数字信号处理教学课件：3-4 Energy Density Spectrum of a Discrete-Time Sequence

1、3.3 DTFT TheoremsThere are a number of important properties of the DTFT that are useful in signal processing applicationsThese are listed here without proofTheir proofs are quite straightforwardWe illustrate the applications of some of the DTFT properties3 Discrete-Time Signals in the Frequency Doma

2、in1/17Symmetry Relation for complex xn3 Discrete-Time Signals in the Frequency Domain2/17Symmetry relations of the DTFT of a real sequence3 Discrete-Time Signals in the Frequency Domain3/17Example3.7 The magnitude and phase of the DTFT X(ej) = 1/(1 0.5e-j) are shown below|X(ej)|= |X(e-j)|()=-(-)3 Di

3、screte-Time Signals in the Frequency Domain4/17Type of Property Sequence DTFT Differentiation ngn jdG(ej)/dFrequency-shifting ej0ngn G(ej(- 0)Time-shifting gn-n0 e-jn0G(ej)Linearity agn+bhn aG(ej)+bH(ej)hn H(ej)gn G(ej)Table 3.43 Discrete-Time Signals in the Frequency Domain5/17Type of Property Sequ

4、ence DTFT Parsevals relationModulation gnhnConvolution gn*hn G(ej)H(ej)Table 3.4(The convolution is caculated in a periodic interval.)periodical convolutionNote:3 Discrete-Time Signals in the Frequency Domain6/17Example 3.11 Determine the DTFT V(ej) of the sequence vn defined by d0vn+d1vn-1 = p0n +

5、p1n-1 Using the time-shifting and linearity property of the DTFT we then obtain the frequency-domain representation3 Discrete-Time Signals in the Frequency Domain7/17Example 3.13 Determine the DTFT Y(ej) of yn=(n+1)nn, |1Let xn=nn, |1We can therefore write yn=nxn + xnThe DTFT of xn is given by3 Disc

6、rete-Time Signals in the Frequency Domain8/17Using the differentiation property of the DTFT, the DTFT of nxn is given byNext using the linearity property of the DTFT we arrive at3 Discrete-Time Signals in the Frequency Domain9/17Supposing We can show that:Example Note: If , then 3 Discrete-Time Sign

7、als in the Frequency Domain10/17Example 3.14 If it is known that ,where,3 Discrete-Time Signals in the Frequency Domain11/17Example: Calculate the integral 3 Discrete-Time Signals in the Frequency Domain12/17So , Recall3.4 Energy Density Spectrum of a Discrete-Time SequenceThe total energy of a fini

8、te-energy sequence gn is given byEE From Parsevals relation we observe that3 Discrete-Time Signals in the Frequency Domain13/17is called the energy density spectrumThe area under this curve in the range - divided by 2 is the energy of the sequenceThe quantity3 Discrete-Time Signals in the Frequency Domain14/17Example 3.15 Compute the energy of the sequence hLPn=sincn/n, -nHerewhere3 Discrete-Time Signals in the Frequency Domain15/17ThereforeHence, hLPn is a finite-energy lowpass sequence3 Discrete