数字信号处理教学课件：3-2 Discrete Time Fourier Translation

1、3.2 Discrete Time Fourier Translation(3.10) 3.2.1 Definition Example 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 sequenceIts DTFT is given byas(3.12) (3.13) 3 Discrete-Time Signals in the Frequency

2、 Domain2/11The 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 Discrete-T

3、ime Signals in the Frequency Domain5/11Sincewe have3 Discrete-Time Signals in the Frequency Domain6/11 X(ej) = Xre(ej) + j Xim(ej)In general, X(ej) is a complex function of the real variable w and can be written in rectangular form asXre(ej) and Xim(ej) are, respectively, the real and imaginary part

4、s of X(ej) , and are real functions of 3.2.2 Basic Properties(3.17) 3 Discrete-Time Signals in the Frequency Domain7/11| X(ej) | is called the magnitude function () is called the phase function X(ej) can alternately be expressed in polar form as X(ej) = | X(ej) |ej() (3.19)where () = argX(ej) (3.20)

5、3 Discrete-Time Signals in the Frequency Domain8/11The 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 the Frequency Domain9/11Note:

6、X(ej) = | X(ej) |ej()+2k = | X(ej) |ej()for any integer kThe 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