Taylor Series Expansions
A Taylor series expansion of a continuous function is a polynomial approximation of . This appendix derives the Taylor series approximation informally, then introduces the remainder term and a formal statement of Taylor's theorem. Finally, a basic result on the completeness of polynomial approximation is stated.
Informal Derivation of Taylor Series
We have a function and we want to approximate it using an th-order polynomial:
Our problem is to find fixed constants so as to obtain the best approximation possible. Let's proceed optimistically as though the approximation will be perfect, and assume for all ( ), given the right values of . Then at we must have
where denotes the th derivative of with respect to , evaluated at . Solving the above relations for the desired constants yields
Thus, defining (as it always is), we have derived the following polynomial approximation:
Taylor Series with Remainder
We repeat the derivation of the preceding section, but this time we treat the error term more carefully.
Again we want to approximate with an th-order polynomial:
Our problem is to find so as to minimize over some interval containing . There are many ``optimality criteria'' we could choose. The one that falls out naturally here is called Padé approximation. Padé approximation sets the error value and its first derivatives to zero at a single chosen point, which we take to be . Since all ``degrees of freedom'' in the polynomial coefficients are used to set derivatives to zero at one point, the approximation is termed maximally flat at that point. In other words, as , the th order polynomial approximation approaches with an error that is proportional to .
Padé approximation comes up elsewhere in signal processing. For example, it is the sense in which Butterworth filters are optimal . (Their frequency responses are maximally flat in the center of the pass-band.) Also, Lagrange interpolation filters (which are nonrecursive, while Butterworth filters are recursive), can be shown to maximally flat at dc in the frequency domain [82,36].
Setting in the above polynomial approximation produces
Differentiating the polynomial approximation and setting gives
In the same way, we find
From this derivation, it is clear that the approximation error (remainder term) is smallest in the vicinity of . All degrees of freedom in the polynomial coefficients were devoted to minimizing the approximation error and its derivatives at . As you might expect, the approximation error generally worsens as gets farther away from 0.
To obtain a more uniform approximation over some interval in , other kinds of error criteria may be employed. Classically, this topic has been called ``economization of series,'' or simply polynomial approximation under different error criteria. In Matlab or Octave, the function polyfit(x,y,n) will find the coefficients of a polynomial of degree n that fits the data y over the points x in a least-squares sense. That is, it minimizes
Formal Statement of Taylor's Theorem
Let be continuous on a real interval containing (and ), and let exist at and be continuous for all . Then we have the following Taylor series expansion:
where is called the remainder term. Then Taylor's theorem [63, pp. 95-96] provides that there exists some between and such that
When , the Taylor series reduces to what is called a Maclaurin series [56, p. 96].
Weierstrass Approximation Theorem
The Weierstrass approximation theorem assures us that polynomial approximation can get arbitrarily close to any continuous function as the polynomial order is increased.
Let be continuous on a real interval . Then for any , there exists an th-order polynomial , where depends on , such that
For a proof, see, e.g., [63, pp. 146-148].
Thus, any continuous function can be approximated arbitrarily well by means of a polynomial. This does not necessarily mean that a Taylor series expansion can be used to find such a polynomial since, in particular, the function must be differentiable of all orders throughout . Furthermore, there can be points, even in infinitely differentiable functions, about which a Taylor expansion will not yield a good approximation, as illustrated in the next section. The main point here is that, thanks to the Weierstrass approximation theorem, we know that good polynomial approximations exist for any continuous function.
Points of Infinite Flatness
Consider the inverted Gaussian pulse,E.1
As mentioned in §E.2, a measure of ``flatness'' is the number of leading zero terms in a function's Taylor expansion (not counting the first (constant) term). Thus, by this measure, the bell curve is ``infinitely flat'' at infinity, or, equivalently, is infinitely flat at .
Another property of is that it has an infinite number of ``zeros'' at . The fact that a function has an infinite number of zeros at can be verified by showing
The reciprocal of a function containing an infinite-order zero at has what is called an essential singularity at [15, p. 157], also called a non-removable singularity. Thus, has an essential singularity at , and has one at .
An amazing result from the theory of complex variables [15, p. 270] is that near an essential singular point (i.e., may be a complex number), the inequality
In summary, a Taylor series expansion about the point will always yield a constant approximation when the function being approximated is infinitely flat at . For this reason, polynomial approximations are often applied over a restricted range of , with constraints added to provide transitions from one interval to the next. This leads to the general subject of splines . In particular, cubic spline approximations are composed of successive segments which are each third-order polynomials. In each segment, four degrees of freedom are available (the four polynomial coefficients). Two of these are usually devoted to matching the amplitude and slope of the polynomial to one side, while the other two are used to maximize some measure of fit across the segment. The points at which adjacent polynomial segments connect are called ``knots'', and finding optimal knot locations is usually a relatively expensive, iterative computation.
Differentiability of Audio Signals
As mentioned in §3.6, every audio signal can be regarded as infinitely differentiable due to the finite bandwidth of human hearing. That is, given any audio signal , its Fourier transform is given by
Since the length of the integral is finite, there is no possibility that it can ``blow up'' due to the weighting by in the frequency domain introduced by differentiation in the time domain.
A basic Fourier property of signals and their spectra is that a signal cannot be both time limited and frequency limited. Therefore, by conceptually ``lowpass filtering'' every audio signal to reject all frequencies above kHz, we implicitly make every audio signal last forever! Another way of saying this is that the ``ideal lowpass filter `rings' forever''. Such fine points do not concern us in practice, but they are important for fully understanding the underlying theory. Since, in reality, signals can be said to have a true beginning and end, we must admit that all signals we really work with in practice have infinite-bandwidth. That is, when a signal is turned on or off, there is a spectral event extending all the way to infinite frequency (while ``rolling off'' with frequency and having a finite total energy).E.2
In summary, audio signals are perceptually equivalent to bandlimited signals, and bandlimited signals are infinitely smooth in the sense that derivatives of all orders exist at all points time .
Logarithms and Decibels