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What is the condition of M, if the structure according to the direct form is as follows?

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What is the condition of M, if the structure according to the direct form is as follows?

(a) M even

(b) M odd

(c) All values of M

(d) Doesn’t depend on value of M

The question was posed to me in exam.

My query is from Structures for FIR Systems topic in division Discrete Time Systems Implementation of Digital Signal Processing

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The correct answer is (b) M odd

The best explanation: When the FIR system has linear phase, the unit sample response of the system satisfies either the symmetry or asymmetry condition, h(n)=±h(M-1-n)

For such a system the number of multiplications is reduced from M to M/2 for M even and to (M-1)/2 for M odd. Thus for the structure given in the question, M is odd.

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