# Other Examples of Complete Orthonormal Systems

I am interested in other examples of finite and complete orthogonal basis. I am not confident in my use of the term “complete”, so what I mean specifically is a set of basis vectors that can be used in a transformation from one domain (or vector space) to another with no loss, duplication or distortion in the transformation. (A constant scaling factor is acceptable, hence not restricted to being “orthonormal”.)

Two examples I am familiar with (which related to communication systems is the difference between OFDMA and CDMA):

OFDMA: The set of exponentials in the Discrete Fourier Transform mapping N samples in time to N samples in frequency using $$e^{j 2\pi n/N}$$ for $$n = 0$$ to $$N-1$$, which has the following resulting code set with $$W_N^{nk} = (e^{j/N})^{nk}$$:

$$W_N = \begin{bmatrix} W_N^0 & W_N^0 & \ldots & W_N^0\\ W_N^0 & W_N^1 & \ldots & W_N^{N-1}\\ \vdots & \vdots & & \vdots \\ W_N^0 & W_N^{N-1} & \ldots & W_N^{(N-1)^2} \\ \end{bmatrix}$$

as a matrix with each row indexed by $$k$$ for $$k=0 \ldots N-1$$ and each column indexed by $$n$$ for $$n = 0 \ldots N-1$$.

Note that $$W_1 = \begin{bmatrix} 1 & 1\\1 & -1\ \end{bmatrix}$$

CDMA: The set of Walsh-Hadamard Codes mapping N samples in time to N samples in a $$W_N$$ code space , with a resulting code set built on the pattern:

$$W_N = \begin{bmatrix} W_{N-1} & W_{N-1}\\W_{N-1} & -W_{N-1}\ \end{bmatrix}$$

With $$W_1 = \begin{bmatrix} 1 & 1\\1 & -1\ \end{bmatrix}$$

Interesting to me is the observation that both start of with the same 2 element form, but the DFT expands using the complex plane into possible elements all on the unit circle, while the Walsh-Hadamard codes commonly used for CDMA are limited to two elements (+1 and -1). In particular, I am looking for the possibility of other known finite and complete orthogonal code sets that also use the complex plane, but more samples beyond the unit circle, or alternatively the reason why such codes can't exist.

• The Walsh-Hadamard Transform does not support the familiar cyclic convolution that the DFT supports, but instead something that can be called Poisson convolution (cf. the latter half of this answer of mine). Commented May 2, 2022 at 15:44
• @DilipSarwate That is interesting! Thanks. I'm sure you must know of other codes, and was hoping you would see this and have an answer for below. (limited to finite ones is important, and ideally but not necessarily from the complex plane) Commented May 2, 2022 at 15:46
• I am thinking of the family of discrete cosine and sine transforms, as well as the Hartley transform. There exist complex versions of Hadamard/Haar matrices. And orthogonal,discrete wavelets Commented May 2, 2022 at 18:03
• @DanBoschen My understanding of a "complete" basis is that it allows representation of a specific set of functions with vanishing error. For example, the FT can represent the set of periodic functions that meet certain mathematical conditions. It's possible the term you're looking for is "lossless", or maybe "energy preserving".
– MBaz
Commented May 2, 2022 at 18:39
• @MarcusMüller my view may be too narrow so open to other thoughts but was considering specifically with the comparison of CDMA (3G) to OFDM (4G+) where in both cases we map N complex symbols using a set of N orthogonal codes- so I wanted to have a better understanding of what other options were out there that would also provide the same purpose and maybe other advantages for comm. Commented May 3, 2022 at 1:10