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Extra info for Answers Sydsaeter & Hammond - Mathematics for Economic Analysis
3z –4 ----------H( z) = Y X(z) (65) If the z-polynomial of Eq. 584j . (Note all quantities have been rounded to 3 the Companion CD DIGITAL COMMUNICATIONS BERNARD SKLAR 2nd Edition 42 Concise DSP Tutorial - Robert W. Stewart, Daniel García-Alís decimal places). The corresponding SFG of the FIR filter written in the zero form of Eq. 3 y(k) The signal flow graph of four first order cascaded filters corresponding to the same impulse response as the 5 weight filter shown above. The first order filter coefficients correspond to the zeroes of the 5 weight filter.
55: x ( k )W 8–3k = – j 6πk --------------x ( k )e 8 = x ( k )( –j ) k e = – j 3πk --------------x ( k )e 4 = x ( k )e π π – j --- + --- k 2 4 = x ( k )e πk πk – j ------ – j -----2e 4 (59) πk – j -----4 = ( – j ) k x ( k )W 8–k This exploitation of the computational redundancy is the basis of the FFT which allows the same result as the DFT to be computed, but with less MACs. To more formally derive one version of the FFT (decimation-in-time radix-2), consider splitting the DFT equation into two “half signals” consisting of the odd numbered and even numbered samples, where the total number of samples is a power of 2 ( K = 2 k ): K⁄2–1 X(n ) = ∑ – j 2πn ( 2k -) -------------------------K x ( 2k )e k=0 K⁄2–1 = ∑ k=0 K⁄2–1 = ∑ K⁄2–1 + ∑ x ( 2k + – j 2πn ( 2k + 1 )-----------------------------------K 1 )e k=0 K⁄2–1 x ( 2k )W K–2kn + ∑ x ( 2k + 1 )W K–( 2k + 1 )n (60) k=0 K⁄2–1 x ( 2k )W K–2kn + W K–n k=0 ∑ x ( 2k + 1 )W K–2 kn k=0 Notice in Eq.
Note of course that for a stable filter, a < 1 . The gain at all frequencies is 1 (a feature of all pass filters of course). One area where fractional delays are useful is in musical instrument synthesis where accurate control of the feedback loop delay is desirable to allow accurate generation of musical notes with rich harmonics using “simple” filters. If a digital audio system is sampling at f s = 48000 Hz then for frequencies up to around 4000 Hz very accurate control is available over the loop delay thus allowing accurate generation of musical note frequencies.