Even and odd signals

Question

We've studied in signal and systems about odd and even signals. But what is the actual implementation or application of this property? like for energy and power signals we know If you're using a signal to communicate, then the more energy it has, the farther it will reach (I'm oversimplifying a bit, but in general this is true). The more energy a signal has when it is received, the easier it is for the receiver to recover the information in the signal without (or with few) errors.

In similar fashion whats with odd and even signal?

Answer 1

I would look for their interest in the mathematical study of even and odd functions, which have their own wikipedia pages. Odd and even functions have inherent properties (symmetry, limit conditions, sum/product) that simplify their analysis. In turn, as every function can be represented uniquely as a sum of an even and an odd function:

$$ f(x) = \frac{f(x)+f(-x)}{2}+\frac{f(x)-f(-x)}{2}$$

this property can be used in more complicated contexts. For instance, if $f$ is differentiable and even, the derivative is odd, and vice versa. Consequently, this decomposition is useful to analyze differential equations. Now, remember that Fourier discovered his series by studying and solving the heat equation. So, it is not surprising to see those odd/even symmetries at play with Fourier related bases: discrete cosine and sine transforms, Hartley transform. The symmetries can be used to reduce complexity and provide faster algorithms.

There is also a more algebraic and statistical derivation. They are detailed from the 1976 papers Properties of the Eigenvectors of Persymmetric Matrices with Applications to Communication Theory and Eigenvalues and Eigenvectors of Symmetric Centrosymmetric Matrices by Cantoni and Butler. They observe that many problems (communication theory, signal processing) lend themselves to using eigenvalues and eigenvectors of certain matrices:

1. Information theory, for example discrete time channel equalisation and maximum likelihood PAM detection
2. Linear system theory, for example, stability of discrete time systems
3. Linear estimation theory (the covariance matrix of a stationary stochastic process belongs to the class), for instance Principal Component Analysis
4. Numerical analysis, for example in the solution of differential equations

Then, there is a nice theorem:

It is proved that the eigenvectors of a symmetric centrosymmetric matrix of order $N$ are either symmetric or skew symmetric, and that there are $\lceil N/2 \rceil$ symmetric and $\lfloor N/2 \rfloor$ skew symmetric eigenvectors. Some previously known but widely scattered facts about symmetric centrosymmetric matrices are presented for completeness. Special cases are considered, in particular tridiagonal matrices of both odd and even order, for which it is shown that the eigenvectors corresponding to the eigenvalues arranged in descending order are alternately symmetric and skew symmetric provided the eigenvalues are distinct.

For instance, a basis derived from a covariance matrix for a signal of length $2K$ will have $K$ odd and $K$ even eigenvectors (under mild conditions). Hence, when designing a basis, it comes as natural to have both odd (skew-symmetric) and even vectors. Interestingly, classical orthogonal dyadic discrete wavelets do not exhibit such a symmetry, inspiring other designs like multi-band wavelets or lapped orthogonal transforms (LOT), that are real, have finite support and odd/even symmetries.

So, among practical applications, you have signal representations (bases, frames) and algorithmic optimization.

Answer 2

I am not sure that you would call this a practical application as much as a practical implication, but knowing (or forcing) a signal to be odd or even enables you to process it in a certain way.

For example, if you know (or force) a signal to be even, then the imaginary part of its Discrete Fourier Transform (DFT) goes to zero and you can represent this signal just as a sum of $\cos(\cdot)$ functions. This relates the DFT to the Discrete Cosine Transform (DCT).

Specifically:

$$X[k] = \sum_{n=0}^{N-1} x[n] \cdot e^{-i \frac{2 \pi k n}{N}}$$

This is a compact way of saying:

$$X[k] = \sum_{n=0}^{N-1} x[n] \cdot \left ( cos\left ( \frac{2 \pi k n}{N} \right ) - i \sin\left ( \frac{2 \pi k n}{N} \right ) \right ) $$

This makes it clear that $X \in \mathbb{C}$ (omplex) and is composed of a real part (where $x[n]$ is multiplied by the $\cos(\cdot)$) and an imaginary part (where $x[n]$ is multiplied by the $\sin(\cdot)$).

But, $\cos(\cdot), \sin(\cdot)$ are even and odd functions respectively and the way their symmetry works when combined with integration means that if a signal is an even periodic function, then it can be represented by a series of cosines.

OK, but I only have a sound recording that could be anything...how do I know (or force) it to be an even function? (or an odd function).

One of the key assumptions of the Discrete Fourier Transform is that $x[n]$ is a periodic function. That is, even if you pass some random recording of a person counting "one two three four", all that the DFT sees at its input is "one two three four one two three four one two three four one two three four...". In other words, once you reach the length of $x[n]$ (which is $N$), you wrap back to its start.

So, how do we make this an even signal?

Notice that in even signals $f(x) = f(-x)$. We could therefore generate a new signal that is double the length of $x[n]$ and once we reach the end of $x[n]$ we repeat it by playing it backwards. In this way, $x[N+n] = x[N-n]$ or "one two three fourouf eerht owt eno one two three fourouf eerht owt eno one two three fourouf eerht owt eno..." and so on.

In terms of a "demo", the DFT of [1,2,3,4] has an imaginary component while the DFT of [1,2,3,4,3,2] doesn't.

I do not think that you will find a direct application as such, it will probably be something that "exploits" the implication that a function is even or odd.

Hope this helps.