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Unformatted text preview: Chapter 13 FREQUENCY RESPONSE Copyright c ° 1996 by Ali H. Sayed. All rights reserved. These notes are distributed only to the students attending the undergraduate DSP course EE113 in the Electrical Engineering Department at UCLA. The notes cannot be reproduced without written consent from the instructor: Prof. A. H. Sayed, Electrical Engineering Department, UCLA, CA 90095, [email protected] The DTFT of a sequence conveys important information about its frequency content. Likewise, the DTFT of the impulse response sequence of a system conveys important infor- mation about the system. The purpose of this chapter is to explore these facts in greater details. Frequency Content of a Sequence Recall the inversion formula x ( n ) = 1 2 π Z π- π X ( e jω ) e jωn dω. We can approximate this integral expression by a finite sum as follows. We divide the interval [- π,π ] into subintervals of small enough width, say Δ w each, and write x ( n ) ≈ 1 2 π N X k =- N X ( e jk Δ ω ) e jk Δ ωn Δ ω = N X k =- N µ 1 2 π X ( e jk Δ ω )Δ ω ¶ e jk Δ ωn Here we are assuming that the interval [- π,π ] is divided into (2 N + 1) subintervals with N large enough (or Δ w small enough). The values ± k Δ ω correspond to the points ± ω . The above expression shows that a generic input sequence x ( n ) can be regarded as a linear combination of exponential sequences e jk Δ ωn . The coefficients of the linear combination are 1 2 π X ( e jk Δ ω )Δ ω . For this reason, we say that the DTFT of x ( n ) provides information about the frequency content of x ( n ). Magnitude and Phase of the DTFT The DTFT of a sequence x ( n ) is a complex function of ω . It therefore has both real and imaginary parts, say X ( e jω ) = X R ( e jω ) + jX I ( e jω ) . 131 132 Frequency Response Chapter 13 We can therefore evaluate the magnitude and phase of a DTFT as follows: | X ( e jω ) | = p | X R ( e jω ) | 2 + | X I ( e jω ) | 2 6 X ( e jω ) = arctan X I ( e jω ) X R ( e jω ) The phase is an angle in radians. We always assume that it is restricted to values in the interval [- π,π ]. For example, consider the DTFT X ( e jω ) = e- j 3 ω . Its magnitude is unity over [- π,π ], while its phase is linear in ω and given by 6 X ( e jω ) =- 3 ω . When we plot this phase as a function of ω however, we obtain the plot in the figure where every time the phase reaches ± π it is corrected by ∓ 2 π in order to guarantee that the phase values are restricted to the interval [- π,π ] in accordance with our convention.-4-3-2-1 1 2 3 4-4-3-2-1 1 2 3 4 Figure 13.1. Plot of the phase of X ( e jω ) = e- j 3 ω over the interval [- π,π ]. As a second example, consider the DTFT X ( e jω ) = 1 + cos ω . It is a real function of ω and, hence, X R ( e jω ) = 1 + cos ω, X I ( e jω ) = 0 ....
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