Sidelobe Roll-Off Rate

In general, when only the first $n$ terms exist in the power-series expansion of a continuous function $w(t)$ (i.e., each term is finite), then the Fourier Transform magnitude $\vert W(\omega)\vert$ is asymptotically proportional to

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

Proof: Papoulis, Signal Analysis, McGraw-Hill, 1977

Thus, we have the following rule-of-thumb:

$$\zbox{\hbox{$n$\ terms} \leftrightarrow -6n \hbox{ dB per octave roll-off rate}}$$

(since $-20\log_{10}(2)=6.0205999\ldots$ ).
This is also $-20n$ 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.

Example Roll-Off Rates:

• Amplitude discontinuity ($n=1$ ) $\leftrightarrow$ $-6$ dB/octave
• Slope discontinuity ($n=2$ ) $\leftrightarrow$ $-12$ dB/octave
• Curvature discontinuity ($n=3$ ) $\leftrightarrow$ $-18$ dB/octave

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':

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

Points to note:

• Zero crossings at integer multiples of $\Omega_M \mathrel{\stackrel{\Delta}{=}}\frac{ 2 \pi}{M}$
  where $\Omega_M \mathrel{\stackrel{\Delta}{=}}\frac{ 2 \pi}{M} =$ frequency sampling interval for a length $M$ DFT
• Main lobe width is $2 \Omega_M = \frac{ 4 \pi}{M}$
• As $M$ gets bigger, the mainlobe narrows
  (better frequency resolution)
• $M$ has no effect on the height of the side lobes
  (Same as the ``Gibbs phenomenon'' for Fourier series)
• First sidelobe only 13 dB down from main-lobe peak
• Side lobes roll off at approximately 6 dB per octave
• A 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., centered about time 0)