Patrick Witomski

Sampling Signals and Poisson’s Formula

We are now going to tackle the problem of sampling analog signals. This operation is a prerequisite of digital signal processing. For example, an analog speech signal must be sampled before it can enter a digital telephone system. A sampler records the level of the signal every a seconds and transforms it into a sequence of impulses (Figure 37.1). An analog-to-digital converter (ADC) codes these impulses as numbers that can be processed digitally.

Filters and Transfer Functions

Systems have properties, at least sometimes. We are going to review several of the more standard properties of systems.

Primitives of a Distribution

When studying physical systems governed by differential equations, physicists often consider the derivatives to be taken in the sense of distributions. This is necessary, for example, when the inputs are discontinuous. This leads one to define generalized solutions of differential equations in terms of distributions.

The Discrete Fourier Transform

We will work with the following assumptions: We know the period a of the function f as well as N of its values that are regularly spaced over one period:

Convolution, Derivation, and Regularization

We saw in Lesson 20 conditions under which the convolution of two functions is well-defined. We turn now to several important properties of the convolution, some of which will be extended to distributions in Lesson 32. In the current lesson we focus on regularization.

Using the FFT for Numerical Computations

We present several examples to indicate the many possible numerical applications of the fast Fourier transform (FFT). It is widely used in signal processing for spectral analysis and for computing convolutions. We will see other important uses in computations involving high-degree polynomials and in interpolation problems.

Convergence of a Sequence of Distributions

We mentioned in Section 26.4 that an essential property of derivation in the sense of distributions is its continuity: We saw in Section 26.2 that from the point of view of physics, the impulse is a limit. For these and other reasons it is important to investigate the notion of limit in D′.

The Space ℒ (ℝ)

We have seen in the last few lessons how it is necessary to restrict the choice of functions in L 1(ℝ) if we wish to use the differentiation formulas and define the inverse Fourier transform. In this lesson, we are going to introduce a subspace of L 1(ℝ) that is invariant under the Fourier transform, differentiation, and multiplication by polynomials.

Convolution of Distributions

We discussed the convolution of functions in Lesson 20. There we saw that it is not always possible to take the convolution of two functions; it is the same for distributions. We will study the convolution of distributions and its basic properties for the more important cases.

Where Functions Prove to Be Inadequate

We are going to take a turn here that will lead to a new environment in which signals are no longer modeled solely by functions. The two themes for this heuristic introduction are impulse and derivation.