Digital Signal Processing Homework Help

DIGITAL SIGNAL PROCESSING ASSIGNMENT HELP ,

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Problems Problem 11.1 Consider the discrete-time system represented by the linear constant11.2 coefficient difference equationy(n) --y(n -1) + . y(n -2) = x(n) (a) Draw a block-diagram representation of the system in terms of adders and delay and coefficient multiplication branches. (b) Draw a linear signal-flow graph representation of the system.

Problem 11.2 In figure P11.2-1 is shown a digital network. (a) For this network, determine the matrices Ft, Bt and Ct as specified on chalkboard (c).

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Figure P11.2-1 (b) Determine the matrices Ect and Fdt in the alternative matrix representation on chalkboard (c).

Problem 11.3 Determine the system functions of the two networks in figure P11.3-1and show that they have the same poles.

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Problem 11.4 For the network shown in figure P11.4-l, determine all possible ordering of the nodes such that the node variables can be computed in sequence.

Figure P11.4-1 Problem 11.5 In figure 11.5-1 is indicated a digital network A with two inputs,x1(n) and x2 (n), and two outputs, y1 (n) and y2 (n).

Figure P11.5-1

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The network A can be described in terms of the two-part set of equations Y1 (z) = H1 (z) + H2(z) X2(z) Y2(z) = H3(z) + H4 (z) X2(z)

(a) Draw a flow-graph implementation of the network. The transmittance of each branch must be a constant or a constant times z . Higher-order functions of z cannot be used as branch transmittances. (b)

We wish now to connect a network B to the network A as indicated in figure P11.5-2. matlabassignmentexperts.com

Figure P11.5-2 Network B has no internal source nodes and is characterized by the system function X2(z) = Y2(z) HB(z). Note that because Of the notation Used

for network A, y2 (n) denotes the input to network B and x2 (n) is the output. By examining the flow graph in part (a) determine a necessary and sufficient condition on HB(z) so that the overall system is possibilities

computable.

If you have difficulty, first try the two

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Solutions Solution 11.1

We have drawn the flow-graph to graphically correspond closely to the block diagram in (a). There are, of course, many other ways of drawing the flow-graph, for example matlabassignmentexperts.com

Solution 11.2 (a) The equations corresponding to this flow graph are: W1 (z) = X(z) W2 (z) = W 1 (z) + z W4 (z) W3 (z) = 2 W2(z) W4(z) = 2 Wl(z) + 3 W3(z) Y(z) = W 3 (z) or, in matrix form

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Solution 11.3 Let us carry this out by obtaining the transfer function for each ofthe networks. For network 1: Y(z) = 2r cos 0 z~ Y(z) -r z Y(z) + X(z) or H1 (z) = 1/[l -2r cos e z~1 + r2 z 2] For network 2: Define W1 (z) as shown in Figure S11.3-1. matlabassignmentexperts.com

Figure S11.3-1 then

W1 (z) = X(z) -r sin 0 z Y(z) + r cos 0 z1 W (z) Y (z) = r sine z1 W1(z) + r cos 6 z 1 Y(z)

Solving for Y(z) in terms of X(z) we obtain Or

Y(z) = X(z) r(sin 0)z~1/f -2r cos e z + rz-21 H2(z) = r r (sine0)) z-/ l -2r cos 0 z -1 + r 2z ]

Thus both networks have the same poles. matlabassignmentexperts.com

Solution 11.4 With the nodes ordered as shown in the figure, the matrix Fct is

Thus with the nodes arranged in the order 1-2-3-4 they can be computed in sequence since the matrix Fct is zero on and above the main diagonal. There is no other ordering possible.

Solution 11.5 (a) A flow-graph in terms of H1, H2, H3 and H can be drawn as

Figure Sll.5-1

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However we want to draw the flow-graph using branch transmittances which are constant or a constant times z 1. Thus we replace H1, H2, H3 and H by their flow-graph implementations to obtain

Figure S11.5-2 (b) We now want to connect a network HB(z) to the right-hand side of the above network. Observe that there is a delay-free path from X2 to Y2'Consequently, if the system H B (z) has a delay-free path from its input to its output, the total system will have a delay-free loop and thus will be non computable. By contrast, if HB(z) does not have delay-free path from its input to its output, the overall system will be computable. A necessary and sufficient condition such that HB(z)is not delay-free from input to output is that hB(n), its unit sample matlabassignmentexperts.com

response be zero at n = 0 (we are, of course, assuming that the system is causal.) This is guaranteed if HB(z) can be written in the form

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