N C F F F F WS p s T p s F F C N 69All right reserved Copyright 2013 Sharifah

N c f f f f ws p s t p s f f c n 69all right reserved

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N C F F F F WS p s T p s F F C N
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69 All right reserved. Copyright © 2013. Sharifah Saon F p and F s are the digital passband frequencies and digital stopband frequencies respectively. The window length depends on the choice of window which dictates the choice of C . The closer the match between the stopband attenuation A s and the stopband attenuation A WS of the windowed spectrum, the smaller is the window length N. In any case, this relation typically overestimates the smallest filter length, and we can often decrease this length and still meet design specifications.
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70 All right reserved. Copyright © 2013. Sharifah Saon The Kaiser Window Empirical relations have been developed to estimate the filter length N of FIR filters based on the Kaiser window. We first compute the peak passband ripple p and the peak stopband ripple s , and choose the smallest of these as the ripple parameter : = min( p , s ) The ripple parameter is used to recompute the actual stopband attenuation A s 0 in decibels: A s 0 = -20log dB
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71 All right reserved. Copyright © 2013. Sharifah Saon Finally, the filter length N is approximated by The Kaiser window parameter is estimated from the actual stopband attenuation A s 0 as follows: dB A dB A F F F F A N s s p s p s s 21 21 , , 1 9222 . 0 1 36 . 14 95 . 7 0 0 0 dB A dB A dB dB A A A A s s s s s s 21 50 21 50 , , , 0 21 0251 . 0 21 186 . 0 7 . 8 0351 . 0 0 0 0 0 4 . 0 0 0
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72 All right reserved. Copyright © 2013. Sharifah Saon Choosing the Cutoff Frequency A common choice for the cutoff frequency (used in the expression for h [ n ]) is F C = 0.5( F p + F s ). The actual frequency that meets specifications for the smallest length is often less than this value. The cutoff frequency is affected by the filter length N . A design that ensures the minimum length N is based on starting with the above value of F C and then reducing the length or tweaking (typically decreasing) F C until we just meet specifications (typically at the passband edge).
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73 All right reserved. Copyright © 2013. Sharifah Saon Spectral Transformations The design of FIR filters other than lowpass starts with a lowpass prototype, followed by appropriate spectral transformations to convert it to the required filter type. These spectral transformations are developed from shifting and modulation properties of the DTFT. Unlike analog and IIR filters, these transformations do not change the filter order (or length). The starting point is an ideal lowpass filter, with unit passband gain and a cutoff frequency of FC whose windowed impulse response hLP [ n ] is given by h LP [ n ] = 2 F C sinc(2 nF C ) w N [ n ], is a window function of length N . n w N 2 1 2 1 N n N
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74 All right reserved. Copyright © 2013. Sharifah Saon Lowpass to Highpass 1 st transformation. is valid only if the filter length N is odd Its cutoff frequency is equals to the cutoff frequency of the lowpass filter.
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75 All right reserved. Copyright © 2013. Sharifah Saon 2 nd transformation is valid for any length N
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76 All right reserved. Copyright © 2013. Sharifah Saon Lowpass to Bandpass/Bandstop
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