figure 4.33. • All Fourier representations share a set of properties that show how different charac-teristics of signals and how different operations upon signals are reflected in their spectra. The exact mathematical descriptions of these properties are different for each representation; however, the underlying concept is the same. 189 Terms and concepts SPECTRA SIGNALS 1 2 p 2 p 2 p N 0 ¥ ¥ – ¥ k= – ¥ x ( t ) = x ( t ) = x [ n ] = ~ ~ N– 1 k= 0 x [ n ] = ~ ~ ~ ~ X (j W )e j W t d W X (e j w )e j w n d w ~ ~ c k e j kn c k e j k W 0 t ¥ − ¥ X (j W ) = x ( t )e – j W t d t x ( t )e – j k W 0 t d t x [ n ]e – j w n X (e j w ) = c k = c k = 1 2 p 1 N 2 p N N– 1 n= 0 x [ n ]e –j kn ¥ n=– ¥ DTFT CTFS CTFT DTFS 1 T 0 T 0 0 Figure 4.33 Summary of four Fourier representations. TERMS AND CONCEPTS Amplitude spectrum A graph of the Fourier series coefficients or transform as a function of frequency when these quantities are real-valued. Analog frequency Represents a number of occurrences of a repeating event per unit time. For sinusoidal signals, the linear frequency, F , is measured in cycles per second (or Hz) while the angular (or radian) frequency, = 2 π F , is measured in radians per second. Autocorrelation sequence A sequence defined by r x [ ] = ∑ ∞ n =−∞ x [ n ] y [ n − ] that measures a degree of similarity between samples of a real-valued sequence x [ n ] at a lag . Continuous-Time Fourier Series (CTFS) Expresses a continuous-time periodic signal x ( t ) as a sum of scaled complex exponentials (or sinusoids) at harmonics kF 0 of the fundamental frequency F 0 of the signal. The scaling factors are called Fourier series coefficients c k . Continuous-Time Fourier Transform (CTFT) Expresses a continuous-time aperiodic signal x ( t ) as an integral of scaled complex exponentials (or sinusoids) of all frequencies.

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  Measure and Integral

  An Introduction to Real Analysis, Second Edition

  • Richard L. Wheeden(Author)
  • 2015(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  13 The Fourier Transform In this chapter, we will study properties of the Fourier transform f ( x ) of a function f on R n , n ≥ 1, defined formally (for the moment) as f ( x ) = 1 ( 2 π ) n R n f ( y ) e − i x · y d y , x ∈ R n . (13.1) Here x · y = ∑ n 1 x k y k is the usual dot product of x = ( x 1 , . . . , x n ) and y = ( y 1 , . . . , y n ) , and i is the complex number i = √ − 1 = e i π / 2 . Both f and f may be complex-valued. Different normalizations of f are common in the literature, such as 1 ( 2 π ) n / 2 R n f ( y ) e − i x · y d y , R n f ( y ) e 2 π i x · y d y , . . . , but the important properties of f are unaffected by normalization, and passing from one normalization to another is easy by scaling. We will often abuse notation by denoting f ( x ) = f ( x ) or f ( x ) = ( f ( x ) ) instead of the more cumbersome notations f ( y )( x ) , ( f ( · )) ( x ) , etc. For exam-ple, we will do this in Theorem 13.8 when computing the Fourier transform of e −| x | 2 since the notations e −| x | 2 and ( e −| x | 2 ) are somewhat simpler than ( e −| · | 2 ) ( x ) . Note that f ( x ) is a formal analogue of the sequence { c j } ∞ −∞ of trigonometric Fourier coefficients of a periodic function on the line, with the continuous variable x now playing the role of j . One of our main goals is to derive an analogue of Parseval’s formula (12.15), that is, to prove that the mapping f → f is essentially an isometry on L 2 ( R n ) . Of course, an important requirement for achieving this is to find an interpretation of f in case f ∈ L 2 ( R n ) . Unlike the formulas for Fourier coef-ficients in the one-dimensional periodic case, the integral in (13.1) may not converge absolutely for every f ∈ L 2 ( R n ) . However, as is easy to see, (13.1) does converge absolutely if f ∈ L 1 ( R n ) . Properties of f when f ∈ L 1 ( R n ) are simpler to derive precisely because of this absolute convergence, and we will 371

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  Window Functions and Their Applications in Signal Processing

  • K. M. M. Prabhu(Author)
  • 2018(Publication Date)
  • CRC Press

    (Publisher)

  After differ-entiation, we find that the higher frequency components of the signal become more pronounced. This property can be stated as follows: dx ( t ) dt F ←→ j X ( j ) . (1.31) In a similar manner, integrating the time-domain signal results in the following: t −∞ x ( t ) dt F ←→ 1 j X ( j ) + π X ( 0 )δ() . (1.32) Integration attenuates the magnitude of the signal at higher fre-quencies and thus acts like a low-pass filter. If X ( 0 ) is nonzero, the signal contains a DC component, which introduces an impulse in the frequency-domain. 5. Time and frequency scaling property: For a scaling factor ‘a’, this property can be given as follows: x ( at ) F ←→ 1 | a | X j a . (1.33) For instance, if a = − 1 then x ( − t ) F ←→ X ( − j ) . (1.34) If the scaling factor has a magnitude greater than unity, then the signal is compressed in the time-domain, while its frequency spectrum gets expanded. For | a | < 1, exactly the converse happens, that is, the time-domain signal is expanded and the spectrum is scaled down. Fourier Analysis Techniques for Signal Processing 9 6. Duality property: This property reveals the effect when we interchange the roles of t and . This helps in finding the CTFTs of some sig-nals directly from a table of transforms. It simply states that every property of CTFT has a dual function, given as follows: X ( t ) F ←→ 2 π x ( − j ) . (1.35) 7. Parseval’s theorem: This gives the relation between the energy ( E ) of a signal in the time-domain and the frequency-domain. We can use this property to easily compute the energy of a signal by integrating the squared magnitude of its FT. E = ∞ −∞ | x ( t ) | 2 dt = 1 2 π ∞ −∞ X ( j ) 2 d . (1.36) 8. Convolution property: Convolution in the time-domain is equivalent to multiplication in the frequency-domain and vice versa. x ( t ) ∗ h ( t ) F ←→ X ( j ) H ( j ) . (1.37) This property is vital especially in the analysis of linear time-invariant (LTI) systems.

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