Mathematics

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By BEERNARD KLAR

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2 1 ------------ secs 10000 time secs time time The impulse responses of low pass filters FIR1, FIR4, and FIR5, all 1dB passband ripple, and transition bandwidths of 500 Hz and stopband attenuation of 40, 60 and 80dB respectively. Clearly the more stringent the filter parameters, the longer the required impulse response. Similarly if the passband ripple parameter is reduced, then a longer impulse response will be required. 1 FIR Filter z-domain Zeroes An important way of representing an FIR digital filter is with a z-domain plot of the filter zeroes.

Therefore an all pass filter is maximum phase. The magnitude frequency response of the pole at z = p i and the zero at z = p i–1 is: 1 – p i–1 z –1 H i ( e jω ) = ---------------------------1 – p i z –1 the Companion CD z = e jω 1= -----pi (96) DIGITAL COMMUNICATIONS BERNARD SKLAR 2nd Edition 58 Concise DSP Tutorial - Robert W. Stewart, Daniel García-Alís If we let p i = x + jy then the frequency response is found by evaluating the transfer function at z = e jω : –j ω 1 – p i–1 e – j ω  1  pi – e 1 - = ---- G ( e jω ) - = ----  -----------------------H i ( e jω ) = ---------------------------jω – jω – p p 1 – pie  i  1 – pi e i (97) where G ( e jω ) = 1 .

The Companion CD DIGITAL COMMUNICATIONS BERNARD SKLAR 2nd Edition 40 Concise DSP Tutorial - Robert W. Stewart, Daniel García-Alís The respective impulse responses of FIR1, FIR2 and FIR3 are respectively 15, 69 and 269 weights long, with group delays of 7, 34 and 134 samples respectively. 2 1 ------------ secs 10000 FIR3, 269 weights time The impulse responses of low pass filters FIR1, FIR2, and FIR3, all with 40 dB stopband attenuation, 1dB passband ripple, but transition bandwidths of 500, 200 and 50 Hz respectively.