Time and frequency representation
Other Transformations

Orthogonal transformations, in general, allow representing any function of time in spectral domain where it’s easier to work with signals and make some mathematical operations (as convolution, reducing redundancy, etc.).

Discrete orthogonal transformations are mainly used in area of data compression, picture recognition, speech synthesis, etc. Our focus is only on one-dimensional orthogonal functions and transformations.

The simplest way how to express one-dimensional signal is as combination of bases functions. The main advantage for linear systems is the superposition principle (or superposition property). It’s desirable for the basis functions u(k, t) to be easily calculated and in simple form. Orthogonal functions fulfill all these requirements.

Mathematically, the orthogonal functions u(0,t), u(1,t),…, u(N – 1,t) on time interval <t1, t2> are defined as:

In case Uk = 1 the functions are orthonormal.

Signal x(t) approximated with basis orthogonal functions using superposition principle in mathematical language is as follows:

Where yk are spectral coefficients defined as:

Please read following example for better understanding.

Example of signal x approximated with basis orthogonal functions using superposition principle

The most common orthogonal functions used in signal processing are Walsh, Haar and Rademacher.

In case of discrete orthogonal functions approximation of signal x(nT), where T is period with M samples, is given as

Optimal coefficients are defined as:

In case of harmonic functions the main parameter of function is frequency. In case of non-harmonic functions the main parameter is sequence. Sequence is given as number of intersections with zero level per second. In case of discrete signals sequence is given as number of changes from negative to positive and vice versa, also per second.

Sampled orthogonal basis functions create system of discrete orthogonal basis functions.

## Walsh basis functions

Walsh functions are set of the square integrable functions on the unit interval. The functions take the values +1 and –1 only. Functions are time (t) and numerical (k) dependent. Generally are marked as wal(k,t) and can be grouped as even (cosines-Walsh) cal(k,t) and odd (sinus-Walsh) sal(k,t). Mathematical definition is also:

Even and odd Walsh functions

Based on ordering of Walsh functions three groups can be distinguished:

1. Walsh (sequence) ordering walw(k,t)
2. Dyadic (Paley) ordering walp(k,t)
3. Hadamard (natural) ordering walh(k,t)

All three groups contain the same functions but in different order. As example we will introduce you natural ordered Hadamard functions.

The main advantage of this ordering is very simple way how to create basis functions of higher dimensions.

In the picture below, there are first eight continuous and discrete Walsh functions, naturally ordered. If we write all values of discrete functions, the values create Hadamard matrix Uh(3) with dimensions 8x8.

In general Hadamard matrix Uh(r) with dimensions MxM, where M=2r, is calculated as Kronecker product of matrixes from Uh(r–1).

where Uh(0)=1

Kronecker product, denoted by , is an operation on two matrices of arbitrary size resulting in a block matrix.

Natural ordering of continuous and discrete Walsh functions and matrix made of function values

There are also other transformations with harmonic basis functions. Here belong (except Discrete Fourier transformation) discrete cosines transformation (DCT), discrete sinus transformation (DST), discrete Hartley transformation (DHYT).