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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 $ 1/\omega$. 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 $ n$th derivative corresponds to a roll-off rate of $ 1/\omega^{n+1}$.

The Fourier transform of an impulse $ x(t)=\delta(t)$ 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
$

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 $ x\;\leftrightarrow\;X$, then

$\displaystyle {\cal F}_\omega\{{\dot x}\} = j\omega X(\omega),
$

where $ {\dot x}(t)\isdef \frac{dx}{dt}(t)$. Consequently, the integral of $ x(t)$ transforms to $ X(\omega)/(j\omega)$:

$\displaystyle \int_{-\infty}^t x(\tau)\,d\tau \;\longleftrightarrow\;\frac{X(\omega)}{j\omega}
$

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.
$

Therefore,B.5

$\displaystyle U(\omega) = \frac{1}{j\omega}.
$

Thus, the unit step function has a roll-off rate of $ -6$ 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)
$

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..
$

Applying the integration theorem again yields

$\displaystyle t\cdot u(t) \;\longleftrightarrow\;\frac{1}{(j\omega)^2}.
$

Thus, the linear ramp has a roll-off rate of $ -12$ dB per octave. Continuing in this way, we obtain the following Fourier pairs:

\begin{eqnarray*}
\delta(t) &\longleftrightarrow& 1\\
u(t) &\longleftrightarro...
...c{1}{n!}t^n u(t) &\longleftrightarrow& \frac{1}{(j\omega)^{n+1}}
\end{eqnarray*}

Now consider the Taylor series expansion of the function $ x(t) = t^n u(t)$ at $ t=0$:

$\displaystyle x(t) = x(0) + {\dot x}(0) x + \frac{1}{2!}{\ddot x}(0) x^2 + \cdots
$

The derivatives up to order $ n-1$ are all zero at $ t=0$. The $ n$th derivative, however, has a discontinuous jump at $ t=0$. Since this is the only ``wideband event'' in the signal, we may conclude that a discontinuity in the $ n$th derivative corresponds to a roll-off rate of $ 1/\omega^{n+1}$. 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 $ n$ of the function $ w(t)$ exist and are of bounded variation (defined below), then its Fourier Transform $ W(\omega)$ is asymptotically of orderB.6 $ 1/\omega^{n+1}$, i.e.,

$\displaystyle W(\omega) = {\cal O}\left(\frac{1}{\omega^{n+1}}\right),
\quad(\hbox{as }\omega\to\infty)
$

Proof: Following [192, p. 95], let $ w(t)$ be any real function of bounded variation on the interval $ (a,b)$ of the real line, and let

$\displaystyle w(t) = w_{\scriptscriptstyle\uparrow}(t) - w_{\scriptscriptstyle\downarrow}(t)
$

denote its decomposition into a nondecreasing part $ w_{\scriptscriptstyle\uparrow}(t)$ and nonincreasing part $ -w_{\scriptscriptstyle\downarrow}(t)$.B.7 Then there exists $ \tau\in(a,b)$ such that

\begin{eqnarray*}
\mbox{re}\left\{W_{\scriptscriptstyle\uparrow}(\omega)\right\}...
... w_{\scriptscriptstyle\uparrow}(b)\int_\tau^b \cos(\omega t) dt
\end{eqnarray*}

Since

$\displaystyle \left\vert\int_a^\tau\cos(\omega t) dt\right\vert = \left\vert\fr...
...ega \tau) - \sin(\omega a)}{\omega}\right\vert \leq \frac{2}{\vert\omega\vert}
$

we conclude

$\displaystyle \left\vert\mbox{re}\left\{W_{\scriptscriptstyle\uparrow}(\omega)\...
...ht\vert\frac{2}{\vert\omega\vert}
\leq \frac{4M}{\left\vert\omega\right\vert}
$

where $ M\isdef \max\{\left\vert w_{\scriptscriptstyle\uparrow}(a)\},\left\vert w_{\scriptscriptstyle\uparrow}(b)\right\vert\right\vert$, which is finite since $ w$ is of bounded variation. Note that the conclusion holds also when $ (a,b)=(-\infty,\infty)$. Analogous conclusions follow for im$ \left\{W_{\scriptscriptstyle\uparrow}(\omega)\right\}$, re$ \left\{w_{\scriptscriptstyle\downarrow}(\omega)\right\}$, and im$ \left\{w_{\scriptscriptstyle\downarrow}(\omega)\right\}$, leading to the result

$\displaystyle \left\vert W(\omega)\right\vert = {\cal O}\left(\frac{1}{\omega}\right).
$

If in addition the derivative $ w^\prime (t)$ is bounded on $ (a,b)$, then the above gives that its transform $ j\omega W(\omega)$ is asymptotically of order $ 1/\omega$, so that $ W(\omega) =
{\cal O}(1/\omega^2)$. Repeating this argument, if the first $ n$ derivatives exist and are of bounded variation on $ (a,b)$, we have $ W(\omega) =
{\cal O}(1/\omega^{n+1})$. $ \Box$

Since spectrum-analysis windows $ w(n)$ are often obtained by sampling continuous time-limited functions $ w(t)$, 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}}
$

where $ \Omega_s=2\pi f_s$ 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 $ \omega=\pm\pi$ (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}}
$

This is also $ -20(n+1)$ 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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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 Associate Professor (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.


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