## 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:

(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

(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.

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Random Variables & Stochastic Processes

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The Uncertainty Principle