## 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 . Consequently, the integral of transforms to :

(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 dB per octave, just like the rectangular window. In fact, the rectangular window can be synthesized as the superposition of two step functions:

(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 dB per octave. Continuing in this way, we obtain the following Fourier pairs:

Now consider the Taylor series expansion of the function at :

(B.78) |

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 order^{B.5}
, *i.e.*,

(B.79) |

**Proof:**Following [202, p. 95], let be any real function of bounded variation on the interval of the real line, and let

(B.80) |

denote its decomposition into a nondecreasing part and nonincreasing part .

^{B.6}Then there exists such that

Since

(B.82) |

we conclude

(B.83) |

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

(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 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:

(B.86) |

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.

**Next Section:**

Random Variables & Stochastic Processes

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The Uncertainty Principle