In Chapter 4, we showed that, in general, the sampling rate of a discrete-time signal ca...

In Chapter 4, we showed that, in general, the sampling rate of a discrete-time signal can be reduced by a combination of linear filtering and time compression. Figure P6.52 shows a block diagram of an M-to-1 decimator that can be used to reduce the sampling rate by an integer factor M. According to the model, the linear filter operates at the high sampling rate. However, if M is large, most of the output samples of the filter will be discarded by the compressor. In some cases, more efficient implementations are possible.

(a) Assume that the filter is an FIR system with impulse response such that h[n] = 0 for n < 0 and for n > 10. Draw the system in Figure P6.52, but replace the filter h[n] with an equivalent signal flow graph based on the given information. Note that it is not possible to implement the M-to-1 compressor using a signal flow graph, so you must leave this as a box, as shown in Figure P6.52.

(b) Note that some of the branch operations can be commuted with the compression operation. Using this fact, draw the flow graph of a more efficient realization of the system of part (a). By what factor has the total number of computations required in obtaining the output y[n] been decreased?

(c) Now suppose that the filter in Figure P6.52 has system function

Draw the flow graph of the direct form realization of the complete system in the figure. With this system for the linear filter, can the total computation per output sample be reduced? If so, by what factor?

(d) Finally, suppose that the filter in Figure P6.52 has system function

Draw the flow graph for the complete system of the figure, using each of the following forms for the linear filter:

(i) direct form I

(ii) direct form II

(iii) transposed direct form I

(iv) transposed direct form II.

For which of the four forms can the system of Figure P6.52 be more efficiently implemented by commuting operations with the compressor?