Transcription of Lecture 11: Discrete-time Fourier transform
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Lecture 11: Discrete-time Fourier transform
TransformThe Discrete-time Fourier transform has essentially the same properties asthe continuous- time Fourier transform , and these properties play parallelroles in continuous time and discrete time . As with the continuous- time Fourier transform , the Discrete-time Fourier transform is a complex-valued func-tion whether or not the sequence is real-valued. Furthermore, as we stressedin Lecture 10, the Discrete-time Fourier transform is always a periodic func-tion of fl. If x(n) is real, then the Fourier transform is corjugate symmetric,which implies that the real part and the magnitude are both even functionsand the imaginary part and phase are both odd functions. Thus for real-valuedsignals the Fourier transform need only be specified for positive frequenciesbecause of the conjugate symmetry. Whether or not a sequence is real, speci-fication of the Fourier transform over a frequency range of 27r specifies it en-tirely. For a real-valued sequence, specification over the frequency rangefrom, for example, 0 to a is sufficient because of conjugate time -shifting property together with the linearity property plays akey role in using the Fourier transform to determine the response of systemscharacterized by linear constant-coefficient difference equations.
Example illustrating the periodicity and symmetry properties. x[n] = anu[n] O<a<1 X(2) =1 1-ae-jQ I X(W) I TRANSPARENCY 11.4 Additional properties of the discrete-time Fourier transform. Time shifting: x[n-n] I Frequency shifting: ejgon x[n] Linearity: ax1 [n] +bx2 [n] Parseval's relation: z + 00 n=- -00 Ix[n] 12 = 27r1 21r R tan1 a2) X(E) X(92 ...
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