## Relation of Smoothness to Roll-Off Rate

In §4.5.1, we found that the side lobes of
the rectangular-window transform ``roll off'' as . In this
section we show that this roll-off rate is due to the *amplitude
discontinuity* at the edges of the window. We also show that, more
generally, a discontinuity in the th derivative corresponds to a
roll-off rate of
.

The Fourier transform of an impulse is simply

*zero*in the Fourier transform of an impulse.

By the *differentiation theorem* for Fourier transforms
(§B.1.2), if
, then

*unit step function*:

^{B.5}

*linear ramp function*:

Now consider the Taylor series expansion of the function at :

**Theorem: **(*Riemann Lemma*):
If the derivatives up to order of the function exist and
are of bounded variation (defined below), then its Fourier Transform
is asymptotically of order^{B.6}
, *i.e.*,

**Proof:**Following [192, p. 95], let be any real function of bounded variation on the interval of the real line, and let

^{B.7}Then there exists such that

Since

**im**,

**re**, and

**im**, leading to the result

If in addition the derivative is bounded on , then the above gives that its transform is asymptotically of order , so that . Repeating this argument, if the first derivatives exist and are of bounded variation on , we have .

Since spectrum-analysis windows are often obtained by
*sampling* continuous time-limited functions , we
normally see these asymptotic roll-off rates in *aliased*
form, *e.g.*,

In summary, we have the following Fourier rule-of-thumb:

*decade*.

To apply this result to estimating FFT window roll-off rate
(as in Chapter 3), we normally only need to look at the window's
*endpoints*. The interior of the window is usually
differentiable of all orders. For discrete-time windows, the roll-off
rate ``slows down'' at high frequencies due to aliasing.

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Random Variables & Stochastic Processes

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Selected Continuous Fourier Theorems