86. 1.2 Basic Building Blocks The computational algorithm of an lti digital filter can be conveniently represented in block diagram form using the basic building blocks shown below A X Multiplier Adder xIn Unit delay Pick-ofif'node
§6.1.2 Basic Building Blocks • The computational algorithm of an LTI digital filter can be conveniently represented in block diagram form using the basic building blocks shown below x[n] y[n] w[n] + A x[n] y[n] y[n] −1 x[n] z x[n] x[n] x[n] Adder Unit delay Multiplier Pick-off node
86. 1.2 Basic Building Blocks Advantages of block diagram representation (1) Easy to write down the computational algorithm by inspection (2)Easy to analyze the block diagram to determine the explicit relation between the output and input
§6.1.2 Basic Building Blocks Advantages of block diagram representation • (1) Easy to write down the computational algorithm by inspection • (2) Easy to analyze the block diagram to determine the explicit relation between the output and input
86. 1.2 Basic Building Blocks (3)Easy to manipulate a block diagram to derive other“ equivalent” block diagrams yielding different computational algorithms (4)Easy to determine the hardware requirements (5)Easier to develop block diagram representations from the transfer function directly
§6.1.2 Basic Building Blocks • (3) Easy to manipulate a block diagram to derive other “equivalent” block diagrams yielding different computational algorithms • (4) Easy to determine the hardware requirements • (5) Easier to develop block diagram representations from the transfer function directly
86.1.3 Analysis of Block Diagrams Carried out by writing down the expressions for the output signals of each adder as a sum of its input signals, and developing a set of equations relating the filter input and output signals in terms of all internal signals Eliminating the unwanted internal variables then results in the expression for the output signal as a function of the input signal and the filter parameters that are the multiplier coefficients
§6.1.3 Analysis of Block Diagrams • Carried out by writing down the expressions for the output signals of each adder as a sum of its input signals, and developing a set of equations relating the filter input and output signals in terms of all internal signals • Eliminating the unwanted internal variables then results in the expression for the output signal as a function of the input signal and the filter parameters that are the multiplier coefficients
86.1.3 Analysis of Block Diagrams Example -Consider the single-loop feedback structure shown below X(z) E(z The output e(z of the adder is E(z=X(z+G2 zY(z · But from the figure Y(z=GZE(Z
§6.1.3 Analysis of Block Diagrams The output E(z) of the adder is E(z) = X(z)+G2 (z)Y(z) • But from the figure Y(z) = G1 (z)E(z) Example - Consider the single-loop feedback structure shown below