Discrete Time Signal Processing. On which we point out that a signal is a function and discuss signals indexed by a single discrete parameter that we call time. We introduce the Discrete Fourier Transform (DFT) and observe that it is a tool that allows us to express a function as a sum of oscillations. This entails a decomposition into different modes of variation. This is important because different aspects of a signal usually manifest as different modes of variation.
- Discrete signals, discrete cosines and sines.
- Inner products and orthogonality. On love, hate, and indifference.
- Discrete complex exponentials.
- Discrete Fourier Transform (DFT) and the inverse (i)DFT, signal reconstruction and spectrum reshaping.
- Linearity of DFTs and energy conservation (Parseval’s theorem).
Continuous Time Signal Processing. Where we cope with the fact that the real world is continuous and that we therefore need to define signals indexed by a continuous parameter that we call time. Some concepts get a little difficult to handle mathematically, but overall the important message here is that the same important properties we observed in discrete time, also hold in continuous time.
- Continuous time signals, complex exponentials.
- Energy, inner products and orthogonality in continuous time.
- Fourier transform (FT) and inverse (i) FT. Frequency decomposition of continuous time signals
- The delta function, generalized Fourier transforms, and orthogonality of complex exponentials
- Linearity, symmetry, and energy conservation
Signal processing applications in Electrical Engineering. Signal Processing is one of the fundamental component disciplines of Electrical Engineering. We explore the applications of that we learned in units 1 and 2 to develop the applications of signal processing to electrical engineering. This units serves to bolster understanding but also sets the stage
- Discrete time Fourier transform (DTFT). Relationship between the DFT, the DTFT, and the FT.
- Sampling. Perfect reconstruction of bandlimited signals. Windowing. Low pass filtering.
- Linear time invariant systems. Duality between convolution and products. FIR filter design.
L’essentiel est invisible pour les yeux. We explore the technical value of this quote from the Little Prince. Many things are invisible to our eyes because we can’t see frequency components. If we were able to see frequency components we would see the decomposition of a signal into different modes of variation. This would make complicated problems such as compression, noise removal, sampling, or the behavior of linear filters, almost trivial to solve. In developing the different versions of the Fourier transforms we have developed a new set of eyes that let us see something essential that is otherwise invisible to our eyes: frequency components. This new set of eyes is designed to understand signals that evolve in time. In the rest of the class we study how different pairs of eyes can be designed to study different types of signals. We will develop eyes to study images, signals with arbitrary correlation structure, and signals defined on top of graphs.
Image Processing. Images are just an easy progression of time where instead of having a single parameter that defines the support of the signal, we have two parameters. Naturally, a somewhat straightforward extension of the DFT provides the tool that we need to remove noise and find borders.
- Images as two dimensional signals.
- Two dimensional (2D)-DFT. Properties. The rate and direction of variability.
- Image filters in the time and frequency domain.
- The discrete cosine transform. Lossy compression.
Principal Components Analysis. Time signals and images are examples of regular signals in which the relationship between components of the signal abides to a regular pattern. Elements of the signal are related because of their proximity in time or space. In signals that have an arbitrary correlation structure, the notion of proximity is not related to the proximity between indexes but by how much the signals tend to move in unison with respect to an underlying probability distribution. In this context the notion of frequency becomes very abstract but it is nonetheless present. It is possible to define a transform that expresses the signal as a sum of components that represent different modes of variability.
- Random signals, probability distributions, mean and covariance matrix
- Eigenvalue decomposition of the covariance matrix.
- The PCA transform. Energy conservation in the PCA and expected energy of PCA transform coefficients
- Dimensionality reduction.
- Face recognition. Discriminative power of principal eigenvectors.
- Interpretation of PCA as a low pass filter to remove noise.
Graph Signal Processing. The progression of this class is towards signals with more complex structure. In this final topic we will study signals in which the relationship between components is given by an arbitrary proximity graph. We will see introduce the notion of graph frequency and see how it generalizes the notions of frequency in the time and spatial domain.
- Graphs. Adjacency matrices, Laplacians, and generic graph shift operators
- The Graph Fourier Transform (GFT) and the inverse (i)GFT
- The DFT, the 2D DFT, and the PCA transform as particular cases of the GFT.
- The GFT modes of variation
- Graph frequency analysis and its use in classification