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Tham khảo tài liệu 'digital filters part 12', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Complex Coefficient IIR Digital Filters 211 z 1 z 1ej z 1 cos9 jsin9 2 to the real transfer function also called real-prototype transfer function thus obtaining the analytical expression of the complex transfer function H z - HComplel z Hr z jH z . 3 HCompIex z is a transfer function with complex coefficients and with the same order of N as the real prototype Hteal z while its real and imaginary parts HR z and Hj z are of doubled order 2N real coefficient transfer functions. When HReal z is an LP transfer function then HR z and Hj z are of BP type. For a highpass HP real prototype transfer function we get HR z and Hj z respectively of BP and bandstop BS types. The substitution 2 is also termed pole rotation because it rotates the poles of the real transfer function to an angle of 9 both clockwise and anti-clockwise simultaneously doubling their number Fig. 2 . Fig. 2. Pole rotation of a first-order real transfer function after applying the substitution 2 . Starting with Y z H z X z 4 and supposing that the quantities in 4 are complex they can be represented by their real and imaginary parts Y z yr z jYI z X z XR z jXI z H z hr z jHI z 5 Then the equation 4 becomes Y z hr z jHI z xr z jXI z HR z XR z - HI z XI z j HI z XR z HR z XI z l 6 212 Digital Filters and its real and imaginary parts respectively are YR Z Hr z Xr z - Hi z X z y z Hi z Xr z Hr z Xi z . 7 According to the equations 7 the block-diagram of a complex filter will be as shown in Fig. 3. Fig. 3. Block-diagram of a complex filter. The synthesis of a complex filter is an important procedure because its sensitivity is influenced by the derived realization. A non-canonic complex filter realization will be obtained if Hr z and Hj z are synthesised individually. The process of synthesising the complex filter can be better understood by examining a particular filter realization - a real LP first-order filter section Fig. 4a with transfer function H z z 1 z-1 1 - a1z-1 8 The complex transfer function obtained
