Discrete-time representations - Signal Processing and Optimization for Transceiver Systems - page 149

4.9 Noble identities and digital design of filters

In a digital communication system, such as the one in Fig. 4.1, the prefilter
F ( jω ) and postfilter G ( jω ) are designed based on a number of considerations
including the minimization of reconstruction error (Chap. 10). These filters
often turn out to be complicated functions of frequency. As such, it is not easy
to design and implement them as continuous-time filters. It turns out, however,
that these filters can be approximated quite well by using digital filters ,which
operate in the sampled data domain. To explain this idea we first introduce a
couple of identities based on the sampling theorem. Shown in Fig. 4.33, these are
called noble identities and they are proved in Appendix G (Sec. G.2). The first
identity says that a digital filter P ( z ) following a C/D converter can be moved
to the left of the C/D converter, as long as ω in P ( e jω ) is replaced with ωT. The
second identity says that a digital filter preceding a D/C converter can be moved
to the right by using the identity shown in Fig. 4.33(b). Now consider Fig.
4.34(a), which shows the digital communication system with a digital prefilter
F d ( e jω ) and a continuous-time prefilter F c ( jω ) . Similarly there are two postfilters
as shown. (Notice the slight difference in notation with respect to Fig. 4.18.)
By using the two noble identities this system can be redrawn as in Fig. 4.34(b)
where the combined prefilter is

F ( jω )= F c ( jω ) F d ( e jωT ) , (4 . 63)

and similarly the combined postfilter is

G ( jω )= G c ( jω ) G d ( e jωT ) .

(4 . 64)

This shows that the digital filter part F d ( e jωT ) (and similarly G d ( e jωT )) can be
completely absorbed into the continuous-time filter. At this point the reader
might wonder why it is necessary to use a digital filter at all. There are some
advantages to this. While the continuous-time filter cannot approximate arbi-
trary shapes easily, the design of digital filters with nearly arbitrary frequency
response shapes is quite straightforward [Oppenheim and Schafer, 1999]. In the
continuous-time case, lowpass filters are easy to design. It is therefore practi-
cable to use G c ( jω ) to approximate a good lowpass filter in | ω | <π/T ,and

G d ( e jω ) to achieve arbitrary shapes within <π/T . This is demonstrated in

Fig. 4.35. Part (a) shows an arbitrary digital filter response G d ( e jω ) and part
(b) shows G d ( e jωT ) , which is exactly the digital filter response, appropriately
scaled. So this is a periodic function with period 2 π/T. If this is multiplied by
the ideal lowpass filter in part (c), then the result is G d ( e jωT ) , restricted to the
first period

| ω |

| ω | < π/T.