Implementation For a causal filter whose impulse obtain has even symmetry:

h(n) = h(N-1-n),         for n = 0, 1,..., (N-1) - a total of N points the transfer function may be written, relaying on whether N is odd or even, as follows.

For even N The difference equation is calculate starting from H(z),

Since Y(z) = H(z) X(z), we can write

Taking the inverse z-transform of the above we get y(n) as

The transferred versions of x(n) are included in pairs and  then multiplied by coefficients h(.). That is given in figure below for N = 8. Note that there are an odd number (= 7) of delay components. There are N/2 = 4 multiplications and (N/2) + 1 = 4 + 1 = 5 adders.

Figure for N = 8

For odd N We need not derive the equations (they would be necessary if we were writing a computer program to automate it). For N = 7, there are N - 1 = 6 delay elements - an even number of delay elements. There are (N + 1)/2 = (7 + 1)/2 = 4 multiplications and 4 adders (the number of two-operand additions is 6).

Figure for N = 7

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