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 .
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.
where . Consequently, the integral of transforms to :
The integral of the impulse is the unit step function:
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:
Integrating the unit step function gives a linear ramp function:
Applying the integration theorem again yields
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 :
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.,
Proof: Following [202, p. 95], let be any real function of bounded variation on the interval of the real line, and let
denote its decomposition into a nondecreasing part and nonincreasing part .B.6 Then there exists such that
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
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 .
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:
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.
Random Variables & Stochastic Processes
The Uncertainty Principle