Three Interpretations of the Fourier Trasform The Fourier transform plays an important role in engineering and science. At one time or another,

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1 hree Interpretations of the Fourier rasform he Fourier transform plays an important role in engineering and science. At one time or another, all electrical engineering students use the transform. However, most elementary discussions do not emphasize the three distinctly different, signal-dependent interpretations that are given the transform. In applications, the Fourier transform integral 1) converges in a point-wise manner, ) converges in a mean-square manner, or 3) must be interpreted as a generalized function (sometimes called a distribution). In a qualitative, non-rigorous manner, these three distinctly different interpretations are discussed below. But first, we provide some relevant vector space definitions. I. Vector Space L p For p > 0, vector space L p [-, ] is the set of all signals f(t), - < t <, for which p f(t) dt. (1.1) In what follows, vector spaces L 1 and L are of interest since they play key roles in many applications. Space L 1 [-, ] (alternatively, L [-, ]) is the space of absolutely integrable (alternatively, square integrable finite energy) functions. As it turns out, neither of these spaces is a subset of the other, and signals exist that are in both L 1 [-, ] and L [-, ]. (However, if we restrict ourselves to functions on a finite-length interval [a, b], then L [a,b] L 1 [a,b]). II. Point-Wise Fourier ransform In most elementary engineering signal and system books, the Fourier transform is defined as j t j t, (1.) F( ) f (t)e dt limit f (t)e dt an improper integral. A sufficient condition for the convergence of improper integral (1.) is that function f L 1 [-, ]; that is, f(t) is absolutely intergrable. When it exists, F() given by (1.) is called 1/7/014 Page 1

2 hree Interpretations of the Fourier rasform the point-wise Fourier transform. As it turns out, for f(t) L 1 [-, ], the integral (1.) converges uniformly at each point, - < <. Hence, we use the name point-wise Fourier transform. Example: Consider f(t) = e -at U(t), a > 0. Clearly f(t) L 1 [-, ]; it is absolutely integrable. he pointwise transform is at j t 1 F( ) limit {e U(t)}e dt. (1.3) a j III. Fourier-Plancherel ransform here are useful finite-energy signals f(t) L [-, ] that are not absolutely integrable (that is, f(t) L 1 [-, ]), and they have no point-wise Fourier transform (f(t) = Sa(t) sin(t)/t is the classic example). We would like to extend definition (1.) of the Fourier transform to include these finite energy signals; we would like to find a Fourier transform operator for all f(t) L [-, ]. Such an extension exists; it is known as the Fourier-Plancherel transform (without any explanation, examples of this transform appear in almost all transform tables). For signals in L 1 [-, ] L [-, ] the Fourier- Plancherel transform will produce the same result as (1.). For f(t) L [-, ], the Fourier-Plancherel transform results from a mean-square interpretation of (1.), not the ordinary (point-wise) convergence of this improper integral. First, define the quantity F( j t ) f(t)e dt, (1.4) a function of and > 0. hen, the Fourier-Plancherel transform is the function F() for which limit F j t ( ) F( ) d limit f (t)e dt F( ) d 0. (1.5) Often, we denote this symbolically as 1/7/014 Page

3 hree Interpretations of the Fourier rasform j t, (1.6) F( ) l.i.m f (t)e dt where l.i.m stands for limit in the mean. Compare (1.6) and (1.); the difference is simple. As, the Fourier-Plancherel transform (1.6) uses a mean-square interpretation of the limit, and the point-wise transform (1.) uses a point-wise interpretation of the limit. Example he function f(t) = Sa(t) = sin(t)/t is the quintessential example where the Fourier- Plancherel transform must be used. For f(t) = Sa(t), the point-wise transform does not exist (the integral (1.) does not converge for f(t) = Sa(t)). Note that almost all Fourier transform tables list F Sa(t) rect( / ), (1.7) a -high rectangular pulse that extends over the interval (-1, 1). his is a Fourier-Plancherel transform. In what follows, an explanation is given that describes how (1.7) is obtained. Consider the integral -jt sin(t) -jt sin(t) F ( ) Sa(t) e dt e dt = costdt t 0 t. (1.8) Using simple identities, Equation (1.8) can be written as sin(1 )t sin(1 )t F( ) dt dt 0 t 0 t (1 ) sin (1 ) sin d d 0 0 Si (1 ) Si (1 ), (1.9) 1/7/014 Page 3

4 hree Interpretations of the Fourier rasform where Si(x) denotes the standard, tabulated Sine Integral x sin Si(x) d 0 (1.10) that appears as an intrinsic function in Matlab, Mathcad, etc. For - < <, F () does not have a (point-wise) limit as. Instead, F displays ripple behavior; as, the ripple increases in frequency while the peak ripple magnitude does not decrease (the peak ripple occurs near =. By considering (1.9) and the figures, study the behavior of F for near (were the ripple phenomenon is very evident). Figures 1 and show plots of F () for = 10 and = 50, respectively, as constructed by using Matlab (the sinint function). As dotted line graphs, both figures contain a plot of (1.7), the Fourier- Plancherel transform listed in commonly available tables. It should be evident that point-wise convergence does not occur. However, as, Equation (1.9) does have the mean square limit F() = rect(/) in that limit F ( ) rect( / ) d 0. (1.11) 4 F 10 () = si(10{+1}) + si({1-}10) F() = rect(/) 4 F 50 () = si(50{+1}) + si({1-}50) F() = rect(/) F() F 10 () 1 F() F 50 () Fig. 1: Case = Fig. : Case = 50. 1/7/014 Page 4

5 hree Interpretations of the Fourier rasform IV. Fourier ransform as a Generalized Function Of interest in electrical engineering are many signals, such as sin(t), that are not absolutely integrable, and they do not have finite energy (that is, they are not in L 1 or L ). However, they do have finite power, and they exhibit sufficient periodic (or other regular) behavior to allow their Fourier transform to be interpreted in terms of a generalized function (in terms of a delta function(s) in most elementary applications). j t e Example: Consider f(t) = 0. Use this signal in (1.4) to write j t j t j( )t F( ) e 0 e dt e 0 dt (1.1) For (1.1), simple manipulation yields sin( 0) F ( ) Sa ( 0) ( 0). (1.13) It is easily shown that there is units of area under F (), < <. Also, as, this area is concentrated in an infinitesimally small interval centered at 0. hat is, one can give the interpretation 0. (1.14) F( ) ( ) j t e he Fourier transform of f(t) = 0 has been interpreted in terms of a delta function (a commonly-used generalized function). Summary In electrical engineering, there are three commonly-used interpretations of the Fourier transform integral. In a given problem, the interpretation that is relevant depends on the nature of the function f(t) that is to be transformed. If f(t) is absolutely integrable (i.e., f L 1 [-, ]), then the Fourier integral can be interpreted in a point-wise manner (convergence of the transform is uniform in frequency ). On the 1/7/014 Page 5

6 hree Interpretations of the Fourier rasform other hand, suppose that f(t) is not absolutely integrable but that it has finite energy (i.e., f L [-, ]). In this case, the Fourier-Plancherel transform must be used, a mean-square interpretation of the Fourier integral. Finally, there are many important signals that are not absolutely integrable, and they do not have a finite energy. For example, consider signals that are composed of sums of sinusoidal functions. hese signals have Fourier transforms that are interpreted in terms of generalized functions (i.e., delta functions). 1/7/014 Page 6