Sidelobe Roll-Off Rate

In general, if the first $n$ derivatives of a continuous function $w(t)$ exist (i.e., they are finite and uniquely defined), then its Fourier Transform magnitude is asymptotically proportional to

$$\vert W(\omega)\vert \to \frac{\hbox{constant}}{\omega^{n+1}} \quad(\hbox{as }\omega\to\infty)$$

Proof: Look up ``roll-off rate'' in text index.

• Thus, we have the following rule-of-thumb:
  $$\zbox{\hbox{$n$\ derivatives} \;\longleftrightarrow\;-6(n+1) \hbox{ dB per octave roll-off rate}}$$
  (since $20\log_{10}(2)=6.0205999\ldots$ ).
  

• This is also $-20(n+1)$ dB per decade.

• To apply this result, we normally only need to look at the window's endpoints. The interior of the window is usually differentiable of all orders.

Examples:

• Amplitude discontinuity $\;\longleftrightarrow\;$ $-6$ dB/octave roll-off
• Slope discontinuity $\;\longleftrightarrow\;$ $-12$ dB/octave roll-off
• Curvature discontinuity $\;\longleftrightarrow\;$ $-18$ dB/octave roll-off

For discrete-time windows, the roll-off rate slows down at high frequencies due to aliasing.

In summary, the DTFT of the $M$ -sample rectangular window is proportional to the `aliased sinc function':

$$\begin{eqnarray*} \hbox{asinc}_M(\omega T) &\mathrel{\stackrel{\Delta}{=}}& \frac{\sin(\omega M T / 2)}{M\sin(\omega T/2)} \\ [0.2in] &\approx& \frac{\sin(\pi fMT)}{M\pi fT} \mathrel{\stackrel{\mathrm{\Delta}}{=}}\mbox{sinc}(fMT) \end{eqnarray*}$$

Some important points (rect window transform):

• Zero crossings at integer multiples of $\zbox{\Omega_M \mathrel{\stackrel{\Delta}{=}}\frac{ 2 \pi}{M}}$
  ($=$ freq. sampling interval used by a length $M$ DFT)
• Main lobe width is $2 \Omega_M = \frac{ 4 \pi}{M}$
• As $M$ gets bigger, the main-lobe narrows
  (better frequency resolution)
• $M$ has no effect on the height of the side lobes
  (Same as the ``Gibbs phenomenon'' for Fourier series)
• First side lobe only 13 dB down from main-lobe peak
• Side lobes roll off at approximately 6dB per octave
• A linear phase term arises when we shift the window to make it causal, while the window transform is real in the zero-centered case (i.e., when the window $w(n)$ is an even function of $n$ )