# An new arranging of the discrete Sine transforms

Let $$n$$ be even and consider the non-normalized discrete Sine transform of type 5 which is

$$S=\left(\sin(k+1)(l+1)\frac{\pi}{n+\frac12}\right)_{k,l=0}^{n-1}$$

Let us denote $$s_{-,l}$$ by the $$l^{th}$$-column of $$S$$. It can be considered as a $$n$$-tuple in $$\mathbb{R}^n$$.

Q. I am looking for $$n$$-tuples $$v=(v_0,\cdots,v_{n-1})$$ in $$\mathbb{R}^n$$ satisfying the following conditions:

1- $$v_j=\sin2(j+1)\frac{\pi}{n+\frac12}$$ if $$j$$ is even.

2- The following are valid concerning inner products: $$\langle v , s_{-,l} \rangle=\left\{ \begin{array}{cl} 1 & l=0 \\ 0 & l\neq 0~,~ l \operatorname{is even} \end{array} \right.$$

• is this a spectral leakage question? Nov 27, 2022 at 10:28

Let's restate your problem as a linear equation

Let $$U_{ee}$$, $$U_{eo}$$, $$U_{oe}$$, $$U_{oo}$$ be the partition of the $$S$$ matrix in even-odd row-column index, similarly $$v_e$$ and $$v_o$$ are the partitions of $$v$$.

Restating your conditions in terms of matrices

1 - $$v_e$$ is fixed

2 - $$v_e U_{ee} + v_o U_{oe} = I_1$$, where $$I_1$$ denotes the first row of the identity matrix.

Placing all the constant terms on the right side of the condition 2, $$v_o U_{oe} = (I_1 - v_e U_{ee})$$, transposing we get a linear equation system in the standard form $$U_{oe}^{T} v_o = (I_1 - v_e U_{ee})^T$$

### Proof that $$U_{oe}$$ has no pair of linearly dependent columns

Assuming

$$s_{2i+1, 2j} = \sin\left( \frac{2\pi (2i+2)(2j+1)}{2n+1} \right)$$

Two columns will be linearly dependent if, and only if, there are two integers $$0 \le j_1, j_2 \le n/2-1$$, for all integer $$0 \le i \le n/2-1$$

$$(2i+2)(2j_1 + 1) \equiv (2i+2)(2j_2 + 1) \operatorname{mod} (2n+1),$$

$$(2j_1 + 1) \equiv (2j_2 + 1) \operatorname{mod} (2n+1)$$

$$j_1 \equiv j_2 \operatorname{mod} (2n+1)$$

since both $$j_1$$ and $$j_2$$ are positive integers, smaller than $$2n+1$$, the condition can only be satisfied with $$j_1 = j_2$$, thus the matrix $$U_{oe}$$ is non-singular.

• That is a nice approach, but seems there is only a gap! How can we sure that the entries in the diagonal of the upper triangular matrix $R$ are all non-zero?
– ABB
Nov 23, 2022 at 8:38
• If the rows of the input matrix are linearly independent, the diagonal elements are non-zero. I will add more details.
– Bob
Nov 23, 2022 at 10:23
• Yes, and so the proof linear independency will be equivalent to find the vector $q$.
– ABB
Nov 23, 2022 at 10:25
• To be honest I couldn't unambiguously parse your equation, do you mean $$s_{k,l} = \sin\left(\frac{(k+1)(l+1)\pi}{n+1}\right)$$
– Bob
Nov 23, 2022 at 10:49
• I expressed in terms of a linear equation, the necessary and sufficient condition for a solution to exist is that $s_{k,l}$ for odd $k$ and even $l$ must be non-singular. Bear with me, and double check if I didn't mess with the indices ;)
– Bob
Nov 23, 2022 at 11:20

1.- for any other reader not familiar with DST type 5 or DST-V, I'd like to include the definition of Discrete Sine Transform and types.

Following a list with DST types 1 to 5, from

https://planetmath.org/discretesinetransform 1D DST types 1st to 8th

2.- Bob's assumption is correct. The question expression actually means and I'd like to add that the question expression for DST-V is this 3.- DST Discrete Sine Transform and the counterpart DCT Discrete Cosine Transform are just partials of the DFT Discrete Fourier Transform.

Like Wolfram concisely explains here

https://mathworld.wolfram.com/DiscreteFourierTransform.html

and here

https://mathworld.wolfram.com/Leakage.html

Unless the signal or signal fragment is periodic in the watched interval, there's always (spectral) leakage.

4.- So, if the unknown input signal were all ones, satisfying the 2 conditions in the question would go the Bob answered.

5.- Yet it is up to x(:)*sin((k+1)*(L+1)*pi/(N+1)) not just the term sin((k+1)*(L+1)*pi/(N+1) to meet the non-leakage condition mentioned in point 2.

And since x is not supplied in the question in any shape or manner. there's no way to tell without knowing x .

5.- Is this a MATLAB question anyway?

I politely suggest this question to be moved to a Stack Overflow Maths page, or to for instance signal analysis or sampling tags in this same Signal Processing page