During the next weeks I will be writing a series of posts about my pre-print with Daniel Abreu, it essentially has to do with Sampling Theory, Frames and Applied Harmonic Analysis. The first post of this series will be a more elementary one, and will be devoted to introducing bandlimited function and presenting one of the classical results in this subject, the Shannon Sampling Theorem.
In Signal Processing it is usual to represent a signal by a function depending on (usually considered to be time). It is often important to consider the same signal on the frequency side, this is achieved by the Fourier Transform of the signal,
Roughly speaking, the value of stands for how much the frequency is present in . The Fourier Transform lies in the heart of the Fourier Analysis, and is a mathematical object with several beautiful (and sometimes amazing) proprieties. Two of them are very important in what follows:
– The Plancharel Theorem that essentially states that the Fourier Transform is an isometry in , meaning that: .
– The inversion formula, that essentially gives the relation
In several applications it is reasonable to assume that a signal can not have frequencies of arbitrarily big absolute value (a simple illustration of this is the fact that the human ear can only listen sounds whose frequency lie on a certain band). For this reason one is interested in studying functions whose frequency is supported on , for some (we will restrict ourselves to to make the exposition cleaner, although all results below can be easily generalized to a general band space). This motivates the next definition.
Definition 1: Bandlimited functions
The space of bandlimited functions is the space of all such that its Fourier Transform is supported on , i.e. such that
The space of bandlimited functions is also tightly connected to a space of entire functions by the Paley-Wiener Theorem.
Now, we are ready to state and prove the Whittaker-Shannon-Kotel’nikov Sampling Theorem.
Theorem 1: Whittaker-Shannon-Kotel’nikov
Let be a Bandlimited function (see Definition 1), i.e.: . Then
Proof: Let be given. By the Plancharel Theorem we have . As is an orthonormal basis of (Fourier Series are based on this fact) and is supported on , then we can write
for some coefficients , where stands for the characteristic function of the set . Using the inversion formula and performing simple calculations we obtain
Setting we obtain giving the first equality of the Theorem.
The second part of the theorem is obtained with a direct application of the Plancharel Theorem and the fact that is an orthonormal basis of .
This theorem shows that is a sequence where we can sample functions in in the sense that, if we know the values of a bandlimited function at the function is uniquely determined and can be reconstructed by the formula in Theorem 1, this sampling rate is known as the Nyquist Rate.
One question that naturally arises is if there exists a sampling sequence “smaller” than (or, in other words, the optimality of the Nyquist Rate). To properly ask this question one needs to define what is smaller than in infinite sets and one needs to define sampling sequence. This will be done in the next post, and an answer to this question, due to Landau, will be discussed in future posts as well.
As this is my first Math blog post I would much appreciate comments about it. Was it too elementary? Did I lost too much time on basic stuff? Was it too fast to follow? too slow? too long? too short? Was something not clear enough? Answers to these questions will help me writing better posts in the future.