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Lesson_14

# Lesson_14 - Discrete-Time Signals and Systems Dr Fred J...

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Discrete-Time Signals and Systems Dr. Fred J. Taylor, Professor Lesson Title: FIR Frequency Response Lesson Number: 14 (Section 6-3 to 6-4) L Background: In Chapter 6, the concept of frequency response for FIRs was introduced. Sections 6-3 and 6-4 examines this concept in increasing detail. Discrete-Time System Frequency Response If an LTI FIR is presented with periodic input data, then using an exponential or trigonometric Fourier series, the input can be partitioned into a collection of discrete input signals at discrete frequencies. The output consists of a like collection of exponential or trigonometric signals at the frequencies of the input signal, but with altered amplitudes and phases. This analysis gives rise to the concept of the LTI’s frequency response. In Chapter 6, the steady-state frequency response of a discrete-time LTI FIR system is determined by evaluating the convolution sum at specific frequency locations. In particular, the frequency response of an N th order FIR, having an impulse response given by h [ k ]={ h 0 , 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 0 1 where ω = ϖ T s is the normalized discrete-time baseband frequency ranging over ω∈ [ - π , π ]. Periodicity Observe that over an extended range ϖ = ω +(2 π )s, for s an integer, that: ( 29 ( 29 ( 29 s j j e H e H π ω ω 2 + = . 2 That is, the FIR frequency response replicates itself on integer multiples of the normalized baseband frequency as motivated in Figure 1. This is a fundamentally important observation in that it characterizes the frequency domain signature of periodically sampled signal processes is likewise periodic. It is implicitly assumed, however, that the sample rate f s was chosen with knowledge that the highest frequency in the signal being sampled and processed is bounded by f s /2. Therefore, the periodically extended frequency responses are devoid of actual signal energy in the ideal case. Figure 1: Baseband and full spectrum of a discrete-time filter. 1 - π 0 π f s /2 0 f s /2 Baseband -4 π -2 π - π 0 π 2 π 4 π -2f s -f s 0 f s 2f s Actual Spectrum

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Discrete-Time Signals and Systems Dr. Fred J. Taylor, Professor The magnitude frequency response is called the filter's gain (defined at frequency ω ) and provides information about how the FIR amplifies or attenuates an input signal. The phase response (defined at frequency ω ) establishes the amount of steady-state phase shift imparted to the input signal. There are a number of techniques that can be used to mathematically or experimentally produce an FIR's frequency response. These concepts will only be superficially motivated at this time. The first method presumes that an FIR's impulse response, h [ k ], has been measured or derived from its underlying equations. The analysis of the impulse response can be accomplished using one or more of the many Fourier-based techniques (e.g., Fourier Series), as suggested in Figure 2. The accuracy of each method is based on the quality of the impulse response definition and the selected Fourier tool. The top method shown in Figure 2, uses a tool
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Lesson_14 - Discrete-Time Signals and Systems Dr Fred J...

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