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# fft - 453.701 Linear Systems S.M Tan The University of...

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453.701 Linear Systems, S.M. Tan, The University of Auckland 9-1 Chapter 9 The Discrete Fourier transform 9.1 De ° nition When computing spectra on a computer it is not possible to carry out the integrals involved in the continuous time Fourier transform. Instead a related transform called the discrete Fourier transform is used. We shall examine the properties of this transform and its relationship to the continuous time Fourier transform. The discrete Fourier transform (also known as the ° nite Fourier transform) relates two ° nite se- quences of length N . Given a sequence with components x [ k ] for k = 0 ; 1 ; :::; N ¡ 1 , the discrete Fourier transform of this sequence is a sequence X [ r ] for r = 0 ; 1 ; :::; N ¡ 1 de ° ned by X [ r ] = 1 N N 1 X k =0 x [ k ] exp ° ¡ j 2 …rk N (9.1) The corresponding inverse transform is x [ k ] = N 1 X r =0 X [ r ] exp ° j 2 …rk N (9.2) Notice that we shall adopt the de ° nition in which there is a factor of 1 =N in front of the forward transform. Other conventions place this factor in front of the inverse transform (this is used by MatLab) or a factor of 1 = p N in front of both transforms. Exercise: Show that these relationships are indeed inverses of each other. It is useful ° rst to establish the relationship N 1 X r =0 exp ° j 2 …rk N = N if k is a multiple of N 0 otherwise (9.3) 9.2 Discrete Fourier transform of a sampled complex exponential A common application of the discrete Fourier transform is to ° nd sinusoidal components within a signal. The continuous Fourier transform of exp( j 2 …” 0 t ) is simply a delta function ( ¡ 0 ) at the frequency 0 . We now calculate the discrete Fourier transform of a sampled complex exponential x [ k ] = A exp ( j 2 …” 0 k¿ ) ; for k = 0 ; 1 ; : : : ; N ¡ 1 (9.4) The sampling interval is ¿ and we denote the duration of the entire sampled signal by T = N¿ . Substituting (9.4) into the de ° nition of the discrete Fourier transform (9.1) yields X [ r ] = 1 N N 1 X k =0 A exp ° j 2 …k N [ 0 T ¡ r ] (9.5) If 0 T is an integer, i.e., if there are an integer number of cycles in the frame of duration T , we see that X [ r ] = A if r ¡ 0 T is a multiple of N 0 otherwise (9.6)

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453.701 Linear Systems, S.M. Tan, The University of Auckland 9-2 In this case, there is only a single non-zero term in the discrete Fourier transform at an index which depends on the frequency 0 . The value of this non-zero term is A , the complex amplitude of the component. If 0 T is not an integer however, we ° nd that X [ r ] = A N exp ( ¡ j… ( N ¡ 1)( r ¡ 0 T ) =N ) sin[ ( r ¡ 0 T )] sin[ ( r ¡ 0 T ) =N ] (9.7) This is nonzero for all values of r . Thus even though there is only a single frequency component in the signal, it can a ff ect all the terms in the discrete Fourier transform. A plot of the last factor in the equation (9.7) is shown in Figure 9.1. Since r only takes on integer values, the terms in the output sequence are samples from this function taken at unit spacing.
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fft - 453.701 Linear Systems S.M Tan The University of...

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