i Show that the transfer function may be written as 2 2 2 cos 2 1 cos 2 z z z z

I show that the transfer function may be written as 2

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i) Show that the transfer function may be written as: 2 2 2 cos 2 1 cos 2 ) ( z z z z z H What is the frequency response at rad/sample? 5 ii) Using the graphical method of “distance to zeros/distance to poles”, or otherwise, show that if ρ = 1- Δ, where Δ is a variable with a small value, the magnitude of the frequency response at ± Δ is approximately . 4 iii) Using the result for part (ii), or otherwise, design a second order infinite impulse response (IIR) notch digital filter to eliminate an unwanted sinusoidal component of a digitised signal sampled at 600 Hz without affecting the magnitudes of other frequency components too severely. The frequency of the unwanted signal is 50 Hz and the 3 dB bandwidth of the notch should be approximately 20 Hz. 10 iv) Draw a realisation for the designed filter in (iii) above. 4 b) What s-domain transformation will transform an analogue low-pass filter with 3 dB frequency of Ω cL to a high- pass filter with 3 dB frequency of Ω cH ? 2 Total 25
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PAPER CODE .................... ELEC309 ............... PAGE .... 5 ........ OF .............. 6 .......................... CONTINUED 4. a) The sequence x [ n ] is related to its discrete time Fourier transform (DTFT), X ( e ), by the expression: d e X e n x j jn ) ( 2 1 ] [ . i) Use this expression to design a 10th order high-pass finite impulse response (FIR) filter with cut-off frequency of 7 kHz for signals sampled at 16 kHz. Perform your design using a rectangular window. 5 ii) State what improvement in the performance of the filter might be obtained by the use of a Hamming window. 3 iii) Sketch a direct form implementation structure for your filter which uses a minimum number of multipliers.
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