Relation of Smoothness to Roll-Off Rate
In §3.1.1, 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
![]() |
(B.70) |
by the sifting property of the impulse under integration. This shows that an impulse consists of Fourier components at all frequencies in equal amounts. The roll-off rate is therefore zero in the Fourier transform of an impulse.
By the differentiation theorem for Fourier transforms
(§B.2), if
, then
![]() |
(B.71) |
where



![]() |
(B.72) |
The integral of the impulse is the unit step function:
![]() |
(B.73) |
Therefore,B.4
![]() |
(B.74) |
Thus, the unit step function has a roll-off rate of

![]() |
(B.75) |
Integrating the unit step function gives a linear ramp function:
![]() |
(B.76) |
Applying the integration theorem again yields
![]() |
(B.77) |
Thus, the linear ramp has a roll-off rate of


Now consider the Taylor series expansion of the function
at
:
![]() |
(B.78) |
The derivatives up to order






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 orderB.5
, i.e.,
![]() |
(B.79) |
Proof: Following [202, p. 95], let


![]() |
(B.80) |
denote its decomposition into a nondecreasing part




Since
![]() |
(B.82) |
we conclude
![]() |
(B.83) |
where






![]() |
(B.84) |
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.,
![]() |
(B.85) |
where


In summary, we have the following Fourier rule-of-thumb:
![]() |
(B.86) |
This is also

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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