1. **Problem Statement:** We need to find a structure realization for the linear phase FIR filter given by the transfer function: $$H(z) = \frac{1}{2} + \frac{1}{4}z^{-1} - \frac{1}{2}z^{-3} + \frac{1}{8}z^{-4}$$ Our goal is to implement this filter with a minimum number of multipliers. 2. **Understanding Linear Phase FIR Filters:** Linear phase FIR filters have symmetric or antisymmetric coefficients. This property allows us to reduce the number of multipliers by grouping symmetric terms. 3. **Identify the coefficients:** The coefficients corresponding to powers of $z^{-n}$ are: $$h_0 = \frac{1}{2}, \quad h_1 = \frac{1}{4}, \quad h_2 = 0, \quad h_3 = -\frac{1}{2}, \quad h_4 = \frac{1}{8}$$ 4. **Check for symmetry:** Compare $h_0$ and $h_4$: $$h_0 = \frac{1}{2}, \quad h_4 = \frac{1}{8}$$ Compare $h_1$ and $h_3$: $$h_1 = \frac{1}{4}, \quad h_3 = -\frac{1}{2}$$ Since $h_0 \neq h_4$ and $h_1 \neq h_3$, the filter is not symmetric or antisymmetric in the usual sense. 5. **Express $H(z)$ grouping terms to minimize multipliers:** Rewrite $H(z)$ grouping terms with common delays: $$H(z) = h_0 + h_1 z^{-1} + h_3 z^{-3} + h_4 z^{-4}$$ Note $h_2=0$ so it is omitted. 6. **Realization with minimum multipliers:** Since no symmetry is present, each coefficient requires a multiplier. However, we can factor common terms if possible. No common factors among coefficients, so the minimum number of multipliers is 4. 7. **Structure realization:** The direct form FIR filter structure is: $$y[n] = \frac{1}{2}x[n] + \frac{1}{4}x[n-1] - \frac{1}{2}x[n-3] + \frac{1}{8}x[n-4]$$ This requires 4 multipliers and 4 delays. **Final answer:** The minimum number of multipliers needed is 4, and the direct form FIR structure is the optimal realization for this filter.