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# 7.10 Proofs

## 7.10.1 Proof of (7.262)

Fourier transform of a time-scaled, delayed, and frequency-shifted signal:

(7.383)

(7.384)

(7.385)

## 7.10.2 Proof of (7.263) and (7.264)

By using (7.261), the result is that

where is the continuous-time infinite average with respect to t, and, in the third equality, the variable change t′ = stda is made, so that

This proof can be equivalently carried out in the FOT approach by simply removing the expectation operator E{ · } in (7.386) and (7.387).

(7.264) is obtained by Fourier transforming both sides of (7.263).

## 7.10.3 Proof of (7.262) and (7.272)

By reasoning as for (7.386), we have

(7.388)

(7.268) is obtained by Fourier transforming both sides of (7.267).

## 7.10.4 Proof of (7.273)

Accounting for (7.262) and (7.272), one obtains

(7.389) ...

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