Models (7.129) and (7.156) are derived in (Van Trees 1971) for the case of relative motion between transmitter and receiver when the relative radial speed can be considered constant within the observation interval (t0, t0 + T). Moreover, in (Van Trees 1971) it is shown that the time–scale factor s can be assumed unity, provided that the condition
is fulfilled, where B is the bandwidth of and T the length of the observation interval. Even if condition (7.209) involves both the signal bandwidth and observation interval, it is generally referred to as narrow-band condition.
In the following, the narrow-band condition is derived for a deterministic signal with Fourier transform approaching zero sufficiently fast. The special case of a strictly band-limited signal is considered. Then, the case of a stochastic process with power spectrum approaching zero sufficiently fast is addressed.
Theorem 7.5.1 Narrow-Band Condition –Deterministic Signals. Let x(t) be a differentiable deterministic signal with Fourier transform X(f) such that X(f) = O(|f|−γ) with γ > 2 for |f| > B (that is, |X(f)| ≤ K|f|−γ for |f| > B). It results that