Digital communication systems and filter banks - Signal Processing and Optimization for Transceiver Systems - page 91

z − z − z −

1 1 1

x ( n )

M

f (1) f ( N )

time

f (0)

T

y ( n )

time

T / M

Figure 3.21 . An FIR interpolation filter implemented directly.

Next consider the interpolation filter shown in Fig. 3.20(a). This is demon-
strated in Fig. 3.21 for the FIR case; in this implementation the zero-valued
samples inserted by the expander enter the multipliers f ( k ) and result in wasted
computation. Furthermore the multipliers operate at the higher rate (they only
have T/M seconds per computation), where T is the separation between input
samples). To avoid such ine cient computations we represent the filter in its
Type 1 polyphase form:

M− 1

z −k R k ( z M ) .

F ( z )=

k =0

Then the system can be redrawn as in Fig. 3.20(b). By using the second noble
identity (Fig. 3.7) this can be simplified to the form shown in Fig. 3.20(c). In
this implementation there are no zero-valued samples entering the filters R k ( z ) .
Furthermore all the computations are taking place at the lower rate (i.e., before
the expanders). Observe that the output y ( n )isthe interleaved version of the
outputs of R k ( z ) . Thus each Type-1 polyphase component R k ( z )ofthefilter
computes the corresponding Type 1 polyphase component of the output y ( n ) ,
and these results are interleaved to get y ( n ) . In order to maximize the e ciency
of computation in multirate systems we move all decimators as far to the left as
possible ,and move all expanders as far to the right as possible. In this way, the
computational building blocks operate at the lowest possible rate.

3.7.2 Transmitting filter banks or synthesis filter banks

We next consider the system shown in Fig. 3.22(a). In this system, there are M
interpolation filters with inputs s k ( n ) , and their outputs are added. The interpo-
lation ratio P is in general different from M (usually P>M in communication
applications). The system is called a synthesis filter bank because it combines a
set of signals s k ( n ) into a single signal x ( n ) . In digital communications, such a
filter bank is used as a transmitting filter bank or transmitter filter bank. This
system will be described in greater detail in Sec. 3.8. We now show how this
filter bank can be expressed in polyphase form.