# 5.11 Proofs for Section 4.10 “Multirate Processing of Discrete-Time Jointly SC Processes”

In this section, proofs of results of Section 4.10 are reported.

For future reference, let us consider the discrete-time sample train with sampling period M, its discrete Fourier series (DFS), and its Fourier transform:

(5.190)

where δ_{m} is the Kronecker delta, that is, δ_{m} = 1 if m = 0 and δ_{m} = 0 if m ≠ 0 and mod M denotes modulo operation with values in {0, 1, …, M − 1}. Thus, δ_{n mod M} = 1 if n = kM for some integer k and δ_{n mod M} = 0 otherwise.

Furthermore, in the sequel, the following identities are used

(5.191)

(5.192)

In the following proofs, the bounded variation assumption (4.192) allows to use the Fubini and Tonelli theorem (Champeney 1990, Chapter 3) to interchange the order of the integrals in ν_{1} and ν_{2}.

## 5.11.1 Expansion (Section 4.10.1): Proof of (4.241), (4.242b), and (4.242c)

From the identity

(5.193)

it follows that the impulse-response function of the LTV system that operates expansion is

(5.194)

Thus, the transmission function is

(5.195)

where in the last equality the Poisson's ...