lecture_17

# 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 ± ² 2 F 2 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
± ² For convenience denote the complete sample set { H k } as H k = H ( e k ) k =1 ,...,N 1 . 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 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 )= ² 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 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 in
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## This note was uploaded on 02/27/2012 for the course MECHANICAL 2.161 taught by Professor Derekrowell during the Fall '08 term at MIT.

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

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