Q5Digital Signal Processing

An FIR lattice filter of order M is characterized by M reflection coefficients K1, K2, ..., KM, and is converted to the equivalent direct-form FIR coefficients using the standard order-recursive lattice-to-direct-form algorithm. Starting with a0(0)=1 (order-0 filter), each successive order m is built from the previous order (m-1) coefficients using:

Order 1 (using K1=0.65): a1(0)=1, a1(1) = K1 = 0.65. So h1(n) = {1, 0.65}.

Order 2 (using K2=0.5): a2(0)=1, a2(2) = K2 = 0.5. For the middle coefficient: a2(1) = a1(1) + K2·a1(1) = a1(1)(1+K2) = 0.65×1.5 = 0.975. So h2(n) = {1, 0.975, 0.5}.

Order 3 (using K3=0.9): starting from a2 = {1, 0.975, 0.5}, apply the same recursion with m=3: a3(0)=1; a3(1) = a2(1) + K3·a2(2) = 0.975 + 0.9(0.5) = 0.975 + 0.45 = 1.425; a3(2) = a2(2) + K3·a2(1) = 0.5 + 0.9(0.975) = 0.5 + 0.8775 = 1.3775; a3(3) = K3 = 0.9. So the final direct-form FIR coefficients (impulse response) are:

Direct form structure: given these 4 direct-form coefficients, the direct form (tapped delay line) FIR realization is simply the standard structure: the input x(n) passes through a chain of 3 unit-delay elements (z^-1) producing x(n-1), x(n-2), x(n-3); each of these four signals (x(n), x(n-1), x(n-2), x(n-3)) is multiplied by its corresponding coefficient (1, 1.425, 1.3775, 0.9 respectively); and all four scaled products are summed to produce the output y(n) = x(n) + 1.425x(n-1) + 1.3775x(n-2) + 0.9x(n-3). This direct form structure is computationally simpler (fewer arithmetic operations per output sample) than the lattice structure, but the lattice structure (from which these coefficients were derived) has the practical advantage of being modular/order-recursive (extending the filter order requires adding only one more lattice stage without recomputing earlier coefficients) and offering better numerical robustness to coefficient quantization, which is why lattice structures are preferred in adaptive filtering applications even though the direct form is more efficient for a fixed, already-designed filter.

Lattice filter structure recap: for completeness, the original lattice structure itself (from which these direct-form coefficients were derived) consists of M=3 identical lattice stages connected in cascade, each stage taking a forward signal f_{m-1}(n) and a backward signal g_{m-1}(n) as inputs and producing f_m(n) = f_{m-1}(n) + K_m·g_{m-1}(n-1) and g_m(n) = K_m·f_{m-1}(n) + g_{m-1}(n-1) as outputs, with f_0(n)=g_0(n)=x(n) initializing the first stage and y(n)=f_M(n) taken as the final filter output. Each stage requires only one multiplier-pair (using the same coefficient Km for both the forward and backward paths) and one delay element, making the lattice structure's total hardware complexity comparable to the direct form (each requires roughly M multipliers) while offering the additional structural advantages of modularity and reduced coefficient-quantization sensitivity noted above.

Stability/invertibility note: all three given reflection coefficients (K1=0.65, K2=0.5, K3=0.9) satisfy |Ki| < 1, which for an all-pole (IIR) lattice would be the standard necessary and sufficient condition for filter stability; for this FIR lattice, the same |Ki| < 1 condition guarantees that the equivalent direct-form filter is minimum-phase (all its zeros lie inside the unit circle), a property that is often desirable in FIR filter design and is a convenient, easily-checked by-product of specifying a lattice filter directly in terms of its reflection coefficients rather than its direct-form tap weights.