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Unformatted text preview: DiscreteTime Signals and Systems Dr. Fred J. Taylor, Professor Lesson Title: FIR Frequency Response Lesson Number: 15 (Section 65 to 66) L Background: In Chapter 6, the concept of frequency response for FIRs was introduced. Sections 65 and 66 examines this concept from a graphical standpoint and specific filter behavior. DiscreteTime System Frequency Response In Chapter 6, the steadystate frequency response of a discretetime 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 N1 }, 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 discretetime baseband frequency ranging over ω∈ [ π , π ]. The frequencydomain 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 linearlinear, log (dB)linear, or loglog (Bode) framework. The phase response is normally plotted using a linearlinear or linearlog 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 unwrapping 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 ππ Unwrapped phase Wrapped phase DiscreteTime Signals and Systems Dr. Fred J. Taylor, Professor As noted in Lesson 13, communications engineers prefer to interpret the phasespace 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 =( N1)/2, which corresponds to the midpoint of the FIR measured in clock delays. Delay System A discretetime 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[nn ]. The frequency response of the delay system is given by: H(e j ω ) = ej ω n 4....
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 Spring '08
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 Digital Signal Processing, Frequency, Signal Processing, Finite impulse response, Dr. Fred J. Taylor, Dr. Fred J., Fred J. Taylor

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