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lecture_17 - MIT OpenCourseWare http/ocw.mit.edu 2.161...

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MIT OpenCourseWare http://ocw.mit.edu 2.161 Signal Processing: Continuous and Discrete Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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1 Massachusetts Institute of Technology Department of Mechanical Engineering 2.161 Signal Processing - Continuous and Discrete Fall Term 2008 Lecture 17 1 Reading: Class Handout: Frequency-Sampling Filters . Proakis and Manolakis: Secs. 10.2.3, 10.2.4 Oppenheim, Schafer and Buck: 7.4 Cartinhour: Ch. 9 Frequency-Sampling Filters In the frequency-sampling filters the parameters that characterize the the filter are the values of the desired frequency response H ( e ) at a discrete set of equally spaced sampling frequencies. In particular, let 2 π ω k = k k = 0 , . . . , N 1 (1) N as shown below for the cases of N even, and N odd. Note that when N is odd there is no sample at the Nyquist frequency, ω = π . The frequency-sampling method guarantees that the resulting filter design will meet the given design specification at each of the sample frequencies. 1 - 1 1 - 1 I m ( z ) R e ( z ) M ± M ² ² 2 F 2 F 1 0 1 1 z - p l a n e R e ( z ) z - p l a n e I m ( z ) N = 1 0 ( e v e n ) N = 1 1 ( o d d ) ( a ) ( b ) 1 copyright c D.Rowell 2008 17–1
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For convenience denote the complete sample set { H k } as H k = H ( e k ) k = 1 , . . . , N 1 . For a filter with a real impulse response { h n } we require conjugate symmetry, that is ¯ H N k = H k and further, for a filter with a real, even impulse response we require { H k } to be real and even, that is H N k = H k . Within these constraints, it is sufficient to specify frequency samples for the upper half of the z -plane, that is for 2 π k = 0 , . . . , N 2 1 N odd ω k = k N N k = 0 , . . . , 2 N even . and use the symmetry constraints to determine the other samples. If we assume that H ( e ) may be recovered from the complete sample set { H k } by the cardinal sinc interpolation method, that is N 1 H ( e ) = H k sin ( ω 2 πk/N ) ω 2 πk/N k =0 then H ( e ) is completely specified by its sample set, and the impulse response, of length N , may be found directly from the inverse DFT, { h n } = IDFT { H k } where N 1 1 j 2 πkn h n = H k e N n = 0 , . . . , N 1 N k =0 As mentioned above, this method guarantees that the resulting FIR filter, represented by { h n } , will meet the specification H ( e ) = H k at ω = ω k = 2 kπ/N . Between the given sampling frequencies the response H ( e ) will be described by the cardinal interpolation. 1.1 Linear-Phase Frequency-Sampling Filter The filter described above is finite, with length N , but is non-causal. To create a causal filter with a linear phase characteristic we require an impulse response that is real and symmetric about its mid-point. This can be done by shifting the computed impulse response to the right by ( N 1) / 2 samples to form H ( z ) = z ( N 1) / 2 H ( z ) but this involves a non-integer shift for even N . Instead, it is more convenient to add the appropriate phase taper to the frequency domain samples H k before taking the IDFT. The non-integer delay then poses no problems: 17–2
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------------------------------------------------------------------------- Apply a phase shift of πk ( N 1) φ k = (2) N to each of the samples
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