"The first week notebook (introduction to Python, Numpy and Matplotlib) can be used as a help.\n",
"\n",
"## Important\n",
"You need to submit *individually* your answers on moodle before the next exercise session. For the theoretical questions you can either fill the notebook or write it on a separate sheet (if you are not comfortable with Markdown/TeX) and upload a scanned version. \n",
"\n",
"## Objective\n",
"\n",
"The end goal is to understand purely algebraic, matrix based, view of a few linear transforms. You will use those linear transform to perform some basic time-frequency analysis of signals."
The first week notebook (introduction to Python, Numpy and Matplotlib) can be used as a help.
## Important
You need to submit *individually* your answers on moodle before the next exercise session. For the theoretical questions you can either fill the notebook or write it on a separate sheet (if you are not comfortable with Markdown/TeX) and upload a scanned version.
## Objective
The end goal is to understand purely algebraic, matrix based, view of a few linear transforms. You will use those linear transform to perform some basic time-frequency analysis of signals.
1. Prove that any set of orthogonal vectors $v_i \in \mathbb{C}^N, \, i=1, \ldots , M \leq N$ such that $v_i^H v_j = C \delta_{i,j}$ is linearly independent (where $C$ is some constant).
4. Implement the function `get_fourier_matrix` that returns a normalized Fourier matrix of size $N\times N$. Do not make use of the builtin DFT/FFT functions in `numpy` or `scipy`. Raise a `ValueError` exception if a ngetive or zero $N$ value is supplied.
The second signal $x_2(t)$ is the sum of the same sinusoids over the complete time interval, with a scaling term s.t. the amplitude of both signals is identical.
In order to have a better compromise between time and frequency, the input signal will be split in smaller non-overlapping blocks of length $p$, and we will perform the DFT of each block.
6. Using the `get_fourier_matrix` implemented previously, fill the `get_block_dft_matrix` function below s.t. it returns a $N\times N$ matrix that will perform the block Fourier transform when applied to an input vector. Raise a `ValueError` if $p$ does not divide $N$.
We will now use this block Fourier transform to how the frequencies of the signal evolve through time.
7.
- Using the `reshape` and `plt.imshow` functions, display the amplitude of the result of the block Fourier transform of $x_1$ and $x_2$. Is the result improved when compared to the one observed in question 5 ?
In this part we will study another approach to study the time/frequency properties of signals.
Let us consider a vector $x\in\mathbb{R}^N$, with $N$ being even. The single-level Haar transform decomposes $x$ into two vectors $a^1$ and $d^1$ belonging to $\mathbb{R}^{\frac{N}{2}}$.
The coefficients of $a^1$ are defined as follows: $a^1_n = \frac{1}{\sqrt{2}}(x_{2n-1} + x_{2n}), n=1, ..., \frac{N}{2}$. $a^1$ is referred to as the *average coefficients vector*.
The coefficients of $d^1$ are defined as follows: $d^1_n = \frac{1}{\sqrt{2}}(x_{2n-1} - x_{2n}), n=1, ..., \frac{N}{2}$. $d^1$ is referred to as the *detail coefficients vector*.
The multi-level Haar transform is defined by recursively applying the single-level transform **to the average coefficient parts**.
For instance constructing 2-level Haar transform over $N$ points start with the previously defined $H_{1,N}$ matrix of size $N\times N$ and the corresponding $\frac{N}{2}\times\frac{N}{2}$ version denoted by $H_{1,\frac{N}{2}}$.
11. Implement a function that returns the $H_{p,N}$ matrix of size $N\times N$ that performs a $p$-level haar transform. Raise a `ValueError` if the size and the level are incompatible, or if the level is smaller than 1.
In order to make the visualization of the Haar decomposition easy, we provide you the `plot_haar_coeffs` function that will display the average and detail coefficients of the different levels.
The noise is usually located in the higher frequencies. However, the signal we created is a bit special as it has two discontinuities which also generate high frequencies components (remember the Fourier transform of a rectangle function is a sinc).
14. Implement a function `denoise_signal` that perform denoising of the input signal by using a supplied orthonormal transform matrix, and by setting the transformed coefficients having an amplitude smaller than a given threshold to 0. You might want to use the [numpy.where](https://numpy.org/doc/stable/reference/generated/numpy.where.html) function for this. When denoising using the Haar transform, you can preform the thresholding only on the detail coefficients. Implement the function `denoise_signal_haar` that performs this operation.
NB: The result returned should be real, in order to be displayed.
15. Compare the three denoising methods (Fourier, block Fourier and Haar). Which one performs better (in terms of noise removal but also in terms of discontinuity preservation) ? Was that expected (justify) ?
