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
![$ {\dot x}(t)\isdef \frac{dx}{dt}(t)$](http://www.dsprelated.com/josimages_new/sasp2/img2575.png)
![$ x(t)$](http://www.dsprelated.com/josimages_new/sasp2/img109.png)
![$ X(\omega)/(j\omega)$](http://www.dsprelated.com/josimages_new/sasp2/img2576.png)
![]() |
(B.72) |
The integral of the impulse is the unit step function:
![]() |
(B.73) |
Therefore,B.4
![]() |
(B.74) |
Thus, the unit step function has a roll-off rate of
![$ -6$](http://www.dsprelated.com/josimages_new/sasp2/img380.png)
![]() |
(B.75) |
Integrating the unit step function gives a linear ramp function:
![]() |
(B.76) |
Applying the integration theorem again yields
![]() |
(B.77) |
Thus, the linear ramp has a roll-off rate of
![$ -12$](http://www.dsprelated.com/josimages_new/sasp2/img2588.png)
![\begin{eqnarray*}
\delta(t) &\longleftrightarrow& 1\\
u(t) &\longleftrightarrow& \frac{1}{j\omega}\\
t\cdot u(t) &\longleftrightarrow& \frac{1}{(j\omega)^2}\\
\frac{1}{2}t^2 u(t) &\longleftrightarrow& \frac{1}{(j\omega)^3}\\
\vdots & \vdots & \vdots \\
\frac{1}{n!}t^n u(t) &\longleftrightarrow& \frac{1}{(j\omega)^{n+1}}
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/sasp2/img2589.png)
Now consider the Taylor series expansion of the function
at
:
![]() |
(B.78) |
The derivatives up to order
![$ n-1$](http://www.dsprelated.com/josimages_new/sasp2/img2592.png)
![$ t=0$](http://www.dsprelated.com/josimages_new/sasp2/img1877.png)
![$ n$](http://www.dsprelated.com/josimages_new/sasp2/img97.png)
![$ t=0$](http://www.dsprelated.com/josimages_new/sasp2/img1877.png)
![$ n$](http://www.dsprelated.com/josimages_new/sasp2/img97.png)
![$ 1/\omega^{n+1}$](http://www.dsprelated.com/josimages_new/sasp2/img247.png)
Theorem: (Riemann Lemma):
If the derivatives up to order
of the function
exist and
are of bounded variation (defined below), then its Fourier Transform
is asymptotically of orderB.5
, i.e.,
![]() |
(B.79) |
Proof: Following [202, p. 95], let
![$ w(t)$](http://www.dsprelated.com/josimages_new/sasp2/img252.png)
![$ (a,b)$](http://www.dsprelated.com/josimages_new/sasp2/img2593.png)
![]() |
(B.80) |
denote its decomposition into a nondecreasing part
![$ w_{\scriptscriptstyle\uparrow}(t)$](http://www.dsprelated.com/josimages_new/sasp2/img2595.png)
![$ -w_{\scriptscriptstyle\downarrow}(t)$](http://www.dsprelated.com/josimages_new/sasp2/img2596.png)
![$ \tau\in(a,b)$](http://www.dsprelated.com/josimages_new/sasp2/img2601.png)
![\begin{eqnarray*}
\mbox{re}\left\{W_{\scriptscriptstyle\uparrow}(\omega)\right\}
&=& \int_a^b w_{\scriptscriptstyle\uparrow}(t)\cos(\omega t) dt \\
&=& w_{\scriptscriptstyle\uparrow}(a)\int_a^\tau \cos(\omega t) dt
+ w_{\scriptscriptstyle\uparrow}(b)\int_\tau^b \cos(\omega t) dt
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/sasp2/img2602.png)
Since
![]() |
(B.82) |
we conclude
![]() |
(B.83) |
where
![$ M\isdef \max\{\left\vert w_{\scriptscriptstyle\uparrow}(a)\},\left\vert w_{\scriptscriptstyle\uparrow}(b)\right\vert\right\vert$](http://www.dsprelated.com/josimages_new/sasp2/img2605.png)
![$ w$](http://www.dsprelated.com/josimages_new/sasp2/img580.png)
![$ (a,b)=(-\infty,\infty)$](http://www.dsprelated.com/josimages_new/sasp2/img2606.png)
![$ \left\{W_{\scriptscriptstyle\uparrow}(\omega)\right\}$](http://www.dsprelated.com/josimages_new/sasp2/img2607.png)
![$ \left\{w_{\scriptscriptstyle\downarrow}(\omega)\right\}$](http://www.dsprelated.com/josimages_new/sasp2/img2608.png)
![$ \left\{w_{\scriptscriptstyle\downarrow}(\omega)\right\}$](http://www.dsprelated.com/josimages_new/sasp2/img2608.png)
![]() |
(B.84) |
If in addition the derivative
is bounded on
, then
the above gives that its transform
is
asymptotically of order
, so that
. Repeating this argument, if the first
derivatives exist and are of bounded variation on
, we have
.
Since spectrum-analysis windows
are often obtained by
sampling continuous time-limited functions
, we
normally see these asymptotic roll-off rates in aliased
form, e.g.,
![]() |
(B.85) |
where
![$ \Omega_s=2\pi f_s$](http://www.dsprelated.com/josimages_new/sasp2/img2615.png)
![$ \omega=\pm\pi$](http://www.dsprelated.com/josimages_new/sasp2/img203.png)
In summary, we have the following Fourier rule-of-thumb:
![]() |
(B.86) |
This is also
![$ -20(n+1)$](http://www.dsprelated.com/josimages_new/sasp2/img2617.png)
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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