discrete time Fourier transform in relation with continuous time Fourier transform

Fourier TransformsDlmfMathworldPlanetmath of Sampled SignalsFernando Diego Sanz Gamiz

Computers are able to handle only finite number of data. Hence, if we are to study and treat real world signals (i.e. functionsMathworldPlanetmath ) in a computer, a way to characterize signals by a finite number of data has to be found.

If we sample the values of the signal x(t) at periodic times t=nT,n we form the sequence xd[n]=x(nT). Does this sequence contain all the information relative to x(t)?

We know from the sampling theoremMathworldPlanetmath that if the signal x(t) is bandlimited, the sampled sequence allows us to recover the original continuousMathworldPlanetmathPlanetmath time signal provided the sampling frequency 1/T is at least twice the maximum frequency of the signal. However, real signals are not of finite bandwidth as this would imply the signal to be of infiniteMathworldPlanetmath time duration. Therefore, a problem arise of how well can we approximate the original signal x(t) by the sampled sequence xd[n]. In fact, we are interested in studying the spectrum of the original signal based upon the samples xd[n].

While the relationMathworldPlanetmath between xd[n] and the spectrum of x(t) is widely used in communication and electronic engineering books, it is difficult to find a rigorous proof. We cover here the gap between engineering daily knowledge and rigorous mathematical proof of the named relations establishing under what assumptionsPlanetmathPlanetmath those relations are valid.

Relation between Discrete Time Fourier Transform and Continuous time Fourier transform

Theorem 1.

Let x:RC be a bounded variationMathworldPlanetmath (it can be, in particular, piecewise smooth) L1 function and let X(ω) be its Fourier transform 11in the present entry we will take the Fourier transform of x(t) to be X(ω)=-+x(t)e-jwt𝑑ω. Let TR. If gn(ω)=k=-n+nX(ω-2πkT) convergesPlanetmathPlanetmath a.e as n+ and is there is M such that |gn(ω)|<M a.e. then

1Tk=-+X(ω-2πkT)=k=-+x(kT+)+x(kT-)2e-jωk in L2([-π,π]) (1)

If additionally k=-+X(ω-2πkT) is continuous, the right hand side of (1) converges uniformly to the left hand side in any closed intervalDlmfMathworldPlanetmath [a,b][-π,π].


By hypothesisMathworldPlanetmath we can form the function


This function is obviously periodic of period 2π and bounded, hence it can be expanded in its Fourier series which converge in L2([-π,π]); the Fourier theory shows that the convergence is uniformly in [a,b][-π,π] if g(ω) is continuous.


where the coefficients ck are given by


As |gn(ω)|<M we can appeal the dominated convergence theorem to write

12π-ππlimn+gn(ω)e-jωkdω = limn+12π-ππgn(ω)e-jωk𝑑ω
= 12πk=-+-π-2kππ+2kπX(ωT)e-jωk𝑑ω
= T2πP.V.-+X(ω)ejω(-kT)𝑑ω

Now, as x(t) is of bounded variation, the Jordan theorem on Fourier transform inversion says that


and the result follows.

There are, however, very which do not satisfy the conditions required in the theoremMathworldPlanetmath above. Such is the case of the pulse function


whose Fourier transform is sin(x/2)x/2 -with the definition made in footnote 1-. sinc(x) behaves as 1x, so k=-+sinc(ω-2πkT) will not converge in general 22for monotone decreasingMathworldPlanetmath functions ”series behaves as integralsDlmfPlanetmath”, that is, if k=0+f(x-kM) converges or diverges so does 0+f; it will converge for those T that make the series an alternating (http://planetmath.org/AlternatingSeries) one, but not for the rest values of T. Therefore we need another result which somehow relates both sides of eq 1.

As we have pointed out, the problem with the pulse function is that its Fourier transform does not decay rapid enough for the series to converge. So we will try smoothing the signal out so that its Fourier transform will decay faster and, hopefully, the series converges. We wish the smoothed version of x(t) to resemble the original signal, so uniform approximation seems reasonable. But, as we will see, for an infinite number of samples {nT,n} each of these might require a different degree of approximation and it could be impossible to find an uniform approximation for all the samples. So, we will focus on time limited signal, for which we have the following result.


𝒟() will denote the set of test functions on and 𝒮 the set of rapidly decreasing functions on -the Schwartz space-. The symbol ^ above a function will denote its Fourier transform. We know that 𝒟()𝒮 and that the Fourier transform is an isomorphismPlanetmathPlanetmathPlanetmath of 𝒮 onto itself. The productPlanetmathPlanetmath of two functions X and ϕ at ω will be denoted by Xϕ(ω). ConvolutionMathworldPlanetmath will be denoted by .

Theorem 2.

Let ϕD(R) be a test function and {ϕj(t)=jϕ(jt),j=1,2,} be an approximate identity. Let x(t) be a time limited bounded variation signal. If k=-+Xϕj^(ω-2πkT) converges for all j=1,2, to a continuous function, then

limj1Tk=-+Xϕj^(ω-2πkT)=k=-+x(kT+)+x(kT-)2e-jωk (2)

Take the signal (xϕj)(t) whose Fourier transform is Xϕ^(ω). This signal satisfies, by hypothesis, the conditions of Theorem 1, so we can write


The right hand side is actually a sum of a finite number of terms since the signal (xϕj)(t) is time limited -being the convolution of two compact supported functions-. This makes the Fourier series a continuous function, which, together with the hypothesis that 1Tk=-+Xϕj^(ω-2πkT) is continuous, shows that equality in the above equation is pointwisePlanetmathPlanetmath.

Now let j and use the fact that {ϕj(t)} is an approximate identity to obtain equation (2). ∎

The rationale behind choosing test functions in the last theorem is that, in most cases,k=-+Xϕj^(ω-2πkT) will converge even though k=-+X(ω-2πkT) do not. This is because rapidly decreasing functions decay faster than 1xα for any α. So, for example, the Fourier transform of the pulse function has been tamed enough to make the series converge.

Remark 1.

When the signal x(t) is continuous, the right hand side of eqs. (1) or (2) reads


where xd[k]=x(kT). This is defined as the (Discrete Time) Fourier Transform, DTFT of the sequence xd[n]=x(nT)

Title discrete time Fourier transform in relation with continuous time Fourier transform
Canonical name DiscreteTimeFourierTransformInRelationWithContinuousTimeFourierTransform
Date of creation 2013-03-22 17:37:45
Last modified on 2013-03-22 17:37:45
Owner fernsanz (8869)
Last modified by fernsanz (8869)
Numerical id 10
Author fernsanz (8869)
Entry type Theorem
Classification msc 94A20
Related topic SamplingTheorem
Related topic ApproximatingFourierIntegralsWithDiscreteFourierTransforms
Related topic Distribution4
Related topic SpaceOfRapidlyDecreasingFunctions
Defines DTFT
Defines Discrete Time Fourier Transform