Lesson_15

# Lesson_15 - Discrete-Time Signals and Systems Dr Fred J...

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Unformatted text preview: Discrete-Time Signals and Systems Dr. Fred J. Taylor, Professor Lesson Title: FIR Frequency Response Lesson Number: 15 (Section 6-5 to 6-6) L Background: In Chapter 6, the concept of frequency response for FIRs was introduced. Sections 6-5 and 6-6 examines this concept from a graphical standpoint and specific filter behavior. Discrete-Time System Frequency Response In Chapter 6, the steady-state frequency response of a discrete-time LTI FIR system was determined by evaluating the convolution sum at specific frequency locations. The steady state assumption is based on the fact that the filter’s input frequency was assumed to be: [ ] n j j e Ae n x ω φ = 1. which is a signal that exists for all time. In particular, the frequency response of an N th order FIR, having an impulse response given by h [ k ]={ h , h 1 , … , h N-1 }, and sampled at a rate f s , can be expressed as: ( 29 ( 29 ( 29 ω ω ω ω φ j j N m m j i j e e H e h e H ∠ = = ∑- =- 1 2. where ω = ϖ T s is the normalized discrete-time baseband frequency ranging over ω∈ [- π , π ]. The frequency-domain behavior of an FIR filter can be graphically interpreted in terms of the filter’s magnitude frequency response (i.e., |H(e j ω )|) and phase response (i.e., ∠ H(e j ω )). The magnitude frequency response maybe presented within a linear-linear, log (dB)-linear, or log-log (Bode) framework. The phase response is normally plotted using a linear-linear or linear-log framework. Plotting the phase response of the filter can, at times, be challenging. The filter’s actual phase response and that produced by a computing machine can, at times, appear to disagree. This is generally due to the fact that a digital computer works only within a range of principal angles (- π , π ) as motivated in Figure 1. Phase un-wrapping is an attempt to overcome this problem. Here the computer attempts reconstruct a continuous phase response from a collection of piecewise continuous segments as suggested in Figure 1. Figure 1. Illustration of phase 'wrapping' and 'unwrapping' with respect to the principal angles ±π . 1 π-π Unwrapped phase Wrapped phase Discrete-Time Signals and Systems Dr. Fred J. Taylor, Professor As noted in Lesson 13, communications engineers prefer to interpret the phase-space in terms of a measurement called group delay . Group delay is formally defined to be: ( 29 ( 29 ω ω φ τ d d- = 3 For an N th order linear phase FIR, is can be shown that τ =( N-1)/2, which corresponds to the mid-point of the FIR measured in clock delays. Delay System A discrete-time system can, in theory, precisely delay a signal by some specified amount of time. For digital systems, delays can be realized using shift registers which create delay paths that are multiplies of some clock period. Consider the simple delay system y[n]=x[n-n ]. The frequency response of the delay system is given by: H(e j ω ) = e-j ω n 4....
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Lesson_15 - Discrete-Time Signals and Systems Dr Fred J...

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