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1、3.2 Discrete Time Fourier Translation(3.10) 3.2.1 Definition nExample 3.5- The DTFT of the unit sample sequence n is given by(3.11) 3 Discrete-Time Signals in the Frequency Domain1/11 Example 3.6- Consider the causal sequencenIts DTFT is given byas(3.12) (3.13) 3 Discrete-Time Signals in the Frequen
2、cy Domain2/11nThe DTFT X(ej ) of a sequence xn is a periodic continuous function of with a period 2For all values of integer k.3 Discrete-Time Signals in the Frequency Domain3/11Given the DTFT X(ejw), how to recover the xn?3 Discrete-Time Signals in the Frequency DomainInverse DTFT4/11Proof:3 Discre
3、te-Time Signals in the Frequency Domain5/11Sincewe have3 Discrete-Time Signals in the Frequency Domain6/11 X(ej ) = Xre(ej ) + j Xim(ej )nIn general, X(ej ) is a complex function of the real variable w and can be written in rectangular form asnXre(ej ) and Xim(ej ) are, respectively, the real and im
4、aginary parts of X(ej ) , and are real functions of 3.2.2 Basic Properties(3.17) 3 Discrete-Time Signals in the Frequency Domain7/11n| X(ej ) | is called the magnitude function n ( ) is called the phase functionn X(ej ) can alternately be expressed in polar form as X(ej ) = | X(ej ) |ej ( ) (3.19)wh
5、ere ( ) = argX(ej ) (3.20)3 Discrete-Time Signals in the Frequency Domain8/11nThe relations between the rectangular and polar forms of X(ej) are given by: Xre(ej) = | X(ej)|cos( ) Xim(ej) = | X(ej)|sin( ) |X(ej)|2=X(ej) X* (ej)=X2re(ej)+X2im(ej) tan( ) = Xim(ej) / Xre(ej) 3 Discrete-Time Signals in
6、the Frequency Domain9/11Note: X(ej ) = | X(ej ) |ej ( )+2 k = | X(ej ) |ej ( )for any integer knThe phase function () cannot be uniquely specified for any DTFT Unless otherwise stated, we shall assume that the phase function () is restricted to the following range of values: - () the principal value (called as wrapped phase)Wrapped Phase and Unwrapped Phase3 Discrete-Time Signals in the Frequency Domain10/11 ()= - 4Example -3 Discrete-Time Signals in the Frequency Domain11/11
