TABLE OF CONTENTS The noble identities are depicted in Figure 9.8. They have to do with the commutation of the filtering and decimation or interpolation operations, and are very useful in analyzing multirate systems and filter banks.
The identity in Figure 9.8a means that to decimate a signal by M and then filter it with H( z) is equivalent to filtering the signal with H( z M) and then decimating the result by M. A filter H( z M) is one whose impulse response is equal to the impulse response of H( z) with ( M 1) zeros inserted between adjacent samples. Mathematically, it can be stated as
| (9.1) |  |
where D M is the decimation-by- M operator.
The identity in Figure 9.8b means that to filter a signal with H( z) and then interpolate it by M is equivalent to interpolating it by M and then filtering it with H( z M). Mathematically, it is stated as
| (9.2) |  |
where I M is the interpolation-by- M operator.
In order to prove the identity in Figure 9.8a, one begins by rewriting equation (8.6), which gives the Fourier transform of the decimated signal x d( n) as a function of the input signal x( m), in the z domain, that is
| (9.3) |  |
For the decimator followed by filter H( z), we have that
