if the region of convergence for X z includes the unit circle C Williams W

If the region of convergence for x z includes the

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if the region of convergence for X ( z ) includes the unit circle. C. Williams & W. Alexander (NCSU) FUNDAMENTAL DSP CONCEPTS ECE 513, Fall 2016 40 / 156
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Frequency Representation Let us multiply X ( ω ) by e j ω m and integrate it over one period to obtain π π X ( ω ) e j ω m d ω = π π n = −∞ x ( n ) e j ω n e j ω m d ω (52) If X ( ω ) exist (implies that the region of convergence for X ( z ) includes the unit circle), then we can change the order of summation and integration. π π X ( ω ) e j ω m d ω = n = −∞ x ( n ) π π e j ω ( m n ) d ω (53) C. Williams & W. Alexander (NCSU) FUNDAMENTAL DSP CONCEPTS ECE 513, Fall 2016 41 / 156 Frequency Representation We observe that π π e j ω ( m n ) d ω = 2 π m = n 0 m = n (54) It follows that π π X ( ω ) e j ω m d ω = 2 π x ( n ) (55) or x ( m ) = 1 2 π π π X ( ω ) e j ω m d ω (56) Thus, the Fourier Transform synthesis, transform pair for discrete–time sequence is given by x ( m ) = 1 2 π π π X ( ω ) e j ω m d ω (57) X ( ω ) = n = −∞ x ( n ) e j ω n (58) C. Williams & W. Alexander (NCSU) FUNDAMENTAL DSP CONCEPTS ECE 513, Fall 2016 42 / 156
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Frequency Representation We can obtain the frequency representation of a sequence by evaluating its Z-Transform on the unit circle as shown in the previous section. We will present an example to illustrate the frequency representation of a discrete–time sequence. C. Williams & W. Alexander (NCSU) FUNDAMENTAL DSP CONCEPTS ECE 513, Fall 2016 43 / 156 Frequency Representation Example Example (1.4) Consider the discrete–time sequence x ( n ) = 2 . 0 ( 0 . 75 ) n u ( n ) (59) The corresponding Z–Transform is given by X ( z ) = 2 . 0 n = 0 ( 0 . 75 ) n z n = 2 . 0 1 0 . 75 z 1 ; | z | > 0 . 75 (60) Since the region of convergence includes the unit circle, X ( ω ) is defined and X ( ω ) = 2 . 0 1 0 . 75 e j ω (61) C. Williams & W. Alexander (NCSU) FUNDAMENTAL DSP CONCEPTS ECE 513, Fall 2016 44 / 156
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Frequency Representation Example Example (1.4) We can simplify X ( ω ) X ( ω ) = 2 . 0 ( 1 0 . 75 e j ω ) ( 1 0 . 75 e j ω )( 1 0 . 75 e j ω ) (62) X ( ω ) = 2 . 0 1 . 5 cos ( ω ) 1 . 5 jsin ( ω ) 1 1 . 5 cos ( ω ) + 0 . 5625 (63) Fig. 9 gives a plot of the magnitude of X ( ω ) and Fig. 10 gives a plot of the phase of X ( ω ) . C. Williams & W. Alexander (NCSU) FUNDAMENTAL DSP CONCEPTS ECE 513, Fall 2016 45 / 156 Frequency Representation Example Example (1.4) 0 0.5 1 1.5 2 2.5 3 3.5 1 2 3 4 5 6 7 8 Frequency Plot Frequency(radians) Magnitude Figure ¹: Magnitude plot of X ( ω ) ! for Example 19. C. Williams & W. Alexander (NCSU) FUNDAMENTAL DSP CONCEPTS ECE 513, Fall 2016 46 / 156
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