## 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

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.1.2), if
, then

where

. Consequently, the integral
of

transforms to

:

The integral of the impulse is the

*unit step function*:

Therefore,

^{B.5}
Thus, the unit step function has a roll-off rate of

dB per
octave, just like the rectangular window. In fact, the rectangular
window can be synthesized as the superposition of two step functions:

Integrating the unit step function gives a

*linear ramp function*:

Applying the integration theorem again yields

Thus, the linear ramp has a roll-off rate of

dB per octave.
Continuing in this way, we obtain the following Fourier pairs:

Now consider the Taylor series expansion of the function
at
:

The derivatives up to order

are all zero at

. The

th
derivative, however, has a discontinuous jump at

. Since this is
the only ``wideband event'' in the

signal, we may conclude that a
discontinuity in the

th derivative corresponds to a roll-off rate
of

. The following theorem generalizes this result to
a wider class of functions which, for our purposes, will be

spectrum
analysis window functions (before

sampling):

**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

denote its decomposition into a nondecreasing part

and
nonincreasing part

.

^{B.7} Then there exists

such that

Since

we conclude

where

, which is finite since

is of bounded variation. Note that the conclusion holds also
when

. Analogous conclusions follow for

**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.*,

where

denotes the

sampling rate in radians per
second. This aliasing normally causes the roll-off rate to ``slow
down'' near half the sampling rate, as shown in
Fig.

4.12
for the rectangular window transform. Every window transform must be
continuous at

(for finite windows), so the roll-off

envelope must reach a slope of zero there.

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

This is also

dB per

*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.

**Previous:** Time-Bandwidth Products
are Unbounded Above**Next:** Beginning Statistical Signal Processing

**About the Author: ** Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and (by courtesy) of Electrical Engineering at

Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See

http://ccrma.stanford.edu/~jos/ for details.