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suppose we have a cascade realization of a transfer function of a higher order.

What difference does it make if the order of the transfer function is an Even or an Odd number?

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If the order $N$ of the system is even, you get $N/2$ second-order sections, where each second-order section combines either a pair of complex conjugate poles and zeros, or two real-valued poles and zeros. If $N$ is odd, you get $(N-1)/2$ second-order sections and one additional first-order section for the remaining real-valued pole and zero.

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