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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
![$\displaystyle X(\omega)\isdef \int_{-\infty}^\infty x(t)e^{-j\omega t}dt = \int_{-\infty}^\infty \delta(t)e^{-j\omega t}dt = 1$](img2563.png) |
(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
![$\displaystyle {\cal F}_\omega\{{\dot x}\} = j\omega X(\omega),$](img2565.png) |
(B.71) |
where
. Consequently, the integral
of
transforms to
:
![$\displaystyle \int_{-\infty}^t x(\tau)\,d\tau \;\longleftrightarrow\;\frac{X(\omega)}{j\omega}$](img2568.png) |
(B.72) |
The integral of the impulse is the unit step function:
![$\displaystyle \int_{-\infty}^t \delta(\tau)\,d\tau = u(t) \isdef \left\{\begin{array}{ll} 1, & t\geq0 \\ [5pt] 0, & t<0 \\ \end{array} \right.$](img2569.png) |
(B.73) |
Therefore,B.4
![$\displaystyle U(\omega) = \frac{1}{j\omega}.$](img2575.png) |
(B.74) |
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:
![$\displaystyle w_R(n) = u\left(n+\frac{M-1}{2}\right) - u\left(n-\frac{M-1}{2}\right)$](img2576.png) |
(B.75) |
Integrating the unit step function gives a linear ramp function:
![$\displaystyle \int_{-\infty}^t u(\tau)d\tau = t \cdot u(t) = \left\{\begin{array}{ll} t, & t\geq0 \\ [5pt] 0, & t<0 \\ \end{array} \right..$](img2577.png) |
(B.76) |
Applying the integration theorem again yields
![$\displaystyle t\cdot u(t) \;\longleftrightarrow\;\frac{1}{(j\omega)^2}.$](img2578.png) |
(B.77) |
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
:
![$\displaystyle x(t) = x(0) + {\dot x}(0) x + \frac{1}{2!}{\ddot x}(0) x^2 + \cdots$](img2582.png) |
(B.78) |
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 orderB.5
, i.e.,
![$\displaystyle W(\omega) = {\cal O}\left(\frac{1}{\omega^{n+1}}\right), \quad(\hbox{as }\omega\to\infty)$](img251.png) |
(B.79) |
Proof: Following
[202, p. 95], let
be any real function of bounded
variation on the interval
of the real line, and let
![$\displaystyle w(t) = w_{\scriptscriptstyle\uparrow}(t) - w_{\scriptscriptstyle\downarrow}(t)$](img2585.png) |
(B.80) |
denote its decomposition into a nondecreasing part
and
nonincreasing part
.B.6 Then there exists
such that
Since
![$\displaystyle \left\vert\int_a^\tau\cos(\omega t) dt\right\vert = \left\vert\frac{\sin(\omega \tau) - \sin(\omega a)}{\omega}\right\vert \leq \frac{2}{\vert\omega\vert}$](img2594.png) |
(B.82) |
we conclude
![$\displaystyle \left\vert\mbox{re}\left\{W_{\scriptscriptstyle\uparrow}(\omega)\right\}\right\vert = \left\vert\int_a^b w_{\scriptscriptstyle\uparrow}(t)\cos(\omega t) dt \right\vert \leq \left\vert w_{\scriptscriptstyle\uparrow}(a)\right\vert\frac{2}{\vert\omega\vert} + \left\vert w_{\scriptscriptstyle\uparrow}(b)\right\vert\frac{2}{\vert\omega\vert} \leq \frac{4M}{\left\vert\omega\right\vert}$](img2595.png) |
(B.83) |
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
![$\displaystyle \left\vert W(\omega)\right\vert = {\cal O}\left(\frac{1}{\omega}\right).$](img2600.png) |
(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.,
![$\displaystyle \hbox{\sc Alias}_{\Omega_s}\left(\frac{1}{w^{n+1}}\right) = \sum_{k=-\infty}^\infty\frac{1}{(w+k\Omega_s)^{n+1}}$](img2605.png) |
(B.85) |
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.3.6
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:
![$\displaystyle \zbox {\hbox{$n$\ derivatives} \;\longleftrightarrow\;-6(n+1) \hbox{ dB per octave roll-off rate}}$](img2607.png) |
(B.86) |
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.
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